I am looking for a Siegel modular form of genus $2$ (living on the Sielge modular 3-fold $A_2=\mathbb{H}/\mathrm{Sp}(4,\mathbb{Z})$) which becomes "roughly" the product of two eta functions on the locus where the parametrized abelian surface is the product of two elliptic curves. Here "roughly" means up to multiplication by something like e^{\pi i \tau/12}.

Does anyone know a reference for this sort of result?