  Utilizing the theory of the Poisson transform, we develop some new concrete
models for the Hecke theory in a space $M_{\lambda}(N)$ of Maass forms with
eigenvalue $1/4-\lambda^2$ on a congruence subgroup $\Gamma_1(N)$. We introduce
the field $F_{\lambda} = {\mathbb Q} (\lambda ,\sqrt{n}, n^{\lambda /2} \mid
\~n\in {\mathbb N})$ so that $F_{\lambda}$ consists entirely of algebraic
numbers if $\lambda = 0$.
  The main result of the paper is the following. For a packet $\Phi = (\nu_p
\mid p\nmid N)$ of Hecke eigenvalues occurring in $M_{\lambda}(N)$ we then have
that either every $\nu_p$ is algebraic over $F_{\lambda}$, or else $\Phi$ will
- for some $m\in {\mathbb N}$ - occur in the first cohomology of a certain
space $W_{\lambda,m}$ which is a space of continuous functions on the unit
circle with an action of $\mathrm{SL}_2({\mathbb R})$ well-known from the
theory of (non-unitary) principal representations of $\mathrm{SL}_2({\mathbb
R})$.

Number Theory;Representation Theory