Space of modular forms of level 126, weight 4, and Character 126.g

• Label: 126.4.g
• Level: $126$
• Weight: $4$
• Character orbit: 126.g
• Rep. character: $\chi_{126}(37,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $20$
• Newform subspaces: $7$
• Sturm bound: $96$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	126.g (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(96\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(126, [\chi])\).

	Total	New	Old
Modular forms	160	20	140
Cusp forms	128	20	108
Eisenstein series	32	0	32

Trace form

20 * q - 40 * q^4 - 18 * q^5 - 8 * q^7 + 8 * q^10 + 6 * q^11 - 192 * q^13 - 72 * q^14 - 160 * q^16 - 150 * q^17 + 46 * q^19 + 144 * q^20 + 256 * q^22 + 210 * q^23 - 68 * q^25 - 120 * q^26 + 40 * q^28 - 336 * q^29 - 138 * q^31 - 512 * q^34 - 426 * q^35 - 242 * q^37 - 192 * q^38 + 32 * q^40 + 2016 * q^41 + 1288 * q^43 + 24 * q^44 + 40 * q^46 + 102 * q^47 + 1412 * q^49 - 1824 * q^50 + 384 * q^52 - 1182 * q^53 - 1916 * q^55 + 288 * q^56 - 136 * q^58 + 90 * q^59 - 294 * q^61 + 3744 * q^62 + 1280 * q^64 + 1524 * q^65 + 1226 * q^67 - 600 * q^68 - 1520 * q^70 - 672 * q^71 + 1142 * q^73 - 312 * q^74 - 368 * q^76 - 2226 * q^77 + 1266 * q^79 - 288 * q^80 + 1200 * q^82 + 2568 * q^83 - 4892 * q^85 - 984 * q^86 - 512 * q^88 - 2754 * q^89 + 2100 * q^91 - 1680 * q^92 - 216 * q^94 - 1974 * q^95 - 160 * q^97 + 2880 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(126, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
126.4.g.a	$126$	$4$	126.g	7.c	$3$	$2$	$1$	$7.434$	\(\Q(\sqrt{-3}) \)	None					42.4.e.b	$2$	$0$	\(-2\)	\(0\)	\(-15\)	\(35\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(-15+\cdots)q^{5}+\cdots\)
126.4.g.b	$126$	$4$	126.g	7.c	$3$	$2$	$1$	$7.434$	\(\Q(\sqrt{-3}) \)	None					42.4.e.a	$2$	$0$	\(-2\)	\(0\)	\(-6\)	\(-7\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(-6+6\zeta_{6})q^{5}+\cdots\)
126.4.g.c	$126$	$4$	126.g	7.c	$3$	$2$	$1$	$7.434$	\(\Q(\sqrt{-3}) \)	None					14.4.c.b	$2$	$0$	\(-2\)	\(0\)	\(7\)	\(-20\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(7-7\zeta_{6})q^{5}+\cdots\)
126.4.g.d	$126$	$4$	126.g	7.c	$3$	$2$	$1$	$7.434$	\(\Q(\sqrt{-3}) \)	None					14.4.c.a	$2$	$0$	\(2\)	\(0\)	\(-9\)	\(-28\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(-9+9\zeta_{6})q^{5}+\cdots\)
126.4.g.e	$126$	$4$	126.g	7.c	$3$	$4$	$2$	$7.434$	\(\Q(\sqrt{-3}, \sqrt{193})\)	None		✓			126.4.g.e	$2$	$0$	\(-4\)	\(0\)	\(7\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\beta _{2}q^{2}+(-4+4\beta _{2})q^{4}+(\beta _{1}+3\beta _{2}+\cdots)q^{5}+\cdots\)
126.4.g.f	$126$	$4$	126.g	7.c	$3$	$4$	$2$	$7.434$	\(\Q(\sqrt{-3}, \sqrt{193})\)	None		✓			126.4.g.e	$2$	$0$	\(4\)	\(0\)	\(-7\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\beta _{2}q^{2}+(-4+4\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
126.4.g.g	$126$	$4$	126.g	7.c	$3$	$4$	$2$	$7.434$	\(\Q(\sqrt{-3}, \sqrt{1345})\)	None					42.4.e.c	$2$	$0$	\(4\)	\(0\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\beta _{2}q^{2}+(-4+4\beta _{2})q^{4}+(\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(126, [\chi])\) into lower level spaces

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