Space of modular forms of level 688, weight 2, and Character 688.x

• Label: 688.2.x
• Level: $688$
• Weight: $2$
• Character orbit: 688.x
• Rep. character: $\chi_{688}(165,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $344$
• Newform subspaces: $1$
• Sturm bound: $176$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	688.x (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 688 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(176\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(688, [\chi])\).

	Total	New	Old
Modular forms	360	360	0
Cusp forms	344	344	0
Eisenstein series	16	16	0

Trace form

\( 344 q - 8 q^{2} - 2 q^{3} - 12 q^{4} - 2 q^{5} - 2 q^{6} - 8 q^{8} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(688, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
688.2.x.a	$688$	$2$	688.x	688.x	$12$	$344$	$86$	$5.494$		None		$4$	$0$	\(-8\)	\(-2\)	\(-2\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$