  We study the geometry of some varieties of sums of powers related to the
Klein quartic. This allows us to describe the birational geometry of certain
moduli spaces of abelian surfaces. In particular we show that the moduli space
$\mathcal{A}_2(1,7)^{-}_{sym}$ of $(1,7)$-polarized abelian surfaces with a
symmetric theta structure and an odd theta characteristic is unirational by
showing that it admits a dominant morphism from a unirational conic bundle.
