A Borcherds Product in S

A Borcherds Product in S₉(K(16))⁻

(Filename: BP-9-16-1-0-8.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(2/ζ^9 - ζ^(-8) - ζ^8 + 2*ζ^9)
+q^3(10/ζ^14 + 10*ζ^14)
+q^4(4/ζ^16 + 4*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) 11

(9,3) 1

(16,4) 1

(17,9) 2
From the singular part above, we compute that the valuation Ord(ψ) is -0.265625, 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)	11
(9,3)	1
(16,4)	1
(17,9)	2

THE VECTOR c (return to top)

The vector c has components given by the following table.

c₁	-778533934574263/608069329798730958
c₂	660607134625553819/1182086777128732982352
c₃	-788990091541240293911/502682401973993700745188
c₄	50080267267335201518/13963400054833158354033
c₅	102915904044250558609/27926800109666316708066
c₆	877034516878225219265/502682401973993700745188
c₇	788990091541240293911/502682401973993700745188
c₈	-33808100906872181/1329847624269824605146
c₉	-289770218778786023/11968628618428421446314
c₁₀	-1986820441283079443/7979085745618947630876
c₁₁	1657595117298415/73880423570545811397
c₁₂	6908205884362750753/17952942927642632169471

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(218 + 2/ζ^9 - ζ^(-8) + 28/ζ^7 - 22/ζ^6 - 61/ζ^5 + 20/ζ^4 + 19/ζ^3 - 106/ζ^2 + 12/ζ + 12*ζ - 106*ζ^2 + 19*ζ^3 + 20*ζ^4 - 61*ζ^5 - 22*ζ^6 + 28*ζ^7 - ζ^8 + 2*ζ^9)
+q^2(1548 + 11/ζ^11 + 10/ζ^10 - 41/ζ^9 + 42/ζ^8 + 297/ζ^7 - 168/ζ^6 - 586/ζ^5 + 208/ζ^4 + 245/ζ^3 - 866/ζ^2 + 74/ζ + 74*ζ - 866*ζ^2 + 245*ζ^3 + 208*ζ^4 - 586*ζ^5 - 168*ζ^6 + 297*ζ^7 + 42*ζ^8 - 41*ζ^9 + 10*ζ^10 + 11*ζ^11)
+q^3(7976 + 10/ζ^14 - 21/ζ^13 - 84/ζ^12 + 109/ζ^11 + 144/ζ^10 - 452/ζ^9 + 380/ζ^8 + 2055/ζ^7 - 964/ζ^6 - 3436/ζ^5 + 1348/ζ^4 + 1400/ζ^3 - 4822/ζ^2 + 345/ζ + 345*ζ - 4822*ζ^2 + 1400*ζ^3 + 1348*ζ^4 - 3436*ζ^5 - 964*ζ^6 + 2055*ζ^7 + 380*ζ^8 - 452*ζ^9 + 144*ζ^10 + 109*ζ^11 - 84*ζ^12 - 21*ζ^13 + 10*ζ^14)
+q^4(33704 + 4/ζ^16 + 8/ζ^15 + 118/ζ^14 - 265/ζ^13 - 721/ζ^12 + 812/ζ^11 + 916/ζ^10 - 2737/ζ^9 + 1960/ζ^8 + 10129/ζ^7 - 4506/ζ^6 - 15607/ζ^5 + 6481/ζ^4 + 6602/ζ^3 - 21104/ζ^2 + 1058/ζ + 1058*ζ - 21104*ζ^2 + 6602*ζ^3 + 6481*ζ^4 - 15607*ζ^5 - 4506*ζ^6 + 10129*ζ^7 + 1960*ζ^8 - 2737*ζ^9 + 916*ζ^10 + 812*ζ^11 - 721*ζ^12 - 265*ζ^13 + 118*ζ^14 + 8*ζ^15 + 4*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 -1

