A Borcherds Product in S

A Borcherds Product in S₉(K(16))⁻

(Filename: BP-9-16-1-0-10.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^(-4) + ζ^(-3) + 2/ζ^2 - ζ^(-1) - ζ + 2*ζ^2 + ζ^3 + ζ^4)
+q(13/ζ^8 + 13*ζ^8)
+q^2(2/ζ^12 + 2*ζ^12)
+q^3(2/ζ^14 + 2*ζ^14)
+q^4(16/ζ^16 + 16*ζ^16)
From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block
φ = TB(9;−1,2,2,3,4)

= 1⁻¹2²3¹4¹

= η(τ)¹⁸(θ(τ,z)/η(τ))⁻¹(θ(τ,2z)/η(τ))²(θ(τ,3z)/η(τ))(θ(τ,4z)/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) 3

(4,14) 3

(9,3) 1

(16,4) 1

(16,12) 2
From the singular part above, we compute that the valuation Ord(ψ) is -0.25, and hence we need to multiply ψ by Δ^1 to get a Jacobi cusp form. The definition of ψ is
ψ = c · θ/ Δ^1.
Here c is a vector of coefficients, θ is a basis of of J_12,16^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)	3
(4,14)	3
(9,3)	1
(16,4)	1
(16,12)	2

THE VECTOR c (return to top)

The vector c has components given by the following table.

c₁	-120523198633507/1520173324496827395
c₂	6780505356200951989/5910433885643664911760
c₃	-4102239450568826234377/2513412009869968503725940
c₄	256353010211796788128/69817000274165791770165
c₅	535706729631925963487/139634000548331583540330
c₆	4149138743663261824759/2513412009869968503725940
c₇	4102239450568826234377/2513412009869968503725940
c₈	-691731165760143211/6649238121349123025730
c₉	-11807658806697117937/59843143092142107231570
c₁₀	-12547909178109949141/39895428728094738154380
c₁₁	730271188563061/73880423570545811397
c₁₂	43067585306449890827/89764714638213160847355

THE VECTOR θ (return to top)

The basis of J_12,16^cusp (with dimension 12) is given by the following theta blocks possibly with index lowering operators denoted by W.

θ₁	TB(12;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4)\|W_2
θ₂	TB(12;1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3)\|W_2
θ₃	TB(12;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,8)\|W_3
θ₄	TB(12;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,4,7)\|W_3
θ₅	TB(12;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,4,4,4,5)\|W_3
θ₆	TB(12;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4)\|W_3
θ₇	TB(12;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,7)\|W_3
θ₈	TB(12;1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5,5)\|W_3
θ₉	TB(12;1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4)\|W_3
θ₁₀	TB(12;1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3)\|W_3
θ₁₁	TB(12;1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,5)\|W_3
θ₁₂	TB(12;1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3)\|W_3

EXPANSION OF ψ (return to top)

The expansion of ψ up to q^(16/4) is

(18 + ζ^(-4) + ζ^(-3) + 2/ζ^2 - ζ^(-1) - ζ + 2*ζ^2 + ζ^3 + ζ^4)
+q(174 + 13/ζ^8 - 6/ζ^7 - 6/ζ^6 + 11/ζ^5 - 100/ζ^4 + 27/ζ^3 + 6/ζ^2 - 32/ζ - 32*ζ + 6*ζ^2 + 27*ζ^3 - 100*ζ^4 + 11*ζ^5 - 6*ζ^6 - 6*ζ^7 + 13*ζ^8)
+q^2(1252 + 2/ζ^12 + 5/ζ^11 + 2/ζ^10 - 19/ζ^9 + 158/ζ^8 - 73/ζ^7 - 16/ζ^6 + 120/ζ^5 - 786/ζ^4 + 217/ζ^3 + 14/ζ^2 - 250/ζ - 250*ζ + 14*ζ^2 + 217*ζ^3 - 786*ζ^4 + 120*ζ^5 - 16*ζ^6 - 73*ζ^7 + 158*ζ^8 - 19*ζ^9 + 2*ζ^10 + 5*ζ^11 + 2*ζ^12)
+q^3(6136 + 2/ζ^14 + 9/ζ^13 - 92/ζ^12 + 61/ζ^11 + 8/ζ^10 - 162/ζ^9 + 1204/ζ^8 - 447/ζ^7 - 60/ζ^6 + 720/ζ^5 - 4180/ζ^4 + 1134/ζ^3 + 50/ζ^2 - 1315/ζ - 1315*ζ + 50*ζ^2 + 1134*ζ^3 - 4180*ζ^4 + 720*ζ^5 - 60*ζ^6 - 447*ζ^7 + 1204*ζ^8 - 162*ζ^9 + 8*ζ^10 + 61*ζ^11 - 92*ζ^12 + 9*ζ^13 + 2*ζ^14)
+q^4(25072 + 16/ζ^16 - 14/ζ^15 - 2/ζ^14 + 87/ζ^13 - 753/ζ^12 + 376/ζ^11 + 44/ζ^10 - 911/ζ^9 + 6008/ζ^8 - 2081/ζ^7 - 146/ζ^6 + 3253/ζ^5 - 17807/ζ^4 + 4786/ζ^3 + 104/ζ^2 - 5496/ζ - 5496*ζ + 104*ζ^2 + 4786*ζ^3 - 17807*ζ^4 + 3253*ζ^5 - 146*ζ^6 - 2081*ζ^7 + 6008*ζ^8 - 911*ζ^9 + 44*ζ^10 + 376*ζ^11 - 753*ζ^12 + 87*ζ^13 - 2*ζ^14 - 14*ζ^15 + 16*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 0

