Monotone Chains in Modulus of Two Classes of Dirichlet Coefficients

Abstract

Let f₁,…,fk and g₁,…,g_k be non-CM newforms of square-free levels. Denote by ${\lambda _{{\rm{sy}}{{\rm{m}}^j}\,{f_i}}}(n)$ the coefficients of the Dirichlet expansion of L(sym^jf_i, s) and v₁,…,v_k the distinct positive integers such that ${\lambda _{{\rm{sy}}{{\rm{m}}^j}\,{f_i}}}({\nu _i}) \ne 0$. In this paper, we obtain that there exist infinitely many positive integers m such that $0 < |{\lambda _{{\rm{sy}}{{\rm{m}}^j}\,{f_1}}}(m + {\nu _1})| < |{\lambda _{{\rm{sy}}{{\rm{m}}^j}\,{f_2}}}(m + {\nu _2})| < \cdots < |{\lambda _{{\rm{sy}}{{\rm{m}}^j}\,{f_k}}}(m + {\nu _k})|$. For coefficients of the Dirichlet expansion of L(sym^j1f × sym^j2g, s), we have a similar result.