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 $k\geq 2$ for the full modular group $SL(2, \mathbb{Z})$, and let $j\geq 3$ be any fixed integer. For $f\in H_k$, we write $\lambda_{{\rm{sym}^j }f}(n)$ for the $n^\textit{th}$ normalized Fourier coefficient of $L(s,{\rm{sym}}^j f)$. In this article, we establish an asymptotic formula for the sum $$\begin{equation} \sum_{\substack{n=a_1^2+a_2^2+\ldots+a_6^2\leq x\ \left(a_1,a_2,\ldots, a_6\right)\in \mathbb{Z}^6}} \lambda_{{\rm{sym}}^j f}^2(n), \end{equation}$$ with an improved error term.