  We derive explicit formulae to compute the $a$ and $c$ central charges of
four dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs) directly
from Coulomb branch related quantities. The formulae apply at arbitrary rank.
We also discover general properties of the low-energy limit behavior of the
flavor symmetry of $\mathcal{N}=2$ SCFTs which culminate with our
$\mathcal{N}=2$ UV-IR simple flavor condition. This is done by determining
precisely the relation between the integrand of the partition function of the
topologically twisted version of the 4d $\mathcal{N}=2$ SCFTs and the singular
locus of their Coulomb branches. The techniques developed here are extensively
applied to many rank-2 SCFTs, including new ones, in a companion paper.
