  The electronic nematic phase is characterized as an ordered state of matter
with rotational symmetry breaking, and has been well studied in the quantum
Hall system and the high-$T_c$ superconductors, regardless of cuprate or
pnictide family. The nematic state in high-$T_c$ systems often relates to the
structural transition or electronic instability in the normal phase.
Nevertheless, the electronic states below the superconducting transition
temperature is still an open question. With high-resolution scanning tunneling
microscope measurements, direct observation of vortex core in FeSe thin films
revealed the nematic superconducting state by Song \emph{et al}. Here,
motivated by the experiment, we construct the extended Ginzburg-Landau free
energy to describe the elliptical vortex, where a mixed \emph{s}-wave and
\emph{d}-wave superconducting order is coupled to the nematic order. The
nematic order induces the mixture of two superconducting orders and enhances
the anisotropic interaction between the two superconducting orders, resulting
in a symmetry breaking from $C_4$ to $C_2$. Consequently, the vortex cores are
stretched into an elliptical shape. In the equilibrium state, the elliptical
vortices assemble a lozenge-like vortex lattice, being well consistent with
experimental results.
