# Properties

 Label 3200.2.f Level $3200$ Weight $2$ Character orbit 3200.f Rep. character $\chi_{3200}(449,\cdot)$ Character field $\Q$ Dimension $72$ Newform subspaces $19$ Sturm bound $960$ Trace bound $49$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3200.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$40$$ Character field: $$\Q$$ Newform subspaces: $$19$$ Sturm bound: $$960$$ Trace bound: $$49$$ Distinguishing $$T_p$$: $$3$$, $$7$$, $$11$$, $$13$$, $$31$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(3200, [\chi])$$.

Total New Old
Modular forms 528 72 456
Cusp forms 432 72 360
Eisenstein series 96 0 96

## Trace form

 $$72 q + 72 q^{9} + O(q^{10})$$ $$72 q + 72 q^{9} - 16 q^{41} - 72 q^{49} + 8 q^{81} - 16 q^{89} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(3200, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
3200.2.f.a $2$ $25.552$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q-q^{3}+4iq^{7}-2q^{9}+3iq^{11}-iq^{17}+\cdots$$
3200.2.f.b $2$ $25.552$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q-q^{3}-4iq^{7}-2q^{9}+3iq^{11}-iq^{17}+\cdots$$
3200.2.f.c $2$ $25.552$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-3q^{9}-4q^{13}+iq^{17}-2iq^{29}+\cdots$$
3200.2.f.d $2$ $25.552$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-3q^{9}+4q^{13}+iq^{17}+2iq^{29}+\cdots$$
3200.2.f.e $2$ $25.552$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+q^{3}+4iq^{7}-2q^{9}-3iq^{11}-iq^{17}+\cdots$$
3200.2.f.f $2$ $25.552$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+q^{3}+4iq^{7}-2q^{9}+3iq^{11}+iq^{17}+\cdots$$
3200.2.f.g $4$ $25.552$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{8}^{3}q^{3}-3\zeta_{8}^{2}q^{7}-q^{9}+2\zeta_{8}^{2}q^{11}+\cdots$$
3200.2.f.h $4$ $25.552$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{3}q^{3}+\zeta_{8}^{2}q^{7}-q^{9}+2\zeta_{8}^{2}q^{11}+\cdots$$
3200.2.f.i $4$ $25.552$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{8}^{3}q^{3}-3\zeta_{8}^{2}q^{7}-q^{9}-4\zeta_{8}^{2}q^{11}+\cdots$$
3200.2.f.j $4$ $25.552$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{8}^{3}q^{3}+3\zeta_{8}^{2}q^{7}-q^{9}-4\zeta_{8}^{2}q^{11}+\cdots$$
3200.2.f.k $4$ $25.552$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{3}q^{3}+\zeta_{8}^{2}q^{7}-q^{9}-2\zeta_{8}^{2}q^{11}+\cdots$$
3200.2.f.l $4$ $25.552$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{8}^{3}q^{3}-3\zeta_{8}^{2}q^{7}-q^{9}-2\zeta_{8}^{2}q^{11}+\cdots$$
3200.2.f.m $4$ $25.552$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}+2q^{9}+\beta _{2}q^{11}-4q^{13}+\cdots$$
3200.2.f.n $4$ $25.552$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}+2q^{9}-\beta _{2}q^{11}+4q^{13}+\cdots$$
3200.2.f.o $4$ $25.552$ $$\Q(\zeta_{8})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{3}q^{3}+5q^{9}-\zeta_{8}^{2}q^{11}-3\zeta_{8}q^{17}+\cdots$$
3200.2.f.p $4$ $25.552$ $$\Q(i, \sqrt{10})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}-\beta _{2}q^{7}+7q^{9}-6q^{13}-\beta _{1}q^{17}+\cdots$$
3200.2.f.q $4$ $25.552$ $$\Q(i, \sqrt{10})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}+\beta _{2}q^{7}+7q^{9}+6q^{13}-\beta _{1}q^{17}+\cdots$$
3200.2.f.r $8$ $25.552$ 8.0.40960000.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{3}-\beta _{2}q^{7}+2q^{9}+\beta _{3}q^{11}+\cdots$$
3200.2.f.s $8$ $25.552$ $$\Q(\zeta_{24})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{24}q^{3}+(2-\zeta_{24}^{3})q^{9}+\zeta_{24}^{6}q^{11}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(3200, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(3200, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(160, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(320, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(640, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(800, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1600, [\chi])$$$$^{\oplus 2}$$