A Borcherds Product in S

A Borcherds Product in S₁₄(K(8))⁺

(Filename: BP-14-8-2-0-1.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 8, such that it and its Borcherds lift Borch(ψ) have the following list of properties.

The singular part of ψ truncated up to q^(8/4) is
(28 + 2/ζ^3 + 2/ζ^2 + 6/ζ + 6*ζ + 2*ζ^2 + 2*ζ^3)
+q(6/ζ^6 + 6*ζ^6)
+q^2(4/ζ^8 + 4*ζ^8)
From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block
φ = TB(14;1,1,1,1,1,1,2,2,3,3)

= 1⁶2²3²

= η(τ)²⁸(θ(τ,z)/η(τ))⁶(θ(τ,2z)/η(τ))²(θ(τ,3z)/η(τ))²
This theta block has q-order of vanishing 2. This theta block has index 2*8, and so Borch(ψ) begins with Jacobi coefficient #2.
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 -1 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) 10

(4,2) 2

(4,6) 6

(9,3) 2
From the singular part above, we compute that the valuation Ord(ψ) is -0.28125, 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_12,8^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)	10
(4,2)	2
(4,6)	6
(9,3)	2

THE VECTOR c (return to top)

The vector c has components given by the following table.

c₁	-194726594477623225/3035084189885318842788
c₂	52033938321882443/758771047471329710697
c₃	846705155078116887/25292368249044323689900
c₄	844960966213608879/25292368249044323689900
c₅	167850035041242746/6323092062261080922475
c₆	15576377048110995/1011694729961772947596

THE VECTOR θ (return to top)

The basis of J_12,8^cusp (with dimension 6) is given by the following theta blocks possibly with index lowering operators denoted by W.

θ₁	TB(12;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,4)\|W_3
θ₂	TB(12;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2)\|W_3
θ₃	TB(12;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,6)\|W_5
θ₄	TB(12;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,4,4,4)\|W_5
θ₅	TB(12;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,6)\|W_5
θ₆	TB(12;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,4)\|W_5

EXPANSION OF ψ (return to top)

The expansion of ψ up to q^(8/4) is

(28 + 2/ζ^3 + 2/ζ^2 + 6/ζ + 6*ζ + 2*ζ^2 + 2*ζ^3)
+q(320 + 6/ζ^6 + 34/ζ^5 - 32/ζ^4 - 94/ζ^3 - 134/ζ^2 + 60/ζ + 60*ζ - 134*ζ^2 - 94*ζ^3 - 32*ζ^4 + 34*ζ^5 + 6*ζ^6)
+q^2(2296 + 4/ζ^8 - 36/ζ^7 + 126/ζ^6 + 406/ζ^5 - 128/ζ^4 - 1058/ζ^3 - 1150/ζ^2 + 688/ζ + 688*ζ - 1150*ζ^2 - 1058*ζ^3 - 128*ζ^4 + 406*ζ^5 + 126*ζ^6 - 36*ζ^7 + 4*ζ^8)

