# Properties

 Label 1350.2.c Level $1350$ Weight $2$ Character orbit 1350.c Rep. character $\chi_{1350}(649,\cdot)$ Character field $\Q$ Dimension $24$ Newform subspaces $12$ Sturm bound $540$ Trace bound $29$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1350.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$12$$ Sturm bound: $$540$$ Trace bound: $$29$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$13$$, $$17$$, $$29$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1350, [\chi])$$.

Total New Old
Modular forms 306 24 282
Cusp forms 234 24 210
Eisenstein series 72 0 72

## Trace form

 $$24 q - 24 q^{4} + O(q^{10})$$ $$24 q - 24 q^{4} + 24 q^{16} - 12 q^{19} + 24 q^{34} + 48 q^{49} - 12 q^{61} - 24 q^{64} + 12 q^{76} + 36 q^{79} + 96 q^{91} - 72 q^{94} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1350, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1350.2.c.a $2$ $10.780$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+2iq^{7}-iq^{8}-3q^{11}+\cdots$$
1350.2.c.b $2$ $10.780$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+iq^{7}-iq^{8}-3q^{11}+\cdots$$
1350.2.c.c $2$ $10.780$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+2iq^{7}+iq^{8}-3q^{11}+\cdots$$
1350.2.c.d $2$ $10.780$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+2iq^{7}+iq^{8}-3q^{11}+\cdots$$
1350.2.c.e $2$ $10.780$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+4iq^{7}+iq^{8}-3q^{11}+\cdots$$
1350.2.c.f $2$ $10.780$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+iq^{7}-iq^{8}+2iq^{13}+\cdots$$
1350.2.c.g $2$ $10.780$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+iq^{7}+iq^{8}+2iq^{13}+\cdots$$
1350.2.c.h $2$ $10.780$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+4iq^{7}-iq^{8}+3q^{11}+\cdots$$
1350.2.c.i $2$ $10.780$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+2iq^{7}-iq^{8}+3q^{11}+\cdots$$
1350.2.c.j $2$ $10.780$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+2iq^{7}-iq^{8}+3q^{11}+\cdots$$
1350.2.c.k $2$ $10.780$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+iq^{7}+iq^{8}+3q^{11}+\cdots$$
1350.2.c.l $2$ $10.780$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+2iq^{7}+iq^{8}+3q^{11}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1350, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1350, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(135, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(270, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(450, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(675, [\chi])$$$$^{\oplus 2}$$