# Properties

 Label 690.2.q Level $690$ Weight $2$ Character orbit 690.q Rep. character $\chi_{690}(11,\cdot)$ Character field $\Q(\zeta_{22})$ Dimension $320$ Newform subspaces $2$ Sturm bound $288$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 690.q (of order $$22$$ and degree $$10$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$69$$ Character field: $$\Q(\zeta_{22})$$ Newform subspaces: $$2$$ Sturm bound: $$288$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(690, [\chi])$$.

Total New Old
Modular forms 1520 320 1200
Cusp forms 1360 320 1040
Eisenstein series 160 0 160

## Trace form

 $$320q + 32q^{4} - 4q^{6} - 4q^{9} + O(q^{10})$$ $$320q + 32q^{4} - 4q^{6} - 4q^{9} - 32q^{16} - 16q^{18} + 132q^{21} + 4q^{24} - 32q^{25} + 84q^{27} - 8q^{31} + 4q^{36} + 16q^{39} + 88q^{43} + 8q^{46} + 140q^{49} + 30q^{54} + 20q^{55} - 132q^{57} - 72q^{58} - 88q^{61} - 220q^{63} + 32q^{64} - 88q^{66} - 88q^{67} - 8q^{69} - 24q^{70} - 72q^{72} - 56q^{73} - 108q^{78} - 88q^{79} - 132q^{81} - 56q^{82} - 22q^{84} + 92q^{87} + 8q^{93} + 48q^{94} - 4q^{96} + 132q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(690, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
690.2.q.a $$160$$ $$5.510$$ None $$0$$ $$0$$ $$-16$$ $$0$$
690.2.q.b $$160$$ $$5.510$$ None $$0$$ $$0$$ $$16$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(690, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(690, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(69, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(138, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(345, [\chi])$$$$^{\oplus 2}$$