Low-lying zeros in a family of holomorphic cusp forms

In a joint work with Daniel Fiorilli and Anders Södergren, we study the low-lying zeros of L-functions attached to holomorphic cusp forms of level 1 as the weight increases. This family was proved to be of orthogonal type (resp. special orthogonal even or odd when the family is separated with respect to sign of the functional equation) by Iwaniec, Luo and Sarnak who obtained the predicted main term for the one-level density of the low-lying zeros for test functions having Fourier transform supported in (-2,2). Building on their work, we obtain lower order terms showing a transition similar to that in the main term when the support of the Fourier transform of the test function reaches the point 1.