I want to know properties of the following sum: $$\sum_{j=0}^{p-1} \omega^{\beta j^2}= ~? $$ where $p$ is a prime, and $\omega^p=1$, is a $p$th root of unity (and $\beta$ is an integer between $0$ and $p-1$). If anyone has any references related to this, let me know.

Also, if someone can tell me about the sum: $$\sum_{j=0}^{p-1} \omega^{\beta j^n}= ~? $$ for higher values of $n$ such as 3, that would also be nice.