Space of modular forms of level 126, weight 6, and trivial character

• Label: 126.6.a
• Level: $126$
• Weight: $6$
• Character orbit: 126.a
• Rep. character: $\chi_{126}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $12$
• Sturm bound: $144$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(144\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(126))\).

	Total	New	Old
Modular forms	128	12	116
Cusp forms	112	12	100
Eisenstein series	16	0	16

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(14\)	\(1\)	\(13\)		\(12\)	\(1\)	\(11\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(18\)	\(1\)	\(17\)		\(16\)	\(1\)	\(15\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(17\)	\(2\)	\(15\)		\(15\)	\(2\)	\(13\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(15\)	\(1\)	\(14\)		\(13\)	\(1\)	\(12\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(16\)	\(1\)	\(15\)		\(14\)	\(1\)	\(13\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(16\)	\(1\)	\(15\)		\(14\)	\(1\)	\(13\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(17\)	\(2\)	\(15\)		\(15\)	\(2\)	\(13\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(15\)	\(3\)	\(12\)		\(13\)	\(3\)	\(10\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(62\)	\(5\)	\(57\)		\(54\)	\(5\)	\(49\)		\(8\)	\(0\)	\(8\)
Minus space			\(-\)		\(66\)	\(7\)	\(59\)		\(58\)	\(7\)	\(51\)		\(8\)	\(0\)	\(8\)

Trace form

12 * q + 8 * q^2 + 192 * q^4 - 138 * q^5 + 128 * q^8 - 312 * q^10 - 36 * q^11 + 1590 * q^13 + 392 * q^14 + 3072 * q^16 - 1236 * q^17 - 4122 * q^19 - 2208 * q^20 + 1872 * q^22 + 4536 * q^23 - 6096 * q^25 - 3512 * q^26 + 14388 * q^29 + 25068 * q^31 + 2048 * q^32 + 11040 * q^34 + 8526 * q^35 + 1668 * q^37 - 1832 * q^38 - 4992 * q^40 + 6804 * q^41 + 3564 * q^43 - 576 * q^44 - 29280 * q^46 - 3780 * q^47 + 28812 * q^49 + 20264 * q^50 + 25440 * q^52 - 7776 * q^53 + 73800 * q^55 + 6272 * q^56 - 2208 * q^58 - 61206 * q^59 - 10542 * q^61 + 27568 * q^62 + 49152 * q^64 - 85692 * q^65 - 144552 * q^67 - 19776 * q^68 - 1176 * q^70 + 63768 * q^71 + 48912 * q^73 + 49888 * q^74 - 65952 * q^76 + 47628 * q^77 + 16920 * q^79 - 35328 * q^80 - 137472 * q^82 + 168402 * q^83 - 110988 * q^85 - 96560 * q^86 + 29952 * q^88 - 172704 * q^89 - 11466 * q^91 + 72576 * q^92 + 82608 * q^94 + 305640 * q^95 - 328020 * q^97 + 19208 * q^98

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(126))\) 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					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	7
126.6.a.a	$126$	$6$	126.a	1.a	$1$	$1$	$1$	$20.208$	\(\Q\)	None	✓				42.6.a.e	$1$	$0$	\(-4\)	\(0\)	\(-76\)	\(-49\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}-76q^{5}-7^{2}q^{7}-2^{6}q^{8}+\cdots\)
126.6.a.b	$126$	$6$	126.a	1.a	$1$	$1$	$1$	$20.208$	\(\Q\)	None	✓		✓	✓	42.6.a.f	$1$	$1$	\(-4\)	\(0\)	\(-24\)	\(49\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}-24q^{5}+7^{2}q^{7}-2^{6}q^{8}+\cdots\)
126.6.a.c	$126$	$6$	126.a	1.a	$1$	$1$	$1$	$20.208$	\(\Q\)	None	✓				14.6.a.b	$1$	$0$	\(-4\)	\(0\)	\(-10\)	\(-49\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}-10q^{5}-7^{2}q^{7}-2^{6}q^{8}+\cdots\)
126.6.a.d	$126$	$6$	126.a	1.a	$1$	$1$	$1$	$20.208$	\(\Q\)	None	✓	✓	✓	✓	126.6.a.d	$1$	$1$	\(-4\)	\(0\)	\(26\)	\(-49\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+26q^{5}-7^{2}q^{7}-2^{6}q^{8}+\cdots\)
126.6.a.e	$126$	$6$	126.a	1.a	$1$	$1$	$1$	$20.208$	\(\Q\)	None	✓	✓	✓	✓	126.6.a.e	$1$	$0$	\(-4\)	\(0\)	\(54\)	\(49\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+54q^{5}+7^{2}q^{7}-2^{6}q^{8}+\cdots\)
126.6.a.f	$126$	$6$	126.a	1.a	$1$	$1$	$1$	$20.208$	\(\Q\)	None	✓				14.6.a.a	$1$	$0$	\(4\)	\(0\)	\(-84\)	\(49\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}-84q^{5}+7^{2}q^{7}+2^{6}q^{8}+\cdots\)
126.6.a.g	$126$	$6$	126.a	1.a	$1$	$1$	$1$	$20.208$	\(\Q\)	None	✓	✓	✓	✓	126.6.a.e	$1$	$1$	\(4\)	\(0\)	\(-54\)	\(49\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}-54q^{5}+7^{2}q^{7}+2^{6}q^{8}+\cdots\)
126.6.a.h	$126$	$6$	126.a	1.a	$1$	$1$	$1$	$20.208$	\(\Q\)	None	✓				42.6.a.b	$1$	$1$	\(4\)	\(0\)	\(-44\)	\(-49\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}-44q^{5}-7^{2}q^{7}+2^{6}q^{8}+\cdots\)
126.6.a.i	$126$	$6$	126.a	1.a	$1$	$1$	$1$	$20.208$	\(\Q\)	None	✓				42.6.a.d	$1$	$1$	\(4\)	\(0\)	\(-26\)	\(-49\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}-26q^{5}-7^{2}q^{7}+2^{6}q^{8}+\cdots\)
126.6.a.j	$126$	$6$	126.a	1.a	$1$	$1$	$1$	$20.208$	\(\Q\)	None	✓	✓	✓	✓	126.6.a.d	$1$	$0$	\(4\)	\(0\)	\(-26\)	\(-49\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}-26q^{5}-7^{2}q^{7}+2^{6}q^{8}+\cdots\)
126.6.a.k	$126$	$6$	126.a	1.a	$1$	$1$	$1$	$20.208$	\(\Q\)	None	✓				42.6.a.a	$1$	$0$	\(4\)	\(0\)	\(54\)	\(49\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+54q^{5}+7^{2}q^{7}+2^{6}q^{8}+\cdots\)
126.6.a.l	$126$	$6$	126.a	1.a	$1$	$1$	$1$	$20.208$	\(\Q\)	None	✓				42.6.a.c	$1$	$0$	\(4\)	\(0\)	\(72\)	\(49\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+72q^{5}+7^{2}q^{7}+2^{6}q^{8}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(126))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(126)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)