# Properties

 Label 300.2.j Level $300$ Weight $2$ Character orbit 300.j Rep. character $\chi_{300}(7,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $36$ Newform subspaces $4$ Sturm bound $120$ Trace bound $4$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.j (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$20$$ Character field: $$\Q(i)$$ Newform subspaces: $$4$$ Sturm bound: $$120$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$7$$, $$19$$

## Dimensions

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

Total New Old
Modular forms 144 36 108
Cusp forms 96 36 60
Eisenstein series 48 0 48

## Trace form

 $$36q + 8q^{6} + 12q^{8} + O(q^{10})$$ $$36q + 8q^{6} + 12q^{8} + 8q^{12} + 4q^{13} - 24q^{16} + 20q^{17} - 12q^{22} - 32q^{26} + 4q^{28} - 20q^{32} - 8q^{33} - 8q^{36} - 4q^{37} - 16q^{38} - 32q^{41} - 20q^{42} - 40q^{46} - 16q^{48} + 8q^{52} - 4q^{53} + 8q^{56} + 20q^{58} + 64q^{61} + 56q^{62} + 48q^{66} + 16q^{68} + 12q^{72} - 44q^{73} - 16q^{76} - 48q^{77} + 24q^{78} - 36q^{81} - 16q^{82} - 8q^{86} - 60q^{88} - 56q^{92} + 16q^{93} + 32q^{96} + 20q^{97} - 24q^{98} + O(q^{100})$$

## 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.j.a $$8$$ $$2.396$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\zeta_{24}-\zeta_{24}^{4}+\zeta_{24}^{5})q^{2}+\zeta_{24}q^{3}+\cdots$$
300.2.j.b $$8$$ $$2.396$$ 8.0.157351936.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{2}+\beta _{4})q^{2}+\beta _{6}q^{3}+(\beta _{3}+\beta _{5})q^{4}+\cdots$$
300.2.j.c $$8$$ $$2.396$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\zeta_{24}-\zeta_{24}^{7})q^{2}-\zeta_{24}^{5}q^{3}+(\zeta_{24}^{2}+\cdots)q^{4}+\cdots$$
300.2.j.d $$12$$ $$2.396$$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{9}q^{3}+(\beta _{2}+\beta _{8}+\beta _{9})q^{4}+\cdots$$

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

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