> restart;
> with(combinat):
> q:=4; d:=2;

                                q := 4


                                d := 2

> for j from 0 to d do
> for theta from 0 to d do
>   print( [j,theta], add( (-q)^u*(q-1)^(j-u)*numbcomb(d-u,j-u)*numbcomb(theta,u),u=0..j ) );
> end do: end do:

                              [0, 0], 1


                              [0, 1], 1


                              [0, 2], 1


                              [1, 0], 6


                              [1, 1], 2


                              [1, 2], -2


                              [2, 0], 9


                              [2, 1], -3


                              [2, 2], 1

> solve(alpha^2+(q-2)*alpha+1,alpha);

                                -1, -1

> q^(d/2)/( 1+(q-1)*alpha )^d;

                                  4
                            --------------
                                         2
                            (1 + 3 alpha)

> alpha:=-1;

                             alpha := -1

> q^(d/2)/( 1+(q-1)*alpha )^d;

                                  1

> solve(alpha_0^3-alpha,alpha_0);

                                 1/2               1/2
                -1, 1/2 + 1/2 I 3   , 1/2 - 1/2 I 3

> 
>