Cusp forms and special values of certain Dirichlet series (Q1183085)

Let \(S_k\) denote the space of elliptic cusp forms on \(\mathrm{SL}_2(\mathbb{Z})\) of even weight \(k\). Given \(f\in S_k\), \(g\in S_l\), \(k>l+2\), \(n\geq 1\), the author constructs a cusp form from special values of the Dirichlet series \[ L_{f,g;n}(s):=\sum_{m\geq 1}\alpha_ f(m+n)\cdot\overline{\alpha_ g(m)}\cdot(m+n)^{-s} \] attached to the Fourier expansions of \(f\) and \(g\). More precisely the adjoint with respect to the Petersson scalar product of the map \(S_{k-l}\to S_ k\), \(h\mapsto gh\), is up to a constant given by \(W_g: S_k\to S_{k- l}\), \(f\mapsto W_ g(f)\), where \[ W_g(f)(z):=\sum_{n\geq 1}n^{k-l- 1} L_{f,g;n}(k-1)e^{2\pi inz}. \]