Siegel modular forms for $M_{k,j}\left(\Gamma_0(2)\right)$

Dimension table of spaces of degree 2 Siegel 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

Enter a new range of weights for dimension table:

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

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