# Properties

 Label 1800.1.bk Level 1800 Weight 1 Character orbit bk Rep. character $$\chi_{1800}(1051,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 10 Newform subspaces 5 Sturm bound 360 Trace bound 6

# Related objects

## Defining parameters

 Level: $$N$$ = $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 1800.bk (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$72$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$5$$ Sturm bound: $$360$$ Trace bound: $$6$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(1800, [\chi])$$.

Total New Old
Modular forms 40 22 18
Cusp forms 16 10 6
Eisenstein series 24 12 12

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 10 0 0 0

## Trace form

 $$10q + q^{2} + q^{3} - 5q^{4} + q^{6} - 2q^{8} + q^{9} + O(q^{10})$$ $$10q + q^{2} + q^{3} - 5q^{4} + q^{6} - 2q^{8} + q^{9} - q^{11} - 2q^{12} - 5q^{16} + 2q^{17} - 2q^{18} + 2q^{19} - q^{22} + q^{24} - 2q^{27} + q^{32} - q^{33} - q^{34} + q^{36} - q^{38} - q^{41} - q^{43} + 2q^{44} + q^{48} - 5q^{49} + 11q^{51} - 5q^{54} - q^{57} + 5q^{59} + 10q^{64} - 4q^{66} - q^{67} - q^{68} + q^{72} + 2q^{73} - q^{76} + q^{81} + 2q^{82} + 2q^{83} + 5q^{86} - q^{88} - 4q^{89} - 2q^{96} - q^{97} - 2q^{98} - 4q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(1800, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field Image CM RM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1800.1.bk.a $$2$$ $$0.898$$ $$\Q(\sqrt{-3})$$ $$D_{3}$$ $$\Q(\sqrt{-2})$$ None $$-1$$ $$-1$$ $$0$$ $$0$$ $$q+\zeta_{6}^{2}q^{2}-\zeta_{6}q^{3}-\zeta_{6}q^{4}+q^{6}+q^{8}+\cdots$$
1800.1.bk.b $$2$$ $$0.898$$ $$\Q(\sqrt{-3})$$ $$D_{3}$$ $$\Q(\sqrt{-2})$$ None $$-1$$ $$2$$ $$0$$ $$0$$ $$q+\zeta_{6}^{2}q^{2}+q^{3}-\zeta_{6}q^{4}+\zeta_{6}^{2}q^{6}+\cdots$$
1800.1.bk.c $$2$$ $$0.898$$ $$\Q(\sqrt{-3})$$ $$D_{3}$$ $$\Q(\sqrt{-2})$$ None $$1$$ $$-2$$ $$0$$ $$0$$ $$q-\zeta_{6}^{2}q^{2}-q^{3}-\zeta_{6}q^{4}+\zeta_{6}^{2}q^{6}+\cdots$$
1800.1.bk.d $$2$$ $$0.898$$ $$\Q(\sqrt{-3})$$ $$D_{3}$$ $$\Q(\sqrt{-2})$$ None $$1$$ $$1$$ $$0$$ $$0$$ $$q-\zeta_{6}^{2}q^{2}-\zeta_{6}^{2}q^{3}-\zeta_{6}q^{4}-\zeta_{6}q^{6}+\cdots$$
1800.1.bk.e $$2$$ $$0.898$$ $$\Q(\sqrt{-3})$$ $$D_{3}$$ $$\Q(\sqrt{-2})$$ None $$1$$ $$1$$ $$0$$ $$0$$ $$q-\zeta_{6}^{2}q^{2}+\zeta_{6}q^{3}-\zeta_{6}q^{4}+q^{6}-q^{8}+\cdots$$

## Decomposition of $$S_{1}^{\mathrm{old}}(1800, [\chi])$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(1800, [\chi]) \cong$$ $$S_{1}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 3}$$

## Hecke Characteristic Polynomials

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