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Half integral weight Hecke operators

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?

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 Thank you, this is exactly what I wanted. Although I am a little surprised that the $T(p^{2l})$ term does not appear in the expansion of $T(p^2l)T(p^2)$. I guess what is happening is that the matrices required for $T(p^{2l})$ term are there in the double coset decomposition, but the second components of the representatives act in such a way that makes it vanish on modular forms. – Eren Mehmet Kiral Oct 9 at 3:30 I am glad I could help. – GH Oct 9 at 15:29