  The Katz-Sarnak philosophy states that statistics of zeros of $L$-function
families near the central point as the conductors tend to infinity agree with
those of eigenvalues of random matrix ensembles as the matrix size tends to
infinity. While numerous results support this conjecture, S. J. Miller observed
that for finite conductors, very different behavior can occur for zeros near
the central point in elliptic curve families. This led to the excised model of
Due\~{n}ez, Huynh, Keating, Miller, and Snaith, whose predictions for quadratic
twists of a given elliptic curve are beautifully fit by the data. The key
ingredients are relating the discretization of central values of the
$L$-functions to excising matrices based on the value of the characteristic
polynomials at 1 and using lower order terms (in statistics such as the
one-level density and pair-correlation) to adjust the matrix size. We discuss
recent successes by the authors in extending this model to a family of
quadratic twists of finite conductor of a given holomorphic cuspidal newform of
level an odd prime level. In particular, we predict very little repulsion for
forms with weight greater than 2.
