On averages of Fourier coefficients of Maass cusp forms

Let f be a holomorphic cusp form of weight k for SL(2, ℤ) with Fourier coefficients λf(n). We study the sum ∑n>0λf(n)ϕ(n/X)e(αn), where [Formula: see text]. It is proved that the sum is rapidly decaying for α close to a rational number a/q where q2 < X1-ε. The main techniques used in this paper include Dirichlet's rational approximation of real numbers, a Voronoi summation formula for SL(2, ℤ) and the asymptotic expansion for Bessel functions.

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Higher Moments of Fourier Coefficients of Cusp Forms

Canadian Mathematical Bulletin ◽

10.4153/cmb-2015-031-1 ◽

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).

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

International Journal of Number Theory ◽

10.1142/s1793042114500626 ◽

2014 ◽

Vol 10 (08) ◽

pp. 1921-1927 ◽

Cited By ~ 2

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.

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ON THE FOURIER COEFFICIENTS OF SIEGEL MODULAR FORMS

Nagoya Mathematical Journal ◽

10.1017/nmj.2016.41 ◽

2016 ◽

Vol 234 ◽

pp. 1-16

Author(s):

SIEGFRIED BÖCHERER ◽

WINFRIED KOHNEN

Keyword(s):

Modular Form ◽

Modular Forms ◽

Fourier Coefficients ◽

Main Tool ◽

Siegel Modular Forms ◽

Cusp Forms ◽

Growth Properties

One can characterize Siegel cusp forms among Siegel modular forms by growth properties of their Fourier coefficients. We give a new proof, which works also for more general types of modular forms. Our main tool is to study the behavior of a modular form for $Z=X+iY$ when $Y\longrightarrow 0$.

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