In this basis, the operator ˆP+=ˆa†1ˆa†2+ˆa†3ˆa†4+ˆa†5ˆa†6 From this we can determine that ˆP−|1,4,6⟩=ˆP−|2,3,6⟩=ˆP−|2,4,5⟩=0 so those states all have eigenvalue 0.