Fourier coefficients of symmetric power L-functions

Abstract

Let f be a Hecke eigencusp form of even integral weight k or Maass cusp form for the full modular group SL2(Z). Denote by λsymmf(n) the nth normalized coefficient of the Dirichlet expansion of the mth symmetric power L-function associated to f. In this paper, we establish some bounds for∑n⩽xλsymmf(n),∑n⩽xλf(nm), which improve the corresponding results of Lau & Lü [8].