On geometric density of Hecke eigenvalues for certain cusp forms

In this paper we prove certain density results for Hecke eigenvalues as well as we give estimates on the length of modules for Hecke algebra acting on the cusp forms constructed out of Poincaré series for a semisimple group G over a number field k. The cusp forms discusses here are taken from Muić (Math Ann 343:207–227, 2009) and they generalize usual cuspidal modular forms Sk(Γ) of weight k ≥ 3 for a Fuchsian group Γ (Muić, in On the cuspidal modular forms for the Fuchsian groups of the first kind).