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.
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