Benford's Law for Coefficients of Newforms

Let $f(z)=\sum_{n=1}^\infty \lambda_f(n)e^{2\pi i n z}\in S_{k}^{new}(\Gamma_0(N))$beanormalizedHeckeeigenformofevenweight$k\geq2$on$\Gamma_0(N)$withoutcomplexmultiplication\mathrm{.}Let$$\mathbb{P}$$denotethesetofallprimes\mathrm{.}Weprovethatthesequence${\lambda_f(p)}_{p\in$\mathbb{P}$}$doesnotsatisfyBenfor{d}^{\mathrm{\prime }}sLawinanybase$b\geq2$\mathrm{.}However,givenabase$b\geq2$andastringofdigits$S$inbase$b, the set \[ A_{\lambda_f}(b,S):=\{\text{p$prime:thefirstdigitsof$\lambda_f(p)$inbase$b$aregivenby$S$}} ] has logarithmic density equal to ${\log }_b(1+S^{-1})$. Thus $\{{\lambda }_f(p){\}}_{p\in \mathbb{P}}$ follows Benford’s Law with respect to logarithmic density. Both results rely on the now-proven Sato-Tate Conjecture.