Pair Sato–Tate equidistribution conjecture for non-CM primitive forms

For $i=1,2$, let $g_i=\sum_{n=1}^{\infty}b_i(n)q^n$ be primitive eigenforms of weight $2k_i$ and level $2N_i$, and define

$$B_i(p)=\frac{b_i(p)}{2p^{k_i-\frac12}}\in[-1,1].$$

Let $g_1,g_2$ be distinct non-CM primitive forms of weights $2k_1,2k_2$ and levels $2N_1,2N_2$, respectively, and assume that they are not twists of each other. Pair Sato–Tate equidistribution conjecture. For any two subintervals $I_1,I_2\subseteq[-1,1]$, define

S(I_1,I_2)=\left\\{p\in\mathbb{P}:p\nmid 2N_1N_2,\\ B_1(p)\in I_1,\\ B_2(p)\in I_2\right\\}, S(I_1,I_2)(x)=\left\\{p\leq x:p\in S(I_1,I_2)\right\\}.

Then

d(S(I_1,I_2))=\lim_{x\to\infty}\frac{\\#S(I_1,I_2)(x)}{\pi(x)}=\mu_{\mathrm{ST}}(I_1)\mu_{\mathrm{ST}}(I_2)=\frac{4}{\pi^2}\int_{I_1}\sqrt{1-s^2}\\,ds\int_{I_2}\sqrt{1-t^2}\\,dt.

In other words, the Fourier coefficients at primes are conjectured to be independently distributed with respect to the Sato–Tate distribution. This is the pair form of the Sato–Tate conjecture and is assumed in the paper to derive the equidistribution result for products of Fourier coefficients; its general validity remains open.