Properties

Label 324.2.e
Level $324$
Weight $2$
Character orbit 324.e
Rep. character $\chi_{324}(109,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $8$
Newform subspaces $4$
Sturm bound $108$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 324.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(108\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(324, [\chi])\).

Total New Old
Modular forms 144 8 136
Cusp forms 72 8 64
Eisenstein series 72 0 72

Trace form

\( 8q - 5q^{7} + O(q^{10}) \) \( 8q - 5q^{7} - 5q^{13} + 22q^{19} + 2q^{25} + 16q^{31} - 2q^{37} + 4q^{43} - 21q^{49} - 72q^{55} - 11q^{61} + q^{67} + 10q^{73} + 37q^{79} + 18q^{85} - 46q^{91} + q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(324, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
324.2.e.a \(2\) \(2.587\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-3\) \(-2\) \(q-3\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(6-6\zeta_{6})q^{11}+\cdots\)
324.2.e.b \(2\) \(2.587\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-5\) \(q+(-5+5\zeta_{6})q^{7}+7\zeta_{6}q^{13}-q^{19}+\cdots\)
324.2.e.c \(2\) \(2.587\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(4\) \(q+(4-4\zeta_{6})q^{7}-2\zeta_{6}q^{13}+8q^{19}+\cdots\)
324.2.e.d \(2\) \(2.587\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(3\) \(-2\) \(q+3\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(-6+6\zeta_{6})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(324, [\chi])\) into lower level spaces

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