# Properties

 Label 390.2.bn Level $390$ Weight $2$ Character orbit 390.bn Rep. character $\chi_{390}(67,\cdot)$ Character field $\Q(\zeta_{12})$ Dimension $56$ Newform subspaces $3$ 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.bn (of order $$12$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$65$$ Character field: $$\Q(\zeta_{12})$$ Newform subspaces: $$3$$ 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 368 56 312
Cusp forms 304 56 248
Eisenstein series 64 0 64

## Trace form

 $$56 q - 4 q^{2} - 28 q^{4} + 8 q^{8} + O(q^{10})$$ $$56 q - 4 q^{2} - 28 q^{4} + 8 q^{8} - 8 q^{11} + 16 q^{13} - 8 q^{15} - 28 q^{16} + 4 q^{17} + 40 q^{19} + 16 q^{21} - 16 q^{23} - 36 q^{25} + 24 q^{31} - 4 q^{32} + 8 q^{33} - 36 q^{34} + 12 q^{37} - 8 q^{39} - 12 q^{41} + 8 q^{43} - 8 q^{44} + 8 q^{45} + 8 q^{46} + 36 q^{49} - 40 q^{50} - 8 q^{52} + 12 q^{53} + 56 q^{55} - 24 q^{56} - 48 q^{58} + 8 q^{59} - 8 q^{60} - 32 q^{61} - 72 q^{62} + 56 q^{64} - 92 q^{65} + 16 q^{66} - 72 q^{67} - 8 q^{68} - 16 q^{69} - 48 q^{70} - 32 q^{71} - 24 q^{73} + 60 q^{74} - 32 q^{76} + 48 q^{77} + 16 q^{78} + 28 q^{81} + 40 q^{82} + 16 q^{84} + 100 q^{85} + 48 q^{87} - 36 q^{89} + 4 q^{90} - 48 q^{91} + 32 q^{92} + 24 q^{94} + 48 q^{95} + 4 q^{98} - 8 q^{99} + O(q^{100})$$

## 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.bn.a $8$ $3.114$ $$\Q(\zeta_{24})$$ None $$4$$ $$0$$ $$0$$ $$-12$$ $$q+\zeta_{24}^{4}q^{2}+\zeta_{24}q^{3}+(-1+\zeta_{24}^{4}+\cdots)q^{4}+\cdots$$
390.2.bn.b $16$ $3.114$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$8$$ $$0$$ $$0$$ $$24$$ $$q+(1+\beta _{14})q^{2}-\beta _{15}q^{3}+\beta _{14}q^{4}+\cdots$$
390.2.bn.c $32$ $3.114$ None $$-16$$ $$0$$ $$0$$ $$-12$$

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

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