  Let $Q$ be a positive-definite quaternary quadratic form with integer
coefficients. We study the problem of giving bounds on the largest positive
integer $n$ that is locally represented by $Q$ but not represented. Assuming
that $n$ is relatively prime to $D(Q)$, the determinant of the Gram matrix of
$Q$, we show that $n$ is represented provided that \[
  n \gg \max \{ N(Q)^{3/2 + \epsilon} D(Q)^{5/4 + \epsilon}, N(Q)^{2 +
\epsilon} D(Q)^{1 + \epsilon} \}. \] Here $N(Q)$ is the level of $Q$. We give
three other bounds that hold under successively weaker local conditions on $n$.
  These results are proven by bounding the Petersson norm of the cuspidal part
of the theta series, which is accomplished using an explicit formula for the
Weil representation due to Scheithauer.
