# Properties

 Label 1000.1.g Level $1000$ Weight $1$ Character orbit 1000.g Rep. character $\chi_{1000}(251,\cdot)$ Character field $\Q$ Dimension $4$ Newform subspaces $1$ Sturm bound $150$ Trace bound $0$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1000 = 2^{3} \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1000.g (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$150$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(1000, [\chi])$$.

Total New Old
Modular forms 18 4 14
Cusp forms 8 4 4
Eisenstein series 10 0 10

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 4 0 0 0

## Trace form

 $$4q - 4q^{4} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} - 4q^{9} - 2q^{11} + 2q^{14} + 4q^{16} + 2q^{19} - 2q^{26} + 4q^{36} - 2q^{41} + 2q^{44} - 2q^{46} - 2q^{49} - 2q^{56} + 2q^{59} - 4q^{64} + 2q^{74} - 2q^{76} + 4q^{81} + 2q^{89} - 4q^{91} + 2q^{94} + 2q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(1000, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field Image CM RM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1000.1.g.a $$4$$ $$0.499$$ $$\Q(i, \sqrt{5})$$ $$D_{5}$$ $$\Q(\sqrt{-10})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{2}-q^{4}-\beta _{1}q^{7}+\beta _{3}q^{8}-q^{9}+\cdots$$

## Decomposition of $$S_{1}^{\mathrm{old}}(1000, [\chi])$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(1000, [\chi]) \cong$$ $$S_{1}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 2}$$