The moduli space of bilevel-6 abelian surfaces
  The moduli space of abelian surfaces with polarisation of type (1,6) and a bilevel structure has positive Kodaira dimension. By contrast, Mukai has shown that the moduli space of bilevel-t abelian sufaces is rational for t=2,3,4,5. 
G.K. Sankaran, J.G. Spandaw