For positive integers \(N\) and \(k\), let \(M_k (\Gamma_1(N))\) denote the space of entire modular forms of weight \(k\) on the group \(\Gamma_1(N)\). This space is the direct sum of the subspaces \(M_k(\Gamma_0 (N),\chi)\), where \(\chi\) runs through the Dirichlet characters modulo \(N\). An explicit basis \({\mathcal B}\) of \(M_k (\Gamma_1(N))\), consisting of eigenvectors of all the Hecke operators \(T_n\) with \(n\) coprime to the level \(N\), can be constructed from newforms of levels \(M\) dividing \(N\) and with characters \(\chi\) whose conductor divides \(M\). (For this material, see chapter 4 in T. Miyake, Modular forms, Springer (1989; Zbl 0701.11014).) The scalar multiples of functions in the basis \({\mathcal B}\) are called almost everywhere eigenforms (a.e. eigenforms).
The author deals with the following problem. Let \(N\) be square-free, and let \(g\in M_l (\Gamma_1(N))\) and \(h\in M_{k-l} (\Gamma_1(N))\) be a.e. eigenforms. When is \(gh\in M_k (\Gamma_1(N))\) an a.e. eigenform? This can happen only rarely. In Theorem 1 the author gives sufficient conditions for \(gh\) not to be an a.e. eigenform, distinguishing the cases that two, one or none of the factors \(g\), \(h\) is a cusp form. His tools in the proof are the Atkin-Lehner involutions and the Rankin-Selberg method. As an application he obtains, in Theorem 5, a criterion for the nonvanishing of the twists of standard \(L\)-functions of modular forms at the center of the critical strip. The paper ends with a list of examples of a.e. eigenforms whose products are a.e. eigenforms. Special cases of the problem were considered previously by the author himself [J. Ramanujan Math. Soc. 15, 71-79 (2000; Zbl 0966.11017)] and by W. Duke [Number Theory in Progress, Vol. 2 (Zakopane 1997), de Gruyter, 737-741 (1999; Zbl 0953.11017)].