Properties

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

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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

\( 40 q + q^{2} + q^{4} - 4 q^{6} + q^{8} + q^{9} + O(q^{10}) \) \( 40 q + q^{2} + q^{4} - 4 q^{6} + q^{8} + q^{9} + 2 q^{13} + 5 q^{16} + 2 q^{17} - 9 q^{18} - 5 q^{20} - 8 q^{21} - 6 q^{26} + 2 q^{29} - 9 q^{32} - 8 q^{34} + q^{36} - 8 q^{37} - 10 q^{41} - 4 q^{46} - 9 q^{49} + 2 q^{52} - 8 q^{53} - 4 q^{56} + 2 q^{58} - 10 q^{61} + q^{64} - 5 q^{65} + 2 q^{68} + q^{72} + 2 q^{73} + 2 q^{74} - 3 q^{81} + 2 q^{82} - 5 q^{85} - 4 q^{86} - 8 q^{89} - 4 q^{96} + 2 q^{97} + q^{98} + O(q^{100}) \)

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 the 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

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