International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 149154 | https://doi.org/10.1155/2012/149154

Hirotaka Kodama, Shoyu Nagaoka, Yoshitsugu Nakamura, "On Level p Siegel Cusp Forms of Degree Two", International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 149154, 9 pages, 2012. https://doi.org/10.1155/2012/149154

On Level p Siegel Cusp Forms of Degree Two

Academic Editor: Aloys Krieg
Received09 Jul 2012
Accepted06 Aug 2012
Published12 Sep 2012

Abstract

We give a simple formula for the Fourier coefficients of some degree-two Siegel cusp form with level p.

1. Introduction

In the previous paper [1], the second and the third authors introduced a simple construction of a Siegel cusp form of degree 2. This construction has an advantage because the Fourier coefficients are explicitly computable. After this work was completed, Kikuta and Mizuno proved that the š‘-adic limit of a sequence of the aforementioned cusp forms becomes a Siegel cusp form of degree 2 with level š‘.

In this paper, we give an explicit description of the Fourier expansion of such a form. This result shows that the cusp form becomes a nonzero cusp form of weight 2 on Ī“20(š‘) if š‘>7 and š‘ā‰”3ā€‰ā€‰(mod4).

2. Siegel Modular Forms of Degree 2

We start by recalling the basic facts of Siegel modular forms.

The Siegel upper half-space of degree 2 is defined by ā„2ī€½āˆ¶=š‘=š‘‹+š‘–š‘ŒāˆˆSym2ī€¾(ā„‚)āˆ£š‘Œ>0(positive-deļ¬nite).(2.1) Then the degree 2 Siegel modular group Ī“2āˆ¶=Sp2(ā„¤) acts on ā„2 discontinuously. For a congruence subgroup Ī“ī…žāŠ‚Ī“2, we denote by š‘€š‘˜(Ī“ī…ž) (resp., š‘†š‘˜(Ī“ī…ž)) the corresponding space of Siegel modular forms (resp., cusp forms) of weight š‘˜.

We will be mainly concerned with the Siegel modular group Ī“2 and the congruence subgroup Ī“20āŽ§āŽŖāŽØāŽŖāŽ©āŽ›āŽœāŽœāŽāŽžāŽŸāŽŸāŽ (š‘)āˆ¶=š“šµš¶š·āˆˆĪ“2āŽ«āŽŖāŽ¬āŽŖāŽ­āˆ£š¶ā‰”š‘‚(modš‘).(2.2) In both cases, š¹āˆˆš‘€š‘˜(Ī“ī…ž) has a Fourier expansion of the form ī“š¹(š‘)=0ā‰¤š‘‡āˆˆĪ›2[],š‘Ž(š‘‡;š¹)exp2šœ‹š‘–tr(š‘‡š‘)(2.3) where Ī›2ī€½ī€·š‘”āˆ¶=š‘‡=š‘–š‘—ī€øāˆˆSym2(ā„š)āˆ£š‘”11,š‘”22,2š‘”12ī€¾,āˆˆā„¤(2.4) and š‘Ž(š‘‡;š¹) is the Fourier coefficient of š¹ at š‘‡.

3. Siegel Cusp Form of Degree 2

In the previous paper [1], we constructed a cusp form š‘“š‘˜āˆˆš‘†š‘˜(Ī“2) whose Fourier coefficients are explicitly computable. We review the result.

First, we recall the definition of Cohenā€™s function. Cohen defined an arithmetical function š»(š‘Ÿ,š‘) (š‘Ÿ,š‘āˆˆā„¤ā‰„0) in [2]. In the case that š‘Ÿ and š‘ satisfy (āˆ’1)š‘Ÿš‘=š·ā‹…š‘“2 where š· is a fundamental discriminant and š‘“āˆˆā„•, the function is given by ī€·š»(š‘Ÿ,š‘)=šæ1āˆ’š‘Ÿ,šœ’š·ī€øī“0<š‘‘|š‘“šœ‡(š‘‘)šœ’š·(š‘‘)š‘‘š‘Ÿāˆ’1šœŽ2š‘Ÿāˆ’1ī‚µš‘“š‘‘ī‚¶.(3.1) Here, šæ(š‘ ,šœ’) is the Dirichlet šæ-function with character šœ’, and šœ‡ is the Mƶbius function. For the precise definition of š»(š‘Ÿ,š‘), see [2, page 272].