[24 -73 224] 47 -1

[4 9 32] 47 1

[16 -55 192] 47 1

[2 4 32] 48 -1

[26 76 224] 48 -1

[6 12 32] 48 -1

[14 36 96] 48 0

[2 3 32] 55 1

[14 -29 64] 55 0

[14 -99 704] 55 0

[8 -35 160] 55 -1

[2 2 32] 60 2

[30 -210 1472] 60 2

[16 -50 160] 60 2

[16 -178 1984] 60 2

[10 -30 96] 60 1

[10 -50 256] 60 2

[8 -114 1632] 60 2

[2 1 32] 63 2

[16 -225 3168] 63 0

[12 -159 2112] 63 0

[8 -97 1184] 63 -2

[2 0 32] 64 0

[2 8 64] 64 2

[2 16 160] 64 0

[4 8 32] 64 2

[20 56 160] 64 0

[10 -24 64] 64 0

[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 0

[134 307 704] 87 -1

[46 -138 416] 92 2

[46 -230 1152] 92 2

[4 6 32] 92 -2

[24 -218 1984] 92 -10

[162 374 864] 92 2

[6 26 128] 92 0

[26 -86 288] 92 -3

[12 -26 64] 92 -10

[12 -122 1248] 92 0

[12 -134 1504] 92 0

[48 -337 2368] 95 5

[4 17 96] 95 -3

[16 -81 416] 95 3

[10 15 32] 95 5

[52 -261 1312] 103 -11

[4 5 32] 103 11

[56 -169 512] 111 14

[120 297 736] 111 3

[6 9 32] 111 -14

[10 23 64] 111 -3

[2 4 64] 112 12

[58 172 512] 112 12

[4 4 32] 112 12

[28 -84 256] 112 6

[28 -140 704] 112 -22

[28 -196 1376] 112 -16

[14 -28 64] 112 16

[8 12 32] 112 12

[16 -84 448] 112 6

[2 3 64] 119 -17

[30 -451 6784] 119 17

[20 -61 192] 119 -21

[90 221 544] 119 18

[12 -61 320] 119 -21

[12 -125 1312] 119 -18

[2 2 64] 124 -26

[62 -434 3040] 124 -26

[4 2 32] 124 -26

[16 -226 3200] 124 2

[8 34 160] 124 -8

[28 -94 320] 124 -20

[10 14 32] 124 -26

[14 -158 1792] 124 -8

[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 -18

[6 8 32] 128 -24

[22 -112 576] 128 18

[22 -288 3776] 128 -2

[18 -256 3648] 128 0

[12 16 32] 128 0

[12 -64 352] 128 2

[68 -885 11520] 135 41

[34 -171 864] 135 3

[24 -267 2976] 135 -45

[8 11 32] 135 41

[72 -793 8736] 143 -27

[42 121 352] 143 -17

[6 7 32] 143 -27

[6 25 128] 143 27

[18 -199 2208] 143 27

[12 -25 64] 143 17

[76 -685 6176] 151 14

[38 -115 352] 151 -7

[8 19 64] 151 7

[10 13 32] 151 -14

[78 -234 704] 156 -28

[78 -390 1952] 156 -28

[40 -202 1024] 156 116

[40 -522 6816] 156 44

[6 6 32] 156 28

[8 10 32] 156 28

[10 22 64] 156 44

[80 -561 3936] 159 -72

[46 -143 448] 159 3

[28 -143 736] 159 3

[12 15 32] 159 -72

[84 -421 2112] 167 34

[42 -379 3424] 167 -46

[28 -421 6336] 167 34

[14 -197 2784] 167 46

[16 -37 96] 167 10

[88 -265 800] 175 -81

[4 9 64] 175 -102

[8 9 32] 175 81

[2 4 96] 176 -49

[90 268 800] 176 -49

[30 -92 288] 176 -7

[30 -332 3680] 176 42

[30 -452 6816] 176 49

[10 12 32] 176 -49

[18 -92 480] 176 -7

[2 3 96] 183 125

[6 3 32] 183 125

[38 -125 416] 183 78

[16 -227 3232] 183 -12

[2 2 96] 188 126

[94 -658 4608] 188 126

[48 -146 448] 188 -74

[48 -530 5856] 188 -10

[6 2 32] 188 126

[6 14 64] 188 -10

[142 350 864] 188 -87

[8 18 64] 188 74

[12 14 32] 188 126

[14 34 96] 188 87

[2 1 96] 191 143

[6 1 32] 191 143

[32 -417 5440] 191 -70

[8 33 160] 191 72

[20 -223 2496] 191 72

[18 -95 512] 191 70

[18 -257 3680] 191 -70

[2 0 96] 192 0

