Higher moments for symmetric powers of modular forms

Let $f$ be a cuspidal eigenform of weight $k$ on $\SL_2(\BZ)$ and let $λ_{\Sym^d f}(n)$ be the normalized Fourier coefficients of its $d$-th symmetric power lift. This paper establishes asymptotic formulas for the moments $\sum_{n\leq x}λ^l_{\Sym^d f}(n)$ for all positive integers $d$ and $l$. We also prove an asymptotic formula for the corresponding sum over the values of any positive definite binary quadratic form $Q$. Our results generalize and improve upon previous work, which was limited to small values of $d$ or $l$. The proofs rely on the decomposition of $\ell$-adic Galois representations and the analytic properties of the associated $L$-functions.

💡 Research Summary

The paper studies the higher moments of Fourier coefficients attached to symmetric power lifts of a cuspidal eigenform (f) of weight (k) on (\SL_2(\mathbb Z)). For any integers (d\ge1) and (l\ge1) with (dl>4), the authors obtain an asymptotic formula for the sum
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