A Borcherds Product in S

A Borcherds Product in S₁₁(K(8))⁻

(Filename: BP-11-8-2-0-1.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
(22 + 2/ζ^3 + ζ^(-2) + 10/ζ + 10*ζ + ζ^2 + 2*ζ^3)
+q(3/ζ^6 + 3*ζ^6)
+q^2(-6/ζ^8 - 6*ζ^8)
From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block
φ = TB(11;1,1,1,1,1,1,1,1,1,1,2,3,3)

= 1¹⁰2¹3²

= η(τ)²²(θ(τ,z)/η(τ))¹⁰(θ(τ,2z)/η(τ))(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) 13

(4,2) 1

(4,6) 3

(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)	13
(4,2)	1
(4,6)	3
(9,3)	2

THE VECTOR c (return to top)

The vector c has components given by the following table.

c₁	-36992823402834263/551833489070057971416
c₂	2422618715133829/137958372267514492854
c₃	111653704715924889/4598612408917149761800
c₄	112410678028990953/4598612408917149761800
c₅	11450234274738761/574826551114643720225
c₆	17272401877694017/919722481783429952360

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

(22 + 2/ζ^3 + ζ^(-2) + 10/ζ + 10*ζ + ζ^2 + 2*ζ^3)
+q(288 + 3/ζ^6 + 30/ζ^5 - 16/ζ^4 - 102/ζ^3 - 131/ζ^2 + 72/ζ + 72*ζ - 131*ζ^2 - 102*ζ^3 - 16*ζ^4 + 30*ζ^5 + 3*ζ^6)
+q^2(2188 - 6/ζ^8 - 32/ζ^7 + 127/ζ^6 + 390/ζ^5 - 64/ζ^4 - 1086/ζ^3 - 1151/ζ^2 + 728/ζ + 728*ζ - 1151*ζ^2 - 1086*ζ^3 - 64*ζ^4 + 390*ζ^5 + 127*ζ^6 - 32*ζ^7 - 6*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 -1

