Let $f$ be a modular form of an even weight $k$ over the modular group $SL_2(Z).$ Denote $\lambda_f(n)$ the $n$th normalized Fourier coefficient of $f.$ I am doing some calculations and I am stack in finding the expression of the sum $\sum_{dn} \lambda_f\left(\frac{n^2}{d^2}\right)$ in terms of $ \lambda_f(n)$ or $ \lambda_f(n)^2.$ Any help is appreciated.
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From the Hecke relation, we get $$ \lambda_f(n)^2 =\sum_{dn} \lambda_f\left(\frac{n^2}{d^2}\right).$$