Properties

Label 500.1
Level 500
Weight 1
Dimension 40
Nonzero newspaces 5
Newforms 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: \( 5 \)
Newforms: \( 6 \)
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

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

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

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