On average behavior of Hecke eigenvalues over arithmetic progressions

Let [Formula: see text] be a given integer. Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group [Formula: see text]. Denote by [Formula: see text] the nth normalized coefficient of the Dirichlet expansion of the jth symmetric power L-function [Formula: see text]. In this paper, we are interested in the average behavior of [Formula: see text] over arithmetic progressions.