Space of modular forms of level 60, weight 4, and trivial character

• Label: 60.4.a
• Level: $60$
• Weight: $4$
• Character orbit: 60.a
• Rep. character: $\chi_{60}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $48$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	60.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 \(M_{4}(\Gamma_0(60))\).

	Total	New	Old
Modular forms	42	2	40
Cusp forms	30	2	28
Eisenstein series	12	0	12

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
\(+\)	\(+\)	\(+\)	\(+\)		\(7\)	\(0\)	\(7\)		\(5\)	\(0\)	\(5\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(4\)	\(0\)	\(4\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(6\)	\(0\)	\(6\)		\(5\)	\(0\)	\(5\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(22\)	\(1\)	\(21\)		\(16\)	\(1\)	\(15\)		\(6\)	\(0\)	\(6\)
Minus space			\(-\)		\(20\)	\(1\)	\(19\)		\(14\)	\(1\)	\(13\)		\(6\)	\(0\)	\(6\)

Trace form

2 * q - 6 * q^3 + 4 * q^7 + 18 * q^9 + 12 * q^11 - 80 * q^13 + 24 * q^17 + 160 * q^19 - 12 * q^21 - 264 * q^23 + 50 * q^25 - 54 * q^27 + 312 * q^29 - 152 * q^31 - 36 * q^33 + 300 * q^35 - 248 * q^37 + 240 * q^39 - 540 * q^41 - 344 * q^43 + 336 * q^47 + 1122 * q^49 - 72 * q^51 - 528 * q^53 + 300 * q^55 - 480 * q^57 - 228 * q^59 - 476 * q^61 + 36 * q^63 + 300 * q^65 - 704 * q^67 + 792 * q^69 + 360 * q^71 - 212 * q^73 - 150 * q^75 + 1824 * q^77 + 184 * q^79 + 162 * q^81 - 1344 * q^83 - 900 * q^85 - 936 * q^87 - 1620 * q^89 + 1640 * q^91 + 456 * q^93 + 600 * q^95 - 92 * q^97 + 108 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(60))\) 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
60.4.a.a	$60$	$4$	60.a	1.a	$1$	$1$	$1$	$3.540$	\(\Q\)	None	✓	✓	✓	✓	60.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(-5\)	\(-28\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}-28q^{7}+9q^{9}-24q^{11}+\cdots\)
60.4.a.b	$60$	$4$	60.a	1.a	$1$	$1$	$1$	$3.540$	\(\Q\)	None	✓	✓	✓	✓	60.4.a.b	$1$	$0$	\(0\)	\(-3\)	\(5\)	\(32\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+2^{5}q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(60))\) into lower level spaces

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