[12 -109 992] 23 0

[6 -13 32] 23 0

[6 -19 64] 23 0

[14 -42 128] 28 0

[14 -70 352] 28 0

[8 -42 224] 28 0

[8 -106 1408] 28 0

[16 -113 800] 31 0

[8 -113 1600] 31 0

[14 -47 160] 31 0

[20 -101 512] 39 1

[10 -91 832] 39 0

[10 -101 1024] 39 0

[8 -27 96] 39 0

[24 -73 224] 47 -1

[4 9 32] 47 1

[16 -55 192] 47 0

[2 4 32] 48 -1

[26 76 224] 48 -1

[6 12 32] 48 -1

[14 36 96] 48 -1

[2 3 32] 55 1

[14 -29 64] 55 0

[14 -99 704] 55 0

[8 -35 160] 55 0

[2 2 32] 60 2

[30 -210 1472] 60 2

[16 -50 160] 60 0

[16 -178 1984] 60 0

[10 -30 96] 60 0

[10 -50 256] 60 0

[8 -114 1632] 60 2

[2 1 32] 63 2

[16 -225 3168] 63 0

[12 -159 2112] 63 1

[8 -97 1184] 63 0

[2 0 32] 64 0

[2 8 64] 64 2

[2 16 160] 64 0

[4 8 32] 64 2

[20 56 160] 64 2

[10 -24 64] 64 -2

[10 -144 2080] 64 0

[36 -469 6112] 71 -3

[18 -235 3072] 71 -2

[120 277 640] 71 3

[10 -53 288] 71 2

[40 -441 4864] 79 2

[190 441 1024] 79 -2

[44 -397 3584] 87 -1

[22 -243 2688] 87 12

[134 307 704] 87 -1

[46 -138 416] 92 2

[46 -230 1152] 92 2

[4 6 32] 92 -2

[24 -218 1984] 92 2

[162 374 864] 92 2

[6 26 128] 92 0

[26 -86 288] 92 0

[12 -26 64] 92 2

[12 -122 1248] 92 -2

[12 -134 1504] 92 0

[48 -337 2368] 95 5

[4 17 96] 95 -11

[16 -81 416] 95 11

[10 15 32] 95 5

[52 -261 1312] 103 -11

[4 5 32] 103 11

[56 -169 512] 111 14

[120 297 736] 111 -9

[6 9 32] 111 -14

[10 23 64] 111 9

[2 4 64] 112 12

[58 172 512] 112 12

[4 4 32] 112 12

[28 -84 256] 112 -4

[28 -140 704] 112 -4

[28 -196 1376] 112 12

[14 -28 64] 112 -12

[8 12 32] 112 12

[16 -84 448] 112 -4

[2 3 64] 119 -17

[30 -451 6784] 119 17

[20 -61 192] 119 -23

[90 221 544] 119 9

[12 -61 320] 119 -23

[12 -125 1312] 119 -9

[2 2 64] 124 -26

[62 -434 3040] 124 -26

[4 2 32] 124 -26

[16 -226 3200] 124 -26

[8 34 160] 124 0

[28 -94 320] 124 0

[10 14 32] 124 -26

[14 -158 1792] 124 0

[2 1 64] 127 -27

[4 1 32] 127 -27

[2 0 64] 128 0

[2 8 96] 128 -24

[2 16 192] 128 0

[4 0 32] 128 0

[4 16 96] 128 -12

[6 8 32] 128 -24

[22 -112 576] 128 12

[22 -288 3776] 128 -12

[18 -256 3648] 128 0

[12 16 32] 128 0

[12 -64 352] 128 12

[68 -885 11520] 135 41

[34 -171 864] 135 31

[24 -267 2976] 135 -53

[8 11 32] 135 41

[72 -793 8736] 143 -27

[42 121 352] 143 -13

[6 7 32] 143 -27

[6 25 128] 143 40

[18 -199 2208] 143 40

[12 -25 64] 143 13

[76 -685 6176] 151 14

[38 -115 352] 151 -19

[8 19 64] 151 19

[10 13 32] 151 -14

[78 -234 704] 156 -28

[78 -390 1952] 156 -28

[40 -202 1024] 156 18

[40 -522 6816] 156 -18

[6 6 32] 156 28

[8 10 32] 156 28

[10 22 64] 156 -18

[80 -561 3936] 159 -72

[46 -143 448] 159 -1

[28 -143 736] 159 -1

[12 15 32] 159 -72

[84 -421 2112] 167 34

[42 -379 3424] 167 -2

[28 -421 6336] 167 34

[14 -197 2784] 167 2

[16 -37 96] 167 -7

[88 -265 800] 175 -81

[4 9 64] 175 -126