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 -21 112] 7 0

[8 -25 80] 15 0

[4 -25 160] 15 0

[2 4 16] 16 0

[2 3 16] 23 0

[6 -13 32] 23 0

[6 -19 64] 23 0

[2 2 16] 28 0

[14 -42 128] 28 0

[4 6 16] 28 0

[8 -42 224] 28 0

[2 1 16] 31 0

[8 -49 304] 31 0

[4 17 80] 31 0

[2 0 16] 32 0

[2 8 48] 32 0

[6 8 16] 32 0

[6 -32 176] 32 0

[20 -101 512] 39 0

[4 5 16] 39 0

[24 -73 224] 47 0

[4 9 32] 47 0

[6 7 16] 47 0

[2 4 32] 48 0

[4 4 16] 48 0

[12 -36 112] 48 0

[6 12 32] 48 0

[2 3 32] 55 0

[14 -99 704] 55 0

[2 2 32] 60 0

[30 -90 272] 60 0

[4 2 16] 60 0

[16 -50 160] 60 1

[6 6 16] 60 0

[10 -30 96] 60 0

[8 -18 48] 60 0

[8 -50 320] 60 1

[2 1 32] 63 0

[4 1 16] 63 0

[12 -63 336] 63 0

[2 0 32] 64 0

[2 8 64] 64 0

[4 0 16] 64 0

[4 8 32] 64 1

[4 16 80] 64 -2

[8 8 16] 64 0

[10 -64 416] 64 -2

[36 -181 912] 71 0

[18 -91 464] 71 -2

[12 -85 608] 71 0

[20 53 144] 71 -2

[40 -121 368] 79 0

[26 73 208] 79 -6

[10 -71 512] 79 0

[2 4 48] 80 0

[14 -100 720] 80 0

[2 3 48] 87 0

[22 -67 208] 87 -2

[6 3 16] 87 0

[26 67 176] 87 -2

[2 2 48] 92 0

[46 -138 416] 92 0

[4 6 32] 92 13

[24 -122 624] 92 10

[6 2 16] 92 0

[6 14 48] 92 -2

[26 70 192] 92 10

[12 -86 624] 92 0

[2 1 48] 95 0

[4 17 96] 95 4

[6 1 16] 95 0

[16 -81 416] 95 10

[8 17 48] 95 10

[10 15 32] 95 4

[2 0 48] 96 0

[2 8 80] 96 0

[6 0 16] 96 0

[10 -72 528] 96 0

[52 -261 1312] 103 0

[4 5 32] 103 -10

[14 -101 736] 103 0

[56 -169 512] 111 0

[4 9 48] 111 -20

[6 9 32] 111 -20

[12 -87 640] 111 0

[2 4 64] 112 0

[4 4 32] 112 -2

[28 -84 256] 112 2

[8 4 16] 112 0

[8 12 32] 112 2

[2 3 64] 119 0

[30 -211 1488] 119 2

[6 13 48] 119 -14

[8 3 16] 119 0

[2 2 64] 124 0

[62 -186 560] 124 0

[4 2 32] 124 3

[32 -98 304] 124 -26

[8 2 16] 124 0

[10 14 32] 124 -26

[14 -102 752] 124 0

[2 1 64] 127 0

[4 1 32] 127 14

[8 1 16] 127 0

[2 0 64] 128 0

[2 8 96] 128 0

[4 0 32] 128 -30

[4 8 48] 128 2

[4 16 96] 128 26

[6 8 32] 128 2

[22 -112 576] 128 6

[8 0 16] 128 0

[8 16 48] 128 6

[12 16 32] 128 26

[12 -88 656] 128 0

[68 -341 1712] 135 0

[34 -171 864] 135 30

[8 11 32] 135 30

[10 5 16] 135 0

[72 -217 656] 143 0

[42 121 352] 143 108

[6 7 32] 143 108

[14 -103 768] 143 0

[2 4 80] 144 0

[6 12 48] 144 190

[10 4 16] 144 0

