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The following question arises while I'm reading a paper of Jerzy Kaczorowski and Alberto Perelli (A correspondence theorem for L-functions and partial differential operators, Publ. Math. Debrecen 79/3-4 (2011), 497-505).

Consider an eigenfunction $f$ of hyperbolic laplacian $\Delta=-y^2(\frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2})$ defined on the upper half-plane $\mathbb{H}$(eigenvalue is real).$f$ satisfies the following conditions:

1)$f(z)=f(z+1),z=x+iy$;

2)$f(z)=o(e^{2\pi y}),y\rightarrow+\infty;$

3)If $z=iy,y>0$,then $z^pf(z)=Kf(-\frac{1}{qz}),p\geq 0,q>0,K\in\mathbb{C}$.

Question:Is it possible that 3) is true for every $z\in\mathbb{H}$?

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The condition (3) implies that if $f\ne 0$, then $K^2=(-1)^pq^{_p}$, for you can apply this condition to $w=-1/(qz)$ in place of $z$.

Then, a necessary condition is that the group of isometries generated by $z\mapsto z+1$ and $z\mapsto -1/(qz)$ is discrete. See Beardon's book on hyperbolic geometry for criteria on this. It is only true for contably many values of $q$. Among them is $q=1$ in which case the group of isometries is $SL_2({\mathbb Z})/\pm 1$.

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