Space of modular forms of level 126, weight 2, and trivial character

• Label: 126.2.a
• Level: $126$
• Weight: $2$
• Character orbit: 126.a
• Rep. character: $\chi_{126}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $48$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(48\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(126))\).

	Total	New	Old
Modular forms	32	2	30
Cusp forms	17	2	15
Eisenstein series	15	0	15

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
\(+\)	\(+\)	\(+\)	\(+\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(5\)	\(1\)	\(4\)		\(3\)	\(1\)	\(2\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(3\)	\(1\)	\(2\)		\(1\)	\(1\)	\(0\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(14\)	\(0\)	\(14\)		\(7\)	\(0\)	\(7\)		\(7\)	\(0\)	\(7\)
Minus space			\(-\)		\(18\)	\(2\)	\(16\)		\(10\)	\(2\)	\(8\)		\(8\)	\(0\)	\(8\)

Trace form

2 * q + 2 * q^4 + 2 * q^5 - 2 * q^10 + 4 * q^11 + 2 * q^13 + 2 * q^14 + 2 * q^16 - 8 * q^17 - 2 * q^19 + 2 * q^20 - 4 * q^22 - 8 * q^23 - 6 * q^25 - 10 * q^26 + 8 * q^29 - 4 * q^31 - 4 * q^34 - 2 * q^35 - 8 * q^37 + 6 * q^38 - 2 * q^40 + 4 * q^43 + 4 * q^44 + 8 * q^46 + 12 * q^47 + 2 * q^49 - 4 * q^50 + 2 * q^52 - 12 * q^53 + 8 * q^55 + 2 * q^56 + 4 * q^58 + 2 * q^59 + 14 * q^61 - 4 * q^62 + 2 * q^64 + 12 * q^65 - 8 * q^68 + 2 * q^70 - 8 * q^71 + 12 * q^73 + 12 * q^74 - 2 * q^76 - 4 * q^77 + 8 * q^79 + 2 * q^80 - 12 * q^82 + 10 * q^83 - 4 * q^85 + 12 * q^86 - 4 * q^88 + 12 * q^89 - 10 * q^91 - 8 * q^92 + 12 * q^94 - 8 * q^95 - 24 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(126))\) 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
126.2.a.a	$126$	$2$	126.a	1.a	$1$	$1$	$1$	$1.006$	\(\Q\)	None	✓		✓	✓	42.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(2\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{5}-q^{7}-q^{8}-2q^{10}+\cdots\)
126.2.a.b	$126$	$2$	126.a	1.a	$1$	$1$	$1$	$1.006$	\(\Q\)	None	✓		✓	✓	14.2.a.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}-4q^{13}+q^{14}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(126))\) into lower level spaces

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