Defining parameters
| Level: | \( N \) | = | \( 512 = 2^{9} \) |
| Weight: | \( k \) | = | \( 1 \) |
| Nonzero newspaces: | \( 3 \) | ||
| Newform subspaces: | \( 4 \) | ||
| Sturm bound: | \(16384\) | ||
| Trace bound: | \(9\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(512))\).
| Total | New | Old | |
|---|---|---|---|
| Modular forms | 397 | 120 | 277 |
| Cusp forms | 13 | 8 | 5 |
| Eisenstein series | 384 | 112 | 272 |
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 \(S_{1}^{\mathrm{new}}(\Gamma_1(512))\)
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 |
|---|---|---|---|---|
| 512.1.c | \(\chi_{512}(511, \cdot)\) | 512.1.c.a | 2 | 1 |
| 512.1.d | \(\chi_{512}(255, \cdot)\) | 512.1.d.a | 2 | 1 |
| 512.1.f | \(\chi_{512}(127, \cdot)\) | 512.1.f.a | 2 | 2 |
| 512.1.f.b | 2 | |||
| 512.1.h | \(\chi_{512}(63, \cdot)\) | None | 0 | 4 |
| 512.1.j | \(\chi_{512}(31, \cdot)\) | None | 0 | 8 |
| 512.1.l | \(\chi_{512}(15, \cdot)\) | None | 0 | 16 |
| 512.1.n | \(\chi_{512}(7, \cdot)\) | None | 0 | 32 |
| 512.1.p | \(\chi_{512}(3, \cdot)\) | None | 0 | 64 |
Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(512))\) into lower level spaces
\( S_{1}^{\mathrm{old}}(\Gamma_1(512)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(128))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(256))\)\(^{\oplus 2}\)