Computations of vector-valued Siegel modular forms

We carry out some computations of vector-valued Siegel modular forms of degree two, weight (k,2) and level one, and highlight three experimental results: (1) we identify a rational eigenform in a three-dimensional space of cusp forms; (2) we observe that non-cuspidal eigenforms of level one are not always rational; (3) we verify a number of cases of conjectures about congruences between classical modular forms and Siegel modular forms. Our approach is based on Satohʼs description of the module of vector-valued Siegel modular forms of weight (k,2) and an explicit description of the Hecke action on Fourier expansions.