# Properties

 Label 390.2.e Level $390$ Weight $2$ Character orbit 390.e Rep. character $\chi_{390}(79,\cdot)$ Character field $\Q$ Dimension $12$ Newform subspaces $5$ Sturm bound $168$ Trace bound $10$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.e (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$168$$ Trace bound: $$10$$ 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 92 12 80
Cusp forms 76 12 64
Eisenstein series 16 0 16

## Trace form

 $$12 q - 12 q^{4} + 8 q^{5} - 4 q^{6} - 12 q^{9} + O(q^{10})$$ $$12 q - 12 q^{4} + 8 q^{5} - 4 q^{6} - 12 q^{9} + 4 q^{10} - 16 q^{11} + 4 q^{15} + 12 q^{16} + 8 q^{19} - 8 q^{20} - 8 q^{21} + 4 q^{24} + 4 q^{25} + 8 q^{30} + 8 q^{31} + 16 q^{34} - 24 q^{35} + 12 q^{36} - 4 q^{40} + 16 q^{41} + 16 q^{44} - 8 q^{45} - 32 q^{46} - 12 q^{49} + 8 q^{51} + 4 q^{54} + 16 q^{55} + 16 q^{59} - 4 q^{60} - 40 q^{61} - 12 q^{64} - 8 q^{65} - 8 q^{69} - 8 q^{70} - 16 q^{74} - 8 q^{76} + 32 q^{79} + 8 q^{80} + 12 q^{81} + 8 q^{84} - 24 q^{85} + 32 q^{86} + 32 q^{89} - 4 q^{90} - 16 q^{94} + 24 q^{95} - 4 q^{96} + 16 q^{99} + O(q^{100})$$

## 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.e.a $2$ $3.114$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+iq^{2}+iq^{3}-q^{4}+(-2-i)q^{5}+\cdots$$
390.2.e.b $2$ $3.114$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+iq^{2}+iq^{3}-q^{4}+(2-i)q^{5}-q^{6}+\cdots$$
390.2.e.c $2$ $3.114$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+iq^{2}-iq^{3}-q^{4}+(2+i)q^{5}+q^{6}+\cdots$$
390.2.e.d $2$ $3.114$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+iq^{2}-iq^{3}-q^{4}+(2-i)q^{5}+q^{6}+\cdots$$
390.2.e.e $4$ $3.114$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{1}q^{3}-q^{4}-\beta _{2}q^{5}-q^{6}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(390, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(65, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(130, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(195, [\chi])$$$$^{\oplus 2}$$