For prime numbers $p$, does the Legendre symbol $(\frac {5}{p})$ depend only on the congruence class of $p$ modulo $5$? For prime numbers $p$, does the Legendre symbol $(\frac {5}{p})$
depend only on the congruence class
of $p$ modulo $5$?
So I know that $(\frac {5}{11}) = 5^5$mod $11$.
I don't think it does, and I know that I need to use Quadratic Reciprocity and the fact that $5$ is not congruent to $11$ (mod $4)$ to show it but I'm not sure how.
How should I approach this?

As $5\not\equiv 4\pmod 4$, we have (for odd $p$) $$(\frac {5}{p})=(\frac {p}{5})=(\frac {p\bmod 5}{5})$$
which shows that the result depends only on $p\bmod 5$ for odd $p$.
In order for $(\frac {5}{p})$ to depend only on $p\bmod 5$ for all primes $p$, however, we need to check additionally that e.g. $(\frac {5}{2})=(\frac {5}{7})$. Does it?
