Defining parameters
| Level: | \( N \) | = | \( 400 = 2^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | = | \( 1 \) |
| Nonzero newspaces: | \( 2 \) | ||
| Newform subspaces: | \( 2 \) | ||
| Sturm bound: | \(9600\) | ||
| Trace bound: | \(1\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(400))\).
| Total | New | Old | |
|---|---|---|---|
| Modular forms | 419 | 97 | 322 |
| Cusp forms | 27 | 5 | 22 |
| Eisenstein series | 392 | 92 | 300 |
The following table gives the dimensions of subspaces with specified projective image type.
| \(D_n\) | \(A_4\) | \(S_4\) | \(A_5\) | |
|---|---|---|---|---|
| Dimension | 5 | 0 | 0 | 0 |
Trace form
Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(400))\)
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 |
|---|---|---|---|---|
| 400.1.b | \(\chi_{400}(351, \cdot)\) | 400.1.b.a | 1 | 1 |
| 400.1.e | \(\chi_{400}(199, \cdot)\) | None | 0 | 1 |
| 400.1.g | \(\chi_{400}(151, \cdot)\) | None | 0 | 1 |
| 400.1.h | \(\chi_{400}(399, \cdot)\) | None | 0 | 1 |
| 400.1.i | \(\chi_{400}(93, \cdot)\) | None | 0 | 2 |
| 400.1.k | \(\chi_{400}(99, \cdot)\) | None | 0 | 2 |
| 400.1.m | \(\chi_{400}(57, \cdot)\) | None | 0 | 2 |
| 400.1.p | \(\chi_{400}(193, \cdot)\) | None | 0 | 2 |
| 400.1.r | \(\chi_{400}(51, \cdot)\) | None | 0 | 2 |
| 400.1.t | \(\chi_{400}(157, \cdot)\) | None | 0 | 2 |
| 400.1.v | \(\chi_{400}(71, \cdot)\) | None | 0 | 4 |
| 400.1.x | \(\chi_{400}(79, \cdot)\) | 400.1.x.a | 4 | 4 |
| 400.1.z | \(\chi_{400}(31, \cdot)\) | None | 0 | 4 |
| 400.1.ba | \(\chi_{400}(39, \cdot)\) | None | 0 | 4 |
| 400.1.bc | \(\chi_{400}(53, \cdot)\) | None | 0 | 8 |
| 400.1.bf | \(\chi_{400}(19, \cdot)\) | None | 0 | 8 |
| 400.1.bg | \(\chi_{400}(17, \cdot)\) | None | 0 | 8 |
| 400.1.bj | \(\chi_{400}(73, \cdot)\) | None | 0 | 8 |
| 400.1.bk | \(\chi_{400}(11, \cdot)\) | None | 0 | 8 |
| 400.1.bn | \(\chi_{400}(13, \cdot)\) | None | 0 | 8 |
Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(400))\) into lower level spaces
\( S_{1}^{\mathrm{old}}(\Gamma_1(400)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 15}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 9}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 5}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(400))\)\(^{\oplus 1}\)