# Properties

 Label 390.2.i Level $390$ Weight $2$ Character orbit 390.i Rep. character $\chi_{390}(61,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $16$ 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.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$7$$ Sturm bound: $$168$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

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

Total New Old
Modular forms 184 16 168
Cusp forms 152 16 136
Eisenstein series 32 0 32

## Trace form

 $$16q - 8q^{4} - 8q^{9} + O(q^{10})$$ $$16q - 8q^{4} - 8q^{9} - 4q^{10} + 12q^{11} - 8q^{13} + 8q^{14} - 8q^{16} + 4q^{19} - 16q^{22} + 16q^{23} + 16q^{25} - 8q^{29} - 4q^{30} + 8q^{31} + 16q^{34} + 4q^{35} - 8q^{36} + 8q^{37} - 12q^{39} + 8q^{40} - 24q^{41} - 8q^{42} + 8q^{43} - 24q^{44} + 4q^{46} + 16q^{47} - 12q^{49} - 8q^{52} + 48q^{53} - 4q^{55} - 4q^{56} - 16q^{57} - 8q^{58} - 16q^{59} - 8q^{61} - 8q^{62} + 16q^{64} + 12q^{65} - 8q^{66} - 24q^{67} - 8q^{69} + 40q^{71} - 48q^{73} + 4q^{74} + 4q^{76} - 16q^{77} + 16q^{78} + 24q^{79} - 8q^{81} - 24q^{82} + 48q^{83} + 48q^{86} - 8q^{87} - 16q^{88} + 36q^{89} + 8q^{90} - 4q^{91} - 32q^{92} - 8q^{94} + 16q^{95} - 16q^{97} + 16q^{98} - 24q^{99} + O(q^{100})$$

## 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.i.a $$2$$ $$3.114$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$1$$ $$-2$$ $$-2$$ $$q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$$
390.2.i.b $$2$$ $$3.114$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$1$$ $$2$$ $$3$$ $$q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$$
390.2.i.c $$2$$ $$3.114$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$-1$$ $$-2$$ $$-2$$ $$q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$$
390.2.i.d $$2$$ $$3.114$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$-1$$ $$-2$$ $$5$$ $$q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$$
390.2.i.e $$2$$ $$3.114$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$1$$ $$-2$$ $$-3$$ $$q+(1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$$
390.2.i.f $$2$$ $$3.114$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$1$$ $$2$$ $$2$$ $$q+(1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$$
390.2.i.g $$4$$ $$3.114$$ $$\Q(\sqrt{-3}, \sqrt{17})$$ None $$-2$$ $$-2$$ $$4$$ $$-3$$ $$q+(-1+\beta _{2})q^{2}+(-1+\beta _{2})q^{3}-\beta _{2}q^{4}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(390, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(26, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(39, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(65, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(78, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(130, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(195, [\chi])$$$$^{\oplus 2}$$