[2 8 128] 192 96

[2 16 224] 192 0

[4 8 64] 192 40

[52 152 448] 192 -184

[6 0 32] 192 0

[6 24 128] 192 -28

[8 8 32] 192 96

[24 -120 608] 192 28

[12 24 64] 192 -184

[14 16 32] 192 0

[100 -1301 16928] 199 -219

[50 -651 8480] 199 42

[10 11 32] 199 -219

[10 21 64] 199 42

[104 -1145 12608] 207 139

[478 1113 2592] 207 -6

[26 -391 5888] 207 -139

[108 -973 8768] 215 -75

[54 -595 6560] 215 179

[36 -397 4384] 215 -32

[6 13 64] 215 179

[12 13 32] 215 75

[110 -330 992] 220 152

[110 -550 2752] 220 152

[4 6 64] 220 88

[56 -506 4576] 220 -144

[28 -422 6368] 220 152

[10 10 32] 220 -152

[170 390 896] 220 -144

[112 -785 5504] 223 412

[4 17 128] 223 -113

[8 17 64] 223 -113

[14 15 32] 223 412

[116 -581 2912] 231 67

[4 5 64] 231 159

[40 -123 384] 231 -105

[30 -453 6848] 231 67

[24 -123 640] 231 -105

[120 -361 1088] 239 243

[248 617 1536] 239 180

[48 -151 480] 239 -21

[30 -151 768] 239 -21

[24 -361 5440] 239 243

[198 457 1056] 239 180

[2 4 128] 240 28

[122 364 1088] 240 28

[4 4 64] 240 110

[60 -180 544] 240 -172

[60 -300 1504] 240 -152

[6 12 64] 240 152

[8 4 32] 240 28

[32 -356 3968] 240 354

[10 20 64] 240 172

[12 12 32] 240 28

[16 -36 96] 240 104

[16 -228 3264] 240 130

[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 -210

[42 -126 384] 252 60

[42 -210 1056] 252 -201

[8 2 32] 252 -210

[24 -126 672] 252 60

[14 14 32] 252 -210

[18 -258 3712] 252 -126

[2 1 128] 255 -334

[4 1 64] 255 -12

[44 -575 7520] 255 -87

[8 1 32] 255 -334

[14 33 96] 255 -87

[2 0 128] 256 0

[2 8 160] 256 -32

[4 0 64] 256 0

[4 16 128] 256 -176

[8 0 32] 256 0

[8 16 64] 256 -176

[8 32 160] 256 -72

[26 -288 3200] 256 -72

[26 -392 5920] 256 32

[16 16 32] 256 0

[20 -288 4160] 256 0

[132 -1717 22336] 263 504

[66 -331 1664] 263 -126

[408 949 2208] 263 -126

[22 -331 4992] 263 -504

[136 -1497 16480] 271 -276

[74 217 640] 271 206

[28 -423 6400] 271 276

[212 487 1120] 271 -206

[140 -1261 11360] 279 151

[70 -211 640] 279 -309

[268 621 1440] 279 -309

[20 -301 4544] 279 151

[142 -426 1280] 284 -334

[142 -710 3552] 284 -334

[72 -362 1824] 284 -678

[72 -938 12224] 284 -462

[450 1046 2432] 284 -678

[6 26 160] 284 34

[58 -182 576] 284 250

[36 -182 928] 284 250

[30 -154 800] 284 -34

[30 -454 6880] 284 -334

[24 -362 5472] 284 -334

[240 554 1280] 284 462

[78 -239 736] 287 290

[48 -241 1216] 287 -1009

[338 785 1824] 287 290

[18 -271 4096] 287 1052

[148 -741 3712] 295 -684

[38 -421 4672] 295 -389

[10 5 32] 295 684

[226 517 1184] 295 -44

[152 -457 1376] 303 -381

[4 9 96] 303 294

[6 9 64] 303 294

[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] 311 641

[10 3 32] 311 1033

[12 29 96] 311 -641

[16 35 96] 311 1214

[240 547 1248] 311 231

[2 2 160] 316 -368

[80 -242 736] 316 480

[80 -882 9728] 316 56

[380 882 2048] 316 -56

[10 2 32] 316 -368

[310 718 1664] 316 480

[20 -302 4576] 316 368

[2 1 160] 319 116

[50 -159 512] 319 -299

