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Keywords:
cusp form; Fourier coefficient; symmetric power $L$-function
cusp form; Fourier coefficient; symmetric power $L$-function
Summary:
Let $j\geq 2$ be a given integer. Let $H_{k}^{*}$ be the set of all normalized primitive holomorphic cusp forms of even integral weight $k\geq 2$ for the full modulo group ${\rm SL}(2,\mathbb {Z})$. For $f\in H_{k}^{*}$, denote by $\lambda _{{\rm sym}^{j}f}(n)$ the $n$th normalized Fourier coefficient of $j$th symmetric power $L$-function ($L(s, {\rm sym}^{j}f)$) attached to $f$. We are interested in the average behaviour of the sum $$ \sum _{n=a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}\leq x \atop (a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})\in \mathbb {Z}^{ 6}} \lambda _{{\rm sym}^{j}f}^{2}(n), $$ where $x$ is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023).
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