Eigenvalues and congruences for the weight $3$ paramodular nonlifts of levels $61$, $73$, and $79$

We use Borcherds products to give a new construction of the weight $3$ paramodular nonlift eigenform $f_N$ for levels $N=61,73,79$. We classify the congruences of $f_N$ to Gritsenko lifts. We provide techniques that compute eigenvalues to support future modularity applications. Our method does not compute Hecke eigenvalues from Fourier coefficients but instead uses elliptic modular forms, specifically the restrictions of Gritsenko lifts and their images under the slash operator to modular curves.