# Properties

 Label 300.9.g Level $300$ Weight $9$ Character orbit 300.g Rep. character $\chi_{300}(101,\cdot)$ Character field $\Q$ Dimension $51$ Newform subspaces $8$ Sturm bound $540$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ 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: $$540$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(300, [\chi])$$.

Total New Old
Modular forms 498 51 447
Cusp forms 462 51 411
Eisenstein series 36 0 36

## Trace form

 $$51 q - 91 q^{3} - 1994 q^{7} - 1939 q^{9} + O(q^{10})$$ $$51 q - 91 q^{3} - 1994 q^{7} - 1939 q^{9} + 54466 q^{13} - 323276 q^{19} - 274396 q^{21} + 93869 q^{27} + 1220744 q^{31} - 1440060 q^{33} - 882062 q^{37} + 3798264 q^{39} + 6461026 q^{43} + 44800155 q^{49} + 13072850 q^{51} - 1576694 q^{57} - 40919376 q^{61} + 15229606 q^{63} - 12796814 q^{67} + 13142600 q^{69} + 13092238 q^{73} + 121145662 q^{79} - 28903279 q^{81} + 40636620 q^{87} + 157439246 q^{91} + 62977666 q^{93} - 31762634 q^{97} + 191259250 q^{99} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(300, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
300.9.g.a $1$ $122.214$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$-81$$ $$0$$ $$-4034$$ $$q-3^{4}q^{3}-4034q^{7}+3^{8}q^{9}+35806q^{13}+\cdots$$
300.9.g.b $1$ $122.214$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$-81$$ $$0$$ $$-239$$ $$q-3^{4}q^{3}-239q^{7}+3^{8}q^{9}+20641q^{13}+\cdots$$
300.9.g.c $1$ $122.214$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$81$$ $$0$$ $$239$$ $$q+3^{4}q^{3}+239q^{7}+3^{8}q^{9}-20641q^{13}+\cdots$$
300.9.g.d $2$ $122.214$ $$\Q(\sqrt{-110})$$ None $$0$$ $$102$$ $$0$$ $$6188$$ $$q+(51+\beta )q^{3}+3094q^{7}+(-1359+\cdots)q^{9}+\cdots$$
300.9.g.e $10$ $122.214$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$0$$ $$-137$$ $$0$$ $$-1048$$ $$q+(-14-\beta _{1})q^{3}+(-104+2\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots$$
300.9.g.f $10$ $122.214$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$0$$ $$-112$$ $$0$$ $$-4148$$ $$q+(-11+\beta _{1})q^{3}+(-417-11\beta _{1}+\cdots)q^{7}+\cdots$$
300.9.g.g $10$ $122.214$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$0$$ $$137$$ $$0$$ $$1048$$ $$q+(14+\beta _{1})q^{3}+(104-2\beta _{1}+\beta _{3})q^{7}+\cdots$$
300.9.g.h $16$ $122.214$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{3}-\beta _{7}q^{7}+(922+\beta _{2})q^{9}-\beta _{3}q^{11}+\cdots$$

## Decomposition of $$S_{9}^{\mathrm{old}}(300, [\chi])$$ into lower level spaces

$$S_{9}^{\mathrm{old}}(300, [\chi]) \simeq$$ $$S_{9}^{\mathrm{new}}(3, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(6, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(12, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 2}$$