Let $d$ be a positive integer such that $-d$ is a fundamental discriminant, and let $K = \mathbb{Q}(\sqrt{-d})$. Let $\mathcal{O}_K$ denote the ring of integers in $K$. It is a well-known theorem of Gauss that the ideal class group of $\mathcal{O}_K$ is parametrized by $\operatorname{SL}_2(\mathbb{Z})$-equivalence classes of integral binary quadratic forms $f$ of discriminant $-d$. Put $[f]$ for the $\operatorname{SL}_2(\mathbb{Z})$-class of $f$.
For each $f$, let $a(f)$ denote the smallest positive integer representable by $f$, or equivalently, the leading coefficient of $f$ if $f$ is reduced (note that $a(f)$ only depends on the class $[f]$ of $f$).
Consider the homomorphism $\rho$ from the ideal class group to itself defined by $[f] \mapsto [f]^4$. The kernel of this map is precisely the 4-part of the ideal class group. Put $C(-d)$ for the class group of $\mathcal{O}_K$, and put $C^{(4)}(-d)$ for the image of $\rho$ in $C(-d)$.
My question is what is the cardinality of the set
$$\displaystyle \{a(f) : [f] \in C^{(4)}(-d), a(f) \leq d^{1/4}\}?$$
