/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 8, No. 3, 2015, 395-416 ISSN 1307-5543 – www.ejpam.com Fourier Coefficients of Some Eta Quotients of Weight 8 Barı̧s Kendirli Department of Mathematics, Art & Sciences Faculty, Aydin University, Istanbul, Turkey Abstract. Recently, Williams[12] and then Yao, Xia, Jin [13] discovered the explicit formulas of the coefficients of Fourier series expansions of a class of eta quotients. Williams expressed all coefficients of 126 eta quotients in terms of σ(n), σ( n 2 ), σ( n 3 ), σ( n 6 ) and Yao, Xia, Jin expressed only even coef- ficients of 104 eta quotients in terms of σ3(n), σ3( n 2 ), σ3( n 3 ), σ3( n 6 ). Here, we will express the odd Fourier coefficients of 64 eta quotients in terms ofσ7 (2n− 1), σ7 � 2n−1 3 � and even Fourier coefficients of 130 eta quotients in terms of σ7 (n), σ7 � n 2 � , σ7 � n 3 � , σ7 � n 4 � , σ7 � n 6 � , σ7 � n 12 � . 2010 Mathematics Subject Classifications: 11F20, 11F30 Key Words and Phrases: Dedekind eta function; eta quotients; Fourier series. 1. Introduction The divisor function σi (n) is defined by σi (n) :=    ∑ d positive integer,d|n d i if n is a positive integer 0 otherwise (1) Dedekind eta function is defined by η (z) := q1/24 ∞ ∏ n=1 � 1− qn � ,q := e2πiz , (2) and an eta quotient of level n is defined by f (z) := ∏ m|n η(mz)am , n, m ∈ N, am ∈ Z . (3) It is interesting and important to determine explicit formulas of the Fourier coefficients of eta quotients since they are the building blocks of modular forms of level n and weight k. The book of Kohler [10] describes such expansions by means of Hecke Theta series and develops Email address: baris.kendirli@gmail.com http://www.ejpam.com 395 c© 2015 EJPAM All rights reserved. B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 395-416 396 algorithms for the determination of suitable eta quotients. One can find more information in [4, 5, 11, 14, 15]. I have determined the Fourier coefficients of the theta series associated to some quadratic forms, see [6–9]. Recently, Williams, see [12] discovered the explicit formulas of the coefficients of Fourier series expansions of a class of 126 eta quotients in terms of σ(n), σ( n 2 ), σ( n 3 ), σ( n 6 ). One example is as follows: η2 (2z)η4 (4z)η6 (6z) η2 (z)η2 (3z)η4 (12z) . Then Yao, Xia, Jin, see [13] expressed even Fourier coefficients of 104 eta quotients in terms of σ3(n), σ3( n 2 ), σ3( n 3 ), σ3( n 6 ). One example is as follows: η25 (2z)η4 (3z) η12 (z)η5 (4z)η3 (6z)η (12z) , where the even coefficients are obtained. Motivated by these two results, we find that we can express the odd Fourier coefficients of 64 eta quotients in terms of σ7(2n−1), σ7( 2n−1 3 ), see Table 1 in the appendix. One example is as follows: η36 (2z)η14 (12z) η18 (4z)η16 (6z) , where the odd coefficients are obtained.We can also express the even Fourier coefficients of 130 eta quotients in terms of σ7 (n) ,σ7 � n 2 � ,σ7 � n 3 � ,σ7 � n 4 � ,σ7 � n 6 � ,σ7 � n 12 � , see Table 2, also in the appendix. One example is as follows: η24 (4z)η8 (12z) η12 (2z)η4 (6z) . Now we can state our main Theorem: Theorem 1. Let b1, b2, . . . , b5 be non-negative integers satisfying b1 + b2 + . . .+ b5 ≤ 16. (4) Define the integers a1, a2, a3, a4, a6, a12 by a1 :=− b1 + 2b2 − 2b3 − 4b4 − b5 + 16 (5) a2 :=3b1 + b2 + 3b3 + 10b4 + b5 − 40 (6) a3 :=3b1 + 2b2 + 6b3 + 4b4 + 3b5 − 48 (7) a4 :=− 2b1 − b2 − b3 − 4b4 + 2b5 + 16 (8) a6 :=− 9b1 − 7b2 − 9b3 − 10b4 − 7b5 + 120 (9) a12 :=6b1 + 3b2 + 3b3 + 4b4 + 2b5 − 48. (10) B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 395-416 397 f1 := ∞ ∑ n=0 f1 (n) = η10 (4z)η28 (6z) η8 (2z)η14 (12z) , f2 := ∞ ∑ n=0 f2 (n) = η5 (4z)η33 (6z) η7 (2z)η15 (12z) , f3 := ∞ ∑ n=0 f3 (n) = η9 (2z)η17 (6z) η3 (4z)η7 (12z) , f4 := ∞ ∑ n=0 f4 (n) = η17 (2z)η9 (6z) η7 (4z)η3 (12z) , f5 := ∞ ∑ n=0 f5 (n) = η25 (2z)η (6z)η (12z) η11 (4z) , f6 := ∞ ∑ n=0 f6 (n) = η33 (2z)η5 (12z) η15 (4z)η7 (6z) f7 := ∞ ∑ n=0 f7 (n) = η6 (4z)η26 (6z) η6 (2z)η10 (12z) , f8 := ∞ ∑ n=0 f8 (n) = η2 (2z)η2 (4z)η18 (6z) η6 (12z) , f9 := ∞ ∑ n=0 f9 (n) = η10 (2z)η10 (6z) η2 (4z)η2 (12z) , f10 := ∞ ∑ n=0 f10 (n) = η18 (2z)η2 (6z)η2 (12z) η6 (4z) , f11 := ∞ ∑ n=0 f11 (n) = η26 (2z)η6 (12z) η10 (4z)η6 (6z) . Now define integers k0, k1, k2, k3, k4, k5, k6, k7, k8, k9, k10, k11, k12, k13, k14, k15 and k16 by 1 2b1+b5 x b1(1− x)b2(1+ x)b3(1+ 2x)b4(2+ x)b5 (11) =k0 + k1 x + k2 x2 + k3 x + k4 x4 + k5 x5 + k6 x6 + k7 x7 + k8 x8 + k9 x9 + k10 x10 + k11 x11 + k12 x12 + k13 x13 + k14 x14 + k15 x15 + k16 x16. (12) Define the rational numbers c1, c2, c3, c4, c6, c12, r1, r2, r3, r4, r5, r6, r7, r8, r9, r10 and r11 by c1 =− 21872/94095k0 + 730/6273k1 − 5488/94095k2 + 2758/94095k3 − 464/31365k4 + 142/18819k5 − 368/94095k6 + 22/10455k7 − 112/94095k8 + 14/18819k9 − 16/31365k10 + 38/94095k11 − 32/94095k12 + 2/6273k13 − 28/94095k14 + 28/94095k15, c2 =5614304/94095k0 − 2807152/94095k1 + 82564/5535k2 B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 