# Properties

 Label 1600.2.f Level $1600$ Weight $2$ Character orbit 1600.f Rep. character $\chi_{1600}(1249,\cdot)$ Character field $\Q$ Dimension $36$ Newform subspaces $10$ Sturm bound $480$ Trace bound $31$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$40$$ Character field: $$\Q$$ Newform subspaces: $$10$$ Sturm bound: $$480$$ Trace bound: $$31$$ Distinguishing $$T_p$$: $$3$$, $$31$$

## Dimensions

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

Total New Old
Modular forms 276 36 240
Cusp forms 204 36 168
Eisenstein series 72 0 72

## Trace form

 $$36 q + 36 q^{9} + O(q^{10})$$ $$36 q + 36 q^{9} + 24 q^{41} - 36 q^{49} + 132 q^{81} + 120 q^{89} + O(q^{100})$$

## 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.f.a $2$ $12.776$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$-4$$ $$0$$ $$0$$ $$q-2q^{3}+q^{9}+3iq^{11}-3iq^{17}+iq^{19}+\cdots$$
1600.2.f.b $2$ $12.776$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$4$$ $$0$$ $$0$$ $$q+2q^{3}+q^{9}-3iq^{11}-3iq^{17}-iq^{19}+\cdots$$
1600.2.f.c $4$ $12.776$ $$\Q(\zeta_{12})$$ None $$0$$ $$-4$$ $$0$$ $$0$$ $$q-q^{3}-\zeta_{12}^{2}q^{7}-2q^{9}+3\zeta_{12}q^{11}+\cdots$$
1600.2.f.d $4$ $12.776$ $$\Q(\zeta_{12})$$ None $$0$$ $$-4$$ $$0$$ $$0$$ $$q+(-1+\zeta_{12})q^{3}+(\zeta_{12}^{2}-\zeta_{12}^{3})q^{7}+\cdots$$
1600.2.f.e $4$ $12.776$ $$\Q(\zeta_{12})$$ None $$0$$ $$-4$$ $$0$$ $$0$$ $$q+(-1+\zeta_{12})q^{3}+(\zeta_{12}^{2}-\zeta_{12}^{3})q^{7}+\cdots$$
1600.2.f.f $4$ $12.776$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$-4$$ $$0$$ $$0$$ $$q+(-1+\beta _{3})q^{3}+(4-2\beta _{3})q^{9}+(3\beta _{1}+\cdots)q^{11}+\cdots$$
1600.2.f.g $4$ $12.776$ $$\Q(\zeta_{12})$$ None $$0$$ $$4$$ $$0$$ $$0$$ $$q+q^{3}+\zeta_{12}^{2}q^{7}-2q^{9}+3\zeta_{12}q^{11}+\cdots$$
1600.2.f.h $4$ $12.776$ $$\Q(\zeta_{12})$$ None $$0$$ $$4$$ $$0$$ $$0$$ $$q+(1-\zeta_{12})q^{3}+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{7}+\cdots$$
1600.2.f.i $4$ $12.776$ $$\Q(\zeta_{12})$$ None $$0$$ $$4$$ $$0$$ $$0$$ $$q+(1-\zeta_{12})q^{3}+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{7}+\cdots$$
1600.2.f.j $4$ $12.776$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$4$$ $$0$$ $$0$$ $$q+(1-\beta _{3})q^{3}+(4-2\beta _{3})q^{9}+(3\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(1600, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(160, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(320, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(800, [\chi])$$$$^{\oplus 2}$$