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 + 681 * q^11 + 1462 * q^15 - 3 * q^17 + 15123 * q^19 - 8755 * q^21 + 12185 * q^23 + 47446 * q^25 - 16068 * q^29 - 110877 * q^31 - 13611 * q^33 + 89187 * q^35 - 1801 * q^37 + 42736 * q^39 + 102004 * q^43 + 228810 * q^45 + 188499 * q^47 - 113666 * q^49 - 301413 * q^51 + 50159 * q^53 + 417850 * q^57 + 664611 * q^59 + 264597 * q^61 + 798434 * q^63 + 103952 * q^65 - 360767 * q^67 - 808028 * q^71 - 385563 * q^73 + 811950 * q^75 - 770393 * q^77 + 572089 * q^79 - 955143 * q^81 + 619338 * q^85 - 3755130 * q^87 - 881499 * q^89 - 3439920 * q^91 + 563149 * q^93 + 2991867 * q^95 + 714148 * q^99

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	Twist minimal	Largest	Maximal	Minimal twist	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					7.7.d.a	$2$	$0$	\(0\)	\(21\)	\(315\)	\(686\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(7+7\zeta_{6})q^{3}+(210-105\zeta_{6})q^{5}+\cdots\)
112.7.s.b	$112$	$7$	112.s	7.d	$6$	$4$	$2$	$25.766$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					7.7.d.b	$2$	$0$	\(0\)	\(-18\)	\(-150\)	\(-280\)	$3$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-6-3\beta _{1}-13\beta _{3})q^{3}+(-5^{2}+5^{2}\beta _{1}+\cdots)q^{5}+\cdots\)
112.7.s.c	$112$	$7$	112.s	7.d	$6$	$8$	$4$	$25.766$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					14.7.d.a	$2$	$0$	\(0\)	\(0\)	\(-336\)	\(-652\)	$2^{8}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{3}+\beta _{7})q^{3}+(-56+28\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
112.7.s.d	$112$	$7$	112.s	7.d	$6$	$8$	$4$	$25.766$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					28.7.h.a	$2$	$0$	\(0\)	\(0\)	\(168\)	\(452\)	$2^{8}\cdot 3^{4}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{3}q^{3}+(28-14\beta _{1}-\beta _{5})q^{5}+(11^{2}+\cdots)q^{7}+\cdots\)
112.7.s.e	$112$	$7$	112.s	7.d	$6$	$24$	$12$	$25.766$		None			✓		56.7.o.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-564\)		$\mathrm{SU}(2)[C_{6}]$

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

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