# Properties

 Label 650.2.n Level $650$ Weight $2$ Character orbit 650.n Rep. character $\chi_{650}(49,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $40$ Newform subspaces $6$ Sturm bound $210$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$650 = 2 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 650.n (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$65$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$6$$ Sturm bound: $$210$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(650, [\chi])$$.

Total New Old
Modular forms 232 40 192
Cusp forms 184 40 144
Eisenstein series 48 0 48

## Trace form

 $$40q - 20q^{4} + 8q^{9} + O(q^{10})$$ $$40q - 20q^{4} + 8q^{9} + 12q^{11} + 24q^{14} - 20q^{16} + 24q^{19} + 12q^{26} + 8q^{36} + 92q^{39} - 96q^{41} - 24q^{46} - 4q^{49} + 48q^{51} - 36q^{54} - 12q^{56} - 84q^{59} + 28q^{61} + 40q^{64} - 48q^{66} + 12q^{69} - 36q^{74} - 24q^{76} - 40q^{79} - 44q^{81} - 36q^{84} + 96q^{89} - 84q^{91} + 12q^{94} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(650, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
650.2.n.a $$4$$ $$5.190$$ $$\Q(\zeta_{12})$$ None $$-2$$ $$-6$$ $$0$$ $$-6$$ $$q+(-1+\zeta_{12}^{2})q^{2}+(-1+\zeta_{12}-\zeta_{12}^{2}+\cdots)q^{3}+\cdots$$
650.2.n.b $$4$$ $$5.190$$ $$\Q(\zeta_{12})$$ None $$2$$ $$6$$ $$0$$ $$6$$ $$q+\zeta_{12}^{2}q^{2}+(2-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots$$
650.2.n.c $$8$$ $$5.190$$ 8.0.22581504.2 None $$-4$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{6}q^{2}+(-1+\beta _{1}+\beta _{2}+\beta _{4}-\beta _{5}+\cdots)q^{3}+\cdots$$
650.2.n.d $$8$$ $$5.190$$ 8.0.22581504.2 None $$-4$$ $$6$$ $$0$$ $$0$$ $$q+(-1-\beta _{6})q^{2}+(1+\beta _{2})q^{3}+\beta _{6}q^{4}+\cdots$$
650.2.n.e $$8$$ $$5.190$$ 8.0.22581504.2 None $$4$$ $$-6$$ $$0$$ $$0$$ $$q+(1-\beta _{2})q^{2}+(\beta _{4}+\beta _{6})q^{3}-\beta _{2}q^{4}+\cdots$$
650.2.n.f $$8$$ $$5.190$$ 8.0.22581504.2 None $$4$$ $$0$$ $$0$$ $$0$$ $$q+(1-\beta _{6})q^{2}+(\beta _{3}+\beta _{5}+\beta _{7})q^{3}-\beta _{6}q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(650, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(650, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(65, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(130, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(325, [\chi])$$$$^{\oplus 2}$$