Squareness in the special L-value

Let N be a prime and let A be a quotient of J0(N) over Q associated to a newform f such that the special L-value of A (at s = 1) is non-zero. Suppose that the algebraic part of special L-value is divisible by an odd prime q such that q does not divide the numerator of N−1 12 . Then the Birch and Swinnerton-Dyer conjecture predicts that q divides the algebraic part of special L value of A, as well as the order of the Shafarevich-Tate group. Under a mod q non-vanishing hypothesis on special L-values of twists of A, we show that q does divide the algebraic part of the special L-value of A and the Birch and Swinnerton-Dyer conjectural order of the Shafarevich-Tate group of A. This gives theoretical evidence towards the second part of the Birch and Swinnerton-Dyer conjecture. We also give a formula for the algebraic part of the special L-value of A over suitable quadratic imaginary fields which shows that this algebraic part is a perfect square away from two.