Denote the function $$ \Psi (x,y)= y^{1/2+ik}+ \sum_{g\in SL(2,\Bbb Z)} \frac{y^{1/2+ik}} {|c_{g}z+d_{g}|^{1/2+ik}}\tag{1}$$
My question is if I can write the wave function in terms of the Eisenstein series
$$ G_{s}(z)= \sum_{(c,d)\in\Bbb Z\setminus(0,0)}|cz+d|^{-s}$$ as the solution
$$\Psi (x,y)= y^{1/2+ik}+ y^{1/2+ik}G_{1/2+ik}(z)$$
Equation $(1)$ is the solution to the Laplace equation $ y^{2}( \partial _{x}^{2}+ \partial _{y}^{2})+(1-s)s=0$ imposing boundary conditions on $SL(2,\Bbb Z)$. [I assume $s=1/2+ik$ and $z=x+iy$, $-$anon]