Properties

Label 500.1.j
Level 500
Weight 1
Character orbit j
Rep. character \(\chi_{500}(51,\cdot)\)
Character field \(\Q(\zeta_{10})\)
Dimension 4
Newforms 1
Sturm bound 75
Trace bound 0

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Defining parameters

Level: \( N \) = \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 500.j (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 100 \)
Character field: \(\Q(\zeta_{10})\)
Newforms: \( 1 \)
Sturm bound: \(75\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(500, [\chi])\).

Total New Old
Modular forms 52 28 24
Cusp forms 12 4 8
Eisenstein series 40 24 16

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 4 0 0 0

Trace form

\( 4q + q^{2} - q^{4} + q^{8} - q^{9} + O(q^{10}) \) \( 4q + q^{2} - q^{4} + q^{8} - q^{9} + 2q^{13} - q^{16} + 2q^{17} - 4q^{18} - 2q^{26} - 2q^{29} - 4q^{32} + 3q^{34} - q^{36} - 3q^{37} - 2q^{41} + 4q^{49} + 2q^{52} - 3q^{53} + 2q^{58} - 2q^{61} - q^{64} + 2q^{68} + q^{72} + 2q^{73} - 2q^{74} - q^{81} + 2q^{82} + 3q^{89} + 2q^{97} + q^{98} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(500, [\chi])\) into irreducible Hecke orbits

Label Dim. \(A\) Field Image CM RM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
500.1.j.a \(4\) \(0.250\) \(\Q(\zeta_{10})\) \(D_{5}\) \(\Q(\sqrt{-1}) \) None \(1\) \(0\) \(0\) \(0\) \(q-\zeta_{10}^{4}q^{2}-\zeta_{10}^{3}q^{4}-\zeta_{10}^{2}q^{8}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(500, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(500, [\chi]) \cong \) \(S_{1}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 2}\)