Secondly, we introduce Kriegā€™s function šŗ(š‘ ,š‘) (š‘ ,š‘āˆˆā„¤ā‰„0) associated with the Gaussian field ā„š(š‘–). Let šœ’āˆ’4 be the Kronecker character associated with ā„š(š‘–). Kriegā€™s function šŗ(š‘ ,š‘)=šŗā„š(š‘–)(š‘ ,š‘) over ā„š(š‘–) is defined by āŽ§āŽŖāŽØāŽŖāŽ©1šŗ(š‘ ,š‘)āˆ¶=||šœ’1+āˆ’4||ī€·šœŽ(š‘)š‘ ,šœ’āˆ’4(š‘)āˆ’ī‚šœŽš‘ ,šœ’āˆ’4ī€øāˆ’šµ(š‘),ifš‘>0,š‘ +1,šœ’āˆ’42(š‘ +1),ifš‘=0,(3.2) where šµš‘š,šœ’ is the generalized Bernoulli number with character šœ’, šœŽš‘ ,šœ’āˆ’4ī“(š‘)āˆ¶=0<š‘‘|š‘šœ’āˆ’4(š‘‘)š‘‘š‘ ,ī‚šœŽš‘ ,šœ’āˆ’4ī“(š‘)āˆ¶=0<š‘‘|š‘šœ’āˆ’4ī‚€š‘š‘‘ī‚š‘‘š‘ .(3.3) This function was introduced by Krieg [3] to describe the Fourier coefficients of Hermitian Eisenstein series of degree 2.

The following theorem is one of the main results in [1].

Theorem 3.1. There exists a Siegel cusp form š‘“š‘˜āˆˆš‘†š‘˜(Ī“2) whose Fourier coefficients š‘Ž(š‘‡;š‘“š‘˜) are given as follows: š‘Žī€·š‘‡;š‘“š‘˜ī€ø=ī“0<š‘‘|šœ€(š‘‡)š‘‘š‘˜āˆ’1š›¼š‘˜ī‚µ4det(š‘‡)š‘‘2ī‚¶,(3.4) where š›¼š‘˜(šµš‘)āˆ¶=š»(š‘˜āˆ’1,š‘)āˆ’2š‘˜āˆ’2šµš‘˜āˆ’1,šœ’āˆ’4ī“š‘ š‘ āˆˆā„¤2ā‰¤š‘šŗī€·š‘˜āˆ’2,š‘āˆ’š‘ 2ī€ø,ī€½šœ€(š‘‡)āˆ¶=maxš‘™āˆˆā„•āˆ£š‘™āˆ’1š‘‡āˆˆĪ›2ī€¾.(3.5) Here, šµš‘š is the š‘šth Bernoulli number.

Remark 3.2. The above result shows that the cusp form š‘“š‘˜ is a form in the Maass space (cf. [1]).

4. š‘-Adic Siegel Modular Forms

The cusp form š‘“š‘˜ introduced in Theorem 3.1 was constructed by the difference between the Siegel Eisenstein series šøš‘˜ and the restriction of the Hermitian Eisenstein series šøš‘˜,ā„š(š‘–): š‘“š‘˜=š‘š‘˜ā‹…ī€·šøš‘˜āˆ’šøš‘˜,ā„š(š‘–)||ā„2ī€ø,(4.1) for some š‘š‘˜āˆˆā„š. The š‘-adic properties of the Eisenstein series šøš‘˜ and šøš‘˜,ā„š(š‘–) are studied by the second author (cf. [4, 5]). After our work [1] was completed, Kikuta and Mizuno studied š‘-adic properties of our form š‘“š‘˜. The following statement is a special case in [6].

