A Borcherds Product in S

A Borcherds Product in S₉(K(16))⁻

(Filename: BP-9-16-1-0-1.html)

properties | c vector | θ vector | ψ expansion | Borch(ψ) coefficients

PROPERTIES OF ψ AND BORCH(ψ)

We will define a weakly holomorphic Jacobi form ψ of weight 0 and index 16, such that it and its Borcherds lift Borch(ψ) have the following list of properties.

The singular part of ψ truncated up to q^(16/4) is
(18 + ζ^(-3) + 7/ζ^2 - 5/ζ - 5*ζ + 7*ζ^2 + ζ^3)
+q(19/ζ^8 + 19*ζ^8)
+q^3(ζ^(-16) + ζ^(-14) + ζ^14 + ζ^16)
+q^4(14/ζ^16 + 14*ζ^16)
From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block
φ = TB(9;−1,−1,−1,−1,−1,2,2,2,2,2,2,2,3)

= 1⁻⁵2⁷3¹

= η(τ)¹⁸(θ(τ,z)/η(τ))⁻⁵(θ(τ,2z)/η(τ))⁷(θ(τ,3z)/η(τ))
This theta block has q-order of vanishing 1. This theta block has index 1*16, and so Borch(ψ) begins with Jacobi coefficient #1.
From the truncated singular part of ψ, we compute that Borch(ψ) is a symmetric form, and so the Fricke sign of Borch(ψ) is −1.
We need to check that certain Fourier coefficients of Borch(ψ) are zero to determine that Borch(ψ) is a cusp form. For this weight and level and Fricke sign, an expansion of Borch(ψ) to 4 Fourier-Jacobi coefficients was needed to show that Borch(ψ) is a cusp form.
From the truncated singular part of ψ, we can compute that all Humbert divisors of Borch(ψ) have nonnegative multiplicities, and hence Borch(ψ) is holomorphic.
(D,r) multiplicity of Hum(D,r)
(1,1) 3

(4,2) 7

(4,14) 1

(9,3) 1

(64,16) 1
From the singular part above, we compute that the valuation Ord(ψ) is -1, and hence we need to multiply ψ by Δ^2 to get a Jacobi cusp form. The definition of ψ is
ψ = c · θ/ Δ^2.
Here c is a vector of coefficients, θ is a basis of of J_24,16^cusp, and Δ is the weight 12 elliptic cusp form. Both c and θ will be given in tables.

(D,r)	multiplicity of Hum(D,r)
(1,1)	3
(4,2)	7
(4,14)	1
(9,3)	1
(64,16)	1

THE VECTOR c (return to top)

The vector c has components given by the following table.

c₁	-201617955840384188883578629947631723836904816799945653966145109599686181357350288533236059616758894168473720971950230385810565498422241148401498845796005663/172485063118103119697103430388102851655120257107904437507915444925427836694546409838175448369055993571001967378012572432568513158291228039522463418846164846889824662540
c₂	481406643522332848078782543576497048193361394384968112930988254214971641614735207676841161566796013022757777209473163798591925221652257422746660944029133268187/1552365568062928077273930873492925664896082313971139937571239004328850530250917688543579035321503942139017706402113151893116618424621052355702170769615483622008421962860
c₃	369705503680212694159945409630951130059695230355050138233744515722021092719610734797811582877386684306546927677602961873597950316920694434124813176538463526843/1552365568062928077273930873492925664896082313971139937571239004328850530250917688543579035321503942139017706402113151893116618424621052355702170769615483622008421962860
c₄	1625045381667623643700454217282149742027846878696910239835152857182279278265574973112077490715946617652425293265904433155192236038627081518620158476242297725688/388091392015732019318482718373231416224020578492784984392809751082212632562729422135894758830375985534754426600528287973279154606155263088925542692403870905502105490715
c₅	-8375604205662644083095506221909133888054793348305179592333935155044724954121028177716133643806126875321066643229982808871733354006522442988338283218557811456651/129363797338577339772827572791077138741340192830928328130936583694070877520909807378631586276791995178251475533509429324426384868718421029641847564134623635167368496905
c₆	-7428418597523142449822177883580384732263320710381858136160647663853593319522030880990519148812400050209703483091656281786176120971166424775623859634547485175133/388091392015732019318482718373231416224020578492784984392809751082212632562729422135894758830375985534754426600528287973279154606155263088925542692403870905502105490715
c₇	-471433744223463542757501606953130682921927315915225662619088487327655676124106532322170072523869129147917611615404429396381438615106152186078118804162846798665297893/10730407572451438291341793267778389208868881735590119810243582366599243312884108250928160127125879229677495077046499731771478683169159151764273621253450648750379717142968750
c₈	-695858464509422810359490271560602595436590099572038593157502672815488280770518752314433607612924577463805833803857696329772090753106497130802719819460448642615993913/5365203786225719145670896633889194604434440867795059905121791183299621656442054125464080063562939614838747538523249865885739341584579575882136810626725324375189858571484375
c₉	-35148135424651907660791493210047591453250627295113278607989908764372485656320350331660613562117841233841897554598532453310703326787940690605561210562776545478085263/499088724300066897271711314780390195761343336539075340011329412399964805250423639578053959401203685101278840792860452640533892240426007058803424244346541802343242657812500
c₁₀	-8560773727812049877705462646770925234699957097836294149559786802154058035414197650886530702561653372546614340465717440430676619839472381788668292993571149273572157/99817744860013379454342262956078039152268667307815068002265882479992961050084727915610791880240737020255768158572090528106778448085201411760684848869308360468648531562500
c₁₁	-5972367343461983597114116706169225068035687547063610976649933077206593472331308763353522166238183061510651433481144508283815107514527179801001183213181199484984949/124772181075016724317927828695097548940335834134768835002832353099991201312605909894513489850300921275319710198215113160133473060106501764700856061086635450585810664453125
c₁₂	-1338899202737268033864925251117457288095751206323056846980815281444200981104295288201076854334724289063862779814460903652156044245453478407194177665638238711275101/249544362150033448635855657390195097880671668269537670005664706199982402625211819789026979700601842550639420396430226320266946120213003529401712122173270901171621328906250
c₁₃	-1539504878291789774767846506564608600946761765422175187952829916988466954878589613676099978002784720533514536097962031216160779803059556516884778328253029706286961/99817744860013379454342262956078039152268667307815068002265882479992961050084727915610791880240737020255768158572090528106778448085201411760684848869308360468648531562500
c₁₄	-741249904846894908462545847337050228078974167182591474814199756838286241226808129771300753896975322385343450341347995509407293483562997505996219058729874894464004/124772181075016724317927828695097548940335834134768835002832353099991201312605909894513489850300921275319710198215113160133473060106501764700856061086635450585810664453125
c₁₅	-6610503143995284665621269110119352834847440263008828656294063888203211232254475611635960683264331169812915815279622169924380763836277624089160054583503684773621671/249544362150033448635855657390195097880671668269537670005664706199982402625211819789026979700601842550639420396430226320266946120213003529401712122173270901171621328906250
c₁₆	-4684814955956064524073131945879347392896821628628499291217894081341520152300435129033257700542662907233440818420630025037102276802869899307498863496030566673927377/499088724300066897271711314780390195761343336539075340011329412399964805250423639578053959401203685101278840792860452640533892240426007058803424244346541802343242657812500
c₁₇	-1089463793103458000984039285135625594259271960471700977486707765034804140003903800209469148573166982956062597654625745095500280593163179975052527127087653289925423/499088724300066897271711314780390195761343336539075340011329412399964805250423639578053959401203685101278840792860452640533892240426007058803424244346541802343242657812500
c₁₈	-1123364445140353372167607219135667124918973992884351851563322769224906153760507232995016539988962637982936605472823033491881988581436324287383656960536471405994693/499088724300066897271711314780390195761343336539075340011329412399964805250423639578053959401203685101278840792860452640533892240426007058803424244346541802343242657812500
c₁₉	-154881870416278772446219151217427363865278285279378983914917372524897120596833329278082818965044865584506642661124384009889145492974447591940401240520040057752733/249544362150033448635855657390195097880671668269537670005664706199982402625211819789026979700601842550639420396430226320266946120213003529401712122173270901171621328906250
c₂₀	-1401915078051752145784300259932709795422495343595465250164298475800662733798264504355555254011794695495572742207236143715624336944450983237453788047746805410743647/499088724300066897271711314780390195761343336539075340011329412399964805250423639578053959401203685101278840792860452640533892240426007058803424244346541802343242657812500
c₂₁	0
c₂₂	0
c₂₃	0
c₂₄	0
c₂₅	0
c₂₆	0
c₂₇	0
c₂₈	0
c₂₉	0