[10 1 32] 319 116

[32 -353 3904] 319 299

[20 -289 4192] 319 143

[2 0 160] 320 0

[2 8 192] 320 -824

[4 8 96] 320 -884

[84 248 736] 320 964

[6 8 64] 320 -884

[54 -272 1376] 320 552

[54 -704 9184] 320 -832

[10 0 32] 320 0

[28 -424 6432] 320 824

[14 32 96] 320 -832

[254 584 1344] 320 -964

[18 -272 4128] 320 0

[164 -2133 27744] 327 -72

[82 -1067 13888] 327 -246

[56 -619 6848] 327 800

[42 -213 1088] 327 -800

[282 651 1504] 327 246

[24 -363 5504] 327 72

[168 -1849 20352] 335 -215

[766 1785 4160] 335 62

[6 7 64] 335 -62

[6 25 160] 335 -1236

[36 -185 960] 335 1236

[30 -455 6912] 335 215

[86 -947 10432] 343 -827

[422 979 2272] 343 827

[22 -333 5056] 343 196

[174 -522 1568] 348 -186

[174 -870 4352] 348 -186

[4 6 96] 348 -142

[6 6 64] 348 -142

[44 -486 5376] 348 -954

[12 6 32] 348 186

[26 -394 5984] 348 -186

[268 614 1408] 348 842

[4 17 160] 351 453

[60 -303 1536] 351 4317

[352 815 1888] 351 -453

[20 -303 4608] 351 -316

[180 -901 4512] 359 1478

[4 5 96] 359 -1148

[60 -901 13536] 359 1148

[12 5 32] 359 -1478

[184 -553 1664] 367 237

[376 937 2336] 367 -527

[28 -425 6464] 367 237

[296 681 1568] 367 -527

[2 4 192] 368 -932

[4 4 96] 368 -1112

[92 -276 832] 368 142

[92 -460 2304] 368 1378

[62 -188 576] 368 -1160

[62 -684 7552] 368 -248

[62 -932 14016] 368 1112

[8 12 64] 368 -1378

[48 -244 1248] 368 248

[12 4 32] 368 -932

[12 28 96] 368 1160

[324 748 1728] 368 142

[24 -364 5536] 368 932

[282 644 1472] 368 -124

[2 3 192] 375 -283

[94 -1411 21184] 375 207

[6 3 64] 375 -207

[12 3 32] 375 -283

[2 2 192] 380 1900

[4 2 96] 380 2116

[6 2 64] 380 2116

[6 14 96] 380 -335

[60 -190 608] 380 1030

[38 -190 960] 380 1030

[12 2 32] 380 1900

[16 34 96] 380 2108

[22 -334 5088] 380 -1900

[20 30 64] 380 -692

[2 1 192] 383 961

[4 1 96] 383 532

[6 1 64] 383 532

[64 -833 10848] 383 847

[12 1 32] 383 961

[14 31 96] 383 847

[2 0 192] 384 0

[2 8 224] 384 1872

[4 0 96] 384 0

[4 16 160] 384 1200

[6 0 64] 384 0

[6 24 160] 384 492

[10 24 96] 384 492

[394 912 2112] 384 -1200

[12 0 32] 384 0

[30 -456 6944] 384 -1872

[20 -304 4640] 384 0

[22 32 64] 384 0

[98 -491 2464] 391 575

[8 11 64] 391 -575

[26 -395 6016] 391 1840

[200 -2201 24224] 399 1771

[50 -551 6080] 399 201

[14 7 32] 399 1771

[310 711 1632] 399 843

[102 -307 928] 407 1371

[68 -749 8256] 407 -1843

[6 13 96] 407 1808

[8 19 96] 407 -1843

[366 845 1952] 407 1371

[24 -365 5568] 407 -1514

[206 -1030 5152] 412 2422

[104 -522 2624] 412 966

[104 -1354 17632] 412 798

[8 10 64] 412 -966

[14 6 32] 412 -2422

[338 778 1792] 412 -798

[28 -426 6496] 412 2422

[110 -335 1024] 415 -1438

[10 15 64] 415 1438

[22 -335 5120] 415 -4630

[212 -1061 5312] 423 190

[72 -219 672] 423 -4842

[12 27 96] 423 4842

[14 5 32] 423 -190

[18 27 64] 423 -768

[4 9 128] 431 392

[80 -247 768] 431 1013

[8 9 64] 431 392

[10 23 96] 431 -1013

[30 -457 6976] 431 323

[2 4 224] 432 -308

