Numerical computation of a certain Dirichlet series attached to Siegel modular forms of degree two

Abstract

The Rankin convolution type Dirichlet series D F,G (s) of Siegel modular forms F and G of degree two, which was introduced by Kohnen and the second author, is computed numerically for various F and G. In particular, we prove that the series D F,G (s), which share the same functional equation and analytic behavior with the spinor L-functions of eigenforms of the same weight are not linear combinations of those. In order to conduct these experiments a numerical method to compute the Petersson scalar products of Jacobi Forms is developed and discussed in detail. Following the terminology of [15] we call a cusp form interesting if it belongs to the orthogonal complement of the Maass spezialschaar.

Weil representations associated with finite quadratic modules

“…However, for the discriminant form of Example 1.1, which is connected to classical Jacobi forms, we have the following corollary. Using elementary properties of Kronecker symbols it is easy to show that the formula in this corollary is equivalent to that of [39,Thm. 3].…”

“…The computational aspects were foremost in mind when we obtained these formulas; we needed efficient algorithms for the Weil representation in order to compute vector-valued Poincaré series [39] and harmonic weak Maass forms [6]. The formula stated in the Main Theorem is implemented as part of a package [1] written in Sage [48] for computing with finite quadratic modules.…”