## 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 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

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