Space of modular forms of level 126, weight 4, and trivial character

• Label: 126.4.a
• Level: $126$
• Weight: $4$
• Character orbit: 126.a
• Rep. character: $\chi_{126}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $8$
• Sturm bound: $96$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(96\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(126))\).

	Total	New	Old
Modular forms	80	8	72
Cusp forms	64	8	56
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
\(+\)	\(+\)	\(+\)	\(+\)		\(12\)	\(1\)	\(11\)		\(10\)	\(1\)	\(9\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(8\)	\(1\)	\(7\)		\(6\)	\(1\)	\(5\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(9\)	\(1\)	\(8\)		\(7\)	\(1\)	\(6\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(11\)	\(2\)	\(9\)		\(9\)	\(2\)	\(7\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(10\)	\(1\)	\(9\)		\(8\)	\(1\)	\(7\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(10\)	\(1\)	\(9\)		\(8\)	\(1\)	\(7\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(9\)	\(1\)	\(8\)		\(7\)	\(1\)	\(6\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(11\)	\(0\)	\(11\)		\(9\)	\(0\)	\(9\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(42\)	\(5\)	\(37\)		\(34\)	\(5\)	\(29\)		\(8\)	\(0\)	\(8\)
Minus space			\(-\)		\(38\)	\(3\)	\(35\)		\(30\)	\(3\)	\(27\)		\(8\)	\(0\)	\(8\)

Trace form

8 * q - 4 * q^2 + 32 * q^4 + 6 * q^5 - 16 * q^8 - 20 * q^10 + 60 * q^11 - 106 * q^13 - 28 * q^14 + 128 * q^16 + 36 * q^17 + 178 * q^19 + 24 * q^20 + 8 * q^22 + 336 * q^23 + 708 * q^25 + 76 * q^26 - 276 * q^29 - 604 * q^31 - 64 * q^32 - 400 * q^34 - 126 * q^35 - 308 * q^37 + 220 * q^38 - 80 * q^40 - 756 * q^41 + 244 * q^43 + 240 * q^44 + 464 * q^46 + 84 * q^47 + 392 * q^49 - 52 * q^50 - 424 * q^52 - 984 * q^53 + 8 * q^55 - 112 * q^56 - 752 * q^58 + 1326 * q^59 - 574 * q^61 - 296 * q^62 + 512 * q^64 + 1620 * q^65 - 2616 * q^67 + 144 * q^68 + 644 * q^70 - 768 * q^71 - 1560 * q^73 - 1712 * q^74 + 712 * q^76 - 84 * q^77 - 520 * q^79 + 96 * q^80 - 1152 * q^82 - 570 * q^83 + 4484 * q^85 - 824 * q^86 + 32 * q^88 - 840 * q^89 + 1106 * q^91 + 1344 * q^92 + 600 * q^94 - 264 * q^95 - 252 * q^97 - 196 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(126))\) 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
126.4.a.a	$126$	$4$	126.a	1.a	$1$	$1$	$1$	$7.434$	\(\Q\)	None	✓				42.4.a.a	$1$	$0$	\(-2\)	\(0\)	\(-18\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-18q^{5}+7q^{7}-8q^{8}+\cdots\)
126.4.a.b	$126$	$4$	126.a	1.a	$1$	$1$	$1$	$7.434$	\(\Q\)	None	✓	✓	✓	✓	126.4.a.b	$1$	$1$	\(-2\)	\(0\)	\(-6\)	\(7\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-6q^{5}+7q^{7}-8q^{8}+\cdots\)
126.4.a.c	$126$	$4$	126.a	1.a	$1$	$1$	$1$	$7.434$	\(\Q\)	None	✓		✓	✓	42.4.a.b	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(-7\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-2q^{5}-7q^{7}-8q^{8}+\cdots\)
126.4.a.d	$126$	$4$	126.a	1.a	$1$	$1$	$1$	$7.434$	\(\Q\)	None	✓				14.4.a.b	$1$	$0$	\(-2\)	\(0\)	\(12\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+12q^{5}+7q^{7}-8q^{8}+\cdots\)
126.4.a.e	$126$	$4$	126.a	1.a	$1$	$1$	$1$	$7.434$	\(\Q\)	None	✓	✓	✓	✓	126.4.a.e	$1$	$0$	\(-2\)	\(0\)	\(22\)	\(-7\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+22q^{5}-7q^{7}-8q^{8}+\cdots\)
126.4.a.f	$126$	$4$	126.a	1.a	$1$	$1$	$1$	$7.434$	\(\Q\)	None	✓	✓	✓	✓	126.4.a.e	$1$	$1$	\(2\)	\(0\)	\(-22\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-22q^{5}-7q^{7}+8q^{8}+\cdots\)
126.4.a.g	$126$	$4$	126.a	1.a	$1$	$1$	$1$	$7.434$	\(\Q\)	None	✓	✓	✓	✓	126.4.a.b	$1$	$0$	\(2\)	\(0\)	\(6\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+6q^{5}+7q^{7}+8q^{8}+\cdots\)
126.4.a.h	$126$	$4$	126.a	1.a	$1$	$1$	$1$	$7.434$	\(\Q\)	None	✓		✓	✓	14.4.a.a	$1$	$0$	\(2\)	\(0\)	\(14\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+14q^{5}-7q^{7}+8q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(126))\) into lower level spaces

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