[8 9 32] 175 81

[2 4 96] 176 -49

[90 268 800] 176 -49

[30 -92 288] 176 -17

[30 -332 3680] 176 17

[30 -452 6816] 176 49

[10 12 32] 176 -49

[18 -92 480] 176 -17

[2 3 96] 183 125

[6 3 32] 183 125

[38 -125 416] 183 -73

[16 -227 3232] 183 12

[2 2 96] 188 126

[94 -658 4608] 188 126

[48 -146 448] 188 -20

[48 -530 5856] 188 20

[6 2 32] 188 126

[6 14 64] 188 20

[142 350 864] 188 -20

[8 18 64] 188 20

[12 14 32] 188 126

[14 34 96] 188 20

[2 1 96] 191 143

[6 1 32] 191 143

[32 -417 5440] 191 -60

[8 33 160] 191 -11

[20 -223 2496] 191 -11

[18 -95 512] 191 60

[18 -257 3680] 191 -14

[2 0 96] 192 0

[2 8 128] 192 96

[2 16 224] 192 0

[4 8 64] 192 -112

[52 152 448] 192 -112

[6 0 32] 192 0

[6 24 128] 192 -8

[8 8 32] 192 96

[24 -120 608] 192 8

[12 24 64] 192 -112

[14 16 32] 192 0

[100 -1301 16928] 199 -219

[50 -651 8480] 199 110

[10 11 32] 199 -219

[10 21 64] 199 110

[104 -1145 12608] 207 139

[478 1113 2592] 207 -94

[26 -391 5888] 207 -139

[108 -973 8768] 215 -75

[54 -595 6560] 215 79

[36 -397 4384] 215 202

[6 13 64] 215 79

[12 13 32] 215 75

[110 -330 992] 220 152

[110 -550 2752] 220 152

[4 6 64] 220 202

[56 -506 4576] 220 -202

[28 -422 6368] 220 152

[10 10 32] 220 -152

[170 390 896] 220 -202

[112 -785 5504] 223 412

[4 17 128] 223 -61

[8 17 64] 223 -61

[14 15 32] 223 412

[116 -581 2912] 231 67

[4 5 64] 231 163

[40 -123 384] 231 58

[30 -453 6848] 231 67

[24 -123 640] 231 58

[120 -361 1088] 239 243

[248 617 1536] 239 340

[48 -151 480] 239 73

[30 -151 768] 239 73

[24 -361 5440] 239 243

[198 457 1056] 239 340

[2 4 128] 240 28

[122 364 1088] 240 28

[4 4 64] 240 -44

[60 -180 544] 240 -196

[60 -300 1504] 240 -196

[6 12 64] 240 196

[8 4 32] 240 28

[32 -356 3968] 240 -84

[10 20 64] 240 196

[12 12 32] 240 28

[16 -36 96] 240 -196

[16 -228 3264] 240 -44

[2 3 128] 247 -499

[62 -931 13984] 247 42

[8 3 32] 247 -499

[2 2 128] 252 -210

[126 -882 6176] 252 -210

[4 2 64] 252 -4

[42 -126 384] 252 -32

[42 -210 1056] 252 -32

[8 2 32] 252 -210

[24 -126 672] 252 -32

[14 14 32] 252 -210

[18 -258 3712] 252 -4

[2 1 128] 255 -334

[4 1 64] 255 -28

[44 -575 7520] 255 -93

[8 1 32] 255 -334

[14 33 96] 255 -93

[2 0 128] 256 0

[2 8 160] 256 -32

[4 0 64] 256 0

[4 16 128] 256 -192

[8 0 32] 256 0

[8 16 64] 256 -192

[8 32 160] 256 176

[26 -288 3200] 256 176

[26 -392 5920] 256 32

[16 16 32] 256 0

[20 -288 4160] 256 0

[132 -1717 22336] 263 504

[66 -331 1664] 263 -442

[408 949 2208] 263 -442

[22 -331 4992] 263 -504

[136 -1497 16480] 271 -276

[74 217 640] 271 126

[28 -423 6400] 271 276

[212 487 1120] 271 -126

[140 -1261 11360] 279 151

[70 -211 640] 279 -101

[268 621 1440] 279 -101

[20 -301 4544] 279 151

[142 -426 1280] 284 -334

[142 -710 3552] 284 -334

[72 -362 1824] 284 66

