  The moduli space of principally polarized abelian surfaces is parametrized by
three Igusa functions. In this article we investigate a new way to evaluate
these functions by using Siegel Eisenstein series. We explain how to compute
the Fourier coefficients of certain Siegel modular forms using classical
modular forms of half-integral weight. One of the results in this paper is an
explicit algorithm to evaluate the Igusa functions to a prescribed precision.