[2 3 80] 151 0

[38 -115 352] 151 58

[10 3 16] 151 0

[10 13 32] 151 58

[2 2 80] 156 0

[78 -234 704] 156 0

[4 6 48] 156 -270

[40 -202 1024] 156 -188

[6 6 32] 156 -270

[8 10 32] 156 -188

[10 2 16] 156 0

[12 6 16] 156 0

[2 1 80] 159 0

[4 17 112] 159 -62

[28 -143 736] 159 -230

[30 81 224] 159 -230

[10 1 16] 159 0

[12 15 32] 159 -62

[2 0 80] 160 0

[2 8 112] 160 0

[10 0 16] 160 0

[84 -421 2112] 167 0

[4 5 48] 167 214

[28 -197 1392] 167 214

[36 101 288] 167 -448

[12 5 16] 167 0

[88 -265 800] 175 0

[4 9 64] 175 414

[8 9 32] 175 414

[2 4 96] 176 0

[4 4 48] 176 56

[44 -132 400] 176 -56

[30 -212 1504] 176 56

[10 12 32] 176 -56

[12 4 16] 176 0

[2 3 96] 183 0

[46 -323 2272] 183 -36

[6 3 32] 183 -36

[12 3 16] 183 0

[2 2 96] 188 0

[94 -282 848] 188 0

[4 2 48] 188 -124

[48 -146 448] 188 259

[6 2 32] 188 -124

[6 14 64] 188 326

[42 118 336] 188 -30

[36 98 272] 188 326

[12 2 16] 188 0

[12 14 32] 188 259

[2 1 96] 191 0

[4 1 48] 191 -264

[6 1 32] 191 -264

[12 1 16] 191 0

[2 0 96] 192 0

[2 8 128] 192 0

[4 0 48] 192 652

[4 8 64] 192 -296

[4 16 112] 192 -60

[6 0 32] 192 652

[8 8 32] 192 -296

[12 0 16] 192 0

[100 -501 2512] 199 0

[50 -251 1264] 199 -120

[10 11 32] 199 -120

[104 -313 944] 207 0

[58 169 496] 207 -762

[6 9 48] 207 910

[26 -183 1296] 207 -762

[2 4 112] 208 0

[2 3 112] 215 0

[54 -163 496] 215 -702

[6 13 64] 215 528

[42 115 320] 215 528

[12 13 32] 215 -702

[2 2 112] 220 0

[110 -330 992] 220 0

[4 6 64] 220 2390

[56 -282 1424] 220 1386

[28 -198 1408] 220 2390

[10 10 32] 220 1386

[2 1 112] 223 0

[4 17 128] 223 310

[2 0 112] 224 0

[6 8 48] 224 204

[116 -581 2912] 231 0

[4 5 64] 231 -1966

[30 -213 1520] 231 -1966

[4 9 80] 239 -3630

[6 7 48] 239 -2506

[24 -169 1200] 239 -3630

[2 4 128] 240 0

[4 4 64] 240 -706

[60 -180 544] 240 706

[6 12 64] 240 -2512

[8 4 32] 240 -706

[8 12 48] 240 -2512

[12 12 32] 240 706

[2 3 128] 247 0

[62 -435 3056] 247 290

[8 3 32] 247 290

[2 2 128] 252 0

[4 2 64] 252 1893

[64 -194 592] 252 -1116

[6 6 48] 252 2278

[8 2 32] 252 1893

[2 1 128] 255 0

[4 1 64] 255 1996

[8 1 32] 255 1996

[2 0 128] 256 0

[4 0 64] 256 -6192

[4 8 80] 256 3504

[4 16 128] 256 -816

[8 0 32] 256 -6192

[26 -184 1312] 256 3504

