## Dimension table of spaces of degree 2Siegel modular forms

The table below lists, for each bold value of $k$ in the header, the dimensions of the following subspaces of $M_{k,j}\left(\Gamma_0(2)\right)$ for $j=2$:

• Total: The full space.
• Non cusp: The codimension of the subspace of cusp forms.
• Cusp: The subspace of cusp forms.
$4$ $5$ $6$ $7$ $8$ $9$ $10$ $11$ $12$ $13$ $14$ $15$ $16$ $17$ $18$ $19$ $20$ $21$ $22$ $23$ $24$
Total 0 0 2 0 4 0 9 1 13 3 22 6 30 10 43 17 56 24 75 35 93
Non cusp 0 0 2 0 2 0 4 0 4 0 6 0 6 0 8 0 8 0 10 0 10
Cusp 0 0 0 0 2 0 5 1 9 3 16 6 24 10 35 17 48 24 65 35 83

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# The ring of Siegel modular forms of degree 2 for $\Gamma_0(2)$

By a result of Tomoyoshi Ibukiyama [MR:1082834, 10.1142/S0129167X9100003X] , the ring $M_{*}(\Gamma_0(2))$ of Siegel modular forms of degree 2 with respect to the group $\Gamma_0(2)$ is generated by the following five generators, which are specified in terms of theta constants.

• $X$, a form of weight 2, with formula $X = ((\theta_{0000})^4+(\theta_{0001})^4+(\theta_{0010})^4+(\theta_{0011})^4)/4.$
• $Y$, a form of weight 4, with formula $Y = (\theta_{0000}\theta_{0001}\theta_{0010}\theta_{0011})^2.$
• $Z$, a form of weight 4, with formula $Z = ((\theta_{0100})^4-(\theta_{0110})^4)^2)/16384.$
• $K$, a form of weight 6, with formula $K = (\theta_{0100}\theta_{0110}\theta_{1000}\theta_{1001}\theta_{1100}\theta_{1111})^2/4096.$
• $\chi_{19}$, a cusp form of weight 19, with formula $\chi_{19} = \theta \theta'(8YZ-X^2T +YT +1024ZT+96T^2 - 8XK)/32,$ where $$\theta=\theta_{0000}\theta_{0001}\theta_{0010}\theta_{0011}\theta_{0100}\theta_{0110}\theta_{1000}\theta_{1001}\theta_{1100}\theta_{1111},\qquad \theta' = ((\theta_{1000})^{12}+(\theta_{1001})^{12}+(\theta_{1100})^{12}+(\theta_{1111})^{12})/1536,\qquad T = (\theta_{0100}\theta_{0110})^{4}/256.$$

The generators $X, Y, Z, K$ are algebraically independent. The ring of modular forms is

$$M(\Gamma_0(2)) = \C[X,Y,Z,K] + \chi_{19}\,\C[X,Y,Z,K].$$