Let $f(x,y) = f_2 x^2 + f_1 xy + f_0 y^2$ be a primitive, positive definite, and reduced binary quadratic form. Put $k_f$ for the fundamental discriminant associated to $f$. That is, $k_f$ is square-free and the splitting field of $f$ is equal to $\mathbb{Q}(\sqrt{k_f})$. Note that $k_f < 0$, by hypothesis.
We say that $f$ is admissible if for all primes $p | k_f$, we have $\left(\frac{f_2}{p} \right) = 1$; that is, the leading coefficient $f_2$ is a quadratic residue for all primes $p$ dividing $k_f$. Put
$$\displaystyle N^-(X) = \{f : f \text{ primitive, reduced, and admissible, with} -X < \Delta(f) < 0\}.$$
Is there an asymptotic formula known for $N^-(X)$? If we drop the condition that $f$ is admissible, then the corresponding asymptotic formula was known to Gauss and was given a modern treatment by Siegel.