Theorem 4.1. Let š‘ be a prime number satisfying š‘ā‰”3(mod4), and {š‘˜š‘š} is the sequence defined by š‘˜š‘š=š‘˜š‘š(š‘)āˆ¶=2+(š‘āˆ’1)š‘š‘šāˆ’1.(4.2) Then there exists the š‘-adic limit š‘“āˆ—š‘āˆ¶=limš‘šā†’āˆžš‘“š‘˜š‘š,(4.3) and š‘“āˆ—š‘ represents a cusp form of weight 2 with level š‘, that is, š‘“āˆ—š‘āˆˆš‘†2ī€·Ī“20ī€ø.(š‘)(4.4)

Remark 4.2. (1) The š‘-adic convergence of modular forms is interpreted as the convergence of the Fourier coefficients.
(2) Kikuta and Mizuno studied a similar problem under more general situation. They noted that if we take the sequence {š‘˜š‘š} with š‘˜š‘š=š‘˜+(š‘āˆ’1)š‘š‘šāˆ’1, š‘˜āˆˆā„• (š‘˜>4), then limš‘šā†’āˆžš‘“š‘˜š‘š is no longer a cusp form [6, Theorem 1.7].
(3) The cuspidality of š‘“āˆ—š‘ essentially results from the fact that there are no nontrivial modular forms of weight 2 on the full modular group Ī“2.

5. Main Result

In this section, we give an explicit formula for the Fourier coefficients of š‘“āˆ—š‘.

To describe š‘Ž(š‘‡;š‘“āˆ—š‘), we will introduce two functions š»āˆ—š‘ and šŗāˆ—š‘.

First, for š‘āˆˆā„• with š‘ā‰”0or3(mod4), we write š‘ as š‘=āˆ’š·ā‹…š‘“2 where š· is a fundamental discriminant and š‘“āˆˆā„•. Then, we define š»āˆ—š‘ī€·(š‘)āˆ¶=āˆ’1āˆ’šœ’š·ī€øšµ(š‘)1,šœ’š·ī“0<š‘‘|š‘“(š‘‘,š‘)=1šœ‡(š‘‘)šœ’š·(š‘‘)šœŽāˆ—1ī‚µš‘“š‘‘ī‚¶,(5.1) where šœŽāˆ—1(ī“š‘š)=0<š‘‘|š‘š(š‘‘,š‘)=1š‘‘.(5.2) Secondly, for š‘āˆˆā„¤ā‰„0, we define šŗāˆ—š‘āŽ§āŽŖāŽØāŽŖāŽ©(š‘)āˆ¶=1āˆ’(āˆ’1)ordš‘(š‘)||šœ’1+āˆ’4||šœŽ(š‘)āˆ—0,šœ’āˆ’41(š‘),ifš‘>0,2,ifš‘=0,(5.3) where šœŽāˆ—0,šœ’āˆ’4(ī“š‘)=0<š‘‘|š‘(š‘‘,š‘)=1šœ’āˆ’4(š‘‘).(5.4)

Remark 5.1. From the definition, the following holds: šŗāˆ—š‘(š‘)=0ifš‘āˆ¤š‘.(5.5)

The main theorem of this paper can be stated as follows.

Theorem 5.2. Let š‘ be a prime number satisfying š‘ā‰”3(mod4). Then the Fourier coefficients š‘Ž(š‘‡;š‘“āˆ—š‘) of š‘“āˆ—š‘āˆˆš‘†2(Ī“20(š‘)) are given by š‘Žī€·š‘‡;š‘“āˆ—š‘ī€ø=ī“0<š‘‘|šœ€(š‘‡)(š‘‘,š‘)=1š‘‘š›¼āˆ—š‘ī‚µ4det(š‘‡)š‘‘2ī‚¶,(5.6) where š›¼āˆ—š‘(š‘)āˆ¶=š»āˆ—š‘(š‘)āˆ’š‘āˆ’16ī“š‘ š‘ āˆˆā„¤2ā‰¤š‘šŗāˆ—š‘ī€·š‘āˆ’š‘ 2ī€ø.(5.7) Here, š»āˆ—š‘ and šŗāˆ—š‘ are the functions defined in (5.1) and (5.3), respectively.

