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 $\mathrm{ SL}_2({ \mathbb R})$.
Categories: math.NT math.RT