Space of modular forms of level 324, weight 2, and Character 324.e

• 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: $7$

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

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

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

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
324.2.e.a	$324$	$2$	324.e	9.c	$3$	$2$	$1$	$2.587$	\(\Q(\sqrt{-3}) \)	None		✓			324.2.a.b	$2$	$0$	\(0\)	\(0\)	\(-3\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-3\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(6-6\zeta_{6})q^{11}+\cdots\)
324.2.e.b	$324$	$2$	324.e	9.c	$3$	$2$	$1$	$2.587$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					108.2.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-5\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+(-5+5\zeta_{6})q^{7}+7\zeta_{6}q^{13}-q^{19}+\cdots\)
324.2.e.c	$324$	$2$	324.e	9.c	$3$	$2$	$1$	$2.587$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					36.2.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+(4-4\zeta_{6})q^{7}-2\zeta_{6}q^{13}+8q^{19}+\cdots\)
324.2.e.d	$324$	$2$	324.e	9.c	$3$	$2$	$1$	$2.587$	\(\Q(\sqrt{-3}) \)	None		✓			324.2.a.b	$2$	$0$	\(0\)	\(0\)	\(3\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+3\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(-6+6\zeta_{6})q^{11}+\cdots\)

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

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