Determining newforms via arithmetic relations among Fourier coefficients

Abstract

We investigate the distribution of primes satisfying arithmetic inequalities involving the Fourier coefficients of two non-CM newforms at prime powers. More precisely, we establish asymptotic formulas for the number of primes for which the differences, products, and ratios of the Fourier coefficients satisfy prescribed inequalities, together with explicit estimates for the corresponding densities. The proofs combine an effective joint Sato--Tate theorem with a geometric analysis of the associated semi-algebraic regions. As applications, we obtain new multiplicity one criteria, improve a theorem of Matomäki on small differences between Fourier coefficients, establish density-one analogues in the spirit of the Atkin--Serre conjecture, and derive a new characterization of twist-equivalence through the distribution of ratios of Fourier coefficients.

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