Shifted convolution sums of coefficients of symmetric power $L$-functions with $k$-full kernels over sums of squares in arithmetic progressions

Let $q$ be an integer and let $f$ be a normalised Hecke eigenform of integral weight for the full modular group.

Let $L(s,\mathrm{sym}^j f)$ denote the $j$-th symmetric power $L$-function associated to $f$, and let $\lambda_{\mathrm{sym}^j f}(n)$ denote its $n$-th coefficient.

We study the behaviour of the partial sum of $\lambda_{\mathrm{sym}^j f}(n)$, and of its second moment, taken over those sums of $m$ squares that are congruent to $1$ modulo $q$.

As an application, we investigate the shifted convolution sum of $\lambda_{\mathrm{sym}^j f}(n)$ against a $k$-full kernel function, for any $k \geq 2$.

We also study the number of sign changes of $\lambda_{\mathrm{sym}^j f}(n)$ twisted with a $k$-full kernel function, again over sums of $m$ squares.

Throughout, $m$ is even with $m \in \{2,4,6,8,10,12\}$.