# Properties

 Label 2850.2.d Level $2850$ Weight $2$ Character orbit 2850.d Rep. character $\chi_{2850}(799,\cdot)$ Character field $\Q$ Dimension $56$ Newform subspaces $24$ Sturm bound $1200$ Trace bound $26$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2850.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$24$$ Sturm bound: $$1200$$ Trace bound: $$26$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(2850, [\chi])$$.

Total New Old
Modular forms 624 56 568
Cusp forms 576 56 520
Eisenstein series 48 0 48

## Trace form

 $$56 q - 56 q^{4} + 4 q^{6} - 56 q^{9} + O(q^{10})$$ $$56 q - 56 q^{4} + 4 q^{6} - 56 q^{9} + 8 q^{11} + 56 q^{16} - 4 q^{19} - 4 q^{24} + 24 q^{26} + 8 q^{29} - 40 q^{31} - 8 q^{34} + 56 q^{36} + 16 q^{39} + 40 q^{41} - 8 q^{44} - 16 q^{46} - 24 q^{49} - 32 q^{51} - 4 q^{54} + 64 q^{59} - 8 q^{61} - 56 q^{64} + 32 q^{69} - 64 q^{71} - 56 q^{74} + 4 q^{76} + 40 q^{79} + 56 q^{81} + 16 q^{86} + 40 q^{89} + 96 q^{91} + 48 q^{94} + 4 q^{96} - 8 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(2850, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
2850.2.d.a $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+iq^{3}-q^{4}-q^{6}+4iq^{7}+\cdots$$
2850.2.d.b $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-iq^{3}-q^{4}-q^{6}+iq^{8}-q^{9}+\cdots$$
2850.2.d.c $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+iq^{3}-q^{4}-q^{6}+4iq^{7}+\cdots$$
2850.2.d.d $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-iq^{3}-q^{4}-q^{6}+2iq^{7}+\cdots$$
2850.2.d.e $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-iq^{3}-q^{4}-q^{6}+2iq^{7}+\cdots$$
2850.2.d.f $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-iq^{3}-q^{4}-q^{6}+4iq^{7}+\cdots$$
2850.2.d.g $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+iq^{3}-q^{4}-q^{6}-iq^{8}-q^{9}+\cdots$$
2850.2.d.h $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-iq^{3}-q^{4}-q^{6}+2iq^{7}+\cdots$$
2850.2.d.i $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-iq^{3}-q^{4}-q^{6}+iq^{8}-q^{9}+\cdots$$
2850.2.d.j $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}+2iq^{7}+\cdots$$
2850.2.d.k $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-iq^{3}-q^{4}+q^{6}+4iq^{7}+\cdots$$
2850.2.d.l $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-iq^{3}-q^{4}+q^{6}+2iq^{7}+\cdots$$
2850.2.d.m $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}+2iq^{7}+\cdots$$
2850.2.d.n $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-iq^{3}-q^{4}+q^{6}+2iq^{7}+\cdots$$
2850.2.d.o $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}+4iq^{7}+\cdots$$
2850.2.d.p $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}+4iq^{7}+\cdots$$
2850.2.d.q $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}+2iq^{7}+\cdots$$
2850.2.d.r $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-iq^{3}-q^{4}+q^{6}+2iq^{7}+\cdots$$
2850.2.d.s $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}+4iq^{7}+\cdots$$
2850.2.d.t $2$ $22.757$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-iq^{3}-q^{4}+q^{6}+2iq^{7}+\cdots$$
2850.2.d.u $4$ $22.757$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{2}+\zeta_{12}q^{3}-q^{4}-q^{6}+(\zeta_{12}+\cdots)q^{7}+\cdots$$
2850.2.d.v $4$ $22.757$ $$\Q(i, \sqrt{10})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{1}q^{3}-q^{4}-q^{6}+\beta _{2}q^{7}+\cdots$$
2850.2.d.w $4$ $22.757$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-\beta _{1}q^{3}-q^{4}+q^{6}+(2\beta _{1}+\cdots)q^{7}+\cdots$$
2850.2.d.x $4$ $22.757$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{2}-\zeta_{12}q^{3}-q^{4}+q^{6}+(\zeta_{12}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(2850, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(2850, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(95, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(190, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(285, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(475, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(570, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(950, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1425, [\chi])$$$$^{\oplus 2}$$