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Last year, I had found a pdf where the expression of the Dirichlet coefficient $\lambda_{\pi}(p^{\nu})$ in terms of the Satake parameters $\alpha_{p,i}(\pi)$ was given. Unfortunately I don't remember neither its title nor its author. Can someone provide a reference of remind me this expression?

Edit: I think it was something like $\lambda_{\pi}(p^{\nu})=\sum_{i=1}^{d}(\alpha_{p,i}(\pi))^{\nu}$ with $d$ the degree of the relevant local factor. Can someone confirm?

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The answer is in Peter Humphries' answer to Rankin-Selberg convolution : normalization issues

I performed calculations this noon leading to the same conclusion.

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