Jeff Ray serves as Vice President of Finance for Wolf Midstream Partners. His responsibilities include managing the firm's capital structure and providing strategic and financial analysis of proposed and existing projects and investments.
Prior to joining Wolf, Jeff was a Senior Analyst with VALUE Incorporated, a premier consulting firm specializing in the valuation of businesses, securities and intellectual property, where he advised clients through mergers and acquisitions, litigation and financial reporting. His engagements have represented clients in a wide range of industries, including banking, energy, financial services, healthcare, hospitality, manufacturing, media, retail, wholesale and various technologies.
Prior to joining VALUE Incorporated, Jeff held internships at a private equity fund focused on debt capital and minority equity investments, as well as the treasury department at Oncor Electric. He previously served as an assistant project manager for Henry & Sons Construction, where he built Schlitterbahn Beach on South Padre Island, Texas.
Jeff is a Chartered Financial Analyst and holds a Master of Business Administration degree from the Cox School of Business at Southern Methodist University, where his studies were concentrated on finance, strategy and entrepreneurship. He received his baccalaureate degree in Construction Science from Texas A&M University.


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text:
\section{Introduction}

In the study of theta lifts (such as Maass lifts and Borcherds products) it is convenient to work with vector-valued modular forms for the dual Weil representation $\rho^{*}$ associated to a finite quadratic module $(A,Q)$. Therefore, it is useful to understand the precise relationship between vector-valued modular forms for $\rho^{*}$ and scalar-valued modular forms for congruence subgroups.

For example, in some cases there are isomorphisms between spaces of vector-valued and scalar-valued modular forms. In \cite{BB}, Bruinier and Bundschuh showed that if $|A| = p$ is an odd prime, then modular forms for $\rho^{*}$ of weight $k \in \mathbb{Z}$ with $k \equiv \mathrm{sig}(A,Q)/2 \pmod 2$ can be identified with certain modular forms of weight $k$ for $\Gamma_{0}(p)$ and Nebentypus $\chi_{p} = \big( \frac{\cdot}{p}\big)$. The isomorphism is given by the component sum
\[
\varphi\left(\sum_{\gamma \in A}F_{\gamma}(\tau)\mathfrak{e}_{\gamma}\right) = \sum_{\gamma \in A}F_{\gamma}(p\tau)
\]
of the vector-valued modular form $F(\tau) =\sum_{\gamma \in A}F_{\gamma}(\tau)\mathfrak{e}_{\gamma}$, where $\mathfrak{e}_{\gamma}$ denotes the standard basis of the group algebra $\mathbb{C}[A]$. Using similar ideas, Y. Zhang constructed isomorphisms between spaces of vector-valued and scalar-valued modular forms for certain classes of finite quadratic modules which do not necessarily have odd prime order (see \cite{Z1,Z2}).

The condition $k\equiv \mathrm{sig}(A,Q)/2 \pmod 2$ turns out to be crucial for the aforementioned result of Bruinier and Bundschuh, since otherwise the components of any modular form for $\rho^{*}$ satisfy $F_{-\gamma} = -F_{\gamma}$ and hence cancel out in pairs in the sum. To obtain a non-zero map in any weight, we twist the component sums of vector-valued modular forms by a Dirichlet character $\chi$ mod $p$ with $\chi(-1) = (-1)^{k+\mathrm{sig}(A,Q)/2}$. Suppose that $|A| = p$ with an odd prime $p$. We define the \textbf{twisted component sum} of a modular form $F(\tau) = \sum_{\gamma \in A}F_{\gamma}(\tau)\mathfrak{e}_{\gamma}$ for $\rho^{*}$ by
\[
\varphi_{\chi}\left(\sum_{\gamma \in A}F_{\gamma}(\tau)\mathfrak{e}_{\gamma} \right) = \sum_{\gamma \in A}\chi(\gamma)F_{\gamma}(p\tau),
\]
where we fix an identification of $A$ with $\mathbb{Z}/p\mathbb{Z}$ to define $\chi(\gamma)$ for $\gamma \in A$. The assumptions on $k$ and $\chi$ imply that $\varphi_{\chi}$ is not trivially zero. Moreover, we have the following result.

\begin{prop}
	The map $\varphi_{\chi}$ defines an injective homomorphism from the space of modular forms of weight $k$ for $\rho^{*}$ to the space of scalar-valued modular forms of weight $k$ for $\Gamma_{0}(p^{2})$ with Nebentypus $\chi\otimes \chi_{p}$.
\end{prop}

For the proof we refer to Proposition~\ref{prop varphi injective} below. It is immediate from the construction that the $n$-th Fourier coefficient of $\varphi_{\chi}(F)$ vanishes unless $n \in p(\mathbb{Z}-Q(\gamma))$ for some $\gamma \in A \setminus \{0\}$. However, $\varphi_{\chi}$ is in general not surjective onto the subspace defined by this vanishing condition. We characterize the image of $\varphi_{\chi}$ in terms of the Atkin-Lehner involution in Proposition~\ref{prop isomorphism p} below.

We construct an analogous map $\varphi_{\chi}$ in the case that $|A| = 2p$ is twice an odd prime $p$, see Proposition~\ref{prop isomorphism 2p}; in this case, $(A,Q)$ must have odd signature and all modular forms are of half-integral weight.

As an application, we compute restrictions of Doi-Naganuma lifts of odd weight to components of Hirzebruch-Zagier curves $T_{\ell}$ of prime index $\ell$. Let $K = \mathbb{Q}(\sqrt{p})$ with a prime $p \equiv 1 \pmod 4$ and let $\mathcal{O}_{K}$ be its ring of integers. Recall that the Doi-Naganuma lift maps a vector-valued cusp form $F$ of weight $k$ for the dual Weil representation associated to the lattice $(\mathcal{O}_{K},-N_{K/\mathbb{Q}})$ to a Hilbert cusp form $\Phi_{F}$ of weight $k$ for $\mathrm{SL}_{2}(\mathcal{O}_{K})$. The restriction of $\Phi_{F}$ to a component of the Hirzebruch-Zagier curve $T_{\ell}$ of prime index $\ell$ is given by the Shimura lift of the vector-valued cusp form of weight $k+1/2$ for the dual Weil representation of the lattice $(\mathbb{Z},-\ell x^{2})$ obtained by the so-called theta contraction of $F$ as defined in \cite{M}, i.e. we have the commutative diagram
\begin{center} \begin{tikzpicture}[node distance=2.5cm, auto]
\node (A) {$S_{k}(-N_{K/\mathbb{Q}})$};
\node (AA) [right of=A] {$ $};
\node (B) [right of=AA] {$S_k(\mathrm{SL}_{2}(\mathcal{O}_{K}))$};
\node (C) [below of=A] {$S_{k+1/2}((-2\ell))$};
\node (D) [below of=B] {$S_{2k}(\Gamma_{0}(\ell))$};
\draw[->] (B) to node {\text{restr. to $T_{\ell}$}} (D);
\draw[->] (A) to node {\text{theta contr.}} (C);
\draw[->] (A) to node {$\mathrm{Doi-Naganuma}$} (B);
\draw[->] (C) to node {$\mathrm{Shimura}$} (D);
\end{tikzpicture}
\end{center}
For a proof, see Lemma~\ref{lem doi-naganuma-shimura}. We show that, on the level of the corresponding scalar-valued modular forms, the theta contraction basically becomes a multiplication by the Jacobi theta function. In this way, passing to scalar-valued modular forms makes it easier to compute the restriction of $\Phi_{F}$ to a component of $T_{\ell}$. To illustrate the result, we consider the case $\ell = p$ in the introduction.

\begin{prop}
Let $\chi$ be a Dirichlet character mod $p$ with $\chi(-1) = (-1)^{k}$. We have the following commutative diagram:
\begin{center} \begin{tikzpicture}[node distance=2.5cm, auto]
\node (A) {$S_{k}(-N_{K/\mathbb{Q}})$};
\node (AA) [right of=A] {$ $};
\node (B) [right of=AA] {$S_k(\Gamma_{0}(p^{2}),\chi\otimes \chi_{p})$};
\node (C) [below of=A] {$S_{k+1/2}((-2\ell))$};
\node (D) [below of=B] {$S_{k+1/2}(\Gamma_{0}(p^{2}),\chi)$};
\draw[->] (B) to node {$G \mapsto \big(G(4p\tau)\cdot \vartheta(\tau)\big)|U_{p}$} (D);
\draw[->] (A) to node {$\mathrm{theta}\, \mathrm{contr.}$} (C);
\draw[->] (A) to node {$\varphi_{\chi}$} (B);
\draw[->] (C) to node {$\varphi_{\chi}$} (D);
\end{tikzpicture}
\end{center}
where $\vartheta(\tau) = \sum_{n \in \mathbb{Z}}q^{n^{2}}$ is the Jacobi theta function and $U_{p}$ is the usual Hecke operator acting on Fourier expansions by $\big(\sum_{n}c(n)q^{n}\big)|U_{p} = \sum_{n}c(pn)q^{n}$.
\end{prop}

We refer to Proposition~\ref{prop theta contraction} for the general statement and its proof. We also give two numerical examples illustrating the use of the above proposition in Section~\ref{sec Doi-Naganuma}.

The work is organized as follows. We start with preliminaries about modular forms for the Weil representation associated to a finite quadratic module. In Section~\ref{sec isomorphisms}, we investigate twisted component sums of vector-valued modular forms and obtain isomorphisms between spaces of vector-valued and scalar-valued modular forms in the case that the underlying finite quadratic module has order $p$ or $2p$, with an odd prime $p$. Finally, in Section~\ref{sec Doi-Naganuma}, we explain how these isomorphisms can be used to compute restrictions of Doi-Naganuma lifts of odd weight to components of Hirzebruch-Zagier curves.

\section{Modular forms for the Weil representation}

A \textbf{finite quadratic module} $(A,Q)$ consists of a finite abelian group $A$ and a nondegenerate $\mathbb{Q}/\mathbb{Z}$-valued quadratic form $Q$ on it. The \textbf{signature} of $(A,Q)$ is the number $\mathrm{sig}(A,Q) \in \mathbb{Z}/8\mathbb{Z}$ defined through the Gauss sum of $A$ by
\begin{align}\label{eq Milgram formula}
\mathbf{e}(\mathrm{sig}(A,Q)/8) = \frac{1}{\sqrt{|A|}} \sum_{\gamma \in A} \mathbf{e}(Q(\gamma)),
\end{align}
where $\mathbf{e}(x) = e^{2\pi i x}$. By Milgram's formula (\cite{MH}, appendix 4), this is also the signature mod $8$ of any even lattice which induces $(A,Q)$ as its discriminant form.

Let $\mathbb{C}[A]$ be the group algebra of $A$ with basis $\mathfrak{e}_{\gamma}$, $\gamma \in A,$ and let $\mathrm{Mp}_{2}(\mathbb{Z})$ be the integral metaplectic group, consisting of pairs $\left(\left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right),\pm \sqrt{c\tau + d}\right)$ with $\left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right) \in \mathrm{SL}_{2}(\mathbb{Z})$. The \textbf{dual Weil representation} $\rho^{*}$ is a unitary representation of $\mathrm{Mp}_{2}(\mathbb{Z})$ on $\mathbb{C}[A]$ which is defined on the generators $S = \left(\begin{psmallmatrix} 0 & -1 \\ 1 & 0 \end{psmallmatrix},\sqrt{\tau}\right)$ and $T =\left( \begin{psmallmatrix} 1 & 1 \\ 0 & 1 \end{psmallmatrix},1\right)$ by
\[
\rho^*(T) \mathfrak{e}_{\gamma} = \mathbf{e}(-Q(\gamma)) \mathfrak{e}_{\gamma}, \qquad
\rho^*(S) \mathfrak{e}_{\gamma} = \frac{\mathbf{e}(\mathrm{sig}(A,Q)/8) }{\sqrt{|A|}} \sum_{\beta \in A} \mathbf{e}(\langle \beta, \gamma \rangle) \mathfrak{e}_{\beta},
\]
where $\langle \beta, \gamma \rangle = Q(\beta + \gamma) - Q(\beta) - Q(\gamma)$ is the bilinear form associated to $Q$. We also write $\rho^{*}_{A}$ if we want to emphasize the dependence on $A$, or $\rho^{*}_{\Lambda}$ if $A$ is the discriminant form of an even lattice $\Lambda$.

A function $F: \H \to \mathbb{C}[A]$ is called a \textbf{weakly holomorphic modular form} of weight $k \in \frac{1}{2}\mathbb{Z}$ for $\rho^{*}$ if it is holomorphic on $\H$, if it satisfies
\[
F\left( \frac{a\tau+b}{c\tau+d}\right) = (c\tau+d)^{k}\rho^{*}\left(\begin{pmatrix}a & b \\ c & d \end{pmatrix},\sqrt{c\tau+d}\right)F(\tau)
\]
for all $\left(\left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right),\sqrt{c\tau+d}\right) \in \mathrm{Mp}_{2}(\mathbb{Z})$, and if it is meromorphic at $\infty$, which means that it has a Fourier expansion of the form
\[
F(\tau) = \sum_{\gamma \in A}\sum_{\substack{n \in \mathbb{Z}-Q(\gamma) \\ n \gg -\infty}}c(n,\gamma)q^{n}\mathfrak{e}_{\gamma},
\]
with coefficients $c(n,\gamma) \in \mathbb{C}$ and $q = e^{2\pi i \tau}$. Following \cite{BB}, we will denote the space of all these functions by $A_{k}(\rho^{*})$ (instead of the more common $M_{k}^{!}(\rho^{*})$). We let $M_{k}(\rho^{*})$ and $S_{k}(\rho^{*})$ be the subspaces of holomorphic modular forms and cusp forms, respectively. If $A$ is the discriminant form of an even lattice $\Lambda$, then we also write $A_{k}(Q)$ or $A_{k}(\mathbf{S})$ for $A_{k}(\rho^{*})$, where $Q$ is the quadratic form on $\Lambda$ and $\mathbf{S}$ is the Gram matrix of $Q$ with respect to some basis of $\Lambda$.

