# Properties

 Label 315.5.e Level $315$ Weight $5$ Character orbit 315.e Rep. character $\chi_{315}(244,\cdot)$ Character field $\Q$ Dimension $78$ Newform subspaces $7$ Sturm bound $240$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 315.e (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$35$$ Character field: $$\Q$$ Newform subspaces: $$7$$ Sturm bound: $$240$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$2$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(315, [\chi])$$.

Total New Old
Modular forms 200 82 118
Cusp forms 184 78 106
Eisenstein series 16 4 12

## Trace form

 $$78 q - 612 q^{4} + O(q^{10})$$ $$78 q - 612 q^{4} - 2 q^{11} - 228 q^{14} + 5220 q^{16} - 846 q^{25} + 742 q^{29} + 1454 q^{35} - 3944 q^{44} + 6504 q^{46} - 1638 q^{49} - 1404 q^{50} + 13968 q^{56} - 20868 q^{64} + 2314 q^{65} + 4236 q^{70} + 9148 q^{71} - 10152 q^{74} - 14190 q^{79} + 18450 q^{85} - 74712 q^{86} - 55302 q^{91} + 31068 q^{95} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(315, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
315.5.e.a $$1$$ $$32.562$$ $$\Q$$ $$\Q(\sqrt{-35})$$ $$0$$ $$0$$ $$-25$$ $$49$$ $$q+2^{4}q^{4}-5^{2}q^{5}+7^{2}q^{7}+73q^{11}+\cdots$$
315.5.e.b $$1$$ $$32.562$$ $$\Q$$ $$\Q(\sqrt{-35})$$ $$0$$ $$0$$ $$25$$ $$-49$$ $$q+2^{4}q^{4}+5^{2}q^{5}-7^{2}q^{7}+73q^{11}+\cdots$$
315.5.e.c $$2$$ $$32.562$$ $$\Q(\sqrt{-6})$$ None $$0$$ $$0$$ $$-10$$ $$70$$ $$q+\beta q^{2}+10q^{4}+(-5-10\beta )q^{5}+(35+\cdots)q^{7}+\cdots$$
315.5.e.d $$2$$ $$32.562$$ $$\Q(\sqrt{-6})$$ None $$0$$ $$0$$ $$10$$ $$-70$$ $$q+\beta q^{2}+10q^{4}+(5+10\beta )q^{5}+(-35+\cdots)q^{7}+\cdots$$
315.5.e.e $$8$$ $$32.562$$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{4}q^{2}+(-22+\beta _{2})q^{4}+(-\beta _{1}-\beta _{7})q^{5}+\cdots$$
315.5.e.f $$32$$ $$32.562$$ None $$0$$ $$0$$ $$0$$ $$0$$
315.5.e.g $$32$$ $$32.562$$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{5}^{\mathrm{old}}(315, [\chi])$$ into lower level spaces

$$S_{5}^{\mathrm{old}}(315, [\chi]) \cong$$ $$S_{5}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$