newton’s method to find fixed points of a jacobi theta function

The first is the answer to the question: fix the nome to be e^{-pi}, what does Newton’s
method to find the fixed point of \theta_{3}(z,e^{-\pi}) produce on the z-plane?

The second is the answer to the question fix $z=0$. Perform Newton’s method on z on the q-plane to find the fixed point.

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