[66 -331 1664] 263 -546

[44 -309 2176] 263 -844

[52 149 432] 263 5134

[8 11 48] 263 5134

[22 -155 1104] 263 -546

[74 217 640] 271 2226

[28 -199 1424] 271 2226

[46 -324 2288] 272 1058

[70 -211 640] 279 4554

[6 3 48] 279 -902

[4 6 80] 284 -11234

[72 -362 1824] 284 -4101

[6 2 48] 284 -1222

[58 166 480] 284 -2158

[8 10 48] 284 -2158

[30 -214 1536] 284 -11234

[24 -170 1216] 284 -4101

[4 17 144] 287 168

[6 1 48] 287 3804

[6 0 48] 288 -4952

[4 5 80] 295 9676

[10 5 32] 295 9676

[4 9 96] 303 16564

[6 9 64] 303 -6846

[8 9 48] 303 -6846

[26 -185 1328] 303 16564

[4 4 80] 304 5208

[76 -228 688] 304 -5208

[10 4 32] 304 5208

[78 -547 3840] 311 -1518

[10 3 32] 311 -1518

[4 2 80] 316 -15214

[80 -242 736] 316 66

[10 2 32] 316 -15214

[4 1 80] 319 -6696

[10 1 32] 319 -6696

[4 0 80] 320 32868

[4 8 96] 320 -19724

[6 8 64] 320 -1848

[8 8 48] 320 -1848

[10 0 32] 320 32868

[28 -200 1440] 320 -19724

[82 -411 2064] 327 7054

[90 265 784] 335 1160

[6 7 64] 335 18634

[42 -295 2080] 335 18634

[30 -215 1552] 335 1160

[86 -259 784] 343 -15888

[4 6 96] 348 26399

[88 -442 2224] 348 -4970

[6 6 64] 348 2214

[44 -310 2192] 348 2214

[12 6 32] 348 26399

[4 5 96] 359 -23870

[60 -421 2960] 359 -17434

[46 -325 2304] 359 -17434

[12 5 32] 359 -23870

[4 9 112] 367 -34082

[4 4 96] 368 -24170

[92 -276 832] 368 24170

[62 -436 3072] 368 -18736

[8 4 48] 368 -18736

[12 4 32] 368 -24170

[94 -659 4624] 375 6734

[6 3 64] 375 17052

[8 3 48] 375 17052

[12 3 32] 375 6734

[4 2 96] 380 70946

[6 2 64] 380 32386

[8 2 48] 380 32386

[12 2 32] 380 70946

[4 1 96] 383 718

[6 1 64] 383 -16848

[8 1 48] 383 -16848

[12 1 32] 383 718

[4 0 96] 384 -100684

[4 8 112] 384 58852

[6 0 64] 384 -19356

[8 0 48] 384 -19356

[12 0 32] 384 -100684

[98 -491 2464] 391 -25866

[106 313 928] 399 -24578

[4 6 112] 412 -6518

[104 -522 2624] 412 73537

[4 5 112] 423 1854

[4 9 128] 431 -28464

[4 4 112] 432 68040

[110 -771 5408] 439 -26132

[4 2 112] 444 -180650

[4 1 112] 447 71084

[4 0 112] 448 143312

[4 8 128] 448 -66304

[122 361 1072] 463 40436

[4 6 128] 476 -118701

[4 5 128] 487 171240

[4 4 128] 496 -81516

	φ	= TB(14;1,1,1,1,1,1,2,2,3,3)
		= 1⁶2²3²
		= η(τ)²⁸(θ(τ,z)/η(τ))⁶(θ(τ,2z)/η(τ))²(θ(τ,3z)/η(τ))²

