Proper circulant weighing matrices of weight $$p^2$$

A circulant weighing matrix $$CW(v,n)$$ is a circulant matrix $$M$$ of order $$v$$ with $$0,\pm 1$$ entries such that $$MM^T=nI_v$$. In this paper, we study proper circulant matrices with $$n=p^2$$ where $$p$$ is an odd prime divisor of $$v$$. For $$p\ge 5$$, it turns out that to search for such circulant matrices leads us to two group ring equations and by studying these two equations, we manage to prove that no proper $$CW(pw,p^2)$$ exists when $$p\equiv 3\pmod {4}$$ or $$p=5$$.