[6 12 96] 432 -4818

[14 4 32] 432 -308

[26 -396 6048] 432 308

[2 3 224] 439 -4140

[70 -221 704] 439 124

[44 -221 1120] 439 124

[14 3 32] 439 -4140

[20 29 64] 439 1675

[2 2 224] 444 -1500

[112 -338 1024] 444 -1572

[112 -1234 13600] 444 -108

[74 -222 672] 444 1319

[74 -370 1856] 444 -3562

[8 18 96] 444 3562

[10 14 64] 444 -108

[12 18 64] 444 1572

[14 2 32] 444 -1500

[14 30 96] 444 -1319

[24 -366 5600] 444 1500

[2 1 224] 447 -1524

[14 1 32] 447 -1524

[16 33 96] 447 -2253

[22 31 64] 447 408

[2 0 224] 448 0

[4 8 128] 448 3296

[8 8 64] 448 3296

[56 -280 1408] 448 -832

[14 0 32] 448 0

[16 8 32] 448 1536

[16 24 64] 448 -288

[22 -336 5152] 448 0

[114 -1483 19296] 455 7

[696 1621 3776] 455 1454

[14 21 64] 455 7

[28 -427 6528] 455 -2281

[1054 2457 5728] 463 6

[58 -871 13088] 463 6

[16 7 32] 463 -1473

[118 -1299 14304] 471 531

[10 13 64] 471 531

[26 -397 6080] 471 2024

[4 6 128] 476 162

[738 1718 4000] 476 -9077

[6 26 192] 476 -706

[90 -278 864] 476 4518

[60 -902 13568] 476 -162

[10 22 96] 476 -4518

[12 26 96] 476 -706

[16 6 32] 476 3762

[18 26 64] 476 -426

[30 -458 7008] 476 -3762

[4 17 192] 479 1217

[80 -401 2016] 479 -8758

[8 17 96] 479 8758

[12 17 64] 479 1217

[24 -367 5632] 479 9085

[4 5 128] 487 2445

[62 -933 14048] 487 -2445

[16 5 32] 487 5535

[6 9 96] 495 3867

[62 -311 1568] 495 2979

[16 23 64] 495 1974

[18 9 32] 495 2193

[4 4 128] 496 2372

[124 -372 1120] 496 3096

[124 -620 3104] 496 -1776

[8 4 64] 496 2372

[64 -708 7840] 496 368

[10 12 64] 496 1776

[50 -252 1280] 496 -368

[14 20 64] 496 -3096

[16 4 32] 496 3880

[20 28 64] 496 -2500

[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	-1
[24 -73 224]	47	-1
[4 9 32]	47	1
[16 -55 192]	47	1
[2 4 32]	48	-1
[26 76 224]	48	-1
[6 12 32]	48	-1
[14 36 96]	48	0
[2 3 32]	55	1
[14 -29 64]	55	0
[14 -99 704]	55	0
[8 -35 160]	55	-1
[2 2 32]	60	2
[30 -210 1472]	60	2
[16 -50 160]	60	2
[16 -178 1984]	60	2
[10 -30 96]	60	1
[10 -50 256]	60	2
[8 -114 1632]	60	2
[2 1 32]	63	2
[16 -225 3168]	63	0
[12 -159 2112]	63	0
[8 -97 1184]	63	-2
[2 0 32]	64	0
[2 8 64]	64	2
[2 16 160]	64	0
[4 8 32]	64	2
[20 56 160]	64	0
[10 -24 64]	64	0
[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	0
[134 307 704]	87	-1
[46 -138 416]	92	2
[46 -230 1152]	92	2
[4 6 32]	92	-2
[24 -218 1984]	92	-10
[162 374 864]	92	2
[6 26 128]	92	0
[26 -86 288]	92	-3
[12 -26 64]	92	-10
[12 -122 1248]	92	0
[12 -134 1504]	92	0
[48 -337 2368]	95	5
[4 17 96]	95	-3
[16 -81 416]	95	3
[10 15 32]	95	5
[52 -261 1312]	103	-11
[4 5 32]	103	11
[56 -169 512]	111	14
[120 297 736]	111	3
[6 9 32]	111	-14
[10 23 64]	111	-3
[2 4 64]	112	12
[58 172 512]	112	12
[4 4 32]	112	12
[28 -84 256]	112	6
[28 -140 704]	112	-22
[28 -196 1376]	112	-16
[14 -28 64]	112	16
[8 12 32]	112	12
[16 -84 448]	112	6
[2 3 64]	119	-17
[30 -451 6784]	119	17
[20 -61 192]	119	-21
[90 221 544]	119	18
[12 -61 320]	119	-21
[12 -125 1312]	119	-18
[2 2 64]	124	-26
