We consider some questions related to the signs of Hecke eigenvalues or
Fourier coefficients of classical modular forms. One problem is to determine to
what extent those signs, for suitable sets of primes, determine uniquely the
modular form, and we give both individual and statistical results. The second
problem, which has been considered by a number of authors, is to determine the
size, in terms of the conductor and weight, of the first sign-change of Hecke
eigenvalues. Here we improve significantly the recent estimate of Iwaniec,
Kohnen and Sengupta.Comment: 23 pages, 1 figure; new version with new coauthor and strong
improvements of two of the two main results
