  We investigate the low-lying zeros in families of $L$-functions attached to
quadratic and cubic twists of elliptic curves defined over $\mathbb{F}_q(T)$.
In particular, we present precise expressions for the expected values of traces
of high powers of the Frobenius class in these families with a focus on the
lower order behavior. As an application we obtain results on one-level
densities and we verify that these elliptic curve families have orthogonal
symmetry type. In the quadratic twist families our results refine previous work
of Comeau-Lapointe. Moreover, in this case we find a lower order term in the
one-level density reminiscent of the deviation term found by Rudnick in the
hyperelliptic ensemble. On the other hand, our investigation is the first to
treat these questions in families of cubic twists of elliptic curves and in
this case it turns out to be more complicated to isolate lower order terms due
to a larger degree of cancellation among lower order contributions.
