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 | \(0\) | \(0\) | \(0\) | \(14\) | \(q+\beta_1 q^{5}+7\beta_{2} q^{7}+(\beta_{3}-\beta_1)q^{11}+\cdots\) |
324.3.g.d | $8$ | $8.828$ | \(\Q(\zeta_{24})\) | None | \(0\) | \(0\) | \(0\) | \(4\) | \(q+\beta_{5} q^{5}+(-\beta_{2}+\beta_1)q^{7}+\beta_{7} q^{11}+\cdots\) |
Decomposition of \(S_{3}^{\mathrm{old}}(324, [\chi])\) into lower level spaces
\( S_{3}^{\mathrm{old}}(324, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 2}\)