# Computing genus 2 Hilbert-Siegel modular forms over $\Q(\sqrt{5})$ via the Jacquet-Langlands correspondence

## TL;DR

This paper introduces an algorithm to compute Hecke eigensystems of Hilbert-Siegel cusp forms over real quadratic fields, with examples over $ ext{Q}( ext{sqrt}(5))$, revealing potential lifts from Hilbert eigenforms.

## Contribution

The paper develops a novel algorithm for computing Hilbert-Siegel cusp forms over real quadratic fields, expanding computational tools in this area.

## Key findings

- Identified Hilbert-Siegel eigenforms that are potential lifts from Hilbert eigenforms.
- Provided explicit examples over $ ext{Q}( ext{sqrt}(5))$ demonstrating the algorithm.
- Enhanced understanding of the structure of Hilbert-Siegel modular forms over quadratic fields.

## Abstract

In this paper we present an algorithm for computing Hecke eigensystems of Hilbert-Siegel cusp forms over real quadratic fields of narrow class number one. We give some illustrative examples using the quadratic field $\Q(\sqrt{5})$. In those examples, we identify Hilbert-Siegel eigenforms that are possible lifts from Hilbert eigenforms.