Defining parameters
| Level: | \( N \) | = | \( 1000 = 2^{3} \cdot 5^{3} \) |
| Weight: | \( k \) | = | \( 1 \) |
| Nonzero newspaces: | \( 2 \) | ||
| Newform subspaces: | \( 3 \) | ||
| Sturm bound: | \(60000\) | ||
| Trace bound: | \(4\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(1000))\).
| Total | New | Old | |
|---|---|---|---|
| Modular forms | 1192 | 264 | 928 |
| Cusp forms | 112 | 8 | 104 |
| Eisenstein series | 1080 | 256 | 824 |
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 \(S_{1}^{\mathrm{new}}(\Gamma_1(1000))\)
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 the newforms together with their dimension.
| Label | \(\chi\) | Newforms | Dimension | \(\chi\) degree |
|---|---|---|---|---|
| 1000.1.b | \(\chi_{1000}(751, \cdot)\) | None | 0 | 1 |
| 1000.1.e | \(\chi_{1000}(499, \cdot)\) | 1000.1.e.a | 2 | 1 |
| 1000.1.e.b | 2 | |||
| 1000.1.g | \(\chi_{1000}(251, \cdot)\) | 1000.1.g.a | 4 | 1 |
| 1000.1.h | \(\chi_{1000}(999, \cdot)\) | None | 0 | 1 |
| 1000.1.i | \(\chi_{1000}(557, \cdot)\) | None | 0 | 2 |
| 1000.1.l | \(\chi_{1000}(57, \cdot)\) | None | 0 | 2 |
| 1000.1.n | \(\chi_{1000}(51, \cdot)\) | None | 0 | 4 |
| 1000.1.p | \(\chi_{1000}(199, \cdot)\) | None | 0 | 4 |
| 1000.1.r | \(\chi_{1000}(151, \cdot)\) | None | 0 | 4 |
| 1000.1.s | \(\chi_{1000}(99, \cdot)\) | None | 0 | 4 |
| 1000.1.u | \(\chi_{1000}(257, \cdot)\) | None | 0 | 8 |
| 1000.1.x | \(\chi_{1000}(93, \cdot)\) | None | 0 | 8 |
| 1000.1.z | \(\chi_{1000}(39, \cdot)\) | None | 0 | 20 |
| 1000.1.ba | \(\chi_{1000}(19, \cdot)\) | None | 0 | 20 |
| 1000.1.bc | \(\chi_{1000}(11, \cdot)\) | None | 0 | 20 |
| 1000.1.bf | \(\chi_{1000}(31, \cdot)\) | None | 0 | 20 |
| 1000.1.bg | \(\chi_{1000}(17, \cdot)\) | None | 0 | 40 |
| 1000.1.bj | \(\chi_{1000}(13, \cdot)\) | None | 0 | 40 |
Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(1000))\) into lower level spaces
\( S_{1}^{\mathrm{old}}(\Gamma_1(1000)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(500))\)\(^{\oplus 2}\)