Defining parameters
| Level: | \( N \) | = | \( 324 = 2^{2} \cdot 3^{4} \) |
| Weight: | \( k \) | = | \( 1 \) |
| Nonzero newspaces: | \( 3 \) | ||
| Newform subspaces: | \( 5 \) | ||
| Sturm bound: | \(5832\) | ||
| Trace bound: | \(4\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(324))\).
| Total | New | Old | |
|---|---|---|---|
| Modular forms | 280 | 64 | 216 |
| Cusp forms | 10 | 8 | 2 |
| Eisenstein series | 270 | 56 | 214 |
The following table gives the dimensions of subspaces with specified projective image type.
| \(D_n\) | \(A_4\) | \(S_4\) | \(A_5\) | |
|---|---|---|---|---|
| Dimension | 8 | 0 | 0 | 0 |
Trace form
Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(324))\)
We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list available newforms together with their dimension.
| Label | \(\chi\) | Newforms | Dimension | \(\chi\) degree |
|---|---|---|---|---|
| 324.1.c | \(\chi_{324}(161, \cdot)\) | None | 0 | 1 |
| 324.1.d | \(\chi_{324}(163, \cdot)\) | 324.1.d.a | 1 | 1 |
| 324.1.d.b | 1 | |||
| 324.1.f | \(\chi_{324}(55, \cdot)\) | 324.1.f.a | 2 | 2 |
| 324.1.f.b | 2 | |||
| 324.1.g | \(\chi_{324}(53, \cdot)\) | 324.1.g.a | 2 | 2 |
| 324.1.j | \(\chi_{324}(19, \cdot)\) | None | 0 | 6 |
| 324.1.k | \(\chi_{324}(17, \cdot)\) | None | 0 | 6 |
| 324.1.n | \(\chi_{324}(7, \cdot)\) | None | 0 | 18 |
| 324.1.o | \(\chi_{324}(5, \cdot)\) | None | 0 | 18 |
Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(324))\) into lower level spaces
\( S_{1}^{\mathrm{old}}(\Gamma_1(324)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 15}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 10}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 5}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(162))\)\(^{\oplus 2}\)