395-416 398 − 701806/94095k3 + 350912/94095k4 − 175462/94095k5 + 87728/94095k6 − 43846/94095k7 + 21872/94095k8 − 10822/94095k9 + 304/5535k10 − 2086/94095k11 + 32/94095k12 + 2018/94095k13 − 5092/94095k14 + 10724/94095k15 − 28672/94095k16, c3 =198992/94095k0 − 12538/6273k1 + 182608/94095k2 − 179878/94095k3 + 59504/31365k4 − 35566/18819k5 + 177488/94095k6 − 19702/10455k7 + 177232/94095k8 − 35438/18819k9 + 59056/31365k10 − 177158/94095k11 + 177152/94095k12 − 11810/6273k13 + 177148/94095k14 − 177148/94095k15, c4 =− 5599232/94095k0 + 2796032/94095k1 − 93184/6273k2 + 698368/94095k3 − 4096/1107k4 + 57344/31365k5 − 16384/18819k6 + 32768/94095k7 − 32768/94095k9 + 16384/18819k10 − 57344/31365k11 + 4096/1107k12 − 698368/94095k13 + 93184/6273k14 − 2796032/94095k15 + 5599232/94095k16, c6 =− 5968544/94095k0 + 2984272/94095k1 − 1580708/94095k2 + 878926/94095k3 − 528032/94095k4 + 352582/94095k5 − 264848/94095k6 + 12998/5535k7 − 198992/94095k8 + 187942/94095k9 − 182288/94095k10 + 179206/94095k11 − 177152/94095k12 + 175102/94095k13 − 172028/94095k14 + 9788/5535k15 + 45371392/94095k16, c12 =50941952/94095k0 − 2796032/94095k1 + 93184/6273k2 − 698368/94095k3 + 4096/1107k4 − 57344/31365k5 + 16384/18819k6 − 32768/94095k7 + 32768/94095k9 − 16384/18819k10 + 57344/31365k11 − 4096/1107k12 + 698368/94095k13 − 93184/6273k14 + 2796032/94095k15 − 50941952/94095k16 r1 =8324272/10455k0 − 1604737/1394k1 + 21432061/20910k2 − 8074478/10455k3 + 1855009/3485k4 − 1445785/4182k5 + 4490171/20910k6 − 445011/3485k7 + 757742/10455k8 − 160223/4182k9 + 122227/6970k10 − 58468/10455k11 − 21683/10455k12 + 7615/1394k13 − 92542/10455k14 + 92542/10455k15, r2 =− 9365552/10455k0 + 3452381/2788k1 − 45845977/41820k2 + 69110609/83640k3 − 7952071/13940k4 + 3105151/8364k5 − 2415053/10455k6 + 3834183/27880k7 − 1632419/20910k8 + 344483/8364k9 − 260849/13940k10 + 481609/83640k11 + 53621/20910k12 − 34879/5576k13 + 415943/41820k14 − 415943/41820k15, r3 =3765428/31365k0 − 1968431/16728k1 + 24665831/250920k2 − 9439003/125460k3 + 1132177/20910k4 − 1855277/50184k5 + 6009511/250920k6 − 202957/13940k7 + 252718/31365k8 − 184453/50184k9 + 67157/83640k10 + 127237/125460k11 − 279703/125460k12 + 47555/16728k13 − 216811/62730k14 + 216811/62730k15, r4 =− 1300196/31365k0 + 619175/16728k1 − 7731047/250920k2 + 764689/31365k3 B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 395-416 399 − 385399/20910k4 + 665093/50184k5 − 2240767/250920k6 + 37917/6970k7 − 85036/31365k8 + 32269/50184k9 + 73171/83640k10 − 123107/62730k11 + 343771/125460k12 − 53657/16728k13 + 115271/31365k14 − 115271/31365k15, r5 =57262/10455k0 − 53537/11152k1 + 678533/167280k2 − 276769/83640k3 + 72257/27880k4 − 64253/33456k5 + 218083/167280k6 − 10389/13940k7 + 10343/41820k8 + 6329/33456k9 − 31429/55760k10 + 73261/83640k11 − 94159/83640k12 + 14645/11152k13 − 31379/20910k14 + 31379/20910k15, r6 =− 23182/94095k0 + 21773/100368k1 − 280913/1505520k2 + 58637/376380k3 − 31307/250920k4 + 28193/301104k5 − 94003/1505520k6 + 2611/83640k7 + 7/376380k8 − 9413/301104k9 + 31369/501840k10 − 8822/94095k11 + 94099/752760k12 − 15683/100368k13 + 70573/376380k14 − 70573/376380k15, r7 =− 5968/31365k0 + 559211/3690k1 − 9498581/62730k2 + 7125649/62730k3 − 2370607/31365k4 + 1480451/31365k5 − 1768421/62730k6 + 1026679/62730k7 − 290002/31365k8 + 9388/1845k9 − 172211/62730k10 + 85609/62730k11 − 21547/31365k12 + 6041/31365k13 − 2038/31365k14 − 5968/31365k15 + 11936/31365k16, r8 =8170304/31365k0 − 553454/1845k1 + 8087054/31365k2 − 6100906/31365k3 + 4250186/31365k4 − 2796223/31365k5 + 1754294/31365k6 − 1053046/31365k7 + 597956/31365k8 − 18419/1845k9 + 139724/31365k10 − 32926/31365k11 − 24124/31365k12 + 65132/31365k13 − 74056/31365k14 + 91904/31365k15 − 183808/31365k16, r9 =− 1116896/6273k0 + 1011709/6273k1 − 836363/6273k2 + 649939/6273k3 − 479906/6273k4 + 336985/6273k5 − 223349/6273k6 + 137143/6273k7 − 74396/6273k8 + 30469/6273k9 − 575/6273k10 − 19181/6273k11 + 30910/6273k12 − 38834/6273k13 + 39148/6273k14 − 39776/6273k15 + 79552/6273k16, r10 =1067584/31365k0 − 933638/31365k1 + 784454/31365k2 − 635366/31365k3 + 494146/31365k4 − 364693/31365k5 + 14642/1845k6 − 147746/31365k7 + 61556/31365k8 + 9607/31365k9 − 66056/31365k10 + 108554/31365k11 − 138764/31365k12 + 160012/31365k13 − 9608/1845k14 + 169984/31365k15 − 339968/31365k16, r11 =− 185456/94095k0 + 326609/188190k1 − 280927/188190k2 + 234563/188190k3 − 93929/94095k4 + 70492/94095k5 − 5531/11070k6 + 47033/188190k7 − 14/94095k8 − 23483/94095k9 + 93923/188190k10 − 140797/188190k11 + 93751/94095k12 − 116933/94095k13 + 8222/535k14 − 185456/94095k15 + 370912/94095k16. B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 395-416 400 Here � f1, f2, . . . , f11 are in S8 � Γ0 (12) � and ηa1 (z)ηa2 (2z)ηa3 (3z)ηa4 (4z)ηa5 (6z)ηa6 (12z) = δ � b1 � + ∞ ∑ n=1 c(n)qn, where for n∈ N, c(n) =c1σ7 (n) + c2σ7 � n 2 � + c3σ7 � n 3 � + c4σ7 � n 4 � + c6σ7 � n 6 � + c12σ7 � n 12 � + r1 f1 (n) + r2 f2 (n) + r3 f3 (n) + r4 f4 (n) + r5 f5 (n) + r6 f6 (n) + r7 f7 (n) + r8 f8 (n) + r9 f9 (n) + r10 f10 (n) + r11 f11 (n) . In particular, c(2n) =c1σ7 (2n) + c2σ7 (n) + c4σ7 � n 2 � + � 129c3 + c6 � σ7 � n 3 � + (c12 − 128c3)σ7 � n 6 � + r7 f7(2n) + r8 f8(2n) + . . .