Siegel Paramodular Forms of Weight 2 and Squarefree Level

TL;DR

This paper computes and analyzes the space of weight 2 Siegel paramodular cusp forms of squarefree levels less than 300, confirming the paramodular conjecture and identifying special cases with nonlift newforms.

Contribution

It provides explicit computations of Siegel paramodular cusp forms for squarefree levels under 300, confirming the paramodular conjecture and identifying exceptions with nonlift newforms.

Findings

Most spaces are generated by Gritsenko lifts.

For N=249,295, nonlift newforms are present.

Euler factors match for specific abelian surfaces.

Abstract

We compute the space $S_2(K(N))$ of weight $2$ Siegel paramodular cusp forms of squarefree level $N<300$. In conformance with the paramodular conjecture of A. Brumer and K. Kramer, the space is only the additive (Gritsenko) lift space of the Jacobi cusp form space $J_{2,N}^{\text{cusp}}$ except for $N=249,295$, when it further contains one nonlift newform. For these two values of $N$, the Hasse-Weil $p$-Euler factors of a relevant abelian surface match the spin $p$-Euler factors of the nonlift newform for the first two primes $p\nmid N$.