# Properties

 Label 300.3.b Level $300$ Weight $3$ Character orbit 300.b Rep. character $\chi_{300}(149,\cdot)$ Character field $\Q$ Dimension $12$ Newform subspaces $4$ Sturm bound $180$ Trace bound $19$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 300.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$15$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$180$$ Trace bound: $$19$$ 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 12 126
Cusp forms 102 12 90
Eisenstein series 36 0 36

## Trace form

 $$12 q + 2 q^{9} + O(q^{10})$$ $$12 q + 2 q^{9} + 18 q^{19} + 98 q^{21} - 102 q^{31} - 62 q^{39} - 30 q^{49} + 330 q^{51} - 342 q^{61} + 60 q^{69} + 384 q^{79} + 262 q^{81} - 558 q^{91} - 270 q^{99} + O(q^{100})$$

## 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.b.a $2$ $8.174$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+2iq^{7}-9q^{9}+22iq^{13}+\cdots$$
300.3.b.b $2$ $8.174$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}-13iq^{7}-9q^{9}-23iq^{13}+\cdots$$
300.3.b.c $4$ $8.174$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{1}+\beta _{3})q^{3}+\beta _{1}q^{7}+(1-2\beta _{2}+\cdots)q^{9}+\cdots$$
300.3.b.d $4$ $8.174$ $$\Q(i, \sqrt{35})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}-8\beta _{2}q^{7}+(9+\beta _{3})q^{9}+(-3+\cdots)q^{11}+\cdots$$

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

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