+ r11 f11(2n), c(2n− 1) =c1σ7 (2n− 1) + c3σ7 � 2n− 1 3 � + r1 f1(2n− 1) + r2 f2(2n− 1) + r3 f3(2n− 1) + . . .+ r6 f6(2n− 1), for n∈ N. Proof. It follows from equations (5)-(10) that a1 :=− b1 + 2b2 − 2b3 − 4b4 − b5 + 16 (13) a2 :=3b1 + b2 + 3b3 + 10b4 + b5 − 40 (14) a3 :=3b1 + 2b2 + 6b3 + 4b4 + 3b5 − 48 (15) a4 :=− 2b1 − b2 − b3 − 4b4 + 2b5 + 16 (16) a6 :=− 9b1 − 7b2 − 9b3 − 10b4 − 7b5 + 120 (17) a12 :=6b1 + 3b2 + 3b3 + 4b4 + 2b5 − 48. (18) a1 + 2a2 + 3a3 + 4a4 + 6a6 + 12a12 = 24b1, a1 + a2 + a3 + a4 + a6 + a12 = 16, (19) a1 6 + a2 3 + a3 6 + 2 a4 3 + a6 3 + 2 a12 3 = b1 + b5. Now we will use p− k parametrization of Alaca, Alaca and Williams, see [1]: p � q � := ϕ2 � q � −ϕ2 � q3 � 2ϕ2 � q3 � , k � q � := ϕ3 � q3 � ϕ � q � , B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 395-416 401 where the theta function ϕ � q � is defined by ϕ � q � = ∞ ∑ −∞ qn2 . Setting x = p in (11), and multiplying both sides by k8, we obtain k8 2b1+b5 pb1(1− p)b2(1+ p)b3(1+ 2p)b4(2+ p)b5 =(k0 + k1p+ k2p2 + k3p3 + k4p4 + k5p5 + k6p6 + k7p7 + k8p8 + k9p9 + k10p10 + k11p11 + k12p12 + k13p13 + k14p14 + k15p15 + k16p16)k8. Alaca, Alaca and Williams [2] have established the following representations in terms of p and k: η � q � =2−1/6p1/24(1− p)1/2(1+ p)1/6(1+ 2p)1/8(2+ p)1/8k1/2, (20) η � q2 � =2−1/3p1/12(1− p)1/4(1+ p)1/12(1+ 2p)1/4(2+ p)1/4k1/2, (21) η � q3 � =2−1/6p1/8(1− p)1/6(1+ p)1/2(1+ 2p)1/24(2+ p)1/24k1/2, (22) η � q4 � =2−2/3p1/6(1− p)1/8(1+ p)1/24(1+ 2p)1/8(2+ p)1/2k1/2, (23) η � q6 � =2−1/3p1/4(1− p)1/12(1+ p)1/4(1+ 2p)1/12(2+ p)1/12k1/2, (24) η � q12 � =2−2/3p1/2(1− p)1/24(1+ p)1/8(1+ 2p)1/24(2+ p)1/6k1/2. (25) Since E8 = E2 4 , we have E8 � q � :=1+ 480 ∞ ∑ n=1 σ7 (n)q n =(1+ 248p+ 17304p2 + 244648p3 + 1628540p4 + 6350520p5 16004776p6 + 27416744p7 + 32723334p8 + 27416744p9 + 16004776p10 + 6350520p11 + 1628540p12 + 244648p13 + 17304p14 + 248p15 + p16)k8, E8(q 2) =(1+ 8p+ 144p2 + 868p3 + 5990p4 + 25020p5 + 63316p6 + 106964p7 + 126819p8 + 106964p9 + 63316p10 + 25020p11 + 5990p12 + 868p13 + 144p14 + 8p15 + p16)k8, E8(q 3) =(1+ 8p+ 24p2 + 88p3 + 380p4 + 840p5 + 1576p6 + 4184p7 + 6534p8 + 4184p9 + 1576p10 + 840p11 + 380p12 + 88p13 + 24p14 + 8p15 + p16)k8, E8(q 4) =(1+ 8p+ 24p2 + 28p3 + 20p4 + 120p5 + 647 2 p6 + 463 2 p7 − 63 8 p8 + 943 2 p9 + 1937 2 p10 B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 395-416 402 + 1155 4 p11 − 5515 16 p12 − 871 8 p13 + 1713 32 p14 − 29 32 p15 + 1 256 p16)k8, E8(q 6) =(1+ 8p+ 24p2 + 28p3 − 10p4 − 60p5 − 44p6 + 44p7 + 99p8 + 44p9 − 44p10 − 60p11 − 10p12 + 28p13 + 24p14 + 8p15 + p16)k8, E8(q 12) =(1+ 8p+ 24p2 + 28p3 − 10p4 − 60p5 − 103 2 p6 + 13 2 p7 + 297 8 p8 + 43 2 p9 + p10 − 15 4 p11 − 25 16 p12 − 1 8 p13 + 3 32 p14 + 1 32 p15 + 1 256 p16)k8. It is easy to check the following expressions by (20)-(25) f1 := ∞ ∑ n=0 f1 (n) = η10 (4z)η28 (6z) η8 (2z)η14 (12z) = ( 1 2 p+ 15 4 p2 + 91 8 p3 + 273 16 p4 + 41 4 p5 − 99 16 p6 − 65 4 p7 − 213 16 p8 − 23 4 p9 − 21 16 p10 − 1 8 p11)k8, f2 := ∞ ∑ n=0 f2 (n) = η5 (4z)η33 (6z) η7 (2z)η15 (12z) = ( 1 2 p+ 15 4 p2 + 45 4 p3 + 65 4 p4 + 33 4 p5 − 33 4 p6 − 65 4 p7 − 45 4 p8 − 15 4 p9 − 1 2 p10)k8, f3 := ∞ ∑ n=0 f3 (n) = η9 (2z)η17 (6z) η3 (4z)η7 (12z) = ( 1 2 p+ 15 4 p2 + 37 4 p3 + 13 4 p4 − 87 4 p5 − 121 4 p6 + 19 4 p7 + 135 4 p8 + 65 4 p9 − 17 2 p10 − 9p11 − 2p12)k8, f4 := ∞ ∑ n=0 f4 (n) = η17 (2z)η9 (6z) η7 (4z)η3 (12z) = ( 1 2 p+ 15 4 p2 + 33 4 p3 − 13 4 p4 − 135 4 p 5 − 99 4 p6 + 171 4 p7 + 225 4 p8 − 63 4 p9 − 44p10 − 6p11 + 12p12 + 4p13)k8, f5 := ∞ ∑ n=0 f5 (n) = η25 (2z)η (6z)η (12z) η11 (4z) = ( 1 2 p+ 15 4 p2 + 29 4 p3 − 39 4 p4 − 175 4 p 5 − 33 4 p6 + 375 4 p7 + 219 4 p8 − 399 4 p9 − 145 2 p10 + 54p11 + 40p12 − 12p13 − 8p14)k8, f6 := ∞ ∑ n=0 f6 (n) = η33 (2z)η5 (12z) η15 (4z)η7 (6z) = ( 1 2 p+ 15 4 p2 + 25 4 p3 − 65 4 p4 − 207 4 p5 + 77 4 p6 + 615 4 p7 + 45 4 p8 − 975 4 p9 − 38p10 + 219p11 + 20p12 − 100p13 + 16p15)k8, f7 := ∞ ∑ n=0 f7 (n) = η6 (4z)η26 (6z) η6 (2z)η10 (12z) = ( 1 4 p2 + 7 4 p3 + 77 16 p4 B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 395-416 403 + 49 8 p5 + 33 16 p6 − 33 8 p7 − 97 16 p8 − 29 8 p9 − 17 16 p10 − 1 8 p11)k8, f8 := ∞ ∑ n=0 f8 (n) = η2 (2z)η2 (4z)η18 (6z) η6 (12z) = ( 1 4 p2 + 7 4 p3 + 69 16 p4 + 25 8 p5 − 73 16 p6 − 39 4 p7 − 73 16 p8 + 25 8 p9 + 69 16 p10 + 7 4 p11 + 1 4 p12)k8, f9 := ∞ ∑ n=0 f9 (n) = η10 (2z)η10 (6z) η2 (4z)η2 (12z) = ( 1 4 p2 + 7 4 p3 + 53 16 p4 − 23 8 p5 − 237 16 p 6 − 6p7 + 339 16 p8 + 141 8 p9 − 183 16 p10 − 29 8 p11 + 1 2 p12 + 4p13 + p14)k8, f10 := ∞ ∑ n=0 f10 (n) = η18 (2z)η2 (6z)η2 (12z) η6 (4z) = ( 1 4 p2 + 7 4 p3 + 61 16 p4 + 1 8 p5 − 163 16 p6 − 83 8 p7 + 83 16 p8 + 103 8 p9 + 59 16 p10 − 31 8 p11 − 11 4 p12 − 1 2 p13)k8, f11 := ∞ ∑ n=0 f11 (n) = η26 (2z)η6 (12z) η10 (4z)η6 (6z) = ( 1 4 p2 + 7 4 p3 + 45 16 p4 − 47 8 p5 − 295 16 p6 + 27 8 p7 + 663 16 p8 + 75 8 p9 − 