Rational period functions on \(G(\sqrt {2})\) and \(G(\sqrt {3})\) with hyperbolic poles are not Hecke eigenfunctions (Q1913502)

Using the existence of a linear map between the space of rational period functions defined on the Hecke group \(G(\sqrt p)\), \(p= 2\) or 3, and the space of rational period functions defined on the modular group it is established that a rational period function defined on \(G(\sqrt p)\) with a pole that is the fixed point of a hyperbolic element of \(G(\sqrt p)\) cannot be an eigenfunction of the induced Hecke operator. This extends the corresponding result for the modular group.