The element $Z = (-I,i) = S^{2}$ acts by $\rho^{*}(Z)\mathfrak{e}_{\gamma} = (-1)^{\mathrm{sig}(A,Q)/2}\mathfrak{e}_{-\gamma}$ which implies that $A_{k}(\rho^{*}) = 0$ if $k + \mathrm{sig}(A,Q)/2$ is not integral, and that the components of any weakly holomorphic modular form $F = \sum_{\gamma \in A}F_{\gamma}\mathfrak{e}_{\gamma} \in A_{k}(\rho^{*})$ satisfy
\[
F_{\gamma} = (-1)^{k+\mathrm{sig}(A,Q)/2}F_{-\gamma}
\]
for all $\gamma \in A$. Therefore we refer to $k$ as a \textbf{symmetric} or \textbf{antisymmetric weight} if $k+\mathrm{sig}(A,Q)/2$ is respectively even or odd.

\section{Vector-valued and scalar-valued modular forms}\label{sec isomorphisms}

In this section, we give isomorphisms between spaces $A_{k}(\rho^{*})$ of vector-valued modular forms for $\rho^{*}$ and scalar-valued modular forms for $\Gamma_{0}(p^{2})$ and $\Gamma_{0}(4p^{2})$ in the cases $|A| = p$ and $|A| = 2p$ with an odd prime $p$, for both symmetric and antisymmetric weights $k \in \frac{1}{2}\mathbb{Z}$.

\subsection{Finite quadratic modules of order $p$} Suppose that $|A| = p$ is an odd prime. Then $A \cong \mathbb{Z}/p\mathbb{Z}$ with $Q(\gamma) = \alpha\gamma^{2}/p$ for some $\alpha \in \mathbb{Z}$ with $p \nmid \alpha$. We put $\epsilon = \chi_{p}(\alpha) = \big( \frac{\alpha}{p}\big)$, and for odd $d \in \mathbb{Z}$ we let
\[
\varepsilon_{d} =
	\begin{cases}
		1, & p \equiv 1 \pmod 4, \\
		i, & p\equiv 3 \pmod 4.
	\end{cases}
\]
The evaluation of the quadratic Gauss sum $\sum_{n(p)}\mathbf{e}(\alpha n^{2}/p) = \chi_{p}(\alpha)\varepsilon_{p}\sqrt{p}$ and Milgram's formula \eqref{eq Milgram formula} show that $\epsilon\varepsilon_{p} = \mathbf{e}(\mathrm{sig}(A,Q)/8)$. Thus the signature $\mathrm{sig}(A,Q) \in \mathbb{Z}/8\mathbb{Z}$ depends on $p$ and $\epsilon$ as shown in the following table:
\begin{align*}
	\begin{array}{c||c|c}
	p \pmod 4 & 1 & 3 \\
	\hline\hline
	\epsilon =+1 & 0 & 2\\
	\hline
	\epsilon =-1 & 4 & 6 \\
	\end{array}
\end{align*}
In particular, $\mathrm{sig}(A,Q)$ is even. Hence we can assume that $k$ is an integer since otherwise $A_{k}(\rho^{*}) = 0$.

Let $\chi$ be a Dirichlet character mod $p$ and let $A_{k}(p^{2},\chi\otimes \chi_{p})$ be the space of scalar-valued weakly holomorphic modular forms of weight $k$ for $\Gamma_{0}(p^{2})$ with character $\chi\otimes \chi_{p}$. We assume that
\begin{align}\label{eq parity chi}
\chi(-1) = (-1)^{k+\mathrm{sig}(A,Q)/2}
\end{align}
since otherwise $A_{k}(p^{2},\chi\otimes\chi_{p}) = 0$. We define the subspace
\begin{align*}
A_{k}^{\epsilon}(p^{2},\chi\otimes\chi_{p})
= \left\{\sum_{\substack{n \in \mathbb{Z} \\ n\gg -\infty}}c(n)q^{n} \in A_{k}(p^{2},\chi\otimes\chi_{p})\ : \ \begin{aligned} & c(n) = 0 \text{ unless } n \in p(\mathbb{Z}-Q(\gamma)) \\ &\text{for some } \gamma \in (\mathbb{Z}/p\mathbb{Z})^{*} \end{aligned}\right\}.
\end{align*}
The condition in the brackets can be restated by saying that $c(n) = 0$ unless $\chi_{p}(-n)=\epsilon$. Hence we also call it the \textbf{$\epsilon$-condition}. Note that, in contrast to the definition of the $\epsilon$-condition in \cite{BB}, we also require that $c(n) = 0$ if $p \mid n$.

We define the \textbf{twisted component sum} of a vector-valued modular form $F(\tau) = \sum_{\gamma (p)}F_{\gamma}(\tau)\mathfrak{e}_{\gamma} \in A_{k}(\rho^{*})$ by
\[
\varphi_{\chi}\left( \sum_{\gamma (p)}F_{\gamma}(\tau)\mathfrak{e}_{\gamma}\right) = \sum_{\gamma (p)^{*}}\chi(\gamma)F_{\gamma}(p\tau).
\]
The parity condition \eqref{eq parity chi} ensures that $\varphi_{\chi}(F)$ is not trivially identically zero. Again, also for symmetric weight $k$ and with $\chi$ the trivial character mod $p$, our twisted component sum differs from the component sum $\sum_{\gamma(p)}F_{\gamma}(p\tau)$ considered in \cite{BB} since we omit the zero component $F_{0}(p\tau)$ in the sum. For antisymmetric weight $k$ we have $F_{0} = 0$.

\begin{prop}\label{prop varphi injective}
	If $F \in A_{k}(\rho^{*})$, then $\varphi_{\chi}(F) \in A_{k}^{\epsilon}(p^{2},\chi\otimes\chi_{p})$. Furthermore, $\varphi_{\chi}$ is injective.
\end{prop}

\begin{proof}
	Let $M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix} \right) \in \Gamma_{0}(p^{2})$ and $N = \left(\begin{smallmatrix}a & bp \\ c/p & d \end{smallmatrix} \right)$. We write
	\begin{align*}
	\varphi_{\chi}(F)(M\tau) = \sum_{\gamma (p)^{*}}\chi(\gamma)F_{\gamma}(p\cdot M\tau) = \sum_{\gamma (p)^{*}}\chi(\gamma)F_{\gamma}(N(p\tau)).
	\end{align*}
	By \cite{BReflection}, Theorem 5.2, we have $\rho^{*}(N)\mathfrak{e}_{\gamma} = \chi_{p}(d)\mathfrak{e}_{d\gamma}$, hence
	\[
	F_{\gamma}(N(p\tau)) =\chi_{p}(d) (c\tau + d)^{k}F_{d^{-1}\gamma}(p\tau),
	\]
	where $d^{-1}$ denotes an inverse of $d$ mod $p$. Thus we find
	\begin{align*}
	\varphi_{\chi}(F)(M\tau) &= \chi_{p}(d)(c\tau + d)^{k}\sum_{\gamma (p)^{*}}\chi(\gamma)F_{d^{-1}\gamma}(p\tau) = \chi\chi_{p}(d)(c\tau+d)^{k}\varphi_{\chi}(F)(\tau).
	\end{align*}
	It is clear that $\varphi_{\chi}(F)$ is holomorphic on $\H$ and meromorphic at the cusps (since $F_{\gamma}|_{k}M$ is a linear combination of components of $F$), and that it satisfies the $\epsilon$-condition, so $\varphi_{\chi}(F) \in A_{k}^{\epsilon}(p^{2},\chi\otimes\chi_{p})$.

	Now suppose that $\varphi_{\chi}(F) = 0$ for some $F \in A_{k}(\rho^{*})$. Since the components $F_{\gamma},F_{\beta}$ for $\beta \neq \pm \gamma$ are supported on disjoint index sets, $\varphi_{\chi}(F) = 0$ implies that $F_{\gamma} = 0$ for all $\gamma \neq 0$. But then the zero component of $F$ satisfies
	\[
	F_{0}|_{k}S = \frac{\mathbf{e}(\mathrm{sig}(A,Q)/8)}{\sqrt{p}}\sum_{\gamma(p)}F_{\gamma} = \frac{\mathbf{e}(\mathrm{sig}(A,Q)/8)}{\sqrt{p}}F_{0},
	\]
	and applying $|_{k}S$ a second time, we find $(-1)^{k}F_{0} = \frac{\mathbf{e}(\mathrm{sig}(A,Q)/4)}{p}F_{0}$, hence $F_{0} = 0$. We have shown that $F = 0$, so $\varphi_{\chi}$ is injective.
\end{proof}

The map $\varphi_{\chi}$ is in general not surjective. Its image can be described in terms of the behaviour of certain twists of $G$ under the Atkin-Lehner involution, which we explain now.

We can split $G(\tau) = \sum_{n \gg -\infty}c(n)q^{n} \in A_{k}^{\epsilon}(p^{2},\chi\otimes\chi_{p})$ into components
\[
G(\tau) = \sum_{\gamma(p)^{*}}G_{\gamma}(\tau), \qquad G_{\gamma}(\tau) = \frac{1}{2}\sum_{\substack{n \in p(\mathbb{Z}-Q(\gamma)) \\ n\gg -\infty}}c(n)q^{n}.
\]
We define the \textbf{component-wise twist} of $G$ by a Dirichlet character $\psi$ mod $p$ by
\[
G_{\psi}(\tau) = \sum_{\gamma (p)^{*}}\psi(\gamma)G_{\gamma}(\tau).
\]
Note that the component-wise twist differs from the usual twist $\sum_{n \gg-\infty}\psi(n)c(n)q^{n}$ of a modular form.

\begin{lem}\label{lem twisting}
	Let $G \in A_{k}^{\epsilon}(p^{2},\chi\otimes \chi_{p})$ and let $\psi$ be a Dirichlet character mod $p$. Then $G_{\psi} \in A_{k}^{\epsilon}(p^{2},\psi\otimes\chi\otimes \chi_{p})$.
\end{lem}

\begin{proof}
	We can write
	\[
	G_{\psi} = \frac{1}{2p}\sum_{\gamma (p)^{*}}\psi(\gamma)\sum_{j(p)}G\bigg|_{k}\begin{pmatrix}1 &j/p \\ 0 &1 \end{pmatrix}\mathbf{e}(jQ(\gamma)).
	\]
	Let $M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right) \in \Gamma_{0}(p^{2})$. We compute
	\[
	\begin{pmatrix}1 &j/p \\ 0 &1 \end{pmatrix}\begin{pmatrix}a & b \\ c & d \end{pmatrix}\begin{pmatrix}1 &-d^{2}j/p \\ 0 &1 \end{pmatrix} = \begin{pmatrix}a + cj/p & b+(1-ad)dj/p-d^{2}cj^{2}/p^{2} \\ c & d-d^{2}cj/p \end{pmatrix} \in \Gamma_{0}(p^{2}).
	\]
	Note that the $d$-entry of this matrix equals $d$ mod $p$. Hence we obtain
	\begin{align*}
	G_{\psi}|_{k}M = \chi\chi_{p}(d)\frac{1}{2p}\sum_{\gamma \in (p)^{*}}\psi(\gamma)\sum_{j(p)}G\bigg|_{k}\begin{pmatrix}1 &d^{2}j/p \\ 0 &1 \end{pmatrix}\mathbf{e}(jQ(\gamma)).
	\end{align*}
	Replacing $d^{2}j$ by $j$ and then $\gamma$ by $d\gamma$ gives a factor $\psi(d)$ and completes the proof.
\end{proof}

We let $W_{p^{2}} = \left(\begin{smallmatrix}0 & - 1 \\ p^{2} & 0 \end{smallmatrix} \right)$ be the  Atkin-Lehner (or Fricke) involution. It maps $A_{k}(p^{2},\chi\otimes\chi_{p})$ to $A_{k}(p^{2},\overline{\chi}\otimes \chi_{p})$, but it does in general not respect the $\epsilon$-condition. We say that $G \in A_{k}^{\epsilon}(p^{2},\chi\otimes\chi_{p})$ satisfies the \textbf{Atkin-Lehner condition} if the twists of $G$ by all Dirichlet characters $\psi$ mod $p$ with $\psi \neq \overline{\chi}$ satisfy
\begin{align}\label{AL}
G_{\psi}|_{k}W_{p^{2}} = \overline{\psi\chi(2\alpha)}\frac{g(\psi\chi)}{\sqrt{p}}\mathbf{e}(\mathrm{sig}(A,Q)/8)G_{\overline{\psi}\overline{\chi}^{2}},
\end{align}
where $g(\psi\chi) = \sum_{n(p)^{*}}\psi\chi(n)\mathbf{e}(n/p)$ is a Gauss sum. We let $A_{k}^{\epsilon,\mathrm{AL}}(p^{2},\chi\otimes\chi_{p})$ be the subspace of $A_{k}^{\epsilon}(p^{2},\chi\otimes \chi_{p})$ consisting of all forms satisfying the Atkin-Lehner condition.

