Space of modular forms of level 688, weight 5, and Character 688.b

• Label: 688.5.b
• Level: $688$
• Weight: $5$
• Character orbit: 688.b
• Rep. character: $\chi_{688}(257,\cdot)$
• Character field: $\Q$
• Dimension: $87$
• Newform subspaces: $6$
• Sturm bound: $440$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	688.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 43 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(440\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(688, [\chi])\).

	Total	New	Old
Modular forms	358	89	269
Cusp forms	346	87	259
Eisenstein series	12	2	10

Trace form

\( 87 q - 2185 q^{9} + O(q^{10}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(688, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
688.5.b.a	$688$	$5$	688.b	43.b	$2$	$1$	$1$	$71.119$	\(\Q\)	\(\Q(\sqrt{-43}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3^{4}q^{9}-199q^{11}-7^{2}q^{13}-497q^{17}+\cdots\)
688.5.b.b	$688$	$5$	688.b	43.b	$2$	$2$	$2$	$71.119$	\(\Q(\sqrt{129}) \)	\(\Q(\sqrt{-43}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3$	$\mathrm{U}(1)[D_{2}]$	\(q+3^{4}q^{9}+(103+7\beta )q^{11}+(33+17\beta )q^{13}+\cdots\)
688.5.b.c	$688$	$5$	688.b	43.b	$2$	$12$	$12$	$71.119$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{4}q^{5}-\beta _{8}q^{7}+(-38+\beta _{2}+\cdots)q^{9}+\cdots\)
688.5.b.d	$688$	$5$	688.b	43.b	$2$	$12$	$12$	$71.119$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{3}+\beta _{7}q^{5}+(\beta _{1}-2\beta _{3}+\beta _{10}+\cdots)q^{7}+\cdots\)
688.5.b.e	$688$	$5$	688.b	43.b	$2$	$16$	$16$	$71.119$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{18}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{8}q^{3}-\beta _{10}q^{5}+(\beta _{8}+\beta _{12})q^{7}+\cdots\)
688.5.b.f	$688$	$5$	688.b	43.b	$2$	$44$	$44$	$71.119$		None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{5}^{\mathrm{old}}(688, [\chi])\) into lower level spaces

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