729 16 p10 − 125 8 p11 + 49 2 p12 + 9p13 − 5p14 − 2p15)k8. We immediately verify that f1, . . . , f11 ∈ S8 � Γ0 (12) � . Now a1 :=− b1 + 2b2 − 2b3 − 4b4 − b5 + 16, (26) a2 :=3b1 + b2 + 3b3 + 10b4 + b5 − 40, (27) a3 :=3b1 + 2b2 + 6b3 + 4b4 + 3b5 − 48, (28) a4 :=− 2b1 − b2 − b3 − 4b4 + 2b5 + 16, (29) a6 :=− 9b1 − 7b2 − 9b3 − 10b4 − 7b5 + 120, (30) a12 := 6b1 + 3b2 + 3b3 + 4b4 + 2b5 − 48. (31) ηa1 (z)ηa2 (2z)ηa3 (3z)ηa4 (4z)ηa6 (6z)ηa12 (12z) =qb1 ∞ ∏ n=1 � 1− qn �a1 � 1− q2n �a2 � 1− q3n �a3 � 1− q4n �a4 � 1− q6n �a6 � 1− q12n �a12 =2− a1 6 − a2 3 − a3 6 −2 a4 3 − a6 3 −2 a12 3 p a1 24+ a2 12+ a3 8 + a4 6 + a6 4 + a12 2 (1− p) a1 2 + a2 4 + a3 6 + a4 8 + a6 12+ a12 24 (1+ p) a1 6 + a2 12+ a3 2 + a4 24+ a6 4 + a12 8 (1+ 2p) a1 8 + a2 4 + a3 24+ a4 8 + a6 12+ a12 24 (2+ p) a1 8 + a2 4 + a3 24+ a4 2 + a6 12+ a12 6 k a1+a2+a3+a4+a6+a12 2 = k8 2b1+b5 pb1(1− p)b2(1+ p)b3(1+ 2p)b4(2+ p)b5 =k8(k0 + k1p+ k2p2 + k3p3 + k4p4 + k5p5 + k6p6 + k7p7 + k8p8 + k9p9 + k10p10 + k11p11 + k12p12) = c1 480 � 1+ 480 ∞ ∑ n=1 σ7 (n)q n � + c2 480 � 1+ 480 ∞ ∑ n=1 σ7 (n)q 2n � B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 395-416 404 + c3 480 � 1+ 480 ∞ ∑ n=1 σ7 (n)q 3n � + c4 480 � 1+ 480 ∞ ∑ n=1 σ7 (n)q 4n � + c6 480 � 1+ 480 ∞ ∑ n=1 σ7 (n)q 6n � + c12 480 � 1+ 480 ∞ ∑ n=1 σ7 (n)q 12n � + r1q ∞ ∏ n=1 � 1− q4n �10 � 1− q6n �28 � 1− q2n �8 � 1− q12n �14 + r2q ∞ ∏ n=1 � 1− q4n �5 � 1− q6n �33 � 1− q2n �7 � 1− q12n �15 + r3q4 ∞ ∏ n=1 � 1− q2n �9 � 1− q6n �17 � 1− q4n �3 � 1− q12n �7 + r4q ∞ ∏ n=1 � 1− q2n �17 � 1− q6n �9 � 1− q4n �7 � 1− q12n �3 + r5.q ∞ ∏ n=1 � 1− q2n �25 � 1− q6n � � 1− q12n � � 1− q4n �11 + r6q ∞ ∏ n=1 � 1− q2n �33 � 1− q12n �5 � 1− q4n �15 � 1− q6n �7 + r7q2 ∞ ∏ n=1 � 1− q4n �6 � 1− q6n �26 � 1− q2n �6 � 1− q12n �10 + r8q2 ∞ ∏ n=1 � 1− q2n �2 � 1− q4n �2 � 1− q6n �18 � 1− q12n �6 + r9q2 ∞ ∏ n=1 � 1− q2n �10 � 1− q6n �10 � 1− q4n �2 � 1− q12n �2 + r10q2 ∞ ∏ n=1 � 1− q2n �18 � 1− q6n �2 � 1− q12n �2 � 1− q4n �6 + r11q2 ∞ ∏ n=1 � 1− q2n �26 � 1− q12n �6 � 1− q4n �10 � 1− q6n �6 =δ(b1) + ∞ ∑ n=1 (c1σ7 (n) + c2σ7 � n 2 � + c3σ7 � n 3 � + c4σ7 � n 4 � + c6σ7 � n 6 � + c12σ7 � n 12 � ) + r1 f1(n) + . . .+ r11 f11(n), B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 395-416 405 where δ(b1) = ¨ 0 if b1 6= 0 1 if b1 = 0 So c(n) =(c1σ7 (n) + c2σ7 � n 2 � + c3σ7 � n 3 � + c4σ7 � n 4 � + c6σ7 � n 6 � + c12σ7 � n 12 � ) + r1 f1(n) + . . .+ r11 f11(n). Therefore, since for n= 1,2, . . ., f1 (2n) = f2 (2n) = . . .= f6 (2n) = 0, f7 (2n− 1) = f8 (2n− 1) = f9 (2n− 1) = . . .= f11 (2n− 1) = 0, we have c(2n) =c1σ7 (2n) + c2σ7 (n) + c4σ7 � n 2 � + � 129c3 + c6 � σ7 � n 3 � + (c12 − 128c3)σ7 � n 6 � + r7 f7(2n) + r8 f8(2n) + . . .+ r11 f11(2n), c(2n− 1) =c1σ7 (2n− 1) + c3σ7 � 2n− 1 3 � + r1 f1(2n− 1) + r2 f2(2n− 1) + r3 f3(2n− 1) + . . .+ r6 f6(2n− 1), by the following Lemma. Lemma 1. σk � 2n 3 � = � 2k + 1 � σk � n 3 � − 2kσk � n 6 � . Proof. If 3 doesn’t divide n, both sides are 0. So it is enough to prove that σk (2n) = � 2k + 1 � σk (n)− 2kσk � n 2 � . If 2 doesn’t divide n, it is obvious. So assume that n= 2l m, l ≥ 1, 2 doesn’t divide m. Then � 2k + 1 � σk (n)− 2kσk � n 2 � = � 2k + 1 � σk � 2l � σk (m)− 2kσk � 2l−1 � σk (m) = �� 2k + 1 � σk � 2l � − 2kσk � 2l−1 �� σk (m) = � � 2k + 1 � � 1+ 2k + � 2k �2 + . . .+ � 2k �l � − 2k � 1+ 2k + � 2k �2 + . . .+ � 2k �l−1 �� σk (m) = � 2k � 2k �l + � 1+ 2k + � 2k �2 + . . .+ � 2k �l �� σk (m) = σk � 2l+1m � . B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 395-416 406 These formulas are valid for 17,346 nontrivial eta quotients. Among them, we have found 64 nontrivial eta quotients, see Table 1, such that c(2n) =c1σ7 (2n) + c2σ7 (n) + c4σ7 � n 2 � + � 129c3 + c6 � σ7 � n 3 � + (c12 − 128c3)σ7 � n 6 � , c(2n− 1) =c1σ7 (2n− 1) + c3σ7 � 2n− 1 3 � + r1 f1(2n− 1) + r2 f2(2n− 1) + r3 f3(2n− 1) + . . .+ r6 f6(2n− 1), and 130 eta quotients, see Table 2 (appendix), such that c(2n− 1) =c1σ7 (2n− 1) + c3σ7 � 2n− 1 3 � = 0, c(2n) =c2σ7 (n) + c4σ7 � n 2 � + c6σ7 � n 3 � + c12σ7 � n 6 � + f7 (2n) + f8 (2n) + . . .+ f11 (2n) . Remark 1. S8(Γ0(12)) is 11 dimensional, see [3, Chapter 3, pg.87 and Chapter 5, pg.197], and generated by ∆2,8, ∆2,8 (2z), ∆2,8 (3z), ∆2,8 (6z), ∆3,8, ∆3,8 (2z), ∆3,8 (4z), ∆6,8, ∆6,8 (2z), ∆12,8,1 (z), ∆12,8,2 (z), where∆2,8 is the unique newform in S8(Γ0(2)), ∆3,8 is the unique new- form in S8(Γ0(3)), ∆6,8 is the unique newform in S8(Γ0(6)). ∆12,8,1, ∆12,8,2 are the newforms in S8(Γ0(12)). By simple calculation, we see that f1 = 11 45 ∆2,8 + 88 45 ∆2,8 (2z) + 19 5 ∆2,8 (3z) + 152 5 ∆2,8 (6z) + 32 135 ∆3,8 − 64 45 ∆3,8 (2z) + 4096 135 ∆3,8 (4z) + 5 27 ∆6,8 − 40 27 ∆6,8 (2z) + 35 162 ∆12,8,1 (z) + 19 162 ∆12,8,2 (z) , f2 = 11 45 ∆2,8 + 88 45 ∆2,8 (2z) + 47 15 ∆2,8 (3z) + 376 15 ∆2,8 (6z) + 32 135 ∆3,8 − 64 45 ∆3,8 (2z) + 4096 135 ∆3,8 (4z) + 5 27 ∆6,8 − 40 27 ∆6,8 (2z) + 17 81 ∆12,8,1 (z) + 10 81 ∆12,8,2 (z) , f3 = 1 5 ∆2,8 + 8 5 ∆2,8 (2z)− 9 5 ∆2,8 (3z)− 72 5 ∆2,8 (6z) + 16 45 ∆3,8 − 32 15 ∆3,8 (2z) + 2048 45 ∆3,8 (4z) + 1 9 ∆6,8 − 8 9 ∆6,8 (2z) + 1 9 ∆12,8,1 (z) + 2 9 ∆12,8,2 (z) , f4 =− 1 15 ∆2,8 − 8 15 ∆2,8 (2z) + 243 5 ∆2,8 (3z) + 1944 5 ∆2,8 (6z) REFERENCES 407 + 16 15 ∆3,8 − 32 5 ∆3,8 (2z) + 2048 15 ∆3,8 (4z)− 1 3 ∆6,8 + 8 3 ∆6,8 (2z)− 1 3 ∆12,8,1 (z) + 2 3 ∆12,8,2 (z) , f5 =− 11 15 ∆2,8 − 88 15 ∆2,8 (2z) + 2673 5 ∆2,8 (3z) + 21384 5 ∆2,8 (6z) + 32 5 ∆3,8 − 192 5 ∆3,8 (2z) + 4096 5 ∆3,8 (4z)− 5∆6,8 + 40∆6,8 (2z) − 13 3 ∆12,8,1 (z) + 14 3 ∆12,8,2 (z) , f6 = 47 5 ∆2,8 + 376 5 ∆2,8 (2z) + 24057 5 ∆2,8 (3z) + 192456 5 ∆2,8 (6z) + 288 5 ∆3,8 − 1728 5 ∆3,8 (2z) + 36864 5 ∆3,8 (4z)− 45∆6,8 + 360∆6,8 (2z)− 51∆12,8,1 (z) + 30∆12,8,2 (z) , f7 = 4 15 ∆2,8 (2z) + 68 5 ∆2,8 (6z) + 13 45 ∆3,8 (2z) + 128 45 ∆3,8 (4z) + 4 9 ∆6,8 (2z) , f8 = 2 5 ∆2,8 (2z)− 18 5 ∆2,8 (6z)− 17 45 ∆3,8 (2z)− 128 45 ∆3,8 (4z) + 2 9 ∆6,8 (2z) , f9 = 8 15 ∆2,8 (2z) + 216 5 ∆2,8 (6z) + 7 15 ∆3,8 (2z)− 128 15 ∆3,8 (4z) , f10 =− 2 5 ∆2,8 (2z) + 1458 5 ∆2,8 (6z) + 17 5 ∆3,8 (2z)− 128 5 ∆3,8 (4z)− 2∆6,8 (2z) , f11 = 68 5 ∆2,8 (2z) + 8748 5 ∆2,8 (6z) + 117 5 ∆3,8 (2z) + 1152 5 ∆3,8 (4z)− 36∆6,8 (2z) . References [1] A.Alaca, S.Alaca, and K. S. Williams. On the two-dimensional theta functions of Borweins. Acta Arith. 124, 177-195. 2006. [2] A.Alaca, S.Alaca, and K. S. Williams. Evaluation of the convolution sums ∑ l+12m=nσ (l)σ (m) and ∑ 3l+4m=nσ (l)σ (m). Adv. Theor. Appl. Math. 1, 27-48. 2006. [3] F. Diamond and J. Shurman. A First Course in Modular Forms. Springer Graduate Texts in Mathematics 228. Springer Science+Business Media, Inc. 2005. [4] B.Gordon. Some identities in combinatorial analysis. Quart. J. Math. Oxford Ser.12, 285- 290. 1961. [5] V. G. Kac. Infinite-dimensional algebras, Dedekind’s η-function, classical Möbius function and the very strange formula. Adv. Math. 30, 85-136. 1978. [6] B. Kendirli. Evaluation of Some Convolution Sums and the Representation numbers. Ars Combinatorica Volume CXVI, July, 65-91. 2014. REFERENCES 408 [7] B. Kendirli. Cusp Forms in S4 � Γ0 (79) � and the number of representations of positive integers by some direct sum of binary quadratic forms with discriminant -79. Bulletin of the Korean Mathematical Society Vol 49/3. 2012. [8] B. Kendirli. Cusp Forms in S4 � Γ0 (47) � and the number of representations of positive integers by some direct sum of binary quadratic forms with discriminant -47. Hindawi, International Journal of Mathematics and Mathematical Sciences Vol 2012, Article ID 303492. http://dx.doi:10.1155/2012/303492. 2012. [9] B. Kendirli. The Bases of M4 � Γ0 (71) � , M6 � Γ0 (71) � and the Number of Representations of Integers. Hindawi, Mathematical Problems in Engineering Vol 2013, Article ID 695265. http://dx.doi.org/10.1155/2013/695265. 2013. [10] G. Köhler. Eta Products and Theta Series Identities. (Springer-Verlag, Berlin). 2011. [11] I. G. Macdonald. Affine root systems and Dedekind’s η-function. Invent. Math. 15, 91- 143. 1972. [12] K. S. Williams. Fourier series of a class of eta quotients, Int. J. Number Theory 8, 993- 1004. 2012. [13] O. X. M. Yao, E. X. W. Xia, and J. Jin. Explicit Formulas for the Fourier coefficients of a class of eta quotients. International Journal of Number Theory Vol. 9, No. 2, 487-503. 2013. [14] I. J. Zucker. A systematic way of converting infinite series into infinite products. J. Phys. A 20, L13-L17. 1987. [15] I. J. Zucker. Further relations amongst infinite series and products:II. The evaluation of three-dimensional lattice sums. J. Phys. A23, 117-132. 1990. R E F E R E N C E S 4 0 9 Appendix: Table 1 No b1 b2 b3 b4 b5 a2 a4 a6 a12 c1 c2 c3 c4 c6 c12 1 1 0 0 0 15 −22 44 6 −12 44287 28549120 − 5713023 28549120 − 45927 28549120 44287 223040 5924583 28549120 − 45927 223040 2 1 0 1 0 13 −21 39 11 −13 19683 14274560 − 2539107 14274560 − 19683 14274560 19683 111520 2539107 14274560 19683 111520 3 1 0 2 0 11 −20 34 16 −14 2187 1784320 − 282123 1784320 − 2187 1784320 2187 13940 282123 1784320 − 2187 13940 4 1 0 3 0 9 −19 29 21 −15 243 223040 − 31347 223040 − 243 223040 486 3485 31347 223040 − 486 3485 5 1 0 4 0 7 −18 24 26 −16 27 27880 − 3483 27880 − 27 27880 432 3485 3483 27880 − 432 3485 6 1 0 5 0 5 −17 19 31 −17 3 3485 − 387 3485 − 3 3485 384 3485 387 3485 101101 6970 7 1 0 6 0 3 −16 14 36 −18 8 10455 − 344 3485 − 8 10455 1024 10455 344 3485 − 1024 10455 8 1 0 7 0 1 −15 9 41 −19 64 94095 − 2752 31365 − 64 94095 8192 94095 2752 31365 − 8192 94095 9 1 1 0 1 13 −13 35 3 −9 − 1 14274560 129 14274560 6561 14274560 − 1 111520 − 846369 14274560 6561 111520 10 1 2 0 2 11 −4 26 0 −6 1 1784320 − 129 1784320 − 6561 1784320 1 13940 846369 1784320 − 6561 13940 11 1 3 0 3 9 5 17 −3 −3 − 1 223040 129 223040 6561 223040 − 2 3485 −846369 223040 13122 3485 12 1 4 0 4 7 14 8 −6 0 1 27880 − 129 27880 − 6561 27880 16 3485 846369 27880 −104976 3485 13 1 5 0 5 5 23 −1 −9 3 − 1 3485 129 3485 6561 3485 − 128 3485 −846369 3485 839808 3485 14 1 6 0 6 3 32 −10 −12 6 8 3485 −1032 3485 −52488 3485 1024 3485 6770952 3485 −6718464 3485 15 1 7 0 7 1 41 −19 −15 9 − 64 3485 8256 3485 