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

• Label: 126.7.b
• Level: $126$
• Weight: $7$
• Character orbit: 126.b
• Rep. character: $\chi_{126}(71,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $2$
• Sturm bound: $168$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	126.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(168\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(5\)

Dimensions

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

	Total	New	Old
Modular forms	152	12	140
Cusp forms	136	12	124
Eisenstein series	16	0	16

Trace form

12 * q - 384 * q^4 + 1248 * q^10 - 6192 * q^13 + 12288 * q^16 - 15360 * q^19 + 17856 * q^22 + 1668 * q^25 + 32736 * q^31 - 93792 * q^34 + 29592 * q^37 - 39936 * q^40 + 21936 * q^43 - 123456 * q^46 + 201684 * q^49 + 198144 * q^52 - 282144 * q^55 + 421536 * q^58 - 96120 * q^61 - 393216 * q^64 - 504816 * q^67 - 197568 * q^70 + 986304 * q^73 + 491520 * q^76 + 684672 * q^79 - 400032 * q^82 - 1799832 * q^85 - 571392 * q^88 - 905520 * q^91 + 2716032 * q^94 + 1304736 * q^97

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.b.a	$126$	$7$	126.b	3.b	$2$	$4$	$4$	$28.987$	\(\Q(\sqrt{-2}, \sqrt{7})\)	None		✓			126.7.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{1}q^{2}-2^{5}q^{4}+(-5^{2}\beta _{1}+5^{2}\beta _{2}+\cdots)q^{5}+\cdots\)
126.7.b.b	$126$	$7$	126.b	3.b	$2$	$8$	$8$	$28.987$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓	✓		126.7.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{4}\cdot 7^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta _{3}q^{2}-2^{5}q^{4}+(7\beta _{3}+\beta _{5})q^{5}+\cdots\)

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

\( S_{7}^{\mathrm{old}}(126, [\chi]) \simeq \) \(S_{7}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(6, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)