  The Katz-Sarnak Density Conjecture states that zeros of families of
$L$-functions are well-modeled by eigenvalues of random matrix ensembles. For
suitably restricted test functions, this correspondence yields upper bounds for
the families' order of vanishing at the central point. We generalize previous
results on the $n$\textsuperscript{th} centered moment of the distribution of
zeros to allow arbitrary test functions. On the computational side, we use our
improved formulas to obtain significantly better bounds on the order of
vanishing for cuspidal newforms, setting world records for the quality of the
bounds. We also discover better test functions that further optimize our
bounds. We see improvement as early as the $5$\textsuperscript{th} order, and
our bounds improve rapidly as the rank grows (more than one order of magnitude
better for rank 10 and more than four orders of magnitude for rank 50).