[4 17 80] 31 1

[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 -1

[6 7 16] 47 0

[2 4 32] 48 0

[4 4 16] 48 0

[12 -36 112] 48 2

[6 12 32] 48 2

[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 8

[6 6 16] 60 0

[10 -30 96] 60 -4

[8 -18 48] 60 4

[8 -50 320] 60 8

[2 1 32] 63 0

[4 1 16] 63 0

[12 -63 336] 63 1

[2 0 32] 64 0

[2 8 64] 64 0

[4 0 16] 64 0

[4 8 32] 64 10

[4 16 80] 64 0

[8 8 16] 64 0

[10 -64 416] 64 0

[36 -181 912] 71 0

[18 -91 464] 71 19

[12 -85 608] 71 0

[20 53 144] 71 19

[40 -121 368] 79 0

[26 73 208] 79 44

[10 -71 512] 79 0

[2 4 48] 80 0

[14 -100 720] 80 0

[2 3 48] 87 0

[22 -67 208] 87 -44

[6 3 16] 87 0

[26 67 176] 87 44

[2 2 48] 92 0

[46 -138 416] 92 0

[4 6 32] 92 112

[24 -122 624] 92 -88

[6 2 16] 92 0

[6 14 48] 92 36

[26 70 192] 92 -88

[12 -86 624] 92 0

[2 1 48] 95 0

[4 17 96] 95 18

[6 1 16] 95 0

[16 -81 416] 95 24

[8 17 48] 95 -24

[10 15 32] 95 -18

[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 -185

[14 -101 736] 103 0

[56 -169 512] 111 0

[4 9 48] 111 -258

[6 9 32] 111 -258

[12 -87 640] 111 0

[2 4 64] 112 0

[4 4 32] 112 220

[28 -84 256] 112 172

[8 4 16] 112 0

[8 12 32] 112 172

[2 3 64] 119 0

[30 -211 1488] 119 221

[6 13 48] 119 -361

[8 3 16] 119 0

[2 2 64] 124 0

[62 -186 560] 124 0

[4 2 32] 124 208

[32 -98 304] 124 88

[8 2 16] 124 0

[10 14 32] 124 88

[14 -102 752] 124 0

[2 1 64] 127 0

[4 1 32] 127 -143

[8 1 16] 127 0

[2 0 64] 128 0

[2 8 96] 128 0

[4 0 32] 128 0

[4 8 48] 128 488

[4 16 96] 128 0

[6 8 32] 128 488

[22 -112 576] 128 -456

[8 0 16] 128 0

[8 16 48] 128 456

[12 16 32] 128 0

[12 -88 656] 128 0

[68 -341 1712] 135 0

[34 -171 864] 135 407

[8 11 32] 135 -407

[10 5 16] 135 0

[72 -217 656] 143 0

[42 121 352] 143 506

[6 7 32] 143 -506

[14 -103 768] 143 0

[2 4 80] 144 0

[6 12 48] 144 1616

[10 4 16] 144 0

[2 3 80] 151 0

[38 -115 352] 151 -198

[10 3 16] 151 0

[10 13 32] 151 -198

[2 2 80] 156 0

[78 -234 704] 156 0

[4 6 48] 156 40

[40 -202 1024] 156 -440

[6 6 32] 156 40

[8 10 32] 156 440

[10 2 16] 156 0

[12 6 16] 156 0

[2 1 80] 159 0

[4 17 112] 159 -1

[28 -143 736] 159 2141

[30 81 224] 159 2141

[10 1 16] 159 0

[12 15 32] 159 1

[2 0 80] 160 0

[2 8 112] 160 0

[10 0 16] 160 0

[84 -421 2112] 167 0

[4 5 48] 167 579

[28 -197 1392] 167 -579

[36 101 288] 167 3820

[12 5 16] 167 0

[88 -265 800] 175 0

[4 9 64] 175 465

[8 9 32] 175 465

[2 4 96] 176 0

[4 4 48] 176 -726

[44 -132 400] 176 -216

[30 -212 1504] 176 726

[10 12 32] 176 -216

[12 4 16] 176 0

[2 3 96] 183 0

[46 -323 2272] 183 -686

[6 3 32] 183 686

[12 3 16] 183 0

[2 2 96] 188 0

[94 -282 848] 188 0

[4 2 48] 188 -1072

[48 -146 448] 188 -376

[6 2 32] 188 -1072

[6 14 64] 188 5308

[42 118 336] 188 -4932

[36 98 272] 188 -5308

[12 2 16] 188 0

[12 14 32] 188 -376

[2 1 96] 191 0

[4 1 48] 191 1120

[6 1 32] 191 1120

[12 1 16] 191 0

[2 0 96] 192 0

[2 8 128] 192 0

[4 0 48] 192 0

[4 8 64] 192 -2672

[4 16 112] 192 0

[6 0 32] 192 0

[8 8 32] 192 -2672

[12 0 16] 192 0

[100 -501 2512] 199 0

[50 -251 1264] 199 -2433

[10 11 32] 199 2433

