Space of modular forms of level 126, weight 6, and Character 126.g

• Label: 126.6.g
• Level: $126$
• Weight: $6$
• Character orbit: 126.g
• Rep. character: $\chi_{126}(37,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $32$
• Newform subspaces: $10$
• Sturm bound: $144$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	126.g (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(144\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(126, [\chi])\).

	Total	New	Old
Modular forms	256	32	224
Cusp forms	224	32	192
Eisenstein series	32	0	32

Trace form

\( 32 q - 256 q^{4} + 72 q^{5} + 236 q^{7} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(126, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
126.6.g.a	$126$	$6$	126.g	7.c	$3$	$2$	$1$	$20.208$	\(\Q(\sqrt{-3}) \)	None		✓			126.6.g.a	$2$	$0$	\(-4\)	\(0\)	\(-12\)	\(119\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-4+4\zeta_{6})q^{2}-2^{4}\zeta_{6}q^{4}+(-12+\cdots)q^{5}+\cdots\)
126.6.g.b	$126$	$6$	126.g	7.c	$3$	$2$	$1$	$20.208$	\(\Q(\sqrt{-3}) \)	None					42.6.e.b	$2$	$0$	\(-4\)	\(0\)	\(86\)	\(49\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-4+4\zeta_{6})q^{2}-2^{4}\zeta_{6}q^{4}+(86-86\zeta_{6})q^{5}+\cdots\)
126.6.g.c	$126$	$6$	126.g	7.c	$3$	$2$	$1$	$20.208$	\(\Q(\sqrt{-3}) \)	None					42.6.e.a	$2$	$0$	\(4\)	\(0\)	\(-6\)	\(119\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4-4\zeta_{6})q^{2}-2^{4}\zeta_{6}q^{4}+(-6+6\zeta_{6})q^{5}+\cdots\)
126.6.g.d	$126$	$6$	126.g	7.c	$3$	$2$	$1$	$20.208$	\(\Q(\sqrt{-3}) \)	None		✓			126.6.g.a	$2$	$0$	\(4\)	\(0\)	\(12\)	\(119\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4-4\zeta_{6})q^{2}-2^{4}\zeta_{6}q^{4}+(12-12\zeta_{6})q^{5}+\cdots\)
126.6.g.e	$126$	$6$	126.g	7.c	$3$	$4$	$2$	$20.208$	\(\Q(\sqrt{-3}, \sqrt{130})\)	None					14.6.c.b	$2$	$0$	\(-8\)	\(0\)	\(-42\)	\(232\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\beta _{1}q^{2}+(-2^{4}-2^{4}\beta _{1})q^{4}+21\beta _{1}q^{5}+\cdots\)
126.6.g.f	$126$	$6$	126.g	7.c	$3$	$4$	$2$	$20.208$	\(\Q(\sqrt{-3}, \sqrt{697})\)	None		✓			126.6.g.f	$2$	$0$	\(-8\)	\(0\)	\(-7\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-4+4\beta _{2})q^{2}-2^{4}\beta _{2}q^{4}+(-6+\cdots)q^{5}+\cdots\)
126.6.g.g	$126$	$6$	126.g	7.c	$3$	$4$	$2$	$20.208$	\(\Q(\sqrt{-3}, \sqrt{505})\)	None					42.6.e.d	$2$	$0$	\(-8\)	\(0\)	\(17\)	\(-408\)	$7$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\beta _{1}q^{2}+(-2^{4}+2^{4}\beta _{1})q^{4}+(-1+\cdots)q^{5}+\cdots\)
126.6.g.h	$126$	$6$	126.g	7.c	$3$	$4$	$2$	$20.208$	\(\Q(\sqrt{-3}, \sqrt{9601})\)	None					42.6.e.c	$2$	$0$	\(8\)	\(0\)	\(-53\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\beta _{2}q^{2}+(-2^{4}+2^{4}\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
126.6.g.i	$126$	$6$	126.g	7.c	$3$	$4$	$2$	$20.208$	\(\Q(\sqrt{-3}, \sqrt{697})\)	None		✓			126.6.g.f	$2$	$0$	\(8\)	\(0\)	\(7\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4-4\beta _{2})q^{2}-2^{4}\beta _{2}q^{4}+(6-5\beta _{1}+\cdots)q^{5}+\cdots\)
126.6.g.j	$126$	$6$	126.g	7.c	$3$	$4$	$2$	$20.208$	\(\Q(\sqrt{-3}, \sqrt{79})\)	None					14.6.c.a	$2$	$0$	\(8\)	\(0\)	\(70\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4+4\beta _{2})q^{2}+2^{4}\beta _{2}q^{4}+(35+4\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(126, [\chi])\) into lower level spaces

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