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	Dim
\(+\)	\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(+\)	\(-\)	$-$	\(1\)
\(+\)	\(-\)	\(+\)	$-$	\(1\)
\(+\)	\(-\)	\(-\)	$+$	\(2\)
\(-\)	\(+\)	\(+\)	$-$	\(1\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
Plus space			\(+\)	\(5\)
Minus space			\(-\)	\(3\)

Trace form

\( 8 q - 4 q^{2} + 32 q^{4} + 6 q^{5} - 16 q^{8} + O(q^{10}) \)

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

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

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