[104 -313 944] 207 0

[58 169 496] 207 -4565

[6 9 48] 207 -3312

[26 -183 1296] 207 -4565

[2 4 112] 208 0

[2 3 112] 215 0

[54 -163 496] 215 2442

[6 13 64] 215 -8113

[42 115 320] 215 8113

[12 13 32] 215 2442

[2 2 112] 220 0

[110 -330 992] 220 0

[4 6 64] 220 -3560

[56 -282 1424] 220 6248

[28 -198 1408] 220 3560

[10 10 32] 220 -6248

[2 1 112] 223 0

[4 17 128] 223 -587

[2 0 112] 224 0

[6 8 48] 224 1344

[116 -581 2912] 231 0

[4 5 64] 231 1423

[30 -213 1520] 231 -1423

[4 9 80] 239 7582

[6 7 48] 239 -578

[24 -169 1200] 239 -7582

[2 4 128] 240 0

[4 4 64] 240 -2980

[60 -180 544] 240 -6068

[6 12 64] 240 6112

[8 4 32] 240 -2980

[8 12 48] 240 6112

[12 12 32] 240 -6068

[2 3 128] 247 0

[62 -435 3056] 247 -5042

[8 3 32] 247 5042

[2 2 128] 252 0

[4 2 64] 252 -536

[64 -194 592] 252 -2192

[6 6 48] 252 -2892

[8 2 32] 252 -536

[2 1 128] 255 0

[4 1 64] 255 -4165

[8 1 32] 255 -4165

[2 0 128] 256 0

[4 0 64] 256 0

[4 8 80] 256 -1888

[4 16 128] 256 0

[8 0 32] 256 0

[26 -184 1312] 256 1888

[66 -331 1664] 263 -3699

[44 -309 2176] 263 -9274

[52 149 432] 263 -4100

[8 11 48] 263 4100

[22 -155 1104] 263 -3699

[74 217 640] 271 6160

[28 -199 1424] 271 6160

[46 -324 2288] 272 11632

[70 -211 640] 279 -726

[6 3 48] 279 9261

[4 6 80] 284 3512

[72 -362 1824] 284 -11792

[6 2 48] 284 -8044

[58 166 480] 284 14168

[8 10 48] 284 -14168

[30 -214 1536] 284 -3512

[24 -170 1216] 284 -11792

[4 17 144] 287 329

[6 1 48] 287 6429

[6 0 48] 288 0

[4 5 80] 295 3975

[10 5 32] 295 3975

[4 9 96] 303 -28451

[6 9 64] 303 14156

[8 9 48] 303 14156

[26 -185 1328] 303 28451

[4 4 80] 304 9544

[76 -228 688] 304 20734

[10 4 32] 304 9544

[78 -547 3840] 311 28816

[10 3 32] 311 -28816

[4 2 80] 316 9848

[80 -242 736] 316 9368

[10 2 32] 316 9848

[4 1 80] 319 17519

[10 1 32] 319 17519

[4 0 80] 320 0

[4 8 96] 320 25048

[6 8 64] 320 -22864

[8 8 48] 320 -22864

[10 0 32] 320 0

[28 -200 1440] 320 -25048

[82 -411 2064] 327 42335

[90 265 784] 335 10563

[6 7 64] 335 45226

[42 -295 2080] 335 -45226

[30 -215 1552] 335 10563

[86 -259 784] 343 -28457

[4 6 96] 348 28944

[88 -442 2224] 348 -24904

[6 6 64] 348 -22296

[44 -310 2192] 348 22296

[12 6 32] 348 28944

[4 5 96] 359 -67387

[60 -421 2960] 359 45766

[46 -325 2304] 359 45766

[12 5 32] 359 -67387

[4 9 112] 367 9047

[4 4 96] 368 28588

[92 -276 832] 368 7228

[62 -436 3072] 368 -52192

[8 4 48] 368 52192

[12 4 32] 368 28588

[94 -659 4624] 375 -51780

[6 3 64] 375 -11595

[8 3 48] 375 -11595

[12 3 32] 375 51780

[4 2 96] 380 -6000

[6 2 64] 380 47360

[8 2 48] 380 47360

[12 2 32] 380 -6000

[4 1 96] 383 -77257

[6 1 64] 383 -87479

[8 1 48] 383 -87479

[12 1 32] 383 -77257

[4 0 96] 384 0

[4 8 112] 384 6672

[6 0 64] 384 0

[8 0 48] 384 0

[12 0 32] 384 0

[98 -491 2464] 391 -64957

[106 313 928] 399 42999

[4 6 112] 412 -15992

[104 -522 2624] 412 66800

[4 5 112] 423 144079

[4 9 128] 431 26471

[4 4 112] 432 -97552

[110 -771 5408] 439 80500

[4 2 112] 444 -16376

[4 1 112] 447 206174

[4 0 112] 448 0

[4 8 128] 448 -136576

[122 361 1072] 463 -281037

[4 6 128] 476 -190128

