Properties

Label 324.3.g
Level $324$
Weight $3$
Character orbit 324.g
Rep. character $\chi_{324}(53,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $16$
Newform subspaces $4$
Sturm bound $162$
Trace bound $7$

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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

\( 16 q + 5 q^{7} - 25 q^{13} - 46 q^{19} + 34 q^{25} - 10 q^{31} + 194 q^{37} + 140 q^{43} + 57 q^{49} - 252 q^{55} - 97 q^{61} - 157 q^{67} + 38 q^{73} - 397 q^{79} + 36 q^{85} + 466 q^{91} + 305 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{3}^{\mathrm{new}}(324, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
324.3.g.a 324.g 9.d $2$ $8.828$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) 108.3.c.a \(0\) \(0\) \(0\) \(-11\) $\mathrm{U}(1)[D_{6}]$ \(q+(-11+11\zeta_{6})q^{7}-23\zeta_{6}q^{13}-37q^{19}+\cdots\)
324.3.g.b 324.g 9.d $2$ $8.828$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) 12.3.c.a \(0\) \(0\) \(0\) \(-2\) $\mathrm{U}(1)[D_{6}]$ \(q+(-2+2\zeta_{6})q^{7}+22\zeta_{6}q^{13}+26q^{19}+\cdots\)
324.3.g.c 324.g 9.d $4$ $8.828$ \(\Q(\zeta_{12})\) None 108.3.c.b \(0\) \(0\) \(0\) \(14\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta_1 q^{5}+7\beta_{2} q^{7}+(\beta_{3}-\beta_1)q^{11}+\cdots\)
324.3.g.d 324.g 9.d $8$ $8.828$ \(\Q(\zeta_{24})\) None 324.3.c.a \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta_{5} q^{5}+(-\beta_{2}+\beta_1)q^{7}+\beta_{7} q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(324, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(324, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 2}\)