Defining parameters
| Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 324.g (of order \(6\) and degree \(2\)) |
| Character conductor: | \(\operatorname{cond}(\chi)\) | \(=\) | \( 9 \) |
| Character field: | \(\Q(\zeta_{6})\) | ||
| Newform subspaces: | \( 4 \) | ||
| Sturm bound: | \(162\) | ||
| Trace bound: | \(7\) | ||
| Distinguishing \(T_p\): | \(5\), \(7\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{3}(324, [\chi])\).
| Total | New | Old | |
|---|---|---|---|
| Modular forms | 252 | 16 | 236 |
| Cusp forms | 180 | 16 | 164 |
| Eisenstein series | 72 | 0 | 72 |
Trace form
Decomposition of \(S_{3}^{\mathrm{new}}(324, [\chi])\) into newform subspaces
| Label | Dim. | \(A\) | Field | CM | Traces | $q$-expansion | |||
|---|---|---|---|---|---|---|---|---|---|
| \(a_2\) | \(a_3\) | \(a_5\) | \(a_7\) | ||||||
| 324.3.g.a | \(2\) | \(8.828\) | \(\Q(\sqrt{-3}) \) | \(\Q(\sqrt{-3}) \) | \(0\) | \(0\) | \(0\) | \(-11\) | \(q+(-11+11\zeta_{6})q^{7}-23\zeta_{6}q^{13}-37q^{19}+\cdots\) |
| 324.3.g.b | \(2\) | \(8.828\) | \(\Q(\sqrt{-3}) \) | \(\Q(\sqrt{-3}) \) | \(0\) | \(0\) | \(0\) | \(-2\) | \(q+(-2+2\zeta_{6})q^{7}+22\zeta_{6}q^{13}+26q^{19}+\cdots\) |
| 324.3.g.c | \(4\) | \(8.828\) | \(\Q(\zeta_{12})\) | None | \(0\) | \(0\) | \(0\) | \(14\) | \(q+\zeta_{12}q^{5}+7\zeta_{12}^{2}q^{7}+(-\zeta_{12}+\zeta_{12}^{3})q^{11}+\cdots\) |
| 324.3.g.d | \(8\) | \(8.828\) | \(\Q(\zeta_{24})\) | None | \(0\) | \(0\) | \(0\) | \(4\) | \(q+\zeta_{24}^{5}q^{5}+(\zeta_{24}-\zeta_{24}^{2})q^{7}+\zeta_{24}^{7}q^{11}+\cdots\) |
Decomposition of \(S_{3}^{\mathrm{old}}(324, [\chi])\) into lower level spaces
\( S_{3}^{\mathrm{old}}(324, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 2}\)