# Properties

 Label 4000.1.b Level 4000 Weight 1 Character orbit b Rep. character $$\chi_{4000}(2751,\cdot)$$ Character field $$\Q$$ Dimension 8 Newform subspaces 2 Sturm bound 600 Trace bound 13

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$4000 = 2^{5} \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 4000.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$4$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$600$$ Trace bound: $$13$$

## Dimensions

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

Total New Old
Modular forms 76 8 68
Cusp forms 36 8 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 0 0 0 8

## Trace form

 $$8q - 4q^{9} + O(q^{10})$$ $$8q - 4q^{9} - 4q^{21} - 8q^{29} + 8q^{41} - 4q^{61} + O(q^{100})$$

## 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.b.a $$4$$ $$1.996$$ $$\Q(i, \sqrt{5})$$ $$A_{5}$$ None None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{1}-\beta _{3})q^{3}-\beta _{3}q^{7}+(-1-\beta _{2}+\cdots)q^{9}+\cdots$$
4000.1.b.b $$4$$ $$1.996$$ $$\Q(i, \sqrt{5})$$ $$A_{5}$$ None None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{3}+\beta _{3}q^{7}+\beta _{2}q^{9}-\beta _{3}q^{11}+\cdots$$

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

$$S_{1}^{\mathrm{old}}(4000, [\chi]) \cong$$ $$S_{1}^{\mathrm{new}}(400, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(500, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(2000, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

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