# Properties

 Label 3600.1.dx Level 3600 Weight 1 Character orbit dx Rep. character $$\chi_{3600}(287,\cdot)$$ Character field $$\Q(\zeta_{20})$$ Dimension 16 Newform subspaces 1 Sturm bound 720 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3600.dx (of order $$20$$ and degree $$8$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$300$$ Character field: $$\Q(\zeta_{20})$$ Newform subspaces: $$1$$ Sturm bound: $$720$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(3600, [\chi])$$.

Total New Old
Modular forms 256 16 240
Cusp forms 64 16 48
Eisenstein series 192 0 192

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 16 0 0 0

## Trace form

 $$16q + O(q^{10})$$ $$16q - 4q^{13} + 4q^{37} + 4q^{73} - 12q^{85} + 4q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(3600, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field Image CM RM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
3600.1.dx.a $$16$$ $$1.797$$ $$\Q(\zeta_{40})$$ $$D_{20}$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{40}^{7}q^{5}+(-\zeta_{40}^{4}+\zeta_{40}^{18})q^{13}+\cdots$$

## Decomposition of $$S_{1}^{\mathrm{old}}(3600, [\chi])$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(3600, [\chi]) \cong$$ $$S_{1}^{\mathrm{new}}(900, [\chi])$$$$^{\oplus 3}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
$7$ $$( 1 + T^{4} )^{8}$$
$11$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4}$$
$13$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$17$ $$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$
$19$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4}$$
$23$ $$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$
$29$ $$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$
$31$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$
$37$ $$( 1 + T^{2} )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$
$41$ $$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$
$43$ $$( 1 + T^{4} )^{8}$$
$47$ $$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$
$53$ $$( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} )$$
$59$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$
$61$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4}$$
$67$ $$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$
$71$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4}$$
$73$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$79$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4}$$
$83$ $$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$
$89$ $$( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} )$$
$97$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$