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Properties

Label 324.3.c
Level $324$
Weight $3$
Character orbit 324.c
Rep. character $\chi_{324}(161,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $2$
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.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(162\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(324, [\chi])\).

Total New Old
Modular forms 126 8 118
Cusp forms 90 8 82
Eisenstein series 36 0 36

Trace form

\( 8 q - 2 q^{7} + O(q^{10}) \) \( 8 q - 2 q^{7} - 14 q^{13} - 26 q^{19} + 2 q^{25} + 46 q^{31} - 8 q^{37} + 40 q^{43} + 18 q^{49} + 126 q^{55} - 146 q^{61} - 212 q^{67} - 170 q^{73} + 142 q^{79} + 252 q^{85} - 82 q^{91} + 232 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
324.3.c.a 324.c 3.b $4$ $8.828$ \(\Q(\sqrt{-2}, \sqrt{3})\) None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{5}+(-1+\beta _{2})q^{7}-\beta _{3}q^{11}+\cdots\)
324.3.c.b 324.c 3.b $4$ $8.828$ \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(0\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{5}+(1-\beta _{2})q^{7}+(-\beta _{1}+\beta _{3})q^{11}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(324, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 2}\)