# Properties

 Label 690.2.h Level $690$ Weight $2$ Character orbit 690.h Rep. character $\chi_{690}(689,\cdot)$ Character field $\Q$ Dimension $48$ Newform subspaces $2$ Sturm bound $288$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 690.h (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$345$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$288$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$47$$

## Dimensions

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

Total New Old
Modular forms 152 48 104
Cusp forms 136 48 88
Eisenstein series 16 0 16

## Trace form

 $$48q + 48q^{4} - 4q^{6} + 12q^{9} + O(q^{10})$$ $$48q + 48q^{4} - 4q^{6} + 12q^{9} + 48q^{16} - 4q^{24} + 24q^{25} - 56q^{31} + 12q^{36} + 8q^{46} - 8q^{49} - 4q^{54} + 8q^{55} + 48q^{64} - 16q^{69} - 16q^{70} + 8q^{75} + 28q^{81} - 88q^{85} - 32q^{94} - 4q^{96} + O(q^{100})$$

## 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.h.a $$24$$ $$5.510$$ None $$-24$$ $$2$$ $$0$$ $$0$$
690.2.h.b $$24$$ $$5.510$$ None $$24$$ $$-2$$ $$0$$ $$0$$

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

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