THE VECTOR θ (return to top)

The basis of J_24,16^cusp (with dimension 29) is given by the following theta blocks possibly with index lowering operators denoted by W.

θ₁	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,4,4,4)\|W_3
θ₂	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,4,5)\|W_3
θ₃	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4)\|W_3
θ₄	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5)\|W_3
θ₅	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3)\|W_3
θ₆	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2)\|W_3
θ₇	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,6,8)\|W_5
θ₈	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,4,5,6,6)\|W_5
θ₉	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,9)\|W_5
θ₁₀	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,4,5,6)\|W_5
θ₁₁	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,4,4,4,5,5)\|W_5
θ₁₂	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,6,6)\|W_5
θ₁₃	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,8)\|W_5
θ₁₄	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6,7)\|W_5
θ₁₅	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4)\|W_5
θ₁₆	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,4,4)\|W_5
θ₁₇	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,8)\|W_5
θ₁₈	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4)\|W_5
θ₁₉	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,6,6)\|W_5
θ₂₀	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,7)\|W_5
θ₂₁	TB(24;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4)\|W_7
θ₂₂	TB(24;1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3)\|W_7
θ₂₃	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,4,4,12)\|W_7
θ₂₄	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,6,8,8)\|W_7
θ₂₅	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5,12)\|W_7
θ₂₆	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,8,10)\|W_7
θ₂₇	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,6,11)\|W_7
θ₂₈	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,4,4,8,9)\|W_7
θ₂₉	TB(24;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,6,6,6,8)\|W_7

EXPANSION OF ψ (return to top)

The expansion of ψ up to q^(16/4) is

(18 + ζ^(-3) + 7/ζ^2 - 5/ζ - 5*ζ + 7*ζ^2 + ζ^3)
+q(33066 + 19/ζ^8 - 202/ζ^7 + 1003/ζ^6 - 3365/ζ^5 + 8088/ζ^4 - 15241/ζ^3 + 23509/ζ^2 - 30344/ζ - 30344*ζ + 23509*ζ^2 - 15241*ζ^3 + 8088*ζ^4 - 3365*ζ^5 + 1003*ζ^6 - 202*ζ^7 + 19*ζ^8)
+q^2(3081440 - 55/ζ^11 + 1037/ζ^10 - 7335/ζ^9 + 32928/ζ^8 - 108809/ζ^7 + 283766/ζ^6 - 610364/ζ^5 + 1112816/ζ^4 - 1750731/ζ^3 + 2402685/ζ^2 - 2896658/ζ - 2896658*ζ + 2402685*ζ^2 - 1750731*ζ^3 + 1112816*ζ^4 - 610364*ζ^5 + 283766*ζ^6 - 108809*ζ^7 + 32928*ζ^8 - 7335*ζ^9 + 1037*ζ^10 - 55*ζ^11)
+q^3(112732110 + ζ^(-16) + ζ^(-14) - 559/ζ^13 + 8104/ζ^12 - 58319/ζ^11 + 283514/ζ^10 - 1042318/ζ^9 + 3082168/ζ^8 - 7616931/ζ^7 + 16134264/ζ^6 - 29802208/ζ^5 + 48582584/ζ^4 - 70497750/ζ^3 + 91615501/ζ^2 - 107054107/ζ - 107054107*ζ + 91615501*ζ^2 - 70497750*ζ^3 + 48582584*ζ^4 - 29802208*ζ^5 + 16134264*ζ^6 - 7616931*ζ^7 + 3082168*ζ^8 - 1042318*ζ^9 + 283514*ζ^10 - 58319*ζ^11 + 8104*ζ^12 - 559*ζ^13 + ζ^14 + ζ^16)
+q^4(2467324388 + 14/ζ^16 - 1434/ζ^15 + 23603/ζ^14 - 197653/ζ^13 + 1112848/ζ^12 - 4727088/ζ^11 + 16136040/ζ^10 - 46039807/ζ^9 + 112727936/ζ^8 - 241236917/ζ^7 + 457198139/ζ^6 - 774921571/ζ^5 + 1183222896/ζ^4 - 1636302390/ζ^3 + 2057392682/ζ^2 - 2358049492/ζ - 2358049492*ζ + 2057392682*ζ^2 - 1636302390*ζ^3 + 1183222896*ζ^4 - 774921571*ζ^5 + 457198139*ζ^6 - 241236917*ζ^7 + 112727936*ζ^8 - 46039807*ζ^9 + 16136040*ζ^10 - 4727088*ζ^11 + 1112848*ζ^12 - 197653*ζ^13 + 23603*ζ^14 - 1434*ζ^15 + 14*ζ^16)

