# Properties

 Label 390.2.b Level $390$ Weight $2$ Character orbit 390.b Rep. character $\chi_{390}(181,\cdot)$ Character field $\Q$ Dimension $12$ Newform subspaces $4$ Sturm bound $168$ Trace bound $13$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$168$$ Trace bound: $$13$$ 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

 $$12 q - 4 q^{3} - 12 q^{4} + 12 q^{9} + O(q^{10})$$ $$12 q - 4 q^{3} - 12 q^{4} + 12 q^{9} + 4 q^{10} + 4 q^{12} + 4 q^{13} + 12 q^{16} + 8 q^{17} - 12 q^{25} + 4 q^{26} - 4 q^{27} + 8 q^{29} + 4 q^{30} + 8 q^{35} - 12 q^{36} - 24 q^{38} + 8 q^{39} - 4 q^{40} - 8 q^{42} + 16 q^{43} - 4 q^{48} - 44 q^{49} + 8 q^{51} - 4 q^{52} - 8 q^{53} + 24 q^{61} - 8 q^{62} - 12 q^{64} - 4 q^{65} + 8 q^{66} - 8 q^{68} - 8 q^{69} + 8 q^{74} + 4 q^{75} - 64 q^{77} + 8 q^{78} + 12 q^{81} + 8 q^{82} + 8 q^{87} + 4 q^{90} + 40 q^{91} + 16 q^{94} + 32 q^{95} + O(q^{100})$$

## 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.b.a $2$ $3.114$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q-iq^{2}-q^{3}-q^{4}+iq^{5}+iq^{6}+iq^{8}+\cdots$$
390.2.b.b $2$ $3.114$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q-iq^{2}-q^{3}-q^{4}+iq^{5}+iq^{6}+iq^{8}+\cdots$$
390.2.b.c $4$ $3.114$ $$\Q(i, \sqrt{13})$$ None $$0$$ $$-4$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-q^{3}-q^{4}+\beta _{1}q^{5}-\beta _{1}q^{6}+\cdots$$
390.2.b.d $4$ $3.114$ $$\Q(i, \sqrt{17})$$ None $$0$$ $$4$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}+q^{3}-q^{4}+\beta _{1}q^{5}-\beta _{1}q^{6}+\cdots$$

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

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