\begin{prop}\label{prop isomorphism p}
	The linear map
	\[
	\varphi_{\chi}: A_{k}(\rho^{*}) \to A_{k}^{\epsilon,\mathrm{AL}}(p^{2},\chi\otimes \chi_{p}), \qquad F(\tau) = \sum_{\gamma(p)}F_{\gamma}(\tau)\mathfrak{e}_{\gamma} \mapsto \sum_{\gamma(p)^{*}}\chi(\gamma)F_{\gamma}(p\tau),
	\]
	is an isomorphism. The inverse map is given by
	\[
	\varphi_{\chi}^{-1}: G(\tau) = \sum_{\gamma(p)^{*}}G_{\gamma}(\tau) \mapsto \sum_{\gamma(p)^{*}}\overline{\chi(\gamma)}G_{\gamma}(\tau/p)\mathfrak{e}_{\gamma} + G_{0}(\tau/p)\mathfrak{e}_{0},
	\]
	where $G_{0}$ is defined by
	\[
	G_{0}(\tau) = \frac{\sqrt{p}}{p-1}\mathbf{e}(-\mathrm{sig}(A,Q)/8)\left(G_{\overline{\chi}}|_{k}W_{p^{2}}\right)(\tau)+\frac{1}{p-1}G_{\overline{\chi}}(\tau).
	\]
\end{prop}

\begin{proof}
	We first show that $G = \varphi_{\chi}(F)$ for $F = \sum_{\gamma(p)}F_{\gamma}\mathfrak{e}_{\gamma} \in A_{k}(\rho^{*})$ satisfies the Atkin-Lehner condition. Let $\psi$ be a Dirichlet character mod $p$ and let $\delta_{\psi,\overline{\chi}} = 1$ if $\psi = \overline{\chi}$ and $\delta_{\psi,\overline{\chi}} = 0$ otherwise. We compute
	\begin{align*}
	\left(G_{\psi}|_{k}W_{p^{2}}\right)(\tau) &= p^{k}(p^{2}\tau )^{-k}G_{\psi}\left(-\frac{1}{p^{2}\tau}\right) \\
	&= \sum_{\gamma(p)^{*}}\psi(\gamma)\chi(\gamma)(p\tau)^{-k}F_{\gamma}\left(-\frac{1}{p\tau} \right) \\
	&= \sum_{\gamma(p)^{*}}\psi(\gamma)\chi(\gamma)\frac{\mathbf{e}(\mathrm{sig}(A,Q)/8)}{\sqrt{p}}\sum_{\beta(p)}\mathbf{e}((\beta,\gamma))F_{\beta}(p\tau) \\
	&= \frac{\mathbf{e}(\mathrm{sig}(A,Q)/8)}{\sqrt{p}}\left(\sum_{\beta(p)^{*}}\left(\sum_{\gamma(p)^{*}}\psi(\gamma)\chi(\gamma)\mathbf{e}(2\alpha \beta \gamma/p) \right) F_{\beta}(p\tau)+ \delta_{\psi,\overline{\chi}}(p-1)F_{0}(p\tau)\right)\\
	&= \frac{\mathbf{e}(\mathrm{sig}(A,Q)/8)}{\sqrt{p}}\left(\sum_{\gamma (p)^{*}}\psi(\gamma)\chi(\gamma)\mathbf{e}(2\alpha \gamma/p)\sum_{\beta(p)^{*}}\overline{\psi(\beta)}\overline{\chi(\beta)} F_{\beta}(p\tau) + \delta_{\psi,\overline{\chi}}(p-1)F_{0}(p\tau)\right)\\
	&= \frac{\mathbf{e}(\mathrm{sig}(A,Q)/8)}{\sqrt{p}}\left(\overline{\psi\chi(2\alpha)}g(\psi\chi)G_{\overline{\psi}\overline{\chi}^{2}}(\tau) + \delta_{\psi,\overline{\chi}}(p-1)F_{0}(p\tau)\right).
	\end{align*}
	This shows that $G$ satisfies the Atkin-Lehner condition, i.e., $G \in A_{k}^{\epsilon,\mathrm{AL}}(p^{2},\chi\otimes \chi_{p})$.

	Conversely, if $G \in A_{k}^{\epsilon,\mathrm{AL}}(p^{2},\chi\otimes\chi_{p})$ and if $G_{0}$ is defined as in the proposition, then we can reverse the above computation (with $F_{0}(p\tau) = G_{0}(\tau)$ and $F_{\gamma}(p\tau) = \overline{\chi(\gamma)}G_{\gamma}(\tau)$ for $\gamma \neq 0$) to see that the second and third line agree for all Dirichlet characters $\psi$ mod $p$. By character orthogonality, we obtain that
	\[
	F_{\gamma}|_{k}S = \frac{\mathbf{e}(\mathrm{sig}(A,Q)/8)}{\sqrt{p}}\sum_{\beta(p)}\mathbf{e}((\beta,\gamma))F_{\beta}
	\]
	for all $\gamma \neq 0$. A short computation shows that this equation also implies
	\[
	F_{0}|_{k}S = \frac{\mathbf{e}(\mathrm{sig}(A,Q)/8)}{\sqrt{p}}\sum_{\beta(p)}F_{\beta}.
	\]
	Furthermore, it is easy to check that $F_{\gamma}$ transforms correctly under $T$. We find that $\varphi_{\chi}^{-1}(G) \in A_{k}(\rho^{*})$. Since $\varphi_{\chi}\circ \varphi_{\chi}^{-1} = \mathrm{id}$ and $\varphi_{\chi}$ is injective, $\varphi_{\chi}$ is an isomorphism.
\end{proof}

\subsection{Finite quadratic modules of order $2p$} Suppose that $|A| = 2p$ with an odd prime $p$. Then $A \cong \mathbb{Z}/2p\mathbb{Z}$ with the quadratic form
\[
Q(p\gamma_{1} + \gamma_{2}) = \delta \gamma_{1}^{2}/4+\alpha \gamma_{2}^{2}/p
\]
for $\gamma_{1} \in \mathbb{Z}/2\mathbb{Z}$ and $\gamma_{2} \in \mathbb{Z}/p\mathbb{Z}$, where $\delta \in \{\pm 1\}$ and $\alpha \in \mathbb{Z}$ with $p \nmid \alpha$. Set $\epsilon = \chi_{p}(\alpha)$. Using the quadratic Gauss sum and Milgram's formula we obtain $(1+\delta i)\epsilon\varepsilon_{p} = \sqrt{2}\mathbf{e}(\mathrm{sig}(A,Q)/8)$, so the signature $\mathrm{sig}(A,Q) \in\mathbb{Z}/8\mathbb{Z}$ is given in terms of $p, \epsilon$ and $\delta$ as follows:
\begin{align*}
	\begin{array}{c||c|c}
	p \pmod 4 & 1 & 3 \\
	\hline\hline
	\epsilon = +1 & \delta & \delta+2\\
	\hline
	\epsilon = -1 & \delta+4 & \delta+6
	\end{array}
\end{align*}
Now $\mathrm{sig}(A,Q)$ is odd. Hence we can assume that $k$ is half-integral since otherwise $A_{k}(\rho^{*}) = 0$.

Let us briefly recall the definition of modular forms of half-integral weight. The theta multiplier is given by
\[
\nu_{\vartheta}(M)= \bigg(\frac{c}{d}\bigg)\varepsilon_{d}^{-1}, \qquad \varepsilon_{d} = \bigg(\frac{2}{d}\bigg)\mathbf{e}((1-d)/8)= \begin{cases} 1, & d \equiv 1(4), \\
i,&  d \equiv 3 (4),  \end{cases}
\]
for $M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right) \in \Gamma_{0}(4)$. A function $G:\H \to \mathbb{C}$ is called a weakly holomorphic modular form of weight $k \in \frac{1}{2}+\mathbb{Z}$ for $\Gamma_{0}(N)$ with $4 \mid N$ and character $\chi$ mod $N$ if it is holomorphic on $\H$ and meromorphic at the cusps, and if it transforms as
\[
G(M\tau) = \chi(M)\nu_{\vartheta}(M)^{\pm 1}(c\tau+d)^{k}G(\tau)
\]
for $M = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right) \in \Gamma_{0}(N)$, where the sign in $\nu_{\vartheta}(M)^{\pm 1}$ is chosen such that $\nu_{\vartheta}(-1)^{\pm 1}\chi(-1)i^{2k} = 1$. We denote the space of all these functions by $A_{k}(N,\chi)$.

We let $\chi$ be a character mod $p$ and we again assume that
\[
\chi(-1) = (-1)^{k+\mathrm{sig}(A,Q)/2}.
\]
We consider the space
\begin{align*}
A_{k}^{\epsilon}(16p^{2},\chi)= \left\{\sum_{n\gg -\infty}c(n)q^{n} \in A_{k}(16p^{2},\chi) \ : \  \begin{aligned} & c(n) = 0 \text{ unless } n \in 4p(\mathbb{Z}-Q(\gamma)) \\ & \text{for some } \gamma \in (\mathbb{Z}/2p\mathbb{Z}) \setminus \{0,p\} \end{aligned}\right\}
\end{align*}
and we let $A_{k}^{\epsilon}(4p^{2},\chi) = A_{k}^{\epsilon}(16p^{2},\chi) \cap A_{k}(4p^{2},\chi)$.

We define the \textbf{twisted component sum} of $F(\tau) = \sum_{\gamma(2p)}F_{\gamma}(\tau)\mathfrak{e}_{\gamma} \in A_{k}(\rho^{*})$ by
\[
\varphi_{\chi}\left(\sum_{\gamma(2p)}F_{\gamma}(\tau)\mathfrak{e}_{\gamma} \right) = \sum_{\gamma(2p)}\chi(\gamma)F_{\gamma}(4p\tau).
\]
Note that, since $\chi$ is a character mod $p$, we discard the components $F_{0}$ and $F_{p}$ in the twisted component sum. If $k$ is an antisymmetric weight, then $F_{0} = F_{p} = 0$ is automatic. This map was already suggested in \cite{EZ}, p. 70, in the context of Jacobi forms.

\begin{prop}
	If $F \in A_{k}(\rho^{*})$, then $\varphi_{\chi}(F) \in A_{k}^{\epsilon}(4p^{2}, \chi)$. Furthermore, $\varphi_{\chi}$ is injective.
\end{prop}

\begin{proof}
	We first show that that $\varphi_{\chi}(F)$ transforms correctly under $\Gamma_{0}(16p^{2})$. For $M = \left( \begin{smallmatrix}a & b \\ c & d  \end{smallmatrix}\right) \in \Gamma_{0}(16p^{2})$ we let $N = \left( \begin{smallmatrix}a & 4pb \\ c/4p & d  \end{smallmatrix}\right)$. Then we compute
	\[
	\varphi_{\chi}(F)(M\tau) = \sum_{\gamma(2p)}\chi(\gamma)F_{\gamma}(4p\cdot M\tau) = \sum_{\gamma(2p)}\chi(\gamma)F_{\gamma}(N(4p\tau)).
	\]
	Using \cite{BReflection}, Theorem~5.2, we obtain
	\begin{align*}
	F_{\gamma}(N(4p\tau)) &= \bigg(\frac{c/4p}{d}\bigg)\bigg( \frac{d}{2p}\bigg)\mathbf{e}((1-d)\delta/8)(c\tau+d)^{k}F_{d^{-1}\gamma}(4p\tau).
	\end{align*}
	We compute
	\begin{align*}
	\bigg(\frac{c/4p}{d}\bigg)\bigg( \frac{d}{2p}\bigg)\mathbf{e}((1-d)\delta/8) &=\bigg(\frac{p}{d}\bigg)\bigg(\frac{d}{p}\bigg)\bigg( \frac{c}{d}\bigg)\bigg( \frac{2}{d}\bigg)\mathbf{e}((1-d)\delta/8) = \bigg(\frac{p}{d}\bigg)\bigg( \frac{d}{p}\bigg)\nu_{\vartheta}(M)^{-\delta}.
	\end{align*}
	By quadratic reciprocity we have $\big(\frac{p}{d}\big)\big( \frac{d}{p}\big) = 1$ if $p \equiv 1 \pmod 4$ and
	\[
	\bigg(\frac{p}{d}\bigg)\bigg( \frac{d}{p}\bigg) = \begin{cases}1, & \text{if } d \equiv 1 \pmod 4, \\
	-1, & \text{if } d\equiv 3 \pmod 4, \end{cases}
	\]
	if $p \equiv 3 \pmod 4$. This gives the stated transformation behaviour under $\Gamma_{0}(16p^{2})$.

	In order to show the transformation behaviour under $\Gamma_{0}(4p^{2})$, it suffices to check the transformation under the matrices $U_{4p^{2}j} = \left(\begin{smallmatrix}1 & 0 \\ 4p^{2}j &1 \end{smallmatrix}\right)$ for $j = 0,1,2,3$ since they represent $\Gamma_{0}(16p^{2})\backslash \Gamma_{0}(4p^{2})$. We compute
	\begin{align*}
	\varphi_{\chi}(F)(U_{4p^{2}j}\tau) &= \sum_{\gamma(2p)}\chi(\gamma)F_{\gamma}(4p\cdot U_{4p^{2}j} \tau) \\
	&=  \sum_{\gamma(2p)}\chi(\gamma)F_{\gamma}(U_{pj}(4p\tau)) \\
	&= \sum_{\gamma(2p)}\chi(\gamma)F_{\gamma}(S^{-1}T^{-pj}S(4p\tau)) \\
	&= (4p^{2}j\tau+1)^{k}\frac{1}{2p}\sum_{\gamma(2p)}\chi(\gamma)\sum_{\beta(2p)}\mathbf{e}(-(\beta,\gamma))\mathbf{e}(jpQ(\beta))\sum_{\mu(2p)}\mathbf{e}((\mu,\beta))F_{\mu}(4p\tau).
	\end{align*}
	If we write $\gamma = p\gamma_{1}+\gamma_{2}$ with $\gamma_{1} \in \mathbb{Z}/2\mathbb{Z}$ and $\gamma_{2} \in \mathbb{Z}/p\mathbb{Z}$, and similarly for $\beta$, and use that $\chi$ only depends on $\gamma_{2}$, we see that the sum over $\gamma_{1}$ vanishes unless $\beta_{2} = 0$. This means that we can replace $\mathbf{e}(jpQ(\beta))$ by $1$ in the above sum. But then the sum over $\beta$ equals $2p$ if $\mu = \gamma$, and vanishes otherwise. Hence we get
	\begin{align*}
	\varphi_{\chi}(F)(U_{4p^{2}j}\tau)&=(4p^{2}j\tau+1)^{k}\sum_{\gamma(2p)}\chi(\gamma)F_{\gamma}(4p\tau) =(4p^{2}j\tau+1)^{k}\varphi_{\chi}(F)(\tau).
	\end{align*}
	This shows that $\varphi_{\chi}(F)$ transforms correctly under $\Gamma_{0}(4p^{2})$. It is easy to see that $\varphi_{\chi}$ is holomorphic on $\H$ and meromorphic at the cusps, and that it satisfies the $\epsilon$-condition.

