  We prove a non-vanishing result of modular L-values with quadratic twists,
where the quadratic discriminants are in a short interval. Using this fact and
Waldspurger's theorem, we improve the results of Balog-Ono[The chebotarev
density theorem in short intervals and some questions of Serre, Journal of
number theory. 91(2):356-371(2001)] on the non-vanishing of Fourier
coefficients of half-integral weight eigenform.
