# Properties

 Label 720.4.f Level $720$ Weight $4$ Character orbit 720.f Rep. character $\chi_{720}(289,\cdot)$ Character field $\Q$ Dimension $44$ Newform subspaces $14$ Sturm bound $576$ Trace bound $19$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 720.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$14$$ Sturm bound: $$576$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(720, [\chi])$$.

Total New Old
Modular forms 456 46 410
Cusp forms 408 44 364
Eisenstein series 48 2 46

## Trace form

 $$44 q + 2 q^{5} + O(q^{10})$$ $$44 q + 2 q^{5} + 64 q^{11} + 104 q^{19} - 96 q^{25} + 172 q^{29} + 56 q^{31} + 184 q^{35} - 28 q^{41} - 2220 q^{49} + 264 q^{55} - 720 q^{59} - 536 q^{61} - 328 q^{65} - 176 q^{71} + 1432 q^{79} - 652 q^{85} - 324 q^{89} - 2000 q^{91} + 1216 q^{95} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(720, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
720.4.f.a $2$ $42.481$ $$\Q(\sqrt{-19})$$ None $$0$$ $$0$$ $$-14$$ $$0$$ $$q+(-7-\beta )q^{5}-\beta q^{7}+20q^{11}+6\beta q^{13}+\cdots$$
720.4.f.b $2$ $42.481$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-10$$ $$0$$ $$q+(-5+5i)q^{5}+2iq^{7}-28q^{11}+\cdots$$
720.4.f.c $2$ $42.481$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+(-2-11i)q^{5}+2iq^{7}+70q^{11}+\cdots$$
720.4.f.d $2$ $42.481$ $$\Q(\sqrt{-5})$$ $$\Q(\sqrt{-15})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-5\beta q^{5}+62\beta q^{17}-164q^{19}-44\beta q^{23}+\cdots$$
720.4.f.e $2$ $42.481$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+(2+11i)q^{5}+10iq^{7}-14q^{11}+\cdots$$
720.4.f.f $2$ $42.481$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$10$$ $$0$$ $$q+(5-5i)q^{5}+13iq^{7}-28q^{11}+6iq^{13}+\cdots$$
720.4.f.g $2$ $42.481$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$20$$ $$0$$ $$q+(10-5i)q^{5}+10iq^{7}-46q^{11}+\cdots$$
720.4.f.h $2$ $42.481$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$20$$ $$0$$ $$q+(10+5i)q^{5}+22iq^{7}-14q^{11}+\cdots$$
720.4.f.i $4$ $42.481$ $$\Q(i, \sqrt{129})$$ None $$0$$ $$0$$ $$-22$$ $$0$$ $$q+(-5-\beta _{1}-\beta _{2})q^{5}+(-2\beta _{1}+3\beta _{2}+\cdots)q^{7}+\cdots$$
720.4.f.j $4$ $42.481$ $$\Q(i, \sqrt{41})$$ None $$0$$ $$0$$ $$-6$$ $$0$$ $$q+(-1-\beta _{2}-\beta _{3})q^{5}+(2\beta _{1}-\beta _{2})q^{7}+\cdots$$
720.4.f.k $4$ $42.481$ $$\Q(\sqrt{10}, \sqrt{-34})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{5}+\beta _{1}q^{7}+(3\beta _{2}-\beta _{3})q^{11}+\cdots$$
720.4.f.l $4$ $42.481$ $$\Q(i, \sqrt{31})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{1}-\beta _{2})q^{5}-\beta _{3}q^{7}+(2\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots$$
720.4.f.m $4$ $42.481$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+(1+\beta _{1}-\beta _{3})q^{5}+(\beta _{1}-2\beta _{2})q^{7}+\cdots$$
720.4.f.n $8$ $42.481$ 8.0.$$\cdots$$.2 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{5}+\beta _{2}q^{7}-\beta _{6}q^{11}-\beta _{7}q^{13}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(720, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(720, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(120, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(180, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(240, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(360, [\chi])$$$$^{\oplus 2}$$