	Now suppose that $\varphi_{\chi}(F) = 0$ for some $F \in A_{k}(\rho^{*})$. By comparing the index sets on which the components $F_{\gamma}$ are supported, we obtain that $F_{\gamma} = 0$ for $\gamma \notin \{0,p\}$. Then the transformation behaviour of $F$ under $S$ implies
	\[
	F_{0}(-1/\tau) = \tau^{k}\frac{\mathbf{e}(\mathrm{sig}(A,Q)/8)}{\sqrt{2p}}(F_{0}(\tau)+F_{p}(\tau)), \qquad 	F_{p}(-1/\tau) = \tau^{k}\frac{\mathbf{e}(\mathrm{sig}(A,Q)/8)}{\sqrt{2p}}(F_{0}(\tau)-F_{p}(\tau)).
	\]
	Applying $\tau \mapsto -1/\tau$ a second time, we get $i^{2k}F_{0}= \frac{\mathbf{e}(\mathrm{sig}(A,Q)/4)}{p}F_{0}$ and $i^{2k}F_{p} = \frac{\mathbf{e}(\mathrm{sig}(A,Q)/4)}{p}F_{p}$, hence $F_{0} = F_{p} = 0$. Thus $F = 0$ and $\varphi_{\chi}$ is injective.
\end{proof}

We split $G = \sum_{n\gg-\infty}c(n)q^{n}\in A_{k}^{\epsilon}(16p^{2},\chi)$ into components
\[
G(\tau) = \sum_{\gamma(2p)}G_{\gamma}(\tau), \qquad G_{\gamma}(\tau) = \frac{1}{2}\sum_{\substack{n \in 4p(\mathbb{Z} - Q(\gamma)) \\ n \gg -\infty}}c(n)q^{n},
\]
and define its \textbf{component-wise twist} by a Dirichlet character $\psi$ mod $p$ by
\[
G_{\psi}(\tau) = \sum_{\gamma(2p)}\psi(\gamma)G_{\gamma}(\tau).
\]

\begin{lem}
	Let $G \in A_{k}^{\epsilon}(16p^{2},\chi)$ and let $\psi$ be a Dirichlet character mod $p$. Then $G_{\psi} \in A_{k}^{\epsilon}(16p^{2},\psi\otimes \chi)$.
\end{lem}

\begin{proof}
	The proof is analogous to the proof of Lemma~\ref{lem twisting}, so we leave the details to the reader.
\end{proof}

In contrast to the case $|A| = p$ we need another notion to describe the image of $\varphi_{\chi}$. We call $\gamma \in \mathbb{Z}/2p\mathbb{Z}$ even if $4pQ(\gamma)$ is even, and odd if $4pQ(\gamma)$ is odd. The \textbf{even} and \textbf{odd parts} of $G \in A_{k}^{\epsilon}(16p^{2},\chi)$ are defined by
\begin{align*}
	G^{\mathrm{even}}(\tau) = \sum_{\substack{\gamma (2p) \\ \gamma \, \mathrm{even}}}G_{\gamma}(\tau), \qquad G^{\mathrm{odd}}(\tau) = \sum_{\substack{\gamma (2p) \\ \gamma \, \mathrm{odd}}}G_{\gamma}(\tau).
\end{align*}
Note that taking the even and odd parts of $G$ commutes with component-wise twisting.

\begin{lem}
	If $G \in A_{k}^{\epsilon}(16p^{2},\chi)$ then $G^{\mathrm{even}},G^{\mathrm{odd}} \in A_{k}^{\epsilon}(16p^{2},\chi)$ as well.
\end{lem}

\begin{proof}
	We can write
	\[
	G^{\mathrm{even}}(\tau) = \frac{1}{2}\left(G(\tau)+ G(\tau+1/2) \right), \qquad G^{\mathrm{odd}}(\tau) = \frac{1}{2}\left(G(\tau)- G(\tau+1/2) \right).
	\]
	For $\left(\begin{smallmatrix}a & b \\ c & d\end{smallmatrix}\right) \in \Gamma_{0}(16p^{2})$ we have
	\[
	\begin{pmatrix}1 & 1/2 \\ 0 & 1\end{pmatrix}\begin{pmatrix}a & b \\ c & d\end{pmatrix}\begin{pmatrix}1 & -1/2 \\ 0 & 1\end{pmatrix} = \begin{pmatrix}a+c/2 & b+(d-a)/2-c/4 \\ c & d-c/2\end{pmatrix} \in \Gamma_{0}(16p^{2}),
	\]
	which easily implies $G(\tau+1/2) \in A_{k}^{\epsilon}(16p^{2},\chi)$. This proves the lemma.
\end{proof}

For $G \in A_{k}(16p^{2},\chi)$ we define the Atkin-Lehner involution
\[
G|_{k}W_{16p^{2}} = (4p)^{-k}(-i\tau)^{-k}G(-1/16p^{2}\tau).
\]
Then $G|_{k}W_{16p^{2}} \in A_{k}(16p^{2},\overline{\chi})$ and $G|_{k}W_{16p^{2}}|_{k}W_{16p^{2}} = G$. In general, the Atkin-Lehner involution does not preserve the $\epsilon$-condition. We say that $G \in A_{k}^{\epsilon}(16p^{2},\chi)$ satisfies the \textbf{Atkin-Lehner condition} if its twists by all Dirichlet characters $\psi$ mod $p$ with $\psi \neq \overline{\chi}$ satisfy
\begin{align*}
	G_{\psi}|_{k}W_{16p^{2}} &= \overline{\psi\chi(2\alpha)}\frac{\sqrt{2}g(\psi\chi)}{\sqrt{p}}\mathbf{e}(\mathrm{sig}(A,Q)/8)i^{k}G_{\overline{\psi}\overline{\chi}^{2}}^{\mathrm{even}},
\end{align*}
where $g(\psi\chi) = \sum_{n (p)^{*}}\psi\chi(n)\mathbf{e}(n/p)$ is a Gauss sum. Let $A_{k}^{\epsilon,\mathrm{AL}}(16p^{2},\chi)$ be the subspace of $A_{k}^{\epsilon}(16p^{2},\chi)$ satisfying the Atkin-Lehner condition. Note that, after applying $W_{16p^{2}}$ and a short calculation, the Atkin-Lehner condition also implies that
\begin{align*}
(G_{\psi}^{\mathrm{even}}-G_{\psi}^{\mathrm{odd}})|_{k}W_{16p^{2}} &= \overline{\psi\chi(2\alpha)}\frac{\sqrt{2}g(\psi\chi)}{\sqrt{p}}\mathbf{e}(\mathrm{sig}(A,Q)/8)i^{k}G_{\overline{\psi}\overline{\chi}^{2}}^{\mathrm{odd}}
\end{align*}
for $\psi \neq \overline{\chi}$.

\begin{prop}\label{prop isomorphism 2p}
	The linear map
	\[
	\varphi_{\chi}: A_{k}(\rho^{*}) \to A_{k}^{\epsilon,\mathrm{AL}}(16p^{2},\chi), \qquad F(\tau) = \sum_{\gamma(2p)}F_{\gamma}(\tau)\mathfrak{e}_{\gamma} \mapsto \sum_{\gamma(2p)}\chi(\gamma)F_{\gamma}(4p\tau),
	\]
	is an isomorphism. The inverse map is given by
	\[
	\varphi_{\chi}^{-1}: G(\tau) = \sum_{\gamma(2p)}G_{\gamma}(\tau) \mapsto \sum_{\substack{\gamma(2p) \\ \gamma \neq 0,p (2p)}}\overline{\chi(\gamma)}G_{\gamma}(\tau/4p)\mathfrak{e}_{\gamma} + G_{0}(\tau/4p)\mathfrak{e}_{0} + G_{p}(\tau/4p)\mathfrak{e}_{p},
	\]
	where $G_{0}$ and $G_{p}$ are defined by
	\begin{align*}
	G_{0}(\tau) &= \frac{\sqrt{p}}{\sqrt{2}(p-1)}\mathbf{e}(-\mathrm{sig}(A,Q)/8)i^{-k}(G_{\overline{\chi}}^{\mathrm{even}}(\tau)+G_{\overline{\chi}}^{\mathrm{odd}}(\tau))|_{k}W_{16p^{2}} +\frac{1}{p-1} G_{\overline{\chi}}^{\mathrm{even}}(\tau),\\
	G_{p}(\tau) &= \frac{\sqrt{p}}{\sqrt{2}(p-1)}\mathbf{e}(-\mathrm{sig}(A,Q)/8)i^{-k}(G_{\overline{\chi}}^{\mathrm{even}}(\tau)-G_{\overline{\chi}}^{\mathrm{odd}}(\tau))|_{k}W_{16p^{2}} +\frac{1}{p-1} G_{\overline{\chi}}^{\mathrm{odd}}(\tau).
	\end{align*}
\end{prop}

\begin{proof}
	The proof is very similar to the proof of Proposition~\ref{prop isomorphism p}, so we omit it for brevity.
\end{proof}

\section{Application: the Doi-Naganuma lift and theta contraction} \label{sec Doi-Naganuma}

Let $K = \mathbb{Q}(\sqrt{p})$ for a prime $p \equiv 1 \pmod 4$ and let $\mathcal{O}_K$ be its ring of integers. Let $\mathcal{O}_K^{\#} = (1 / \sqrt{p}) \mathcal{O}_K$ be the dual lattice of $\mathcal{O}_K$ with respect to the trace and consider the finite quadratic module $(\mathcal{O}_K^{\#}/\mathcal{O}_K,-N_{K/\mathbb{Q}})$. It has order $p$ and signature $0$ mod $8$. Let $\lambda \in \mathcal{O}_K$ be any totally positive prime with norm $\ell = N_{K/\mathbb{Q}}(\lambda)$, and let $b \in \mathbb{Z}$ be any integer with $b^2 \equiv p \pmod {4\ell}$ (which exists by quadratic reciprocity). Since $\lambda$ is prime, one of $(b + \sqrt{p}) / 2\lambda, (b - \sqrt{p})/2\lambda$ is integral; it is then straightforward to show that $\{\lambda', (b \pm \sqrt{p})/2\lambda\}$ is a $\mathbb{Z}$-basis of $\mathcal{O}_K$ and
\[
	\mathbf{S} = \begin{pmatrix} -2\ell & -b \\ -b & \frac{p - b^2}{2\ell} \end{pmatrix}
\]
is the Gram matrix of $-N_{K/\mathbb{Q}}$ in that basis.

If $\ell \neq p$, then for each $r \in \mathbb{Z}/p\mathbb{Z}$ there exists a unique $a \in \mathbb{Z}/2\ell\mathbb{Z}$ with $r \equiv ab \pmod {2\ell}$. We fix a bijection $\mathcal{O}_{K}^{\#}/\mathcal{O}_{K}\cong \mathbb{Z}/p\mathbb{Z}$ by sending $r \in \mathbb{Z}/p\mathbb{Z}$ to the element
\[
\gamma_{a,r}+\mathcal{O}_{K} = \frac{a \pm r/\sqrt{p}}{2 \lambda}+\mathcal{O}_{K} \in \mathcal{O}_{K}^{\#}/\mathcal{O}_{K}.
\]
If $\ell = p$, then we fix the bijection which identifies $a \in \mathbb{Z}/p\mathbb{Z}$ with $\gamma_a + \mathcal{O}_K = \frac{a(1 + \sqrt{p})}{2 \lambda}+\mathcal{O}_{K}$ instead. (In this case, one can always take $b = p$ and $\lambda = \varepsilon \sqrt{p}$ if $\varepsilon$ is the fundamental unit of $\mathcal{O}_K$.) \\

The \textbf{theta decomposition} identifies vector-valued modular forms for the dual Weil representation attached to $(\mathcal{O}_K^{\#}/\mathcal{O}_K,-N_{K/\mathbb{Q}})$ with vector-valued Jacobi forms of fractional index $p/4\ell$ for the dual Weil representation attached to the discriminant form with Gram matrix $(-2\ell)$ and a particular representation of the Heisenberg group, see e.g. \cite{W}. By setting the Heisenberg variable of those Jacobi forms equal to zero one obtains the \textbf{theta contraction}, a graded homomorphism between the modular forms $M_*(-N_{K/\mathbb{Q}})$ and $M_{*+1/2}((-2\ell))$ as graded modules over the ring $M_*(\mathrm{SL}_2(\mathbb{Z}))$ of scalar-valued modular forms. This was introduced by Ma \cite{M} in order to study the quasi-pullback of Borcherds products. Explicitly in terms of Fourier coefficients, it is the map
\[
	\Theta : M_k(-N_{K/\mathbb{Q}}) \to M_{k+1/2}((-2\ell)), \quad \sum_{\gamma \in \mathcal{O}_K^{\#}/\mathcal{O}_K} \sum_{n \in \mathbb{Z} + N_{K/\mathbb{Q}}(\gamma)} c(n,\gamma) q^n \mathfrak{e}_{\gamma} \mapsto \sum_{a \in \mathbb{Z}/2\ell \mathbb{Z}} \sum_{n \in \mathbb{Z} + a^2 / 4 \ell} \tilde c(n,a) q^n \mathfrak{e}_a,
\]
where
\[
	\tilde c(n,a) = \sum_{r \equiv ab \, (2 \ell)} c\left( n - \frac{r^2}{4 \ell p}, \gamma_{a,r}\right).
\]

Possibly the most important aspect of the theta contraction (which can be defined more generally) is that it fits into a commutative diagram involving the additive theta lift (of Oda and Rallis-Schiffmann) and restriction to Heegner divisors: letting $\Lambda$ be an even lattice of type $(2,b^-)$ such that $b^{-} \geq 2$ is greater than the Witt rank of $\Lambda$, and $\lambda^{\perp}$ the orthogonal complement of a primitive, negative-norm vector $\lambda \in \Lambda$, the natural pullback map $\mathrm{Res}$ for orthogonal modular forms satisfies
\begin{center} \begin{tikzpicture}[node distance=2.5cm, auto]
\node (A) {$S_{k + 1 - b^-/2}(\rho^*_{\Lambda})$};
\node (AA) [right of=A] {$ $};
\node (B) [right of=AA] {$S_{k}(O(\Lambda))$};
\node (C) [below of=A] {$S_{k+1 - (b^- - 1)/2}(\rho^*_{\lambda^{\perp}})$};
\node (D) [below of=B] {$S_{k}(O(\lambda^{\perp}))$};
\draw[->] (B) to node {$\text{Res}$} (D);
\draw[->] (A) to node {$\Theta$} (C);
\draw[->] (A) to node {$\text{Theta lift}$} (B);
\draw[->] (C) to node {$\text{Theta lift}$} (D);
\end{tikzpicture}
\end{center}
for $k \geq 2$.