COEFFICIENTS OF Borch(ψ) (return to top)

Use the notation

[a b c ]

a	b
b	c

Here is a list of some Fourier coefficients of Borch(ψ)

2u det(2u) a(u, Borch(ψ))

[4 -53 704] 7 0

[8 -89 992] 15 0

[4 -25 160] 15 1

[12 -109 992] 23 0

[6 -13 32] 23 0

[6 -19 64] 23 0

[14 -42 128] 28 0

[14 -70 352] 28 1

[8 -42 224] 28 0

[8 -106 1408] 28 1

[16 -113 800] 31 0

[8 -113 1600] 31 0

[14 -47 160] 31 0

[20 -101 512] 39 5

[10 -91 832] 39 -7

[10 -101 1024] 39 7

[8 -27 96] 39 0

[24 -73 224] 47 -1

[4 9 32] 47 1

[16 -55 192] 47 0

[2 4 32] 48 -8

[26 76 224] 48 0

[6 12 32] 48 0

[14 36 96] 48 -56

[2 3 32] 55 1

[14 -29 64] 55 21

[14 -99 704] 55 -21

[8 -35 160] 55 0

[2 2 32] 60 19

[30 -210 1472] 60 0

[16 -50 160] 60 0

[16 -178 1984] 60 19

[10 -30 96] 60 19

[10 -50 256] 60 0

[8 -114 1632] 60 0

[2 1 32] 63 22

[16 -225 3168] 63 35

[12 -159 2112] 63 74

[8 -97 1184] 63 0

[2 0 32] 64 0

[2 8 64] 64 12

[2 16 160] 64 0

[4 8 32] 64 12

[20 56 160] 64 -12

[10 -24 64] 64 12

[10 -144 2080] 64 0

[36 -469 6112] 71 -7

[18 -235 3072] 71 5

[120 277 640] 71 7

[10 -53 288] 71 -5

[40 -441 4864] 79 38

[190 441 1024] 79 -38

[44 -397 3584] 87 -21

[22 -243 2688] 87 -107

[134 307 704] 87 -21

[46 -138 416] 92 51

[46 -230 1152] 92 -48

[4 6 32] 92 48

[24 -218 1984] 92 -51

[162 374 864] 92 51

[6 26 128] 92 0

[26 -86 288] 92 157

[12 -26 64] 92 -51

[12 -122 1248] 92 -176

[12 -134 1504] 92 0

[48 -337 2368] 95 -35

[4 17 96] 95 -74

[16 -81 416] 95 74

[10 15 32] 95 -35

[52 -261 1312] 103 -19

[4 5 32] 103 19

[56 -169 512] 111 118

[120 297 736] 111 39

[6 9 32] 111 -118

[10 23 64] 111 -39

[2 4 64] 112 -24

[58 172 512] 112 104

[4 4 32] 112 -24

[28 -84 256] 112 -64

[28 -140 704] 112 -144

[28 -196 1376] 112 -104

[14 -28 64] 112 104

[8 12 32] 112 104

[16 -84 448] 112 -64

[2 3 64] 119 -81

[30 -451 6784] 119 81

[20 -61 192] 119 -107

[90 221 544] 119 -798

[12 -61 320] 119 -107

[12 -125 1312] 119 798

[2 2 64] 124 -144

[62 -434 3040] 124 -98

[4 2 32] 124 -144

[16 -226 3200] 124 98

[8 34 160] 124 0

[28 -94 320] 124 -722

[10 14 32] 124 -98

[14 -158 1792] 124 0

[2 1 64] 127 -127

[4 1 32] 127 -127

[2 0 64] 128 0

[2 8 96] 128 -160

[2 16 192] 128 0

[4 0 32] 128 0

[4 16 96] 128 736

[6 8 32] 128 -160

[22 -112 576] 128 -736

[22 -288 3776] 128 -736

[18 -256 3648] 128 0

[12 16 32] 128 0

[12 -64 352] 128 736

[68 -885 11520] 135 185

[34 -171 864] 135 4

[24 -267 2976] 135 1619

[8 11 32] 135 185

[72 -793 8736] 143 -239

[42 121 352] 143 111

[6 7 32] 143 -239

[6 25 128] 143 642

[18 -199 2208] 143 642

[12 -25 64] 143 -111

[76 -685 6176] 151 138

[38 -115 352] 151 799

[8 19 64] 151 -799

[10 13 32] 151 -138

[78 -234 704] 156 -400

[78 -390 1952] 156 237

[40 -202 1024] 156 989

[40 -522 6816] 156 128

[6 6 32] 156 -237

[8 10 32] 156 400

[10 22 64] 156 128

[80 -561 3936] 159 28

[46 -143 448] 159 1891

[28 -143 736] 159 1891

[12 15 32] 159 28

[84 -421 2112] 167 -106

[42 -379 3424] 167 109

[28 -421 6336] 167 -106

[14 -197 2784] 167 -109

[16 -37 96] 167 1257

[88 -265 800] 175 -865

[4 9 64] 175 424

[8 9 32] 175 865

[2 4 96] 176 544

[90 268 800] 176 -680

[30 -92 288] 176 1152

[30 -332 3680] 176 -5064

[30 -452 6816] 176 -544

[10 12 32] 176 -680

[18 -92 480] 176 1152

[2 3 96] 183 693

[6 3 32] 183 693

[38 -125 416] 183 8193

[16 -227 3232] 183 -114

[2 2 96] 188 518

[94 -658 4608] 188 608

[48 -146 448] 188 -1834

[48 -530 5856] 188 2704

[6 2 32] 188 518

[6 14 64] 188 2704

[142 350 864] 188 1706

[8 18 64] 188 1834

[12 14 32] 188 608

[14 34 96] 188 -1706

[2 1 96] 191 203

[6 1 32] 191 203

[32 -417 5440] 191 2266

[8 33 160] 191 -2027

[20 -223 2496] 191 -2027

[18 -95 512] 191 -2266

[18 -257 3680] 191 406

[2 0 96] 192 0

[2 8 128] 192 832

[2 16 224] 192 0

[4 8 64] 192 384

[52 152 448] 192 -1280

[6 0 32] 192 0

[6 24 128] 192 -5312

[8 8 32] 192 832

[24 -120 608] 192 5312

[12 24 64] 192 -1280

[14 16 32] 192 0

[100 -1301 16928] 199 -1247

