# Properties

 Label 1600.2.d Level $1600$ Weight $2$ Character orbit 1600.d Rep. character $\chi_{1600}(801,\cdot)$ Character field $\Q$ Dimension $38$ Newform subspaces $9$ Sturm bound $480$ Trace bound $17$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$9$$ Sturm bound: $$480$$ Trace bound: $$17$$ Distinguishing $$T_p$$: $$3$$, $$7$$, $$17$$

## 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 38 166
Eisenstein series 72 0 72

## Trace form

 $$38q - 38q^{9} + O(q^{10})$$ $$38q - 38q^{9} + 12q^{17} + 24q^{33} - 36q^{41} + 22q^{49} - 8q^{57} + 28q^{73} + 110q^{81} - 36q^{89} - 52q^{97} + O(q^{100})$$

## 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.d.a $$2$$ $$12.776$$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-q^{9}+3iq^{11}+6q^{17}-iq^{19}+\cdots$$
1600.2.d.b $$4$$ $$12.776$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$-12$$ $$q-\zeta_{12}^{3}q^{3}+(-3+\zeta_{12})q^{7}+(-1+\cdots)q^{9}+\cdots$$
1600.2.d.c $$4$$ $$12.776$$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}+(-4-\beta _{3})q^{9}+(\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots$$
1600.2.d.d $$4$$ $$12.776$$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}+(-4-\beta _{3})q^{9}+(-\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots$$
1600.2.d.e $$4$$ $$12.776$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{3}-\zeta_{12}^{3}q^{7}+2q^{9}+3\zeta_{12}q^{11}+\cdots$$
1600.2.d.f $$4$$ $$12.776$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{3}+\zeta_{12}^{3}q^{7}+2q^{9}-3\zeta_{12}q^{11}+\cdots$$
1600.2.d.g $$4$$ $$12.776$$ $$\Q(i, \sqrt{5})$$ $$\Q(\sqrt{-10})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{7}+3q^{9}-\beta _{1}q^{11}+\beta _{2}q^{13}+\cdots$$
1600.2.d.h $$4$$ $$12.776$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$12$$ $$q-\zeta_{12}^{3}q^{3}+(3-\zeta_{12})q^{7}+(-1+2\zeta_{12}+\cdots)q^{9}+\cdots$$
1600.2.d.i $$8$$ $$12.776$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{24}^{5}q^{3}-\zeta_{24}^{4}q^{7}-3q^{9}-\zeta_{24}q^{11}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(1600, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(64, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(160, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(320, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(800, [\chi])$$$$^{\oplus 2}$$