Properties

Label 1000.1.e
Level $1000$
Weight $1$
Character orbit 1000.e
Rep. character $\chi_{1000}(499,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $2$
Sturm bound $150$
Trace bound $2$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1000.e (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 40 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(150\)
Trace bound: \(2\)

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

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

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

Label Char Prim Dim $A$ Field Image CM RM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1000.1.e.a 1000.e 40.e $2$ $0.499$ \(\Q(\sqrt{5}) \) $D_{5}$ \(\Q(\sqrt{-10}) \) None \(-2\) \(0\) \(0\) \(1\) \(q-q^{2}+q^{4}+(1-\beta )q^{7}-q^{8}+q^{9}+\cdots\)
1000.1.e.b 1000.e 40.e $2$ $0.499$ \(\Q(\sqrt{5}) \) $D_{5}$ \(\Q(\sqrt{-10}) \) None \(2\) \(0\) \(0\) \(-1\) \(q+q^{2}+q^{4}+(-1+\beta )q^{7}+q^{8}+q^{9}+\cdots\)

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

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