# Properties

 Label 390.2.t Level $390$ Weight $2$ Character orbit 390.t Rep. character $\chi_{390}(307,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $28$ Newform subspaces $2$ Sturm bound $168$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.t (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$65$$ Character field: $$\Q(i)$$ Newform subspaces: $$2$$ Sturm bound: $$168$$ Trace bound: $$1$$ 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 28 156
Cusp forms 152 28 124
Eisenstein series 32 0 32

## Trace form

 $$28 q - 28 q^{4} - 4 q^{5} + O(q^{10})$$ $$28 q - 28 q^{4} - 4 q^{5} + 8 q^{11} + 12 q^{13} - 8 q^{15} + 28 q^{16} + 4 q^{17} + 4 q^{18} - 8 q^{19} + 4 q^{20} - 16 q^{21} - 16 q^{23} + 12 q^{25} + 24 q^{31} - 12 q^{34} - 48 q^{37} - 8 q^{39} + 12 q^{41} + 32 q^{43} - 8 q^{44} + 16 q^{46} + 48 q^{47} + 60 q^{49} + 16 q^{50} - 12 q^{52} - 60 q^{53} - 32 q^{55} - 40 q^{58} + 8 q^{59} + 8 q^{60} - 16 q^{61} - 28 q^{64} - 60 q^{65} - 16 q^{66} - 4 q^{68} - 16 q^{69} - 56 q^{70} - 16 q^{71} - 4 q^{72} + 8 q^{76} + 48 q^{77} + 16 q^{78} - 4 q^{80} - 28 q^{81} + 4 q^{82} + 48 q^{83} + 16 q^{84} - 28 q^{85} + 48 q^{87} + 12 q^{89} + 4 q^{90} + 48 q^{91} + 16 q^{92} - 48 q^{95} - 8 q^{99} + O(q^{100})$$

## 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.t.a $12$ $3.114$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q-\beta _{4}q^{2}-\beta _{7}q^{3}-q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots$$
390.2.t.b $16$ $3.114$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{4}q^{2}+\beta _{8}q^{3}-q^{4}-\beta _{10}q^{5}+\beta _{5}q^{6}+\cdots$$

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

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