For the Doi-Naganuma lift (i.e. $b^{-} =2$) this can be made very explicit. Recall that for a cusp form
\[
	F(\tau) = \sum_{\gamma \in \mathcal{O}_K^{\#} / \mathcal{O}_K} \sum_{n \in \mathbb{Z} + N_{K/\mathbb{Q}}(\gamma)} c(n,\gamma) q^n \mathfrak{e}_{\gamma} \in S_k(-N_{K/\mathbb{Q}})
\]
of weight $k \ge 2$, the Doi-Naganuma lift is a Hilbert cusp form $\Phi_F$ of weight $k$ for $\mathrm{SL}_2(\mathcal{O}_K)$ with Fourier expansion
\[
	\Phi_F(\tau_1,\tau_2) = \sum_{\substack{\nu \in \mathcal{O}_K^{\#} \\ \nu,\nu' > 0}}\sum_{n=1}^{\infty} c(\nu \nu', \nu) n^{k-1} \mathbf{q}^{n \nu}, \quad \text{where} \; \; \mathbf{q}^{\nu} = e^{2\pi i (\nu \tau_1 + \nu' \tau_2)}.
\]
Moreover $\Phi_F$ satisfies the graded symmetry $\Phi_F(\tau_1,\tau_2) = (-1)^k \Phi_F(\tau_2,\tau_1)$.

With $\lambda$ as above, there is a natural restriction map onto a component of the Hirzebruch-Zagier curve $T_{\ell}$:
\[
	\mathrm{Res} : S_k(\mathrm{SL}_2(\mathcal{O}_K)) \to S_{2k}(\Gamma_0(\ell)), \quad f(\tau_1,\tau_2) \mapsto f(\lambda \tau, \lambda' \tau).
\]
It turns out that $\mathrm{Res}(\Phi_{F})$ equals the Shimura lift
\[
\sum_{a=1}^{\infty}\sum_{n=1}^{\infty} \tilde c(a^2 / 4\ell, a) n^{k-1} q^{na}
\]
of the contracted form $\Theta F = \sum_{a,n}\tilde c(n,a)q^{n}\mathfrak{e}_{a} \in S_{k+1/2}((-2\ell))$.

\begin{lem}\label{lem doi-naganuma-shimura}
	We have the following commutative diagram:
\begin{center} \begin{tikzpicture}[node distance=2.5cm, auto]
\node (A) {$S_{k}(-N_{K/\mathbb{Q}})$};
\node (AA) [right of=A] {$ $};
\node (B) [right of=AA] {$S_k(\mathrm{SL}_{2}(\mathcal{O}_{K}))$};
\node (C) [below of=A] {$S_{k+1/2}((-2\ell))$};
\node (D) [below of=B] {$S_{2k}(\Gamma_{0}(\ell))$};
\draw[->] (B) to node {$\mathrm{Res}$} (D);
\draw[->] (A) to node {$\Theta$} (C);
\draw[->] (A) to node {$\mathrm{Doi-Naganuma}$} (B);
\draw[->] (C) to node {$\mathrm{Shimura}$} (D);
\end{tikzpicture}
\end{center}
\end{lem}

\begin{proof} Since the elements $\nu \in \mathcal{O}_K^{\#}$ with $\mathrm{Tr}(\nu \lambda) = a \in \mathbb{N}$ are exactly those of the form $\gamma_{a,r} = \frac{a \pm r / \sqrt{p}}{2 \lambda}$ with $r \equiv ab$ mod $2\ell$, we find
\begin{align*}
	\Phi_F(\lambda \tau, \lambda' \tau) &= \sum_{n=1}^{\infty} \sum_{\mathrm{Tr}(\nu \lambda) = a} c(\nu \nu', \nu) n^{k-1} q^{na} \\ &= \sum_{n=1}^{\infty} \sum_{a=1}^{\infty} \sum_{r \equiv ab \, (2\ell)} c \left( \frac{a^2}{4\ell} - \frac{r^2}{4 \ell p}, \frac{a \pm r / \sqrt{p}}{2 \lambda} \right) n^{k-1} q^{na} \\ &= \sum_{n,a=1}^{\infty} \tilde c(a^2 / 4\ell, a) n^{k-1} q^{na},
\end{align*}
i.e. $\mathrm{Res}(\Phi_{F})(\tau) = \Phi_F(\lambda \tau, \lambda' \tau)$ is the Shimura lift of the contracted form $\Theta F \in S_{k+1/2}((-2\ell)).$
\end{proof}

In this section we observe that this relationship takes a simple form in terms of twisted component sums of $F$ and $\Theta F$. Recall that for a $q$-series $f(\tau) = \sum_n c(n) q^n$ the Hecke operator $U_p$ is defined by
\[
	f|U_p(\tau) = \sum_n c(pn) q^n = \sum_{n \equiv 0 \, (p)} c(n) q^{n/p}.
\]

\begin{prop}\label{prop theta contraction}~
	\begin{enumerate}
		\item[(i)] Suppose $\ell \ne p$. Let  $\psi_{\ell}$ and $\psi_p$ be Dirichlet characters modulo $\ell$ and $p$ with $\psi_{\ell}(-1) = \psi_{p}(-1) = (-1)^{k}$ and let $\chi$ be the Dirichlet character modulo $\ell p$ defined by $\chi(r) = \psi_{\ell}(r) \psi_p(r)$ for all $r \in \mathbb{Z}$. By abuse of notation, let $\psi_p$ denote the ``character" on the cosets $\mathcal{O}_K^{\#} / \mathcal{O}_K$ defined by setting $\psi_p((a \pm r / \sqrt{p}) / 2 \lambda) = \psi_p(r)$ for any $a,r \in \mathbb{Z}$ satisfying $r \equiv ab$ mod $2 \ell$. Then
		\[
			\varphi_{\psi_{\ell}} (\Theta F)(\tau) = \frac{1}{2\psi_{\ell}(b)} \cdot \Big( \varphi_{\overline{\psi_p}} (F)(4\ell \tau) \cdot \vartheta_{\chi}(\tau) \Big) \Big| U_p
		\]
		where $\vartheta_{\chi}(\tau) = \sum_{r \in \mathbb{Z}} \chi(r) q^{r^2}$ is the twisted Jacobi theta series.
	\item[(ii)] Suppose $\ell = p$, and let $\psi_p$ be a Dirichlet character mod $p$ with $\psi_{p}(-1) = (-1)^{k}$. By abuse of notation, define $\psi_p$ on $\mathcal{O}_K^{\#}/\mathcal{O}_K$ by $\psi_p(a / \lambda) = \psi_p(2a)$, $a \in \mathbb{Z}/p\mathbb{Z}$, where $\lambda = \varepsilon \sqrt{p}$ and $\varepsilon$ is the fundamental unit of $\mathcal{O}_K$. Then $$\varphi_{\psi_p} (\Theta F)(\tau) = \Big( \varphi_{\psi_p} (F)(4p\tau) \cdot \vartheta(\tau) \Big) \Big| U_p,$$ where $\vartheta(\tau) = \sum_{r \in \mathbb{Z}} q^{r^2}.$
	\end{enumerate}
\end{prop}

\begin{proof} ~
	\begin{enumerate}
		\item[(i)] Write $f(\tau) = \varphi_{\overline{\psi_p}} (F)(\tau) = \sum_{\gamma \in \mathcal{O}_K^{\#} / \mathcal{O}_K} \sum_{n \in \mathbb{Z} + N_{K/\mathbb{Q}}(\gamma)} \overline{\psi_p(\gamma)} c(n,\gamma) q^{pn}$. In the product
		\begin{align*}
			f(4 \ell \tau) \vartheta_{\chi}(\tau) &= \Big( \sum_{\gamma,n} \overline{\psi_p(\gamma)} c(n,\gamma) q^{4\ell pn} \Big) \Big( \sum_{r=-\infty}^{\infty} \chi(r) q^{r^2} \Big) \\ &= \sum_{\gamma} \sum_{n,r} \overline{\psi_p(\gamma)} \chi(r) c(n, \gamma) q^{4 \ell pn + r^2},
			\end{align*}
			we get exponents which are divisible by $p$ only when $n \in \mathbb{Z} + a^2 / 4\ell$ and $r \equiv \pm ab \, (2\ell)$ for some $a \in \mathbb{N}$ (which is uniquely determined mod $2\ell$). In this case $\gamma \in \gamma_{a,r} + \mathcal{O}_K$ with $\gamma_{a,r} = \frac{a \pm r/\sqrt{p}}{2 \lambda}$ as before. Applying the $U_p$ operator yields
			\begin{align*}
			\Big( f(4 \ell \tau) \vartheta_{\chi}(\tau) \Big) \Big| U_p &= 2 \sum_{a \in \mathbb{Z}/2\ell \mathbb{Z}} \sum_{n \in \mathbb{Z} + a^2 / 4\ell} \sum_{r \equiv ab \, (2 \ell)} \overline{\psi_p(\gamma_{a,r})} \chi(r) c(n - r^2 / 4p \ell, \gamma_{a,r}) q^{4 \ell n} \\ &= 2\sum_{a,n,r} \psi_{\ell}(ab) c(n-r^2 / 4p \ell, \gamma_{a,r}) q^{4 \ell n} \\ &= 2\psi_{\ell}(b) \sum_{a,n} \psi_{\ell}(a) \tilde c(n,a) q^{4 \ell n} \\ &= 2\psi_{\ell}(b) \varphi_{\psi_{\ell}} (\Theta F)(\tau).
			\end{align*}
	\item[(ii)] This is proved similarly to part (i). We do not need to divide by two, since the sum over $r$ runs through only one congruence class (namely, $r \equiv ap \, (2p)$). The definition of $\psi_p$ on $\mathcal{O}_K^{\#}/\mathcal{O}_K$ is such that $\psi_p(\gamma_a) = \psi_p(\frac{a (1+p)}{2\lambda}) = \psi_p(a)$ for all $a \in \mathbb{Z}/2p\mathbb{Z}$. \qedhere
	\end{enumerate}
\end{proof}

If we abbreviate
\[
\Theta_{\chi} G = \Big( G(4\ell \tau) \cdot \vartheta_{\chi}(\tau) \Big) \Big| U_p
\]
for a scalar valued modular form $G$, then the first item of the proposition (the case $\ell \neq p$) can be illustrated by the diagram
\begin{center} \begin{tikzpicture}[node distance=2.5cm, auto]
\node (A) {$S_{k}(-N_{K/\mathbb{Q}})$};
\node (AA) [right of=A] {$ $};
\node (B) [right of=AA] {$S_k^{\epsilon,\mathrm{AL}}(p^{2},\overline{\psi_{p}}\otimes \chi_{p})$};
\node (C) [below of=A] {$S_{k+1/2}((-2\ell))$};
\node (D) [below of=B] {$S_{k+1/2}^{\epsilon,\mathrm{AL}}(\ell^{2},\psi_{\ell})$};
\draw[->] (B) to node {$\Theta_{\chi}$} (D);
\draw[->] (A) to node {$\Theta$} (C);
\draw[->] (A) to node {$\varphi_{\overline{\psi_{p}}}$} (B);
\draw[->] (C) to node {$\varphi_{\psi_{\ell}}$} (D);
\end{tikzpicture}
\end{center}
which commutes up to a constant factor. Note that the horizontal arrows are isomorphisms.

\begin{ex} Let $p = 5$. Fix the element $\lambda = 4+\sqrt{5}$ of norm $\ell = 11$, and fix $b = 7$. We fix the Dirichlet characters $\psi_{11}$ and $\psi_5$ by specifying $\psi_{11}(2) = e^{\pi i / 5}$ and $\psi_5(2) = i$. Up to scalar multiples there is a unique cusp form of (antisymmetric) weight $5$ for the dual Weil representation attached to $(\mathcal{O}_K,-N_{K/\mathbb{Q}})$ with $K = \mathbb{Q}(\sqrt{5})$, and it is
\begin{align*}
	F(\tau) &= ( q^{1/5} + 42q^{6/5} - 108q^{11/5} \- 4q^{16/5} - 378q^{21/5} \pm \ldots ) (\mathfrak{e}_{3 / \sqrt{5}} - \mathfrak{e}_{2/\sqrt{5}}) \\ &\quad+ (26q^{4/5} + 39q^{9/5} - 378q^{14/5} + 140q^{19/5} + 420q^{24/5} \pm \ldots) (\mathfrak{e}_{4/\sqrt{5}} - \mathfrak{e}_{1/\sqrt{5}}).
\end{align*}
One can compute $F$ using, for example, the algorithm described in \cite{W2} (compare the example of Section 7 there); and after enough coefficients have been computed, one can identify its twisted component sum in $S_5(\Gamma_1(25))$ using standard methods for computing scalar-valued modular forms. The Doi-Naganuma lift of $F$ is, up to a multiple, the well-known product $s_5$ of theta constants for $\mathbb{Q}(\sqrt{5})$ constructed by Gundlach (\cite{Gu}; see also the example of Section 4 of \cite{BB}). The character $\psi_5$ on $\mathcal{O}_K^{\#}/\mathcal{O}_K$ is defined such that e.g. \[
	\psi_5\left(1/\sqrt{5} + \mathcal{O}_K\right) = \psi_5 \left( \frac{1 - 7/\sqrt{5}}{2\lambda} + \mathcal{O}_K \right) = \psi_5(7) = i.
\]
Therefore the twisted component sum of $F$ by $\overline{\psi_{5}}$ is the cusp form
\[
	\varphi_{\overline{\psi_5}} (F)(\tau) = 2q + 52i q^4 + 84q^6 + 78iq^9 - 216q^{11} - 756iq^{14} - 8q^{16} \pm \ldots \in S_5(\Gamma_0(25),\overline{\psi_5}\otimes \chi_5) = S_5(\Gamma_0(25),\psi_5).
\]
After multiplying
\begin{align*}
	\varphi_{\overline{\psi_5}}(F)(44\tau)\vartheta_{\chi}(\tau) &= \Big( 2q^{44} + 52i q^{176} \pm \ldots\Big) \Big( 2q + 2 \zeta_{20}^7 q^{4} - 2\zeta_{20}^9 q^{9} \pm \ldots \Big) \\ &= 4q^{45} + 4\zeta_{20}^7 q^{48} - 4\zeta_{20} q^{53} - 4 \zeta_{20}^4 q^{60} \pm \ldots
	\end{align*}
and applying $U_{5}$ we get the series
\[
	4q^9 - 4 \zeta_{20}^4 q^{12} + 4 \zeta_{20}^{18} q^{16}+ 4 \zeta_{20}^2 q^{25} - 104 \zeta_{20}^2 q^{36} \pm \ldots
\]
Dividing by $2 \psi_{11}(b) = -2\zeta_{20}^4$ yields the twisted component sum of the theta contraction $\Theta F$:
\[
	\varphi_{\psi_{11}}(\Theta F)(\tau) = 2\zeta_{20}^6 q^9 + 2q^{12} + 2 \zeta_{20}^4 q^{16} - 2\zeta_{20}^{18} q^{25} + 52 \zeta_{20}^{18} q^{36} \pm \ldots
\]
From this we can read off the Shimura lift of the underlying vector-valued modular form $\Theta F$: the coefficient of $q^n$ is zero if $11 | n$, and otherwise $\sum_{d | n} \frac{1}{2\psi_{11}(d)} (n/d)^{5-1} c(d^2)$ if $c(n)$ is the coefficient of $q^n$ in $\varphi_{\psi_{11}}(\Theta F)(\tau)$, so
\begin{align*} s_5(\lambda \tau, \lambda' \tau)
&= -q^{3} + q^{4} + q^{5} + 10q^{6} - 10q^{8} - 121q^{9} + 98q^{10} + 275q^{12} + 32q^{13} + 140q^{14} \pm \ldots \in S_{10}(\Gamma_0(11)).
\end{align*}
\end{ex}

