= Space-time average of periodic modes
{title2=$\langle\cdot\rangle_{x,t}$}
A space-time average over periodic modes integrates over the spatial periodic cell and one temporal period. For real spatial <Fourier modes> sharing a <wavevector>, $\langle\operatorname{Re}(\mathbf v e^{i\mathbf k\cdot\mathbf x})\times\operatorname{Re}(\mathbf w e^{i\mathbf k\cdot\mathbf x})\rangle_x=\operatorname{Re}(\mathbf v^*\times\mathbf w)/2$. A matching sine or cosine temporal factor supplies another $1/2$. Opposite wavevectors can have nonzero spatial cross averages, so their temporal coefficients must be checked rather than discarded merely because the wavevectors are distinct.
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