# On the computation of algebraic modular forms on compact inner forms of $\mathrm{GSp}_4$

## TL;DR

This paper presents an algorithm to compute algebraic modular forms on compact inner forms of GSp_4 over totally real fields, enabling the calculation of Hecke eigensystems of Hilbert-Siegel modular forms of genus 2.

## Contribution

It introduces a novel algorithm for computing algebraic modular forms on GSp_4 inner forms, extending computational capabilities for Hilbert-Siegel modular forms.

## Key findings

- Successfully computed Hecke eigensystems over Q(√2)
- Demonstrated the algorithm's effectiveness on specific examples
- Extended computational methods to totally real fields

## Abstract

In this paper, we describe an algorithm for computing algebraic modular forms on compact inner forms of $\mathrm{GSp}_4$ over totally real number fields. By analogues of the Jacquet-Langlands correspondence for $\mathrm{GL}_2$, this algorithm in fact computes Hecke eigensystems of Hilbert-Siegel modular forms of genus 2. We give some examples of such eigensystems over $\Q(\sqrt{2})$.