Distinguishing newforms by their Hecke eigenvalues
In the paper under review, the authors concern the question of distinguishing modular forms by their Hecke eigenvalues. Let \(p\) be a prime, \(f\) and \(g\) be distinct normalized Hecke eigenforms of weights \(k_1\) and \(k_2\) with complex multiplication lying in the subspace of newforms with Fourier coefficients \(\{ n^{k_1-1/2} \lambda_f(n) \}_{n \in \mathbb{N}}\) and \(\{ n^{k_2-1/2} \lambda_g(n) \}_{n \in \mathbb{N}}\) respectively. Let us define the following set: \[ S:=\{ p : \lambda_f(p) = \lambda_g(p) \} .\] Then, in the paper under review, the authors calculate the natural density of \(S\). More precisely, they find that the upper natural density of \(S\) is \(\leq 3/4\) if \(f \neq g\) and it is equal to \(1/2\) when \(f\) and \(g\) have different weights and have the same associated CM (quadratic) field. Further, \(f\) and \(g\) have different associated CM (quadratic) fields if and only if the natural density of S is \(1/4\). For another problem where at least one of \(f, g\) is a non-CM form, define the following sets: \[ S_+(x, \alpha):= \{ p \leq x : \theta_f(p)+\theta_g(p) = \alpha \} \] and \[ S_-(x, \beta):= \{p \leq x : \theta_f(p)-\theta_g(p) = \beta \}. \] Then, the authors study also natural density of sets \(S_+(x, \alpha)\) and \(S_-(x, \beta)\). Here \(\theta_f(p), \theta_g(p) \in [0, \pi]\) are the angles associated to the \(p\)-th Hecke eigenvalues of \(f\), \(g\) respectively and \(\alpha \in [0, 2\pi]\) and \(\beta \in [-\pi, \pi]\). In this case, the authors show that sets \(S_+(x, \alpha)\) and \(S_-(x, \beta)\) have natural density zero when \(f\) and \(g\) are distinct and not twists of each other. Finally, they establish an explicit link between the elements of these sets and sign changes of Fourier coefficients at prime powers and this connection allows them to improve some previous results in the literature. Proofs are based on calculations on analytic number theory objects.
signs of Fourier coefficients of newforms
scientific article; zbMATH DE number 7381867
scientific article