[50 -651 8480] 199 1238

[10 11 32] 199 -1247

[10 21 64] 199 1238

[104 -1145 12608] 207 319

[478 1113 2592] 207 1028

[26 -391 5888] 207 -319

[108 -973 8768] 215 -159

[54 -595 6560] 215 772

[36 -397 4384] 215 7897

[6 13 64] 215 772

[12 13 32] 215 159

[110 -330 992] 220 1065

[110 -550 2752] 220 80

[4 6 64] 220 519

[56 -506 4576] 220 -1408

[28 -422 6368] 220 80

[10 10 32] 220 -1065

[170 390 896] 220 -1408

[112 -785 5504] 223 1632

[4 17 128] 223 -5007

[8 17 64] 223 -5007

[14 15 32] 223 1632

[116 -581 2912] 231 967

[4 5 64] 231 -1174

[40 -123 384] 231 -7212

[30 -453 6848] 231 967

[24 -123 640] 231 -7212

[120 -361 1088] 239 1895

[248 617 1536] 239 -4828

[48 -151 480] 239 -13932

[30 -151 768] 239 -13932

[24 -361 5440] 239 1895

[198 457 1056] 239 -4828

[2 4 128] 240 -1976

[122 364 1088] 240 456

[4 4 64] 240 848

[60 -180 544] 240 3848

[60 -300 1504] 240 5368

[6 12 64] 240 -5368

[8 4 32] 240 -1976

[32 -356 3968] 240 35024

[10 20 64] 240 -3848

[12 12 32] 240 456

[16 -36 96] 240 -4872

[16 -228 3264] 240 2368

[2 3 128] 247 -2339

[62 -931 13984] 247 -3675

[8 3 32] 247 -2339

[2 2 128] 252 -1760

[126 -882 6176] 252 151

[4 2 64] 252 -841

[42 -126 384] 252 2432

[42 -210 1056] 252 12344

[8 2 32] 252 -1760

[24 -126 672] 252 2432

[14 14 32] 252 151

[18 -258 3712] 252 -4864

[2 1 128] 255 -474

[4 1 64] 255 -340

[44 -575 7520] 255 179

[8 1 32] 255 -474

[14 33 96] 255 179

[2 0 128] 256 0

[2 8 160] 256 -1920

[4 0 64] 256 0

[4 16 128] 256 7744

[8 0 32] 256 0

[8 16 64] 256 7744

[8 32 160] 256 12928

[26 -288 3200] 256 12928

[26 -392 5920] 256 1920

[16 16 32] 256 0

[20 -288 4160] 256 0

[132 -1717 22336] 263 2916

[66 -331 1664] 263 -13667

[408 949 2208] 263 -13667

[22 -331 4992] 263 -2916

[136 -1497 16480] 271 500

[74 217 640] 271 -5562

[28 -423 6400] 271 -500

[212 487 1120] 271 5562

[140 -1261 11360] 279 -841

[70 -211 640] 279 -10539

[268 621 1440] 279 -10539

[20 -301 4544] 279 -841

[142 -426 1280] 284 -880

[142 -710 3552] 284 -2203

[72 -362 1824] 284 10336

[72 -938 12224] 284 -15211

[450 1046 2432] 284 10336

[6 26 160] 284 -27371

[58 -182 576] 284 4960

[36 -182 928] 284 4960

[30 -154 800] 284 27371

[30 -454 6880] 284 -2203

[24 -362 5472] 284 -880

[240 554 1280] 284 15211

[78 -239 736] 287 11723

[48 -241 1216] 287 -14794

[338 785 1824] 287 11723

[18 -271 4096] 287 7152

[148 -741 3712] 295 -3320

[38 -421 4672] 295 -69521

[10 5 32] 295 3320

[226 517 1184] 295 13358

[152 -457 1376] 303 359

[4 9 96] 303 3847

[6 9 64] 303 3847

[26 -393 5952] 303 359

[2 4 160] 304 3016

[154 460 1376] 304 6304

[10 4 32] 304 3016

[22 -332 5024] 304 -6304

[2 3 160] 311 3373

[52 -157 480] 311 -1567

[10 3 32] 311 3373

[12 29 96] 311 1567

[16 35 96] 311 -8097

[240 547 1248] 311 -14797

[2 2 160] 316 5130

[80 -242 736] 316 6288

[80 -882 9728] 316 -854

[380 882 2048] 316 854

[10 2 32] 316 5130

[310 718 1664] 316 6288

[20 -302 4576] 316 9424

[2 1 160] 319 2872

[50 -159 512] 319 39867

[10 1 32] 319 2872

[32 -353 3904] 319 -39867

[20 -289 4192] 319 15760

[2 0 160] 320 0

[2 8 192] 320 1200

[4 8 96] 320 13552

[84 248 736] 320 11152

[6 8 64] 320 13552

[54 -272 1376] 320 10112

[54 -704 9184] 320 12288

[10 0 32] 320 0

[28 -424 6432] 320 -1200

[14 32 96] 320 12288

[254 584 1344] 320 -11152

[18 -272 4128] 320 0

[164 -2133 27744] 327 1060

[82 -1067 13888] 327 30278

[56 -619 6848] 327 -33698

[42 -213 1088] 327 33698

[282 651 1504] 327 -30278

[24 -363 5504] 327 -1060

[168 -1849 20352] 335 -295

[766 1785 4160] 335 35693

[6 7 64] 335 -35693

[6 25 160] 335 50906

[36 -185 960] 335 -50906

[30 -455 6912] 335 295

[86 -947 10432] 343 18860

[422 979 2272] 343 -18860

[22 -333 5056] 343 2884

[174 -522 1568] 348 -703

[174 -870 4352] 348 816

[4 6 96] 348 37184

[6 6 64] 348 37184

[44 -486 5376] 348 64881

[12 6 32] 348 -816

[26 -394 5984] 348 -703

[268 614 1408] 348 -42673

[4 17 160] 351 -13413

[60 -303 1536] 351 49198

[352 815 1888] 351 13413

