Space of modular forms of level 688, weight 3, and Character 688.b

• Label: 688.3.b
• Level: $688$
• Weight: $3$
• Character orbit: 688.b
• Rep. character: $\chi_{688}(257,\cdot)$
• Character field: $\Q$
• Dimension: $43$
• Newform subspaces: $7$
• Sturm bound: $264$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	688.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 43 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(264\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(688, [\chi])\).

	Total	New	Old
Modular forms	182	45	137
Cusp forms	170	43	127
Eisenstein series	12	2	10

Trace form

\( 43 q - 133 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(688, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
688.3.b.a	$688$	$3$	688.b	43.b	$2$	$1$	$1$	$18.747$	\(\Q\)	\(\Q(\sqrt{-43}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+9q^{9}+21q^{11}-17q^{13}-9q^{17}+\cdots\)
688.3.b.b	$688$	$3$	688.b	43.b	$2$	$2$	$2$	$18.747$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+3\beta q^{5}-9\beta q^{7}+7q^{9}-11q^{11}+\cdots\)
688.3.b.c	$688$	$3$	688.b	43.b	$2$	$2$	$2$	$18.747$	\(\Q(\sqrt{129}) \)	\(\Q(\sqrt{-43}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+9q^{9}+(-11+\beta )q^{11}+(7+3\beta )q^{13}+\cdots\)
688.3.b.d	$688$	$3$	688.b	43.b	$2$	$4$	$4$	$18.747$	\(\Q(\sqrt{-2}, \sqrt{21})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots\)
688.3.b.e	$688$	$3$	688.b	43.b	$2$	$6$	$6$	$18.747$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{3}+\beta _{2}q^{5}+(-\beta _{1}+\beta _{2}+\beta _{5})q^{7}+\cdots\)
688.3.b.f	$688$	$3$	688.b	43.b	$2$	$6$	$6$	$18.747$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-\beta _{5}q^{5}+\beta _{2}q^{7}+(-6-2\beta _{3}+\cdots)q^{9}+\cdots\)
688.3.b.g	$688$	$3$	688.b	43.b	$2$	$22$	$22$	$18.747$		None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{3}^{\mathrm{old}}(688, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(688, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(43, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(86, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(172, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(344, [\chi])\)\(^{\oplus 2}\)