Hilbert surfaces, modular forms, and Siegel-Veech constants

We give the values of the Siegel-Veech constants associated with saddle connections having distinct endpoints on translation surfaces in Prym eigenform loci in $\Omega \mathcal{M}_3(2,2)^{\rm odd}$. In particular, we show that these constants are actually the same for all of these loci. As a by-product, we show that the Euler characteristic of the Hilbert modular surfaces which parametrize Abelian surfaces with $(1,2)$-polarization admitting a real multiplication and the Euler characteristic of their product locus are related by a simple formula. For principally polarized Abelian surfaces, a similar phenomenon has been observed by Bainbridge.