Effective Joint Sato-Tate Distribution and Sign Change of Symmetric Power Coefficients

We prove an unconditional, effective joint Sato-Tate distribution for the Fourier coefficients of two twist-inequivalent, non-CM newforms $f$ and $f^{\mathrm{\prime }}$. Our result generalises a result of Thorner, which holds for rectangular regions, by extending it to a wide range of measurable subsets of $[-2,2{]}^2$. Indeed, our theorem applies to any measurable region whose boundary consists of a finite number of continuous curves of finite length. As a consequence, we develop a unified framework to study various arithmetic properties of Fourier coefficients of symmetric power $L$-functions attached to $f$ and $f^{\mathrm{\prime }}$. In particular, for these coefficients (and their polynomial expressions), we obtain effective distribution results, quantitative statements on simultaneous sign behaviour, and bounds for the first sign change.