# Properties

 Label 3850.2.c Level $3850$ Weight $2$ Character orbit 3850.c Rep. character $\chi_{3850}(1849,\cdot)$ Character field $\Q$ Dimension $92$ Newform subspaces $29$ Sturm bound $1440$ Trace bound $29$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3850.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$29$$ Sturm bound: $$1440$$ Trace bound: $$29$$ Distinguishing $$T_p$$: $$3$$, $$13$$, $$17$$, $$19$$, $$37$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(3850, [\chi])$$.

Total New Old
Modular forms 744 92 652
Cusp forms 696 92 604
Eisenstein series 48 0 48

## Trace form

 $$92 q - 92 q^{4} - 108 q^{9} + O(q^{10})$$ $$92 q - 92 q^{4} - 108 q^{9} + 4 q^{11} - 8 q^{14} + 92 q^{16} + 24 q^{26} - 40 q^{29} + 24 q^{31} - 40 q^{34} + 108 q^{36} + 48 q^{39} - 40 q^{41} - 4 q^{44} - 92 q^{49} + 16 q^{51} + 8 q^{56} + 88 q^{59} + 8 q^{61} - 92 q^{64} + 16 q^{69} - 8 q^{71} - 8 q^{74} - 32 q^{79} + 140 q^{81} + 24 q^{86} - 80 q^{89} + 8 q^{91} - 52 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(3850, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
3850.2.c.a $$2$$ $$30.742$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+2iq^{3}-q^{4}-2q^{6}+iq^{7}+\cdots$$
3850.2.c.b $$2$$ $$30.742$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+2iq^{3}-q^{4}-2q^{6}+iq^{7}+\cdots$$
3850.2.c.c $$2$$ $$30.742$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+2iq^{3}-q^{4}-2q^{6}-iq^{7}+\cdots$$
3850.2.c.d $$2$$ $$30.742$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+2iq^{3}-q^{4}-2q^{6}+iq^{7}+\cdots$$
3850.2.c.e $$2$$ $$30.742$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+2iq^{3}-q^{4}-2q^{6}+iq^{7}+\cdots$$
3850.2.c.f $$2$$ $$30.742$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+iq^{3}-q^{4}-q^{6}+iq^{7}-iq^{8}+\cdots$$
3850.2.c.g $$2$$ $$30.742$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+iq^{3}-q^{4}-q^{6}+iq^{7}-iq^{8}+\cdots$$
3850.2.c.h $$2$$ $$30.742$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+iq^{7}-iq^{8}+3q^{9}+\cdots$$
3850.2.c.i $$2$$ $$30.742$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}-iq^{7}+iq^{8}+3q^{9}+\cdots$$
3850.2.c.j $$2$$ $$30.742$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}-iq^{7}+iq^{8}+3q^{9}+\cdots$$
3850.2.c.k $$2$$ $$30.742$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+iq^{7}+iq^{8}+3q^{9}+\cdots$$
3850.2.c.l $$2$$ $$30.742$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}-iq^{7}-iq^{8}+3q^{9}+\cdots$$
3850.2.c.m $$2$$ $$30.742$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}-iq^{7}-iq^{8}+3q^{9}+\cdots$$
3850.2.c.n $$2$$ $$30.742$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}-iq^{7}+iq^{8}+\cdots$$
3850.2.c.o $$2$$ $$30.742$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+2iq^{3}-q^{4}+2q^{6}+iq^{7}+\cdots$$
3850.2.c.p $$2$$ $$30.742$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+2iq^{3}-q^{4}+2q^{6}-iq^{7}+\cdots$$
3850.2.c.q $$4$$ $$30.742$$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}+(-\beta _{1}+\beta _{2})q^{3}-q^{4}+(-1+\cdots)q^{6}+\cdots$$
3850.2.c.r $$4$$ $$30.742$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{2}+(\zeta_{12}-\zeta_{12}^{2})q^{3}-q^{4}+\cdots$$
3850.2.c.s $$4$$ $$30.742$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{2}+(\zeta_{12}-\zeta_{12}^{2})q^{3}-q^{4}+\cdots$$
3850.2.c.t $$4$$ $$30.742$$ $$\Q(i, \sqrt{10})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{3}-q^{4}-\beta _{3}q^{6}+\beta _{1}q^{7}+\cdots$$
3850.2.c.u $$4$$ $$30.742$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{8}q^{2}+\zeta_{8}^{2}q^{3}-q^{4}+\zeta_{8}^{3}q^{6}+\cdots$$
3850.2.c.v $$4$$ $$30.742$$ $$\Q(i, \sqrt{7})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}+\beta _{3}q^{3}-q^{4}+\beta _{2}q^{6}+\beta _{1}q^{7}+\cdots$$
3850.2.c.w $$4$$ $$30.742$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{8}q^{2}+\zeta_{8}^{2}q^{3}-q^{4}+\zeta_{8}^{3}q^{6}+\cdots$$
3850.2.c.x $$4$$ $$30.742$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{12}q^{2}+(\zeta_{12}-\zeta_{12}^{2})q^{3}-q^{4}+\cdots$$
3850.2.c.y $$4$$ $$30.742$$ $$\Q(i, \sqrt{33})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-2\beta _{1}q^{3}-q^{4}+2q^{6}+\beta _{1}q^{7}+\cdots$$
3850.2.c.z $$6$$ $$30.742$$ 6.0.3182656.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{4}q^{2}+\beta _{3}q^{3}-q^{4}+\beta _{2}q^{6}-\beta _{4}q^{7}+\cdots$$
3850.2.c.ba $$6$$ $$30.742$$ 6.0.399424.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{2}+(\beta _{2}+\beta _{5})q^{3}-q^{4}+(1+\beta _{1}+\cdots)q^{6}+\cdots$$
3850.2.c.bb $$6$$ $$30.742$$ 6.0.399424.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{2}+(-\beta _{2}+\beta _{5})q^{3}-q^{4}+(1+\cdots)q^{6}+\cdots$$
3850.2.c.bc $$6$$ $$30.742$$ 6.0.3534400.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{4}q^{2}+(\beta _{4}+\beta _{5})q^{3}-q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(3850, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(3850, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(55, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(110, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(275, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(350, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(385, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(550, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(770, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1925, [\chi])$$$$^{\oplus 2}$$