Set as usual $e(z):=\exp(2\pi iz)$. Having two series that are related for all $z$ in the upper halfplane via a transformation formula $$ \sum_{n=1}^{\infty}{\alpha_n e\Big(n\frac{az+b}{cz+d}\Big)}=p(z)+q(z)\sum_{n=1}^{\infty}{\beta_n e(nz)} $$ where $p(z)$ and $q(z)$ are polynomials and $ad-bc=1$, can I deduce some relation between $\alpha_n$ and $\beta_n$?



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