# Properties

 Label 3200.2.d Level $3200$ Weight $2$ Character orbit 3200.d Rep. character $\chi_{3200}(1601,\cdot)$ Character field $\Q$ Dimension $76$ Newform subspaces $23$ Sturm bound $960$ Trace bound $33$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3200.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$23$$ Sturm bound: $$960$$ Trace bound: $$33$$ Distinguishing $$T_p$$: $$3$$, $$7$$, $$11$$, $$13$$, $$17$$

## Dimensions

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

Total New Old
Modular forms 528 76 452
Cusp forms 432 76 356
Eisenstein series 96 0 96

## Trace form

 $$76 q - 76 q^{9} + O(q^{10})$$ $$76 q - 76 q^{9} - 8 q^{17} - 16 q^{33} + 24 q^{41} + 44 q^{49} + 48 q^{57} + 56 q^{73} + 28 q^{81} + 56 q^{89} - 8 q^{97} + O(q^{100})$$

## 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.d.a $2$ $25.552$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+iq^{3}-4q^{7}+2q^{9}-3iq^{11}-q^{17}+\cdots$$
3200.2.d.b $2$ $25.552$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+iq^{3}-4q^{7}+2q^{9}+3iq^{11}+q^{17}+\cdots$$
3200.2.d.c $2$ $25.552$ $$\Q(\sqrt{-2})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta q^{3}-5q^{9}-\beta q^{11}-6q^{17}+3\beta q^{19}+\cdots$$
3200.2.d.d $2$ $25.552$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+3q^{9}-3iq^{13}-8q^{17}-2iq^{29}+\cdots$$
3200.2.d.e $2$ $25.552$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+3q^{9}-iq^{13}+2q^{17}+iq^{29}-3iq^{37}+\cdots$$
3200.2.d.f $2$ $25.552$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+3q^{9}+3iq^{13}+8q^{17}-2iq^{29}+\cdots$$
3200.2.d.g $2$ $25.552$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q+iq^{3}+4q^{7}+2q^{9}-3iq^{11}-q^{17}+\cdots$$
3200.2.d.h $2$ $25.552$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q+iq^{3}+4q^{7}+2q^{9}+3iq^{11}+q^{17}+\cdots$$
3200.2.d.i $4$ $25.552$ $$\Q(\sqrt{2}, \sqrt{-5})$$ $$\Q(\sqrt{-5})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}-3\beta _{1}q^{7}-7q^{9}+3\beta _{3}q^{21}+\cdots$$
3200.2.d.j $4$ $25.552$ $$\Q(i, \sqrt{10})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}-\beta _{3}q^{7}-7q^{9}+3\beta _{1}q^{13}+\cdots$$
3200.2.d.k $4$ $25.552$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}-\beta _{3}q^{7}-3q^{9}-2\beta _{2}q^{11}+\cdots$$
3200.2.d.l $4$ $25.552$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}+\beta _{3}q^{7}-3q^{9}+2\beta _{2}q^{11}+\cdots$$
3200.2.d.m $4$ $25.552$ $$\Q(\sqrt{2}, \sqrt{-5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}+\beta _{1}q^{7}-2q^{9}+\beta _{3}q^{11}+\cdots$$
3200.2.d.n $4$ $25.552$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}+(-2+\beta _{3})q^{9}+(\beta _{1}+2\beta _{2}+\cdots)q^{11}+\cdots$$
3200.2.d.o $4$ $25.552$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{3}-2q^{9}+\beta _{1}q^{11}-\beta _{2}q^{13}+\cdots$$
3200.2.d.p $4$ $25.552$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}-2q^{9}+\beta _{1}q^{11}-\beta _{2}q^{13}+\cdots$$
3200.2.d.q $4$ $25.552$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}+(-2+\beta _{3})q^{9}+(-\beta _{1}-2\beta _{2}+\cdots)q^{11}+\cdots$$
3200.2.d.r $4$ $25.552$ $$\Q(\sqrt{2}, \sqrt{-5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}-\beta _{1}q^{7}-2q^{9}-\beta _{3}q^{11}+\cdots$$
3200.2.d.s $4$ $25.552$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{2}q^{3}+3\zeta_{8}^{3}q^{7}+q^{9}+4\zeta_{8}^{2}q^{11}+\cdots$$
3200.2.d.t $4$ $25.552$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{2}q^{3}-\zeta_{8}^{3}q^{7}+q^{9}-2\zeta_{8}^{2}q^{11}+\cdots$$
3200.2.d.u $4$ $25.552$ $$\Q(\sqrt{-2}, \sqrt{-5})$$ $$\Q(\sqrt{-5})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+\beta _{2}q^{7}+q^{9}+\beta _{3}q^{21}-3\beta _{2}q^{23}+\cdots$$
3200.2.d.v $4$ $25.552$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{2}q^{3}-\zeta_{8}^{3}q^{7}+q^{9}+2\zeta_{8}^{2}q^{11}+\cdots$$
3200.2.d.w $4$ $25.552$ $$\Q(\sqrt{2}, \sqrt{-5})$$ $$\Q(\sqrt{-10})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{7}+3q^{9}-\beta _{2}q^{11}+\beta _{3}q^{13}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(3200, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(64, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(128, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(160, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(320, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(640, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(800, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1600, [\chi])$$$$^{\oplus 2}$$