2u	det(2u)	a(u, Borch(ψ))
[4 -21 112]	7	0
[8 -25 80]	15	0
[4 -25 160]	15	0
[2 4 16]	16	0
[2 3 16]	23	0
[6 -13 32]	23	0
[6 -19 64]	23	0
[2 2 16]	28	0
[14 -42 128]	28	0
[4 6 16]	28	0
[8 -42 224]	28	0
[2 1 16]	31	0
[8 -49 304]	31	0
[4 17 80]	31	0
[2 0 16]	32	0
[2 8 48]	32	0
[6 8 16]	32	0
[6 -32 176]	32	0
[20 -101 512]	39	0
[4 5 16]	39	0
[24 -73 224]	47	0
[4 9 32]	47	0
[6 7 16]	47	0
[2 4 32]	48	0
[4 4 16]	48	0
[12 -36 112]	48	0
[6 12 32]	48	0
[2 3 32]	55	0
[14 -99 704]	55	0
[2 2 32]	60	0
[30 -90 272]	60	0
[4 2 16]	60	0
[16 -50 160]	60	1
[6 6 16]	60	0
[10 -30 96]	60	0
[8 -18 48]	60	0
[8 -50 320]	60	1
[2 1 32]	63	0
[4 1 16]	63	0
[12 -63 336]	63	0
[2 0 32]	64	0
[2 8 64]	64	0
[4 0 16]	64	0
[4 8 32]	64	1
[4 16 80]	64	-2
[8 8 16]	64	0
[10 -64 416]	64	-2
[36 -181 912]	71	0
[18 -91 464]	71	-2
[12 -85 608]	71	0
[20 53 144]	71	-2
[40 -121 368]	79	0
[26 73 208]	79	-6
[10 -71 512]	79	0
[2 4 48]	80	0
[14 -100 720]	80	0
[2 3 48]	87	0
[22 -67 208]	87	-2
[6 3 16]	87	0
[26 67 176]	87	-2
[2 2 48]	92	0
[46 -138 416]	92	0
[4 6 32]	92	13
[24 -122 624]	92	10
[6 2 16]	92	0
[6 14 48]	92	-2
[26 70 192]	92	10
[12 -86 624]	92	0
[2 1 48]	95	0
[4 17 96]	95	4
[6 1 16]	95	0
[16 -81 416]	95	10
[8 17 48]	95	10
[10 15 32]	95	4
[2 0 48]	96	0
[2 8 80]	96	0
[6 0 16]	96	0
[10 -72 528]	96	0
[52 -261 1312]	103	0
[4 5 32]	103	-10
[14 -101 736]	103	0
[56 -169 512]	111	0
[4 9 48]	111	-20
[6 9 32]	111	-20
[12 -87 640]	111	0
[2 4 64]	112	0
[4 4 32]	112	-2
[28 -84 256]	112	2
[8 4 16]	112	0
[8 12 32]	112	2
[2 3 64]	119	0
[30 -211 1488]	119	2
[6 13 48]	119	-14
[8 3 16]	119	0
[2 2 64]	124	0
[62 -186 560]	124	0
[4 2 32]	124	3
[32 -98 304]	124	-26
[8 2 16]	124	0
[10 14 32]	124	-26
[14 -102 752]	124	0
[2 1 64]	127	0
[4 1 32]	127	14
[8 1 16]	127	0
[2 0 64]	128	0
[2 8 96]	128	0
[4 0 32]	128	-30
[4 8 48]	128	2
[4 16 96]	128	26
[6 8 32]	128	2
[22 -112 576]	128	6
[8 0 16]	128	0
[8 16 48]	128	6
[12 16 32]	128	26
[12 -88 656]	128	0
[68 -341 1712]	135	0
[34 -171 864]	135	30
[8 11 32]	135	30
[10 5 16]	135	0
[72 -217 656]	143	0
[42 121 352]	143	108
[6 7 32]	143	108
[14 -103 768]	143	0
[2 4 80]	144	0
[6 12 48]	144	190
[10 4 16]	144	0
[2 3 80]	151	0
[38 -115 352]	151	58
[10 3 16]	151	0
[10 13 32]	151	58
[2 2 80]	156	0
[78 -234 704]	156	0
[4 6 48]	156	-270
[40 -202 1024]	156	-188
[6 6 32]	156	-270
[8 10 32]	156	-188
[10 2 16]	156	0
[12 6 16]	156	0
[2 1 80]	159	0
[4 17 112]	159	-62
[28 -143 736]	159	-230
[30 81 224]	159	-230
[10 1 16]	159	0
[12 15 32]	159	-62
[2 0 80]	160	0
[2 8 112]	160	0
[10 0 16]	160	0
[84 -421 2112]	167	0
[4 5 48]	167	214
[28 -197 1392]	167	214
[36 101 288]	167	-448
[12 5 16]	167	0
[88 -265 800]	175	0
[4 9 64]	175	414
[8 9 32]	175	414
[2 4 96]	176	0
[4 4 48]	176	56
[44 -132 400]	176	-56
[30 -212 1504]	176	56
[10 12 32]	176	-56
[12 4 16]	176	0
[2 3 96]	183	0
[46 -323 2272]	183	-36
[6 3 32]	183	-36
[12 3 16]	183	0
[2 2 96]	188	0
[94 -282 848]	188	0
[4 2 48]	188	-124
[48 -146 448]	188	259
[6 2 32]	188	-124
[6 14 64]	188	326
[42 118 336]	188	-30
[36 98 272]	188	326
[12 2 16]	188	0
[12 14 32]	188	259
[2 1 96]	191	0
[4 1 48]	191	-264
[6 1 32]	191	-264
[12 1 16]	191	0
[2 0 96]	192	0
[2 8 128]	192	0
[4 0 48]	192	652
[4 8 64]	192	-296
[4 16 112]	192	-60
[6 0 32]	192	652
