Defining parameters
| Level: | \( N \) | = | \( 675 = 3^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | = | \( 1 \) |
| Nonzero newspaces: | \( 4 \) | ||
| Newform subspaces: | \( 7 \) | ||
| Sturm bound: | \(32400\) | ||
| Trace bound: | \(4\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(675))\).
| Total | New | Old | |
|---|---|---|---|
| Modular forms | 870 | 350 | 520 |
| Cusp forms | 30 | 22 | 8 |
| Eisenstein series | 840 | 328 | 512 |
The following table gives the dimensions of subspaces with specified projective image type.
| \(D_n\) | \(A_4\) | \(S_4\) | \(A_5\) | |
|---|---|---|---|---|
| Dimension | 14 | 0 | 0 | 8 |
Trace form
Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(675))\)
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 |
|---|---|---|---|---|
| 675.1.c | \(\chi_{675}(26, \cdot)\) | 675.1.c.a | 1 | 1 |
| 675.1.c.b | 1 | |||
| 675.1.c.c | 2 | |||
| 675.1.d | \(\chi_{675}(674, \cdot)\) | 675.1.d.a | 2 | 1 |
| 675.1.g | \(\chi_{675}(82, \cdot)\) | 675.1.g.a | 4 | 2 |
| 675.1.g.b | 4 | |||
| 675.1.i | \(\chi_{675}(224, \cdot)\) | None | 0 | 2 |
| 675.1.j | \(\chi_{675}(251, \cdot)\) | None | 0 | 2 |
| 675.1.m | \(\chi_{675}(134, \cdot)\) | None | 0 | 4 |
| 675.1.o | \(\chi_{675}(161, \cdot)\) | 675.1.o.a | 8 | 4 |
| 675.1.p | \(\chi_{675}(118, \cdot)\) | None | 0 | 4 |
| 675.1.s | \(\chi_{675}(74, \cdot)\) | None | 0 | 6 |
| 675.1.t | \(\chi_{675}(101, \cdot)\) | None | 0 | 6 |
| 675.1.v | \(\chi_{675}(28, \cdot)\) | None | 0 | 8 |
| 675.1.x | \(\chi_{675}(71, \cdot)\) | None | 0 | 8 |
| 675.1.z | \(\chi_{675}(44, \cdot)\) | None | 0 | 8 |
| 675.1.bb | \(\chi_{675}(7, \cdot)\) | None | 0 | 12 |
| 675.1.be | \(\chi_{675}(37, \cdot)\) | None | 0 | 16 |
| 675.1.bf | \(\chi_{675}(11, \cdot)\) | None | 0 | 24 |
| 675.1.bh | \(\chi_{675}(14, \cdot)\) | None | 0 | 24 |
| 675.1.bj | \(\chi_{675}(13, \cdot)\) | None | 0 | 48 |
Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(675))\) into lower level spaces
\( S_{1}^{\mathrm{old}}(\Gamma_1(675)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 9}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(75))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(135))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(225))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(675))\)\(^{\oplus 1}\)