Consider the following binary quadratic form:

$$\begin{array}{r}f(x,y)=x^2+5xy+3y^2\end{array}$$

A positive integer $q$ has a primitive representation if there exist positive integers $x$ and $y$ such that $q=f(x,y)$ and $gcd(x,y)=1$.

We are interested in primitive representations of perfect squares. For example:
${17}^2=f(1,9)$
${87}^2=f(13,40)=f(46,19)$

Define $C(N)$ as the total number of primitive representations of $z^2$ for $0<z\le N$.
Multiple representations are counted separately, so for example $z=87$ is counted twice.

You are given $C({10}^3)=142$ and $C({10}^6)=142463$.

Find $C({10}^{14})$.