Degree of a Siegel modular form (awaiting review)

A Siegel modular form on an arithmetic subgroup of $\mathrm{GSp}(2g)$ is said to have degree $g$ (their L-functions then have degree $2g$). Under the paramodular conjecture, genus 2 curves $C/\Q$ with $\mathrm{End}(\mathrm{Jac}(C))=\Z$ correspond to weight 2 Siegel modular forms of degree $g=2$.