[62 -434 3040]	124	-26
[4 2 32]	124	-26
[16 -226 3200]	124	2
[8 34 160]	124	-8
[28 -94 320]	124	-20
[10 14 32]	124	-26
[14 -158 1792]	124	-8
[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	-18
[6 8 32]	128	-24
[22 -112 576]	128	18
[22 -288 3776]	128	-2
[18 -256 3648]	128	0
[12 16 32]	128	0
[12 -64 352]	128	2
[68 -885 11520]	135	41
[34 -171 864]	135	3
[24 -267 2976]	135	-45
[8 11 32]	135	41
[72 -793 8736]	143	-27
[42 121 352]	143	-17
[6 7 32]	143	-27
[6 25 128]	143	27
[18 -199 2208]	143	27
[12 -25 64]	143	17
[76 -685 6176]	151	14
[38 -115 352]	151	-7
[8 19 64]	151	7
[10 13 32]	151	-14
[78 -234 704]	156	-28
[78 -390 1952]	156	-28
[40 -202 1024]	156	116
[40 -522 6816]	156	44
[6 6 32]	156	28
[8 10 32]	156	28
[10 22 64]	156	44
[80 -561 3936]	159	-72
[46 -143 448]	159	3
[28 -143 736]	159	3
[12 15 32]	159	-72
[84 -421 2112]	167	34
[42 -379 3424]	167	-46
[28 -421 6336]	167	34
[14 -197 2784]	167	46
[16 -37 96]	167	10
[88 -265 800]	175	-81
[4 9 64]	175	-102
[8 9 32]	175	81
[2 4 96]	176	-49
[90 268 800]	176	-49
[30 -92 288]	176	-7
[30 -332 3680]	176	42
[30 -452 6816]	176	49
[10 12 32]	176	-49
[18 -92 480]	176	-7
[2 3 96]	183	125
[6 3 32]	183	125
[38 -125 416]	183	78
[16 -227 3232]	183	-12
[2 2 96]	188	126
[94 -658 4608]	188	126
[48 -146 448]	188	-74
[48 -530 5856]	188	-10
[6 2 32]	188	126
[6 14 64]	188	-10
[142 350 864]	188	-87
[8 18 64]	188	74
[12 14 32]	188	126
[14 34 96]	188	87
[2 1 96]	191	143
[6 1 32]	191	143
[32 -417 5440]	191	-70
[8 33 160]	191	72
[20 -223 2496]	191	72
[18 -95 512]	191	70
[18 -257 3680]	191	-70
[2 0 96]	192	0
[2 8 128]	192	96
[2 16 224]	192	0
[4 8 64]	192	40
[52 152 448]	192	-184
[6 0 32]	192	0
[6 24 128]	192	-28
[8 8 32]	192	96
[24 -120 608]	192	28
[12 24 64]	192	-184
[14 16 32]	192	0
[100 -1301 16928]	199	-219
[50 -651 8480]	199	42
[10 11 32]	199	-219
[10 21 64]	199	42
[104 -1145 12608]	207	139
[478 1113 2592]	207	-6
[26 -391 5888]	207	-139
[108 -973 8768]	215	-75
[54 -595 6560]	215	179
[36 -397 4384]	215	-32
[6 13 64]	215	179
[12 13 32]	215	75
[110 -330 992]	220	152
[110 -550 2752]	220	152
[4 6 64]	220	88
[56 -506 4576]	220	-144
[28 -422 6368]	220	152
[10 10 32]	220	-152
[170 390 896]	220	-144
[112 -785 5504]	223	412
[4 17 128]	223	-113
[8 17 64]	223	-113
[14 15 32]	223	412
[116 -581 2912]	231	67
[4 5 64]	231	159
[40 -123 384]	231	-105
[30 -453 6848]	231	67
[24 -123 640]	231	-105
[120 -361 1088]	239	243
[248 617 1536]	239	180
[48 -151 480]	239	-21
[30 -151 768]	239	-21
[24 -361 5440]	239	243
[198 457 1056]	239	180
[2 4 128]	240	28
[122 364 1088]	240	28
[4 4 64]	240	110
[60 -180 544]	240	-172
[60 -300 1504]	240	-152
[6 12 64]	240	152
[8 4 32]	240	28
[32 -356 3968]	240	354
[10 20 64]	240	172
[12 12 32]	240	28
[16 -36 96]	240	104
[16 -228 3264]	240	130
[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	-210
[42 -126 384]	252	60
[42 -210 1056]	252	-201
[8 2 32]	252	-210
[24 -126 672]	252	60
[14 14 32]	252	-210
[18 -258 3712]	252	-126
[2 1 128]	255	-334
[4 1 64]	255	-12
