For a non-negative integer $k$, the triple $(p,q,r)$ of positive integers is called a $k$-shifted Pythagorean triple if $$p^2+q^2+k=r^2$$

$(p,q,r)$ is said to be primitive if $gcd(p,q,r)=1$.

Let $P_k(n)$ be the number of primitive $k$-shifted Pythagorean triples such that $1\le p\le q\le r$ and $p+q+r\le n$.
For example, $P_0({10}^4)=703$ and $P_{20}({10}^4)=1979$.

Define $${\displaystyle S(m,n)=\sum _{k=0}^m{P}_k(n).}$$ You are given that $S(10,{10}^4)=10956$.

Find $S({10}^2,{10}^8)$.