Characterization of Fourier Jacobi expansions of Paramodular forms

We give linear equations that characterize the Fourier Jacobi expansions of paramodular forms from among all convergent series of Jacobi forms. We suspect that these linear equations in fact characterize the Fourier-Jacobi expansions of paramodular forms from among all formal series of Jacobi forms. We use these linear equations to compute small eigenvalues of possible weight two paramodular cusp forms up to level 1000. We compare this data with the Paramodular Conjecture for modularity in genus two using the work on rational abelian surfaces of A. Brumer and K. Kramer.