Question 73
Let $\zeta = \cos \frac{2\pi}{13} + i \sin \frac{2\pi}{13}$. Suppose $a \gt b \gt c \gt d$ are positive integers satisfying
$$
\left|\zeta^{a} + \zeta^{b} + \zeta^{c} + \zeta^{d}\right| = \sqrt{3}
$$
Compute the smallest possible value of $1000a + 100b + 10c + d$.
$$
\left|\zeta^{a} + \zeta^{b} + \zeta^{c} + \zeta^{d}\right| = \sqrt{3}
$$
Compute the smallest possible value of $1000a + 100b + 10c + d$.