Generalization on the higher moments of the Fourier coefficients of symmetric power $L$-functions

Abstract

For an even integer $k\geq 2$, let $f$ be a primitive holomorphic cusp form of weight $k$ for the full modular group $SL(2,\mathbb{Z})$ and let $\lambda_{{\rm{sym}}^jf}(n)$ denote the $n^\text{th}$ normalized Fourier coefficient of the $j^{\text{th}}$ symmetric power $L$-function $L(s,{\rm{sym}}^j f)$. It has been an interesting problem to study the average behaviour of $\lambda_{{\rm{sym}}^jf}(n)$ and their higher powers, and many researchers in the literature have studied the sum \begin{equation*} \sum_{n\leq x} \lambda_{{\rm{sym}}^j}^l(n), \end{equation*} for various values of $l$ and $j$. In this paper, we improve and generalize previously known results concerning the sum above for positive integers $l$ and $j$ such that $lj\geq 4$.