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?



Half integral weight Hecke operators
I think you can find the answer [here](https://arxiv.org/abs/1208.4326).


