# Properties

 Label 500.1.d Level $500$ Weight $1$ Character orbit 500.d Rep. character $\chi_{500}(499,\cdot)$ Character field $\Q$ Dimension $4$ Newform subspaces $2$ Sturm bound $75$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$500 = 2^{2} \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 500.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$20$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$75$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(500, [\chi])$$.

Total New Old
Modular forms 14 4 10
Cusp forms 4 4 0
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

 $$4 q + 4 q^{4} - 2 q^{6} + 2 q^{9} + O(q^{10})$$ $$4 q + 4 q^{4} - 2 q^{6} + 2 q^{9} - 2 q^{14} + 4 q^{16} - 4 q^{21} - 2 q^{24} - 2 q^{29} + 2 q^{36} - 2 q^{41} - 2 q^{46} + 2 q^{49} - 4 q^{54} - 2 q^{56} - 2 q^{61} + 4 q^{64} - 4 q^{69} - 4 q^{84} - 2 q^{86} - 2 q^{89} - 2 q^{94} - 2 q^{96} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(500, [\chi])$$ into newform subspaces

Label Dim $A$ Field Image CM RM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
500.1.d.a $2$ $0.250$ $$\Q(\sqrt{5})$$ $D_{5}$ $$\Q(\sqrt{-5})$$ None $$-2$$ $$1$$ $$0$$ $$1$$ $$q-q^{2}+(1-\beta )q^{3}+q^{4}+(-1+\beta )q^{6}+\cdots$$
500.1.d.b $2$ $0.250$ $$\Q(\sqrt{5})$$ $D_{5}$ $$\Q(\sqrt{-5})$$ None $$2$$ $$-1$$ $$0$$ $$-1$$ $$q+q^{2}+(-1+\beta )q^{3}+q^{4}+(-1+\beta )q^{6}+\cdots$$