# Properties

 Label 360.4.f Level $360$ Weight $4$ Character orbit 360.f Rep. character $\chi_{360}(289,\cdot)$ Character field $\Q$ Dimension $22$ Newform subspaces $6$ Sturm bound $288$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$360 = 2^{3} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 360.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$288$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(360, [\chi])$$.

Total New Old
Modular forms 232 22 210
Cusp forms 200 22 178
Eisenstein series 32 0 32

## Trace form

 $$22 q - 4 q^{5} + O(q^{10})$$ $$22 q - 4 q^{5} - 52 q^{11} - 80 q^{19} - 126 q^{25} + 16 q^{29} - 384 q^{31} - 292 q^{35} + 20 q^{41} - 1614 q^{49} + 88 q^{55} + 1284 q^{59} - 916 q^{61} + 356 q^{65} - 376 q^{71} - 2320 q^{79} + 108 q^{85} - 1452 q^{89} + 1304 q^{91} - 2704 q^{95} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(360, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
360.4.f.a $2$ $21.241$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-10$$ $$0$$ $$q+(-5-5i)q^{5}+2iq^{7}+28q^{11}+\cdots$$
360.4.f.b $2$ $21.241$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+(2-11i)q^{5}+10iq^{7}+14q^{11}+\cdots$$
360.4.f.c $2$ $21.241$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$20$$ $$0$$ $$q+(10+5i)q^{5}+10iq^{7}+46q^{11}+\cdots$$
360.4.f.d $4$ $21.241$ $$\Q(i, \sqrt{129})$$ None $$0$$ $$0$$ $$-22$$ $$0$$ $$q+(-5+\beta _{1}-\beta _{3})q^{5}+(-2\beta _{1}+3\beta _{2}+\cdots)q^{7}+\cdots$$
360.4.f.e $4$ $21.241$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+(1+\beta _{2}-\beta _{3})q^{5}+(\beta _{1}-2\beta _{2})q^{7}+\cdots$$
360.4.f.f $8$ $21.241$ 8.0.$$\cdots$$.2 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{4}q^{5}+\beta _{2}q^{7}+\beta _{6}q^{11}+\beta _{7}q^{13}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(360, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(360, [\chi]) \simeq$$ $$S_{4}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(120, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(180, [\chi])$$$$^{\oplus 2}$$