Space of modular forms of level 126, weight 8, and Character 126.g

• Label: 126.8.g
• Level: $126$
• Weight: $8$
• Character orbit: 126.g
• Rep. character: $\chi_{126}(37,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $48$
• Newform subspaces: $11$
• Sturm bound: $192$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	126.g (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(192\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(126, [\chi])\).

	Total	New	Old
Modular forms	352	48	304
Cusp forms	320	48	272
Eisenstein series	32	0	32

Trace form

48 * q - 1536 * q^4 - 248 * q^5 + 228 * q^7 + 2208 * q^10 + 3560 * q^11 - 26880 * q^13 + 11584 * q^14 - 98304 * q^16 - 11008 * q^17 + 51960 * q^19 + 31744 * q^20 - 38976 * q^22 + 14948 * q^23 - 402528 * q^25 - 188576 * q^26 + 175872 * q^28 + 851416 * q^29 - 46524 * q^31 - 428160 * q^34 - 1442776 * q^35 - 514908 * q^37 - 349344 * q^38 + 141312 * q^40 + 880992 * q^41 - 1467456 * q^43 + 227840 * q^44 - 712704 * q^46 - 1742004 * q^47 + 754572 * q^49 + 1818496 * q^50 + 860160 * q^52 + 1423344 * q^53 + 2181864 * q^55 - 112640 * q^56 - 397152 * q^58 - 4155992 * q^59 + 8660676 * q^61 + 535744 * q^62 + 12582912 * q^64 + 4247964 * q^65 + 2027016 * q^67 - 704512 * q^68 + 8114400 * q^70 + 11325224 * q^71 - 10053108 * q^73 + 550368 * q^74 - 6650880 * q^76 - 8833664 * q^77 + 2624364 * q^79 - 1015808 * q^80 - 7956480 * q^82 + 25231616 * q^83 + 26363928 * q^85 - 1560096 * q^86 + 1247232 * q^88 - 6556788 * q^89 + 6780456 * q^91 - 1913344 * q^92 + 13904832 * q^94 - 18457324 * q^95 - 77037000 * q^97 + 4018368 * q^98

Decomposition of \(S_{8}^{\mathrm{new}}(126, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
126.8.g.a	$126$	$8$	126.g	7.c	$3$	$2$	$1$	$39.361$	\(\Q(\sqrt{-3}) \)	None					42.8.e.b	$2$	$0$	\(-8\)	\(0\)	\(-290\)	\(1477\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-8+8\zeta_{6})q^{2}-2^{6}\zeta_{6}q^{4}+(-290+\cdots)q^{5}+\cdots\)
126.8.g.b	$126$	$8$	126.g	7.c	$3$	$2$	$1$	$39.361$	\(\Q(\sqrt{-3}) \)	None					42.8.e.a	$2$	$0$	\(-8\)	\(0\)	\(165\)	\(-343\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-8+8\zeta_{6})q^{2}-2^{6}\zeta_{6}q^{4}+(165+\cdots)q^{5}+\cdots\)
126.8.g.c	$126$	$8$	126.g	7.c	$3$	$4$	$2$	$39.361$	\(\Q(\sqrt{-3}, \sqrt{5497})\)	None		✓			126.8.g.c	$2$	$0$	\(-16\)	\(0\)	\(-455\)	\(1554\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-8+8\beta _{2})q^{2}-2^{6}\beta _{2}q^{4}+(-230+\cdots)q^{5}+\cdots\)
126.8.g.d	$126$	$8$	126.g	7.c	$3$	$4$	$2$	$39.361$	\(\Q(\sqrt{-3}, \sqrt{949})\)	None					14.8.c.b	$2$	$0$	\(-16\)	\(0\)	\(-14\)	\(-1848\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-8\beta _{1}q^{2}+(-2^{6}+2^{6}\beta _{1})q^{4}+(-7\beta _{1}+\cdots)q^{5}+\cdots\)
126.8.g.e	$126$	$8$	126.g	7.c	$3$	$4$	$2$	$39.361$	\(\Q(\sqrt{-3}, \sqrt{2389})\)	None					14.8.c.a	$2$	$0$	\(16\)	\(0\)	\(-238\)	\(168\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(8-8\beta _{1})q^{2}-2^{6}\beta _{1}q^{4}+(-119+\cdots)q^{5}+\cdots\)
126.8.g.f	$126$	$8$	126.g	7.c	$3$	$4$	$2$	$39.361$	\(\Q(\sqrt{-3}, \sqrt{2881})\)	None					42.8.e.c	$2$	$0$	\(16\)	\(0\)	\(309\)	\(868\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+8\beta _{1}q^{2}+(-2^{6}+2^{6}\beta _{1})q^{4}+(156\beta _{1}+\cdots)q^{5}+\cdots\)
126.8.g.g	$126$	$8$	126.g	7.c	$3$	$4$	$2$	$39.361$	\(\Q(\sqrt{-3}, \sqrt{5497})\)	None		✓			126.8.g.c	$2$	$0$	\(16\)	\(0\)	\(455\)	\(1554\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(8-8\beta _{2})q^{2}-2^{6}\beta _{2}q^{4}+(230-5\beta _{1}+\cdots)q^{5}+\cdots\)
126.8.g.h	$126$	$8$	126.g	7.c	$3$	$6$	$3$	$39.361$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None					42.8.e.e	$2$	$0$	\(-24\)	\(0\)	\(-110\)	\(635\)	$2^{2}\cdot 3\cdot 7^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-8+8\beta _{1})q^{2}-2^{6}\beta _{1}q^{4}+(-37+\cdots)q^{5}+\cdots\)
126.8.g.i	$126$	$8$	126.g	7.c	$3$	$6$	$3$	$39.361$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		✓			126.8.g.i	$2$	$0$	\(-24\)	\(0\)	\(718\)	\(-1471\)	$2^{2}\cdot 3\cdot 7$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-8-8\beta _{3})q^{2}+2^{6}\beta _{3}q^{4}+(239+\cdots)q^{5}+\cdots\)
126.8.g.j	$126$	$8$	126.g	7.c	$3$	$6$	$3$	$39.361$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		✓			126.8.g.i	$2$	$0$	\(24\)	\(0\)	\(-718\)	\(-1471\)	$2^{2}\cdot 3\cdot 7$	$\mathrm{SU}(2)[C_{3}]$	\(q+(8+8\beta _{3})q^{2}+2^{6}\beta _{3}q^{4}+(-239+\cdots)q^{5}+\cdots\)
126.8.g.k	$126$	$8$	126.g	7.c	$3$	$6$	$3$	$39.361$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None					42.8.e.d	$2$	$0$	\(24\)	\(0\)	\(-70\)	\(-895\)	$2^{2}\cdot 3\cdot 7$	$\mathrm{SU}(2)[C_{3}]$	\(q+8\beta _{1}q^{2}+(-2^{6}+2^{6}\beta _{1})q^{4}+(-23\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(126, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(126, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)