Using Katsurada's Determination of the Eisenstein Series to Compute Siegel Eigenforms

TL;DR

This paper computes bases of level one, degree three Siegel modular forms and their 2-Euler factors using Katsurada's explicit Eisenstein series Fourier coefficients, supporting Miyawaki's conjecture.

Contribution

It introduces a novel computational approach leveraging Katsurada's Fourier coefficients and Garrett's decomposition to explicitly determine Siegel eigenforms and their properties.

Findings

Computed Hecke eigenform bases up to weight 22

Provided evidence supporting Miyawaki's conjectural lift

Identified examples of eigenforms that are congruence neighbors

Abstract

We compute Hecke eigenform bases of spaces of level one, degree~three Siegel modular forms and 2-Euler factors of the eigenforms through weight 22. Our method uses the Fourier coefficients of Siegel Eisenstein series, which are fully known and computationally tractable by the work of H. Katsurada; we also use P. Garrett's decomposition of the pullback of the Eisenstein series through the Witt map. Our results support I. Miyawaki's conjectural lift, and they give examples of eigenforms that are congruence neighbors.