Space of modular forms of level 112, weight 7, and Character 112.s

• Label: 112.7.s
• Level: $112$
• Weight: $7$
• Character orbit: 112.s
• Rep. character: $\chi_{112}(17,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $46$
• Newform subspaces: $5$
• Sturm bound: $112$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	112.s (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(112\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(112, [\chi])\).

	Total	New	Old
Modular forms	204	50	154
Cusp forms	180	46	134
Eisenstein series	24	4	20

Trace form

\( 46 q + 3 q^{3} - 3 q^{5} - 358 q^{7} + 5102 q^{9} + O(q^{10}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(112, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
112.7.s.a	$112$	$7$	112.s	7.d	$6$	$2$	$1$	$25.766$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(21\)	\(315\)	\(686\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(7+7\zeta_{6})q^{3}+(210-105\zeta_{6})q^{5}+\cdots\)
112.7.s.b	$112$	$7$	112.s	7.d	$6$	$4$	$2$	$25.766$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$2$	$0$	\(0\)	\(-18\)	\(-150\)	\(-280\)	$3$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-6-3\beta _{1}-13\beta _{3})q^{3}+(-5^{2}+5^{2}\beta _{1}+\cdots)q^{5}+\cdots\)
112.7.s.c	$112$	$7$	112.s	7.d	$6$	$8$	$4$	$25.766$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-336\)	\(-652\)	$2^{8}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{3}+\beta _{7})q^{3}+(-56+28\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
112.7.s.d	$112$	$7$	112.s	7.d	$6$	$8$	$4$	$25.766$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(168\)	\(452\)	$2^{8}\cdot 3^{4}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{3}q^{3}+(28-14\beta _{1}-\beta _{5})q^{5}+(11^{2}+\cdots)q^{7}+\cdots\)
112.7.s.e	$112$	$7$	112.s	7.d	$6$	$24$	$12$	$25.766$		None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-564\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{7}^{\mathrm{old}}(112, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(112, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 2}\)