quadratic Gauss sum over a power of 2 Is there a general formula for the generalized quadratic Gauss sum defined by
$$
G(a,b,c)=\frac{1}{c}\sum_{n=0}^{c-1}e\left(\frac{an^2+bn}{c}\right)
$$
where $e(x)=\exp(2\pi ix)$ and $c$ is a power of 2?

You could probably use the generalized quadratic Gauss sum reciprocity formula, defined for integers $a, b, c$ satisfying $ac \neq 0$ and $ac+b$ even. In this case, we set $\displaystyle S(a,b,c) = \sum_{x=0}^{|c|-1} e\left(\frac{ax^2+bx}{2c}\right)$, and the theorem states
\begin{eqnarray*}
S(a,b,c) = \left|\frac{c}a\right| e\left(\frac{sgn(ac)-(b^2/ac)}8\right)S(-c,-b,a).
\end{eqnarray*}
See Gauss and Jacobi Sums by Berndt, Evans and Williams.
Observe that we can complete the square $ax^2+bx = a(x+\frac{b}{2a})^2-\frac{b^2}{4a}$, so if $2 \mid b$ and $a$ is odd we have $ax^2+bx \equiv a(x+a^{-1}\frac{b}2)^2 - a^{-1}(\frac{b}2)^2 \mod c$. Under these assumptions, we have that
\begin{align*}
G(a,b,c) &= \sum_{x=0}^{|c|-1} e\left(\frac{a(x+a^{-1}\frac{b}2)^2}c\right) e \left(\frac{- a^{-1}(\frac{b}2)^2}c\right)\\
&= e \left(\frac{- a^{-1}(\frac{b}2)^2}c\right) G(a;|c|)
\end{align*}
where $G(a;|c|) = (1+\imath^a)\left(\frac{|c|}a\right) \sqrt{|c|}$ is the quadratic Gauss sum when $c = 2^r$ for some integer $r > 1$.
