The sum of the values of the character of a given group representation

Question

Let $G$ be a finite group, $\rho: G \rightarrow GL(V)$ a finite dimensional representation over $\mathbb{C}$, and $\chi = tr(\rho)$ be the character of $\rho$.

Recall that since $G$ is finite, $|\chi(g)| \leq \dim(V)$ for any $g\in G$. While

$$ \Sigma_{g\in G} |\chi(g)| \leq |G|\dim(V), $$

I feel that there might be a cancelling phenomenon in $\Sigma_{g\in G} \chi(g)$ when $\rho$ is not trivial.

Question

1. To what extent do the $\chi(g)$ cancel each other?
2. Can we say anything about the sum?
3. Can we generalize the result to any compact (or say unimodular) Lie group?