  We compare the L-Function Ratios Conjecture's prediction with number theory
for the family of quadratic twists of a fixed elliptic curve with prime
conductor, and show agreement in the 1-level density up to an error term of
size X^{-(1-sigma)/2} for test functions supported in (-sigma, sigma); this
gives us a power-savings for \sigma<1. This test of the Ratios Conjecture
introduces complications not seen in previous cases (due to the level of the
elliptic curve). Further, the results here are one of the key ingredients in
the companion paper [DHKMS2], where they are used to determine the effective
matrix size for modeling zeros near the central point for this family. The
resulting model beautifully describes the behavior of these low lying zeros for
finite conductors, explaining the data observed by Miller in [Mil3].
  A key ingredient in our analysis is a generalization of Jutila's bound for
sums of quadratic characters with the additional restriction that the
fundamental discriminant be congruent to a non-zero square modulo a square-free
integer M. This bound is needed for two purposes. The first is to analyze the
terms in the explicit formula corresponding to characters raised to an odd
power. The second is to determine the main term in the 1-level density of
quadratic twists of a fixed form on GL_n. Such an analysis was performed by
Rubinstein [Rub], who implicitly assumed that Jutila's bound held with the
additional restriction on the fundamental discriminants; in this paper we show
that assumption is justified.
