On Hecke eigenvalues of newforms
Let \(f\) and \(g\) be holomorphic cuspidal Hecke eigenforms of fixed level \(N\) and even integral weights \(k_1\) and \(k_2\), respectively which are different normalized newforms. Given \(A\in \mathbb N\), the strong multiplicity one theorem for \(\text{GL}_2\) implies that there are infinitly many primes \(p\) with \((p,A)=1\) such that \(\lambda_f(p)\neq\lambda_g(p)\), where \(\lambda_f(p)\) resp. \(\lambda_g(p)\) denote the normalized eigenvalues of \(f\) resp. \(g\). A natural question then is to ask for the smallest such prime \(p\). In this paper the author proves the following Theorem: Suppose that \(k_1,\;k_2\geq 2\) and \(k_1\neq k_2\). Let \(A\in \mathbb N\) with \(N| A\). Then there exists a prime \(p\) with \((p,A)=1\) and \(p \ll_A \max k_1,k_2\cdot| k_1-k_2|\;\) such that \(\lambda_f(p)\neq\lambda_g(p)\). The constant implied in \(\ll_A\) is effective. The proof is mainly based on the theory of modular forms modulo \(l\) due to Serre and Katz.
scientific article; zbMATH DE number 2123985