# Properties

 Label 1210.2.b Level $1210$ Weight $2$ Character orbit 1210.b Rep. character $\chi_{1210}(969,\cdot)$ Character field $\Q$ Dimension $54$ Newform subspaces $13$ Sturm bound $396$ Trace bound $6$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1210 = 2 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1210.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$13$$ Sturm bound: $$396$$ Trace bound: $$6$$ Distinguishing $$T_p$$: $$3$$, $$13$$, $$19$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1210, [\chi])$$.

Total New Old
Modular forms 222 54 168
Cusp forms 174 54 120
Eisenstein series 48 0 48

## Trace form

 $$54q - 54q^{4} + 4q^{6} - 54q^{9} + O(q^{10})$$ $$54q - 54q^{4} + 4q^{6} - 54q^{9} - 4q^{10} + 4q^{14} + 4q^{15} + 54q^{16} - 16q^{19} - 4q^{24} - 12q^{25} - 4q^{26} - 4q^{29} + 4q^{30} + 24q^{31} + 8q^{34} + 8q^{35} + 54q^{36} - 16q^{39} + 4q^{40} + 36q^{41} + 32q^{45} + 12q^{46} - 82q^{49} - 24q^{50} - 48q^{51} - 16q^{54} - 4q^{56} - 4q^{59} - 4q^{60} + 36q^{61} - 54q^{64} - 16q^{65} + 48q^{69} - 24q^{70} - 24q^{75} + 16q^{76} + 16q^{79} + 54q^{81} - 8q^{85} - 8q^{86} + 12q^{89} + 20q^{90} - 8q^{91} + 20q^{94} + 4q^{96} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1210, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1210.2.b.a $$2$$ $$9.662$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+(-2+i)q^{5}+\cdots$$
1210.2.b.b $$2$$ $$9.662$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q+iq^{2}+3iq^{3}-q^{4}+(-1-2i)q^{5}+\cdots$$
1210.2.b.c $$2$$ $$9.662$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q-iq^{2}+3iq^{3}-q^{4}+(-1-2i)q^{5}+\cdots$$
1210.2.b.d $$2$$ $$9.662$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+iq^{2}+2iq^{3}-q^{4}+(1+2i)q^{5}+\cdots$$
1210.2.b.e $$2$$ $$9.662$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q-iq^{2}+3iq^{3}-q^{4}+(2-i)q^{5}+3q^{6}+\cdots$$
1210.2.b.f $$4$$ $$9.662$$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$-8$$ $$0$$ $$q-\beta _{3}q^{2}+2\beta _{1}q^{3}-q^{4}+(-2-\beta _{3})q^{5}+\cdots$$
1210.2.b.g $$4$$ $$9.662$$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$-8$$ $$0$$ $$q+\beta _{3}q^{2}+2\beta _{1}q^{3}-q^{4}+(-2-\beta _{3})q^{5}+\cdots$$
1210.2.b.h $$4$$ $$9.662$$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{3}-q^{4}-\beta _{2}q^{5}-\beta _{3}q^{6}+\cdots$$
1210.2.b.i $$4$$ $$9.662$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+\zeta_{12}^{3}q^{2}+(1-2\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots$$
1210.2.b.j $$4$$ $$9.662$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q-\zeta_{12}^{3}q^{2}+(1-2\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots$$
1210.2.b.k $$8$$ $$9.662$$ 8.0.$$\cdots$$.1 None $$0$$ $$0$$ $$2$$ $$0$$ $$q-\beta _{3}q^{2}+(-\beta _{3}+\beta _{4})q^{3}-q^{4}+(1+\cdots)q^{5}+\cdots$$
1210.2.b.l $$8$$ $$9.662$$ 8.0.$$\cdots$$.1 None $$0$$ $$0$$ $$2$$ $$0$$ $$q+\beta _{3}q^{2}+(-\beta _{3}+\beta _{4})q^{3}-q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots$$
1210.2.b.m $$8$$ $$9.662$$ 8.0.303595776.1 None $$0$$ $$0$$ $$6$$ $$0$$ $$q-\beta _{5}q^{2}+(\beta _{2}-\beta _{4}+\beta _{6})q^{3}-q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1210, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1210, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(55, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(110, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(605, [\chi])$$$$^{\oplus 2}$$