# Properties

 Label 900.3.p Level $900$ Weight $3$ Character orbit 900.p Rep. character $\chi_{900}(101,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $76$ Newform subspaces $6$ Sturm bound $540$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 900.p (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$6$$ Sturm bound: $$540$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$7$$

## Dimensions

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

Total New Old
Modular forms 756 76 680
Cusp forms 684 76 608
Eisenstein series 72 0 72

## Trace form

 $$76q + q^{3} - q^{7} + 19q^{9} + O(q^{10})$$ $$76q + q^{3} - q^{7} + 19q^{9} + 5q^{13} + 2q^{19} - 29q^{21} - 45q^{23} - 2q^{27} + 9q^{29} + 23q^{31} - 42q^{33} + 20q^{37} + 9q^{39} + 54q^{41} - 46q^{43} - 135q^{47} - 255q^{49} + 115q^{51} - 7q^{57} + 324q^{59} - 55q^{61} + 215q^{63} + 38q^{67} - 77q^{69} + 86q^{73} + 153q^{77} - 49q^{79} - 65q^{81} + 279q^{83} + 213q^{87} - 134q^{91} + 53q^{93} + 98q^{97} + 137q^{99} + O(q^{100})$$

## 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.p.a $$4$$ $$24.523$$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$0$$ $$-3$$ $$0$$ $$1$$ $$q+(\beta _{2}-\beta _{3})q^{3}+(-\beta _{1}+3\beta _{3})q^{7}+(-1+\cdots)q^{9}+\cdots$$
900.3.p.b $$4$$ $$24.523$$ $$\Q(\sqrt{-3}, \sqrt{-5})$$ None $$0$$ $$6$$ $$0$$ $$-8$$ $$q+3\beta _{1}q^{3}+(-4\beta _{1}-\beta _{2})q^{7}+(-9+\cdots)q^{9}+\cdots$$
900.3.p.c $$12$$ $$24.523$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$-2$$ $$0$$ $$6$$ $$q-\beta _{1}q^{3}+(-\beta _{1}-\beta _{3}+\beta _{4}+\beta _{6})q^{7}+\cdots$$
900.3.p.d $$16$$ $$24.523$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$-2$$ $$0$$ $$1$$ $$q+(\beta _{1}-\beta _{4})q^{3}-\beta _{10}q^{7}+(1-\beta _{3}+\beta _{15})q^{9}+\cdots$$
900.3.p.e $$16$$ $$24.523$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$2$$ $$0$$ $$-1$$ $$q+(-\beta _{1}+\beta _{4})q^{3}+\beta _{10}q^{7}+(1-\beta _{3}+\cdots)q^{9}+\cdots$$
900.3.p.f $$24$$ $$24.523$$ None $$0$$ $$0$$ $$0$$ $$0$$

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

$$S_{3}^{\mathrm{old}}(900, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(9, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(180, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(450, [\chi])$$$$^{\oplus 2}$$