Almost all Fourier coefficients of symmetric power \(L\)-functions are small (Q6164942)

Let \(f\) be a Hecke eigenform of level \(k\) with respect to \(\Gamma_0(N)\) with a trivial central character and \(\pi=\pi_f\) be the cuspidal representation attached to \(f\). Let \(L(s,\mathrm{Sym}^m\pi)\) be the symmetric power \(L\)-functions. Write \[ L(s,\pi)=\sum_{n=1}^{\infty} a_f(n) n^{-s}\text{ and } L(s,\mathrm{Sym}^m \pi)=\sum_{n=1}^{\infty} a_{\mathrm{Sym}^m\pi}(n) n^{-s}. \] Let \(\varepsilon>0\) be fixed. \textit{F. Luca} et al. proved that \(|a_f(n)|\le (\log n)^{-1/2+\varepsilon}\) holds for almost all positive integers \(n\) [Math. Proc. Camb. Philos. Soc. 166, No. 1, 173--189 (2019; Zbl 1450.11100)]. The author proves that \(|a_{\mathrm{Sym}^m\pi}(n)|\le (\log n)^{-1/2+\varepsilon}\) holds for almost all positive integers \(n\). The method of proof follows closely that of \textit{F. Luca} et al. [Math. Proc. Camb. Philos. Soc. 166, No. 1, 173--189 (2019; Zbl 1450.11100)].