# Properties

 Label 170.2.g Level $170$ Weight $2$ Character orbit 170.g Rep. character $\chi_{170}(89,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $16$ Newform subspaces $6$ Sturm bound $54$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$170 = 2 \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 170.g (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$85$$ Character field: $$\Q(i)$$ Newform subspaces: $$6$$ Sturm bound: $$54$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$3$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(170, [\chi])$$.

Total New Old
Modular forms 64 16 48
Cusp forms 48 16 32
Eisenstein series 16 0 16

## Trace form

 $$16q + 16q^{4} + 4q^{5} + O(q^{10})$$ $$16q + 16q^{4} + 4q^{5} - 4q^{10} - 16q^{11} + 16q^{16} + 4q^{20} - 32q^{21} + 8q^{29} - 8q^{34} - 40q^{35} - 24q^{39} - 4q^{40} - 16q^{44} + 12q^{45} - 24q^{46} - 8q^{50} - 24q^{51} - 24q^{54} - 8q^{55} + 8q^{61} + 16q^{64} + 32q^{65} + 80q^{69} + 48q^{71} + 16q^{74} - 56q^{75} + 40q^{79} + 4q^{80} + 48q^{81} - 32q^{84} + 36q^{85} - 32q^{86} + 8q^{89} + 28q^{90} + 40q^{91} + 48q^{95} + 40q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(170, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
170.2.g.a $$2$$ $$1.357$$ $$\Q(\sqrt{-1})$$ None $$-2$$ $$2$$ $$2$$ $$-6$$ $$q-q^{2}+(1-i)q^{3}+q^{4}+(1-2i)q^{5}+\cdots$$
170.2.g.b $$2$$ $$1.357$$ $$\Q(\sqrt{-1})$$ None $$-2$$ $$2$$ $$2$$ $$2$$ $$q-q^{2}+(1-i)q^{3}+q^{4}+(1+2i)q^{5}+\cdots$$
170.2.g.c $$2$$ $$1.357$$ $$\Q(\sqrt{-1})$$ None $$2$$ $$-2$$ $$-4$$ $$6$$ $$q+q^{2}+(-1+i)q^{3}+q^{4}+(-2+i)q^{5}+\cdots$$
170.2.g.d $$2$$ $$1.357$$ $$\Q(\sqrt{-1})$$ None $$2$$ $$-2$$ $$4$$ $$-2$$ $$q+q^{2}+(-1+i)q^{3}+q^{4}+(2+i)q^{5}+\cdots$$
170.2.g.e $$4$$ $$1.357$$ $$\Q(\zeta_{8})$$ None $$-4$$ $$-4$$ $$0$$ $$4$$ $$q-q^{2}+(-1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+q^{4}+\cdots$$
170.2.g.f $$4$$ $$1.357$$ $$\Q(\zeta_{8})$$ None $$4$$ $$4$$ $$0$$ $$-4$$ $$q+q^{2}+(1-\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+q^{4}+(\zeta_{8}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(170, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(170, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(85, [\chi])$$$$^{\oplus 2}$$