Recovering modular forms from squares

The purpose of this appendix is to provide a proof of the fact that a holomorphic newform f of weight 2k, level N and trivial character, with Hecke eigenvalues {ap | (p, N) = 1}, is determined up to a quadratic twist, in fact on the nose if N is square-free, by the knowledge of ap for all primes p in a set of sufficiently large density. We will in fact prove a more general statement below, including the case of odd weight and non-trivial character, and also establish a mod l analog. We found this result in the summer of 94, and we have since learned that it has also been known to others, including Don Blasius and J.-P. Serre. Also, Siman Wong has recently come up with a different proof in the weight 2 case (with trivial character). So we do not intend any display of great achievement by this write-up, and we give all the details for ease of use by those working in classical modular forms and number theory. We have also found a non-trivial extension of this result (in characteristic zero) to Maass forms using an array of results on automorphic L−functions, and this is the subject matter of a paper under preparation. This work was partially supported by an NSF grant. We thank Serre for his helpful comments on an earlier version which led to a finer result.