# Properties

 Label 300.3.g Level $300$ Weight $3$ Character orbit 300.g Rep. character $\chi_{300}(101,\cdot)$ Character field $\Q$ Dimension $13$ Newform subspaces $8$ Sturm bound $180$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 300.g (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$3$$ Character field: $$\Q$$ Newform subspaces: $$8$$ Sturm bound: $$180$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

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

Total New Old
Modular forms 138 13 125
Cusp forms 102 13 89
Eisenstein series 36 0 36

## Trace form

 $$13 q - q^{3} - 6 q^{7} - 13 q^{9} + O(q^{10})$$ $$13 q - q^{3} - 6 q^{7} - 13 q^{9} + 6 q^{13} + 48 q^{19} + 32 q^{21} + 71 q^{27} + 52 q^{31} - 60 q^{33} - 138 q^{37} - 132 q^{39} + 6 q^{43} + 225 q^{49} - 10 q^{51} + 214 q^{57} - 28 q^{61} - 14 q^{63} - 186 q^{67} - 280 q^{69} + 258 q^{73} + 114 q^{79} + 23 q^{81} + 180 q^{87} - 162 q^{91} - 194 q^{93} - 246 q^{97} - 290 q^{99} + O(q^{100})$$

## 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.g.a $1$ $8.174$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$-3$$ $$0$$ $$-13$$ $$q-3q^{3}-13q^{7}+9q^{9}+23q^{13}+11q^{19}+\cdots$$
300.3.g.b $1$ $8.174$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$3$$ $$0$$ $$-2$$ $$q+3q^{3}-2q^{7}+9q^{9}+22q^{13}+26q^{19}+\cdots$$
300.3.g.c $1$ $8.174$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$3$$ $$0$$ $$13$$ $$q+3q^{3}+13q^{7}+9q^{9}-23q^{13}+11q^{19}+\cdots$$
300.3.g.d $2$ $8.174$ $$\Q(\sqrt{-5})$$ None $$0$$ $$-4$$ $$0$$ $$-4$$ $$q+(-2+\beta )q^{3}-2q^{7}+(-1-4\beta )q^{9}+\cdots$$
300.3.g.e $2$ $8.174$ $$\Q(\sqrt{-5})$$ None $$0$$ $$-4$$ $$0$$ $$16$$ $$q+(-2+\beta )q^{3}+8q^{7}+(-1-4\beta )q^{9}+\cdots$$
300.3.g.f $2$ $8.174$ $$\Q(\sqrt{-35})$$ None $$0$$ $$-1$$ $$0$$ $$-16$$ $$q-\beta q^{3}-8q^{7}+(-9+\beta )q^{9}+(-3+\cdots)q^{11}+\cdots$$
300.3.g.g $2$ $8.174$ $$\Q(\sqrt{-35})$$ None $$0$$ $$1$$ $$0$$ $$16$$ $$q+\beta q^{3}+8q^{7}+(-9+\beta )q^{9}+(-3+\cdots)q^{11}+\cdots$$
300.3.g.h $2$ $8.174$ $$\Q(\sqrt{-5})$$ None $$0$$ $$4$$ $$0$$ $$-16$$ $$q+(2+\beta )q^{3}-8q^{7}+(-1+4\beta )q^{9}+\cdots$$

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

$$S_{3}^{\mathrm{old}}(300, [\chi]) \simeq$$ $$S_{3}^{\mathrm{new}}(12, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 2}$$