  We compute Hecke eigenform bases of spaces of level one, degree~three Siegel
modular forms and 2-Euler factors of the eigenforms through weight 22. Our
method uses the Fourier coefficients of Siegel Eisenstein series, which are
fully known and computationally tractable by the work of H. Katsurada; we also
use P. Garrett's decomposition of the pullback of the Eisenstein series through
the Witt map. Our results support I. Miyawaki's conjectural lift, and they give
examples of eigenforms that are congruence neighbors.