419904 3485 −8192 3485 −54167616 3485 −407889 3485 16 3 0 0 0 13 −18 36 2 −4 4921 28549120 − 634809 28549120 − 6561 28549120 4921 223040 846369 28549120 − 6561 223040 17 3 0 1 0 11 −17 31 7 −5 2187 14274560 − 282123 14274560 − 2187 14274560 2187 111520 282123 14274560 − 2187 111520 18 3 0 2 0 9 −16 26 12 −6 243 1784320 − 31347 1784320 − 243 1784320 243 13940 31347 1784320 − 243 13940 19 3 0 3 0 7 −15 21 17 −7 27 223040 − 3483 223040 − 27 223040 54 385 3483 223040 − 54 3485 20 3 0 4 0 5 −14 16 22 −8 3 27880 − 387 27880 − 3 27880 48 3485 387 27880 − 48 3485 21 3 0 5 0 3 −13 11 27 −9 1 10455 − 43 3485 − 1 10455 128 10455 43 3485 − 128 10455 22 3 0 6 0 1 −12 6 32 −10 8 94095 − 344 31365 − 8 94095 1024 94095 344 31365 − 1024 94095 23 3 1 0 1 11 −9 27 −1 −1 − 1 14274560 129 14274560 6561 14274560 − 1 111520 − 846369 14274560 6561 111520 24 3 2 0 2 9 0 18 −4 2 1 1784320 − 129 1784320 −6561 1784320 1 13940 846369 1784320 − 6561 13940 25 3 3 0 3 7 9 9 −7 5 − 1 223040 129 223040 6561 223040 − 2 3485 −846369 223040 13122 3485 26 3 4 0 4 5 18 0 −10 8 1 27880 − 129 27880 − 6561 27880 16 3485 846369 27880 −104976 3485 R E F E R E N C E S 4 1 0 No b1 b2 b3 b4 b5 a2 a4 a6 a12 c1 c2 c3 c4 c6 c12 27 3 5 0 5 3 27 −9 −13 11 − 1 3485 129 3485 6561 3485 − 128 3485 −846369 3485 839808 3485 28 3 6 0 6 1 36 −18 −16 14 8 3485 −1032 3485 −52488 3485 1024 3485 6770952 3485 −6718464 3485 29 5 0 0 0 11 −14 28 −2 4 547 28549120 − 70563 28549120 − 2187 28549120 547 223040 282123 28549120 − 2187 223040 30 5 0 1 0 9 −13 23 3 3 243 14274560 − 31347 14274560 − 243 14274560 243 14274560 31347 14274560 − 243 111520 31 5 0 2 0 7 −12 18 8 2 27 1784320 − 3483 1784320 − 27 1784320 27 13940 3483 1784320 − 27 13940 32 5 0 4 0 3 −10 8 18 0 1 83640 − 43 27880 − 1 83640 16 10455 43 27880 − 16 10455 33 5 0 5 0 1 −9 3 23 −1 1 94095 − 43 31365 − 1 94095 128 94095 43 31365 − 128 94095 34 5 1 0 1 9 −5 19 −5 7 − 1 14274560 129 14274560 6561 14274560 − 1 111520 − 846369 14274560 6561 111520 35 5 2 0 2 7 4 10 −8 10 1 1784320 − 129 1784320 − 6561 1784320 1 13940 846369 1784320 − 6561 13940 36 5 3 0 3 5 13 1 −11 13 − 1 223040 129 223040 6561 223040 − 2 3485 −846369 223040 13122 3485 37 5 4 0 4 3 22 −8 −14 16 1 27880 − 129 27880 − 6561 27880 16 3485 846369 27780 −104976 3485 38 5 5 0 5 1 31 −17 −17 19 − 1 3485 129 3485 6561 3485 − 128 3485 −846369 3485 839808 3485 39 7 0 0 0 9 −10 20 −6 12 61 28549120 − 7869 28549120 − 1701 28549120 61 223040 219429 28549120 − 1701 223040 40 7 0 1 0 7 −9 15 −1 11 27 14274560 − 3483 14274560 − 27 14274560 27 111520 3483 14274560 − 27 111520 41 7 0 2 0 5 −8 10 4 10 3 1784320 − 387 1784320 − 3 1784320 3 13940 387 1784320 − 3 13940 42 7 0 3 0 3 −7 5 9 9 1 669120 − 43 223040 − 1 669120 2 10455 43 223040 − 2 10455 43 7 0 4 0 1 −6 0 14 8 1 752760 − 43 250920 − 1 752760 16 94095 43 250920 − 16 94095 44 7 1 0 1 7 −1 11 −9 15 − 1 14274560 129 14274560 6561 14274560 − 1 111520 − 846369 14274560 6561 111520 45 7 2 0 2 5 8 2 −12 18 1 1784320 − 129 1784320 − 6561 1784320 1 13940 846369 1784320 − 6561 13940 46 7 3 0 3 3 17 −7 −15 21 − 1 223040 129 223040 6561 223040 − 2 3485 −846369 223040 −132122 3485 47 7 4 0 4 1 26 −16 −18 24 1 27880 − 129 27880 − 6561 27880 16 3485 846369 27880 −104976 3485 48 9 0 0 0 7 −6 12 −10 20 7 28549120 − 903 28549120 − 1647 28549120 7 223040 212463 28549120 − 1647 223040 49 9 0 1 0 5 −5 7 −5 19 3 14274560 − 387 14274560 − 3 14274560 3 111520 387 14274560 − 3 111520 50 9 0 2 0 3 −4 2 0 18 1 5352960 − 43 1784320 − 1 5352960 1 41820 43 1784320 − 1 41820 51 9 0 3 0 1 −3 −3 5 17 1 6022080 − 43 2007360 − 1 6022080 2 94095 43 2007360 − 2 94095 52 9 1 0 1 5 3 3 −13 23 − 1 14274560 129 14274560 6561 14274560 − 1 111520 − 846369 14274560 6561 111520 53 9 2 0 2 3 12 −6 −16 26 1 1784320 − 129 1784320 −6561 1784320 1 13940 846369 1784320 − 6561 13940 R E F E R E N C E S 4 1 1 No b1 b2 b3 b4 b5 a2 a4 a6 a12 c1 c2 c3 c4 c6 c12 54 9 3 0 3 1 21 −15 −19 29 − 1 223040 129 223040 6561 223040 − 2 3485 −846369 223040 13122 3485 55 11 0 0 0 5 −2 4 −14 28 1 28549120 − 129 28549120 − 1641 28549120 1 223040 211689 28549120 − 1641 223040 56 11 0 1 0 3 −1 −1 −9 27 1 42823680 − 43 14274560 − 1 42823680 1 334560 43 14274560 − 1 334560 57 11 0 2 0 1 0 −6 −4 26 1 48176640 −43 16058880 − 1 48176640 1 376380 43 16058880 − 1 376380 58 11 2 0 2 1 16 −14 −20 34 1 1784320 − 129 1784320 −6561 1784320 1 13940 846369 1784320 − 6561 13940 59 13 0 0 0 3 2 −4 −18 36 1 85647360 − 43 28549120 − 4921 85647360 1 669120 211603 28549120 − 4921 669120 60 5 0 3 0 5 −11 13 13 1 1 223040 − 387 223040 − 3 223040 6 3485 387 223040 − 6 3485 61 11 1 0 1 3 7 −5 −17 31 − 1 14274560 129 14274560 6561 14274560 − 1 111520 − 846369 14274560 6561 111520 62 13 0 1 0 1 3 −9 −13 35 1 385413120 − 43 128471040 − 1 385413120 1 3011040 43 128471040 − 1 30110940 63 13 1 0 1 1 11 −13 −21 39 − 1 14274560 129 14274560 6561 14274560 − 1 111520 − 846369 14274560 6561 111520 64 15 0 0 0 1 6 −12 −22 44 7 770826240 − 301 2569942080 − 44287 770826240 7 6022080 1904341 