In a passage I'm reading, the authors suppose $G$ is a finite group, $e$ is a primitive idempotent of $\mathbb{C}G$, and $n$ is a primitive idempotent of $e\mathbb{C}Ge$. They mention that $ne\mathbb{C}Gen=\mathbb{C}n$, so that the left multiplication map $N:=e\mathbb{C}Gen\to ne\mathbb{C}Gen=\mathbb{C}n$, and thus $\psi(n)=1$, where $\psi$ is the character afforded by $N$.
Why is this? I know $ne=en=n$, but I don't see why $ngn$ should be a scalar multiple of $n$ for any $g\in G$, and how this implies the trace of the map induced by $n$.