# Properties

 Label 1600.2.c Level $1600$ Weight $2$ Character orbit 1600.c Rep. character $\chi_{1600}(449,\cdot)$ Character field $\Q$ Dimension $34$ Newform subspaces $15$ Sturm bound $480$ Trace bound $19$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$15$$ Sturm bound: $$480$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$3$$, $$7$$, $$11$$, $$19$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1600, [\chi])$$.

Total New Old
Modular forms 276 38 238
Cusp forms 204 34 170
Eisenstein series 72 4 68

## Trace form

 $$34 q - 26 q^{9} + O(q^{10})$$ $$34 q - 26 q^{9} + 16 q^{21} - 4 q^{29} + 4 q^{41} - 18 q^{49} - 28 q^{61} + 64 q^{69} - 30 q^{81} + 44 q^{89} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1600, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1600.2.c.a $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}-2iq^{7}-6q^{9}-q^{11}-4iq^{13}+\cdots$$
1600.2.c.b $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}-2iq^{7}-6q^{9}+q^{11}+4iq^{13}+\cdots$$
1600.2.c.c $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+iq^{7}-q^{9}-4q^{11}-3iq^{13}+\cdots$$
1600.2.c.d $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-iq^{7}-q^{9}-iq^{13}+3iq^{17}+\cdots$$
1600.2.c.e $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-iq^{7}-q^{9}+iq^{13}-3iq^{17}+\cdots$$
1600.2.c.f $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+iq^{7}-q^{9}+4q^{11}+3iq^{13}+\cdots$$
1600.2.c.g $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-2iq^{7}+2q^{9}-5q^{11}+5iq^{17}+\cdots$$
1600.2.c.h $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+2iq^{7}+2q^{9}-3q^{11}+4iq^{13}+\cdots$$
1600.2.c.i $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+2iq^{7}+2q^{9}+3q^{11}-4iq^{13}+\cdots$$
1600.2.c.j $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-2iq^{7}+2q^{9}+5q^{11}-5iq^{17}+\cdots$$
1600.2.c.k $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-2iq^{7}+3q^{9}-4q^{11}-iq^{13}+iq^{17}+\cdots$$
1600.2.c.l $2$ $12.776$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+3q^{9}-3iq^{13}-iq^{17}-10q^{29}+\cdots$$
1600.2.c.m $2$ $12.776$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{7}+3q^{9}+4q^{11}-iq^{13}+iq^{17}+\cdots$$
1600.2.c.n $4$ $12.776$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{2}q^{3}-\zeta_{8}^{2}q^{7}-5q^{9}-\zeta_{8}^{3}q^{11}+\cdots$$
1600.2.c.o $4$ $12.776$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}+2\beta _{2}q^{7}-2q^{9}+\beta _{3}q^{11}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1600, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1600, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(160, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(320, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(400, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(800, [\chi])$$$$^{\oplus 2}$$