The sixth and eighth moments of Fourier coefficients of cusp forms

Abstract

Let λ ( n ) be the nth normalized Fourier coefficient of a holomorphic Hecke eigencuspform f ( z ) of even integral weight k for the full modular group. In this paper we are able to prove the following results. (i) For any ε > 0 , we have ∑ n ⩽ x λ 6 ( n ) = x P 1 ( log x ) + O f , ε ( x 31 32 + ε ) , where P 1 ( x ) is a polynomial of degree 4. (ii) For any ε > 0 , we have ∑ n ⩽ x λ 8 ( n ) = x P 2 ( log x ) + O f , ε ( x 127 128 + ε ) , where P 2 ( x ) is a polynomial of degree 13.