[72 -938 12224] 284 -66

[450 1046 2432] 284 66

[6 26 160] 284 -50

[58 -182 576] 284 50

[36 -182 928] 284 50

[30 -154 800] 284 50

[30 -454 6880] 284 -334

[24 -362 5472] 284 -334

[240 554 1280] 284 66

[78 -239 736] 287 282

[48 -241 1216] 287 -416

[338 785 1824] 287 282

[18 -271 4096] 287 1052

[148 -741 3712] 295 -684

[38 -421 4672] 295 -392

[10 5 32] 295 684

[226 517 1184] 295 -528

[152 -457 1376] 303 -381

[4 9 96] 303 482

[6 9 64] 303 482

[26 -393 5952] 303 -381

[2 4 160] 304 365

[154 460 1376] 304 365

[10 4 32] 304 365

[22 -332 5024] 304 -365

[2 3 160] 311 1033

[52 -157 480] 311 422

[10 3 32] 311 1033

[12 29 96] 311 -422

[16 35 96] 311 956

[240 547 1248] 311 443

[2 2 160] 316 -368

[80 -242 736] 316 110

[80 -882 9728] 316 -110

[380 882 2048] 316 110

[10 2 32] 316 -368

[310 718 1664] 316 110

[20 -302 4576] 316 368

[2 1 160] 319 116

[50 -159 512] 319 -89

[10 1 32] 319 116

[32 -353 3904] 319 89

[20 -289 4192] 319 -425

[2 0 160] 320 0

[2 8 192] 320 -824

[4 8 96] 320 416

[84 248 736] 320 416

[6 8 64] 320 416

[54 -272 1376] 320 384

[54 -704 9184] 320 -384

[10 0 32] 320 0

[28 -424 6432] 320 824

[14 32 96] 320 -384

[254 584 1344] 320 -416

[18 -272 4128] 320 0

[164 -2133 27744] 327 -72

[82 -1067 13888] 327 -618

[56 -619 6848] 327 -200

[42 -213 1088] 327 200

[282 651 1504] 327 618

[24 -363 5504] 327 72

[168 -1849 20352] 335 -215

[766 1785 4160] 335 1074

[6 7 64] 335 -1074

[6 25 160] 335 227

[36 -185 960] 335 -227

[30 -455 6912] 335 215

[86 -947 10432] 343 -603

[422 979 2272] 343 603

[22 -333 5056] 343 196

[174 -522 1568] 348 -186

[174 -870 4352] 348 -186

[4 6 96] 348 -998

[6 6 64] 348 -998

[44 -486 5376] 348 170

[12 6 32] 348 186

[26 -394 5984] 348 -186

[268 614 1408] 348 998

[4 17 160] 351 465

[60 -303 1536] 351 2528

[352 815 1888] 351 -465

[20 -303 4608] 351 -316

[180 -901 4512] 359 1478

[4 5 96] 359 -1072

[60 -901 13536] 359 1072

[12 5 32] 359 -1478

[184 -553 1664] 367 237

[376 937 2336] 367 -1543

[28 -425 6464] 367 237

[296 681 1568] 367 -1543

[2 4 192] 368 -932

[4 4 96] 368 228

[92 -276 832] 368 788

[92 -460 2304] 368 788

[62 -188 576] 368 -636

[62 -684 7552] 368 636

[62 -932 14016] 368 -228

[8 12 64] 368 -788

[48 -244 1248] 368 -636

[12 4 32] 368 -932

[12 28 96] 368 636

[324 748 1728] 368 788

[24 -364 5536] 368 932

[282 644 1472] 368 -228

[2 3 192] 375 -283

[94 -1411 21184] 375 -733

[6 3 64] 375 733

[12 3 32] 375 -283

[2 2 192] 380 1900

[4 2 96] 380 -16

[6 2 64] 380 -16

[6 14 96] 380 -314

[60 -190 608] 380 142

[38 -190 960] 380 142

[12 2 32] 380 1900

[16 34 96] 380 -314

[22 -334 5088] 380 -1900

[20 30 64] 380 -16

[2 1 192] 383 961

[4 1 96] 383 624

[6 1 64] 383 624

[64 -833 10848] 383 -672

[12 1 32] 383 961

[14 31 96] 383 -672

