On Signs of Hecke eigenvalues of Ikeda lifts

Abstract

Let $F$ be an Ikeda lift, ${\lambda }_F(m)$ be the eigenvalue corresponding to the Hecke operator $T(m)$. We show that ${\lambda }_F(p)$ is positive for all large enough primes $p$. This is proved for Ikeda lifts of all genus. The second result is specific for genus 4 Ikeda lifts. If $F$ is genus 4 Ikeda lift, we show that, given $r$ there exists a constant $c_r$ such that ${\lambda }_F(p^r)$ is positive for all $p>c_r$.