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If d is square-free or is a fundamental discriminant, then let χd = χD denote the Kronecker character for the quadratic field Q( √ d) whose fundamental discriminant is D. Throughout D shall denote a fundamental discriminant. The D-quadratic twist of F , denoted F ⊗ χD, is the newform corresponding to the twist of F by the character χD. In particular, if gcd(M,D) = 1, then (F ⊗ χD)(z) = ∑∞ n=1 χD(n)a(n)q n and

Nonvanishing of quadratic twists of modular L -functions and applications to elliptic curves