Space of modular forms of level 112, weight 5, and Character 112.c

• Label: 112.5.c
• Level: $112$
• Weight: $5$
• Character orbit: 112.c
• Rep. character: $\chi_{112}(97,\cdot)$
• Character field: $\Q$
• Dimension: $15$
• Newform subspaces: $4$
• Sturm bound: $80$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	112.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(80\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(112, [\chi])\).

	Total	New	Old
Modular forms	70	17	53
Cusp forms	58	15	43
Eisenstein series	12	2	10

Trace form

15 * q - 15 * q^7 - 353 * q^9 - 46 * q^11 - 160 * q^15 - 224 * q^21 - 1150 * q^23 - 641 * q^25 - 626 * q^29 - 1632 * q^35 - 914 * q^37 - 352 * q^39 - 4206 * q^43 + 2159 * q^49 + 10368 * q^51 + 1390 * q^53 + 2144 * q^57 + 7009 * q^63 + 3360 * q^65 + 4530 * q^67 + 13922 * q^71 + 5422 * q^77 - 23422 * q^79 + 7695 * q^81 - 1408 * q^85 - 2976 * q^91 - 6400 * q^93 + 14304 * q^95 - 30286 * q^99

Decomposition of \(S_{5}^{\mathrm{new}}(112, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
112.5.c.a	$112$	$5$	112.c	7.b	$2$	$1$	$1$	$11.577$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓				7.5.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-49\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-7^{2}q^{7}+3^{4}q^{9}+206q^{11}+734q^{23}+\cdots\)
112.5.c.b	$112$	$5$	112.c	7.b	$2$	$2$	$2$	$11.577$	\(\Q(\sqrt{-3}) \)	None					28.5.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(14\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{3}+3\beta q^{5}+(-7\beta+7)q^{7}+\cdots\)
112.5.c.c	$112$	$5$	112.c	7.b	$2$	$4$	$4$	$11.577$	\(\Q(\sqrt{-36 +3 \sqrt{2}})\)	None					14.5.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(76\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{2})q^{5}+(19-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
112.5.c.d	$112$	$5$	112.c	7.b	$2$	$8$	$8$	$11.577$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None			✓		56.5.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-56\)	$2^{20}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(-7-\beta _{4})q^{7}+(-31+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(112, [\chi])\) into lower level spaces

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