Summary: Harvey Cohn’s problem on character sums [see H. L. Montgomery, “Ten lectures on the interface between analytic number theory and harmonic analysis”, Regional Conf. Ser. Math. 84, Amer. Math. Soc. (1994; Zbl 0814.11001), p. 202] asks whether a multiplicative character on a finite field can be characterized by a kind of two level autocorrelation property. Let \(f\) be a map from a finite field \(F\) to the complex plane such that \(f(0)=0\), \(f(1)=1\), and \(|f(\alpha)|=1\) for all \(\alpha\neq 0\). In this paper we show that if for all \(a,b\in F^*\), we have \[ (q-1) \sum_{\alpha\in F} f(b\alpha) \overline{f(\alpha+a)}= -\sum_{\alpha\in F}f(b\alpha) \overline{f(\alpha)}, \] then \(f\) is a multiplicative character of \(F\). We also prove that if \(F\) is a prime field and \(f\) is a real valued function on \(F\) with \(f(0)=0\), \(f(1)=1\), and \(|f(\alpha)|=1\) for all \(\alpha\neq 0\), then \(\sum_{\alpha\in F}f(\alpha) f(\alpha+a)= -1\) for all \(a\neq 0\) if and only if \(f\) is the Legendre symbol. These results partially answer Cohn’s problem.