On a problem of Hasse and Ramachandra

Let $K$ be an imaginary quadratic field, and let $\mathfrak{f}$ be a nontrivial integral ideal of $K$. Hasse and Ramachandra asked whether the ray class field of $K$ modulo $\mathfrak{f}$ can be generated by a single value of the Weber function. We completely resolve this question when $\mathfrak{f}=(N)$ for an integer $N>1$.