[20 -303 4608] 351 -5272

[180 -901 4512] 359 5126

[4 5 96] 359 -7087

[60 -901 13536] 359 7087

[12 5 32] 359 -5126

[184 -553 1664] 367 -3695

[376 937 2336] 367 8872

[28 -425 6464] 367 -3695

[296 681 1568] 367 8872

[2 4 192] 368 -3768

[4 4 96] 368 -30328

[92 -276 832] 368 48464

[92 -460 2304] 368 68672

[62 -188 576] 368 4232

[62 -684 7552] 368 64888

[62 -932 14016] 368 30328

[8 12 64] 368 -68672

[48 -244 1248] 368 -64888

[12 4 32] 368 -3768

[12 28 96] 368 -4232

[324 748 1728] 368 48464

[24 -364 5536] 368 16440

[282 644 1472] 368 10120

[2 3 192] 375 -227

[94 -1411 21184] 375 -22778

[6 3 64] 375 22778

[12 3 32] 375 -227

[2 2 192] 380 -6208

[4 2 96] 380 -46032

[6 2 64] 380 -46032

[6 14 96] 380 110717

[60 -190 608] 380 -42224

[38 -190 960] 380 -42224

[12 2 32] 380 -6208

[16 34 96] 380 -5376

[22 -334 5088] 380 -23357

[20 30 64] 380 -72141

[2 1 192] 383 -3551

[4 1 96] 383 51579

[6 1 64] 383 51579

[64 -833 10848] 383 -49798

[12 1 32] 383 -3551

[14 31 96] 383 -49798

[2 0 192] 384 0

[2 8 224] 384 192

[4 0 96] 384 0

[4 16 160] 384 58752

[6 0 64] 384 0

[6 24 160] 384 -28544

[10 24 96] 384 -28544

[394 912 2112] 384 -58752

[12 0 32] 384 0

[30 -456 6944] 384 -192

[20 -304 4640] 384 0

[22 32 64] 384 0

[98 -491 2464] 391 -79607

[8 11 64] 391 79607

[26 -395 6016] 391 14000

[200 -2201 24224] 399 -1765

[50 -551 6080] 399 -16596

[14 7 32] 399 -1765

[310 711 1632] 399 85524

[102 -307 928] 407 -42787

[68 -749 8256] 407 21523

[6 13 96] 407 -116965

[8 19 96] 407 21523

[366 845 1952] 407 -42787

[24 -365 5568] 407 -3326

[206 -1030 5152] 412 14641

[104 -522 2624] 412 47905

[104 -1354 17632] 412 -34112

[8 10 64] 412 -47905

[14 6 32] 412 -14641

[338 778 1792] 412 34112

[28 -426 6496] 412 4016

[110 -335 1024] 415 60636

[10 15 64] 415 -60636

[22 -335 5120] 415 -25514

[212 -1061 5312] 423 5230

[72 -219 672] 423 -15527

[12 27 96] 423 15527

[14 5 32] 423 -5230

[18 27 64] 423 -34294

[4 9 128] 431 -50775

[80 -247 768] 431 119975

[8 9 64] 431 -50775

[10 23 96] 431 -119975

[30 -457 6976] 431 -4537

[2 4 224] 432 2656

[6 12 96] 432 14888

[14 4 32] 432 2656

[26 -396 6048] 432 -4864

[2 3 224] 439 -9748

[70 -221 704] 439 -64296

[44 -221 1120] 439 -64296

[14 3 32] 439 -9748

[20 29 64] 439 127866

[2 2 224] 444 -5435

[112 -338 1024] 444 -18715

[112 -1234 13600] 444 41968

[74 -222 672] 444 -130475

[74 -370 1856] 444 126880

[8 18 96] 444 -126880

[10 14 64] 444 41968

[12 18 64] 444 18715

[14 2 32] 444 -5435

[14 30 96] 444 130475

[24 -366 5600] 444 1520

[2 1 224] 447 -11704

[14 1 32] 447 -11704

[16 33 96] 447 13455

[22 31 64] 447 105608

[2 0 224] 448 0

[4 8 128] 448 152448

[8 8 64] 448 152448

[56 -280 1408] 448 32000

[14 0 32] 448 0

[16 8 32] 448 13312

[16 24 64] 448 125824

[22 -336 5152] 448 0

[114 -1483 19296] 455 105159

[696 1621 3776] 455 -87098

[14 21 64] 455 105159

[28 -427 6528] 455 -8857

[1054 2457 5728] 463 182713

[58 -871 13088] 463 182713

[16 7 32] 463 -4141

[118 -1299 14304] 471 76457

[10 13 64] 471 76457

[26 -397 6080] 471 -2896

[4 6 128] 476 169270

[738 1718 4000] 476 184394

[6 26 192] 476 17424

[90 -278 864] 476 -251770

[60 -902 13568] 476 -169270

[10 22 96] 476 251770

[12 26 96] 476 17424

[16 6 32] 476 25872

[18 26 64] 476 152896

[30 -458 7008] 476 -11238

[4 17 192] 479 -159715

[80 -401 2016] 479 -120260

[8 17 96] 479 120260

[12 17 64] 479 -159715

[24 -367 5632] 479 47393

[4 5 128] 487 -69508

[62 -933 14048] 487 69508

[16 5 32] 487 37195

[6 9 96] 495 161890

[62 -311 1568] 495 42064

[16 23 64] 495 -39154

[18 9 32] 495 1685

[4 4 128] 496 -97408

[124 -372 1120] 496 199536

[124 -620 3104] 496 158992

[8 4 64] 496 -97408

[64 -708 7840] 496 -222832

[10 12 64] 496 -158992

[50 -252 1280] 496 222832

[14 20 64] 496 -199536

[16 4 32] 496 23216

[20 28 64] 496 -137952

[28 -428 6560] 496 -17328

	φ	= TB(9;−1,−1,−1,−1,−1,2,2,2,2,2,2,2,3)
		= 1⁻⁵2⁷3¹
		= η(τ)¹⁸(θ(τ,z)/η(τ))⁻⁵(θ(τ,2z)/η(τ))⁷(θ(τ,3z)/η(τ))

