Polynomial numerical indices of Banach spaces with 1-unconditional bases

The only infinite-dimensional complex space with 1-unconditional basis which has polynomial numerical index of order 2 equal to 1 is c0. In the real case, there is no space of this type. We also show that, in the complex case, if X is an infinite-dimensional Banach sequence space with absolute norm whose Köthe dual is norming and has polynomial numerical index of order 2 equal to 1, then c0⊂X⊂ℓ∞. In the real case, again there is no space of this type.