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

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 ${\lambda }_{{\mathrm{s}\mathrm{y}\mathrm{m}}^{j}f}(n)$ for the $n^{\mathit{\text{th}}}$ normalized Fourier coefficient of $L(s,{\mathrm{s}\mathrm{y}\mathrm{m}}^{j}f)$. In this article, we establish an asymptotic formula for the sum $$\begin{array}{cccc}& {\displaystyle \sum _{\begin{array}{c}n=a_1^2+a_2^2+\dots +a_6^2\le x\\ (a_1,a_2,\dots ,a_6)\in {\mathbb{Z}}^6\end{array}}{\lambda }_{{\mathrm{s}\mathrm{y}\mathrm{m}}^{j}f}^2(n),}& & \end{array}$$ with an improved error term.