The quantitative distribution of Hecke eigenvalues of cusp forms: Sato-Tate, Lang-Trotter and all that.

Abstract: For holomorphic cusp forms of weight $k$ (even), level $q$ one
knows that the Hecke eigenvalues (unnormalised)  are all algebraic
integers belonging to a fixed number field $K$ say. This immediately
implies that the number of primes $p$,  $(p , q )= 1$ such the normalised Hecke eigenvalues $\lambda ( p ) = a ( p ) / p^{ ( k - 1 ) /2 }$ where $k$ is the weight $= \alpha, \alpha \in [ -2, 2 ] , {\alpha}$ algebraic is finite. However the number of the unnormalised $a ( p )$ 's with this property i.e $a ( p ) = \beta, \beta \in O_K$ fixed could a priori be infinite and is the subject matter of the Lang-Trotter conjecture. We will try to pose these questions in the case of non-holomorphic Maass cusp forms.