# Properties

 Label 1600.2.s Level $1600$ Weight $2$ Character orbit 1600.s Rep. character $\chi_{1600}(207,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $68$ Newform subspaces $5$ Sturm bound $480$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.s (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$80$$ Character field: $$\Q(i)$$ Newform subspaces: $$5$$ Sturm bound: $$480$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

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

Total New Old
Modular forms 528 76 452
Cusp forms 432 68 364
Eisenstein series 96 8 88

## Trace form

 $$68q - 4q^{3} - 4q^{7} + 60q^{9} + O(q^{10})$$ $$68q - 4q^{3} - 4q^{7} + 60q^{9} + 4q^{11} + 4q^{17} - 16q^{19} - 4q^{21} - 4q^{23} - 16q^{27} + 4q^{33} + 24q^{47} + 20q^{51} + 4q^{53} + 12q^{57} + 32q^{59} - 36q^{61} - 12q^{63} + 20q^{69} + 72q^{71} + 8q^{73} + 32q^{77} + 28q^{81} + 36q^{83} + 52q^{87} + 36q^{91} + 4q^{97} + 76q^{99} + O(q^{100})$$

## 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.s.a $$2$$ $$12.776$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-4$$ $$0$$ $$-6$$ $$q-2q^{3}+(-3-3i)q^{7}+q^{9}+(1-i)q^{11}+\cdots$$
1600.2.s.b $$8$$ $$12.776$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\zeta_{24}+\zeta_{24}^{5}+\zeta_{24}^{7})q^{3}+(-\zeta_{24}+\cdots)q^{7}+\cdots$$
1600.2.s.c $$16$$ $$12.776$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{4}q^{3}-\beta _{7}q^{7}+(1-\beta _{12})q^{9}+(-1+\cdots)q^{11}+\cdots$$
1600.2.s.d $$18$$ $$12.776$$ $$\mathbb{Q}[x]/(x^{18} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+\beta _{1}q^{3}+\beta _{11}q^{7}+(1+\beta _{3})q^{9}+(-\beta _{13}+\cdots)q^{11}+\cdots$$
1600.2.s.e $$24$$ $$12.776$$ None $$0$$ $$0$$ $$0$$ $$0$$

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

$$S_{2}^{\mathrm{old}}(1600, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(320, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(400, [\chi])$$$$^{\oplus 3}$$