  We study black hole - host galaxy correlations, and the relation between the
over-massiveness (the distance from the average $M_{BH}-\sigma$ relation) of
super-massive black holes (SMBHs) and star formation histories of their host
galaxies in the Illustris and TNG100 simulations. We find that both simulations
are able to produce black hole scaling relations in general agreement with
observations at $z=0$, but with noticeable discrepancies. Both simulations show
an offset from the observations for the $M_{BH}-\sigma$ relation, and the
relation between $M_{BH}$ and the Sersic index. The relation between $M_{BH}$
and stellar mass $M_*$ is tighter than the observations, especially for TNG100.
For massive galaxies in both simulations, the hosts of over-massive SMBHs
(those above the mean $M_{BH}-\sigma$ relation) tend to have larger Sersic
indices and lower baryon conversion efficiency, suggesting a multidimensional
link between SMBHs and properties of their hosts. In Illustris, the hosts of
over-massive SMBHs have formed earlier and have lower present-day star
formation rates, in qualitative agreement with the observations for massive
galaxies with $\sigma>100 \rm km/s$. For low-mass galaxies, such a correlation
still holds in Illustris but does not exist in the observed data. For TNG100,
the correlation between SMBH over-massiveness and star formation history is
much weaker. The hosts of over-massive SMBHs generally have consistently larger
star formation rates throughout history. These galaxies have higher stellar
mass as well, due to the strong $M_{BH}-M_*$ correlation. Our findings show
that simulated SMBH scaling relations and correlations are sensitive to
features in the modeling of SMBHs.
