Space of modular forms of level 126, weight 7, and Character 126.n

• Label: 126.7.n
• Level: $126$
• Weight: $7$
• Character orbit: 126.n
• Rep. character: $\chi_{126}(19,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $40$
• Newform subspaces: $4$
• Sturm bound: $168$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	126.n (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(168\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(126, [\chi])\).

	Total	New	Old
Modular forms	304	40	264
Cusp forms	272	40	232
Eisenstein series	32	0	32

Trace form

40 * q - 640 * q^4 - 336 * q^5 + 20 * q^7 + 2016 * q^10 - 2508 * q^11 - 4272 * q^14 - 20480 * q^16 + 18984 * q^17 + 6300 * q^19 - 5088 * q^22 - 13776 * q^23 + 83584 * q^25 + 55776 * q^26 + 10624 * q^28 - 2520 * q^29 - 74508 * q^31 - 180900 * q^35 + 71980 * q^37 + 155568 * q^38 - 64512 * q^40 + 13888 * q^43 - 80256 * q^44 + 87216 * q^46 + 64428 * q^47 - 137216 * q^49 + 236928 * q^50 + 67200 * q^52 - 300396 * q^53 - 24576 * q^56 + 142752 * q^58 + 676536 * q^59 - 401268 * q^61 + 1310720 * q^64 - 511956 * q^65 + 598040 * q^67 - 607488 * q^68 + 24240 * q^70 + 127344 * q^71 - 1697220 * q^73 - 371904 * q^74 + 2105376 * q^77 + 1085096 * q^79 + 344064 * q^80 + 1324512 * q^82 - 2056920 * q^85 - 507648 * q^86 + 81408 * q^88 - 5283180 * q^89 - 5404056 * q^91 + 881664 * q^92 + 1279152 * q^94 + 1550472 * q^95 + 3375648 * q^98

Decomposition of \(S_{7}^{\mathrm{new}}(126, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
126.7.n.a	$126$	$7$	126.n	7.d	$6$	$8$	$4$	$28.987$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					42.7.g.a	$2$	$0$	\(0\)	\(0\)	\(-462\)	\(580\)	$2^{8}\cdot 3^{2}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{2}q^{2}+(-2^{5}+2^{5}\beta _{1})q^{4}+(-78+\cdots)q^{5}+\cdots\)
126.7.n.b	$126$	$7$	126.n	7.d	$6$	$8$	$4$	$28.987$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					42.7.g.b	$2$	$0$	\(0\)	\(0\)	\(-210\)	\(-608\)	$2^{6}\cdot 3^{2}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{5}q^{2}+(-2^{5}+2^{5}\beta _{1})q^{4}+(-35+\cdots)q^{5}+\cdots\)
126.7.n.c	$126$	$7$	126.n	7.d	$6$	$8$	$4$	$28.987$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					14.7.d.a	$2$	$0$	\(0\)	\(0\)	\(336\)	\(652\)	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{2}q^{2}+(-2^{5}+2^{5}\beta _{1})q^{4}+(56-28\beta _{1}+\cdots)q^{5}+\cdots\)
126.7.n.d	$126$	$7$	126.n	7.d	$6$	$16$	$8$	$28.987$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓	✓		126.7.n.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-604\)	$2^{20}\cdot 3^{12}\cdot 7^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{3}q^{2}+(-2^{5}-2^{5}\beta _{1})q^{4}+(\beta _{2}-3\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(126, [\chi])\) into lower level spaces

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