# Properties

 Label 360.2.f Level $360$ Weight $2$ Character orbit 360.f Rep. character $\chi_{360}(289,\cdot)$ Character field $\Q$ Dimension $8$ Newform subspaces $4$ Sturm bound $144$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$360 = 2^{3} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 360.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$144$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(360, [\chi])$$.

Total New Old
Modular forms 88 8 80
Cusp forms 56 8 48
Eisenstein series 32 0 32

## Trace form

 $$8 q - 2 q^{5} + O(q^{10})$$ $$8 q - 2 q^{5} + 4 q^{11} + 8 q^{19} + 12 q^{25} + 20 q^{29} + 4 q^{35} - 8 q^{41} - 24 q^{49} - 16 q^{55} - 36 q^{59} - 8 q^{61} - 20 q^{65} - 8 q^{71} + 16 q^{79} + 12 q^{85} + 24 q^{89} - 56 q^{91} + 40 q^{95} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(360, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
360.2.f.a $2$ $2.875$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+(-2+i)q^{5}+2iq^{7}-2q^{11}+2iq^{13}+\cdots$$
360.2.f.b $2$ $2.875$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+(-2+i)q^{5}-4iq^{7}+4q^{11}-4iq^{13}+\cdots$$
360.2.f.c $2$ $2.875$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+(1+i)q^{5}-iq^{7}+4q^{11}+2iq^{13}+\cdots$$
360.2.f.d $2$ $2.875$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+(2+i)q^{5}+4iq^{7}-4q^{11}+4iq^{13}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(360, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(360, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(120, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(180, [\chi])$$$$^{\oplus 2}$$