Fourier Expansions of Gl(2) Newforms at Various Cusps

This paper studies the Fourier expansion of Hecke-Maass eigenforms for GL(2,Q) of arbitrary weight, level, and character at various cusps. It is shown that the Fourier coefficients at a cusp satisfy certain very explicit multiplicativity relations. As an application, it is proved that a local representation of GL(2,Qp) which is isomorphic to a local factor of a global cuspidal automorphic representation generated by the adelic lift of a newform of arbitrary weight, level N , and character χ (mod N) cannot be supercuspidal if χ is primitive. Furthermore, it is supercuspidal if and only if at every cusp (of width m and cusp parameter = 0) the mp` Fourier coefficient, at that cusp, vanishes for all sufficiently large positive integers `. In the last part of this paper a three term identity involving the Fourier expansion at three different cusps is derived.