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

Abstract

We prove an unconditional, effective joint Sato--Tate distribution for the Fourier coefficients of two twist-inequivalent, non-CM newforms $f$ and $f^{\prime }$. Our result generalises a result of Thorner, which holds for rectangular regions, by extending it to any measurable region of $[-2,2{]}^2$ whose boundary consists of finitely many 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 $f^{\prime }$. 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.

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