# Properties

 Label 1600.1.bd Level $1600$ Weight $1$ Character orbit 1600.bd Rep. character $\chi_{1600}(31,\cdot)$ Character field $\Q(\zeta_{10})$ Dimension $16$ Newform subspaces $2$ Sturm bound $240$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1600.bd (of order $$10$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$200$$ Character field: $$\Q(\zeta_{10})$$ Newform subspaces: $$2$$ Sturm bound: $$240$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(1600, [\chi])$$.

Total New Old
Modular forms 64 16 48
Cusp forms 16 16 0
Eisenstein series 48 0 48

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 0 0 0 16

## Trace form

 $$16 q + 8 q^{9} + O(q^{10})$$ $$16 q + 8 q^{9} - 12 q^{17} + 4 q^{25} + 12 q^{33} - 4 q^{41} + 16 q^{57} + 4 q^{65} - 8 q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(1600, [\chi])$$ into newform subspaces

Label Dim $A$ Field Image CM RM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1600.1.bd.a $8$ $0.799$ $$\Q(\zeta_{20})$$ $A_{5}$ None None $$0$$ $$-6$$ $$0$$ $$0$$ $$q+(-1+\zeta_{20}^{6})q^{3}-\zeta_{20}^{3}q^{5}+\zeta_{20}^{5}q^{7}+\cdots$$
1600.1.bd.b $8$ $0.799$ $$\Q(\zeta_{20})$$ $A_{5}$ None None $$0$$ $$6$$ $$0$$ $$0$$ $$q+(1-\zeta_{20}^{6})q^{3}-\zeta_{20}^{3}q^{5}-\zeta_{20}^{5}q^{7}+\cdots$$

This is the first newspace containing multiple newforms with projective image $A_5$.