\begin{ex} Let $p = 13$. Fix the totally positive element $\lambda = \frac{13 + 3\sqrt{13}}{2}$ of norm $\ell = 13$ and fix $b=13$. We fix an odd Dirichlet character $\psi_{13}$ mod $13$ by specifying $\psi_{13}(2) = \zeta_{12} = e^{\pi i / 6}$. The dual Weil representation attached to $(\mathcal{O}_K,-N_{K/\mathbb{Q}})$, $K = \mathbb{Q}(\sqrt{13})$ admits up to scalar multiples a unique cusp form of weight $3$: \begin{align*} F(\tau) &= (q^{1/13} - 33q^{14/13} + 27q^{27/13} + 33q^{40/13} \pm ...) (\mathfrak{e}_{1/\lambda} - \mathfrak{e}_{12/\lambda}) \\ &+ (3q^{3/13} + 5q^{16/13} + 42q^{29/13} - 99q^{42/13} \pm ...) (\mathfrak{e}_{4/\lambda} - \mathfrak{e}_{9/\lambda}) \\ &+ (-7q^{4/13} - 3q^{17/13} - 33q^{30/13} + 49q^{43/13} \pm ...) (\mathfrak{e}_{2/\lambda} - \mathfrak{e}_{11/\lambda}) \\ &+ (0q^{9/13} - 22q^{22/13} + 33q^{35/13} + 15q^{48/13} \pm ...) (\mathfrak{e}_{3/\lambda} - \mathfrak{e}_{10/\lambda}) \\ &+ (11q^{10/13} - 12q^{23/13} + 0q^{36/13} + 50q^{49/13} \pm ...) (\mathfrak{e}_{6/\lambda} - \mathfrak{e}_{7/\lambda}) \\ &+ (21q^{12/13} + 14q^{25/13} - 66q^{38/13} + 9q^{51/13} \pm ...) (\mathfrak{e}_{5/\lambda} - \mathfrak{e}_{8/\lambda}). \end{align*} Under the Doi-Naganuma lift, $F$ is mapped to the cusp form $\omega_3$ used by van der Geer and Zagier to compute the ring of Hilbert modular forms for $\mathcal{O}_K$ (\cite{GZ}, Section 10). The twisted component sum of $F$ by $\psi_{13}$ is $$\varphi_{\psi_{13}}(F)(\tau) = 2\zeta_{12}q + 6\zeta_{12}^3 q^3 - 14\zeta_{12}^2 q^4 - 22q^{10} - 42\zeta_{12}^4 q^{12} \pm ... \in S_3(\Gamma_0(169),\psi_{13} \otimes \chi_{13}).$$ With this we can compute $\omega_3(\lambda \tau, \lambda'\tau)$ as follows: multiply $$\varphi_{\psi_{13}}(F)(52\tau) \cdot \vartheta(\tau) = 2\zeta_{12} q^{52} + 4\zeta_{12} q^{53} + 4\zeta_{12}q^{56} \pm ...$$ and apply the Hecke operator $U_{13}$ to obtain $$\varphi_{\psi_{13}}(\Theta F)(\tau) = 2\zeta_{12} q^4 + 6\zeta_{12}^3 q^{12} - 14\zeta_{12}^2 q^{16} + 4\zeta_{12}q^{17} + 12\zeta_{12}^3 q^{25} \pm ...$$  and therefore the Shimura lift $$\omega_3(\lambda \tau, \lambda' \tau) = q^2 - 3q^4 - 6q^5 + 9q^6 - q^8 + 6q^9 + 57q^{10} \pm ... \in S_6(\Gamma_0(13)).$$
\end{ex}

\bibliographystyle{plainnat}
\bibliofont

redpajama_set_name: RedPajamaArXiv


text:
The Fire in Water
by V. Radhika
"Where is the house of male-widows?" asks the eight-year-old, her white saree and recently tonsured head indicating her initiation into widowhood. The silent gaze of the older women sequestered in a widow's hostel says it all: the callousness of a patriarchal society that exiles widows to its fringes.
Deepa Mehta's `Water', which recently opened the 30th Toronto International Film Festival, holds a mirror to a society that judges a woman's worth by the life and death of her husband. Though set in 1938, the film is equally relevant today, as thousands of despondent Hindu widows continue to lead a cloistered existence. All it takes is a visit to the byzantine lanes of the two holy cities for Hindus - Brindavan and Benares (now called Varanasi) - to see the white-robed widows engaged in a daily fight with poverty. Bent with age and solitude, their heads shaven, their foreheads streaked with sandalwood, most of them have been abandoned by their families and have to beg for a living. Chanting devotional songs the whole day is, for most, not just a route for spiritual salvation but also the only means to earn a few pennies.
While they have been the subject of many an article and documentary, `Water' brings them face-to-face with a large mainstream audience. Examining their plight through the historical lens of patriarchy, the film raises questions that move beyond spatial and temporal boundaries and, thereby, the individuals featured in it.
While her earlier film, `Fire' boldly tackled the issue of women's sexuality, `Water' ventures into the turbulent waters of religion/religious scriptures that are used to bolster patriarchy. `Water' begins with an extract from Manusmriti (ancient Hindu text), which details the obligation of a wife to her husband. Sans husband, she has no worth and can only hope for salvation through penance and barricading herself against social spaces so as not to pollute them.
The social barricade obviously has economic overtones too. As one of the film's male protagonists points out, it is an effective measure to keep the widow from claiming a share in the family's property, apart from shirking the financial responsibility of providing for her.
Set in a widow ashram (refuge), `Water' pivots on three principal characters, the middle-aged Shakuntala, the young and beautiful Kalyani and child-widow Chuiya (played by debutant Sri Lankan child actor Sarala). All that Chuiya, like many inmates, remembers of her 'marriage' is the feast. Optimistic that her mother will take her home, she refuses to abide by the ashram's prescribed norms and in the process forces others to confront their own dormant questions and beliefs.
The two women she befriends are polar opposites: young Kalyani, who is forced into prostitution by the ashram head to meet the hostel expenses, and middle-aged Shakuntala, who is caught between her hatred of being a widow and fear of not being a 'proper' one. It is Shakuntala who emerges as the quiet rebel by freeing Kalyani from captivity so she could marry the man of her choice and also Chuiya. All of them break tradition in their own ways.
"The nucleus of `Water' is the conflict between faith (as in a particular religion) and our conscience," says Mehta and this conflict is played out best by Shakuntala (actor Seema Biswas in a riveting performance). The film has bleak and sombre look but that does not take away its anti-establishment punch. The story is a necklace that strings politics, religion, spirituality and women's status.
`Water' ends on an optimistic note with the child widow Chuiya's rescue but only after she has been sexually abused (perhaps an attempt at being realistic).
It is the exploration of multitude dimensions of women's lives that has characterized Mehta's film trilogy 'Fire', `Earth' and `Water'. Each of them questions the predominantly male-defined norms for women. Perhaps that is why she prefers to call them political trilogy rather than elemental trilogy as they are commonly referred to. "`Fire' is about the politics of sexuality, `Earth' is about politics of nationalism and `Water' is about politics of religion," says the Toronto-based filmmaker who has been hounded by Hindu fundamentalists for raising the curtain on hitherto taboo topics. Shiv Sena activists went on a rampage following `Fire's screening (for portraying a lesbian relationship), and later the self-anointed cultural gatekeepers sabotaged the shooting of `Water' five years ago by burning down the sets in Varanasi and issuing death threats to Mehta.
"They (fundamentalist Hindus) didn't know what it was really about. They felt in some way Hindu widows really have an opportunity to do something good by being segregated. By questioning that, I'm (considered to be) defaming Hindu culture. It's not true at all because this is not what pure Hinduism is all about," she says and adds, "just as this protest by a miniscule minority is also not what Hindus or Indians are all about. In these intervening years many Indians have asked 'When are you making `Water'. We want to see it'."
Regarding the powerful portrayal of women in her films, she says, "Women's focus is important. There are not many realistic films on women. I feel women are very strong but we don't have our voices yet. If I can give a voice to women, I would be glad because I think it is important."
More by : V. Radhika

redpajama_set_name: RedPajamaCommonCrawl


Refrigerated couriers and temperature controlled transport.
National same day and overnight express refrigerated couriers and ice suppliers. Whether you are looking for ice cubes or ice sculptures or whether you need a express chilled or freezer delivery, we can help. Our vans are all less than two years old and can deliver your consignment with temperatures ranging from ambient, through chilled, down to -30c.
Ice cubes delivered 24 hours a day, 7 days a week.
Bags of party ice for events, for any occasion.
You can't have a party without ice!


Build in-house tools to help with the simulator research, data collection, testing and optimization of various systems.
BS/MS in Electronics/Mechanical/Control or equivalent.
Experience in AI or Machine Learning a plus.
Experience working with open source software, including OpenCV, OpenDS, ROS, etc.
Broad knowledge of all sensor systems and sensor fusion, including cameras, LIDAR, RADAR, etc.


text:
The Mama Pasar movement in Merauke, Indonesia: claiming space for indigenous women small traders
by GRAIN / SKP KAME | 5 May 2016 | Corporations | Supermarket Watch Asia
Numerous Papua indigenous women travel daily from surrounding areas, bringing vegetables, fish and forest products to sell in the town of Merauke. But they face difficulties due to the other traders—mostly settlers from other islands or more established locals from the town—who do not wish to give them space to trade in the markets. In the rainy season their pop-up stalls are damaged and often flooded, and in the dry season they have to endure the intense heat because there is no roof and they are placed at the back end of the market. Consequently, the women attempt to sell in locations where they are not allowed, such as parking lots or nearby sheds.
Click here to read this full issue of Supermarketwatch Asia
In 2013, the Advocacy Group for Women (eL_AdPPer) and the Secretariat for Justice and Peace of Merauke's archdiocese (SKP KAME) started to organise and advocate for these women. SKP KAME and eL-AdPPer established a strategic plan in order to enhance democracy for the women of Merauke. Together with the indigenous women traders they have implemented a number of activities since 2013, including: 1) capacity building, 2) networking, 3) organising focus group discussions in Wamanggu and Mopah markets, 5) establishing a community forum in solidarity with indigenous women traders, and 6) participating in a talk show on national radio and local television stations. The activities also involved local authorities, district market managers, academics and the media.
Through this organising process Mama Pasar identified several critical issues, such as: limited access to trading space and transportation; lack of access to small business loans; unequal price competition; high monthly fees; exclusion from the small traders groups; and marginalisation in market revitalisation processes.
The role of SKP KAME and el_AdPPer in this process is to listen and facilitate understanding of the issues; assist the Mama Pasar women in understanding their rights and the role of indigenous women in the community and in democracy; help establish civil society organisations (i.e. the Mama Pasar Association) in collaboration with Merauke local authorities; and push for regulations that protect Mama Pasar.
Local authorities have begun to acknowledge the issues that exist in the markets—though they are still not acting quickly to address the problems. The growing advocacy campaign has been key to pressuring local authorities and shifting public opinion regarding the rights of indigenous women to trade in the markets. To be sure, it hasn't been an easy process. Community organisers have faced intimidation; they have been prohibited from gathering in the market; they were inaccurately branded as part of a separatist movement; and they have been followed by police.
Further work is needed to increase advocacy efforts and broaden the campaign to democratise market spaces and create regulations that protect the right of indigenous women to sustain themselves by selling their produce.
Beatrix Gebze
SKP KAME
See also the social media campaign on twitter: #bangunpasarmamapapua
This article is from Supermarketwatch Asia, a quarterly email bulletin for social movements about developments in food retail and distribution in Asia produced by GRAIN (Issue No. 2, April 2016) Click here to subscribe.


