Fourier expansions of GL(2) newforms at various cusps

This paper studies the Fourier expansion of Hecke-Maass eigenforms for $GL(2, \mathbb Q)$ of arbitrary weight, level, and character at various cusps. Translating well known results in the theory of adelic automorphic representations into classical language, a multiplicative expression for the Fourier coefficients at any cusp is derived. In general, this expression involves Fourier coefficients at several different cusps. A sufficient condition for the existence of multiplicative relations among Fourier coefficients at a single cusp is given. It is shown that if the level is 4 times (or in some cases 8 times) an odd squarefree number then there are multiplicative relations at every cusp. We also show that a local representation of $GL(2,{\mathbb{Q}}_p)$ 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 $\chi (\mathrm{m}\mathrm{o}\mathrm{d}N)$ cannot be supercuspidal if $\chi$ is primitive. Furthermore, it is supercuspidal if and only if at every cusp (of width $m$ and cusp parameter = 0) the $m{p}^{\mathrm{\ell }}$ Fourier coefficient, at that cusp, vanishes for all sufficiently large positive integers $\mathrm{\ell }$. In the last part of this paper a three term identity involving the Fourier expansion at three different cusps is derived.