Fourier expansions of GL(2) newforms at various cusps

• Published in: The Ramanujan journal Vol. 36; no. 1-2; pp. 3 - 42
• Main Authors: Goldfeld, Dorian, Hundley, Joseph, Lee, Min
• Format: Journal Article
• Language: English
• Published: Boston Springer US 27.02.2015
• Subjects: Combinatorics; Field Theory and Polynomials; Fourier Analysis; Functions of a Complex Variable; Mathematics; Mathematics and Statistics; Number Theory; 11F30; Hecke–Maass newform; Fourier coefficients; 11F70
• ISSN: 1382-4090; 1572-9303
• DOI: 10.1007/s11139-012-9411-9

This paper studies the Fourier expansion of Hecke–Maass eigenforms for GL (2,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,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 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 ℓ Fourier coefficient, at that cusp, vanishes for all sufficiently large positive integers ℓ . In the last part of this paper, a three term identity involving the Fourier expansion at three different cusps is derived.