I would like to find a source giving the exact formula for the product of two Hecke operators $T_{\kappa}(n^2)$ and $T_\kappa(m^2)$ of half integral weight. That is, $\kappa \in \frac 12 \mathbb{Z} - \mathbb{Z}$. I am searching for a formula which is similar to $$ T_k(m)T_k(n) = \sum_{d|(m,n)} d^{k-1}T_k\left(\frac{mn}{d^2}\right) $$ in the case of full integral weight $k$.

I suspect the answer to be exactly the same, since double coset decompositions in $\operatorname{GL}_2(\mathbb{R})^+$, and in its double cover should correspond to each other in a one-to-one fashion. The only difference being that the half integral Hecke operators are supported only on the squares.

Am I missing anything?