# Properties

 Label 650.2.o Level $650$ Weight $2$ Character orbit 650.o Rep. character $\chi_{650}(399,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $44$ Newform subspaces $8$ Sturm bound $210$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$650 = 2 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 650.o (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$65$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$8$$ Sturm bound: $$210$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$3$$, $$7$$

## Dimensions

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

Total New Old
Modular forms 236 44 192
Cusp forms 188 44 144
Eisenstein series 48 0 48

## Trace form

 $$44q + 22q^{4} + 34q^{9} + O(q^{10})$$ $$44q + 22q^{4} + 34q^{9} + 12q^{11} - 24q^{14} - 22q^{16} + 16q^{19} + 8q^{21} - 14q^{26} + 22q^{29} + 32q^{31} + 52q^{34} - 34q^{36} - 20q^{39} + 10q^{41} + 24q^{44} + 14q^{49} - 16q^{51} - 12q^{54} - 12q^{56} - 20q^{59} - 2q^{61} - 44q^{64} + 16q^{66} - 36q^{69} - 48q^{71} - 6q^{74} - 16q^{76} - 56q^{79} - 46q^{81} + 4q^{84} - 24q^{86} + 20q^{89} + 76q^{91} + 28q^{94} + 64q^{99} + O(q^{100})$$

## 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.o.a $$4$$ $$5.190$$ $$\Q(\zeta_{12})$$ None $$0$$ $$-6$$ $$0$$ $$12$$ $$q+\zeta_{12}q^{2}+(-1+\zeta_{12}-\zeta_{12}^{2})q^{3}+\cdots$$
650.2.o.b $$4$$ $$5.190$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{2}-2\zeta_{12}q^{3}+\zeta_{12}^{2}q^{4}-2\zeta_{12}^{2}q^{6}+\cdots$$
650.2.o.c $$4$$ $$5.190$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(-4\zeta_{12}+4\zeta_{12}^{3})q^{7}+\cdots$$
650.2.o.d $$4$$ $$5.190$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(3\zeta_{12}-3\zeta_{12}^{3})q^{7}+\cdots$$
650.2.o.e $$4$$ $$5.190$$ $$\Q(\zeta_{12})$$ None $$0$$ $$6$$ $$0$$ $$-12$$ $$q+\zeta_{12}q^{2}+(1+\zeta_{12}+\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots$$
650.2.o.f $$8$$ $$5.190$$ 8.0.49787136.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{1}-\beta _{3})q^{2}+(\beta _{1}-\beta _{2}+\beta _{3}+\beta _{7})q^{3}+\cdots$$
650.2.o.g $$8$$ $$5.190$$ 8.0.3317760000.2 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{3})q^{2}+\beta _{5}q^{3}+(1-\beta _{2})q^{4}+\cdots$$
650.2.o.h $$8$$ $$5.190$$ 8.0.592240896.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{2}-\beta _{1}q^{3}+(1-\beta _{3})q^{4}+(-\beta _{3}+\cdots)q^{6}+\cdots$$

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

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