Corrigendum. Further refinements of the GL(2) converse theorem

Lemma 2.4 of [1] is true, but its proof is not correct in all cases. Precisely, in the proof we defined an n x (n + 1) real matrix M and a vector x(.). is an element of Rn+1 associated to an id`ele class character., and claimed that Mx(.). is an element of Q(n). However, that need not be the case if the number field F is not totally real. To correct for this, we define b(.) = (b(1),..., b(n)), where