From Theorems 3.1 and 4.1, the proof of Theorem 5.2 is reduced to show that limš‘šā†’āˆžš›¼š‘˜š‘š(š‘)=š›¼āˆ—š‘(š‘).(5.8)

We proceed the proof of (5.8) step by step.

Lemma 5.3. Consider the following: limš‘šā†’āˆžš»ī€·š‘˜š‘šī€øāˆ’1,š‘=š»āˆ—š‘(š‘).(5.9)

Proof. Under the description š‘=āˆ’š·ā‹…š‘“2, we can write š»(š‘˜š‘šāˆ’1,š‘) as š»ī€·š‘˜š‘šī€øšµāˆ’1,š‘=āˆ’š‘˜š‘šāˆ’1,šœ’š·š‘˜š‘šī“āˆ’10<š‘‘|š‘“šœ‡(š‘‘)šœ’š·(š‘‘)š‘‘š‘˜š‘šāˆ’2šœŽ2š‘˜š‘šāˆ’3ī‚µš‘“š‘‘ī‚¶,(5.10) (cf. (3.1)).
Using Kummerā€™s congruence, we obtain limš‘šā†’āˆžšµš‘˜š‘šāˆ’1,šœ’š·š‘˜š‘š=ī€·āˆ’11āˆ’šœ’š·ī€øšµ(š‘)1,šœ’š·.(5.11) On the other hand, we have limš‘šā†’āˆžī“0<š‘‘|š‘“šœ‡(š‘‘)šœ’š·(š‘‘)š‘‘š‘˜š‘šāˆ’2šœŽ2š‘˜š‘šāˆ’3ī‚µš‘“š‘‘ī‚¶=ī“0<š‘‘|š‘“(š‘‘,š‘)=1šœ‡(š‘‘)šœ’š·(š‘‘)šœŽāˆ—1ī‚µš‘“š‘‘ī‚¶,(5.12) because limš‘šā†’āˆžš‘‘š‘˜š‘šāˆ’2=ī‚»1,ifš‘āˆ¤š‘‘,0,ifš‘āˆ£š‘‘,limš‘šā†’āˆžšœŽ2š‘˜š‘šāˆ’3(š‘™)=limš‘šā†’āˆžī“0<š‘‘|š‘™š‘‘1+2(š‘āˆ’1)š‘š‘šāˆ’1=ī“0<š‘‘|š‘™(š‘‘,š‘)=1š‘‘=šœŽāˆ—1(š‘™),(š‘™āˆˆā„•).(5.13) This proves (5.9).

Lemma 5.4. Consider the following: limš‘šā†’āˆžšµ2š‘˜š‘šāˆ’2šµš‘˜š‘šāˆ’1,šœ’āˆ’4ī“š‘ š‘ āˆˆā„¤2ā‰¤š‘šŗī€·š‘˜š‘šāˆ’2,š‘āˆ’š‘ 2ī€ø=š‘āˆ’16ī“š‘ š‘ āˆˆā„¤2ā‰¤š‘šŗāˆ—š‘ī€·š‘āˆ’š‘ 2ī€ø.(5.14)

