  We study the nonvanishing of twists of automorphic L-functions at the centre
of the critical strip. Given a primitive character \chi modulo D satisfying
some technical conditions, we prove that the twisted L-functions L(f.\chi,s) do
not vanish at s=1/2 for a positive proportion of primitive forms of weight 2
and level q, for large prime q. We also investigate the central values of high
derivatives of L(f.\chi,s), and from that derive an upper bound for the average
analytic rank of the studied L-functions.
