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.

scholarly journals Estimates for fourier coefficients of siegel cusp forms of degree two, II

1992 ◽  
Vol 128 ◽  
pp. 171-176 ◽  
Author(s):  
Winfried Kohnen
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Positive Definite ◽  
Cusp Forms ◽  
Integral Weight ◽  
Siegel Cusp Form

Let F be a Siegel cusp form of integral weight k on Γ2: = Sp2(Z) and denote by a(T) (T a positive definite symmetric half-integral (2,2)-matrix) its Fourier coefficients. In [2] Kitaoka proved that(1)(the result is actually stated only under the assumption that k is even). In our previous paper [3] it was shown that one can attain(2)

scholarly journals THE SIGN OF FOURIER COEFFICIENTS OF HALF-INTEGRAL WEIGHT CUSP FORMS

2012 ◽  
Vol 08 (03) ◽  
pp. 749-762 ◽  
Author(s):  
THOMAS A. HULSE ◽  
E. MEHMET KIRAL ◽  
CHAN IEONG KUAN ◽  
LI-MEI LIM
Keyword(s):  
Cusp Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Square Root ◽  
Critical Strip ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Finite Set ◽  
Invent Math

From a result of Waldspurger [W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math.64 (1981) 175–198], it is known that the normalized Fourier coefficients a(m) of a half-integral weight holomorphic cusp eigenform 𝔣 are, up to a finite set of factors, one of [Formula: see text] when m is square-free and f is the integral weight cusp form related to 𝔣 by the Shimura correspondence [G. Shimura, On modular forms of half-integral weight, Ann. of Math.97 (1973) 440–481]. In this paper we address a question posed by Kohnen: which square root is a(m)? In particular, if we look at the set of a(m) with m square-free, do these Fourier coefficients change sign infinitely often? By partially analytically continuing a related Dirichlet series, we are able to show that this is so.

scholarly journals Hecke structure of spaces of half-integral weight cusp forms

2000 ◽  
Vol 159 ◽  
pp. 53-85 ◽  
Author(s):  
Sharon M. Frechette
Keyword(s):  
Cusp Form ◽  
Modular Forms ◽  
Dirichlet Character ◽  
Cusp Forms ◽  
Half Integral Weight ◽  
Shimura Lift ◽  
Integral Weight ◽  
Structure Theorems ◽  
Space Of Cusp Forms ◽  
Shimura Correspondence

We investigate the connection between integral weight and half-integral weight modular forms. Building on results of Ueda [14], we obtain structure theorems for spaces of half-integral weight cusp forms Sk/2(4N,χ) where k and N are odd nonnegative integers with k ≥ 3, and χ is an even quadratic Dirichlet character modulo 4N. We give complete results in the case where N is a power of a single prime, and partial results in the more general case. Using these structure results, we give a classical reformulation of the representation-theoretic conditions given by Flicker [5] and Waldspurger [17] in results regarding the Shimura correspondence. Our version characterizes, in classical terms, the largest possible image of the Shimura lift given our restrictions on N and χ, by giving conditions under which a newform has an equivalent cusp form in Sk/2(4N, χ). We give examples (computed using tables of Cremona [4]) of newforms which have no equivalent half-integral weight cusp forms for any such N and χ. In addition, we compare our structure results to Ueda’s [14] decompositions of the Kohnen subspace, illustrating more precisely how the Kohnen subspace sits inside the full space of cusp forms.

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.

scholarly journals CONSTRUCTING SIMULTANEOUS HECKE EIGENFORMS

2010 ◽  
Vol 06 (05) ◽  
pp. 1117-1137 ◽  
Author(s):  
T. SHEMANSKE ◽  
S. TRENEER ◽  
L. WALLING
Keyword(s):  
Hecke Operators ◽  
Cusp Forms ◽  
Hecke Eigenforms ◽  
Linearly Independent ◽  
Integral Weight ◽  
Trivial Character ◽  
Free Level ◽  
Fixed Space ◽  
Space Of Cusp Forms

It is well known that newforms of integral weight are simultaneous eigenforms for all the Hecke operators, and that the converse is not true. In this paper, we give a characterization of all simultaneous Hecke eigenforms associated to a given newform, and provide several applications. These include determining the number of linearly independent simultaneous eigenforms in a fixed space which correspond to a given newform, and characterizing several situations in which the full space of cusp forms is spanned by a basis consisting of such eigenforms. Part of our results can be seen as a generalization of results of Choie–Kohnen who considered diagonalization of "bad" Hecke operators on spaces with square-free level and trivial character. Of independent interest, but used herein, is a lower bound for the dimension of the space of newforms with arbitrary character.

Exponential sums twisted by Fourier coefficients of automorphic cusp forms for SL(2, ℤ)

2014 ◽  
Vol 11 (01) ◽  
pp. 39-49 ◽  
Author(s):  
Bin Wei
Keyword(s):  
Asymptotic Expansion ◽  
Cusp Form ◽  
Rational Number ◽  
Fourier Coefficients ◽  
Bessel Functions ◽  
Exponential Sums ◽  
Summation Formula ◽  
Cusp Forms ◽  
Real Numbers ◽  
Voronoi Summation

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.

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).

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.

scholarly journals Fourier coefficients of Siegel cusp forms of degree two

1984 ◽  
Vol 93 ◽  
pp. 149-171 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Siegel Cusp Form

Our purpose is to prove the followingTheorem. Let k be an even integer ≥ 6. Letbe a Siegel cusp form of degree two, weight k. Then we have

Cancellation of Cusp Forms Coefficients over Beatty Sequences on GL(m)

2011 ◽  
Vol 54 (4) ◽  
pp. 757-762
Author(s):  
Qingfeng Sun
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Maass Cusp Form ◽  
Beatty Sequences

AbstractLet A(n1, n2, … , nm–1) be the normalized Fourier coefficients of a Maass cusp form on GL(m). In this paper, we study the cancellation of A(n1, n2, … , nm–1) over Beatty sequences.

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