# Properties

 Label 1800.2.k Level $1800$ Weight $2$ Character orbit 1800.k Rep. character $\chi_{1800}(901,\cdot)$ Character field $\Q$ Dimension $92$ Newform subspaces $21$ Sturm bound $720$ Trace bound $17$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1800.k (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$21$$ Sturm bound: $$720$$ Trace bound: $$17$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1800, [\chi])$$.

Total New Old
Modular forms 384 98 286
Cusp forms 336 92 244
Eisenstein series 48 6 42

## Trace form

 $$92 q + 2 q^{4} + 4 q^{7} + O(q^{10})$$ $$92 q + 2 q^{4} + 4 q^{7} - 6 q^{16} - 4 q^{17} - 4 q^{22} - 12 q^{23} + 12 q^{28} + 8 q^{31} + 20 q^{32} - 34 q^{34} - 12 q^{38} + 4 q^{41} + 50 q^{44} + 32 q^{46} - 4 q^{47} + 68 q^{49} + 40 q^{52} - 8 q^{56} - 28 q^{58} + 36 q^{62} - 22 q^{64} + 8 q^{68} + 40 q^{71} + 20 q^{74} + 38 q^{76} + 8 q^{79} + 4 q^{82} + 40 q^{86} + 16 q^{88} - 20 q^{89} - 60 q^{92} + 80 q^{94} - 16 q^{97} + 12 q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1800, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1800.2.k.a $2$ $14.373$ $$\Q(\sqrt{-1})$$ None $$-2$$ $$0$$ $$0$$ $$4$$ $$q+(-1-i)q^{2}+2iq^{4}+2q^{7}+(2-2i)q^{8}+\cdots$$
1800.2.k.b $2$ $14.373$ $$\Q(\sqrt{-1})$$ None $$-2$$ $$0$$ $$0$$ $$4$$ $$q+(-1-i)q^{2}+2iq^{4}+2q^{7}+(2-2i)q^{8}+\cdots$$
1800.2.k.c $2$ $14.373$ $$\Q(\sqrt{-1})$$ None $$-2$$ $$0$$ $$0$$ $$8$$ $$q+(-1-i)q^{2}+2iq^{4}+4q^{7}+(2-2i)q^{8}+\cdots$$
1800.2.k.d $2$ $14.373$ $$\Q(\sqrt{-7})$$ None $$-1$$ $$0$$ $$0$$ $$8$$ $$q+(-1+\beta )q^{2}+(-1-\beta )q^{4}+4q^{7}+\cdots$$
1800.2.k.e $2$ $14.373$ $$\Q(\sqrt{-2})$$ $$\Q(\sqrt{-6})$$ $$0$$ $$0$$ $$0$$ $$-4$$ $$q-\beta q^{2}-2q^{4}-2q^{7}+2\beta q^{8}-4\beta q^{11}+\cdots$$
1800.2.k.f $2$ $14.373$ $$\Q(\sqrt{-7})$$ None $$1$$ $$0$$ $$0$$ $$-8$$ $$q+\beta q^{2}+(-2+\beta )q^{4}-4q^{7}+(-2+\cdots)q^{8}+\cdots$$
1800.2.k.g $2$ $14.373$ $$\Q(\sqrt{-1})$$ None $$2$$ $$0$$ $$0$$ $$-4$$ $$q+(1-i)q^{2}-2iq^{4}-2q^{7}+(-2-2i)q^{8}+\cdots$$
1800.2.k.h $2$ $14.373$ $$\Q(\sqrt{-1})$$ None $$2$$ $$0$$ $$0$$ $$-4$$ $$q+(1+i)q^{2}+2iq^{4}-2q^{7}+(-2+2i)q^{8}+\cdots$$
1800.2.k.i $2$ $14.373$ $$\Q(\sqrt{-1})$$ None $$2$$ $$0$$ $$0$$ $$8$$ $$q+(1-i)q^{2}-2iq^{4}+4q^{7}+(-2-2i)q^{8}+\cdots$$
1800.2.k.j $4$ $14.373$ $$\Q(\zeta_{12})$$ None $$-2$$ $$0$$ $$0$$ $$4$$ $$q+(-\zeta_{12}+\zeta_{12}^{2})q^{2}+(\zeta_{12}+\zeta_{12}^{3})q^{4}+\cdots$$
1800.2.k.k $4$ $14.373$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ $$\Q(\sqrt{-6})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}-2q^{4}-\beta _{3}q^{7}+2\beta _{1}q^{8}+\cdots$$
1800.2.k.l $4$ $14.373$ $$\Q(\sqrt{3}, \sqrt{-5})$$ $$\Q(\sqrt{-15})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta q^{2}+\beta ^{2}q^{4}-\beta ^{3}q^{8}+(-4-\beta ^{2}+\cdots)q^{16}+\cdots$$
1800.2.k.m $4$ $14.373$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}+(1+\beta _{3})q^{4}+(-2\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$$
1800.2.k.n $4$ $14.373$ $$\Q(i, \sqrt{7})$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-2q^{7}+(\beta _{1}+\beta _{3})q^{8}+\cdots$$
1800.2.k.o $4$ $14.373$ $$\Q(\sqrt{2}, \sqrt{-5})$$ $$\Q(\sqrt{-30})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+2q^{4}+2\beta _{1}q^{8}+\beta _{3}q^{11}+\cdots$$
1800.2.k.p $6$ $14.373$ 6.0.399424.1 None $$2$$ $$0$$ $$0$$ $$-4$$ $$q-\beta _{1}q^{2}+\beta _{2}q^{4}+(-1-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$$
1800.2.k.q $8$ $14.373$ 8.0.214798336.3 None $$-2$$ $$0$$ $$0$$ $$-8$$ $$q+\beta _{3}q^{2}+(-\beta _{4}-\beta _{5}+\beta _{6})q^{4}+(-1+\cdots)q^{7}+\cdots$$
1800.2.k.r $8$ $14.373$ 8.0.$$\cdots$$.29 None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-1-\beta _{2}+\beta _{5}+\cdots)q^{7}+\cdots$$
1800.2.k.s $8$ $14.373$ 8.0.$$\cdots$$.29 None $$0$$ $$0$$ $$0$$ $$8$$ $$q+\beta _{6}q^{2}-\beta _{5}q^{4}+(1+\beta _{2}-\beta _{5})q^{7}+\cdots$$
1800.2.k.t $8$ $14.373$ 8.0.214798336.3 None $$2$$ $$0$$ $$0$$ $$8$$ $$q+\beta _{2}q^{2}+(1+\beta _{1}-\beta _{3}-\beta _{6})q^{4}+(1+\cdots)q^{7}+\cdots$$
1800.2.k.u $12$ $14.373$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{4}q^{4}+(-\beta _{1}-\beta _{8})q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1800, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1800, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(24, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(120, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(360, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(600, [\chi])$$$$^{\oplus 2}$$