Proof. First, we calculate the factor of Bernoulli numbers. Again by Kummerā€™s congruence, we obtain limš‘šā†’āˆžšµ2š‘˜š‘šāˆ’2šµš‘˜š‘šāˆ’1,šœ’āˆ’4=2limš‘šā†’āˆžšµ2š‘˜š‘šāˆ’22š‘˜š‘šā‹…š‘˜āˆ’2š‘šāˆ’1šµš‘˜š‘šāˆ’1,šœ’āˆ’4šµ=2ā‹…(1āˆ’š‘)ā‹…22ā‹…1ī€·1āˆ’šœ’āˆ’4ī€øšµ(š‘)1,šœ’āˆ’4=š‘āˆ’16.(5.15) Here, we used the facts that šœ’āˆ’4(š‘)=āˆ’1 and šµ1,šœ’āˆ’4=āˆ’1/2.
Next we calculate limš‘šā†’āˆžī“š‘ š‘ āˆˆā„¤2ā‰¤š‘šŗī€·š‘˜š‘šāˆ’2,š‘āˆ’š‘ 2ī€ø.(5.16) If š‘ā€²āˆ¶=š‘āˆ’š‘ 2>0, then šŗī€·š‘˜š‘šāˆ’2,š‘ī…žī€ø=1||šœ’1+āˆ’4ī€·š‘ī…žī€ø||ī€·šœŽš‘˜š‘šāˆ’2,šœ’āˆ’4ī€·š‘ī…žī€øāˆ’ī‚šœŽš‘˜š‘šāˆ’2,šœ’āˆ’4ī€·š‘ī…ž,ī€øī€ø(5.17) (cf. (3.2)). Therefore, we need to calculate limš‘šā†’āˆžšœŽš‘˜š‘šāˆ’2,šœ’āˆ’4ī€·š‘ī…žī€ø,limš‘šā†’āˆžī‚šœŽš‘˜š‘šāˆ’2,šœ’āˆ’4ī€·š‘ī…žī€ø.(5.18) We have limš‘šā†’āˆžšœŽš‘˜š‘šāˆ’2,šœ’āˆ’4ī€·š‘ī…žī€ø=limš‘šā†’āˆžī“0<š‘‘|š‘ā€²šœ’āˆ’4(š‘‘)š‘‘(š‘āˆ’1)š‘š‘šāˆ’1=ī“0<š‘‘|š‘ā€²(š‘‘,š‘)=1šœ’āˆ’4(š‘‘).(5.19) To calculate limš‘šā†’āˆžī‚šœŽš‘˜š‘šāˆ’2,šœ’āˆ’4(š‘ī…ž), we write š‘ā€² as š‘ī…ž=š‘š‘’ā‹…š‘ī…žī…ž, (š‘,š‘ī…žī…ž)=1, namely, š‘’=ordš‘(š‘ī…ž). Then we have limš‘šā†’āˆžī‚šœŽš‘˜š‘šāˆ’2,šœ’āˆ’4ī€·š‘ī…žī€ø=limš‘šā†’āˆžī“0<š‘‘|š‘ā€²šœ’āˆ’4ī‚µš‘ī…žš‘‘ī‚¶š‘‘(š‘āˆ’1)š‘š‘šāˆ’1=ī“0<š‘‘|š‘ā€²ā€²šœ’āˆ’4ī€·š‘š‘’ā‹…š‘ī…žī…žī€ø=ī€·šœ’āˆ’4ī€ø(š‘)š‘’ī“0<š‘‘|š‘ā€²(š‘‘,š‘)=1ī“šœ’āˆ’4(š‘‘).(5.20) Combining these formulas, we obtain limš‘šā†’āˆžī€·šœŽš‘˜š‘šāˆ’2,šœ’āˆ’4ī€·š‘ī…žī€øāˆ’ī‚šœŽš‘˜š‘šāˆ’2,šœ’āˆ’4ī€·š‘ī…ž=ī‚€ī€·šœ’ī€øī€ø1āˆ’āˆ’4ī€ø(š‘)ordš‘(š‘ā€²)ī‚ī“0<š‘‘|š‘ā€²(š‘‘,š‘)=1šœ’āˆ’4=ī‚€(š‘‘)1āˆ’(āˆ’1)ordš‘(š‘ā€²)ī‚šœŽāˆ—0,šœ’āˆ’4ī€·š‘ī…žī€ø.(5.21) If š‘ā€²=š‘āˆ’š‘ 2=0, then šŗī€·š‘˜š‘šī€øšµāˆ’2,0=āˆ’š‘˜š‘šāˆ’1,šœ’āˆ’42ī€·š‘˜š‘šī€ø.āˆ’1(5.22) Thus, we get limš‘šā†’āˆžšŗī€·š‘˜š‘šī€øī€·āˆ’2,0=āˆ’1āˆ’šœ’āˆ’4ī€øšµ(š‘)1,šœ’āˆ’42=12.(5.23) Consequently, limš‘šā†’āˆžī“š‘ š‘ āˆˆā„¤2ā‰¤š‘šŗī€·š‘˜š‘šāˆ’2,š‘āˆ’š‘ 2ī€ø=ī“š‘ š‘ āˆˆā„¤2ā‰¤š‘šŗāˆ—š‘ī€·š‘āˆ’š‘ 2ī€ø.(5.24) The identity (5.14) immediately follows due to these formulas.