[2 0 192] 384 0

[2 8 224] 384 1872

[4 0 96] 384 0

[4 16 160] 384 832

[6 0 64] 384 0

[6 24 160] 384 -848

[10 24 96] 384 -848

[394 912 2112] 384 -832

[12 0 32] 384 0

[30 -456 6944] 384 -1872

[20 -304 4640] 384 0

[22 32 64] 384 0

[98 -491 2464] 391 2031

[8 11 64] 391 -2031

[26 -395 6016] 391 1840

[200 -2201 24224] 399 1771

[50 -551 6080] 399 170

[14 7 32] 399 1771

[310 711 1632] 399 339

[102 -307 928] 407 43

[68 -749 8256] 407 1470

[6 13 96] 407 -3226

[8 19 96] 407 1470

[366 845 1952] 407 43

[24 -365 5568] 407 -1514

[206 -1030 5152] 412 2422

[104 -522 2624] 412 -310

[104 -1354 17632] 412 310

[8 10 64] 412 310

[14 6 32] 412 -2422

[338 778 1792] 412 -310

[28 -426 6496] 412 2422

[110 -335 1024] 415 -1106

[10 15 64] 415 1106

[22 -335 5120] 415 -4630

[212 -1061 5312] 423 190

[72 -219 672] 423 -3484

[12 27 96] 423 3484

[14 5 32] 423 -190

[18 27 64] 423 -2788

[4 9 128] 431 212

[80 -247 768] 431 3081

[8 9 64] 431 212

[10 23 96] 431 -3081

[30 -457 6976] 431 323

[2 4 224] 432 -308

[6 12 96] 432 -1589

[14 4 32] 432 -308

[26 -396 6048] 432 308

[2 3 224] 439 -4140

[70 -221 704] 439 1651

[44 -221 1120] 439 1651

[14 3 32] 439 -4140

[20 29 64] 439 2111

[2 2 224] 444 -1500

[112 -338 1024] 444 -666

[112 -1234 13600] 444 666

[74 -222 672] 444 222

[74 -370 1856] 444 222

[8 18 96] 444 -222

[10 14 64] 444 666

[12 18 64] 444 666

[14 2 32] 444 -1500

[14 30 96] 444 -222

[24 -366 5600] 444 1500

[2 1 224] 447 -1524

[14 1 32] 447 -1524

[16 33 96] 447 2892

[22 31 64] 447 2248

[2 0 224] 448 0

[4 8 128] 448 128

[8 8 64] 448 128

[56 -280 1408] 448 320

[14 0 32] 448 0

[16 8 32] 448 1536

[16 24 64] 448 128

[22 -336 5152] 448 0

[114 -1483 19296] 455 543

[696 1621 3776] 455 -3634

[14 21 64] 455 543

[28 -427 6528] 455 -2281

[1054 2457 5728] 463 -4342

[58 -871 13088] 463 -4342

[16 7 32] 463 -1473

[118 -1299 14304] 471 739

[10 13 64] 471 739

[26 -397 6080] 471 2024

[4 6 128] 476 1310

[738 1718 4000] 476 -1692

[6 26 192] 476 574

[90 -278 864] 476 -574

[60 -902 13568] 476 -1310

[10 22 96] 476 574

[12 26 96] 476 574

[16 6 32] 476 3762

[18 26 64] 476 1310

[30 -458 7008] 476 -3762

[4 17 192] 479 357

[80 -401 2016] 479 -3727

[8 17 96] 479 3727

[12 17 64] 479 357

[24 -367 5632] 479 9085

[4 5 128] 487 1329

[62 -933 14048] 487 -1329

[16 5 32] 487 5535

[6 9 96] 495 3566

[62 -311 1568] 495 852

[16 23 64] 495 -1894

[18 9 32] 495 2193

[4 4 128] 496 -1240

[124 -372 1120] 496 1304

[124 -620 3104] 496 1304

[8 4 64] 496 -1240

[64 -708 7840] 496 -712

[10 12 64] 496 -1304

[50 -252 1280] 496 712

[14 20 64] 496 -1304

[16 4 32] 496 3880

[20 28 64] 496 -1240

[28 -428 6560] 496 -3880

	φ	= TB(9;−1,2,2,3,4)
		= 1⁻¹2²3¹4¹
		= η(τ)¹⁸(θ(τ,z)/η(τ))⁻¹(θ(τ,2z)/η(τ))²(θ(τ,3z)/η(τ))(θ(τ,4z)/η(τ))

