# Properties

 Label 4000.1.g Level $4000$ Weight $1$ Character orbit 4000.g Rep. character $\chi_{4000}(751,\cdot)$ Character field $\Q$ Dimension $4$ Newform subspaces $1$ Sturm bound $600$ Trace bound $0$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$4000 = 2^{5} \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 4000.g (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$600$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(4000, [\chi])$$.

Total New Old
Modular forms 72 4 68
Cusp forms 32 4 28
Eisenstein series 40 0 40

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

 $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} + 2q^{11} - 2q^{19} - 2q^{41} - 2q^{49} - 2q^{59} + 4q^{81} + 2q^{89} + 4q^{91} - 2q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(4000, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field Image CM RM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
4000.1.g.a $$4$$ $$1.996$$ $$\Q(i, \sqrt{5})$$ $$D_{5}$$ $$\Q(\sqrt{-10})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{3})q^{7}-q^{9}-\beta _{2}q^{11}-\beta _{1}q^{13}+\cdots$$

## Decomposition of $$S_{1}^{\mathrm{old}}(4000, [\chi])$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(4000, [\chi]) \cong$$ $$S_{1}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(800, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(1000, [\chi])$$$$^{\oplus 3}$$