Properties

Label 2000.1.bp
Level $2000$
Weight $1$
Character orbit 2000.bp
Rep. character $\chi_{2000}(79,\cdot)$
Character field $\Q(\zeta_{50})$
Dimension $20$
Newform subspaces $1$
Sturm bound $300$
Trace bound $0$

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) \(=\) \( 2000 = 2^{4} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2000.bp (of order \(50\) and degree \(20\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 500 \)
Character field: \(\Q(\zeta_{50})\)
Newform subspaces: \( 1 \)
Sturm bound: \(300\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(2000, [\chi])\).

Total New Old
Modular forms 140 20 120
Cusp forms 20 20 0
Eisenstein series 120 0 120

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 20 0 0 0

Trace form

\( 20q + O(q^{10}) \) \( 20q + 5q^{37} + 5q^{49} + 5q^{53} + 5q^{65} - 5q^{85} - 5q^{89} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(2000, [\chi])\) into newform subspaces

Label Dim. \(A\) Field Image CM RM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
2000.1.bp.a \(20\) \(0.998\) \(\Q(\zeta_{50})\) \(D_{50}\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{50}^{9}q^{5}+\zeta_{50}^{7}q^{9}+(\zeta_{50}^{3}-\zeta_{50}^{11}+\cdots)q^{13}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \)
$5$ \( 1 - T^{5} + T^{10} - T^{15} + T^{20} \)
$7$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{5} \)
$11$ \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
$13$ \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
$17$ \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
$19$ \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
$23$ \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \)
$29$ \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
$31$ \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
$37$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
$41$ \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2} \)
$43$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{5} \)
$47$ \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \)
$53$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
$59$ \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
$61$ \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2} \)
$67$ \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \)
$71$ \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
$73$ \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
$79$ \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
$83$ \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \)
$89$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
$97$ \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
show more
show less