On the fourth power moment of Fourier coefficients of cusp form

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)=Σ_{n ≤ x} a(n). In this paper, we establish an asymptotic formula of the fourth power moment of A(x) and prove that
∫₁^T A⁴(x)dx=（3）/（64κπ⁴） s_4;2(ã)T^2κ + O(T^2κ-δ₄+ε)
with δ₄=1/8, which improves the previous result.