 Solution

The unique order-two character is ${\chi }_3$, the Legendre symbolIt is $1$ on $1,2,4$, is $-1$ on $3,5,6$, and is zero on multiples of $7$. Since the unique quadratic subfield of $\mathbb{Q}({\zeta }_7)$ is $\mathbb{Q}(\sqrt{-7})$, this is the character corresponding to $K$.