# Imprimitive Dirichlet characters as sum of additive characters

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?