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 $\kappa =2n⩾12$ for the full modular group and $A(x)=\sum _{n⩽x}a(n)$. In this paper, we establish an asymptotic formula of the fourth power moment of $A(x)$ and prove that \begin{equation*} \int_1^TA^4(x)$\mathrm{d}$x=\frac{3}{64\kappa\pi^4}s_{4;2}(\tilde{a}) T^{2\kappa}+O\big(T^{2\kappa-\delta_4+\varepsilon}\big) \end{equation*} with ${\delta }_4=1\mathrm{/}8$, which improves the previous result.