2u	det(2u)	a(u, Borch(ψ))
[4 -53 704]	7	0
[8 -89 992]	15	0
[4 -25 160]	15	1
[12 -109 992]	23	0
[6 -13 32]	23	0
[6 -19 64]	23	0
[14 -42 128]	28	0
[14 -70 352]	28	1
[8 -42 224]	28	0
[8 -106 1408]	28	1
[16 -113 800]	31	0
[8 -113 1600]	31	0
[14 -47 160]	31	0
[20 -101 512]	39	5
[10 -91 832]	39	-7
[10 -101 1024]	39	7
[8 -27 96]	39	0
[24 -73 224]	47	-1
[4 9 32]	47	1
[16 -55 192]	47	0
[2 4 32]	48	-8
[26 76 224]	48	0
[6 12 32]	48	0
[14 36 96]	48	-56
[2 3 32]	55	1
[14 -29 64]	55	21
[14 -99 704]	55	-21
[8 -35 160]	55	0
[2 2 32]	60	19
[30 -210 1472]	60	0
[16 -50 160]	60	0
[16 -178 1984]	60	19
[10 -30 96]	60	19
[10 -50 256]	60	0
[8 -114 1632]	60	0
[2 1 32]	63	22
[16 -225 3168]	63	35
[12 -159 2112]	63	74
[8 -97 1184]	63	0
[2 0 32]	64	0
[2 8 64]	64	12
[2 16 160]	64	0
[4 8 32]	64	12
[20 56 160]	64	-12
[10 -24 64]	64	12
[10 -144 2080]	64	0
[36 -469 6112]	71	-7
[18 -235 3072]	71	5
[120 277 640]	71	7
[10 -53 288]	71	-5
[40 -441 4864]	79	38
[190 441 1024]	79	-38
[44 -397 3584]	87	-21
[22 -243 2688]	87	-107
[134 307 704]	87	-21
[46 -138 416]	92	51
[46 -230 1152]	92	-48
[4 6 32]	92	48
[24 -218 1984]	92	-51
[162 374 864]	92	51
[6 26 128]	92	0
[26 -86 288]	92	157
[12 -26 64]	92	-51
[12 -122 1248]	92	-176
[12 -134 1504]	92	0
[48 -337 2368]	95	-35
[4 17 96]	95	-74
[16 -81 416]	95	74
[10 15 32]	95	-35
[52 -261 1312]	103	-19
[4 5 32]	103	19
[56 -169 512]	111	118
[120 297 736]	111	39
[6 9 32]	111	-118
[10 23 64]	111	-39
[2 4 64]	112	-24
[58 172 512]	112	104
[4 4 32]	112	-24
[28 -84 256]	112	-64
[28 -140 704]	112	-144
[28 -196 1376]	112	-104
[14 -28 64]	112	104
[8 12 32]	112	104
[16 -84 448]	112	-64
[2 3 64]	119	-81
[30 -451 6784]	119	81
[20 -61 192]	119	-107
[90 221 544]	119	-798
[12 -61 320]	119	-107
[12 -125 1312]	119	798
[2 2 64]	124	-144
[62 -434 3040]	124	-98
[4 2 32]	124	-144
[16 -226 3200]	124	98
[8 34 160]	124	0
[28 -94 320]	124	-722
[10 14 32]	124	-98
[14 -158 1792]	124	0
[2 1 64]	127	-127
[4 1 32]	127	-127
[2 0 64]	128	0
[2 8 96]	128	-160
[2 16 192]	128	0
[4 0 32]	128	0
[4 16 96]	128	736
[6 8 32]	128	-160
[22 -112 576]	128	-736
[22 -288 3776]	128	-736
[18 -256 3648]	128	0
[12 16 32]	128	0
[12 -64 352]	128	736
[68 -885 11520]	135	185
[34 -171 864]	135	4
[24 -267 2976]	135	1619
[8 11 32]	135	185
[72 -793 8736]	143	-239
[42 121 352]	143	111
[6 7 32]	143	-239
[6 25 128]	143	642
[18 -199 2208]	143	642
[12 -25 64]	143	-111
[76 -685 6176]	151	138
[38 -115 352]	151	799
[8 19 64]	151	-799
[10 13 32]	151	-138
[78 -234 704]	156	-400
[78 -390 1952]	156	237
[40 -202 1024]	156	989
[40 -522 6816]	156	128
[6 6 32]	156	-237
[8 10 32]	156	400
[10 22 64]	156	128
[80 -561 3936]	159	28
[46 -143 448]	159	1891
[28 -143 736]	159	1891
[12 15 32]	159	28
[84 -421 2112]	167	-106
[42 -379 3424]	167	109
[28 -421 6336]	167	-106
[14 -197 2784]	167	-109
[16 -37 96]	167	1257
[88 -265 800]	175	-865
[4 9 64]	175	424
[8 9 32]	175	865
[2 4 96]	176	544
[90 268 800]	176	-680
[30 -92 288]	176	1152
[30 -332 3680]	176	-5064
[30 -452 6816]	176	-544
[10 12 32]	176	-680
[18 -92 480]	176	1152
[2 3 96]	183	693
[6 3 32]	183	693
[38 -125 416]	183	8193
[16 -227 3232]	183	-114
[2 2 96]	188	518
[94 -658 4608]	188	608
[48 -146 448]	188	-1834
[48 -530 5856]	188	2704
[6 2 32]	188	518
[6 14 64]	188	2704
[142 350 864]	188	1706
[8 18 64]	188	1834
[12 14 32]	188	608
[14 34 96]	188	-1706
[2 1 96]	191	203
[6 1 32]	191	203
[32 -417 5440]	191	2266
[8 33 160]	191	-2027
[20 -223 2496]	191	-2027
[18 -95 512]	191	-2266
[18 -257 3680]	191	406
[2 0 96]	192	0
[2 8 128]	192	832
[2 16 224]	192	0
[4 8 64]	192	384
[52 152 448]	192	-1280
[6 0 32]	192	0
[6 24 128]	192	-5312
[8 8 32]	192	832
[24 -120 608]	192	5312
[12 24 64]	192	-1280
[14 16 32]	192	0
[100 -1301 16928]	199	-1247
[50 -651 8480]	199	1238
[10 11 32]	199	-1247
[10 21 64]	199	1238
[104 -1145 12608]	207	319
[478 1113 2592]	207	1028
[26 -391 5888]	207	-319
[108 -973 8768]	215	-159
[54 -595 6560]	215	772