[4 5 128] 487 -12900

[4 4 128] 496 -51800

	φ	= TB(11;1,1,1,1,1,1,1,1,1,1,2,3,3)
		= 1¹⁰2¹3²
		= η(τ)²²(θ(τ,z)/η(τ))¹⁰(θ(τ,2z)/η(τ))(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	-1
[4 17 80]	31	1
[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	-1
[6 7 16]	47	0
[2 4 32]	48	0
[4 4 16]	48	0
[12 -36 112]	48	2
[6 12 32]	48	2
[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	8
[6 6 16]	60	0
[10 -30 96]	60	-4
[8 -18 48]	60	4
[8 -50 320]	60	8
[2 1 32]	63	0
[4 1 16]	63	0
[12 -63 336]	63	1
[2 0 32]	64	0
[2 8 64]	64	0
[4 0 16]	64	0
[4 8 32]	64	10
[4 16 80]	64	0
[8 8 16]	64	0
[10 -64 416]	64	0
[36 -181 912]	71	0
[18 -91 464]	71	19
[12 -85 608]	71	0
[20 53 144]	71	19
[40 -121 368]	79	0
[26 73 208]	79	44
[10 -71 512]	79	0
[2 4 48]	80	0
[14 -100 720]	80	0
[2 3 48]	87	0
[22 -67 208]	87	-44
[6 3 16]	87	0
[26 67 176]	87	44
[2 2 48]	92	0
[46 -138 416]	92	0
[4 6 32]	92	112
[24 -122 624]	92	-88
[6 2 16]	92	0
[6 14 48]	92	36
[26 70 192]	92	-88
[12 -86 624]	92	0
[2 1 48]	95	0
[4 17 96]	95	18
[6 1 16]	95	0
[16 -81 416]	95	24
[8 17 48]	95	-24
[10 15 32]	95	-18
[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	-185
[14 -101 736]	103	0
[56 -169 512]	111	0
[4 9 48]	111	-258
[6 9 32]	111	-258
[12 -87 640]	111	0
[2 4 64]	112	0
[4 4 32]	112	220
[28 -84 256]	112	172
[8 4 16]	112	0
[8 12 32]	112	172
[2 3 64]	119	0
[30 -211 1488]	119	221
[6 13 48]	119	-361
[8 3 16]	119	0
[2 2 64]	124	0
[62 -186 560]	124	0
[4 2 32]	124	208
[32 -98 304]	124	88
[8 2 16]	124	0
[10 14 32]	124	88
[14 -102 752]	124	0
[2 1 64]	127	0
[4 1 32]	127	-143
[8 1 16]	127	0
[2 0 64]	128	0
[2 8 96]	128	0
[4 0 32]	128	0
[4 8 48]	128	488
[4 16 96]	128	0
[6 8 32]	128	488
[22 -112 576]	128	-456
[8 0 16]	128	0
[8 16 48]	128	456
[12 16 32]	128	0
[12 -88 656]	128	0
[68 -341 1712]	135	0
[34 -171 864]	135	407
[8 11 32]	135	-407
[10 5 16]	135	0
[72 -217 656]	143	0
[42 121 352]	143	506
[6 7 32]	143	-506
[14 -103 768]	143	0
[2 4 80]	144	0
[6 12 48]	144	1616
[10 4 16]	144	0
[2 3 80]	151	0
[38 -115 352]	151	-198
[10 3 16]	151	0
[10 13 32]	151	-198
[2 2 80]	156	0
[78 -234 704]	156	0
[4 6 48]	156	40
[40 -202 1024]	156	-440
[6 6 32]	156	40
[8 10 32]	156	440
[10 2 16]	156	0
[12 6 16]	156	0
[2 1 80]	159	0
[4 17 112]	159	-1
[28 -143 736]	159	2141
[30 81 224]	159	2141
[10 1 16]	159	0
[12 15 32]	159	1
[2 0 80]	160	0
[2 8 112]	160	0
[10 0 16]	160	0
[84 -421 2112]	167	0
[4 5 48]	167	579
[28 -197 1392]	167	-579
[36 101 288]	167	3820
[12 5 16]	167	0
[88 -265 800]	175	0
[4 9 64]	175	465
[8 9 32]	175	465
[2 4 96]	176	0
[4 4 48]	176	-726
[44 -132 400]	176	-216
[30 -212 1504]	176	726
[10 12 32]	176	-216
[12 4 16]	176	0
[2 3 96]	183	0
[46 -323 2272]	183	-686
[6 3 32]	183	686
[12 3 16]	183	0
[2 2 96]	188	0
[94 -282 848]	188	0
[4 2 48]	188	-1072
[48 -146 448]	188	-376
[6 2 32]	188	-1072
[6 14 64]	188	5308
[42 118 336]	188	-4932
[36 98 272]	188	-5308
[12 2 16]	188	0
[12 14 32]	188	-376
[2 1 96]	191	0
[4 1 48]	191	1120
[6 1 32]	191	1120
[12 1 16]	191	0
[2 0 96]	192	0
[2 8 128]	192	0
[4 0 48]	192	0
[4 8 64]	192	-2672
[4 16 112]	192	0
[6 0 32]	192	0
[8 8 32]	192	-2672
[12 0 16]	192	0
[100 -501 2512]	199	0
[50 -251 1264]	199	-2433
