Space of modular forms of level 500 and weight 1

• Label: 500.1
• Level: 500
• Weight: 1
• Dimension: 40
• Nonzero newspaces: 5
• Newform subspaces: 6
• Sturm bound: 15000
• Trace bound: 4

Defining parameters

Level:	\( N \)	=	\( 500 = 2^{2} \cdot 5^{3} \)
Weight:	\( k \)	=	\( 1 \)
Nonzero newspaces:			\( 5 \)
Newform subspaces:			\( 6 \)
Sturm bound:			\(15000\)
Trace bound:			\(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(500))\).

	Total	New	Old
Modular forms	498	168	330
Cusp forms	48	40	8
Eisenstein series	450	128	322

The following table gives the dimensions of subspaces with specified projective image type.

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	40	0	0	0

Trace form

\( 40 q + q^{2} + q^{4} - 4 q^{6} + q^{8} + q^{9} + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(500))\)

We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list available newforms together with their dimension.

Label	\(\chi\)	Newforms	Dimension	\(\chi\) degree
500.1.b	\(\chi_{500}(251, \cdot)\)	500.1.b.a	4	1
500.1.d	\(\chi_{500}(499, \cdot)\)	500.1.d.a	2	1
		500.1.d.b	2
500.1.f	\(\chi_{500}(57, \cdot)\)	None	0	2
500.1.h	\(\chi_{500}(99, \cdot)\)	500.1.h.a	8	4
500.1.j	\(\chi_{500}(51, \cdot)\)	500.1.j.a	4	4
500.1.k	\(\chi_{500}(93, \cdot)\)	None	0	8
500.1.n	\(\chi_{500}(19, \cdot)\)	None	0	20
500.1.p	\(\chi_{500}(11, \cdot)\)	500.1.p.a	20	20
500.1.q	\(\chi_{500}(13, \cdot)\)	None	0	40

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(500))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(500)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 2}\)