# Properties

 Label 450.3.g Level $450$ Weight $3$ Character orbit 450.g Rep. character $\chi_{450}(307,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $30$ Newform subspaces $10$ Sturm bound $270$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 450.g (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q(i)$$ Newform subspaces: $$10$$ Sturm bound: $$270$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(450, [\chi])$$.

Total New Old
Modular forms 408 30 378
Cusp forms 312 30 282
Eisenstein series 96 0 96

## Trace form

 $$30 q + 2 q^{2} - 20 q^{7} - 4 q^{8} + O(q^{10})$$ $$30 q + 2 q^{2} - 20 q^{7} - 4 q^{8} - 64 q^{11} - 6 q^{13} - 120 q^{16} + 58 q^{17} + 16 q^{22} - 20 q^{23} + 84 q^{26} + 40 q^{28} + 24 q^{31} - 8 q^{32} + 126 q^{37} - 104 q^{38} - 208 q^{41} - 108 q^{43} - 272 q^{46} + 60 q^{47} - 12 q^{52} + 206 q^{53} + 160 q^{56} + 8 q^{58} + 688 q^{61} - 184 q^{62} - 284 q^{67} - 116 q^{68} + 376 q^{71} + 62 q^{73} - 112 q^{76} - 160 q^{77} + 232 q^{82} + 228 q^{83} - 192 q^{86} - 32 q^{88} - 600 q^{91} - 40 q^{92} + 222 q^{97} - 206 q^{98} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(450, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
450.3.g.a $2$ $12.262$ $$\Q(\sqrt{-1})$$ None $$-2$$ $$0$$ $$0$$ $$-16$$ $$q+(-1-i)q^{2}+2iq^{4}+(-8-8i)q^{7}+\cdots$$
450.3.g.b $2$ $12.262$ $$\Q(\sqrt{-1})$$ None $$-2$$ $$0$$ $$0$$ $$-4$$ $$q+(-1-i)q^{2}+2iq^{4}+(-2-2i)q^{7}+\cdots$$
450.3.g.c $2$ $12.262$ $$\Q(\sqrt{-1})$$ None $$-2$$ $$0$$ $$0$$ $$6$$ $$q+(-1-i)q^{2}+2iq^{4}+(3+3i)q^{7}+\cdots$$
450.3.g.d $2$ $12.262$ $$\Q(\sqrt{-1})$$ None $$2$$ $$0$$ $$0$$ $$-16$$ $$q+(1+i)q^{2}+2iq^{4}+(-8-8i)q^{7}+\cdots$$
450.3.g.e $2$ $12.262$ $$\Q(\sqrt{-1})$$ None $$2$$ $$0$$ $$0$$ $$-6$$ $$q+(1+i)q^{2}+2iq^{4}+(-3-3i)q^{7}+\cdots$$
450.3.g.f $4$ $12.262$ $$\Q(i, \sqrt{6})$$ None $$-4$$ $$0$$ $$0$$ $$0$$ $$q+(-1-\beta _{2})q^{2}+2\beta _{2}q^{4}+\beta _{1}q^{7}+\cdots$$
450.3.g.g $4$ $12.262$ $$\Q(i, \sqrt{6})$$ None $$-4$$ $$0$$ $$0$$ $$24$$ $$q+(-1+\beta _{2})q^{2}-2\beta _{2}q^{4}+(6-6\beta _{2}+\cdots)q^{7}+\cdots$$
450.3.g.h $4$ $12.262$ $$\Q(i, \sqrt{6})$$ None $$4$$ $$0$$ $$0$$ $$-24$$ $$q+(1-\beta _{2})q^{2}-2\beta _{2}q^{4}+(-6+6\beta _{2}+\cdots)q^{7}+\cdots$$
450.3.g.i $4$ $12.262$ $$\Q(i, \sqrt{6})$$ None $$4$$ $$0$$ $$0$$ $$0$$ $$q+(1+\beta _{2})q^{2}+2\beta _{2}q^{4}+\beta _{1}q^{7}+(-2+\cdots)q^{8}+\cdots$$
450.3.g.j $4$ $12.262$ $$\Q(i, \sqrt{6})$$ None $$4$$ $$0$$ $$0$$ $$16$$ $$q+(1+\beta _{2})q^{2}+2\beta _{2}q^{4}+(4+\beta _{1}+4\beta _{2}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(450, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(450, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 2}$$