Space of modular forms of level 60, weight 6, and trivial character

• Label: 60.6.a
• Level: $60$
• Weight: $6$
• Character orbit: 60.a
• Rep. character: $\chi_{60}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $4$
• Sturm bound: $72$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	60.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(72\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(60))\).

	Total	New	Old
Modular forms	66	4	62
Cusp forms	54	4	50
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	Dim
\(-\)	\(+\)	\(+\)	$-$	\(1\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(1\)
Plus space			\(+\)	\(2\)
Minus space			\(-\)	\(2\)

Trace form

\( 4 q - 160 q^{7} + 324 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(60))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
60.6.a.a	$60$	$6$	60.a	1.a	$1$	$1$	$1$	$9.623$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-9\)	\(-25\)	\(44\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}-5^{2}q^{5}+44q^{7}+3^{4}q^{9}+6^{3}q^{11}+\cdots\)
60.6.a.b	$60$	$6$	60.a	1.a	$1$	$1$	$1$	$9.623$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-9\)	\(25\)	\(-16\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+5^{2}q^{5}-2^{4}q^{7}+3^{4}q^{9}-564q^{11}+\cdots\)
60.6.a.c	$60$	$6$	60.a	1.a	$1$	$1$	$1$	$9.623$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(9\)	\(-25\)	\(-244\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-5^{2}q^{5}-244q^{7}+3^{4}q^{9}+\cdots\)
60.6.a.d	$60$	$6$	60.a	1.a	$1$	$1$	$1$	$9.623$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(9\)	\(25\)	\(56\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}+5^{2}q^{5}+56q^{7}+3^{4}q^{9}+156q^{11}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(60))\) into lower level spaces

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