2569942080 − 44287 6022080 R E F E R E N C E S 4 1 2 Appendix: Table 2 No b1 b2 b3 b4 b5 a2 a4 a6 a12 c1 c2 c3 c4 c6 c12 1 0 0 0 0 16 −24 48 8 −16 0 33231 55760 0 −37311 55760 −137781 55760 26906661 55760 2 0 0 1 0 14 −23 43 13 −17 0 1441 2720 0 −67241 2720 53806761 111520 − 433647 71372800 3 0 0 2 0 12 −22 38 18 −18 0 1313 2788 0 −6561 2788 32805 68 1974861 69632 4 0 0 3 0 10 −21 33 23 −19 0 1459 3485 0 −1714 3485 −8019 3485 1681074 3485 5 0 0 4 0 8 −20 28 28 −20 0 1297 3485 0 −1552 3485 −7857 3485 1680912 3485 6 0 0 5 0 6 −19 23 33 −21 0 1153 3485 0 −1408 3485 −7713 3485 1680768 3485 7 0 0 6 0 4 18 18 38 −22 0 5 17 0 −256 697 −37 17 336128 697 8 0 0 7 0 2 −17 13 43 −23 0 8201 31365 0 −256 765 −67241 31365 368896 765 9 0 0 8 0 0 −16 8 48 −24 0 21872 94095 0 −28672 94095 −198992 94095 45371392 94095 10 0 1 0 1 14 −15 39 5 −13 0 33 111520 0 − 8193 111520 −216513 111520 53754273 111520 11 0 1 1 1 12 −14 34 10 −14 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 12 0 1 2 1 10 −13 29 15 −15 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 13 0 1 3 1 8 −12 24 20 −16 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 14 0 1 4 1 6 −11 19 25 −17 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 15 0 1 5 1 4 −10 14 30 −18 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 16 0 1 6 1 2 −9 9 35 −19 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 17 0 1 7 1 0 −8 4 40 −20 0 8 31365 0 − 2048 31365 −59048 31365 15116288 31365 18 0 2 0 2 12 −6 30 2 −10 0 3 13940 0 − 1023 13940 −19683 13940 6711903 13940 19 0 2 1 2 10 −5 25 7 −11 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 20 0 2 2 2 8 −4 20 12 −12 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 21 0 2 3 2 6 −3 15 17 −13 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 22 0 2 4 2 4 −2 10 22 −14 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 23 0 2 5 2 2 −1 5 27 −15 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 24 0 2 6 2 0 0 0 32 −16 0 0 0 0 −32 17 8192 17 25 0 3 0 3 10 3 21 −1 −7 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 26 0 3 1 3 8 4 16 4 −8 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 R E F E R E N C E S 4 1 3 No b1 b2 b3 b4 b5 a2 a4 a6 a12 c1 c2 c3 c4 c6 c12 27 0 3 2 3 6 5 11 9 −9 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 28 0 3 3 3 4 6 6 14 −10 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 29 0 3 4 3 2 7 1 19 −11 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 30 0 3 5 3 0 8 −4 24 −12 0 − 8 3485 0 −2048 3485 −6552 3485 1677312 3485 31 0 4 0 4 8 12 12 −4 −4 0 − 3 697 0 − 48 697 19683 697 19683 6970 32 0 4 1 4 6 13 7 1 −5 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 33 0 4 2 4 4 14 2 6 −6 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 34 0 4 3 4 2 15 −3 11 −7 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 35 0 4 4 4 0 16 −8 16 −8 0 − 16 697 0 4096 697 −1296 697 331776 697 36 0 5 0 5 6 21 3 −7 −1 0 129 3485 0 − 384 3485 −846369 3485 2519424 3485 37 0 5 1 5 4 22 −2 −2 −2 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 38 0 5 2 5 2 23 −7 3 −3 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 39 0 5 3 5 0 24 −12 8 −4 0 − 728 3485 0 186368 3485 −5832 3485 1492992 3485 40 0 6 0 6 4 30 −6 −10 2 0 −1023 3485 0 768 3485 6711903 3485 −5038848 3485 41 0 6 1 6 2 31 −11 −5 1 0 1 3485 0 − 256 3485 −6561 3485 1679616 3485 42 0 6 2 6 0 32 −16 0 0 0 −32 7 0 8192 17 0 0 43 0 7 0 7 2 39 −15 −13 5 0 8193 3485 0 −8448 3485 −53754273 3485 55427328 3485 44 0 7 1 7 0 40 −20 −8 4 0 −59048 3485 0 15116288 3485 52488 3485 −13436928 3485 45 0 8 0 8 0 48 −24 −16 8 0 −596976 3485 0 136114176 3485 430506576 3485 −564350976 3485 46 2 0 0 0 14 −20 40 4 −8 0 7381 111520 0 − 7381 111520 − 6561 111520 6561 111520 47 2 0 1 0 12 −19 35 9 −9 0 6561 111520 0 − 6561 111520 − 6561 111520 6561 111520 48 2 0 2 0 10 −18 30 14 −10 0 729 13940 0 − 729 13940 − 729 13940 729 13940 49 2 0 3 0 8 −17 25 19 −11 0 162 3485 0 − 162 3485 − 162 3485 162 3485 50 2 0 4 0 6 −16 20 24 −12 0 144 3485 0 − 144 3485 − 144 3485 144 3485 51 2 0 5 0 4 −15 15 29 −13 0 128 3485 0 − 128 3485 − 128 3485 128 3485 52 2 0 6 0 2 −14 10 34 −14 0 1024 31365 0 − 1024 31365 − 1024 31365 1024 31365 53 2 0 7 0 0 −13 5 39 −15 0 2731 94095 0 − 2816 94095 − 2731 94095 2816 94095 R E F E R E N C E S 4 1 4 No b1 b2 b3 b4 b5 a2 a4 a6 a12 c1 c2 c3 c4 c6 c12 54 2 1 0 1 12 −11 31 1 −5 0 1 111520 0 − 1 111520 − 6561 111520 6561 111520 55 2 1 6 1 0 −5 1 31 −11 0 1 31365 0 − 256 31365 − 1 31365 256 31365 56 2 2 0 2 10 −2 22 −2 −2 0 − 1 13940 0 1 13940 6561 13940 − 6561 13940 57 2 2 5 2 0 3 −3 23 −7 0 1 3485 0 − 256 3485 − 1 3485 256 3485 58 2 3 0 3 8 7 13 −5 1 0 2 3485 0 − 2 3485 −13122 3485 13122 3485 59 2 3 4 3 0 11 −7 15 −3 0 9 3485 0 −2304 3485 − 9 3485 2304 3485 60 2 4 0 4 6 16 4 −8 4 0 − 16 3485 0 16 3485 104976 3485 −104976 3485 61 2 4 3 4 0 19 −11 7 1 0 81 3485 0 −20736 3485 − 81 3485 20736 3485 62 2 5 0 5 4 25 −5 −11 7 0 128 3485 0 − 128 3485 −839808 3485 839808 3485 63 2 5 2 5 