Space of modular forms of level 120, weight 2, and trivial character

• Label: 120.2.a
• Level: $120$
• Weight: $2$
• Character orbit: 120.a
• Rep. character: $\chi_{120}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $48$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	120.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(48\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(120))\).

	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.

\(2\)	\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(6\)	\(1\)	\(5\)		\(4\)	\(1\)	\(3\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(3\)	\(1\)	\(2\)		\(1\)	\(1\)	\(0\)		\(2\)	\(0\)	\(2\)
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^3 + 4 * q^7 + 2 * q^9 - 4 * q^11 - 8 * q^17 + 4 * q^21 - 8 * q^23 + 2 * q^25 + 2 * q^27 - 8 * q^29 - 8 * q^31 - 4 * q^33 - 4 * q^35 - 8 * q^37 + 4 * q^41 + 8 * q^43 + 16 * q^47 + 2 * q^49 - 8 * q^51 + 16 * q^53 - 4 * q^55 + 12 * q^59 + 20 * q^61 + 4 * q^63 + 12 * q^65 - 8 * q^69 + 8 * q^71 - 20 * q^73 + 2 * q^75 + 8 * q^79 + 2 * q^81 - 4 * q^85 - 8 * q^87 + 12 * q^89 - 24 * q^91 - 8 * q^93 - 8 * q^95 + 4 * q^97 - 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(120))\) 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
120.2.a.a	$120$	$2$	120.a	1.a	$1$	$1$	$1$	$0.958$	\(\Q\)	None	✓	✓	✓	✓	120.2.a.a	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+4q^{7}+q^{9}-6q^{13}-q^{15}+\cdots\)
120.2.a.b	$120$	$2$	120.a	1.a	$1$	$1$	$1$	$0.958$	\(\Q\)	None	✓	✓	✓	✓	120.2.a.b	$1$	$0$	\(0\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+q^{9}-4q^{11}+6q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(120))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(120)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)