  Let $Q(x,y,z)$ be an integral quadratic form with determinant coprime to some
modulus $q$. We show that $q\mid Q$ for some non-zero integer vector $(x,y,z)$
of length $O(q^{5/8+\varepsilon})$, for any fixed $\varepsilon>0$. Without the
coprimality condition on the determinant one could not achieve an exponent
below $2/3$. The proof uses a bound for short character sums involving binary
quadratic forms, which extends a result of Chang.
