Abstract: Modularity in degree one involves weight two elliptic modular forms. Modularity in degree two involves weight two Siegel paramodular forms.

The Paramodular Conjecture of Brumer and Kramer tells us where to look for Siegel modular forms that correspond to certain abelian surfaces defined over the rationals.

This talk focuses on computations, and we consider several ways of computing Siegel modular forms over the paramodular group in degree two. We compile, in conjunction with Brumer and Kramer’s data on abelian surfaces, substantial evidence for the Paramodular Conjecture. We discuss strong evidence for existence and nonexistence of nonlift paramodular eigenforms of weight two in general levels N up to N < 1000.

We will show how to construct a nonlift paramodular eigenform of level 277 that conjecturally corresponds to the abelian surface of conductor 277. We discuss recent work that proves that, indeed, these two objects have the same L-function, and thus the abelian surface of conductor 277 is modular.