The ring of Siegel modular forms of degree 2 with respect to the full modular group (awaiting review)

By a classical result of Jun-Ichi Igusa [MR:0141643], the ring $M_{2}({\rm Sp}(4,\mathbb{Z}))$ of Siegel modular forms of degree 2 with even weights with respect to the full modular group $\mathrm{Sp}(4,{\Bbb Z})$ is generated by the four algebraically independent Eisenstein series: $\psi_4$, $\psi_6$, $\psi_{10}$, $\psi_{12}$.

Alternatively, $\psi_{10}$ can be replaced by the cusp form $$\chi_{10} =-43867\cdot 2^{-12}3^{-5}5^{-2}7^{-1}53^{-1}(\psi_4\psi_6-\psi_{10})$$ and $\psi_{12}$ can be replaced by the cusp form $$\chi_{12}=131\cdot593\cdot2^{-13}3^{-7}5^{-3}7^{-2}337^{-1}(441\psi_4^3+250\psi_6^2-691\psi_{12}).$$

Note that $\chi_{10}$ is a Saito-Kurokawa lifting coming from the cuspform of weight 18 for ${\rm SL}(2,\mathbb{Z})$, and $\chi_{12}$ is a Saito-Kurokawa lifting coming from the cuspform of weight 22.

The even weight forms are: $${\Bbb C}[\psi_4,\psi_6,\chi_{10},\chi_{12}],$$ with the cusp forms being the ideal generated by $\chi_{10}, \chi_{12}$.

To generate the odd weight forms, there is one more generator, namely $\chi_{35}$

The odd weight forms are: $$\chi_{35}\,{\Bbb C}[\psi_4,\psi_6,\chi_{10},\chi_{12}],$$ where $\chi_{35}$ is a weight 35 cusp form.

We write - $A=\psi_4$, - $B=\psi_6$, - $C=-4\chi_{10}$, - $D=12\chi_{12}$, for short in the modular form pages.