# Properties

 Label 500.1.h Level $500$ Weight $1$ Character orbit 500.h Rep. character $\chi_{500}(99,\cdot)$ Character field $\Q(\zeta_{10})$ Dimension $8$ Newform subspaces $1$ Sturm bound $75$ Trace bound $0$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$500 = 2^{2} \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 500.h (of order $$10$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$100$$ Character field: $$\Q(\zeta_{10})$$ Newform subspaces: $$1$$ Sturm bound: $$75$$ Trace bound: $$0$$

## Dimensions

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

Total New Old
Modular forms 48 32 16
Cusp forms 8 8 0
Eisenstein series 40 24 16

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 8 0 0 0

## Trace form

 $$8 q + 2 q^{4} + 2 q^{9} + O(q^{10})$$ $$8 q + 2 q^{4} + 2 q^{9} - 2 q^{16} - 4 q^{26} + 4 q^{29} - 6 q^{34} - 2 q^{36} - 4 q^{41} - 8 q^{49} - 4 q^{61} + 2 q^{64} + 4 q^{74} - 2 q^{81} - 6 q^{89} + O(q^{100})$$

## 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.h.a $8$ $0.250$ $$\Q(\zeta_{20})$$ $D_{5}$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{20}q^{2}+\zeta_{20}^{2}q^{4}-\zeta_{20}^{3}q^{8}-\zeta_{20}^{4}q^{9}+\cdots$$