The hecke operators on the cusp forms of Γ0(N) with nebentype

1994 ◽  
pp. 95-108
Author(s):  
Wang Xiangdong
Keyword(s):  
Hecke Operators ◽  
Cusp Forms

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.

scholarly journals On Dirichlet series attached to cusp forms and the Siegel-zero

1984 ◽  
Vol 25 (1) ◽  
pp. 107-119 ◽  
Author(s):  
F. Grupp
Keyword(s):  
Dirichlet Series ◽  
Cusp Form ◽  
Fourier Expansion ◽  
Dirichlet Character ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Siegel Zero ◽  
Image Position

Let k be an even integer greater than or equal to 12 and f an nonzero cusp form of weight k on SL(2, Z). We assume, further, that f is an eigenfunction for all Hecke-Operators and has the Fourier expansionFor every Dirichlet character xmod Q we define

scholarly journals Period polynomials and explicit formulas for Hecke operators on Γ0(2)

2009 ◽  
Vol 146 (2) ◽  
pp. 321-350 ◽  
Author(s):  
SHINJI FUKUHARA ◽  
YIFAN YANG
Keyword(s):  
Vector Space ◽  
Explicit Form ◽  
Congruence Subgroup ◽  
Bernoulli Polynomials ◽  
Affirmative Answer ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Isomorphism Theorem ◽  
Explicit Formulas ◽  
Space Of Cusp Forms

AbstractLet Sw+2(Γ0(N)) be the vector space of cusp forms of weight w + 2 on the congruence subgroup Γ0(N). We first determine explicit formulas for period polynomials of elements in Sw+2(Γ0(N)) by means of Bernoulli polynomials. When N = 2, from these explicit formulas we obtain new bases for Sw+2(Γ0(2)), and extend the Eichler–Shimura–Manin isomorphism theorem to Γ0(2). This implies that there are natural correspondences between the spaces of cusp forms on Γ0(2) and the spaces of period polynomials. Based on these results, we will find explicit form of Hecke operators on Sw+2(Γ0(2)). As an application of main theorems, we will also give an affirmative answer to a speculation of Imamoglu and Kohnen on a basis of Sw+2(Γ0(2)).

scholarly journals Generators for modules of vector-valued Picard modular forms

2013 ◽  
Vol 212 ◽  
pp. 19-57 ◽  
Author(s):  
Fabien Cléry ◽  
Gerard Van Der Geer
Keyword(s):  
Modular Forms ◽  
Unitary Group ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Picard Modular Forms ◽  
Vector Valued

AbstractWe construct generators for modules of vector-valued Picard modular forms on a unitary group of type (2, 1) over the Eisenstein integers. We also calculate eigenvalues of Hecke operators acting on cusp forms.

scholarly journals Generators for modules of vector-valued Picard modular forms

2013 ◽  
Vol 212 ◽  
pp. 19-57
Author(s):  
Fabien Cléry ◽  
Gerard Van Der Geer
Keyword(s):  
Modular Forms ◽  
Unitary Group ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Picard Modular Forms ◽  
Vector Valued

AbstractWe construct generators for modules of vector-valued Picard modular forms on a unitary group of type (2, 1) over the Eisenstein integers. We also calculate eigenvalues of Hecke operators acting on cusp forms.

scholarly journals Hecke operators on Drinfeld cusp forms

2008 ◽  
Vol 128 (7) ◽  
pp. 1941-1965 ◽  
Author(s):  
Wen-Ching Winnie Li ◽  
Yotsanan Meemark
Keyword(s):  
Hecke Operators ◽  
Cusp Forms

On Hilbert and quaternionic cusp forms

2006 ◽  
Vol 136 (1) ◽  
pp. 1-6
Author(s):  
A. Arenas
Keyword(s):  
Quaternion Algebra ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Natural Manner ◽  
Langlands Correspondence ◽  
Hilbert Modular

The aim of this paper is to determine in a natural manner the subspace of the space of Hilbert modular newforms of level n which correspond to eigenforms of an appropriate quaternion algebra, in the sense of having the same eigenvalues with respect to the corresponding Hecke operators. This study may be seen as a particular case of the Jacquet–Langlands correspondence.

scholarly journals Products of vector valued Eisenstein series

2017 ◽  
Vol 29 (1) ◽  
Author(s):  
Martin Westerholt-Raum
Keyword(s):  
Eisenstein Series ◽  
Modular Forms ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Congruence Subgroups ◽  
Level 1 ◽  
Vector Valued ◽  
Level Aspect

AbstractWe prove that products of at most two vector valued Eisenstein series that originate in level 1 span all spaces of cusp forms for congruence subgroups. This can be viewed as an analogue in the level aspect to a result that goes back to Rankin, and Kohnen and Zagier, which focuses on the weight aspect. The main feature of the proof are vector valued Hecke operators. We recover several classical constructions from them, including classical Hecke operators, Atkin–Lehner involutions, and oldforms. As a corollary to our main theorem, we obtain a vanishing condition for modular forms reminiscent of period relations deduced by Kohnen and Zagier in the context their previously mentioned result.

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