The proof of Theorem 5.2 is completed by combining Lemmas 5.3 and 5.4.

An advantage of the formula (5.6) is that we can prove the nonvanishing property for the cusp form š‘“āˆ—š‘ for š‘>7.

Corollary 5.5. Assume that š‘ā‰”3(mod4). If š‘>7, then š‘“āˆ—š‘ does not vanish identically.

Proof. We calculate the Fourier coefficient š‘Ž(š‘‡;š‘“āˆ—š‘) at ī€·š‘‡=1001ī€ø. From the theorem, we have š‘ŽāŽ›āŽœāŽœāŽāŽ›āŽœāŽœāŽāŽžāŽŸāŽŸāŽ 1001;š‘“āˆ—š‘āŽžāŽŸāŽŸāŽ =š›¼āˆ—š‘(4)=š»āˆ—š‘(4)āˆ’š‘āˆ’16ī€·šŗāˆ—š‘(4)+2šŗāˆ—š‘(3)+2šŗāˆ—š‘ī€ø.(0)(5.25) The assumption š‘ā‰”3(mod4) implies that š»āˆ—š‘ī€·(4)=āˆ’1āˆ’šœ’āˆ’4ī€øšµ(š‘)1,šœ’āˆ’4=1.(5.26) On the other hand, šŗāˆ—š‘(3)=šŗāˆ—š‘(4)=0 (because š‘āˆ¤3,4) and šŗāˆ—š‘(0)=1/2. Hence, š‘āˆ’16ī€·šŗāˆ—š‘(4)+2šŗāˆ—š‘(3)+2šŗāˆ—š‘ī€ø=(0)š‘āˆ’16.(5.27) Consequently, we obtain š‘ŽāŽ›āŽœāŽœāŽāŽ›āŽœāŽœāŽāŽžāŽŸāŽŸāŽ 1001;š‘“āˆ—š‘āŽžāŽŸāŽŸāŽ =š›¼āˆ—š‘(4)=1āˆ’š‘āˆ’16=7āˆ’š‘6<0(5.28) if š‘>7.

Remark 5.6. We have š‘“āˆ—3=š‘“āˆ—7=0. These identities are consistent with the fact that dimš‘†2(Ī“20(3))=dimš‘†2(Ī“20(7))=0 (see [7]).

6. Numerical Examples

In this section, we present numerical examples concerning our Siegel cusp forms. To begin with, we recall the theta series associated with quadratic forms.

Let š‘†=š‘†(2š‘š) be a half-integral, positive-definite symmetric matrix of rank 2š‘š.

We associate the theta series ī“šœ—(š‘†,š‘)=š‘‹āˆˆš‘€2š‘š,2(ā„¤)ī€ŗī€·exp2šœ‹š‘–trš‘”š‘‹š‘†š‘‹š‘ī€øī€»,š‘āˆˆā„2.(6.1) If we take a symmetric š‘†=š‘†(2š‘š)>0 with level š‘, then šœ—(š‘†,š‘)āˆˆš‘€š‘šī€·Ī“20ī€ø.(š‘)(6.2) In some cases, we can construct cusp forms by taking a linear combination of theta series.

The Caseā€‰ā€‰š‘=11. Set š‘„1(11)=āŽ›āŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽ110201010212010302āŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽ 03,š‘„2(11)=āŽ›āŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽ1121212121102121042212āŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽ 24,š‘„3(11)=āŽ›āŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽ12121211202121021212āŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽ 12,(6.3) and šœ—š‘–=šœ—(š‘„š‘–(11),š‘). It is known that dimš‘†2(Ī“20(11))=1 (cf. [7]). We can take a nonzero element of š‘†2(Ī“20(11)) as š¶2(11)=3šœ—1āˆ’2šœ—2āˆ’šœ—3(6.4) (Yoshidaā€™s cusp form cf. [8]).

