Let $N$ and $a$ be positive integers. Consider the Kronecker symbol $\left( \frac{n}{m} \right)$, which is a character modulo $m$. I have seen it several times that 'by Polya-Vinogradov inequality', we have $$ \sum_{0<n\leq Y, n \equiv a (N)} \left(\frac{n}{m}\right) \ll \sqrt m \log m. $$
However, the sum is not over consecutive integers, so I think this is not a trivial fact. How can I prove this?