Available samples
The following table shows the Galois orbits of all vector valued cusp forms of weights 10,2 to 30,2 on the full modular group $\mathrm{Sp}(4,\mathbb{Z})$.
| Weight | Galois orbits (number of forms) |
|---|---|
| 10 | 10_E (1) |
| 14 | 14_C (1) |
| 16 | 16_C (2) |
| 18 | 18_C (2) |
| 20 | 20_C1 (1) 20_C2 (2) 20_E (1) |
| 22 | 22_C (5) |
| 24 | 24_C (5) |
| 26 | 26_C (8) |
| 28 | 28_C (10) |
| 30 | 30_C (11) |
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 $M_{k,2}\left(\textrm{Sp}(4,\mathbb{Z})\right)$:
- Total: The full space.
- Non cusp: The subspace of non 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 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 2 | 0 | 3 | 0 | 3 | 0 | 4 | 1 | 7 | 1 | 6 |
| Non cusp | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 2 | 0 | 1 |
| Cusp | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 2 | 0 | 2 | 0 | 3 | 1 | 5 | 1 | 5 |
Enter a new range of weights for dimension table:
Vector valued Siegel modular forms of degree 2 taking values in three-dimensional space
The module of vector-valued Siegel modular forms of degree 2, taking values in a three-dimensional space, with respect to the full modular group.
Let $\psi_4$, $\psi_6$, $\chi_{10}$, $\chi_{12}$ be generators of $M(\Sp(4,\Z))$, the Siegel modular forms of degree 2 with respect to the full modular group. We will write $\Gamma_2=\Sp(4,\Z)$ for short. For $f\in M_k(\Gamma_2)$ and $g\in M_j(\Gamma_2)$, define the Satoh bracket [MR:0816719] $$[f,g] = \frac 1{2\pi i}\left(\frac1k g\frac{d}{dZ}f-\frac1j f\frac{d}{dZ}g\right).$$ Then $[f,g]\in M_{k,2}(\Gamma_2)$, and for even integers $k$, $$ \begin{aligned} M_{k,2}(\Gamma_2) =& M_{k-10}(\Gamma_2) [\psi_4,\psi_6] \oplus M_{k-14}(\Gamma_2) [\psi_4,\chi_{10}] \\ &\oplus M_{k-16}(\Gamma_2) [\psi_4,\chi_{12}] \oplus V_{k-16}(\Gamma_2) [\psi_6,\chi_{10}]\\ &\oplus V_{k-18}(\Gamma_2) [\psi_6,\chi_{12}]\oplus V_{k-22}(\Gamma_2) [\chi_{10},\chi_{12}], \end{aligned} $$ where $$V_k(\Gamma_2)=M_k(\Gamma_2)\cap \C[\psi_6,\chi_{10},\chi_{12}],$$ $$W_k(\Gamma_2)=M_k(\Gamma_2)\cap \C[\chi_{10},\chi_{12}].$$
We write - $A=\psi_4$, - $B=\psi_6$, - $C=\chi_{10}$, - $D=\chi_{12}$, for short in the modular form sample pages.
Dimension tables are taken from Tsushima [MR:0816719].