Let \(f=\sum_{n>0}a_nq^n\) be a normalized newform of level \(N\), weight \(k\) with nebentypus \(\varepsilon\) and without CM. The paper deals with the relation between the distribution of coefficients of \(f\) and the Sato-Tate measure. Starting with the formula \[ \begin{split} \lim_{x\to +\infty} \frac{\# \left\{ p\in\mathcal{P}(x): p\equiv m\pmod M , \frac{a_p}{2\zeta p^{(k-1)/2}}\leq a\right\} }{\# \left\{ p\in\mathcal{P}(x)\,:\,p\equiv m\pmod M \right\} }\\ = \frac{2}{\pi} \int_{-1}^a \sqrt{1-x^2}\,dx\quad a\in[-1,1] \end{split} \] (where \(\zeta\) is a root of unity, \(\varepsilon(m)=\zeta^2\,\), \(M\) is a multiple of the conductor of \(\varepsilon\) and \(\mathcal{P}(x)\) is the set of primes \(p\leq x\) such that \(p\nmid N\)), the authors present numerical evidence for a series of conjectures aimed at the generalization of the above result. In particular they consider a set of algebraically independent newforms \(f_1,\dots,f_d\) (i.e., there is no Dirichlet character \(\chi\) such that \(f_i=f_j\otimes \chi\)) and relate the distribution of their Fourier coefficients belonging to some fixed arithmetic progressions to the Sato-Tate measure \(\left(\frac{2}{\pi}\right)^d \int_{-1}^{a_1}\cdots \int_{-1}^{a_d} \sqrt{1-x_1^2}\cdots \sqrt{1-x_d^2}\,dx_1\cdots dx_d\,\). Special attention is dedicated to the case \(d=1\) for newforms with rational coefficients (where the generalization only refers to the fact that they are imposing congruence conditions on the coefficients): in that case the authors show that the conjecture (with some additional conditions) would yield a proof of the Lang-Trotter conjecture for the elliptic curve associated to \(f\) and a complete characterization for the set of vanishing Fourier coefficients of \(f\) (for \(k\geq 4\)).