  The Katz-Sarnak philosophy predicts that the behavior of zeros near the
central point in families of $L$-functions agrees with that of eigenvalues near
1 of random matrix ensembles. Under GRH, Iwaniec, Luo and Sarnak showed
agreement in the one-level densities for cuspidal newforms with the support of
the Fourier transform of the test function in $(-2, 2)$. They increased the
support further under a square-root cancellation conjecture, showing that a
${\rm GL}(1)$ estimate led to additional agreement between number theory and
random matrix theory. We formulate a two-dimensional analog and show it leads
to improvements in the two-level density. Specifically, we show that a
square-root cancellation of certain classical exponential sums over primes
increases the support of the test functions such that the main terms in the
$1$- and $2$-level densities of cuspidal newforms averaged over bounded weight
$k$ (and fixed level $1$) converge to their random matrix theory predictions.
We also conjecture a broad class of such exponential sums where we expect
improvement in the case of arbitrary $n$-level densities, and note that the
arguments in [ILS] yield larger support than claimed.
