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

• Label: 112.7.c
• Level: $112$
• Weight: $7$
• Character orbit: 112.c
• Rep. character: $\chi_{112}(97,\cdot)$
• Character field: $\Q$
• Dimension: $23$
• Newform subspaces: $5$
• Sturm bound: $112$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	112.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(112\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	102	25	77
Cusp forms	90	23	67
Eisenstein series	12	2	10

Trace form

23 * q + 361 * q^7 - 5105 * q^9 - 678 * q^11 - 1456 * q^15 + 8752 * q^21 + 14530 * q^23 - 83233 * q^25 - 33738 * q^29 + 82032 * q^35 + 1798 * q^37 - 249040 * q^39 - 211078 * q^43 + 9767 * q^49 - 273984 * q^51 - 232442 * q^53 + 6992 * q^57 - 828431 * q^63 - 456080 * q^65 + 410666 * q^67 - 604270 * q^71 - 380890 * q^77 + 1766978 * q^79 + 1398999 * q^81 + 356544 * q^85 + 1190928 * q^91 - 863872 * q^93 + 2829840 * q^95 + 5472170 * 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.c.a	$112$	$7$	112.c	7.b	$2$	$1$	$1$	$25.766$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓				7.7.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(343\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+7^{3}q^{7}+3^{6}q^{9}-1962q^{11}+22734q^{23}+\cdots\)
112.7.c.b	$112$	$7$	112.c	7.b	$2$	$2$	$2$	$25.766$	\(\Q(\sqrt{-510}) \)	None					7.7.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-266\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+\beta q^{5}+(-133+7\beta )q^{7}-1311q^{9}+\cdots\)
112.7.c.c	$112$	$7$	112.c	7.b	$2$	$4$	$4$	$25.766$	4.0.211968.1	None					14.7.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-308\)	$2^{7}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+5\beta _{2}q^{5}+(-77+14\beta _{1}+\cdots)q^{7}+\cdots\)
112.7.c.d	$112$	$7$	112.c	7.b	$2$	$4$	$4$	$25.766$	4.0.903168.1	None					28.7.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(28\)	$2^{7}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}-\beta _{1}q^{5}+(7-\beta _{1}+2\beta _{2}+\beta _{3})q^{7}+\cdots\)
112.7.c.e	$112$	$7$	112.c	7.b	$2$	$12$	$12$	$25.766$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None			✓		56.7.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(564\)	$2^{51}\cdot 7^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}-\beta _{8}q^{5}+(47+\beta _{4}-\beta _{5})q^{7}+\cdots\)

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

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