When is $2023$ a cubic residue residue mod $p$ that is $1\pmod{3}$?
I do know about quadratic reciprocity, since $x^{2}\equiv2023\pmod{p}$ is only solvable if $p\equiv\pm1\pm3\pm9\pmod{28}$, but I don’t know when $x^{3}\equiv2023\pmod{p}$ is solvable.
I know about cubic reciprocity, but I don’t know if it will work in the similar way as Legendre symbol.
For instance, there is 14 prime number years in this century, but $2089$ is the only prime number year in this century, where $x^{3}\equiv2023\pmod{2089}$ is solvable. $2089$ is a prime that is $1\pmod{3}$.
Hint: $p$ must be $1\pmod{3}$, and $2023=17^{2}\cdot7.$