  Assuming the Generalized Riemann Hypothesis, the non-trivial zeros of
$L$-functions lie on the critical line with the real part $1/2$. We find an
upper bound of the lowest first zero in families of even cuspidal newforms of
prime level tending to infinity. We obtain explicit bounds using the $n$-level
densities and results towards the Katz-Sarnak density conjecture. We prove that
as the level tends to infinity, there is at least one form with a normalized
zero within $1/4$ of the average spacing. We also obtain the first-ever bounds
on the percentage of forms in these families with a fixed number of zeros
within a small distance near the central point.
