Space of modular forms of level 324, weight 1, and Character 324.d

• Label: 324.1.d
• Level: $324$
• Weight: $1$
• Character orbit: 324.d
• Rep. character: $\chi_{324}(163,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $54$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	324.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(54\)
Trace bound:			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(324, [\chi])\).

	Total	New	Old
Modular forms	14	6	8
Cusp forms	2	2	0
Eisenstein series	12	4	8

The following table gives the dimensions of subspaces with specified projective image type.

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	2	0	0	0

Trace form

\( 2 q + 2 q^{4} + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(324, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
324.1.d.a	$324$	$1$	324.d	4.b	$2$	$1$	$1$	$0.162$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-1}) \)	None	✓	$2$	$0$	\(-1\)	\(0\)	\(1\)	\(0\)	$1$		\(q-q^{2}+q^{4}+q^{5}-q^{8}-q^{10}-q^{13}+\cdots\)
324.1.d.b	$324$	$1$	324.d	4.b	$2$	$1$	$1$	$0.162$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-1}) \)	None	✓	$2$	$0$	\(1\)	\(0\)	\(-1\)	\(0\)	$1$		\(q+q^{2}+q^{4}-q^{5}+q^{8}-q^{10}-q^{13}+\cdots\)