The sixth and eighth moments of Fourier coefficients of cusp forms

Let lambda(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 epsilon > 0, we have Sigma(n <= x) lambda(6)(n) = xP(1)(log x) + O(f,epsilon)(x(31/32+epsilon)), where P(1)(x) is a polynomial of degree 4. (ii) For any epsilon > 0, we have Sigma(n <= x)lambda(8)(n) = xP(2)(log x) + O(f,epsilon)(x(127/128+epsilon)), where P(2)(x) is a polynomial of degree 13 (C) 2009 Elsevier Inc All rights reserved