scholarly journals A short note on sign changes and non-vanishing of Fourier coefficients of half-integral weight cusp forms

2021 ◽  
Author(s):  
Winfried Kohnen
Keyword(s):  
Fourier Coefficients ◽  
Short Note ◽  
Double Sequence ◽  
Cusp Forms ◽  
Sign Changes ◽  
Half Integral Weight ◽  
Integral Weight

AbstractWe study sign changes and non-vanishing of a certain double sequence of Fourier coefficients of cusp forms of half-integral weight.

A SHORT NOTE ON FOURIER COEFFICIENTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS

2010 ◽  
Vol 06 (06) ◽  
pp. 1255-1259 ◽  
Author(s):  
WINFRIED KOHNEN
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Short Note ◽  
Cusp Forms ◽  
Sign Changes ◽  
Half Integral Weight ◽  
Integral Weight

We give an unconditional proof of a result on sign changes of Fourier coefficients of cusp forms of half-integral weight that before was proved only under the hypothesis of Chowla's conjecture.

Some remarks on the Fourier coefficients of cusp forms

2020 ◽  
Vol 16 (09) ◽  
pp. 1935-1943
Author(s):  
Balesh Kumar ◽  
Jay Mehta ◽  
G. K. Viswanadham
Keyword(s):  
Fourier Coefficients ◽  
Cusp Forms ◽  
Modular Functions ◽  
Sign Changes ◽  
Half Integral Weight ◽  
Integral Weight

In this paper, we consider the angular changes of Fourier coefficients of half integral weight cusp forms and sign changes of [Formula: see text]-exponents of generalized modular functions.

Sign changes of Fourier coefficients of half-integral weight cusp forms

2014 ◽  
Vol 10 (04) ◽  
pp. 905-914 ◽  
Author(s):  
Jaban Meher ◽  
M. Ram Murty
Keyword(s):  
Fourier Coefficients ◽  
Quantitative Result ◽  
Cusp Forms ◽  
Real Numbers ◽  
Sign Changes ◽  
Half Integral Weight ◽  
Integral Weight

We prove a quantitative result for the number of sign changes of the Fourier coefficients of half-integral weight cusp forms in the Kohnen plus space, provided the Fourier coefficients are real numbers.

Sign changes of Fourier coefficients of cusp forms supported on prime power indices

2014 ◽  
Vol 10 (08) ◽  
pp. 1921-1927 ◽  
Author(s):  
Winfried Kohnen ◽  
Yves Martin
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Power Indices ◽  
Hecke Operator ◽  
Sign Changes ◽  
Integral Weight ◽  
Hecke Eigenform ◽  
Almost All

Let f be an even integral weight, normalized, cuspidal Hecke eigenform over SL2(ℤ) with Fourier coefficients a(n). Let j be a positive integer. We prove that for almost all primes p the sequence (a(pjn))n≥0 has infinitely many sign changes. We also obtain a similar result for any cusp form with real Fourier coefficients that provide the characteristic polynomial of some generalized Hecke operator is irreducible over ℚ.

Fourier Coefficients of Modular Forms of Half-Integral Weight in Arithmetic Progressions

2020 ◽  
Author(s):  
Corentin Darreye
Keyword(s):  
Gaussian Distribution ◽  
Modular Forms ◽  
Prime Power ◽  
Fourier Coefficients ◽  
Arithmetic Progressions ◽  
Cusp Forms ◽  
Half Integral Weight ◽  
Mixed Distribution ◽  
Integral Weight ◽  
Holomorphic Cusp Forms

Abstract We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progressions. We prove a result analogous to previous work of Fouvry–Ganguly–Kowalski–Michel and Kowalski–Ricotta in the context of half-integral weight holomorphic cusp forms and for prime power modulus. We actually show that these sums follow in a suitable range a mixed Gaussian distribution that comes from the asymptotic mixed distribution of Salié sums.

scholarly journals SIGN CHANGES OF FOURIER COEFFICIENTS OF ENTIRE MODULAR INTEGRALS

2012 ◽  
Vol 54 (2) ◽  
pp. 355-358
Author(s):  
YOUNGJU CHOIE ◽  
WINFRIED KOHNEN
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Congruence Subgroup ◽  
Short Note ◽  
Finitely Generated ◽  
Positive Real ◽  
Sign Changes ◽  
Multiplier System ◽  
Integral Weight ◽  
Modular Integrals

AbstractLet f be a non-zero cusp form with real Fourier coefficients a(n) (n ≥ 1) of positive real weight k and a unitary multiplier system v on a subgroup Γ ⊂ SL2(ℝ) that is finitely generated and of Fuchsian type of the first kind. Then, it is known that the sequence (a(n))(n ≥ 1) has infinitely many sign changes. In this short note, we generalise the above result to the case of entire modular integrals of non-positive integral weight k on the group Γ0*(N) (N ∈ ℕ) generated by the Hecke congruence subgroup Γ0(N) and the Fricke involution $W_N:= \big(\scriptsize\begin{array}{c@{}c} 0 & -{1/\sqrt N} \\[3pt] \sqrt N & 0\\ \end{array}\big)$ provided that the associated period functions are polynomials.

Sign in / Sign up

Export Citation Format

Share Document