On the first negative Hecke eigenvalue of an automorphic representation of GL2($${\mathbb{A}_\mathbb{Q}}$$)

Let π be a self-dual irreducible cuspidal automorphic representation of GL2( $${\mathbb{A}_\mathbb{Q}}$$ ) with trivial central character. Its Hecke eigenvalue ⁁π (n) is a real multiplicative function in n. We show that λπ (n) < 0 for some $$n \ll Q_\pi ^{2/5}$$ , where Qπ denotes (a special value of) the analytic conductor. The value $${2 \over 5}$$ is the first explicit exponent for Hecke-Maass newforms.