# Properties

 Label 450.2.e Level $450$ Weight $2$ Character orbit 450.e Rep. character $\chi_{450}(151,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $38$ Newform subspaces $14$ Sturm bound $180$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 450.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$14$$ Sturm bound: $$180$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(450, [\chi])$$.

Total New Old
Modular forms 204 38 166
Cusp forms 156 38 118
Eisenstein series 48 0 48

## Trace form

 $$38q - q^{2} + q^{3} - 19q^{4} - 5q^{6} - 2q^{7} + 2q^{8} - 5q^{9} + O(q^{10})$$ $$38q - q^{2} + q^{3} - 19q^{4} - 5q^{6} - 2q^{7} + 2q^{8} - 5q^{9} - q^{11} + 4q^{12} - 2q^{13} + 2q^{14} - 19q^{16} + 18q^{17} + 2q^{18} - 2q^{19} - 2q^{21} + 3q^{22} + 6q^{23} + q^{24} - 20q^{26} + 16q^{27} + 4q^{28} + 18q^{29} - 8q^{31} - q^{32} + 15q^{33} + 3q^{34} - 5q^{36} + 16q^{37} - 11q^{38} - 24q^{39} - 13q^{41} - 20q^{42} + q^{43} + 2q^{44} + 12q^{46} - 18q^{47} - 5q^{48} - 27q^{49} + 37q^{51} - 2q^{52} - 48q^{53} - 17q^{54} + 2q^{56} + 23q^{57} + 6q^{58} + 43q^{59} - 20q^{61} + 16q^{62} - 28q^{63} + 38q^{64} + 26q^{66} - 17q^{67} - 9q^{68} - 8q^{69} - 24q^{71} - 13q^{72} + 22q^{73} + 8q^{74} + q^{76} - 42q^{77} + 10q^{78} - 8q^{79} - 41q^{81} - 30q^{82} + 24q^{83} - 14q^{84} + 35q^{86} - 54q^{87} + 3q^{88} + 92q^{89} + 56q^{91} + 6q^{92} + 20q^{93} - 6q^{94} + 4q^{96} + 7q^{97} + 42q^{98} - 88q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(450, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
450.2.e.a $$2$$ $$3.593$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$-3$$ $$0$$ $$-4$$ $$q+(-1+\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$$
450.2.e.b $$2$$ $$3.593$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$-3$$ $$0$$ $$1$$ $$q+(-1+\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$$
450.2.e.c $$2$$ $$3.593$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$0$$ $$-2$$ $$q+(-1+\zeta_{6})q^{2}+(-1+2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$$
450.2.e.d $$2$$ $$3.593$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$3$$ $$0$$ $$-4$$ $$q+(-1+\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$$
450.2.e.e $$2$$ $$3.593$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$-3$$ $$0$$ $$-1$$ $$q+(1-\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$$
450.2.e.f $$2$$ $$3.593$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$0$$ $$2$$ $$q+(1-\zeta_{6})q^{2}+(1-2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$$
450.2.e.g $$2$$ $$3.593$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$3$$ $$0$$ $$-1$$ $$q+(1-\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$$
450.2.e.h $$2$$ $$3.593$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$3$$ $$0$$ $$4$$ $$q+(1-\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$$
450.2.e.i $$2$$ $$3.593$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$3$$ $$0$$ $$2$$ $$q+(1-\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$$
450.2.e.j $$4$$ $$3.593$$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$-2$$ $$-2$$ $$0$$ $$1$$ $$q+(-1+\beta _{2})q^{2}+(-\beta _{1}+\beta _{3})q^{3}-\beta _{2}q^{4}+\cdots$$
450.2.e.k $$4$$ $$3.593$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$-2$$ $$2$$ $$0$$ $$4$$ $$q-\beta _{2}q^{2}+(-\beta _{1}+\beta _{2}+\beta _{3})q^{3}+(-1+\cdots)q^{4}+\cdots$$
450.2.e.l $$4$$ $$3.593$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$-2$$ $$4$$ $$0$$ $$4$$ $$q+(-1+\beta _{2})q^{2}+(1-\beta _{3})q^{3}-\beta _{2}q^{4}+\cdots$$
450.2.e.m $$4$$ $$3.593$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$2$$ $$-4$$ $$0$$ $$-4$$ $$q+(1-\beta _{2})q^{2}+(-1+\beta _{3})q^{3}-\beta _{2}q^{4}+\cdots$$
450.2.e.n $$4$$ $$3.593$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$2$$ $$-2$$ $$0$$ $$-4$$ $$q+\beta _{2}q^{2}+(-\beta _{1}-\beta _{2}+\beta _{3})q^{3}+(-1+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(450, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(450, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 2}$$