A positive integer matrix is a matrix whose elements are all positive integers.
Some positive integer matrices can be expressed as a square of a positive integer matrix in two different ways. Here is an example:

$$\left(\begin{array}{cc}40& 12\\ 48& 40\end{array}\right)={\left(\begin{array}{cc}2& 3\\ 12& 2\end{array}\right)}^2={\left(\begin{array}{cc}6& 1\\ 4& 6\end{array}\right)}^2$$

We define $F(N)$ as the number of the $2\times 2$ positive integer matrices which have a tracethe sum of the elements on the main diagonal less than $N$ and which can be expressed as a square of a positive integer matrix in two different ways.
We can verify that $F(50)=7$ and $F(1000)=1019$.

Find $F({10}^7)$.