The average behavior of coefficients of symmetric power $L$-functions over a certain sequence

summary:Let $f$ be a Hecke eigenform of even weight for the full modular group $SL(2,\mathbb {Z})$, and $L(s,{\rm sym}^{j} f)$, $j \ge 2,$ be the $j$th symmetric power $L$-function associated to $f$. Denote by $\lambda _{{\rm sym}^{j} f}(n)$ the $n$th normalized coefficient of the Dirichlet series of $L(s,{\rm sym}^{j} f)$. We study the average behavior of $\lambda _{{\rm sym}^{j} f}(n)$ and $\lambda _{{\rm sym}^{j} f}^{2}(n)$ over sums of squares of eight integers, i.e., $$\sum _{\substack {n=a_{1}^{2}+a_{2}^{2}+\cdots +a_{8}^{2} \leq x \\(a_{1}, a_{2}, \cdots , a_{8}) \in \mathbb {Z}^{8}}}\lambda _{{\rm sym}^{j} f}(n)\quad \text {and}\quad \sum _{\substack {n=a_{1}^{2}+a_{2}^{2}+\cdots +a_{8}^{2} \leq x \\(a_{1}, a_{2}, \cdots , a_{8}) \in \mathbb {Z}^{8}}}\lambda _{{\rm sym}^{j} f}^{2}(n),$$ and obtain the corresponding asymptotic formulas.