Small spanning sets of cusp forms (Q1906494)

Let \(\Gamma\) be a Fuchsian group of the first kind with cusps. In the paper it is shown how to pass from a finite set of generators for \(\Gamma\) including a parabolic element to finite spanning sets of the spaces \(S_{2k} (\Gamma)\) of cusp forms on \(\Gamma\) of weight \(2k\) for arbitrary integers \(k\geq 2\). The spanning sets consist of hyperbolic Poincaré series or of Poincaré series associated to pairs of cusps. The cardinalities of the sets grow linearly with \(k\), i.e. linearly with the dimension of \(S_{2k} (\Gamma)\).