Dear Loic, I made all of the changes and additions you suggested, which includes giving a proof of my Lemma. I then read through the whole paper and made some additional changes. 1) Added the sentence "Let~$E$ be an elliptic curve over $\B Q(\mu_p)$, such that the points of order~$p$ of $E(\bar\B Q)$ are all $\B Q(\mu_p)$-rational." to statement of Proposition 1, because I became confused while reading prop 1. 2) I changed the title of section 2 from "A lemma about elliptic curves" to "A proposition about elliptic curves", since section 2 begins with a proposition. 3) Second paragraph of proof of Proposition 2: We write "(E,\pi(t))", but we mean "(E, cyclic subgroup generated by \pi(t))". I suggest changing "(E,\pi(t))" to "(E,<\pi(t)>)". I did *not* make this change. 4) In the case p=389, the prime 389 is *not* a congruence prime; it is only responsible for ramification in the number field generated by the winding quotient of J_0(389). Thus P_1 + P_2 = (1), in the notation of our paper. Does this imply that both P_1 and P_2 are direct factors of T/pT? 5) I added a reference to Amod's paper, where he discusses the fact that the Hecke algebra T is generated by T_1,T_2,...,T_b, where b = ceil((p+1)/6). 6) I added a remark about L(f,chi,1) = 0 ==/==> L(f^sigma,chi,1)=0. I searched in a different way than before and found a counterexample with f on X_1(13) and chi:(Z/7Z)^*-->C of order 3. Maybe this is the counterexample you really originally had in mind?? -- William