Defining parameters
| Level: | \( N \) | = | \( 768 = 2^{8} \cdot 3 \) |
| Weight: | \( k \) | = | \( 1 \) |
| Nonzero newspaces: | \( 2 \) | ||
| Newform subspaces: | \( 5 \) | ||
| Sturm bound: | \(32768\) | ||
| Trace bound: | \(1\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(768))\).
| Total | New | Old | |
|---|---|---|---|
| Modular forms | 732 | 116 | 616 |
| Cusp forms | 28 | 12 | 16 |
| Eisenstein series | 704 | 104 | 600 |
The following table gives the dimensions of subspaces with specified projective image type.
| \(D_n\) | \(A_4\) | \(S_4\) | \(A_5\) | |
|---|---|---|---|---|
| Dimension | 12 | 0 | 0 | 0 |
Trace form
Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(768))\)
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 |
|---|---|---|---|---|
| 768.1.b | \(\chi_{768}(127, \cdot)\) | None | 0 | 1 |
| 768.1.e | \(\chi_{768}(257, \cdot)\) | 768.1.e.a | 1 | 1 |
| 768.1.e.b | 1 | |||
| 768.1.e.c | 2 | |||
| 768.1.g | \(\chi_{768}(511, \cdot)\) | None | 0 | 1 |
| 768.1.h | \(\chi_{768}(641, \cdot)\) | None | 0 | 1 |
| 768.1.i | \(\chi_{768}(65, \cdot)\) | 768.1.i.a | 4 | 2 |
| 768.1.i.b | 4 | |||
| 768.1.l | \(\chi_{768}(319, \cdot)\) | None | 0 | 2 |
| 768.1.m | \(\chi_{768}(31, \cdot)\) | None | 0 | 4 |
| 768.1.p | \(\chi_{768}(161, \cdot)\) | None | 0 | 4 |
| 768.1.q | \(\chi_{768}(17, \cdot)\) | None | 0 | 8 |
| 768.1.t | \(\chi_{768}(79, \cdot)\) | None | 0 | 8 |
| 768.1.u | \(\chi_{768}(7, \cdot)\) | None | 0 | 16 |
| 768.1.x | \(\chi_{768}(41, \cdot)\) | None | 0 | 16 |
| 768.1.y | \(\chi_{768}(5, \cdot)\) | None | 0 | 32 |
| 768.1.bb | \(\chi_{768}(19, \cdot)\) | None | 0 | 32 |
Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(768))\) into lower level spaces
\( S_{1}^{\mathrm{old}}(\Gamma_1(768)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 18}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 16}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 9}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 14}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 7}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 10}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(48))\)\(^{\oplus 5}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(64))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(128))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(192))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(256))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(384))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(768))\)\(^{\oplus 1}\)