# Properties

 Label 300.3.f Level $300$ Weight $3$ Character orbit 300.f Rep. character $\chi_{300}(199,\cdot)$ Character field $\Q$ Dimension $36$ Newform subspaces $3$ Sturm bound $180$ Trace bound $4$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 300.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$20$$ Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$180$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(300, [\chi])$$.

Total New Old
Modular forms 132 36 96
Cusp forms 108 36 72
Eisenstein series 24 0 24

## Trace form

 $$36q + 4q^{4} - 12q^{6} + 108q^{9} + O(q^{10})$$ $$36q + 4q^{4} - 12q^{6} + 108q^{9} + 44q^{14} + 116q^{16} + 36q^{24} - 212q^{26} + 40q^{29} - 24q^{34} + 12q^{36} + 168q^{41} - 328q^{44} - 40q^{46} + 76q^{49} - 36q^{54} - 112q^{56} + 24q^{61} - 68q^{64} + 48q^{66} + 192q^{69} + 56q^{74} - 96q^{76} + 324q^{81} + 456q^{84} - 308q^{86} - 744q^{89} - 312q^{94} - 348q^{96} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(300, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
300.3.f.a $$4$$ $$8.174$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\zeta_{12}+\zeta_{12}^{3})q^{2}+\zeta_{12}^{3}q^{3}+(2+2\zeta_{12}^{2}+\cdots)q^{4}+\cdots$$
300.3.f.b $$16$$ $$8.174$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{12}q^{2}-\beta _{9}q^{3}+(-1+\beta _{4})q^{4}+\cdots$$
300.3.f.c $$16$$ $$8.174$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{2}-\beta _{4}q^{3}+(1+\beta _{5})q^{4}+(-1+\cdots)q^{6}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(300, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(300, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 2}$$