scholarly journals Values of L-series of Hecke eigenforms

2020 ◽  
Vol 211 ◽  
pp. 28-42
Author(s):  
Kamal Khuri-Makdisi ◽  
Winfried Kohnen ◽  
Wissam Raji
Keyword(s):  
Hecke Eigenforms

scholarly journals CONGRUENCES FOR ANDREWS' SMALLEST PARTS PARTITION FUNCTION AND NEW CONGRUENCES FOR DYSON'S RANK

2010 ◽  
Vol 06 (02) ◽  
pp. 281-309 ◽  
Author(s):  
F. G. GARVAN
Keyword(s):  
Partition Function ◽  
Rank Function ◽  
Maass Forms ◽  
Hecke Eigenforms ◽  
Half Integer ◽  
Integer Weight

Let spt (n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt (n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain explicit Ramanujan-type congruences for spt (n) mod ℓ for ℓ = 11, 17, 19, 29, 31 and 37. Recently, Bringmann and Ono proved that Dyson's rank function has infinitely many Ramanujan-type congruences. Their proof is non-constructive and utilizes the theory of weak Maass forms. We construct two explicit nontrivial examples mod 11 using elementary congruences between rank moments and half-integer weight Hecke eigenforms.

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 Hecke eigenforms as products of eigenforms

2013 ◽  
Vol 133 (7) ◽  
pp. 2339-2362 ◽  
Author(s):  
Matthew Leander Johnson
Keyword(s):  
Hecke Eigenforms

scholarly journals Mass equidistribution for Hecke eigenforms

2010 ◽  
Vol 172 (2) ◽  
pp. 1517 ◽  
Author(s):  
Roman Holowinsky ◽  
Kannan Soundararajan
Keyword(s):  
Hecke Eigenforms

Polynomial identities between Hecke eigenforms

2019 ◽  
Vol 15 (10) ◽  
pp. 2135-2150
Author(s):  
Dianbin Bao
Keyword(s):  
Eisenstein Series ◽  
Product Formula ◽  
Inner Product ◽  
Polynomial Identities ◽  
Number Of Solutions ◽  
Hecke Eigenforms

In this paper, we study solutions to [Formula: see text], where [Formula: see text] are Hecke newforms with respect to [Formula: see text] of weight [Formula: see text] and [Formula: see text]. We show that the number of solutions is finite for all [Formula: see text]. Assuming Maeda’s conjecture, we prove that the Petersson inner product [Formula: see text] is nonzero, where [Formula: see text] and [Formula: see text] are any nonzero cusp eigenforms for [Formula: see text] of weight [Formula: see text] and [Formula: see text], respectively. As a corollary, we obtain that, assuming Maeda’s conjecture, identities between cusp eigenforms for [Formula: see text] of the form [Formula: see text] all are forced by dimension considerations. We also give a proof using polynomial identities between eigenforms that the [Formula: see text]-function is algebraic on zeros of Eisenstein series of weight [Formula: see text].

scholarly journals On lifting Hecke eigenforms

1991 ◽  
Vol 328 (2) ◽  
pp. 881-896
Author(s):  
Lynne H. Walling
Keyword(s):  
Hecke Eigenforms

Unimodularity of zeros of period polynomials of Hecke eigenforms

2014 ◽  
Vol 46 (3) ◽  
pp. 528-536 ◽  
Author(s):  
Ahmad El-Guindy ◽  
Wissam Raji
Keyword(s):  
Hecke Eigenforms

Ramanujan-type congruences for ℓ-regular partitions modulo 3, 5, 11 and 13

2017 ◽  
Vol 13 (08) ◽  
pp. 1995-2006 ◽  
Author(s):  
Hai-Tao Jin ◽  
Li Zhang
Keyword(s):  
Infinite Family ◽  
Hecke Eigenforms ◽  
Regular Partitions

Let [Formula: see text] be the number of [Formula: see text]-regular partitions of [Formula: see text]. Recently, Hou et al. established several infinite families of congruences for [Formula: see text] modulo [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, using the vanishing property given by Hou et al., we prove an infinite family of congruence for [Formula: see text] modulo [Formula: see text]. Moreover, for [Formula: see text] and [Formula: see text], we obtain three infinite families of congruences for [Formula: see text] modulo [Formula: see text] and [Formula: see text] respectively using the theory of Hecke eigenforms.

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