Let $\displaystyle \chi$ be the primitive character modulo 12.

I need to find all possible values of $\displaystyle \underset{m\leq x}{\sum}\chi(m)$ for varying $\displaystyle x$ (not necessarily an integer).

I have an answer which shows that the sum is equal to either 1, 2 or 0, but am not entirely sure it is correct.

Could anybody help please?