  We study new families of curves that are suitable for efficiently
parametrizing their moduli spaces. We explicitly construct such families for
smooth plane quartics in order to determine unique representatives for the
isomorphism classes of smooth plane quartics over finite fields. In this way,
we can visualize the distributions of their traces of Frobenius. This leads to
new observations on fluctuations with respect to the limiting symmetry imposed
by the theory of Katz and Sarnak.
