On the transcendence of certain Petersson inner products

Abstract

‎We show that for all normalized Hecke eigenforms $f$‎ ‎with weight one and of CM type‎, ‎the number $(f,f)$ where $(cdot‎, ‎cdot )$ denotes‎ ‎the Petersson inner product‎, ‎is a linear form in logarithms and‎ ‎hence transcendental‎.