Let $\chi$ be a Dirichlet character modulo $m$ induced by $\chi'$ modulo $m'$. We define $$ \tau(\chi):= \sum_{a(mod \ m)} \chi(a) e(a/m). $$
Could someone please show me how to derive the formula: $$ \tau(\chi) = \mu(m/m') \chi'(m/m') \tau(\chi')? $$ Thank you very much!
PS By $e(z)$, I mean $e^{2 \pi i z}$ and $\mu$ is the Mobius function. And the sum $\sum_{a (mod \ m)}$ means I am summing over all the residue class modulo $m$.