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

TL;DR

This paper introduces a new construction method for certain weight 3 paramodular nonlift eigenforms at levels 61, 73, and 79, using Borcherds products, and analyzes their congruences and eigenvalues.

Contribution

It provides a novel approach to construct and analyze weight 3 paramodular nonlifts using Borcherds products and elliptic modular forms, bypassing traditional Fourier coefficient methods.

Findings

Constructed explicit eigenforms for levels 61, 73, 79

Classified congruences to Gritsenko lifts

Developed techniques to compute eigenvalues without Fourier coefficients

Abstract

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.