Siegel modular forms of level (4,8) and weight two

We consider the space of Siegel modular forms of genus $g$ of weight two relative to the main congruence subgroup of level 2 and to Igusa’s group ${\mathrm{\Gamma }}_g(4,8)$ and ${\mathrm{\Gamma }}_g(2,4)$.One of the main results of this paper is that in the case $g\ge 8$ the space $[{\mathrm{\Gamma }}_g[4,8],2]$ is generated by the products of 4 theta nullwerte. Thus this note can be considered as a completion of the example at the end of [Fr].