O14-915
> restart:

> p:=10;s:=z->sum(a[n]*z^n,n=1..p);

                               p := 10


> sys:={l[0]+l[1]-1,l[0]-l[1]}:sol:=solve(sys);

                   sol := {l[1] = 1/2, l[0] = 1/2}

> s1:=subs(sol,l[0]*s(z)+l[1]*s(-z));

                   2         4         6         8          10
       s1 := a[2] z  + a[4] z  + a[6] z  + a[8] z  + a[10] z

> sys2:={l[0]+l[1],l[0]-l[1]-1}:sol2:=solve(sys2);s2:=subs(sol2,l[0]*s(z)+l[1]*s(-z));

                  sol2 := {l[0] = 1/2, l[1] = -1/2}


                              3         5         7         9
         s2 := a[1] z + a[3] z  + a[5] z  + a[7] z  + a[9] z


> r:=-1/2+I*sqrt(3)/2;

> sys1:={l[0]+l[1]+l[2]-1,l[0]+r*l[1]+r^2*l[2],l[0]+r^2*l[1]+r*l[2]}:sol1:=solve(sys1);

             sol1 := {l[2] = 1/3, l[1] = 1/3, l[0] = 1/3}

> s1:=expand(subs(sol1,l[0]*s(z)+l[1]*s(r*z)+l[2]*s(r^2*z)));

                              6         9         3
                  s1 := a[6] z  + a[9] z  + a[3] z

> sys2:={l[0]+l[1]+l[2],l[0]+r*l[1]+r^2*l[2]-1,l[0]+r^2*l[1]+r*l[2]}:sol2:=solve(sys2);

> s2:=expand(subs(sol2,l[0]*s(z)+l[1]*s(r*z)+l[2]*s(r^2*z)));

                          10         4         7
             s2 := a[10] z   + a[4] z  + a[7] z  + a[1] z

> sys3:={l[0]+l[1]+l[2],l[0]+r*l[1]+r^2*l[2],l[0]+r^2*l[1]+r*l[2]-1}:sol3:=solve(sys3);

> s3:=expand(subs(sol3,l[0]*s(z)+l[1]*s(r*z)+l[2]*s(r^2*z)));

                              8         2         5
                  s3 := a[8] z  + a[2] z  + a[5] z