Space of modular forms of level 120, weight 4, and trivial character

• Label: 120.4.a
• Level: $120$
• Weight: $4$
• Character orbit: 120.a
• Rep. character: $\chi_{120}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $6$
• Sturm bound: $96$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	120.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(96\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(120))\).

	Total	New	Old
Modular forms	80	6	74
Cusp forms	64	6	58
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.

\(2\)	\(3\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)	\(1\)
Plus space			\(+\)	\(4\)
Minus space			\(-\)	\(2\)

Trace form

\( 6 q - 6 q^{3} + 36 q^{7} + 54 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(120))\) 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	5
120.4.a.a	$120$	$4$	120.a	1.a	$1$	$1$	$1$	$7.080$	\(\Q\)	None	✓	✓	✓	✓	120.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(-5\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+4q^{7}+9q^{9}+72q^{11}+\cdots\)
120.4.a.b	$120$	$4$	120.a	1.a	$1$	$1$	$1$	$7.080$	\(\Q\)	None	✓	✓	✓	✓	120.4.a.b	$1$	$1$	\(0\)	\(-3\)	\(-5\)	\(20\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+20q^{7}+9q^{9}-56q^{11}+\cdots\)
120.4.a.c	$120$	$4$	120.a	1.a	$1$	$1$	$1$	$7.080$	\(\Q\)	None	✓	✓	✓	✓	120.4.a.c	$1$	$1$	\(0\)	\(-3\)	\(5\)	\(-16\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}-2^{4}q^{7}+9q^{9}-28q^{11}+\cdots\)
120.4.a.d	$120$	$4$	120.a	1.a	$1$	$1$	$1$	$7.080$	\(\Q\)	None	✓	✓	✓	✓	120.4.a.d	$1$	$0$	\(0\)	\(-3\)	\(5\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+9q^{9}+4q^{11}+54q^{13}+\cdots\)
120.4.a.e	$120$	$4$	120.a	1.a	$1$	$1$	$1$	$7.080$	\(\Q\)	None	✓	✓	✓	✓	120.4.a.e	$1$	$0$	\(0\)	\(3\)	\(-5\)	\(20\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+20q^{7}+9q^{9}+2^{4}q^{11}+\cdots\)
120.4.a.f	$120$	$4$	120.a	1.a	$1$	$1$	$1$	$7.080$	\(\Q\)	None	✓	✓	✓	✓	120.4.a.f	$1$	$0$	\(0\)	\(3\)	\(5\)	\(8\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+8q^{7}+9q^{9}+20q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(120))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(120)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 2}\)