Solution
= Solution
For each $x$, let $h(x)$ be an effectively obtained index of the partial function
$$
y\longmapsto
\begin{cases}
0,&y<x^2,\\
\text{undefined},&y\geq x^2.
\end{cases}
$$
The existence of the total computable index transformation $h$ follows from the <S-m-n theorem>. By the <recursion theorem>, some $m$ satisfies $f_{m,1}\simeq f_{h(m),1}$. Therefore its domain is
$$
W_m=\{0,1,\ldots,m^2-1\},
$$
and hence
$$
\boxed{|W_m|=m^2.}
$$