Patterns of primes in joint Sato--Tate distributions

For $j=1,2$, let $f_j(z)={\sum }_{n=1}^{\mathrm{\infty }}{a}_j(n)e^{2\pi inz}$ be a holomorphic, non-CM cuspidal newform of even weight $k_j\ge 2$ with trivial nebentypus. For each prime $p$, let ${\theta }_j(p)\in [0,\pi ]$ be the angle such that $a_j(p)=2p^{(k-1)\mathrm{/}2}\cos {\theta }_j(p)$. The now-proven Sato–Tate conjecture states that the angles $({\theta }_j(p))$ equidistribute with respect to the measure $d{\mu }_{\mathrm{S}T}=\frac{2}{\pi }{\sin }^2\theta d\theta$. We show that, if $f_1$ is not a character twist of $f_2$, then for subintervals $I_1,I_2\subset [0,\pi ]$, there exist infinitely many bounded gaps between the primes $p$ such that ${\theta }_1(p)\in I_1$ and ${\theta }_2(p)\in I_2$. We also prove a common generalization of the bounded gaps with the Green–Tao theorem.