# Properties

 Label 900.3.l Level $900$ Weight $3$ Character orbit 900.l Rep. character $\chi_{900}(757,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $30$ Newform subspaces $8$ Sturm bound $540$ Trace bound $31$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 900.l (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q(i)$$ Newform subspaces: $$8$$ Sturm bound: $$540$$ Trace bound: $$31$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(900, [\chi])$$.

Total New Old
Modular forms 792 30 762
Cusp forms 648 30 618
Eisenstein series 144 0 144

## Trace form

 $$30q - 2q^{7} + O(q^{10})$$ $$30q - 2q^{7} + 8q^{11} + 6q^{13} - 38q^{17} - 86q^{23} + 36q^{31} + 54q^{37} + 104q^{41} + 126q^{43} - 78q^{47} - 214q^{53} + 124q^{61} + 190q^{67} + 52q^{71} + 182q^{73} - 340q^{77} - 366q^{83} + 588q^{91} - 90q^{97} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(900, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
900.3.l.a $$2$$ $$24.523$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$14$$ $$q+(7+7i)q^{7}-10q^{11}+(-9+9i)q^{13}+\cdots$$
900.3.l.b $$4$$ $$24.523$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$-20$$ $$q+(-5+5\beta _{1}+\beta _{2}-\beta _{3})q^{7}+(4+5\beta _{3})q^{11}+\cdots$$
900.3.l.c $$4$$ $$24.523$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{7}-6q^{11}+5\beta _{3}q^{13}-6\beta _{1}q^{17}+\cdots$$
900.3.l.d $$4$$ $$24.523$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3\beta _{1}q^{7}-6q^{11}-5\beta _{3}q^{13}-18\beta _{1}q^{17}+\cdots$$
900.3.l.e $$4$$ $$24.523$$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+3\beta _{1}q^{7}+7\beta _{3}q^{13}+11\beta _{2}q^{19}+\cdots$$
900.3.l.f $$4$$ $$24.523$$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{7}-\beta _{3}q^{13}+26\beta _{2}q^{19}+46q^{31}+\cdots$$
900.3.l.g $$4$$ $$24.523$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4\beta _{1}q^{7}+15q^{11}-2\beta _{3}q^{13}-3\beta _{1}q^{17}+\cdots$$
900.3.l.h $$4$$ $$24.523$$ $$\Q(i, \sqrt{10})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q+(1-\beta _{1})q^{7}-\beta _{3}q^{11}+(6+6\beta _{1})q^{13}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(900, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(900, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(180, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(300, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(450, [\chi])$$$$^{\oplus 2}$$