Average of Dirichlet coefficients of cuspidal representations related to GL(2)

Abstract

Let \(\pi \) be a cuspidal representation on \({{\,\mathrm{GL}\,}}(2,\mathbb {A}_{\mathbb {Q}}).\) We give nontrivial lower and upper bounds for average of absolute values of Dirichlet coefficients associated to \(\pi ;\) and nontrivial upper bound in the case of \({\text {Sym}}^k\pi ,\) \(k=2, 3.\) These bounds generalize the known estimates in holomorphic case to Maass forms, without assuming the Ramanujan–Petersson conjecture.