0 27 −15 −1 5 0 729 3485 0 −186624 3485 − 729 3485 186624 3485 64 2 6 0 6 2 34 −14 −14 10 0 −1024 3485 0 1024 3485 6718464 3485 −6718464 3485 65 2 6 1 6 0 35 −19 −9 9 0 6561 3485 0 −1679616 3485 −6561 3485 1679616 3485 66 2 7 0 7 0 43 −23 −17 13 0 67241 3485 0 −368896 85 −53806761 3485 1679616 3485 67 4 0 0 0 12 −16 32 0 0 0 1 136 0 1 136 0 0 68 4 0 1 0 10 −15 27 5 −1 0 729 111520 0 − 729 111520 − 729 111520 729 111520 69 4 0 2 0 8 −14 22 10 −2 0 81 13940 0 − 81 13940 − 81 13940 81 13940 70 4 0 3 0 6 −13 17 15 −3 0 18 3485 0 − 18 3485 − 18 3485 18 3485 71 4 0 4 0 4 −12 12 20 −4 0 16 3485 0 − 16 3485 − 16 3485 16 3485 72 4 0 5 0 2 −11 7 25 −5 0 128 31365 0 − 128 31365 − 128 31365 128 31365 73 4 0 6 0 0 −10 2 30 −6 0 341 94095 0 − 256 94095 − 341 94095 256 94095 74 4 1 0 1 10 −7 23 −3 3 0 1 111520 0 − 1 111520 − 6561 111520 6561 111520 75 4 1 5 1 0 −2 −2 22 −2 0 − 1 31365 0 256 31365 1 31365 − 256 31365 76 4 2 0 2 8 2 14 −6 6 0 − 1 13940 0 1 13940 6561 13940 − 6561 13940 77 4 2 4 2 0 6 −6 14 2 0 − 1 3485 0 256 3485 1 3485 − 256 3485 78 4 3 0 3 6 11 5 −9 9 0 2 3485 0 − 2 3485 −13122 3485 13122 3485 79 4 3 3 3 0 14 −10 6 6 0 − 9 3485 0 2304 3485 9 3485 −2304 3485 80 4 4 0 4 4 20 −4 −12 12 0 − 16 3485 0 16 3485 104976 3485 −104976 3485 R E F E R E N C E S 4 1 5 No b1 b2 b3 b4 b5 a2 a4 a6 a12 c1 c2 c3 c4 c6 c12 81 4 4 2 4 0 22 −14 −2 10 0 − 81 3485 0 20736 3485 81 3485 −20736 3485 82 4 5 0 5 2 29 −13 −15 15 0 128 3485 0 − 128 3485 −839808 3485 839808 3485 83 4 5 1 5 0 30 −18 −10 14 0 − 729 3485 0 186624 3485 729 3485 −186624 3485 84 4 6 0 6 0 38 −22 −18 18 0 −37 17 0 336128 697 32805 17 −1679616 697 85 6 0 0 0 10 −12 24 −4 8 0 91 111520 0 − 91 111520 729 111520 − 729 111520 86 6 0 1 0 8 −11 19 1 7 0 81 111520 0 − 81 111520 − 81 111520 81 111520 87 6 0 2 0 6 −10 14 6 6 0 9 13940 0 − 9 13940 − 9 13940 9 13940 88 6 0 3 0 4 −9 9 11 5 0 2 3485 0 − 2 3485 − 2 3485 2 3485 89 6 0 4 0 2 −8 4 16 4 0 16 31365 0 − 16 31365 − 16 31365 16 31365 90 6 0 5 0 0 −7 −1 21 3 0 43 94095 0 − 128 94095 − 43 94095 128 94095 91 6 1 0 1 8 −3 15 −7 11 0 1 111520 0 − 1 111520 − 6561 111520 6561 111520 92 6 1 4 1 0 1 −5 13 7 0 1 31365 0 − 256 31365 − 1 31365 256 31365 93 6 2 0 2 6 6 6 −10 14 0 − 1 13940 0 1 13940 6561 13940 − 6561 13940 94 6 2 3 2 0 9 −9 5 11 0 1 3485 0 − 256 3485 − 1 3485 256 3485 95 6 3 0 3 4 15 −3 −13 17 0 2 3485 0 − 2 3485 −13122 3485 13122 3485 96 6 3 2 3 0 17 −13 −3 15 0 9 3485 0 −2304 3485 − 9 3485 2304 3485 97 6 4 0 4 2 24 −12 −16 20 0 − 16 3485 0 16 3485 104976 3485 104976 3485 98 6 4 1 4 0 25 −17 −11 19 0 81 3485 0 −20736 3485 − 81 3485 20736 3485 99 8 0 0 0 8 −8 16 −8 16 0 − 1 11152 0 1 11152 81 11152 − 81 11152 100 6 5 0 5 0 33 −21 −19 23 0 857 3485 0 −186752 3485 −840537 3485 1026432 3485 101 8 0 1 0 6 −7 11 −3 15 0 9 111520 0 − 9 111520 − 9 111520 9 111520 102 8 0 2 0 4 −6 6 2 14 0 1 13940 0 − 1 13940 − 1 13940 1 13940 103 8 0 3 0 2 −5 1 7 13 0 2 31365 0 − 2 31365 − 2 31365 2 31365 104 8 0 4 0 0 −4 −4 12 12 0 1 18819 0 16 18819 − 1 18819 − 16 18819 105 8 1 0 1 6 1 7 −11 19 0 1 111520 0 − 1 111520 − 6561 111520 6561 111520 106 8 1 3 1 0 4 −8 4 16 0 − 1 31365 0 256 31365 1 31365 − 256 31365 107 8 2 0 2 4 10 −2 −14 22 0 − 1 13940 0 1 13940 6561 13940 − 6561 13940 R E F E R E N C E S 4 1 6 No b1 b2 b3 b4 b5 a2 a4 a6 a12 c1 c2 c3 c4 c6 c12 108 8 2 2 2 0 12 −12 −4 20 0 − 1 3485 0 256 3485 1 3485 − 256 3485 109 8 3 0 3 2 19 −11 −17 25 0 2 3485 0 − 2 3485 −13122 3485 13122 3485 110 8 3 1 3 0 20 −16 −12 24 0 − 9 3485 0 2304 3485 9 3485 −2304 3485 111 8 4 0 4 0 28 −20 −20 28 0 − 97 3485 0 20752 3485 105057 3485 −125712 3485 112 10 0 1 0 4 −3 3 −7 23 0 1 111520 0 − 1 111520 − 1 111520 1 111520 113 10 0 0 0 6 −4 8 −12 24 0 1 111520 0 − 1 111520 819 111520 − 819 111520 114 10 0 2 0 2 −2 −2 −2 22 0 1 125460 0 − 1 125460 − 1 125460 1 125460 115 10 0 3 0 0 −1 −7 3 21 0 1 94095 0 − 86 94095 − 1 94095 86 94095 116 10 1 0 1 4 5 −1 −15 22 0 1 111520 0 − 1 111520 − 6561 111520 6561 111520 117 10 1 2 1 0 7 −11 −5 25 0 1 31365 0 − 256 31365 − 1 31365 256 31365 118 10 2 0 2 2 14 −10 −18 30 0 − 1 13940 0 1 13940 6561 13940 − 6561 13940 119 10 2 1 2 0 15 −15 −13 29 0 1 3485 0 − 256 3485 − 1 3485 256 3485 120 10 3 0 3 0 23 −19 −21 33 0 11 3485 0 −2306 3485 −13131 3485 15426 3485 121 12 0 0 0 4 0 0 −16 32 0 0 0 0 1 136 − 1 136 122 12 0 1 0 2 1 −5 −11 31 0 1 1003680 0 − 1 1003680 − 1 1003680 1 1003680 123 12 0 2 0 0 2 −10 −6 30 0 − 1 376380 0 341 376380 1 376380 − 341 376380 124 12 1 0 1 2 9 −9 −19 35 0 1 111520 0 − 1 111520 − 6561 111520 6561 111520 125 12 1 1 1 0 10 −14 −14 34 0 − 1 31365 0 256 31365 1 31365 − 256 31365 126 12 2 0 2 0 18 −18 −22 38 0 − 1 2788 0 5 68 1313 2788 −37 68 127 14 0 0 0 2 4 −8 −20 40 0 − 1 1003680 0 1 1003680 7381 1003680 − 7381 1003680 128 14 0 1 0 0 5 −13 −15 39 0 11 3011040 0 − 2731 3011040 − 11 3011040 2731 3011040 129 14 1 0 1 0 13 −17 −23 43 0 1 24480 0 − 8201 24480 − 1441 24480 67241 24480 130 16 0 0 0 0 8 −16 −24 48 0 − 7 1505520 0 1367 1505520 11077 1505520 − 12437 1505520