2u	det(2u)	a(u, Borch(ψ))
[4 -53 704]	7	0
[8 -89 992]	15	0
[4 -25 160]	15	0
[12 -109 992]	23	0
[6 -13 32]	23	0
[6 -19 64]	23	0
[14 -42 128]	28	0
[14 -70 352]	28	0
[8 -42 224]	28	0
[8 -106 1408]	28	0
[16 -113 800]	31	0
[8 -113 1600]	31	0
[14 -47 160]	31	0
[20 -101 512]	39	1
[10 -91 832]	39	0
[10 -101 1024]	39	0
[8 -27 96]	39	0
[24 -73 224]	47	-1
[4 9 32]	47	1
[16 -55 192]	47	0
[2 4 32]	48	-1
[26 76 224]	48	-1
[6 12 32]	48	-1
[14 36 96]	48	-1
[2 3 32]	55	1
[14 -29 64]	55	0
[14 -99 704]	55	0
[8 -35 160]	55	0
[2 2 32]	60	2
[30 -210 1472]	60	2
[16 -50 160]	60	0
[16 -178 1984]	60	0
[10 -30 96]	60	0
[10 -50 256]	60	0
[8 -114 1632]	60	2
[2 1 32]	63	2
[16 -225 3168]	63	0
[12 -159 2112]	63	1
[8 -97 1184]	63	0
[2 0 32]	64	0
[2 8 64]	64	2
[2 16 160]	64	0
[4 8 32]	64	2
[20 56 160]	64	2
[10 -24 64]	64	-2
[10 -144 2080]	64	0
[36 -469 6112]	71	-3
[18 -235 3072]	71	-2
[120 277 640]	71	3
[10 -53 288]	71	2
[40 -441 4864]	79	2
[190 441 1024]	79	-2
[44 -397 3584]	87	-1
[22 -243 2688]	87	12
[134 307 704]	87	-1
[46 -138 416]	92	2
[46 -230 1152]	92	2
[4 6 32]	92	-2
[24 -218 1984]	92	2
[162 374 864]	92	2
[6 26 128]	92	0
[26 -86 288]	92	0
[12 -26 64]	92	2
[12 -122 1248]	92	-2
[12 -134 1504]	92	0
[48 -337 2368]	95	5
[4 17 96]	95	-11
[16 -81 416]	95	11
[10 15 32]	95	5
[52 -261 1312]	103	-11
[4 5 32]	103	11
[56 -169 512]	111	14
[120 297 736]	111	-9
[6 9 32]	111	-14
[10 23 64]	111	9
[2 4 64]	112	12
[58 172 512]	112	12
[4 4 32]	112	12
[28 -84 256]	112	-4
[28 -140 704]	112	-4
[28 -196 1376]	112	12
[14 -28 64]	112	-12
[8 12 32]	112	12
[16 -84 448]	112	-4
[2 3 64]	119	-17
[30 -451 6784]	119	17
[20 -61 192]	119	-23
[90 221 544]	119	9
[12 -61 320]	119	-23
[12 -125 1312]	119	-9
[2 2 64]	124	-26
[62 -434 3040]	124	-26
[4 2 32]	124	-26
[16 -226 3200]	124	-26
[8 34 160]	124	0
[28 -94 320]	124	0
[10 14 32]	124	-26
[14 -158 1792]	124	0
[2 1 64]	127	-27
[4 1 32]	127	-27
[2 0 64]	128	0
[2 8 96]	128	-24
[2 16 192]	128	0
[4 0 32]	128	0
[4 16 96]	128	-12
[6 8 32]	128	-24
[22 -112 576]	128	12
[22 -288 3776]	128	-12
[18 -256 3648]	128	0
[12 16 32]	128	0
[12 -64 352]	128	12
[68 -885 11520]	135	41
[34 -171 864]	135	31
[24 -267 2976]	135	-53
[8 11 32]	135	41
[72 -793 8736]	143	-27
[42 121 352]	143	-13
[6 7 32]	143	-27
[6 25 128]	143	40
[18 -199 2208]	143	40
[12 -25 64]	143	13
[76 -685 6176]	151	14
[38 -115 352]	151	-19
[8 19 64]	151	19
[10 13 32]	151	-14
[78 -234 704]	156	-28
[78 -390 1952]	156	-28
[40 -202 1024]	156	18
[40 -522 6816]	156	-18
[6 6 32]	156	28
[8 10 32]	156	28
[10 22 64]	156	-18
[80 -561 3936]	159	-72
[46 -143 448]	159	-1
[28 -143 736]	159	-1
[12 15 32]	159	-72
[84 -421 2112]	167	34
[42 -379 3424]	167	-2
[28 -421 6336]	167	34
[14 -197 2784]	167	2
[16 -37 96]	167	-7
[88 -265 800]	175	-81
[4 9 64]	175	-126
[8 9 32]	175	81
[2 4 96]	176	-49
[90 268 800]	176	-49
[30 -92 288]	176	-17
[30 -332 3680]	176	17
[30 -452 6816]	176	49
[10 12 32]	176	-49
[18 -92 480]	176	-17
[2 3 96]	183	125
[6 3 32]	183	125
[38 -125 416]	183	-73
[16 -227 3232]	183	12
[2 2 96]	188	126
[94 -658 4608]	188	126
[48 -146 448]	188	-20
[48 -530 5856]	188	20
[6 2 32]	188	126
[6 14 64]	188	20
[142 350 864]	188	-20
[8 18 64]	188	20
[12 14 32]	188	126
[14 34 96]	188	20
[2 1 96]	191	143
[6 1 32]	191	143
[32 -417 5440]	191	-60
[8 33 160]	191	-11
[20 -223 2496]	191	-11
[18 -95 512]	191	60
[18 -257 3680]	191	-14
[2 0 96]	192	0
[2 8 128]	192	96
[2 16 224]	192	0
[4 8 64]	192	-112
[52 152 448]	192	-112
[6 0 32]	192	0
[6 24 128]	192	-8
[8 8 32]	192	96
[24 -120 608]	192	8
[12 24 64]	192	-112
[14 16 32]	192	0
[100 -1301 16928]	199	-219
[50 -651 8480]	199	110
[10 11 32]	199	-219
