Space of modular forms of level 60, weight 8, and trivial character

• Label: 60.8.a
• Level: $60$
• Weight: $8$
• Character orbit: 60.a
• Rep. character: $\chi_{60}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $4$
• Sturm bound: $96$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	60.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(96\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(60))\).

	Total	New	Old
Modular forms	90	4	86
Cusp forms	78	4	74
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
\(+\)	\(+\)	\(+\)	\(+\)		\(13\)	\(0\)	\(13\)		\(11\)	\(0\)	\(11\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(10\)	\(0\)	\(10\)		\(8\)	\(0\)	\(8\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(11\)	\(0\)	\(11\)		\(9\)	\(0\)	\(9\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(12\)	\(0\)	\(12\)		\(10\)	\(0\)	\(10\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(11\)	\(1\)	\(10\)		\(10\)	\(1\)	\(9\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(11\)	\(1\)	\(10\)		\(10\)	\(1\)	\(9\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(10\)	\(1\)	\(9\)		\(9\)	\(1\)	\(8\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(12\)	\(1\)	\(11\)		\(11\)	\(1\)	\(10\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(46\)	\(2\)	\(44\)		\(40\)	\(2\)	\(38\)		\(6\)	\(0\)	\(6\)
Minus space			\(-\)		\(44\)	\(2\)	\(42\)		\(38\)	\(2\)	\(36\)		\(6\)	\(0\)	\(6\)

Trace form

4 * q - 1120 * q^7 + 2916 * q^9 + 5664 * q^11 - 22000 * q^13 + 50640 * q^17 - 27040 * q^19 - 40824 * q^21 + 6720 * q^23 + 62500 * q^25 - 157296 * q^29 + 324176 * q^31 - 184680 * q^33 - 420000 * q^35 + 238400 * q^37 + 524880 * q^39 + 572040 * q^41 - 1576960 * q^43 + 408000 * q^47 + 445764 * q^49 + 1251936 * q^51 - 1276800 * q^53 - 930000 * q^55 - 1516320 * q^57 + 1729824 * q^59 + 2207528 * q^61 - 816480 * q^63 - 645000 * q^65 + 2594240 * q^67 - 3681936 * q^69 - 819360 * q^71 - 160840 * q^73 + 6568800 * q^77 - 5218672 * q^79 + 2125764 * q^81 - 8690880 * q^83 + 1485000 * q^85 + 5171040 * q^87 + 4408920 * q^89 + 1720480 * q^91 + 7698240 * q^93 + 4740000 * q^95 - 26632600 * q^97 + 4129056 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(60))\) 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
60.8.a.a	$60$	$8$	60.a	1.a	$1$	$1$	$1$	$18.743$	\(\Q\)	None	✓	✓	✓	✓	60.8.a.a	$1$	$1$	\(0\)	\(-27\)	\(-125\)	\(1028\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-5^{3}q^{5}+1028q^{7}+3^{6}q^{9}+\cdots\)
60.8.a.b	$60$	$8$	60.a	1.a	$1$	$1$	$1$	$18.743$	\(\Q\)	None	✓	✓	✓	✓	60.8.a.b	$1$	$0$	\(0\)	\(-27\)	\(125\)	\(-832\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+5^{3}q^{5}-832q^{7}+3^{6}q^{9}+\cdots\)
60.8.a.c	$60$	$8$	60.a	1.a	$1$	$1$	$1$	$18.743$	\(\Q\)	None	✓	✓	✓	✓	60.8.a.c	$1$	$0$	\(0\)	\(27\)	\(-125\)	\(92\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-5^{3}q^{5}+92q^{7}+3^{6}q^{9}+\cdots\)
60.8.a.d	$60$	$8$	60.a	1.a	$1$	$1$	$1$	$18.743$	\(\Q\)	None	✓	✓	✓	✓	60.8.a.d	$1$	$1$	\(0\)	\(27\)	\(125\)	\(-1408\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+5^{3}q^{5}-1408q^{7}+3^{6}q^{9}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(60))\) into lower level spaces

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