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

• Label: 112.5.s
• Level: $112$
• Weight: $5$
• Character orbit: 112.s
• Rep. character: $\chi_{112}(17,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $30$
• Newform subspaces: $4$
• Sturm bound: $80$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	112.s (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(80\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	140	34	106
Cusp forms	116	30	86
Eisenstein series	24	4	20

Trace form

30 * q + 3 * q^3 - 3 * q^5 + 18 * q^7 + 350 * q^9 + 49 * q^11 + 166 * q^15 - 3 * q^17 + 1251 * q^19 + 221 * q^21 - 1007 * q^23 + 1742 * q^25 + 1916 * q^29 + 4419 * q^31 - 2163 * q^33 + 2499 * q^35 + 911 * q^37 - 4640 * q^39 + 36 * q^43 - 1446 * q^45 + 4323 * q^47 + 2302 * q^49 - 45 * q^51 + 1055 * q^53 - 7430 * q^57 - 9789 * q^59 - 6195 * q^61 - 21886 * q^63 - 5088 * q^65 + 9345 * q^67 - 2108 * q^71 - 5523 * q^73 - 35274 * q^75 + 11567 * q^77 - 431 * q^79 - 11247 * q^81 - 6566 * q^85 + 34374 * q^87 + 10221 * q^89 + 35424 * q^91 - 14243 * q^93 - 3501 * q^95 + 16324 * 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.s.a	$112$	$5$	112.s	7.d	$6$	$4$	$2$	$11.577$	\(\Q(\sqrt{-3}, \sqrt{22})\)	None					7.5.d.a	$2$	$0$	\(0\)	\(-6\)	\(-30\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+\beta _{1}-\beta _{2}-\beta _{3})q^{3}+(-5+2\beta _{1}+\cdots)q^{5}+\cdots\)
112.5.s.b	$112$	$5$	112.s	7.d	$6$	$4$	$2$	$11.577$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					14.5.d.a	$2$	$0$	\(0\)	\(18\)	\(54\)	\(28\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{6}]$	\(q+(6+3\beta _{1}+\beta _{3})q^{3}+(9-9\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
112.5.s.c	$112$	$5$	112.s	7.d	$6$	$6$	$3$	$11.577$	6.0.11337408.1	None					28.5.h.a	$2$	$0$	\(0\)	\(-9\)	\(-27\)	\(-66\)	$2^{2}\cdot 3^{3}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+\beta _{1}-\beta _{3})q^{3}+(-3-3\beta _{1}+\cdots)q^{5}+\cdots\)
112.5.s.d	$112$	$5$	112.s	7.d	$6$	$16$	$8$	$11.577$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None			✓		56.5.o.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(56\)	$2^{38}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{2}+\beta _{6})q^{3}+\beta _{4}q^{5}+(5+3\beta _{1}+\cdots)q^{7}+\cdots\)

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

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