Twisted moments of automorphic \(L\)-functions (Q710513)

Let \(k\geq 2\) be an even integer. Let \(N\) be a squarefree integer such that the set \(H_k^*(N)\) of primitive forms of weight \(k\) over \(\Gamma_0(N)\) is not empty. To any \(f\in H_k^*(N)\) it is associated an automorphic \(L\)-function \(L(s,f)\), arising from the standard representation of \(\text{SU}(2)\). Denote by \(\Sigma^h\) the harmonic average. The authors present formulas for \(\Sigma_{f\in H_k^*(N)}^h L(\frac12,f)^2L(1,\text{Sym}^m f)^z\), \(z\in{\mathbb C}\), \(N\) sufficiently large.