Space of modular forms of level 126, weight 12, and Character 126.g

• Label: 126.12.g
• Level: $126$
• Weight: $12$
• Character orbit: 126.g
• Rep. character: $\chi_{126}(37,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $72$
• Newform subspaces: $8$
• Sturm bound: $288$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	126.g (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(288\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(126, [\chi])\).

	Total	New	Old
Modular forms	544	72	472
Cusp forms	512	72	440
Eisenstein series	32	0	32

Trace form

72 * q - 36864 * q^4 - 1188 * q^5 + 76732 * q^7 + 281728 * q^10 + 504864 * q^11 - 3595760 * q^13 - 2318976 * q^14 - 37748736 * q^16 - 11747772 * q^17 + 8284944 * q^19 + 2433024 * q^20 - 33541120 * q^22 - 41214276 * q^23 - 400076888 * q^25 + 9705600 * q^26 - 126734336 * q^28 - 268502184 * q^29 + 19864708 * q^31 - 245653504 * q^34 + 1587212784 * q^35 - 486903912 * q^37 - 29716992 * q^38 + 288489472 * q^40 - 521358768 * q^41 + 2222261472 * q^43 + 516980736 * q^44 + 1368609920 * q^46 + 2145466044 * q^47 + 3886445100 * q^49 + 2930351616 * q^50 + 1841029120 * q^52 - 809975196 * q^53 + 13537149464 * q^55 - 2452488192 * q^56 - 1912848512 * q^58 + 4679224632 * q^59 - 13851344096 * q^61 + 19522377216 * q^62 + 77309411328 * q^64 + 4417190964 * q^65 - 1643771080 * q^67 - 12029718528 * q^68 + 11581187840 * q^70 + 18019769256 * q^71 - 9695233384 * q^73 - 12761993088 * q^74 - 16967565312 * q^76 - 8375598540 * q^77 + 52736876932 * q^79 - 1245708288 * q^80 + 83185552128 * q^82 + 62061234624 * q^83 - 79019697712 * q^85 + 2408846976 * q^86 + 17173053440 * q^88 - 243056384424 * q^89 + 288297981592 * q^91 + 84406837248 * q^92 - 9132695424 * q^94 + 248107817196 * q^95 - 71019081544 * q^97 - 383791094784 * q^98

Decomposition of \(S_{12}^{\mathrm{new}}(126, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
126.12.g.a	$126$	$12$	126.g	7.c	$3$	$6$	$3$	$96.811$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None					42.12.e.b	$2$	$0$	\(-96\)	\(0\)	\(-1331\)	\(-83545\)	$2^{4}\cdot 3\cdot 7^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2^{5}-2^{5}\beta _{1})q^{2}+2^{10}\beta _{1}q^{4}+(-443+\cdots)q^{5}+\cdots\)
126.12.g.b	$126$	$12$	126.g	7.c	$3$	$6$	$3$	$96.811$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None					42.12.e.a	$2$	$0$	\(96\)	\(0\)	\(-1045\)	\(45731\)	$2^{8}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2^{5}+2^{5}\beta _{2})q^{2}+2^{10}\beta _{2}q^{4}+(-350+\cdots)q^{5}+\cdots\)
126.12.g.c	$126$	$12$	126.g	7.c	$3$	$8$	$4$	$96.811$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					14.12.c.b	$2$	$0$	\(-128\)	\(0\)	\(-3808\)	\(110328\)	$2^{12}\cdot 3^{3}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2^{5}-2^{5}\beta _{2})q^{2}+2^{10}\beta _{2}q^{4}+(-952+\cdots)q^{5}+\cdots\)
126.12.g.d	$126$	$12$	126.g	7.c	$3$	$8$	$4$	$96.811$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					42.12.e.d	$2$	$0$	\(-128\)	\(0\)	\(1420\)	\(-66362\)	$2^{8}\cdot 3^{3}\cdot 7^{6}$	$\mathrm{SU}(2)[C_{3}]$	\(q-2^{5}\beta _{1}q^{2}+(-2^{10}+2^{10}\beta _{1})q^{4}+\cdots\)
126.12.g.e	$126$	$12$	126.g	7.c	$3$	$8$	$4$	$96.811$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					14.12.c.a	$2$	$0$	\(128\)	\(0\)	\(-7504\)	\(-42224\)	$2^{12}\cdot 3^{3}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+2^{5}\beta _{2}q^{2}+(-2^{10}+2^{10}\beta _{2})q^{4}+\cdots\)
126.12.g.f	$126$	$12$	126.g	7.c	$3$	$8$	$4$	$96.811$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					42.12.e.c	$2$	$0$	\(128\)	\(0\)	\(11080\)	\(31758\)	$2^{6}\cdot 3^{3}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+2^{5}\beta _{1}q^{2}+(-2^{10}+2^{10}\beta _{1})q^{4}+\cdots\)
126.12.g.g	$126$	$12$	126.g	7.c	$3$	$14$	$7$	$96.811$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None		✓			126.12.g.g	$2$	$0$	\(-224\)	\(0\)	\(7527\)	\(40523\)	$2^{18}\cdot 3^{17}\cdot 7^{11}$	$\mathrm{SU}(2)[C_{3}]$	\(q+2^{5}\beta _{2}q^{2}+(-2^{10}-2^{10}\beta _{2})q^{4}+\cdots\)
126.12.g.h	$126$	$12$	126.g	7.c	$3$	$14$	$7$	$96.811$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None		✓			126.12.g.g	$2$	$0$	\(224\)	\(0\)	\(-7527\)	\(40523\)	$2^{18}\cdot 3^{17}\cdot 7^{11}$	$\mathrm{SU}(2)[C_{3}]$	\(q-2^{5}\beta _{2}q^{2}+(-2^{10}-2^{10}\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(126, [\chi])\) into lower level spaces

\( S_{12}^{\mathrm{old}}(126, [\chi]) \simeq \) \(S_{12}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)