I'm just getting to know the Fourier transform. I computed $64$ values of the periodic sequence $[1, 2, 4, 8, 16, 32, 29, 23, 11, 22, 9, 18, 1, 2, ...]$ with period length of $12$. Using a machine, I computed
x = [1, 2, 4, 8, 16, 32, 29, 23, 11, 22, 9, 18, 1, 2, ...]
y = fft(x)
Looking at $y$, I see the following complex numbers.
[ 890. +0.j , -44.38255743 -5.25243373j,
-48.01118625 -11.76210086j, -56.39223493 -22.07948054j,
-78.85632437 -45.70630911j, -237.45489019-204.72085508j, ...]
Computing the inverse, I find
array([ 1.-9.99200722e-16j, 2.+2.77555756e-16j, 4.-1.06143818e-15j,
8.+1.52655666e-15j, 16.-6.66133815e-16j, 32.+1.27675648e-15j,
29.-6.03896362e-16j, 23.+1.13797860e-15j, 11.+6.58557145e-16j,
22.+2.62402416e-16j, 9.+1.44243956e-15j, 18.+4.08756964e-16j,
1.-1.44328993e-15j, 2.-2.72004641e-15j, ...]
So I can see my list is back in the real parts, but that's not quite an inverse. What am I missing here?