Example 1 Calculate $\left(\frac{137}{227}\right)$.
To determine if 227 is a prime number, by Theorem 1 we get
$$\begin{aligned}
\left(\frac{137}{227}\right) & =\left(\frac{-90}{227}\right)=\left(\frac{-1}{227}\right)\left(\frac{2 \cdot 3^{2} \cdot 5}{227}\right) \\
& =(-1)\left(\frac{2}{227}\right)\left(\frac{3^{2}}{227}\right)\left(\frac{5}{227}\right) \\
& =(-1)\left(\frac{2}{227}\right)\left(\frac{5}{227}\right) .
\end{aligned}$$

By Theorem 3, we get
$$\left(\frac{2}{227}\right)=-1$$

By Theorem 5, Theorem 1, and Theorem 3, we get
$$\left(\frac{5}{227}\right)=\left(\frac{227}{5}\right)=\left(\frac{2}{5}\right)=-1$$

From the above three equations, we get
$$\left(\frac{137}{227}\right)=-1$$

This indicates that the congruence equation $x^{2} \equiv 137(\bmod 227)$ has no solution.