On the fourth power moment of Fourier coefficients of cusp form

Abstract

Let a(n) be the Fourier coefficients of a holomorphic cusp form of weight κ = 2n 12 for the full modular group and A(x) = n x a(n). In this paper, we establish an asymptotic formula of the fourth power moment of A(x) and proveA 4 (x)dx = 3 64κπ 4 s 4;2 (ã)T 2κ + O T 2κ−δ 4 +ε with δ 4 = 1/8, which improves the previous result.