Utilizing the theory of the Poisson transform, we develop some new concrete models for the Hecke theory in a space $M_{\lambda }\left(N\right)$ of Maass forms with eigenvalue $1/4-{\lambda }^2$ on a congruence subgroup ${\Gamma }_1\left(N\right)$. 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 $M_{\lambda }\left(N\right)$ 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...