text:
European Parliament working group refused entry to Gaza by Israel
Commenting on the Israeli refusal to allow an EU Parliament working group entry into Gaza during a visit to the region from 27th to 30th October, GUE/NGL MEP Martina Anderson said:
"I endorse the sentiments expressed by the Chairperson of the European Parliament's Delegation to the Palestinian Legislative Council (DPLC), Emer Costello, when she said that 'she is both astounded and dismayed at the decision of the Israeli authorities to refuse an official visit of the delegation to Gaza'.
"This visit had the full support of Martin Schulz, the President of the European Parliament and despite his intervention and those of other distinguished Parliamentarians, the Israeli authorities decided to block elected members of an official EU delegation from paying a humanitarian visit to this ravaged and impoverished region.
"The official reason given for refusal of access to Gaza was – 'in line with the policy according to which Israel does not facilitate visits to the Gaza Strip which will strengthen Hamas, a terror organization designated as such also by the European Union.'
"This explanation is deplorable and without foundation. Our visit was due to take place under the auspices of UNRWA, and clearly focused on vital public services such as schools, health centres and food distribution centres. Furthermore, we had arranged to visit EU-funded job programmes and hold meetings on education with students from the Christian community. Given the worrying current developments in the region, the programme had a purely social, economic and humanitarian focus.
"This week at the European Parliament we voted on the budget line 'financial assistance to Palestine, the peace process and UNRWA', granting 300 million Euro in funding to the region for 2014. This makes the decision to refuse access to elected Members of the European Parliament even more disgraceful and nonsensical.
"We have requested that President Schulz and High Representative Catherine Ashton take up the issue with the Israeli authorities and that we continue with the mission. In the event that we are refused access to Gaza, it is envisaged that we meet with UNRWA in the West Bank."
GUE/NGL Press
Gay Kavanagh +32 473 84 23 20
left.eu
European United Left/Nordic Green Left
European Parliamentary Group


I am pleased to inform you that after careful review, Starfleet Command and the Starfleet Academy Commandant have decided to accept your application to Starfleet Academy, San Francisco campus, to study in our Law program.
If you choose to accept this offer, please reply within two weeks. You will report for duty at the San Francisco campus on Stardate 239004.01. You will find logistics attached to this letter, as well as required reading to be completed prior to your arrival.
B. Livingston: John! What have you done!?
J. Livingston: Ha ha! You should have seen the look on your face when you read that, Bennie boy! Oh, wait! You can.
B. Livingston: You listen to me. You cannot accept that offer. And don't tell Dad about it, he'll blow a gasket real quick. You know how he reacted when I applied, and that must have been much easier for him. At least I'd already become an engineer – Starfleet was less of a fall.
J. Livingston: ::adopting a particularly sober expression, John nodded slowly.:: You're right, Ben, you're right. Dad would be upset. It's just that, well … well, I already accepted the offer, and I've already told him. ::John smiled broadly and slapped the desk.:: And you're right, he was so mad … oh, jeez, you should have heard him.
B. Livingston: John, why did you apply?
J. Livingston: What do you mean? I wanted to follow in the footsteps of my big brother!
B. Livingston: No you didn't. ::beat:: Why'd you do it?
B. Livingston: I've heard the pitch, John.
J. Livingston: ::shifting in his seat.:: I thought you'd be happy. Or at least happy for me.
B. Livingston: How'd you get in with your record?
J. Livingston: Top 5%, you know that.
B. Livingston: Not that record.
J. Livingston: Ben, remember … I wasn't convicted. You know that.
B. Livingston: And you remember. About what you put me through. All of us through. ::He shook his head slowly.:: Everything those families had to go through.
Bailiff: Do you swear to tell the truth, the whole truth, and nothing but the truth, so help you God?
Judge: Please state your name for the record.
Judge: Your FULL name, Mr. Livingston.
Judge: And your relation to the defendant?
Livingston: He is my brother.
Prosecutor: And on the day that … ::sniff:: ahem, that this tragedy occurred, Mr. Livingston. Where were you?
Livingston: I was at home.
Prosecutor: And your home is where?
Prosecutor: So you were not with the defendant in New York City on that day.
Livingston: No. But I did speak to him that morning.
Prosecutor: But to be clear, you were not WITH him, Mr. Livingston?
Judge: I remind you that you are under oath to tell the truth.
Livingston: ::beat:: No. I was not.
J. Livingston: Ben … I don't know what to say.
J. Livingston: I guess I'd better go.
B. Livingston: John, wait -- ::no words seemed to express what he wanted to say.:: oO don't go? I'm sorry? I know all of that wasn't your fault? I'm proud of you? What do I say here? Oo Just … promise me something. Let's talk again soon, huh? It's been too long.
This page was last modified on 15 May 2015, at 04:54.


text:
Q: Code Igniter Cant upload File i want to update my data on mysql with structure
| nip | nama | path1 | path2 |
i want to change by uploaded file name
this my view code
<form class="form-horizontal" method="post" action="<?php echo base_url()?>index.php/c_main/uploadFile" enctype="multipart/form-data">

    Berkas 1
            <input type="hidden" name="nip" id="nipUpload" value="">
            <input type="file" class="filestyle" name="berkas[]" data-buttonText="Find file" id="berkas[]">

Example block-level help text here.

            Berkas 2
            <input type="file" class="filestyle" name="berkas[]" data-buttonText="Find file" id="berkas[]">

Example block-level help text here.

                <button type="submit" class="btn btn-default">Upload File</button>

</form>

this is my controller code

function uploadFile()
        {
            $nip = $_POST['nip'];

            $number_of_files = sizeof($_FILES['berkas']['tmp_name']);

            $cek = 0;
            for($i=0;$i<$number_of_files;$i++)
            {
                $namaBerkas[$i] = $nip."-".($i+1).".jpg";

                $config['upload_path']          = base_url().'assets/upload/';
                $config['allowed_types']        = 'gif|jpg|png';
                $config['max_size']             = 1000;
                $config['max_width']            = 2024;
                $config['max_height']           = 1200;
                $config['file_name']            = $namaBerkas;

                $this->load->library('upload', $config);
                // $this->upload->initialize($config);

                if ($this->upload->do_upload()){
                    $cek = $cek +1;
                }
                print_r($config);
                echo $cek;
                die();

            }

            if($cek == 2){
                if($this->m_main->uploadFile($nip,$namaBerkas[0],[1])){
                    echo "";
                    redirect('admin/C_main/beranda');
                }
            }

        }

this is my model
function upload_file($nip, $berkas1, $berkas2){
            $sql = "update daftar set PATH1 = $berkas1, PATH2 = $berkas2 WHERE NIP = $nip";
            return $this->db2->query($sql);
        }

the output of code above is: You did not select a file to upload.

A: I think the problem is you have two same form fields. Try this
<form class="form-horizontal" method="post" action="<?php echo base_url()?>index.php/c_main/uploadFile" enctype="multipart/form-data">

Berkas 1
<input type="hidden" name="nip" id="nipUpload" value="">
<input type="file" class="filestyle" name="berkas[]" data-buttonText="Find file" id="berkas[]">

Example block-level help text here.

<button type="submit" class="btn btn-default">Upload File</button>

</form>

A: <form class="form-horizontal" method="post" action="<?php echo base_url()?>index.php/c_main/uploadFile" enctype="multipart/form-data">

    Berkas 1
            <input type="hidden" name="nip" id="nipUpload" value="">
            <input type="file" class="filestyle" name="berkas" data-buttonText="Find file" id="berkas">

Example block-level help text here.

            Berkas 2
            <input type="file" class="filestyle" name="berkas1" data-buttonText="Find file" id="berkas1">

Example block-level help text here.

                <button type="submit" class="btn btn-default">Upload File</button>

</form>

Conrtoller
function uploadFile()
        {
            $nip = $_POST['nip'];

            $number_of_files = sizeof($_FILES['berkas']['tmp_name']);

            $cek = 0;
            $namaBerkas = $nip."-".1.".jpg";

                $config['upload_path']          = base_url().'assets/upload/';
                $config['allowed_types']        = 'gif|jpg|png';
                $config['max_size']             = 1000;
                $config['max_width']            = 2024;
                $config['max_height']           = 1200;

                $config['file_name']            = $namaBerkas;

                $this->load->library('upload', $config);
                // $this->upload->initialize($config);

                if ($this->upload->do_upload('berkas')){
                    $cek = $cek +1;
                }

                $namaBerkas1 = $nip."-".2.".jpg";
                $config['file_name'] = $namaBerkas1;
                if ($this->upload->do_upload('berkas1')){
                    $cek = $cek +1;
                }

                print_r($config);
                echo $cek;
                die();

            if($cek == 2){
                if($this->m_main->uploadFile($nip,$namaBerkas[0],[1])){
                    echo "";
                    redirect('admin/C_main/beranda');
                }
            }

        }

redpajama_set_name: RedPajamaStackExchange


Expert quilter, Gina Perkes, shows off the beauty of gridwork in her new book.
Use the grid intersections as visual guides to improve free-motion skills.
Discover how to create elaborate designs quickly and easily using grids as guides for even placement.
Learn how to execute beautiful crosshatching designs without backtracking.
Explore the possibilities of freestyle machine embroidery and its applications for quilting and fiber art with this inspirational guidebook for all skill levels.
This overview of all the basic methods and stitches for each technique includes step-by-step photographs and illustrations, diagrams, close-ups of sample swatches, troubleshooting tips, and design guidance, altogether elevating crafters' skills to an artistic level.
Beginning with an introduction that includes a list of necessary supplies, chapters cover adjusting thread tension, understanding fabric distortion as it relates to stitching direction, and caring for a sewing machine.
This comprehensive guide is sure to move crafters beyond the basics of the stitching process to using color and texture creatively in unique designs and professional-grade finishes.
The Guild has an extensive collection of books and magazines. These books can be signed out by members in good standing at the monthly guild meetings.
Books are borrowed for 1 month. All materials are due back at the next Guild meeting. If you are unable to return them, please try to get a friend to bring them back for you.
Please let us know if any books are damaged or out-dated so they can be repaired or discarded.
Enjoy & be sure to share any of the quilts, or items you have made using our library materials.
Click on the above link, it will take you directly to the library on another web page.
Images of the books appear with a small icon (looks like a page with text on it), below the image. Hover over the icon and it states "read reviews".
Click on this icon and it takes you to a written description of the book. At the beginning of each description there is a "B" followed by a number -.the number of the book in the actual library.
If you see something you would like to borrow, write down the name and number and bring to the next meeting and search the shelves for the book.


The 2018 Eurovision Young Musicians takes place in Edinburgh. The Semi-Finals took place on the 18th and 19th of August, the final on the 23rd of August.
We went backstage at the Semi-Finals and met some of the contestants.


As we wind down on this years 52-week blog challenge, this week's writing prompt asks about what participants wear when they write.
The flippant answer is: "Clothes."
Seriously, though, I work from home. Nobody sees me, except for the occasional delivery person. The cats certainly don't care about my attire. The dog loves me regardless of how or whether I am clothed. The horses haven't expressed an opinion: they just want their treats and maybe a moment's attention. My husband generally doesn't comment: he knows better.
I wear what's comfortable. In warm weather, that's usually a loose, flowing dress or shorts and tee shirt. In the winter, it's jeans and a sweater. If the weather suits, I'll go barefoot; if not, then slippers and/or socks. Sometimes I even wear shoes all day.
I share the preference to go without shoes with my younger son. This is the kid who took Mama's words to heart when I told him about my own barefoot adventures with livestock: "It's easier to hose off your feet than to clean your shoes."
Unfortunately, a sedentary lifestyle, age, and a sluggish thyroid mean weight gain. So, many of my outfits don't fit any more. Last weekend I finally decided to start clearing out closets. I sighed over the pretty clothes I wore in my "skinny" days even as I acknowledged that I'll never again fit into them. It's a difficult and painful realization to admit to corpulence.
Which leads me to the medical profession and my distaste for its judgmental practitioners. A long while ago our family doctor shut down her small practice. I transferred to another physician who had the audacity to tell me to eat fewer cheeseburgers. I know I'm fat. That remark was really unnecessary.
So, I've abandoned fashion. I'm not going to lose weight. And I prize comfort above all when it comes to clothing.
Dressing to impress is for other people.
It's hard to believe the year is winding down. I've missed so many topics. When I returned this week, I was shocked to see Week 40. I prefer to be comfortable when writing. There is no one here to complain.


Ever wonder where that signature Spirit of '98 sound comes from? That's OK, I'll tell you anyway. Here's a look at some of the instruments and devices that brought the album to life, along with store locations and pricing info, just in case you want to recreate the magic yourself at home.
Tascam 488 cassette 8-track. The Rolls-Royce of cassette multitrack recorders. Purchased on ebay, broken of course. $200, plus probably another $200 in repairs courtesy of the clueless repair guy in Portland. But she purrs like a kitten now.
Fisher Price children's cassette recorder. Gift from Matt Stefanik. Free!


Clevedon, UK, 1st September 2016 – For print service providers looking to build an unbeatable business, Roland DG has announced today the launch of two new wide-format eco-solvent inkjet printer/cutters to its TrueVIS range; the SG-300 and SG-540.
Roland DG introduced integrated print and cut technology in the mid-1990's and the TrueVIS series represents Roland's most advanced print and cut products to date. With the SG-300/540, two Roland DG FlexFire printheads fire precision droplets in three sizes and a wide print pass, for high-quality output and efficient ink usage. Every drop of ink falls exactly where it is intended, delivering beautiful images and uniform colours. The combination of TrueVIS Ink, engineered specifically for the FlexFire printhead, and Roland VersaWorks Dual RIP software results in stunning fine details and vibrant colours at production speeds. The SG series' added productivity – up to 60% faster than previous comparable models in standard mode – enables increased profitability.
TrueVIS Ink is the next evolution for eco-solvent ink technology, meeting user demands for competitively priced ink with low consumption and a wide colour gamut while also meeting strict environmental standards. TrueVIS Ink has achieved GREENGUARD Gold certification for indoor air quality, which includes safety factors to ensure a product is acceptable for use in environments like schools and hospitals. The SG also features a brilliant 'clean hands' ink system with 500ml pouches which fit neatly into reusable holders for no mess and less waste.
The SG-300/540 is incredibly user-friendly for all operators; from novices to seasoned professionals. New Roland VersaWorks Dual RIP software is included and features a new core engine that quickly and accurately processes both native PDFs as well as legacy PostScript files, including files with transparencies. In addition to a built-in control panel, a new Roland DG Mobile Panel enables users to manage the printer functions remotely using their smartphone or tablet with a Bluetooth connection*.
With an impressive price/performance ratio, the SG series includes the same built-to-last reliability and industry-leading service and support that Roland devices are famous for..