[36 -397 4384]	215	7897
[6 13 64]	215	772
[12 13 32]	215	159
[110 -330 992]	220	1065
[110 -550 2752]	220	80
[4 6 64]	220	519
[56 -506 4576]	220	-1408
[28 -422 6368]	220	80
[10 10 32]	220	-1065
[170 390 896]	220	-1408
[112 -785 5504]	223	1632
[4 17 128]	223	-5007
[8 17 64]	223	-5007
[14 15 32]	223	1632
[116 -581 2912]	231	967
[4 5 64]	231	-1174
[40 -123 384]	231	-7212
[30 -453 6848]	231	967
[24 -123 640]	231	-7212
[120 -361 1088]	239	1895
[248 617 1536]	239	-4828
[48 -151 480]	239	-13932
[30 -151 768]	239	-13932
[24 -361 5440]	239	1895
[198 457 1056]	239	-4828
[2 4 128]	240	-1976
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[4 4 64]	240	848
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[60 -300 1504]	240	5368
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[8 4 32]	240	-1976
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[16 -228 3264]	240	2368
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[62 -931 13984]	247	-3675
[8 3 32]	247	-2339
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[126 -882 6176]	252	151
[4 2 64]	252	-841
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[42 -210 1056]	252	12344
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[24 -126 672]	252	2432
[14 14 32]	252	151
[18 -258 3712]	252	-4864
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[4 1 64]	255	-340
[44 -575 7520]	255	179
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[8 32 160]	256	12928
[26 -288 3200]	256	12928
[26 -392 5920]	256	1920
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[132 -1717 22336]	263	2916
[66 -331 1664]	263	-13667
[408 949 2208]	263	-13667
[22 -331 4992]	263	-2916
[136 -1497 16480]	271	500
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[28 -423 6400]	271	-500
[212 487 1120]	271	5562
[140 -1261 11360]	279	-841
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[268 621 1440]	279	-10539
[20 -301 4544]	279	-841
[142 -426 1280]	284	-880
[142 -710 3552]	284	-2203
[72 -362 1824]	284	10336
[72 -938 12224]	284	-15211
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[6 26 160]	284	-27371
[58 -182 576]	284	4960
[36 -182 928]	284	4960
[30 -154 800]	284	27371
[30 -454 6880]	284	-2203
[24 -362 5472]	284	-880
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[78 -239 736]	287	11723
[48 -241 1216]	287	-14794
[338 785 1824]	287	11723
[18 -271 4096]	287	7152
[148 -741 3712]	295	-3320
[38 -421 4672]	295	-69521
[10 5 32]	295	3320
[226 517 1184]	295	13358
[152 -457 1376]	303	359
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[6 9 64]	303	3847
[26 -393 5952]	303	359
[2 4 160]	304	3016
[154 460 1376]	304	6304
[10 4 32]	304	3016
[22 -332 5024]	304	-6304
[2 3 160]	311	3373
[52 -157 480]	311	-1567
[10 3 32]	311	3373
[12 29 96]	311	1567
[16 35 96]	311	-8097
[240 547 1248]	311	-14797
[2 2 160]	316	5130
[80 -242 736]	316	6288
[80 -882 9728]	316	-854
[380 882 2048]	316	854
[10 2 32]	316	5130
[310 718 1664]	316	6288
[20 -302 4576]	316	9424
[2 1 160]	319	2872
[50 -159 512]	319	39867
[10 1 32]	319	2872
[32 -353 3904]	319	-39867
[20 -289 4192]	319	15760
[2 0 160]	320	0
[2 8 192]	320	1200
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[6 8 64]	320	13552
[54 -272 1376]	320	10112
[54 -704 9184]	320	12288
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[14 32 96]	320	12288
[254 584 1344]	320	-11152
[18 -272 4128]	320	0
[164 -2133 27744]	327	1060
[82 -1067 13888]	327	30278
[56 -619 6848]	327	-33698
[42 -213 1088]	327	33698
[282 651 1504]	327	-30278
[24 -363 5504]	327	-1060
[168 -1849 20352]	335	-295
[766 1785 4160]	335	35693
[6 7 64]	335	-35693
[6 25 160]	335	50906
[36 -185 960]	335	-50906
[30 -455 6912]	335	295
[86 -947 10432]	343	18860
[422 979 2272]	343	-18860
[22 -333 5056]	343	2884
[174 -522 1568]	348	-703
[174 -870 4352]	348	816
[4 6 96]	348	37184
[6 6 64]	348	37184
[44 -486 5376]	348	64881
[12 6 32]	348	-816
