The sixth and eighth moments of Fourier coefficients of cusp forms

Let λ(n)λ(n) be the n  th normalized Fourier coefficient of a holomorphic Hecke eigencuspform f(z)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>0ε>0, we have∑n⩽xλ6(n)=xP1(logx)+Of,ε(x3132+ε), where P1(x)P1(x) is a polynomial of degree 4.(ii)For any ε>0ε>0, we have∑n⩽xλ8(n)=xP2(logx)+Of,ε(x127128+ε), where P2(x)P2(x) is a polynomial of degree 13.