<s>[INST] how did we find the Legendre symbol? The Legendre symbol from a prime $p>2$ and $(a,p)=1$ is:
$$\left ( \frac{a}{p} \right )=\left\{\begin{matrix}
1, & \text{if a is a quadratic residue} \\ 
 -1& \text{if a is a non-quadratic residue}
\end{matrix}\right.$$
According to my notes for $p=11$:
$$\left ( \frac{1}{11} \right )=\left ( \frac{3}{11} \right )=\left ( \frac{4}{11} \right )=\left ( \frac{5}{11} \right )=\left ( \frac{9}{11} \right )=1$$
$$\left ( \frac{2}{11} \right )=\left ( \frac{6}{11} \right )=\left ( \frac{7}{11} \right )=\left ( \frac{8}{11} \right )=\left ( \frac{10}{11} \right )=-1$$
How did we find the Legendre symbol?
 [/INST] Compute all the squares $1^2,2^2,\ldots ,10^2$ modulo $11$. Then you will see that $1,3,4,5,9$ are quadratic residues, i.e., appearing in your list, and the other ones not. For example, $4^2\equiv 5$ mod $11$, hence $5$ is a quadratic residue, i.e., $(5/11)=1$.
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