Space of modular forms of level 688, weight 6, and trivial character

• Label: 688.6.a
• Level: $688$
• Weight: $6$
• Character orbit: 688.a
• Rep. character: $\chi_{688}(1,\cdot)$
• Character field: $\Q$
• Dimension: $105$
• Newform subspaces: $12$
• Sturm bound: $528$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	688.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(528\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(688))\).

	Total	New	Old
Modular forms	446	105	341
Cusp forms	434	105	329
Eisenstein series	12	0	12

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
\(+\)	\(+\)	$+$	\(25\)
\(+\)	\(-\)	$-$	\(28\)
\(-\)	\(+\)	$-$	\(26\)
\(-\)	\(-\)	$+$	\(26\)
Plus space		\(+\)	\(51\)
Minus space		\(-\)	\(54\)

Trace form

\( 105 q + 38 q^{5} + 8505 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(688))\) 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
688.6.a.a	$688$	$6$	688.a	1.a	$1$	$3$	$3$	$110.344$	3.3.159992.1	None	✓	$1$	$1$	\(0\)	\(-8\)	\(-14\)	\(74\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{2})q^{3}+(-5+2\beta _{1}-\beta _{2})q^{5}+\cdots\)
688.6.a.b	$688$	$6$	688.a	1.a	$1$	$3$	$3$	$110.344$	3.3.146508.1	None	✓	$1$	$0$	\(0\)	\(28\)	\(-14\)	\(182\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(9-\beta _{2})q^{3}+(-5+2\beta _{1}-3\beta _{2})q^{5}+\cdots\)
688.6.a.c	$688$	$6$	688.a	1.a	$1$	$5$	$5$	$110.344$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(61\)	\(-122\)		$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(12-2\beta _{1}-\beta _{2}-\beta _{3})q^{5}+\cdots\)
688.6.a.d	$688$	$6$	688.a	1.a	$1$	$6$	$6$	$110.344$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-17\)	\(61\)	\(-210\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{3}+(10-\beta _{1}+\beta _{4}-\beta _{5})q^{5}+\cdots\)
688.6.a.e	$688$	$6$	688.a	1.a	$1$	$8$	$8$	$110.344$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(26\)	\(-212\)	\(136\)		$-$	$-$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{4})q^{3}+(-3^{3}+\beta _{4}-\beta _{6}+\beta _{7})q^{5}+\cdots\)
688.6.a.f	$688$	$6$	688.a	1.a	$1$	$8$	$8$	$110.344$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(26\)	\(-40\)	\(60\)		$-$	$+$	$-$	$2^{5}\cdot 3$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{3}+(-5+\beta _{7})q^{5}+(8-\beta _{1}+\cdots)q^{7}+\cdots\)
688.6.a.g	$688$	$6$	688.a	1.a	$1$	$9$	$9$	$110.344$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-28\)	\(-40\)	\(-136\)		$-$	$-$	$+$	$2^{8}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{3}+(-4+\beta _{4})q^{5}+(-15+\cdots)q^{7}+\cdots\)
688.6.a.h	$688$	$6$	688.a	1.a	$1$	$10$	$10$	$110.344$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-28\)	\(138\)	\(-60\)		$-$	$+$	$-$	$2^{10}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{4})q^{3}+(14+\beta _{7})q^{5}+(-6+\cdots)q^{7}+\cdots\)
688.6.a.i	$688$	$6$	688.a	1.a	$1$	$11$	$11$	$110.344$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(24\)	\(100\)		$+$	$+$	$+$	$2^{13}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(2+\beta _{2})q^{5}+(9+\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
688.6.a.j	$688$	$6$	688.a	1.a	$1$	$13$	$13$	$110.344$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(18\)	\(-50\)	\(232\)		$+$	$-$	$-$	$2^{17}\cdot 3$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(-4+\beta _{3})q^{5}+(18-\beta _{1}+\cdots)q^{7}+\cdots\)
688.6.a.k	$688$	$6$	688.a	1.a	$1$	$14$	$14$	$110.344$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-27\)	\(25\)	\(-160\)		$+$	$+$	$+$	$2^{19}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}+(2+\beta _{4})q^{5}+(-11+\cdots)q^{7}+\cdots\)
688.6.a.l	$688$	$6$	688.a	1.a	$1$	$15$	$15$	$110.344$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(9\)	\(99\)	\(-96\)		$+$	$-$	$-$	$2^{23}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(7-\beta _{4})q^{5}+(-6-\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(688))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(688)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(43))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(86))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(172))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(344))\)\(^{\oplus 2}\)