# Properties

 Label 4000.1.g Level 4000 Weight 1 Character orbit g Rep. character $$\chi_{4000}(751,\cdot)$$ Character field $$\Q$$ Dimension 4 Newform subspaces 1 Sturm bound 600 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$4000 = 2^{5} \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 4000.g (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$600$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(4000, [\chi])$$.

Total New Old
Modular forms 72 4 68
Cusp forms 32 4 28
Eisenstein series 40 0 40

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

 $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} + 2q^{11} - 2q^{19} - 2q^{41} - 2q^{49} - 2q^{59} + 4q^{81} + 2q^{89} + 4q^{91} - 2q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(4000, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field Image CM RM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
4000.1.g.a $$4$$ $$1.996$$ $$\Q(i, \sqrt{5})$$ $$D_{5}$$ $$\Q(\sqrt{-10})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{3})q^{7}-q^{9}-\beta _{2}q^{11}-\beta _{1}q^{13}+\cdots$$

## Decomposition of $$S_{1}^{\mathrm{old}}(4000, [\chi])$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(4000, [\chi]) \cong$$ $$S_{1}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(800, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(1000, [\chi])$$$$^{\oplus 3}$$

## Hecke characteristic polynomials

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