Computing Genus-$2$ Hilbert--Siegel Modular Forms over $\Q(\sqrt{5})$ via the Jacquet--Langlands Correspondence

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.