Imprimitive Dirichlet characters as sum of additive characters

Question

Let $\chi$ be a Dirichlet character modulo $q$. Let $e(x)=\text{exp}(2\pi i x)$ and $\tau(\chi)$ be the Gauss sum $\tau(\chi)=\sum_{m \,\text{mod}\, q}\chi(m)e(m/q)$. For any $n$ with $\gcd(n,q)=1$, we know that $$ \chi(n) = \frac{1}{\tau(\overline{\chi})} \sum_{m \,\text{mod}\, q} \overline{\chi}(m)\,e\!\left(\frac{nm}{q}\right). $$ If $\chi$ is primitive, then the equation holds for all $n\in\mathbb{Z}$. If $\chi$ is not primitive and $\gcd(n,q) > 1$, then $\chi(n)$ is zero while the right hand side may not be. Is there a formula similar to this one, that writes a multiplicative character as a sum of additive characters, that holds for all multiplicative characters, not just the primitive ones?