Space of modular forms of level 712, weight 1, and Character 712.l

• Label: 712.1.l
• Level: $712$
• Weight: $1$
• Character orbit: 712.l
• Rep. character: $\chi_{712}(123,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $2$
• Newform subspaces: $1$
• Sturm bound: $90$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 712 = 2^{3} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	712.l (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 712 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(90\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(712, [\chi])\).

	Total	New	Old
Modular forms	6	6	0
Cusp forms	2	2	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	2	0	0	0

Trace form

\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(712, [\chi])\) 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
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
712.1.l.a	$712$	$1$	712.l	712.l	$4$	$2$	$1$	$0.355$	\(\Q(\sqrt{-1}) \)	$D_{4}$	\(\Q(\sqrt{-2}) \)	None		$4$	$0$	\(-2\)	\(2\)	\(0\)	\(0\)	$1$		\(q-q^{2}+(1-i)q^{3}+q^{4}+(-1+i)q^{6}+\cdots\)