Table 1 gives a first few examples for the Fourier coefficient of š‘“āˆ—11 and š¶2(11).


š‘‡ āŽ› āŽœ āŽœ āŽ āŽž āŽŸ āŽŸ āŽ  1 1 / 2 1 / 2 1 āŽ› āŽœ āŽœ āŽ āŽž āŽŸ āŽŸ āŽ  1 0 0 1 āŽ› āŽœ āŽœ āŽ āŽž āŽŸ āŽŸ āŽ  2 1 / 2 1 / 2 1 āŽ› āŽœ āŽœ āŽ āŽž āŽŸ āŽŸ āŽ  2 0 0 1 āŽ› āŽœ āŽœ āŽ āŽž āŽŸ āŽŸ āŽ  3 1 / 2 1 / 2 1
š‘Ž ( š‘‡ ; š‘“ āˆ— 1 1 ) 2 / 3 āˆ’ 2 / 3 0 0 āˆ’ 2 / 3
š‘Ž ( š‘‡ ; š¶ 2 ( 1 1 ) ) āˆ’ 2 4 2 4 0 0 2 4

š‘‡ āŽ› āŽœ āŽœ āŽ āŽž āŽŸ āŽŸ āŽ  3 0 0 1 āŽ› āŽœ āŽœ āŽ āŽž āŽŸ āŽŸ āŽ  2 1 1 2 āŽ› āŽœ āŽœ āŽ āŽž āŽŸ āŽŸ āŽ  2 1 / 2 1 / 2 2 āŽ› āŽœ āŽœ āŽ āŽž āŽŸ āŽŸ āŽ  2 0 0 2 āŽ› āŽœ āŽœ āŽ āŽž āŽŸ āŽŸ āŽ  4 0 0 1
š‘Ž ( š‘‡ ; š‘“ āˆ— 1 1 ) āˆ’ 2 / 3 2 / 3 2 / 3 0 4 / 3
š‘Ž ( š‘‡ ; š¶ 2 ( 1 1 ) ) 2 4 āˆ’ 2 4 āˆ’ 2 4 0 āˆ’ 4 8

The relation between š‘“āˆ—11 and š¶2(11) is š‘“āˆ—111=āˆ’š¶362(11).(6.5) Further examples of the Fourier coefficients of š‘“āˆ—11 can be obtained from Table 2.


š‘ 3 4 7 8 11 12 15 16 19 20 23 24 27 28 31
š›¼ āˆ— 1 1 ( š‘ ) 2 / 3 āˆ’ 2 / 3 0 0 āˆ’ 2 / 3 āˆ’ 2 / 3 2 / 3 4 / 3 0 2 / 3 āˆ’ 2 / 3 0 āˆ’ 2 / 3 0 āˆ’ 2 / 3

š‘ 32 35 36 39 40 43 44 47 48 51 52 55 56 59 60
š›¼ āˆ— 1 1 ( š‘ ) 0 0 0 0 0 0 2 / 3 0 0 0 0 2 / 3 4 / 3 āˆ’ 2 / 3 āˆ’ 2

š‘ 63 64 67 68 71 72 75 76 79 80 83 84 87 88 91
š›¼ āˆ— 1 1 ( š‘ ) 0 āˆ’ 4 / 3 2 0 2 / 3 0 4 / 3 0 0 āˆ’ 4 / 3 0 0 0 0 āˆ’ 8 / 3

š‘ 92 95 96 99 100ā€‰ā€‰ā€‰ā€‰ā€‰ā€‰ā€‰ā€‰ā€‰ā€‰

š›¼ āˆ— 1 1 ( š‘ ) 2 0 0 4 / 3 0 ā€‰ā€‰ā€‰ā€‰ā€‰ā€‰ā€‰ā€‰ā€‰ā€‰

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Copyright © 2012 Hirotaka Kodama et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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