On signs of Fourier coefficients of cusp forms 209
نویسنده
چکیده
We consider two problems concerning signs of Fourier coefficients of classical modular forms, or equivalently Hecke eigenvalues: first, we give an upper bound for the size of the first sign-change of Hecke eigenvalues in terms of conductor and weight; second, we investigate to what extent the signs of Fourier coefficients determine an unique modular form. In both cases we improve recent results of Kowalski, Lau, Soundararajan and Wu. A part of the paper is also devoted to generalized rearrangement inequalities which are utilized in an alternative treatment of the second question.
منابع مشابه
Non-vanishing and sign changes of Hecke eigenvalues for half-integral weight cusp forms
In this paper, we consider three problems about signs of the Fourier coefficients of a cusp form f with half-integral weight: – The first negative coefficient of the sequence {af(tn)}n∈N, – The number of coefficients af(tn ) of same signs, – Non-vanishing of coefficients af(tn ) in short intervals and in arithmetic progressions, where af(n) is the n-th Fourier coefficient of f and t is a square...
متن کاملFourier Coefficients of Hilbert Cusp Forms Associated with Mixed Hilbert Cusp Forms
We express the Fourier coefficients of the Hilbert cusp form Lhf associated with mixed Hilbert cusp forms f and h in terms of the Fourier coefficients of a certain periodic function determined by f and h. We also obtain an expression of each Fourier coefficient of Lhf as an infinite series involving the Fourier coefficients of f and h.
متن کاملA Relation between Fourier Coefficients of Holomorphic Cusp Forms and Exponential Sums
We consider certain specific exponential sums related to holomorphic cusp forms, give a reformulation for the Lehmer conjecture and prove that certain exponential sums of Fourier coefficients of holomorphic cusp forms contain information on other similar non-overlapping exponential sums. Also, we prove an Omega result for short sums of Fourier coefficients.
متن کاملSign Changes of Coefficients of Half Integral Weight Modular Forms
For a half integral weight modular form f we study the signs of the Fourier coefficients a(n). If f is a Hecke eigenform of level N with real Nebentypus character, and t is a fixed square-free positive integer with a(t) 6= 0, we show that for all but finitely many primes p the sequence (a(tp2m))m has infinitely many signs changes. Moreover, we prove similar (partly conditional) results for arbi...
متن کاملApplications of a Pre–trace Formula to Estimates on Maass Cusp Forms
By using spectral expansions in global automorphic Levi–Sobolev spaces, we estimate an average of the first Fourier coefficients of Maass cusp forms for SL2(Z), producing a soft estimate on the first numerical Fourier coefficients of Maass cusp forms, which is an example of a general technique for estimates on compact periods via application of a pre–trace formula. Incidentally, this shows that...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2012