We consider the fluctuations of regular functions $f$ of a Wigner matrix $W$
viewed as an entire matrix $f(W)$. Going beyond the well studied tracial mode,
$\mathrm{Tr}[f(W)]$, which is equivalent to the customary linear statistics of
eigenvalues, we show that $\mathrm{Tr}[f(W)]$ is asymptotically normal for any
non-trivial bounded deterministic matrix $A$. We identify three different and
asymptotically independent modes of this fluctuation, corresponding to the
tracial part, the traceless diagonal part and the off-diagonal part of $f(W)$
in the entire mesoscopic regime, where we find that the off-diagonal modes
fluctuate on a much smaller scale than the tracial mode. In addition, we
determine the fluctuations in the Eigenstate Thermalisation Hypothesis [Deutsch
1991], i.e. prove that the eigenfunction overlaps with any deterministic matrix
are asymptotically Gaussian after a small spectral averaging. In particular, in
the macroscopic regime our result generalises [Lytova 2013] to complex $W$ and
to all crossover ensembles in between. The main technical inputs are the recent
multi-resolvent local laws with traceless deterministic matrices from the
companion paper [Cipolloni, Erd\H{o}s, Schr\”oder 2020].
