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

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

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

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