Zero density estimate for modular form $L$-functions in weight aspect

Abstract: Considering the family of $L$-functions ${L(s,f)}_{f \in H_k}$ where $H_k$ is the set of weight $k$ Hecke-eigen cusp forms for $SL_2(\mathbb{Z})$, we prove a zero density estimate near the central point, valid as the weight $k \to \infty$. This is an ingredient in the author's related paper, which gives an unconditional upper bound on the distribution of the central values.