# Properties

 Label 390.2.y Level $390$ Weight $2$ Character orbit 390.y Rep. character $\chi_{390}(139,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $32$ Newform subspaces $7$ Sturm bound $168$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.y (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$65$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$7$$ Sturm bound: $$168$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(390, [\chi])$$.

Total New Old
Modular forms 184 32 152
Cusp forms 152 32 120
Eisenstein series 32 0 32

## Trace form

 $$32q + 16q^{4} + 16q^{9} + O(q^{10})$$ $$32q + 16q^{4} + 16q^{9} - 4q^{11} - 8q^{14} + 4q^{15} - 16q^{16} + 28q^{19} + 16q^{21} + 8q^{25} + 24q^{29} - 24q^{31} - 20q^{35} - 16q^{36} - 12q^{39} + 24q^{41} - 8q^{44} + 4q^{46} + 36q^{49} + 8q^{50} - 12q^{55} - 4q^{56} - 48q^{59} + 8q^{60} - 16q^{61} - 32q^{64} - 4q^{65} + 40q^{66} - 48q^{70} - 72q^{71} - 4q^{74} - 8q^{75} - 28q^{76} - 56q^{79} - 16q^{81} + 8q^{84} + 4q^{85} - 48q^{86} + 12q^{89} + 12q^{91} + 24q^{94} - 24q^{95} - 8q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(390, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
390.2.y.a $$4$$ $$3.114$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+\zeta_{12}q^{2}-\zeta_{12}q^{3}+\zeta_{12}^{2}q^{4}+(-1+\cdots)q^{5}+\cdots$$
390.2.y.b $$4$$ $$3.114$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$-2$$ $$-6$$ $$q+\zeta_{12}q^{2}-\zeta_{12}q^{3}+\zeta_{12}^{2}q^{4}+(-1+\cdots)q^{5}+\cdots$$
390.2.y.c $$4$$ $$3.114$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$-2$$ $$6$$ $$q+\zeta_{12}q^{2}-\zeta_{12}q^{3}+\zeta_{12}^{2}q^{4}+(2\zeta_{12}+\cdots)q^{5}+\cdots$$
390.2.y.d $$4$$ $$3.114$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+\zeta_{12}q^{2}+\zeta_{12}q^{3}+\zeta_{12}^{2}q^{4}+(1+\cdots)q^{5}+\cdots$$
390.2.y.e $$4$$ $$3.114$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$8$$ $$0$$ $$q+\zeta_{12}q^{2}-\zeta_{12}q^{3}+\zeta_{12}^{2}q^{4}+(2+\cdots)q^{5}+\cdots$$
390.2.y.f $$4$$ $$3.114$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$8$$ $$0$$ $$q+\zeta_{12}q^{2}+\zeta_{12}q^{3}+\zeta_{12}^{2}q^{4}+(2+\cdots)q^{5}+\cdots$$
390.2.y.g $$8$$ $$3.114$$ 8.0.303595776.1 None $$0$$ $$0$$ $$-12$$ $$0$$ $$q-\beta _{5}q^{2}-\beta _{5}q^{3}+(1+\beta _{4})q^{4}+(-2+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(390, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(390, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(65, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(130, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(195, [\chi])$$$$^{\oplus 2}$$