Primitive roots and Legendre symbol

Question

One knows that if $a$ is a primitive root modulo $p$, then $(\frac{a}{p})=-1$.
Thus, if $(\frac{a}{p})=1$ we know that $a$ is not a primitive root modulo $p$.

Are there any other properties between Legendre symbol and primitive roots?

Answer 1

Suppose that $p-1$ is divisible by $3$. If $a$ is a cubic residue of $p$, then $a$ is not a primitive root of $p$. Similarly for other prime divisors $q$ of $p-1$.