[44 -575 7520]	255	-87
[8 1 32]	255	-334
[14 33 96]	255	-87
[2 0 128]	256	0
[2 8 160]	256	-32
[4 0 64]	256	0
[4 16 128]	256	-176
[8 0 32]	256	0
[8 16 64]	256	-176
[8 32 160]	256	-72
[26 -288 3200]	256	-72
[26 -392 5920]	256	32
[16 16 32]	256	0
[20 -288 4160]	256	0
[132 -1717 22336]	263	504
[66 -331 1664]	263	-126
[408 949 2208]	263	-126
[22 -331 4992]	263	-504
[136 -1497 16480]	271	-276
[74 217 640]	271	206
[28 -423 6400]	271	276
[212 487 1120]	271	-206
[140 -1261 11360]	279	151
[70 -211 640]	279	-309
[268 621 1440]	279	-309
[20 -301 4544]	279	151
[142 -426 1280]	284	-334
[142 -710 3552]	284	-334
[72 -362 1824]	284	-678
[72 -938 12224]	284	-462
[450 1046 2432]	284	-678
[6 26 160]	284	34
[58 -182 576]	284	250
[36 -182 928]	284	250
[30 -154 800]	284	-34
[30 -454 6880]	284	-334
[24 -362 5472]	284	-334
[240 554 1280]	284	462
[78 -239 736]	287	290
[48 -241 1216]	287	-1009
[338 785 1824]	287	290
[18 -271 4096]	287	1052
[148 -741 3712]	295	-684
[38 -421 4672]	295	-389
[10 5 32]	295	684
[226 517 1184]	295	-44
[152 -457 1376]	303	-381
[4 9 96]	303	294
[6 9 64]	303	294
[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	641
[10 3 32]	311	1033
[12 29 96]	311	-641
[16 35 96]	311	1214
[240 547 1248]	311	231
[2 2 160]	316	-368
[80 -242 736]	316	480
[80 -882 9728]	316	56
[380 882 2048]	316	-56
[10 2 32]	316	-368
[310 718 1664]	316	480
[20 -302 4576]	316	368
[2 1 160]	319	116
[50 -159 512]	319	-299
[10 1 32]	319	116
[32 -353 3904]	319	299
[20 -289 4192]	319	143
[2 0 160]	320	0
[2 8 192]	320	-824
[4 8 96]	320	-884
[84 248 736]	320	964
[6 8 64]	320	-884
[54 -272 1376]	320	552
[54 -704 9184]	320	-832
[10 0 32]	320	0
[28 -424 6432]	320	824
[14 32 96]	320	-832
[254 584 1344]	320	-964
[18 -272 4128]	320	0
[164 -2133 27744]	327	-72
[82 -1067 13888]	327	-246
[56 -619 6848]	327	800
[42 -213 1088]	327	-800
[282 651 1504]	327	246
[24 -363 5504]	327	72
[168 -1849 20352]	335	-215
[766 1785 4160]	335	62
[6 7 64]	335	-62
[6 25 160]	335	-1236
[36 -185 960]	335	1236
[30 -455 6912]	335	215
[86 -947 10432]	343	-827
[422 979 2272]	343	827
[22 -333 5056]	343	196
[174 -522 1568]	348	-186
[174 -870 4352]	348	-186
[4 6 96]	348	-142
[6 6 64]	348	-142
[44 -486 5376]	348	-954
[12 6 32]	348	186
[26 -394 5984]	348	-186
[268 614 1408]	348	842
[4 17 160]	351	453
[60 -303 1536]	351	4317
[352 815 1888]	351	-453
[20 -303 4608]	351	-316
[180 -901 4512]	359	1478
[4 5 96]	359	-1148
[60 -901 13536]	359	1148
[12 5 32]	359	-1478
[184 -553 1664]	367	237
[376 937 2336]	367	-527
[28 -425 6464]	367	237
[296 681 1568]	367	-527
[2 4 192]	368	-932
[4 4 96]	368	-1112
[92 -276 832]	368	142
[92 -460 2304]	368	1378
[62 -188 576]	368	-1160
[62 -684 7552]	368	-248
[62 -932 14016]	368	1112
[8 12 64]	368	-1378
[48 -244 1248]	368	248
[12 4 32]	368	-932
[12 28 96]	368	1160
[324 748 1728]	368	142
[24 -364 5536]	368	932
[282 644 1472]	368	-124
[2 3 192]	375	-283
[94 -1411 21184]	375	207
[6 3 64]	375	-207
[12 3 32]	375	-283
[2 2 192]	380	1900