More than 250 printing professionals attended PrintForum 2018, held last week at the International Centre in Mississauga, Ont. The conference program hit on a range of topics for commercial printers, including sheetfed offset, digital enhancement, wide-format inkjet, and production inkjet. Now in its third year, the 2018 edition was our largest show yet.
Greg Neath, General Manager, In-Store Marketing Products for TC Transcontinental Printing, kicked off the day with a dynamic keynote on the company's shift into flexible packaging. Neath discussed the company's shift into flexible packaging as well as the five lessons learned when dealing with constant change: Decide who you want to be; Implementation is key; Figure out what not to do; Change fast; and Culture is critical.
Pazazz; and Deanne Sinclair, Co-owner and General Manager, Cambridge Label – addressed the top opportunities and challenges facing today's printing companies.
After lunch, an energetic panel of technology leaders addressed the growing use of business intelligence to provide more complete transparency into the overall operating efficiency of the printing plant. The speakers included: Sean Springett, CEO, Manroland Sheetfed North America; Walter Chmura, Vice President of Technical Sales, KBA North America; and Anthony Thirlby, Global SVP Commercial Segment & Head Of Subscription Sales, Heidelberger Druckmaschinen AG.
The technology panel was followed by back-to-back impactful presentations from CMD Insight and Canon Canada focused on what they feel to be some of the biggest trends and opportunities in the world of digital printing. The sessions were presented by Christian Knapp, President, CMD Insight, on the profit effects of digital additive and subtractive print manufacturing, and Jim Stalford, Senior Director of Sales, Wide Format Printing Systems, Canon Canada, on diversifying commercial print with wide format.
The day concluded with a spirited closing keynote by Brent Moncrief, Vice President, Strategic Marketing and Brand Management, the Americas, Fujifilm North America Corporation, Graphic Systems Division, with in-depth look at the state of production inkjet.


Business company ZMIK INVESTMENTS, LLC is a legal entity registered under the law of State Nevada. Company is located in the register with the Company number E0027762007-2 and with the national number of State Nevada NV20071599850. This legal entity was firstly registered on 10th January 2007 under the legal form of Domestic Limited-Liability Company. Its registered agent is JULIUS B. GEORGE with the seat at 3813 WISTERIA SHADE AVE, LAS VEGAS, 89115, NV licensed as Noncommercial Registered Agent. Business activities of this company are managed by Managers. Current company's status is Permanently Revoked.
The company ZMIK INVESTMENTS, LLC is managed by 2 persons in total. The persons responsible for business activities are JULIUS B GEORGE with the seat at PO BOX 150877, ELY, 89315, NV as Manager , JULIUS B GEORGE with the seat at PO BOX 150877, ELY, 89315, NV as Manager .
FIVE STAR GROUP INVESTMENTS, INC.
R GOODLOE & ASSOCIATES INC.
DISHNET SATELLITE INTERNET BY CYH INC.


text:
M/W/DVBE Definitions
Superior Communications recognizes and values the importance of diversity, therefore, it is the policy of Superior Communications to provide minority, women, and disabled veterans' business enterprises (M/W/DVBEs) equal opportunity for participating in all aspects of the company's contracting and procurement programs. Where commercially feasible, Superior Communications looks to M/W/DVBEs for its OEM and aftermarket wireless accessories needs.
Companies certified by qualified organizations, including the Women and Minority Business Enterprise Clearinghouse and the National Minority Supplier Development Council, are solicited to submit proposals to Superior Communications. Superior Communications utilizes the California Public Utilities Commission database, the Veterans Corporation database, the MWBE Directory, and other publicly accessible databases to identify qualified businesses. Qualifying companies with a desire to work with Superior Communications which are not properly certified are encouraged to become certified. The Resources page contains links to certifying organizations.
Women Business Enterprise (WBE)
A business which has been certified by a Superior Communications approved agency that is at least 51% owned, operated and controlled by a woman or women.
Minority Business Enterprise (MBE)
A business that has been certified by a Superior Communications approved agency that is at least 51% owned, operated and controlled by a minority individual or a minority group. Minorities include but are not limited to:
African American - A U.S. citizen whose origin is from any of the Black racial groups of Africa.
A U.S. citizen of true-born Hispanic heritage (true-born meaning "authentically" or "genuinely" as per Webster), from any of the Spanish speaking areas of Latin America of the following regions: Mexico, Central America, South America and the Caribbean basin.
A person who is an American Indian, Eskimo, Aleut, or Native Hawaiian and regarded as such by the community of which the person claims to be a part. Native Americans must be documented members of a North American tribe, band, or otherwise organized group of native people who are indigenous to the continental United States and proof can be provided through a Native American Blood Degree Certificate (for example, a tribal registry letter, tribal roll register number).
Asian-Pacific American
A U.S. citizen whose country of origin is Japan, China, Taiwan, Korea, Vietnam, Laos, Cambodia, the Philippines, Samoa, Guam, the U.S. Trust Territories of the Pacific of the Northern Marianas, Indonesia, Micronesia, or Polynesia.
Asian-Indian American
A U.S. citizen whose country of origin is India, Pakistan, or Bangladesh.
Disabled Veteran Business Enterprise (DVBE)
A business that has been certified by a Superior Communications approved agency that is at least 51% owned, operated and controlled by one or more disabled veterans.
LGBT Business Enterprise (LGBTBE)
A business which has been certified by a Superior Communications approved agency that is at least 51% owned, operated, managed, and controlled by an LGBT person or persons who are either U.S. citizens or lawful permanent residents.
Does Superior Communications guarantee work for M/W/DVBEs?
Superior Communications does not guarantee work for M/W/DVBEs. If opportunities exist however, they will be considered as part of Superior Communications' supply chain process.
How can suppliers register with Superior Communications?
M/W/DVBE suppliers who want to register with Superior Communications Supplier Diversity should submit a completed Supplier Application Form . If there are any opportunities to do business you will be contacted and asked to fax or e-mail current M/W/DVBE certification documentation and company brochures that briefly describe your core competencies.
What are the next steps after a company registers with Supplier Diversity?
Once the Supplier Application Form has been evaluated it will be entered into our Supplier Information Database. You will be contacted only if there is an opportunity for you to do business with Superior Communications. However, this is not a guarantee of work. If our staff does not contact you, it means there are currently no opportunities to do business in your area. Your company's information will remain in our database and when opportunities become available the appropriate person from Superior Communications will contact you regarding next steps.
How can a company become M/W/DVBE certified?
Visit the links on the Resources page to be directed to the appropriate certifying organization.
North American Industry Classification System (NAICS)
http://www.census.gov/epcd/www/naics.html
http://www.naics.com/
National Minority Supplier Development Council (NMSDC)
http://www.nmsdc.org/
Minority Business Development Agency (MBDA)
http://www.mbda.gov/
Women's Business Enterprise National Council
http://www.wbenc.org/
Association for Service Disabled Veterans
http://www.asdv.org/
http://www.gsa.gov/
Small Business Administration (SBA)
http://www.sba.gov/
CPUC WMBE Clearinghouse - Administered by Asian, Inc.
http://www.asianinc.org/about/organizational-highlights
US Pan Asian Chamber of Commerce
http://www.uspaacc.com/


James Baker is Principal Percussionist of the New York City Ballet Orchestra. He is Music Director and Conductor of the Composers Conference at Wellesley College and Director of the Percussion Ensemble at the Mannes College of Music. He regularly conducts and plays for both the New York New Music Ensemble and Speculum Musicae.


text:
Q: How to look up items with id for one controller from another controller in AngularJS I have two controllers where the first one contains a list of items:
$scope.items = [{id:1, desc:'desc1'},{id:2, desc:'desc2'}...];

The second one binds to a template which displays a list of items selected:
$scope.myitems = [1,2,3,...]; // ids only

   {{item.id}} / {{item.desc}}

How can I look up the item desc in the ng-repeat of the second controller from the item list of the first controller?

A: You can try to use inheritance concept:
var app = angular.module(....);

app.controller('FirstCtrl ', function($scope) {
  $scope.items = [{id:1, desc:'desc1'},{id:2, desc:'desc2'}...];
});

app.controller('SecondCtrl', function($scope, $controller) {
  $controller('FirstCtrl', {$scope: $scope}); // $scope.items now available

});

Advantage over the $rootScope solution:

*

*second controller will always have access to items, even if first controller is not instantiated

*items array may be changed only by these two controllers, and not anyone other

Main flaw of this approach is that SecondCtrl will have access to any scope variable of the FirstCtrl and not only items array
EDIT. IMHO Factory/Service approach mentioned by @NexusDuck is the best one (composition over inheritance).

A: Sharing data between controllers is best achieved via services/factories, an alternative is to use scope inheritance.
1. Factory/Service
angular.module('yourModule').factory('itemService', itemService);

    function itemService() {
         var items =  [{id:1, desc:'desc1'},{id:2, desc:'desc2'}...];
         return {
             findItemById: findItemById
         }

        function findItemById(id) {
            //logic to find item by id
        }
    }

Inject this factory in your controllers and add more functions if needed.
2. Scope inheritance
The key here is to nest your childcontroller, which I presume is the one with the ids.

         {{item.id}} / {{item.desc}}

With this option, any controller that is nested within the topCtrl in the view has access to the topCtrl scoped variables.
A third option would be to store the data in the $rootScope, which is actually also a sort of scope inheritance (all scopes except for isolated directive scopes inherit from the rootScope), but that's probably not a good idea for your usecase.

A: Can you assign values to $rootScope.items in first controller, then try to access in second controller.

A: You can use $emit and $on.
EX:
In First controller add
$scope.$emit('eventName', $scope.items);
};

And also pass $scope.items.
In Second Controller Add
$scope.$on('eventName', function (event, args) {
    $scope.items = args;
});

args hold $scope.items value and is assign to $scope.items in second controlle, so now ng-repeat display value of $scope.items of First Controller.


My ex started dating one of my friends, no one told me about it until I saw them holding hands walking down the street some 4 months after the breakup. Although if the break up was nasty or your friend still had feeling a for said ex, as a friend, it's pretty low to go behind your friends back knowing there are still feelings. A good friend used to say to me, "You're on your own road. And now she's dating him. I love this, I agree: They will support you more than you could have possibly imagined.
Wish them well and find a new friend.
I agree the "girl code" rule here should expire after a few years, because I don't give a crap about my exes from that long ago, but this was my most recent ex. Whether this is a deal breaker and you want to cut them out is entirely your call. I hope you understand that we don't want to hurt you. But from her responses, it appears that wasn't it. Yes it was wrong of them to hide it from you but frankly he is not your property and can date whoever he wants. I'm sorry that you are going through this. People in this subreddit are so immature.


text:
Във философията "Абсурдното", се отнася за конфликтът между човешката склонност да търси ценности и смисъл в живота и човешката неспособност да намери такива в безцелната, безсмислена или хаотична и ирационална вселена. Вселената и човешкият ум не предизвикват Абсурдното поотделно, а вместо това, Абсурдното се появява поради противоречивата природа на двете, съществуващи едновременно.

Като философия, абсурдизмът изследва фундаменталната природа на Абсурдното и как човек трябва да реагира, след като го осъзнае. Абсурдисткият философ Албер Камю смята, че хората трябва да насърчават абсурдното състояние на човешкото съществуване, като същевременно продължават да изследват и търсят смисъл.

Абсурдизмът споделя някои концепции и общи теоретични виждания с Екзистенциализмът и Нихилизмът. Началото му дава Данският Философ Сьорен Киркегор, който се изправя срещу кризата, която хората срещат в Абсурдното, създавайки своя собствена екзистенциална философия. Абсурдизмът, като система от вярвания се ражда от Европейското екзистенциалистко движение, което последва, особено когато Камю отхвърля някои аспекти от тази философска линия на мисълта и публикува есето си Митът за Сизиф. Последиците от Втората световна война, осигуряват социална среда, която стимулира абсурдистките възгледи и позволява тяхното развитие, особено във Франция.

Обзор
В абсурдистката философия, Абсурдното е резултат на фундаменталната дисхармония между индивидуалното търсене на смисъл и безмислието на вселената. Като същества търсещи смисъл в безсмисления свят, хората имат три начина за справяне с дилемата. Сьорен Киркегор и Албер Камю описват решенията в трудовете си. Те са респективно:

Самоубийство (или "избягване на съществуването"): решение при което човек приключва живота си. И Киркегор и Камю отричат легитимността на този вариант. Камю казва, че тя не се противопоставя на Абсурдното. Вместо това с приключването на собственото си съществуване, човек само прави съществуването си по-абсурдно.

Религиозно, духовно или абстрактно вярване в трансцендентно пространство, съшество или идея: решение включващо вярата в съществуването на реалност отвъд Абсурдното, придаваща смисъл на съществуването. Киркегор казва, че вярата във всичко отвъд Абсурдното изисква ирационален, но вероятно необходим "наклон" към нематериалното и емпирично недоказуемото. Въпреки това, Камю се отнася към това решение като към "философско самоубийство".

Приемане на Абсурдното: решение в което човек приема абсурдното и продължава живота си въпреки него. Камю подкрепя това решение, вярвайки че с приемането на Абсурдното, човек може да постигне най-голяма свобода. С непризнаването на всякакви религиозни или морални ограничения и с противопоставянето си срещу Абсурдното (чрез създаване на смисъл), докато същевременно осъзнава неспособността си да го спре, човек може да намери удовлетворение, чрез преходния личен смисъл, конструиран в процеса. Киркегор, от друга страна нарича това решение "демонична лудост".

Връзка с екзистенциализма и нихилизма
Абсурдизмът произхожда от разклоненията на екзистенциализма и нихилизма, възникнали през 20 век; споделя някои важни отправни точни и с двете движения, но също така достига до различни заключения. И трите философски движения се появяват в отговор на човешкото страдание и объркване при сблъсъка с Абсурдното: нуждата от намирането или създаването на смисъл в безсмисления свят. Това е единствената прилика между трите движения.

Източници

Философия

redpajama_set_name: RedPajamaWikipedia