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Refinements of strong multiplicity one for $\mathrm{GL}(2)$

For distinct unitary cuspidal automorphic representations $π_1$ and $π_2$ for $\mathrm{GL}(2)$ over a number field $F$ and any $α\in\Bbb{R}$, let $\mathcal{S}_α$ be the set of primes $v$ of $F$ for which $λ_{π_1}(v)\neq e^{iα} λ_{π_2}(v)$, where $λ_{π_i}(v)$ is the Fourier coefficient of $π_i$ at $v$. In this article, we show that the lower Dirichlet density of $\mathcal{S}_α$ is at least $\frac{1}{16}$. Moreover, if $π_1$ and $π_2$ are not twist-equivalent, we show that the lower Dirichlet densities of $\mathcal{S}_α$ and $ \cap_α\mathcal{S}_α$ are at least $\frac{2}{13}$ and $\frac{1}{11}$, respectively. Furthermore, for non-twist-equivalent $π_1$ and $π_2$, if each $π_i$ corresponds to a non-CM newform of weight $k_i\ge 2$ and with trivial nebentypus, we obtain various upper bounds for the number of primes $p\le x$ such that $λ_{π_1}(p)^2 = λ_{π_2}(p)^2$. These present refinements of the works of Murty-Pujahari, Murty-Rajan, Ramakrishnan, and Walji.