A Borcherds Product in S₈(K(16))⁺

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c vector |

θ vector |

ψ expansion |

Borch(ψ) coefficients

• The singular part of ψ truncated up to q^(16/4) is
  (16 + ζ^(-5) + ζ^(-4) + 3/ζ^2 + 11/ζ + 11*ζ + 3*ζ^2 + ζ^4 + ζ^5)
  +q(3/ζ^8 + 3*ζ^8)
  +q^3(ζ^(-14) + ζ^14)
  +q^4(-2/ζ^16 - 2*ζ^16)
• From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block

  	φ	= TB(8;1,1,1,1,1,1,1,1,1,1,1,2,2,2,4,5)
  		= 111234151
  		= η(τ)16(θ(τ,z)/η(τ))11(θ(τ,2z)/η(τ))3(θ(τ,4z)/η(τ))(θ(τ,5z)/η(τ))

  This theta block has q-order of vanishing 2. This theta block has index 2*16, 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 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)	16
  (4,2)	4
  (4,14)	2
  (16,4)	1
  (25,5)	1

• From the singular part above, we compute that the valuation Ord(ψ) is -0.390625, 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.

The expansion of ψ up to q^(16/4) is

(16 + ζ^(-5) + ζ^(-4) + 3/ζ^2 + 11/ζ + 11*ζ + 3*ζ^2 + ζ^4 + ζ^5)
+q(410 + 3/ζ^8 + 25/ζ^7 + 47/ζ^6 + 6/ζ^5 - 144/ζ^4 - 252/ζ^3 - 111/ζ^2 + 221/ζ + 221*ζ - 111*ζ^2 - 252*ζ^3 - 144*ζ^4 + 6*ζ^5 + 47*ζ^6 + 25*ζ^7 + 3*ζ^8)
+q^2(5148 - 5/ζ^11 - 51/ζ^10 - 106/ζ^9 + 50/ζ^8 + 631/ζ^7 + 1098/ζ^6 + 213/ζ^5 - 2112/ζ^4 - 3566/ζ^3 - 1559/ζ^2 + 2833/ζ + 2833*ζ - 1559*ζ^2 - 3566*ζ^3 - 2112*ζ^4 + 213*ζ^5 + 1098*ζ^6 + 631*ζ^7 + 50*ζ^8 - 106*ζ^9 - 51*ζ^10 - 5*ζ^11)
+q^3(41064 + ζ^(-14) + 27/ζ^13 + 80/ζ^12 - 152/ζ^11 - 1106/ζ^10 - 1928/ζ^9 + 396/ζ^8 + 7113/ζ^7 + 11500/ζ^6 + 2721/ζ^5 - 18192/ζ^4 - 30184/ζ^3 - 13211/ζ^2 + 22403/ζ + 22403*ζ - 13211*ζ^2 - 30184*ζ^3 - 18192*ζ^4 + 2721*ζ^5 + 11500*ζ^6 + 7113*ζ^7 + 396*ζ^8 - 1928*ζ^9 - 1106*ζ^10 - 152*ζ^11 + 80*ζ^12 + 27*ζ^13 + ζ^14)
+q^4(250996 - 2/ζ^16 - 22/ζ^15 + 115/ζ^14 + 875/ζ^13 + 1599/ζ^12 - 1517/ζ^11 - 11516/ζ^10 - 18223/ζ^9 + 1992/ζ^8 + 52818/ζ^7 + 82459/ζ^6 + 20323/ζ^5 - 116799/ζ^4 - 191311/ζ^3 - 83346/ζ^2 + 137057/ζ + 137057*ζ - 83346*ζ^2 - 191311*ζ^3 - 116799*ζ^4 + 20323*ζ^5 + 82459*ζ^6 + 52818*ζ^7 + 1992*ζ^8 - 18223*ζ^9 - 11516*ζ^10 - 1517*ζ^11 + 1599*ζ^12 + 875*ζ^13 + 115*ζ^14 - 22*ζ^15 - 2*ζ^16)

