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

• Label: 712.1.s
• Level: $712$
• Weight: $1$
• Character orbit: 712.s
• Rep. character: $\chi_{712}(67,\cdot)$
• Character field: $\Q(\zeta_{22})$
• Dimension: $10$
• 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.s (of order \(22\) and degree \(10\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 712 \)
Character field:			\(\Q(\zeta_{22})\)
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	30	30	0
Cusp forms	10	10	0
Eisenstein series	20	20	0

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

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	10	0	0	0

Trace form

\( 10 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{6} - q^{8} - 3 q^{9} + O(q^{10}) \)

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.s.a	$712$	$1$	712.s	712.s	$22$	$10$	$1$	$0.355$	\(\Q(\zeta_{22})\)	$D_{11}$	\(\Q(\sqrt{-2}) \)	None		$4$	$0$	\(-1\)	\(-2\)	\(0\)	\(0\)	$1$		\(q-\zeta_{22}^{5}q^{2}+(\zeta_{22}^{4}-\zeta_{22}^{9})q^{3}+\zeta_{22}^{10}q^{4}+\cdots\)