Asymptotics for the second moment of the Dirichlet coefficients of symmetric power \(L\)-functions (Q6592720)

Let \( m\geq 2\) be an integer. Let \(f\) be a holomorphic Hecke eigenform of even weight \(k\) for the full modular group \(\mathrm{SL}(2,\mathbb {Z})\). Let denote by \(\lambda_{\mathrm{Sym}^m f}(n)\) the \(n\)th normalized Dirichlet coefficient of the corresponding symmetric power \(L\)-function \(L(s,\mathrm{Sym}^m f )\) related to \(f\). The purpose of this paper is to prove the following asymptotic formula involving the weight \(k\) \N\[\N\sum_{n \leq x} \lambda^2_{\mathrm{Sym}^m f}(n)=C x+ O\left( k^{(m+1)^2/4}x^{1/2+\varepsilon}+ x^{((m+1)^2-1)/((m+1)^2+1)+\varepsilon} \right), \N\]\Nwhere \(C\) is a suitable constant, and the implied constant in the \(O\)-term depends on \(\varepsilon\) and \(m\).