Let $E$ be an optimal elliptic curve over $\Q$ of conductor $N$ having
analytic rank zero, i.e., such that the $L$-function $L_E(s)$ of $E$ does not
vanish at $s=1$. Suppose there is another optimal elliptic curve over $\Q$ of
the same conductor $N$ whose Mordell-Weil rank is greater than zero and whose
associated newform is congruent to the newform associated to $E$ modulo an
integer $r$.