# Time Average Spin Angular Momentum

I’m a bit confused about time averages in electromagnetism using complex amplitudes. Specifically about the angular momentum of the fields- the spin part is proportional to $$\bf{E} \times \bf{A}$$. In the Coulomb gauge in the absence of any sources, $$\bf{E} = -c^{-1} \partial_t \bf{A}$$, and so the electric field is proportial to the vector potential for a monochromatic plane wave, and the spin part seems to vanish identically. However, using the time average of the angular momentum we have that $$, in which case the spin is not necessarily zero. How is the time average of zero a non zero number?