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 $M_k\left(\Gamma_0(3)\right)$:

$0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $11$ $12$ $13$ $14$ $15$ $16$ $17$ $18$ $19$ $20$
Total 1 0 1 0 4 0 7 0 10 0 17 0 25 0 32 1 46 3 60 4 74

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The ring of Siegel modular forms of degree 2 with respect to $\Gamma_0(3)$

By a result of Tomoyoshi Ibukiyama [MR:1082834, 10.1142/S0129167X9100003X] , the ring $M_{*}(\Gamma_0(3))$ of Siegel modular forms of degree 2 with respect to the group $\Gamma_0(3)$ is generated by functions that may be specified in terms of theta series.

Let $$ A_2\left(\begin{smallmatrix}2&1\\1&2\end{smallmatrix}\right),\qquad E_6\left(\begin{smallmatrix} 2&-1&0&0&0&0\\ -1&2&-1&0&0&0\\ 0&-1&2&-1&0&-1\\ 0&0&-1&2&-1&0\\ 0&0&0&-1&2&0\\ 0&0&-1&0&0&2 \end{smallmatrix}\right),\qquad E_6^* = 3E6^{-1},\qquad S\left(\begin{smallmatrix} 1&0&3/2&0\\ 0&1&0&3/2\\ 3/2&0&3&0\\ 0&3/2&0&3 \end{smallmatrix}\right), $$ and let $P:{\Bbb C}^4\times{\Bbb C}^4\to{\Bbb C}$ be a pluriharmonic polynomial defined by $$ P(x,y)=(x_1y_3-x_3y_1)+(x_2y_4-x_4y_2)^2-(x_1y_4-x_4y_1 + x_3y_2-x_2y_3 + x_1y_2-x_2y_1 )^2. $$

We now define the generators

where the character $\psi_3:\Gamma_0(3)\to\{\pm1\}$ is defined by $$\psi_3(\left(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\right)) =\left(\frac{-3}{\det D}\right).$$

The generators $\alpha_1, \beta_3, \gamma_4, \delta_3$ are algebraically independent. Let $$B = {\Bbb C}[\alpha_1, \beta_3, \gamma_4, \delta_3],\qquad C = {\Bbb C}[\alpha_1^2, \beta_3^2, \gamma_4, \delta_3^2].$$ We then have the ring $$ M_*(\Gamma_0(3)) = B^{\rm{even}} \oplus C\alpha_1\chi_{14}\oplus C\beta_3\chi_{14} \oplus C\delta_3\chi_{14}\oplus C\alpha_1\beta_3\delta3\chi_{14}. $$