[8 8 32]	192	-296
[12 0 16]	192	0
[100 -501 2512]	199	0
[50 -251 1264]	199	-120
[10 11 32]	199	-120
[104 -313 944]	207	0
[58 169 496]	207	-762
[6 9 48]	207	910
[26 -183 1296]	207	-762
[2 4 112]	208	0
[2 3 112]	215	0
[54 -163 496]	215	-702
[6 13 64]	215	528
[42 115 320]	215	528
[12 13 32]	215	-702
[2 2 112]	220	0
[110 -330 992]	220	0
[4 6 64]	220	2390
[56 -282 1424]	220	1386
[28 -198 1408]	220	2390
[10 10 32]	220	1386
[2 1 112]	223	0
[4 17 128]	223	310
[2 0 112]	224	0
[6 8 48]	224	204
[116 -581 2912]	231	0
[4 5 64]	231	-1966
[30 -213 1520]	231	-1966
[4 9 80]	239	-3630
[6 7 48]	239	-2506
[24 -169 1200]	239	-3630
[2 4 128]	240	0
[4 4 64]	240	-706
[60 -180 544]	240	706
[6 12 64]	240	-2512
[8 4 32]	240	-706
[8 12 48]	240	-2512
[12 12 32]	240	706
[2 3 128]	247	0
[62 -435 3056]	247	290
[8 3 32]	247	290
[2 2 128]	252	0
[4 2 64]	252	1893
[64 -194 592]	252	-1116
[6 6 48]	252	2278
[8 2 32]	252	1893
[2 1 128]	255	0
[4 1 64]	255	1996
[8 1 32]	255	1996
[2 0 128]	256	0
[4 0 64]	256	-6192
[4 8 80]	256	3504
[4 16 128]	256	-816
[8 0 32]	256	-6192
[26 -184 1312]	256	3504
[66 -331 1664]	263	-546
[44 -309 2176]	263	-844
[52 149 432]	263	5134
[8 11 48]	263	5134
[22 -155 1104]	263	-546
[74 217 640]	271	2226
[28 -199 1424]	271	2226
[46 -324 2288]	272	1058
[70 -211 640]	279	4554
[6 3 48]	279	-902
[4 6 80]	284	-11234
[72 -362 1824]	284	-4101
[6 2 48]	284	-1222
[58 166 480]	284	-2158
[8 10 48]	284	-2158
[30 -214 1536]	284	-11234
[24 -170 1216]	284	-4101
[4 17 144]	287	168
[6 1 48]	287	3804
[6 0 48]	288	-4952
[4 5 80]	295	9676
[10 5 32]	295	9676
[4 9 96]	303	16564
[6 9 64]	303	-6846
[8 9 48]	303	-6846
[26 -185 1328]	303	16564
[4 4 80]	304	5208
[76 -228 688]	304	-5208
[10 4 32]	304	5208
[78 -547 3840]	311	-1518
[10 3 32]	311	-1518
[4 2 80]	316	-15214
[80 -242 736]	316	66
[10 2 32]	316	-15214
[4 1 80]	319	-6696
[10 1 32]	319	-6696
[4 0 80]	320	32868
[4 8 96]	320	-19724
[6 8 64]	320	-1848
[8 8 48]	320	-1848
[10 0 32]	320	32868
[28 -200 1440]	320	-19724
[82 -411 2064]	327	7054
[90 265 784]	335	1160
[6 7 64]	335	18634
[42 -295 2080]	335	18634
[30 -215 1552]	335	1160
[86 -259 784]	343	-15888
[4 6 96]	348	26399
[88 -442 2224]	348	-4970
[6 6 64]	348	2214
[44 -310 2192]	348	2214
[12 6 32]	348	26399
[4 5 96]	359	-23870
[60 -421 2960]	359	-17434
[46 -325 2304]	359	-17434
[12 5 32]	359	-23870
[4 9 112]	367	-34082
[4 4 96]	368	-24170
[92 -276 832]	368	24170
[62 -436 3072]	368	-18736
[8 4 48]	368	-18736
[12 4 32]	368	-24170
[94 -659 4624]	375	6734
[6 3 64]	375	17052
[8 3 48]	375	17052
[12 3 32]	375	6734
[4 2 96]	380	70946
[6 2 64]	380	32386
[8 2 48]	380	32386
[12 2 32]	380	70946
[4 1 96]	383	718
[6 1 64]	383	-16848
[8 1 48]	383	-16848
[12 1 32]	383	718
[4 0 96]	384	-100684
[4 8 112]	384	58852
[6 0 64]	384	-19356
[8 0 48]	384	-19356
[12 0 32]	384	-100684
[98 -491 2464]	391	-25866
[106 313 928]	399	-24578
[4 6 112]	412	-6518
[104 -522 2624]	412	73537
[4 5 112]	423	1854
[4 9 128]	431	-28464
[4 4 112]	432	68040
[110 -771 5408]	439	-26132
[4 2 112]	444	-180650
[4 1 112]	447	71084
[4 0 112]	448	143312
[4 8 128]	448	-66304
[122 361 1072]	463	40436
[4 6 128]	476	-118701
[4 5 128]	487	171240
[4 4 128]	496	-81516

A Borcherds Product in S14(K(8))+

A Borcherds Product in S₁₄(K(8))⁺