# Properties

 Label 300.2.i Level $300$ Weight $2$ Character orbit 300.i Rep. character $\chi_{300}(257,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $12$ Newform subspaces $3$ Sturm bound $120$ Trace bound $21$

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## Defining parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.i (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$15$$ Character field: $$\Q(i)$$ Newform subspaces: $$3$$ Sturm bound: $$120$$ Trace bound: $$21$$ Distinguishing $$T_p$$: $$7$$

## Dimensions

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

Total New Old
Modular forms 156 12 144
Cusp forms 84 12 72
Eisenstein series 72 0 72

## Trace form

 $$12 q - 2 q^{3} + 4 q^{7} + O(q^{10})$$ $$12 q - 2 q^{3} + 4 q^{7} + 12 q^{13} + 8 q^{21} - 14 q^{27} - 12 q^{31} - 20 q^{33} + 12 q^{37} + 12 q^{43} - 20 q^{51} + 4 q^{57} + 28 q^{61} - 8 q^{63} + 4 q^{67} - 4 q^{73} - 68 q^{81} - 20 q^{87} - 108 q^{91} - 8 q^{93} - 36 q^{97} + O(q^{100})$$

## 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.i.a $4$ $2.396$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$-2$$ $$0$$ $$4$$ $$q+(-1+\beta _{3})q^{3}+(1-\beta _{2})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots$$
300.2.i.b $4$ $2.396$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+2\beta _{3}q^{7}+3\beta _{2}q^{9}+4\beta _{1}q^{13}+\cdots$$
300.2.i.c $4$ $2.396$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}-3\beta _{3}q^{7}+3\beta _{2}q^{9}-\beta _{1}q^{13}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(300, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 2}$$