Let f be a Hecke eigenform of even weight for the full modular group SL(2, ℤ), and L(s, symjf), j ⩾ 2, be the jth symmetric power L-function associated to f. Denote by \(\lambda_{{\rm sym}^{j} f}(n)\) the nth normalized coefficient of the Dirichlet series of L(s, symjf). 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.
\(\sum_{\substack{n=a_{1}^{2}+a_{2}^{2}+\ldots+a_{8}^{2} \leqslant x \newline(a_{1}, a_{2}, \ldots, a_{8}) \in\mathbb{Z}^{8}}}\lambda_{{\rm sym}^{j} f}(n)\quad\text{and}\quad\sum_{\substack{n=a_{1}^{2}+a_{2}^{2}+\ldots+a_{8}^{2} \leqslant x \newline(a_{1}, a_{2}, \ldots, a_{8}) \in\mathbb{Z}^{8}}}\lambda_{{\rm sym}^{j} f}^{2}(n),\)
and obtain the corresponding asymptotic formulas.