[26 -394 5984]	348	-703
[268 614 1408]	348	-42673
[4 17 160]	351	-13413
[60 -303 1536]	351	49198
[352 815 1888]	351	13413
[20 -303 4608]	351	-5272
[180 -901 4512]	359	5126
[4 5 96]	359	-7087
[60 -901 13536]	359	7087
[12 5 32]	359	-5126
[184 -553 1664]	367	-3695
[376 937 2336]	367	8872
[28 -425 6464]	367	-3695
[296 681 1568]	367	8872
[2 4 192]	368	-3768
[4 4 96]	368	-30328
[92 -276 832]	368	48464
[92 -460 2304]	368	68672
[62 -188 576]	368	4232
[62 -684 7552]	368	64888
[62 -932 14016]	368	30328
[8 12 64]	368	-68672
[48 -244 1248]	368	-64888
[12 4 32]	368	-3768
[12 28 96]	368	-4232
[324 748 1728]	368	48464
[24 -364 5536]	368	16440
[282 644 1472]	368	10120
[2 3 192]	375	-227
[94 -1411 21184]	375	-22778
[6 3 64]	375	22778
[12 3 32]	375	-227
[2 2 192]	380	-6208
[4 2 96]	380	-46032
[6 2 64]	380	-46032
[6 14 96]	380	110717
[60 -190 608]	380	-42224
[38 -190 960]	380	-42224
[12 2 32]	380	-6208
[16 34 96]	380	-5376
[22 -334 5088]	380	-23357
[20 30 64]	380	-72141
[2 1 192]	383	-3551
[4 1 96]	383	51579
[6 1 64]	383	51579
[64 -833 10848]	383	-49798
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[22 32 64]	384	0
[98 -491 2464]	391	-79607
[8 11 64]	391	79607
[26 -395 6016]	391	14000
[200 -2201 24224]	399	-1765
[50 -551 6080]	399	-16596
[14 7 32]	399	-1765
[310 711 1632]	399	85524
[102 -307 928]	407	-42787
[68 -749 8256]	407	21523
[6 13 96]	407	-116965
[8 19 96]	407	21523
[366 845 1952]	407	-42787
[24 -365 5568]	407	-3326
[206 -1030 5152]	412	14641
[104 -522 2624]	412	47905
[104 -1354 17632]	412	-34112
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[14 6 32]	412	-14641
[338 778 1792]	412	34112
[28 -426 6496]	412	4016
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[10 15 64]	415	-60636
[22 -335 5120]	415	-25514
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[12 27 96]	423	15527
[14 5 32]	423	-5230
[18 27 64]	423	-34294
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[80 -247 768]	431	119975
[8 9 64]	431	-50775
[10 23 96]	431	-119975
[30 -457 6976]	431	-4537
[2 4 224]	432	2656
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[14 4 32]	432	2656
[26 -396 6048]	432	-4864
[2 3 224]	439	-9748
[70 -221 704]	439	-64296
[44 -221 1120]	439	-64296
[14 3 32]	439	-9748
[20 29 64]	439	127866
[2 2 224]	444	-5435
[112 -338 1024]	444	-18715
[112 -1234 13600]	444	41968
[74 -222 672]	444	-130475
[74 -370 1856]	444	126880
[8 18 96]	444	-126880
[10 14 64]	444	41968
[12 18 64]	444	18715
[14 2 32]	444	-5435
[14 30 96]	444	130475
[24 -366 5600]	444	1520
[2 1 224]	447	-11704
[14 1 32]	447	-11704
[16 33 96]	447	13455
[22 31 64]	447	105608
[2 0 224]	448	0
[4 8 128]	448	152448
[8 8 64]	448	152448
[56 -280 1408]	448	32000
[14 0 32]	448	0
[16 8 32]	448	13312
[16 24 64]	448	125824
[22 -336 5152]	448	0
[114 -1483 19296]	455	105159
[696 1621 3776]	455	-87098
[14 21 64]	455	105159
[28 -427 6528]	455	-8857
[1054 2457 5728]	463	182713
[58 -871 13088]	463	182713
[16 7 32]	463	-4141
[118 -1299 14304]	471	76457
[10 13 64]	471	76457
[26 -397 6080]	471	-2896
[4 6 128]	476	169270
[738 1718 4000]	476	184394
[6 26 192]	476	17424
[90 -278 864]	476	-251770
[60 -902 13568]	476	-169270
[10 22 96]	476	251770
[12 26 96]	476	17424
[16 6 32]	476	25872
[18 26 64]	476	152896
[30 -458 7008]	476	-11238
[4 17 192]	479	-159715
[80 -401 2016]	479	-120260
[8 17 96]	479	120260
[12 17 64]	479	-159715
[24 -367 5632]	479	47393
[4 5 128]	487	-69508
[62 -933 14048]	487	69508
[16 5 32]	487	37195
[6 9 96]	495	161890
[62 -311 1568]	495	42064
[16 23 64]	495	-39154
[18 9 32]	495	1685
[4 4 128]	496	-97408
[124 -372 1120]	496	199536
[124 -620 3104]	496	158992
[8 4 64]	496	-97408
[64 -708 7840]	496	-222832
[10 12 64]	496	-158992
[50 -252 1280]	496	222832
[14 20 64]	496	-199536
[16 4 32]	496	23216
[20 28 64]	496	-137952
[28 -428 6560]	496	-17328

A Borcherds Product in S9(K(16))−

A Borcherds Product in S₉(K(16))⁻