Moments and sign changes of symmetric power $L$-function coefficients over sums of squares

Abstract

Let $f$ be a normalised Hecke eigenform of even integral weight for the full modular group $\mathrm{SL}(2,\mathbb{Z})$, let $L(s,\mathrm{sym}^{j}f)$ be the $j$th symmetric power $L$-function attached to $f$, and let $λ_{\mathrm{sym}^{j}f}(n)$ denote its $n$th Dirichlet coefficient. For each even integer $m$ with $2 \le m \le 12$, we establish upper bounds for the partial sums of $λ_{\mathrm{sym}^{j}f}(n)$ and asymptotic formulas for those of $λ_{\mathrm{sym}^{j}f}^{2}(n)$ taken over integers represented as a sum of $m$ squares. As an application, we obtain lower bounds for the number of sign changes of $λ_{\mathrm{sym}^{j}f}(n)$ along these sums of $m$ squares.