[10 11 32]	199	2433
[104 -313 944]	207	0
[58 169 496]	207	-4565
[6 9 48]	207	-3312
[26 -183 1296]	207	-4565
[2 4 112]	208	0
[2 3 112]	215	0
[54 -163 496]	215	2442
[6 13 64]	215	-8113
[42 115 320]	215	8113
[12 13 32]	215	2442
[2 2 112]	220	0
[110 -330 992]	220	0
[4 6 64]	220	-3560
[56 -282 1424]	220	6248
[28 -198 1408]	220	3560
[10 10 32]	220	-6248
[2 1 112]	223	0
[4 17 128]	223	-587
[2 0 112]	224	0
[6 8 48]	224	1344
[116 -581 2912]	231	0
[4 5 64]	231	1423
[30 -213 1520]	231	-1423
[4 9 80]	239	7582
[6 7 48]	239	-578
[24 -169 1200]	239	-7582
[2 4 128]	240	0
[4 4 64]	240	-2980
[60 -180 544]	240	-6068
[6 12 64]	240	6112
[8 4 32]	240	-2980
[8 12 48]	240	6112
[12 12 32]	240	-6068
[2 3 128]	247	0
[62 -435 3056]	247	-5042
[8 3 32]	247	5042
[2 2 128]	252	0
[4 2 64]	252	-536
[64 -194 592]	252	-2192
[6 6 48]	252	-2892
[8 2 32]	252	-536
[2 1 128]	255	0
[4 1 64]	255	-4165
[8 1 32]	255	-4165
[2 0 128]	256	0
[4 0 64]	256	0
[4 8 80]	256	-1888
[4 16 128]	256	0
[8 0 32]	256	0
[26 -184 1312]	256	1888
[66 -331 1664]	263	-3699
[44 -309 2176]	263	-9274
[52 149 432]	263	-4100
[8 11 48]	263	4100
[22 -155 1104]	263	-3699
[74 217 640]	271	6160
[28 -199 1424]	271	6160
[46 -324 2288]	272	11632
[70 -211 640]	279	-726
[6 3 48]	279	9261
[4 6 80]	284	3512
[72 -362 1824]	284	-11792
[6 2 48]	284	-8044
[58 166 480]	284	14168
[8 10 48]	284	-14168
[30 -214 1536]	284	-3512
[24 -170 1216]	284	-11792
[4 17 144]	287	329
[6 1 48]	287	6429
[6 0 48]	288	0
[4 5 80]	295	3975
[10 5 32]	295	3975
[4 9 96]	303	-28451
[6 9 64]	303	14156
[8 9 48]	303	14156
[26 -185 1328]	303	28451
[4 4 80]	304	9544
[76 -228 688]	304	20734
[10 4 32]	304	9544
[78 -547 3840]	311	28816
[10 3 32]	311	-28816
[4 2 80]	316	9848
[80 -242 736]	316	9368
[10 2 32]	316	9848
[4 1 80]	319	17519
[10 1 32]	319	17519
[4 0 80]	320	0
[4 8 96]	320	25048
[6 8 64]	320	-22864
[8 8 48]	320	-22864
[10 0 32]	320	0
[28 -200 1440]	320	-25048
[82 -411 2064]	327	42335
[90 265 784]	335	10563
[6 7 64]	335	45226
[42 -295 2080]	335	-45226
[30 -215 1552]	335	10563
[86 -259 784]	343	-28457
[4 6 96]	348	28944
[88 -442 2224]	348	-24904
[6 6 64]	348	-22296
[44 -310 2192]	348	22296
[12 6 32]	348	28944
[4 5 96]	359	-67387
[60 -421 2960]	359	45766
[46 -325 2304]	359	45766
[12 5 32]	359	-67387
[4 9 112]	367	9047
[4 4 96]	368	28588
[92 -276 832]	368	7228
[62 -436 3072]	368	-52192
[8 4 48]	368	52192
[12 4 32]	368	28588
[94 -659 4624]	375	-51780
[6 3 64]	375	-11595
[8 3 48]	375	-11595
[12 3 32]	375	51780
[4 2 96]	380	-6000
[6 2 64]	380	47360
[8 2 48]	380	47360
[12 2 32]	380	-6000
[4 1 96]	383	-77257
[6 1 64]	383	-87479
[8 1 48]	383	-87479
[12 1 32]	383	-77257
[4 0 96]	384	0
[4 8 112]	384	6672
[6 0 64]	384	0
[8 0 48]	384	0
[12 0 32]	384	0
[98 -491 2464]	391	-64957
[106 313 928]	399	42999
[4 6 112]	412	-15992
[104 -522 2624]	412	66800
[4 5 112]	423	144079
[4 9 128]	431	26471
[4 4 112]	432	-97552
[110 -771 5408]	439	80500
[4 2 112]	444	-16376
[4 1 112]	447	206174
[4 0 112]	448	0
[4 8 128]	448	-136576
[122 361 1072]	463	-281037
[4 6 128]	476	-190128
[4 5 128]	487	-12900
[4 4 128]	496	-51800

A Borcherds Product in S11(K(8))−

A Borcherds Product in S₁₁(K(8))⁻