It is well known that every column sum in the character table of a finite group is an integer. Nevertheless, it is not easy to find an example where a column sum is negative. The smallest example has order 96. For example, one can take SmallGroup(96,3) in the notation of GAP, whose character table is presented under https://people.maths.bris.ac.uk/~matyd/GroupNames/73/C2%5E3.3A4.html The column sum for the involution class 2B is negative.
On the other hand, it is also well known that not every element in the commutator subgroup of a group is a commutator. It is not easy to find a counterexample either and the smallest example has also order 96. Moreover, one can take the same group SmallGroup(96,3), and it is the same conjugacy class 2B which lies in the commutator subgroup, but does not consist of commutators! The reason is the well known criterion of Burnside that an element is a commutator if and only if $\sum_{\chi\in Irr(G)}\frac{\chi(g)}{\chi(1)}\ne 0$.
I cannot believe myself that this is a pure coincidence. Do you see an explanation for it?