[4 2 96]	380	2116
[6 2 64]	380	2116
[6 14 96]	380	-335
[60 -190 608]	380	1030
[38 -190 960]	380	1030
[12 2 32]	380	1900
[16 34 96]	380	2108
[22 -334 5088]	380	-1900
[20 30 64]	380	-692
[2 1 192]	383	961
[4 1 96]	383	532
[6 1 64]	383	532
[64 -833 10848]	383	847
[12 1 32]	383	961
[14 31 96]	383	847
[2 0 192]	384	0
[2 8 224]	384	1872
[4 0 96]	384	0
[4 16 160]	384	1200
[6 0 64]	384	0
[6 24 160]	384	492
[10 24 96]	384	492
[394 912 2112]	384	-1200
[12 0 32]	384	0
[30 -456 6944]	384	-1872
[20 -304 4640]	384	0
[22 32 64]	384	0
[98 -491 2464]	391	575
[8 11 64]	391	-575
[26 -395 6016]	391	1840
[200 -2201 24224]	399	1771
[50 -551 6080]	399	201
[14 7 32]	399	1771
[310 711 1632]	399	843
[102 -307 928]	407	1371
[68 -749 8256]	407	-1843
[6 13 96]	407	1808
[8 19 96]	407	-1843
[366 845 1952]	407	1371
[24 -365 5568]	407	-1514
[206 -1030 5152]	412	2422
[104 -522 2624]	412	966
[104 -1354 17632]	412	798
[8 10 64]	412	-966
[14 6 32]	412	-2422
[338 778 1792]	412	-798
[28 -426 6496]	412	2422
[110 -335 1024]	415	-1438
[10 15 64]	415	1438
[22 -335 5120]	415	-4630
[212 -1061 5312]	423	190
[72 -219 672]	423	-4842
[12 27 96]	423	4842
[14 5 32]	423	-190
[18 27 64]	423	-768
[4 9 128]	431	392
[80 -247 768]	431	1013
[8 9 64]	431	392
[10 23 96]	431	-1013
[30 -457 6976]	431	323
[2 4 224]	432	-308
[6 12 96]	432	-4818
[14 4 32]	432	-308
[26 -396 6048]	432	308
[2 3 224]	439	-4140
[70 -221 704]	439	124
[44 -221 1120]	439	124
[14 3 32]	439	-4140
[20 29 64]	439	1675
[2 2 224]	444	-1500
[112 -338 1024]	444	-1572
[112 -1234 13600]	444	-108
[74 -222 672]	444	1319
[74 -370 1856]	444	-3562
[8 18 96]	444	3562
[10 14 64]	444	-108
[12 18 64]	444	1572
[14 2 32]	444	-1500
[14 30 96]	444	-1319
[24 -366 5600]	444	1500
[2 1 224]	447	-1524
[14 1 32]	447	-1524
[16 33 96]	447	-2253
[22 31 64]	447	408
[2 0 224]	448	0
[4 8 128]	448	3296
[8 8 64]	448	3296
[56 -280 1408]	448	-832
[14 0 32]	448	0
[16 8 32]	448	1536
[16 24 64]	448	-288
[22 -336 5152]	448	0
[114 -1483 19296]	455	7
[696 1621 3776]	455	1454
[14 21 64]	455	7
[28 -427 6528]	455	-2281
[1054 2457 5728]	463	6
[58 -871 13088]	463	6
[16 7 32]	463	-1473
[118 -1299 14304]	471	531
[10 13 64]	471	531
[26 -397 6080]	471	2024
[4 6 128]	476	162
[738 1718 4000]	476	-9077
[6 26 192]	476	-706
[90 -278 864]	476	4518
[60 -902 13568]	476	-162
[10 22 96]	476	-4518
[12 26 96]	476	-706
[16 6 32]	476	3762
[18 26 64]	476	-426
[30 -458 7008]	476	-3762
[4 17 192]	479	1217
[80 -401 2016]	479	-8758
[8 17 96]	479	8758
[12 17 64]	479	1217
[24 -367 5632]	479	9085
[4 5 128]	487	2445
[62 -933 14048]	487	-2445
[16 5 32]	487	5535
[6 9 96]	495	3867
[62 -311 1568]	495	2979
[16 23 64]	495	1974
[18 9 32]	495	2193
[4 4 128]	496	2372
[124 -372 1120]	496	3096
[124 -620 3104]	496	-1776
[8 4 64]	496	2372
[64 -708 7840]	496	368
[10 12 64]	496	1776
[50 -252 1280]	496	-368
[14 20 64]	496	-3096
[16 4 32]	496	3880
[20 28 64]	496	-2500
[28 -428 6560]	496	-3880

A Borcherds Product in S9(K(16))−

A Borcherds Product in S₉(K(16))⁻