[10 21 64]	199	110
[104 -1145 12608]	207	139
[478 1113 2592]	207	-94
[26 -391 5888]	207	-139
[108 -973 8768]	215	-75
[54 -595 6560]	215	79
[36 -397 4384]	215	202
[6 13 64]	215	79
[12 13 32]	215	75
[110 -330 992]	220	152
[110 -550 2752]	220	152
[4 6 64]	220	202
[56 -506 4576]	220	-202
[28 -422 6368]	220	152
[10 10 32]	220	-152
[170 390 896]	220	-202
[112 -785 5504]	223	412
[4 17 128]	223	-61
[8 17 64]	223	-61
[14 15 32]	223	412
[116 -581 2912]	231	67
[4 5 64]	231	163
[40 -123 384]	231	58
[30 -453 6848]	231	67
[24 -123 640]	231	58
[120 -361 1088]	239	243
[248 617 1536]	239	340
[48 -151 480]	239	73
[30 -151 768]	239	73
[24 -361 5440]	239	243
[198 457 1056]	239	340
[2 4 128]	240	28
[122 364 1088]	240	28
[4 4 64]	240	-44
[60 -180 544]	240	-196
[60 -300 1504]	240	-196
[6 12 64]	240	196
[8 4 32]	240	28
[32 -356 3968]	240	-84
[10 20 64]	240	196
[12 12 32]	240	28
[16 -36 96]	240	-196
[16 -228 3264]	240	-44
[2 3 128]	247	-499
[62 -931 13984]	247	42
[8 3 32]	247	-499
[2 2 128]	252	-210
[126 -882 6176]	252	-210
[4 2 64]	252	-4
[42 -126 384]	252	-32
[42 -210 1056]	252	-32
[8 2 32]	252	-210
[24 -126 672]	252	-32
[14 14 32]	252	-210
[18 -258 3712]	252	-4
[2 1 128]	255	-334
[4 1 64]	255	-28
[44 -575 7520]	255	-93
[8 1 32]	255	-334
[14 33 96]	255	-93
[2 0 128]	256	0
[2 8 160]	256	-32
[4 0 64]	256	0
[4 16 128]	256	-192
[8 0 32]	256	0
[8 16 64]	256	-192
[8 32 160]	256	176
[26 -288 3200]	256	176
[26 -392 5920]	256	32
[16 16 32]	256	0
[20 -288 4160]	256	0
[132 -1717 22336]	263	504
[66 -331 1664]	263	-442
[408 949 2208]	263	-442
[22 -331 4992]	263	-504
[136 -1497 16480]	271	-276
[74 217 640]	271	126
[28 -423 6400]	271	276
[212 487 1120]	271	-126
[140 -1261 11360]	279	151
[70 -211 640]	279	-101
[268 621 1440]	279	-101
[20 -301 4544]	279	151
[142 -426 1280]	284	-334
[142 -710 3552]	284	-334
[72 -362 1824]	284	66
[72 -938 12224]	284	-66
[450 1046 2432]	284	66
[6 26 160]	284	-50
[58 -182 576]	284	50
[36 -182 928]	284	50
[30 -154 800]	284	50
[30 -454 6880]	284	-334
[24 -362 5472]	284	-334
[240 554 1280]	284	66
[78 -239 736]	287	282
[48 -241 1216]	287	-416
[338 785 1824]	287	282
[18 -271 4096]	287	1052
[148 -741 3712]	295	-684
[38 -421 4672]	295	-392
[10 5 32]	295	684
[226 517 1184]	295	-528
[152 -457 1376]	303	-381
[4 9 96]	303	482
[6 9 64]	303	482
[26 -393 5952]	303	-381
[2 4 160]	304	365
[154 460 1376]	304	365
[10 4 32]	304	365
[22 -332 5024]	304	-365
[2 3 160]	311	1033
[52 -157 480]	311	422
[10 3 32]	311	1033
[12 29 96]	311	-422
[16 35 96]	311	956
[240 547 1248]	311	443
[2 2 160]	316	-368
[80 -242 736]	316	110
[80 -882 9728]	316	-110
[380 882 2048]	316	110
[10 2 32]	316	-368
[310 718 1664]	316	110
[20 -302 4576]	316	368
[2 1 160]	319	116
[50 -159 512]	319	-89
[10 1 32]	319	116
[32 -353 3904]	319	89
[20 -289 4192]	319	-425
[2 0 160]	320	0
[2 8 192]	320	-824
[4 8 96]	320	416
[84 248 736]	320	416
[6 8 64]	320	416
[54 -272 1376]	320	384
[54 -704 9184]	320	-384
[10 0 32]	320	0
[28 -424 6432]	320	824
[14 32 96]	320	-384
[254 584 1344]	320	-416
[18 -272 4128]	320	0
[164 -2133 27744]	327	-72
[82 -1067 13888]	327	-618
[56 -619 6848]	327	-200
[42 -213 1088]	327	200
[282 651 1504]	327	618
[24 -363 5504]	327	72
[168 -1849 20352]	335	-215
[766 1785 4160]	335	1074
[6 7 64]	335	-1074
[6 25 160]	335	227
[36 -185 960]	335	-227
[30 -455 6912]	335	215
[86 -947 10432]	343	-603
[422 979 2272]	343	603
[22 -333 5056]	343	196
[174 -522 1568]	348	-186
[174 -870 4352]	348	-186
[4 6 96]	348	-998
[6 6 64]	348	-998
[44 -486 5376]	348	170
[12 6 32]	348	186
[26 -394 5984]	348	-186
[268 614 1408]	348	998
[4 17 160]	351	465
[60 -303 1536]	351	2528
