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 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.

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.

scholarly journals Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications

2017 ◽  
Vol 15 (1) ◽  
pp. 304-316
Author(s):  
SoYoung Choi ◽  
Chang Heon Kim
Keyword(s):  
Fourier Coefficients ◽  
Hecke Algebra ◽  
Cusp Forms ◽  
Arithmetic Groups ◽  
Multiplicity One ◽  
Integral Weight ◽  
Hecke Eigenform

Abstract For an odd and squarefree level N, Kohnen proved that there is a canonically defined subspace $S_{\kappa+\frac{1}{2}}^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)\subset S_{\kappa+\frac{1}{2}}(N),\,\,{\text{and}}\,\,S_{\kappa+\frac{1}{2}}^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)\,\,{\text{and}}\,\,S_{2k}^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)$ are isomorphic as modules over the Hecke algebra. Later he gave a formula for the product $a_{g}(m)\overline{a_{g}(n)}$ of two arbitrary Fourier coefficients of a Hecke eigenform g of halfintegral weight and of level 4N in terms of certain cycle integrals of the corresponding form f of integral weight. To this end he first constructed Shimura and Shintani lifts, and then combining these lifts with the multiplicity one theorem he deduced the formula in [2, Theorem 3]. In this paper we will prove that there is a Hecke equivariant isomorphism between the spaces $S_{2k}^{+}(p)\,\,{\text{and}}\,\,\mathbb{S}_{k+\frac{1}{2}}(p).$ We will also construct Shintani and Shimura lifts for these spaces, and prove a result analogous to [2, Theorem 3].

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.

scholarly journals Sign changes of Fourier coefficients of Siegel cusp forms of degree two on Hecke congruence subgroups

2017 ◽  
Vol 13 (10) ◽  
pp. 2597-2625 ◽  
Author(s):  
S. Gun ◽  
J. Sengupta
Keyword(s):  
Cusp Form ◽  
Upper Bound ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Congruence Subgroups ◽  
Sign Changes ◽  
Sign Change ◽  
Integral Weight ◽  
Zero Degree ◽  
Siegel Cusp Form

In this paper, we give a lower bound on the number of sign changes of Fourier coefficients of a non-zero degree two Siegel cusp form of even integral weight on a Hecke congruence subgroup. We also provide an explicit upper bound for the first sign change of Fourier coefficients of such Siegel cusp forms. Explicit upper bound on the first sign change of Fourier coefficients of a non-zero Siegel cusp form of even integral weight on the Siegel modular group for arbitrary genus was dealt in an earlier work of Choie, the first author and Kohnen.

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.

Higher Moments of Fourier Coefficients of Cusp Forms

2015 ◽  
Vol 58 (3) ◽  
pp. 548-560
Author(s):  
Guangshi Lü ◽  
Ayyadurai Sankaranarayanan
Keyword(s):  
Modular Group ◽  
Fourier Coefficients ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Cusp Forms ◽  
Higher Moments ◽  
Integral Weight ◽  
Holomorphic Cusp Forms

AbstractLet Sk(Γ) be the space of holomorphic cusp forms of even integral weight k for the full modular group SL(z, ℤ). Let be the n-th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms , and h(z) ∊ Sk3 (Γ), respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as twisted by the arithmetic function λf(n).

scholarly journals Petersson products of bases of spaces of cusp forms and estimates for Fourier coefficients

2018 ◽  
Vol 14 (08) ◽  
pp. 2277-2290 ◽  
Author(s):  
Rainer Schulze-Pillot ◽  
Abdullah Yenirce
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Orthogonal Basis ◽  
Cusp Forms ◽  
Integral Weight ◽  
Space Of Cusp Forms

We prove a bound for the Fourier coefficients of a cusp form of integral weight which is not a newform by computing an explicit orthogonal basis for the space of cusp forms of given integral weight and level.

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