  We calculate the eigenmodes of the Highly Improved Staggered Quark (HISQ)
matrix near the chiral crossover transition in QCD with $2+1$ flavors with the
aim to gain more insights into its temperature dependence. On performing the
continuum extrapolation, we do not observe any gap opening up in the infrared
part of the eigenvalue density of the QCD Dirac operator; instead we observe a
peak. The existence of the peak and oscillations of the infrared eigenmodes can
be understood in terms of an interacting ensemble of instantons. From the
properties of the continuum extrapolated eigenspectrum we further show that the
anomalous $U_A(1)$ part of the chiral symmetry is not effectively restored
simultaneously along with its non-singlet counterpart. We provide an
explanation for this observation, further showing interesting connections
between the anomalous $U_A(1)$ restoration and the change in the infrared part
of the eigenvalue distribution.
