Irreducibility of polynomials related to quadratic residues

Regarding the Galois group of the factor $g(x)=g(q,x)$of these Fekete polynomials that is conjectured to be irreducible, here is PARI code that counts, for each prime $q equiv 1 bmod 4$, $17 leq q leq 100$, the number of primes $p$ for which $g(q,x)$ modulo $p$, has $n$ linear factors, for primes $p$ up to half a million, stored in the matrix $c[q,n+1]$, done for $0leq nleq 14$.

f(m,x)=sum(i=1,m-1,kronecker(i,m)*x^i))

g(m,x)=f(m,x)/(x*(x-1)^2*(x+1))

for(n=0,14,forprime(q=17,100,if(q%4==1,forprime(p=2,500000,if(matsize(polrootsmod(g(q,x),p))==[n,1], c[q,n+1]++)))))

If I did not make any mistakes, then, for these nine primes $q=17,29,37,41,53,61,73,89,97$, here is the $9$ by $15$ matrix showing $n=0$ to $14$.

[25204 3 12634 0 3084 0 538 0 72 0 3 0 0 0 0]

[25360 2 12486 0 3101 0 530 0 50 0 9 0 0 0 0]

[25133 1 12637 0 3187 0 519 0 59 0 2 0 0 0 0]

[25320 2 12519 1 3081 0 543 0 65 0 6 0 1 0 0]

[25293 2 12449 0 3206 0 500 0 81 0 6 0 1 0 0]

[25176 2 12603 0 3157 0 527 0 69 0 4 0 0 0 0]

[25110 3 12564 0 3274 0 516 0 65 0 5 0 1 0 0]

[25286 3 12554 0 3116 0 502 0 71 0 6 0 0 0 0]

[25112 3 12711 0 3132 0 526 0 49 0 5 0 0 0 0]