# Properties

 Label 960.4.f Level $960$ Weight $4$ Character orbit 960.f Rep. character $\chi_{960}(769,\cdot)$ Character field $\Q$ Dimension $72$ Newform subspaces $21$ Sturm bound $768$ Trace bound $11$

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## Defining parameters

 Level: $$N$$ $$=$$ $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 960.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$21$$ Sturm bound: $$768$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(960, [\chi])$$.

Total New Old
Modular forms 600 72 528
Cusp forms 552 72 480
Eisenstein series 48 0 48

## Trace form

 $$72 q - 648 q^{9} + O(q^{10})$$ $$72 q - 648 q^{9} - 88 q^{25} + 944 q^{41} - 3528 q^{49} + 912 q^{61} - 560 q^{65} - 528 q^{69} + 5832 q^{81} - 2352 q^{85} + 176 q^{89} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(960, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
960.4.f.a $2$ $56.642$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-10$$ $$0$$ $$q-3iq^{3}+(-5+10i)q^{5}+4iq^{7}+\cdots$$
960.4.f.b $2$ $56.642$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-10$$ $$0$$ $$q-3iq^{3}+(-5-10i)q^{5}+4iq^{7}+\cdots$$
960.4.f.c $2$ $56.642$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+3iq^{3}+(-2+11i)q^{5}+2iq^{7}+\cdots$$
960.4.f.d $2$ $56.642$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+3iq^{3}+(-2-11i)q^{5}+2iq^{7}+\cdots$$
960.4.f.e $2$ $56.642$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+3iq^{3}+(2+11i)q^{5}+10iq^{7}-9q^{9}+\cdots$$
960.4.f.f $2$ $56.642$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+3iq^{3}+(2-11i)q^{5}+10iq^{7}-9q^{9}+\cdots$$
960.4.f.g $2$ $56.642$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$20$$ $$0$$ $$q+3iq^{3}+(10-5i)q^{5}+10iq^{7}-9q^{9}+\cdots$$
960.4.f.h $2$ $56.642$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$20$$ $$0$$ $$q-3iq^{3}+(10-5i)q^{5}+18iq^{7}-9q^{9}+\cdots$$
960.4.f.i $2$ $56.642$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$20$$ $$0$$ $$q-3iq^{3}+(10+5i)q^{5}+22iq^{7}-9q^{9}+\cdots$$
960.4.f.j $2$ $56.642$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$20$$ $$0$$ $$q-3iq^{3}+(10-5i)q^{5}+22iq^{7}-9q^{9}+\cdots$$
960.4.f.k $2$ $56.642$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$20$$ $$0$$ $$q-3iq^{3}+(10+5i)q^{5}+18iq^{7}-9q^{9}+\cdots$$
960.4.f.l $2$ $56.642$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$20$$ $$0$$ $$q+3iq^{3}+(10+5i)q^{5}+10iq^{7}-9q^{9}+\cdots$$
960.4.f.m $4$ $56.642$ $$\Q(i, \sqrt{89})$$ None $$0$$ $$0$$ $$-24$$ $$0$$ $$q-3\beta _{1}q^{3}+(-6-\beta _{2})q^{5}-22\beta _{1}q^{7}+\cdots$$
960.4.f.n $4$ $56.642$ $$\Q(i, \sqrt{129})$$ None $$0$$ $$0$$ $$-22$$ $$0$$ $$q-3\beta _{1}q^{3}+(-5+5\beta _{1}-\beta _{2})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots$$
960.4.f.o $4$ $56.642$ $$\Q(i, \sqrt{129})$$ None $$0$$ $$0$$ $$-22$$ $$0$$ $$q-3\beta _{1}q^{3}+(-5-6\beta _{1}+\beta _{3})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots$$
960.4.f.p $4$ $56.642$ $$\Q(i, \sqrt{41})$$ None $$0$$ $$0$$ $$-6$$ $$0$$ $$q+\beta _{2}q^{3}+(-1+\beta _{1}+2\beta _{2}-\beta _{3})q^{5}+\cdots$$
960.4.f.q $4$ $56.642$ $$\Q(i, \sqrt{41})$$ None $$0$$ $$0$$ $$-6$$ $$0$$ $$q+\beta _{2}q^{3}+(-1-\beta _{2}-\beta _{3})q^{5}+(3\beta _{1}+\cdots)q^{7}+\cdots$$
960.4.f.r $6$ $56.642$ 6.0.$$\cdots$$.1 None $$0$$ $$0$$ $$-10$$ $$0$$ $$q+3\beta _{1}q^{3}+(-2+\beta _{1}-\beta _{2})q^{5}+(10\beta _{1}+\cdots)q^{7}+\cdots$$
960.4.f.s $6$ $56.642$ 6.0.$$\cdots$$.1 None $$0$$ $$0$$ $$-10$$ $$0$$ $$q+3\beta _{1}q^{3}+(-2-\beta _{1}-\beta _{4})q^{5}+(10\beta _{1}+\cdots)q^{7}+\cdots$$
960.4.f.t $8$ $56.642$ 8.0.$$\cdots$$.9 None $$0$$ $$0$$ $$-12$$ $$0$$ $$q-3\beta _{1}q^{3}+(-2+\beta _{4})q^{5}+(4\beta _{1}-\beta _{5}+\cdots)q^{7}+\cdots$$
960.4.f.u $8$ $56.642$ 8.0.$$\cdots$$.3 None $$0$$ $$0$$ $$12$$ $$0$$ $$q-3\beta _{2}q^{3}+(1+\beta _{3})q^{5}+(2^{4}\beta _{2}+\beta _{5}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(960, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(960, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 7}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(120, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(160, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(240, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(320, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(480, [\chi])$$$$^{\oplus 2}$$