An algorithm to compute upper bounds of dimensions for Siegel Modular Forms of Prime Level and Arbitrary Nebentypus

TL;DR

This paper presents an algorithm and Java implementation to compute upper bounds for the dimensions of Siegel modular forms of genus two with arbitrary level, character, and weight, including low weights.

Contribution

It introduces a novel algorithmic method and software to determine restriction map images for Siegel modular forms with arbitrary characters and weights, providing new dimension bounds.

Findings

Computed upper bounds for non-trivial character spaces

Enabled analysis for low weight forms (k ≤ 4)

Provided a practical tool for modular form dimension estimation

Abstract

We describe an algorithmic method to determine the image of restriction maps for Siegel modular forms with \textit{arbitrary} characters and arbitrary weight. A program has been implemented in the mathematical software \texttt{Java} to compute the Fourier expansion of the image of these restriction maps for Siegel modular forms of genus two. This approach allows us to compute an upper bound for the space of Siegel modular forms with {\it non-trivial} character (which has not been previously known) and arbitrary weights (including low weight $k \leq 4$).