# Properties

 Label 1850.2.b Level $1850$ Weight $2$ Character orbit 1850.b Rep. character $\chi_{1850}(149,\cdot)$ Character field $\Q$ Dimension $54$ Newform subspaces $16$ Sturm bound $570$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$16$$ Sturm bound: $$570$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$3$$, $$7$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1850, [\chi])$$.

Total New Old
Modular forms 298 54 244
Cusp forms 274 54 220
Eisenstein series 24 0 24

## Trace form

 $$54 q - 54 q^{4} + 4 q^{6} - 78 q^{9} + O(q^{10})$$ $$54 q - 54 q^{4} + 4 q^{6} - 78 q^{9} + 8 q^{11} + 54 q^{16} + 20 q^{19} + 8 q^{21} - 4 q^{24} + 4 q^{29} - 8 q^{31} + 78 q^{36} + 24 q^{39} + 8 q^{41} - 8 q^{44} + 24 q^{46} - 30 q^{49} - 28 q^{51} + 20 q^{54} + 8 q^{59} + 20 q^{61} - 54 q^{64} - 28 q^{66} + 72 q^{69} - 8 q^{71} + 10 q^{74} - 20 q^{76} + 16 q^{79} + 134 q^{81} - 8 q^{84} - 28 q^{86} + 32 q^{89} - 8 q^{91} + 24 q^{94} + 4 q^{96} - 8 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1850, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1850.2.b.a $2$ $14.772$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+3iq^{3}-q^{4}-3q^{6}-iq^{8}+\cdots$$
1850.2.b.b $2$ $14.772$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+2iq^{3}-q^{4}-2q^{6}+2iq^{7}+\cdots$$
1850.2.b.c $2$ $14.772$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+2iq^{3}-q^{4}-2q^{6}-iq^{7}+\cdots$$
1850.2.b.d $2$ $14.772$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+iq^{8}+3q^{9}-4q^{11}+\cdots$$
1850.2.b.e $2$ $14.772$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}+4iq^{7}+\cdots$$
1850.2.b.f $2$ $14.772$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+2iq^{3}-q^{4}+2q^{6}-4iq^{7}+\cdots$$
1850.2.b.g $2$ $14.772$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+2iq^{3}-q^{4}+2q^{6}-iq^{7}+\cdots$$
1850.2.b.h $2$ $14.772$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+2iq^{3}-q^{4}+2q^{6}+iq^{8}+\cdots$$
1850.2.b.i $4$ $14.772$ $$\Q(i, \sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{2}+(\beta _{1}-\beta _{2})q^{3}-q^{4}+(-2+\cdots)q^{6}+\cdots$$
1850.2.b.j $4$ $14.772$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{2}+\beta _{1}q^{3}-q^{4}+\beta _{2}q^{6}-2\beta _{1}q^{7}+\cdots$$
1850.2.b.k $4$ $14.772$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{3}-q^{4}+(1+\beta _{3})q^{6}+\cdots$$
1850.2.b.l $4$ $14.772$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{12}q^{2}+(\zeta_{12}-\zeta_{12}^{2})q^{3}-q^{4}+\cdots$$
1850.2.b.m $4$ $14.772$ $$\Q(i, \sqrt{33})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{2}+2\beta _{2}q^{3}-q^{4}+2q^{6}+(\beta _{1}+\cdots)q^{7}+\cdots$$
1850.2.b.n $6$ $14.772$ 6.0.350464.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{2}+(\beta _{3}-\beta _{5})q^{3}-q^{4}+(-1+\cdots)q^{6}+\cdots$$
1850.2.b.o $6$ $14.772$ 6.0.3182656.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{4}q^{2}-\beta _{3}q^{3}-q^{4}+\beta _{2}q^{6}-\beta _{5}q^{7}+\cdots$$
1850.2.b.p $6$ $14.772$ 6.0.37161216.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{4}q^{2}+\beta _{1}q^{3}-q^{4}+\beta _{2}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1850, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1850, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(185, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(370, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(925, [\chi])$$$$^{\oplus 2}$$