Asymptotics for the second moment of the Dirichlet coefficients of symmetric power L-functions

Abstract

Let m ≥ 2 be an integer. Let f be a holomorphic Hecke eigenform of even weight k for the full modular group SL(2, ℤ). Denote by λ_Sym^m _f (n) the nth normalized Dirichlet coefficient of the corresponding symmetric power L-function L(s, Sym^m f) related to f. In this paper, we study the average behavior of the second moment of the Dirichlet coefficients λ_Sym^m _f (n) and establish its asymptotic formula.