Space of modular forms of level 648, weight 1, and Character 648.b

• Label: 648.1.b
• Level: $648$
• Weight: $1$
• Character orbit: 648.b
• Rep. character: $\chi_{648}(163,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $108$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	648.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(108\)
Trace bound:			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(648, [\chi])\).

	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 \(S_{1}^{\mathrm{new}}(648, [\chi])\) 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
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
648.1.b.a	$648$	$1$	648.b	8.d	$2$	$1$	$1$	$0.323$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-2}) \)	None	✓	$2$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)	$1$		\(q-q^{2}+q^{4}-q^{8}+q^{11}+q^{16}+q^{17}+\cdots\)
648.1.b.b	$648$	$1$	648.b	8.d	$2$	$1$	$1$	$0.323$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-2}) \)	None	✓	$2$	$0$	\(1\)	\(0\)	\(0\)	\(0\)	$1$		\(q+q^{2}+q^{4}+q^{8}-q^{11}+q^{16}-q^{17}+\cdots\)