# 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

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