Space of modular forms of level 688, weight 2, and trivial character

• Label: 688.2.a
• Level: $688$
• Weight: $2$
• Character orbit: 688.a
• Rep. character: $\chi_{688}(1,\cdot)$
• Character field: $\Q$
• Dimension: $21$
• Newform subspaces: $10$
• Sturm bound: $176$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	688.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(176\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(688))\).

	Total	New	Old
Modular forms	94	21	73
Cusp forms	83	21	62
Eisenstein series	11	0	11

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(43\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(4\)
\(+\)	\(-\)	$-$	\(7\)
\(-\)	\(+\)	$-$	\(5\)
\(-\)	\(-\)	$+$	\(5\)
Plus space		\(+\)	\(9\)
Minus space		\(-\)	\(12\)

Trace form

\( 21 q - 2 q^{5} + 21 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(688))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	43
688.2.a.a	$688$	$2$	688.a	1.a	$1$	$1$	$1$	$5.494$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+2q^{7}-3q^{9}-q^{11}-q^{13}+\cdots\)
688.2.a.b	$688$	$2$	688.a	1.a	$1$	$1$	$1$	$5.494$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(-4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-4q^{5}+q^{9}-3q^{11}-5q^{13}+\cdots\)
688.2.a.c	$688$	$2$	688.a	1.a	$1$	$1$	$1$	$5.494$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+4q^{7}+q^{9}+3q^{11}-q^{13}+\cdots\)
688.2.a.d	$688$	$2$	688.a	1.a	$1$	$2$	$2$	$5.494$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{3}+\beta q^{5}-\beta q^{7}+(3-4\beta )q^{9}+\cdots\)
688.2.a.e	$688$	$2$	688.a	1.a	$1$	$2$	$2$	$5.494$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-1-\beta )q^{5}+(-2+4\beta )q^{7}+\cdots\)
688.2.a.f	$688$	$2$	688.a	1.a	$1$	$2$	$2$	$5.494$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(4\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(2-\beta )q^{5}+(2-\beta )q^{7}-q^{9}+\cdots\)
688.2.a.g	$688$	$2$	688.a	1.a	$1$	$2$	$2$	$5.494$	\(\Q(\sqrt{21}) \)	None	✓	$1$	$0$	\(0\)	\(1\)	\(3\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1+\beta )q^{5}-2q^{7}+(2+\beta )q^{9}+\cdots\)
688.2.a.h	$688$	$2$	688.a	1.a	$1$	$2$	$2$	$5.494$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-1+\beta )q^{5}+(1-\beta )q^{7}+\cdots\)
688.2.a.i	$688$	$2$	688.a	1.a	$1$	$3$	$3$	$5.494$	3.3.229.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(1\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}-\beta _{2}q^{5}-2q^{7}+(1+\cdots)q^{9}+\cdots\)
688.2.a.j	$688$	$2$	688.a	1.a	$1$	$5$	$5$	$5.494$	5.5.7998268.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{4}q^{5}+(-1+\beta _{2}-\beta _{4})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(688))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(688)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(43))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(86))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(172))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(344))\)\(^{\oplus 2}\)