# Properties

 Label 315.2.j Level $315$ Weight $2$ Character orbit 315.j Rep. character $\chi_{315}(46,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $28$ Newform subspaces $7$ Sturm bound $96$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 315.j (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$7$$ Sturm bound: $$96$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(315, [\chi])$$.

Total New Old
Modular forms 112 28 84
Cusp forms 80 28 52
Eisenstein series 32 0 32

## Trace form

 $$28q - 2q^{2} - 18q^{4} + 2q^{5} + 6q^{7} + 12q^{8} + O(q^{10})$$ $$28q - 2q^{2} - 18q^{4} + 2q^{5} + 6q^{7} + 12q^{8} - 2q^{10} + 16q^{13} + 8q^{14} - 22q^{16} + 12q^{17} - 4q^{19} - 20q^{20} + 24q^{22} - 10q^{23} - 14q^{25} - 4q^{26} - 14q^{28} + 12q^{29} + 8q^{31} - 14q^{32} - 40q^{34} - 4q^{35} - 12q^{37} - 24q^{38} - 6q^{40} + 12q^{41} - 36q^{43} - 16q^{44} + 22q^{46} + 12q^{47} - 10q^{49} + 4q^{50} - 28q^{52} + 8q^{53} + 24q^{55} + 18q^{56} + 10q^{58} - 12q^{59} + 46q^{61} + 116q^{64} + 8q^{65} + 22q^{67} + 8q^{68} + 14q^{70} - 16q^{71} - 32q^{73} + 4q^{74} - 48q^{76} - 72q^{77} - 4q^{79} + 22q^{80} + 2q^{82} - 12q^{83} + 8q^{85} - 14q^{86} - 76q^{88} - 22q^{89} - 24q^{91} + 76q^{92} + 36q^{94} - 24q^{97} + 14q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(315, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
315.2.j.a $$2$$ $$2.515$$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$1$$ $$-1$$ $$q-2\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{4}+\zeta_{6}q^{5}+\cdots$$
315.2.j.b $$2$$ $$2.515$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$5$$ $$q+(2-2\zeta_{6})q^{4}-\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots$$
315.2.j.c $$4$$ $$2.515$$ $$\Q(\zeta_{12})$$ None $$-2$$ $$0$$ $$2$$ $$0$$ $$q+(\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(-2+2\zeta_{12}+\cdots)q^{4}+\cdots$$
315.2.j.d $$4$$ $$2.515$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$-2$$ $$-2$$ $$q+\beta _{1}q^{2}+(-1-\beta _{2})q^{5}+(-1-\beta _{1}+\cdots)q^{7}+\cdots$$
315.2.j.e $$4$$ $$2.515$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$2$$ $$0$$ $$2$$ $$2$$ $$q+(1+\beta _{1}+\beta _{2})q^{2}+(2\beta _{1}+\beta _{2}+2\beta _{3})q^{4}+\cdots$$
315.2.j.f $$6$$ $$2.515$$ 6.0.4406832.1 None $$-2$$ $$0$$ $$-3$$ $$1$$ $$q+(-\beta _{1}+\beta _{2}-\beta _{4})q^{2}+(-2+\beta _{1}+\cdots)q^{4}+\cdots$$
315.2.j.g $$6$$ $$2.515$$ 6.0.4406832.1 None $$2$$ $$0$$ $$3$$ $$1$$ $$q+(\beta _{1}-\beta _{2}+\beta _{4})q^{2}+(-2+\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(315, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(315, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$