n=2: mu ok=True  charpoly ok=True  beta-gap ok=True  tau formula (both forms) ok=True  derivative ok=True   tau(p) = -1/(p - 2)
n=3: mu ok=True  charpoly ok=True  beta-gap ok=True  tau formula (both forms) ok=True  derivative ok=True   tau(p) = -2*(p + 1)/(p - 3)
n=4: mu ok=True  charpoly ok=True  beta-gap ok=True  tau formula (both forms) ok=True  derivative ok=True   tau(p) = -3*(2*p + 1)/(p - 4)
n=5: mu ok=True  charpoly ok=True  beta-gap ok=True  tau formula (both forms) ok=True  derivative ok=True   tau(p) = -4*(3*p + 1)/(p - 5)
n=6: mu ok=True  charpoly ok=True  beta-gap ok=True  tau formula (both forms) ok=True  derivative ok=True   tau(p) = -5*(4*p + 1)/(p - 6)
n=7: mu ok=True  charpoly ok=True  beta-gap ok=True  tau formula (both forms) ok=True  derivative ok=True   tau(p) = -6*(5*p + 1)/(p - 7)
n=8: mu ok=True  charpoly ok=True  beta-gap ok=True  tau formula (both forms) ok=True  derivative ok=True   tau(p) = -7*(6*p + 1)/(p - 8)
all exact checks passed: True
