Defining parameters
| Level: | \( N \) | = | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | = | \( 8 \) |
| Nonzero newspaces: | \( 6 \) | ||
| Newform subspaces: | \( 25 \) | ||
| Sturm bound: | \(3200\) | ||
| Trace bound: | \(1\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(75))\).
| Total | New | Old | |
|---|---|---|---|
| Modular forms | 1456 | 987 | 469 |
| Cusp forms | 1344 | 947 | 397 |
| Eisenstein series | 112 | 40 | 72 |
Trace form
Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(75))\)
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 |
|---|---|---|---|---|
| 75.8.a | \(\chi_{75}(1, \cdot)\) | 75.8.a.a | 1 | 1 |
| 75.8.a.b | 1 | |||
| 75.8.a.c | 1 | |||
| 75.8.a.d | 2 | |||
| 75.8.a.e | 2 | |||
| 75.8.a.f | 2 | |||
| 75.8.a.g | 3 | |||
| 75.8.a.h | 3 | |||
| 75.8.a.i | 4 | |||
| 75.8.a.j | 4 | |||
| 75.8.b | \(\chi_{75}(49, \cdot)\) | 75.8.b.a | 2 | 1 |
| 75.8.b.b | 2 | |||
| 75.8.b.c | 2 | |||
| 75.8.b.d | 4 | |||
| 75.8.b.e | 4 | |||
| 75.8.b.f | 6 | |||
| 75.8.e | \(\chi_{75}(32, \cdot)\) | 75.8.e.a | 4 | 2 |
| 75.8.e.b | 4 | |||
| 75.8.e.c | 16 | |||
| 75.8.e.d | 24 | |||
| 75.8.e.e | 32 | |||
| 75.8.g | \(\chi_{75}(16, \cdot)\) | 75.8.g.a | 68 | 4 |
| 75.8.g.b | 68 | |||
| 75.8.i | \(\chi_{75}(4, \cdot)\) | 75.8.i.a | 144 | 4 |
| 75.8.l | \(\chi_{75}(2, \cdot)\) | 75.8.l.a | 544 | 8 |
Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(75))\) into lower level spaces
\( S_{8}^{\mathrm{old}}(\Gamma_1(75)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 2}\)