# Properties

 Label 390.2.x Level $390$ Weight $2$ Character orbit 390.x Rep. character $\chi_{390}(49,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $24$ Newform subspaces $2$ Sturm bound $168$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.x (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$65$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$2$$ Sturm bound: $$168$$ Trace bound: $$2$$ 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 24 160
Cusp forms 152 24 128
Eisenstein series 32 0 32

## Trace form

 $$24 q - 12 q^{4} + 12 q^{9} + O(q^{10})$$ $$24 q - 12 q^{4} + 12 q^{9} + 2 q^{10} + 12 q^{11} + 8 q^{14} + 12 q^{15} - 12 q^{16} - 12 q^{19} + 6 q^{20} - 20 q^{25} - 4 q^{26} + 28 q^{29} + 4 q^{35} + 12 q^{36} - 4 q^{39} - 4 q^{40} - 36 q^{41} - 6 q^{45} - 12 q^{46} + 16 q^{49} - 30 q^{50} + 32 q^{51} - 20 q^{55} - 4 q^{56} - 72 q^{59} + 20 q^{61} + 24 q^{64} - 38 q^{65} - 24 q^{66} + 32 q^{69} - 24 q^{71} - 24 q^{74} + 8 q^{75} + 12 q^{76} + 8 q^{79} - 6 q^{80} - 12 q^{81} + 24 q^{84} + 66 q^{85} + 36 q^{89} + 4 q^{90} + 4 q^{91} - 16 q^{94} - 24 q^{95} + O(q^{100})$$

## 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.x.a $12$ $3.114$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$-6$$ $$0$$ $$-2$$ $$-2$$ $$q+(-1-\beta _{6})q^{2}+\beta _{4}q^{3}+\beta _{6}q^{4}+(\beta _{3}+\cdots)q^{5}+\cdots$$
390.2.x.b $12$ $3.114$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$6$$ $$0$$ $$2$$ $$2$$ $$q+(1+\beta _{6})q^{2}-\beta _{4}q^{3}+\beta _{6}q^{4}+(\beta _{3}+\cdots)q^{5}+\cdots$$

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

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