  Every double coset in $\text{GL}_m(k[[z]])\backslash
\text{GL}_m(k((z)))/\text{GL}_m(k((z^2)))$ is uniquely represented by a block
diagonal matrix with diagonal blocks in $\{1,z, \begin{pmatrix} 1& z\\ 0 &z^i
  \end{pmatrix} (i>1)\}$ if $char(k) \neq 2$ and $k$ is a finite field. These
cosets form a (spherical) Hecke module $\mathcal{H}(G,H,K)$ over the
(spherical) Hecke algebra $\mathcal{H}(G,K)$ of double cosets in $K\backslash
G/H$, where $K=\text{GL}_m(k[[z]])$ and $H=\text{GL}_m(k((z^2)))$ and
$G=\text{GL}_m(k((z)))$. Similarly to Hall polynomial $h_{\lambda,\nu}^{\mu}$
from the Hecke algebra $\mathcal{H}(G,K)$, coefficients $h_{\lambda,\nu}^{\mu}$
arise from the Hecke module. We will provide a closed formula for
$h_{\lambda,\nu}^\mu$, under some restrictions over ${\lambda,\nu,\mu}$.