[352 815 1888]	351	-465
[20 -303 4608]	351	-316
[180 -901 4512]	359	1478
[4 5 96]	359	-1072
[60 -901 13536]	359	1072
[12 5 32]	359	-1478
[184 -553 1664]	367	237
[376 937 2336]	367	-1543
[28 -425 6464]	367	237
[296 681 1568]	367	-1543
[2 4 192]	368	-932
[4 4 96]	368	228
[92 -276 832]	368	788
[92 -460 2304]	368	788
[62 -188 576]	368	-636
[62 -684 7552]	368	636
[62 -932 14016]	368	-228
[8 12 64]	368	-788
[48 -244 1248]	368	-636
[12 4 32]	368	-932
[12 28 96]	368	636
[324 748 1728]	368	788
[24 -364 5536]	368	932
[282 644 1472]	368	-228
[2 3 192]	375	-283
[94 -1411 21184]	375	-733
[6 3 64]	375	733
[12 3 32]	375	-283
[2 2 192]	380	1900
[4 2 96]	380	-16
[6 2 64]	380	-16
[6 14 96]	380	-314
[60 -190 608]	380	142
[38 -190 960]	380	142
[12 2 32]	380	1900
[16 34 96]	380	-314
[22 -334 5088]	380	-1900
[20 30 64]	380	-16
[2 1 192]	383	961
[4 1 96]	383	624
[6 1 64]	383	624
[64 -833 10848]	383	-672
[12 1 32]	383	961
[14 31 96]	383	-672
[2 0 192]	384	0
[2 8 224]	384	1872
[4 0 96]	384	0
[4 16 160]	384	832
[6 0 64]	384	0
[6 24 160]	384	-848
[10 24 96]	384	-848
[394 912 2112]	384	-832
[12 0 32]	384	0
[30 -456 6944]	384	-1872
[20 -304 4640]	384	0
[22 32 64]	384	0
[98 -491 2464]	391	2031
[8 11 64]	391	-2031
[26 -395 6016]	391	1840
[200 -2201 24224]	399	1771
[50 -551 6080]	399	170
[14 7 32]	399	1771
[310 711 1632]	399	339
[102 -307 928]	407	43
[68 -749 8256]	407	1470
[6 13 96]	407	-3226
[8 19 96]	407	1470
[366 845 1952]	407	43
[24 -365 5568]	407	-1514
[206 -1030 5152]	412	2422
[104 -522 2624]	412	-310
[104 -1354 17632]	412	310
[8 10 64]	412	310
[14 6 32]	412	-2422
[338 778 1792]	412	-310
[28 -426 6496]	412	2422
[110 -335 1024]	415	-1106
[10 15 64]	415	1106
[22 -335 5120]	415	-4630
[212 -1061 5312]	423	190
[72 -219 672]	423	-3484
[12 27 96]	423	3484
[14 5 32]	423	-190
[18 27 64]	423	-2788
[4 9 128]	431	212
[80 -247 768]	431	3081
[8 9 64]	431	212
[10 23 96]	431	-3081
[30 -457 6976]	431	323
[2 4 224]	432	-308
[6 12 96]	432	-1589
[14 4 32]	432	-308
[26 -396 6048]	432	308
[2 3 224]	439	-4140
[70 -221 704]	439	1651
[44 -221 1120]	439	1651
[14 3 32]	439	-4140
[20 29 64]	439	2111
[2 2 224]	444	-1500
[112 -338 1024]	444	-666
[112 -1234 13600]	444	666
[74 -222 672]	444	222
[74 -370 1856]	444	222
[8 18 96]	444	-222
[10 14 64]	444	666
[12 18 64]	444	666
[14 2 32]	444	-1500
[14 30 96]	444	-222
[24 -366 5600]	444	1500
[2 1 224]	447	-1524
[14 1 32]	447	-1524
[16 33 96]	447	2892
[22 31 64]	447	2248
[2 0 224]	448	0
[4 8 128]	448	128
[8 8 64]	448	128
[56 -280 1408]	448	320
[14 0 32]	448	0
[16 8 32]	448	1536
[16 24 64]	448	128
[22 -336 5152]	448	0
[114 -1483 19296]	455	543
[696 1621 3776]	455	-3634
[14 21 64]	455	543
[28 -427 6528]	455	-2281
[1054 2457 5728]	463	-4342
[58 -871 13088]	463	-4342
[16 7 32]	463	-1473
[118 -1299 14304]	471	739
[10 13 64]	471	739
[26 -397 6080]	471	2024
[4 6 128]	476	1310
[738 1718 4000]	476	-1692
[6 26 192]	476	574
[90 -278 864]	476	-574
[60 -902 13568]	476	-1310
[10 22 96]	476	574
[12 26 96]	476	574
[16 6 32]	476	3762
[18 26 64]	476	1310
[30 -458 7008]	476	-3762
[4 17 192]	479	357
[80 -401 2016]	479	-3727
[8 17 96]	479	3727
[12 17 64]	479	357
[24 -367 5632]	479	9085
[4 5 128]	487	1329
[62 -933 14048]	487	-1329
[16 5 32]	487	5535
[6 9 96]	495	3566
[62 -311 1568]	495	852
[16 23 64]	495	-1894
[18 9 32]	495	2193
[4 4 128]	496	-1240
[124 -372 1120]	496	1304
[124 -620 3104]	496	1304
[8 4 64]	496	-1240
[64 -708 7840]	496	-712
[10 12 64]	496	-1304
[50 -252 1280]	496	712
[14 20 64]	496	-1304
[16 4 32]	496	3880
[20 28 64]	496	-1240
[28 -428 6560]	496	-3880

A Borcherds Product in S9(K(16))−

A Borcherds Product in S₉(K(16))⁻