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We obtain a nontrivial upper bound for almost all elements of the sequencesof real numbers which are multiplicative and at the prime indices aredistributed according to the Sato--Tate density. Examples of such sequencescome from coefficients of several $L$-functions of elliptic curves and modularforms. In particular, we show that $|\tau(n)|\le n^{11/2} (\log n)^{-1/2+o(1)}$for a set of $n$ of asymptotic density 1, where $\tau(n)$ is the Ramanujan$\tau$ function while the standard argument yields $\log 2$ instead of $-1/2$in the power of the logarithm. Another consequence of our result is that in thenumber of representations of $n$ by a binary quadratic form one has slightlymore than square-root cancellations for almost all integers $n$.  In addition we obtain a central limit theorem for such sequences, assuming aweak hypothesis on the rate of convergence to the Sato--Tate law. For Fouriercoefficients of primitive holomorphic cusp forms such a hypothesis is knownconditionally assuming the automorphy of all symmetric powers of the form andseems to be within reach unconditionally using the currently establishedpotential automorphy.

On the Typical Size and Cancelations Among the Coefficients of Some Modular Forms