# Properties

 Label 450.2.h Level $450$ Weight $2$ Character orbit 450.h Rep. character $\chi_{450}(91,\cdot)$ Character field $\Q(\zeta_{5})$ Dimension $52$ Newform subspaces $7$ 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.h (of order $$5$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$25$$ Character field: $$\Q(\zeta_{5})$$ Newform subspaces: $$7$$ Sturm bound: $$180$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

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

Total New Old
Modular forms 392 52 340
Cusp forms 328 52 276
Eisenstein series 64 0 64

## Trace form

 $$52q - q^{2} - 13q^{4} - 11q^{5} - q^{8} + O(q^{10})$$ $$52q - q^{2} - 13q^{4} - 11q^{5} - q^{8} + q^{10} + 8q^{11} + 6q^{13} + 4q^{14} - 13q^{16} + 16q^{17} - 14q^{19} + 4q^{20} - 8q^{22} + 30q^{23} + 7q^{25} - 30q^{26} - 10q^{28} + 12q^{29} - 18q^{31} + 4q^{32} + 5q^{34} + 14q^{35} - 13q^{37} + 4q^{38} + q^{40} + 20q^{41} + 28q^{43} - 2q^{44} - 16q^{46} - 30q^{47} + 68q^{49} - q^{50} + 6q^{52} + 39q^{53} + 38q^{55} + 4q^{56} - 18q^{58} - 40q^{61} - 8q^{62} - 13q^{64} - 47q^{65} + 2q^{67} - 54q^{68} - 12q^{70} - 2q^{71} - 62q^{73} - 54q^{74} + 16q^{76} - 40q^{77} - 40q^{79} - q^{80} - 86q^{82} - 18q^{83} + 13q^{85} - 2q^{86} + 2q^{88} - q^{89} - 36q^{91} - 30q^{92} + 8q^{94} + 44q^{95} + 14q^{97} + 31q^{98} + O(q^{100})$$

## 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.h.a $$4$$ $$3.593$$ $$\Q(\zeta_{10})$$ None $$-1$$ $$0$$ $$-5$$ $$-12$$ $$q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots$$
450.2.h.b $$4$$ $$3.593$$ $$\Q(\zeta_{10})$$ None $$-1$$ $$0$$ $$-5$$ $$8$$ $$q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots$$
450.2.h.c $$4$$ $$3.593$$ $$\Q(\zeta_{10})$$ None $$1$$ $$0$$ $$-5$$ $$6$$ $$q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}-\zeta_{10}^{3}q^{4}+\cdots$$
450.2.h.d $$8$$ $$3.593$$ 8.0.1064390625.3 None $$-2$$ $$0$$ $$4$$ $$-2$$ $$q-\beta _{3}q^{2}+(-1-\beta _{2}+\beta _{3}-\beta _{5})q^{4}+\cdots$$
450.2.h.e $$8$$ $$3.593$$ 8.0.58140625.2 None $$2$$ $$0$$ $$0$$ $$4$$ $$q+\beta _{6}q^{2}-\beta _{3}q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots$$
450.2.h.f $$12$$ $$3.593$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$-3$$ $$0$$ $$1$$ $$-2$$ $$q-\beta _{7}q^{2}+(-1+\beta _{4}-\beta _{6}+\beta _{7})q^{4}+\cdots$$
450.2.h.g $$12$$ $$3.593$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$3$$ $$0$$ $$-1$$ $$-2$$ $$q+\beta _{7}q^{2}+(-1+\beta _{4}-\beta _{6}+\beta _{7})q^{4}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(450, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 2}$$