Space of modular forms of level 312, weight 1, and Character 312.b

• Label: 312.1.b
• Level: $312$
• Weight: $1$
• Character orbit: 312.b
• Rep. character: $\chi_{312}(77,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $1$
• Sturm bound: $56$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	312.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 312 \)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(56\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(312, [\chi])\).

	Total	New	Old
Modular forms	8	8	0
Cusp forms	4	4	0
Eisenstein series	4	4	0

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

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	4	0	0	0

Trace form

\( 4 q - 4 q^{9} + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(312, [\chi])\) 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
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
312.1.b.a	$312$	$1$	312.b	312.b	$2$	$4$	$4$	$0.156$	\(\Q(\zeta_{8})\)	$D_{4}$	\(\Q(\sqrt{-39}) \)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q-\zeta_{8}q^{2}-\zeta_{8}^{2}q^{3}+\zeta_{8}^{2}q^{4}+(-\zeta_{8}+\cdots)q^{5}+\cdots\)