A generalized Shimura correspondence for newforms (Q2472413)

Let \(N\) be an odd number and let \(S_{2k}(N)\) be the space of weight \(2k\), level \(N\) cusp forms. Let \[ f(z)=\sum_{n=1}^\infty a(n)e^{2\pi i nz}\in S_{2k}(N) \] be a newform. For a fundamental discriminant \(D\) the twisted \(L\)-function of \(f\) is \[ L(f,D,s)=\sum_{n=1}^\infty\left(\frac{D}{n}\right) a(n) n^{-s}. \] The author attaches to \(f(z)\) a set of one-dimensional spaces \(S^+(f,\chi,S)\), where each space consists of forms \(g(z)=\sum_{n=1}^\infty c(n)e^{2\pi i nz}\) that are Shimura correspondences of \(f(z)\) in a generalized sense. The set of fundamental discriminants is partitioned into finitely many subsets \(\Delta_{\eta,S}\). For each subset \(\Delta_{\eta,S}\) it is associated one \(S^+(f,\chi,S)\) such that for \(g(z)\in S^+(f,\chi,S)\) and \(D\in \Delta_{\eta,S} \) the following formula holds: \[ \kappa\frac{|c(|D|)|^2}{\langle g,g\rangle}=\frac{L(f,D,k)}{\langle f,f\rangle}|D|^{k-1/2}\frac{(k-1)!}{\pi^k}, \] with some explicit constant \(\kappa\).