Higher symmetric power $L$-functions and their Fourier coefficients

Abstract

Let $H_k$ be the set of all normalized primitive holomorphic cusp forms of even integral weight $k\ge 2$ for the full modular group $SL(2,\mathbb{Z})$, and let $j\ge 3$ be any fixed integer. For $f\in H_k$, we write ${\lambda }_{{\mathrm{sym}}^{\mathrm{j}}f}(n)$ for the $n^{\text{th}}$ normalized Fourier coefficient of $L(s,{\mathrm{sym}}^{j}f)$. In this article, we establish an asymptotic formula for the sum $$\begin{array}{c}\sum _{\begin{array}{c}n=a_1^2+a_2^2+\dots +a_6^2\le x\\ \left(a_1,a_2,\dots ,a_6\right)\in {\mathbb{Z}}^6\end{array}}{\lambda }_{{\mathrm{sym}}^{\mathrm{j}}\mathrm{f}}^2(n),\end{array}$$ with an improved error term.