Variants of recognition problems for modular forms

We consider the problem of distinguishing two modular forms, or two elliptic curves, by looking at the coefficients of their L-functions for small primes (compared to their conductor). Using analytic methods based on large-sieve type inequalities we give various upper bounds on the number of forms having the first few coefficients equal to those of a fixed one. In addition, we consider similar questions of recognizing symmetric squares and CM forms from the behavior of small primes.Received: 30 August 2004