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

• Label: 688.2.i
• Level: $688$
• Weight: $2$
• Character orbit: 688.i
• Rep. character: $\chi_{688}(49,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $42$
• Newform subspaces: $9$
• Sturm bound: $176$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	688.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 43 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(176\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

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

	Total	New	Old
Modular forms	188	46	142
Cusp forms	164	42	122
Eisenstein series	24	4	20

Trace form

\( 42 q + 3 q^{3} - q^{5} - 3 q^{7} - 24 q^{9} + O(q^{10}) \)

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.i.a	$688$	$2$	688.i	43.c	$3$	$2$	$1$	$5.494$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(0\)	\(-3\)	\(-1\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-3+3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{5}+\zeta_{6}q^{7}+\cdots\)
688.2.i.b	$688$	$2$	688.i	43.c	$3$	$2$	$1$	$5.494$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-1\)	\(1\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{5}+5\zeta_{6}q^{7}+\cdots\)
688.2.i.c	$688$	$2$	688.i	43.c	$3$	$2$	$1$	$5.494$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(1\)	\(-3\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+(-3+3\zeta_{6})q^{5}-\zeta_{6}q^{7}+\cdots\)
688.2.i.d	$688$	$2$	688.i	43.c	$3$	$2$	$1$	$5.494$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(1\)	\(1\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+(1-\zeta_{6})q^{5}+3\zeta_{6}q^{7}+\cdots\)
688.2.i.e	$688$	$2$	688.i	43.c	$3$	$4$	$2$	$5.494$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(-3\)	\(2\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{1}-\beta _{3})q^{3}+(2-2\beta _{1}+2\beta _{3})q^{5}+\cdots\)
688.2.i.f	$688$	$2$	688.i	43.c	$3$	$4$	$2$	$5.494$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None		$2$	$0$	\(0\)	\(1\)	\(-3\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{3}+(\beta _{1}-2\beta _{2})q^{5}+(2+\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
688.2.i.g	$688$	$2$	688.i	43.c	$3$	$8$	$4$	$5.494$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(-1\)	\(0\)	\(-4\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{4}-\beta _{6})q^{3}+\beta _{5}q^{5}+(-1+\beta _{2}+\cdots)q^{7}+\cdots\)
688.2.i.h	$688$	$2$	688.i	43.c	$3$	$8$	$4$	$5.494$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(3\)	\(0\)	\(-2\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{1}+\beta _{2})q^{3}-\beta _{5}q^{5}+(\beta _{4}+\beta _{7})q^{7}+\cdots\)
688.2.i.i	$688$	$2$	688.i	43.c	$3$	$10$	$5$	$5.494$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		$2$	$0$	\(0\)	\(5\)	\(2\)	\(-4\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{1}+\beta _{7})q^{3}+(\beta _{1}-\beta _{6})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(688, [\chi])\) into lower level spaces

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