Defining parameters
| Level: | \( N \) | = | \( 56 = 2^{3} \cdot 7 \) |
| Weight: | \( k \) | = | \( 8 \) |
| Nonzero newspaces: | \( 6 \) | ||
| Newform subspaces: | \( 13 \) | ||
| Sturm bound: | \(1536\) | ||
| Trace bound: | \(2\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(56))\).
| Total | New | Old | |
|---|---|---|---|
| Modular forms | 708 | 370 | 338 |
| Cusp forms | 636 | 350 | 286 |
| Eisenstein series | 72 | 20 | 52 |
Trace form
Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(56))\)
We only show spaces with even 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 |
|---|---|---|---|---|
| 56.8.a | \(\chi_{56}(1, \cdot)\) | 56.8.a.a | 1 | 1 |
| 56.8.a.b | 1 | |||
| 56.8.a.c | 2 | |||
| 56.8.a.d | 3 | |||
| 56.8.a.e | 3 | |||
| 56.8.b | \(\chi_{56}(29, \cdot)\) | 56.8.b.a | 20 | 1 |
| 56.8.b.b | 22 | |||
| 56.8.e | \(\chi_{56}(27, \cdot)\) | 56.8.e.a | 2 | 1 |
| 56.8.e.b | 52 | |||
| 56.8.f | \(\chi_{56}(55, \cdot)\) | None | 0 | 1 |
| 56.8.i | \(\chi_{56}(9, \cdot)\) | 56.8.i.a | 14 | 2 |
| 56.8.i.b | 14 | |||
| 56.8.l | \(\chi_{56}(31, \cdot)\) | None | 0 | 2 |
| 56.8.m | \(\chi_{56}(3, \cdot)\) | 56.8.m.a | 108 | 2 |
| 56.8.p | \(\chi_{56}(37, \cdot)\) | 56.8.p.a | 108 | 2 |
Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(56))\) into lower level spaces
\( S_{8}^{\mathrm{old}}(\Gamma_1(56)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 2}\)