2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/124133Utilizing the theory of the Poisson transform, we develop some new concrete models for the Hecke theory in a space $M_λ(N)$ of Maass forms with eigenvalue $1/4-λ^2$ on a congruence subgroup $Γ_1(N)$. We introduce the field $F_λ = {\mathbb Q} (λ,\sqrt{n}, n^{λ/2} \mid ñ\in {\mathbb N})$ so that $F_λ$ consists entirely of algebraic numbers if $λ= 0$. The main result of the paper is the following. For a packet $Φ= (ν_p \mid p\nmid N)$ of Hecke eigenvalues occurring in $M_λ(N)$ we then have that either every $ν_p$ is algebraic over $F_λ$, or else $Φ$ will - for some $m\in {\mathbb N}$ - occur in the first cohomology of a certain space $W_{λ,m}$ 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 $\mathrm{SL}_2({\mathbb R})$.To appear in Ann. Inst. Fourier (Grenoble)Number TheoryRepresentation Theory11F70; 11R39; 22E50text