Why is this statement not compiling correctly?

For some reason, the following excerpt from my code is not compiling correctly and I can't figure out why.

    $$\begin{split} \frac{100}{-6} &= -16(-6) + 4\\ &\implies z^100 = z^4\\ \\ z^4 &= \left(e^{\frac{-i\pi}{3}\right)^4 \\ &=e^{\frac{-4\pi}{3}} \\ \\ &\cos\left(\frac{-4\pi}{3}\right) = -\frac{1}{2}, ~\sin\left(\frac{-4\pi}{3}\right) = \frac{\sqrt{3}}{2} \\ &\implies z = -\frac{1}{2} + \frac{\sqrt{3}}{2} i \\ \end{split}$$


The error I get is that paragraph ended before split was complete and that a missing endgroup has been inserted. Any help would be greatly appreciated.

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Welcome to TeX.SX! Please make your code compilable (if possible), or at least complete it with \documentclass{...}, the required \usepackage's, \begin{document}, and \end{document}. That may seem tedious to you, but think of the extra work it represents for TeX.SX users willing to give you a hand. Help them help you: remove that one hurdle between you and a solution to your problem. –  Martin Schröder Feb 15 '14 at 13:52

You are missing a closing brace in line z^4:
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