# Properties

 Label 300.2.e Level $300$ Weight $2$ Character orbit 300.e Rep. character $\chi_{300}(251,\cdot)$ Character field $\Q$ Dimension $32$ Newform subspaces $5$ Sturm bound $120$ Trace bound $12$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.e (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$12$$ Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$120$$ Trace bound: $$12$$ Distinguishing $$T_p$$: $$7$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(300, [\chi])$$.

Total New Old
Modular forms 72 44 28
Cusp forms 48 32 16
Eisenstein series 24 12 12

## Trace form

 $$32q + 2q^{6} + 4q^{9} + O(q^{10})$$ $$32q + 2q^{6} + 4q^{9} - 4q^{12} + 8q^{13} + 8q^{16} - 16q^{18} - 8q^{21} + 4q^{22} - 2q^{24} - 28q^{28} + 16q^{33} + 28q^{34} - 46q^{36} - 8q^{37} + 12q^{42} - 40q^{46} + 36q^{48} - 8q^{49} + 32q^{52} - 20q^{54} - 24q^{57} + 36q^{58} - 24q^{61} - 24q^{64} + 18q^{66} - 40q^{69} - 24q^{72} + 60q^{76} - 40q^{78} - 28q^{81} - 40q^{82} + 56q^{84} - 44q^{88} - 32q^{93} - 56q^{94} - 58q^{96} + 48q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(300, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
300.2.e.a $$4$$ $$2.396$$ $$\Q(\sqrt{3}, \sqrt{-5})$$ $$\Q(\sqrt{-15})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(\beta _{1}-\beta _{3})q^{3}+\beta _{2}q^{4}+(2+\cdots)q^{6}+\cdots$$
300.2.e.b $$4$$ $$2.396$$ $$\Q(\sqrt{2}, \sqrt{-5})$$ $$\Q(\sqrt{-5})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{2}+(-\beta _{1}-\beta _{2})q^{3}+2q^{4}+(-1+\cdots)q^{6}+\cdots$$
300.2.e.c $$8$$ $$2.396$$ 8.0.342102016.5 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{5}q^{2}+(-\beta _{6}+\beta _{7})q^{3}+(\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots$$
300.2.e.d $$8$$ $$2.396$$ 8.0.4521217600.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{6}q^{2}-\beta _{2}q^{3}+(\beta _{1}-\beta _{2}-\beta _{3})q^{4}+\cdots$$
300.2.e.e $$8$$ $$2.396$$ 8.0.4521217600.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{6}q^{2}+\beta _{3}q^{3}+(\beta _{1}-\beta _{2}-\beta _{3})q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(300, [\chi])$$ into lower level spaces

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