Finite field with $3$rd primitive root of unity.

Question

I like to find those finite field $GF(p^n)$ which contains primitive $3$rd root of unity. One thing is clear that $GF(p^n)^*$ is cyclic group of size $p^n-1. $ So for $3$rd primitive root we must have $3/(p^n-1).$ i.e. $$p^n-1\equiv 0\mod3$$ so for primes of the form $p=3n+1$ finite field $GF(p^n)$ must have $3$rd primitive root of unity. Also i noticed that for any prime $p$ and even $n$ $p^n\equiv 1\mod3$ i.e. finite field $GF(p^n)$ always has $3$rd primitive root of unity if $n$ is even. Please help me to find all prime $p$ so that $GF(p^n)$ has $3$rd primitive root of unity. Thanks.