Computations of vector-valued Siegel modular forms

TL;DR

This paper computes vector-valued Siegel modular forms of degree two, revealing new rational eigenforms, non-rational non-cuspidal forms, and supporting conjectures on modular form congruences.

Contribution

It provides explicit computations and experimental evidence for properties of vector-valued Siegel modular forms, including eigenforms and congruences.

Findings

Identified a rational eigenform in a three-dimensional cusp form space

Observed non-cuspidal eigenforms of level one are not always rational

Verified cases of conjectures on congruences between classical and Siegel modular forms

Abstract

We carry out some computations of vector valued Siegel modular forms of degree two, weight (k,2) and level one. Our approach is based on Satoh's description of the module of vector-valued Siegel modular forms of weight (k, 2) and an explicit description of the Hecke action on Fourier expansions. We highlight three experimental results: (1) we identify a rational eigenform in a three dimensional space of cusp forms, (2) we observe that non-cuspidal eigenforms of level one are not always rational and (3) we verify a number of cases of conjectures about congruences between classical modular forms and Siegel modular forms.