Properties

Label 1000.1.g
Level $1000$
Weight $1$
Character orbit 1000.g
Rep. character $\chi_{1000}(251,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $1$
Sturm bound $150$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1000.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(150\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(1000, [\chi])\).

Total New Old
Modular forms 18 4 14
Cusp forms 8 4 4
Eisenstein series 10 0 10

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

\( 4q - 4q^{4} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} - 4q^{9} - 2q^{11} + 2q^{14} + 4q^{16} + 2q^{19} - 2q^{26} + 4q^{36} - 2q^{41} + 2q^{44} - 2q^{46} - 2q^{49} - 2q^{56} + 2q^{59} - 4q^{64} + 2q^{74} - 2q^{76} + 4q^{81} + 2q^{89} - 4q^{91} + 2q^{94} + 2q^{99} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(1000, [\chi])\) into newform subspaces

Label Dim. \(A\) Field Image CM RM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1000.1.g.a \(4\) \(0.499\) \(\Q(i, \sqrt{5})\) \(D_{5}\) \(\Q(\sqrt{-10}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{2}-q^{4}-\beta _{1}q^{7}+\beta _{3}q^{8}-q^{9}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(1000, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(1000, [\chi]) \cong \) \(S_{1}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)