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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(12\)	\(1\)	\(11\)		\(10\)	\(1\)	\(9\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(9\)	\(1\)	\(8\)		\(7\)	\(1\)	\(6\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(8\)	\(0\)	\(8\)		\(6\)	\(0\)	\(6\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(11\)	\(1\)	\(10\)		\(9\)	\(1\)	\(8\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(10\)	\(1\)	\(9\)		\(8\)	\(1\)	\(7\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(9\)	\(1\)	\(8\)		\(7\)	\(1\)	\(6\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(10\)	\(1\)	\(9\)		\(8\)	\(1\)	\(7\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(11\)	\(0\)	\(11\)		\(9\)	\(0\)	\(9\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(42\)	\(4\)	\(38\)		\(34\)	\(4\)	\(30\)		\(8\)	\(0\)	\(8\)
Minus space			\(-\)		\(38\)	\(2\)	\(36\)		\(30\)	\(2\)	\(28\)		\(8\)	\(0\)	\(8\)

Trace form

6 * q - 6 * q^3 + 36 * q^7 + 54 * q^9 + 28 * q^11 + 16 * q^13 + 8 * q^17 + 112 * q^19 + 60 * q^21 + 152 * q^23 + 150 * q^25 - 54 * q^27 - 280 * q^29 + 8 * q^31 + 132 * q^33 - 260 * q^35 + 568 * q^37 + 432 * q^39 - 244 * q^41 - 120 * q^43 - 1168 * q^47 - 922 * q^49 + 120 * q^51 - 1264 * q^53 - 180 * q^55 + 144 * q^57 + 12 * q^59 + 300 * q^61 + 324 * q^63 + 420 * q^65 - 64 * q^67 - 600 * q^69 - 1784 * q^71 + 196 * q^73 - 150 * q^75 + 96 * q^77 + 1784 * q^79 + 486 * q^81 + 672 * q^83 + 340 * q^85 + 408 * q^87 + 228 * q^89 + 8 * q^91 + 888 * q^93 - 40 * q^95 - 916 * q^97 + 252 * q^99

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

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