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	1
[16 -113 800]	31	0
[8 -113 1600]	31	0
[14 -47 160]	31	0
[20 -101 512]	39	0
[10 -91 832]	39	-1
[10 -101 1024]	39	-1
[8 -27 96]	39	-1
[24 -73 224]	47	0
[12 -73 448]	47	1
[4 9 32]	47	0
[8 -73 672]	47	1
[16 -55 192]	47	1
[2 4 32]	48	0
[26 76 224]	48	0
[6 12 32]	48	0
[14 36 96]	48	-1
[2 3 32]	55	0
[14 -29 64]	55	3
[14 -99 704]	55	3
[8 -35 160]	55	3
[2 2 32]	60	0
[30 -210 1472]	60	0
[16 -50 160]	60	-1
[16 -178 1984]	60	0
[10 -30 96]	60	-3
[10 -50 256]	60	-1
[8 -50 320]	60	-1
[8 -114 1632]	60	0
[2 1 32]	63	0
[16 -225 3168]	63	-2
[4 33 288]	63	-2
[12 -159 2112]	63	5
[8 -97 1184]	63	-2
[2 0 32]	64	0
[2 8 64]	64	0
[2 16 160]	64	0
[4 8 32]	64	0
[20 56 160]	64	-3
[10 -24 64]	64	-3
[10 -64 416]	64	2
[10 -144 2080]	64	0
[36 -469 6112]	71	0
[18 -181 1824]	71	-5
[18 -235 3072]	71	-11
[120 277 640]	71	0
[10 -43 192]	71	-5
[10 -53 288]	71	-11
[40 -441 4864]	79	0
[20 -121 736]	79	2
[190 441 1024]	79	0
[10 -121 1472]	79	2
[44 -397 3584]	87	0
[22 -45 96]	87	9
[22 -243 2688]	87	1
[12 -51 224]	87	9
[134 307 704]	87	0
[46 -138 416]	92	0
[46 -230 1152]	92	0
[4 6 32]	92	0
[24 -218 1984]	92	13
[26 70 192]	92	0
[162 374 864]	92	0
[6 26 128]	92	13
[26 -86 288]	92	-9
[12 -26 64]	92	13
[12 -38 128]	92	0
[12 -122 1248]	92	0
[12 -134 1504]	92	13
[48 -337 2368]	95	0
[24 -337 4736]	95	-5
[4 17 96]	95	10
[16 -81 416]	95	10
[12 -145 1760]	95	-5
[60 145 352]	95	-30
[10 15 32]	95	0
[52 -261 1312]	103	0
[4 5 32]	103	0
[26 -261 2624]	103	-1
[14 -59 256]	103	-1
[56 -169 512]	111	0
[28 -169 1024]	111	-6
[120 297 736]	111	23
[6 9 32]	111	0
[14 -169 2048]	111	-6
[14 -183 2400]	111	-41
[10 23 64]	111	23
[2 4 64]	112	0
[58 172 512]	112	0
[4 4 32]	112	0
[28 -84 256]	112	55
[28 -140 704]	112	-11
[28 -196 1376]	112	-20
[14 -28 64]	112	-20
[14 -84 512]	112	28
[8 12 32]	112	0
[16 -84 448]	112	55
[16 -148 1376]	112	28
[16 -212 2816]	112	69
[2 3 64]	119	0
[30 -61 128]	119	-19
[30 -451 6784]	119	0
[20 -61 192]	119	6
[90 221 544]	119	-28
[16 -67 288]	119	-19
[12 -61 320]	119	6
[24 61 160]	119	1
[12 -125 1312]	119	-28
[2 2 64]	124	0
[62 -434 3040]	124	0
[4 2 32]	124	0
[32 -226 1600]	124	-14
[16 -98 608]	124	0
[16 -226 3200]	124	-14
[8 34 160]	124	-5
[28 -94 320]	124	-9
[10 14 32]	124	0
[14 -46 160]	124	33
[14 -130 1216]	124	0
[14 -158 1792]	124	-5
[2 1 64]	127	0
[4 1 32]	127	0
[32 -449 6304]	127	17
[16 -193 2336]	127	17
[2 0 64]	128	0
[2 8 96]	128	0
[2 16 192]	128	0
[4 0 32]	128	0
[4 16 96]	128	-41
[6 8 32]	128	0
[22 -112 576]	128	-41
[22 -288 3776]	128	7
[18 -40 96]	128	22
[18 -112 704]	128	-50
[18 -256 3648]	128	46
[12 16 32]	128	0
[12 -64 352]	128	7
[68 -885 11520]	135	0
[34 -171 864]	135	52
[34 -341 3424]	135	35
[24 -267 2976]	135	28
[8 11 32]	135	0
[12 -75 480]	135	55
[72 -793 8736]	143	0
[36 -217 1312]	143	-9
[42 121 352]	143	-44
[6 7 32]	143	0
[6 25 128]	143	-49
[18 -199 2208]	143	-49
[12 -25 64]	143	-44
[14 -89 576]	143	-31
[76 -685 6176]	151	0
[38 -77 160]	151	-74
[38 -115 352]	151	-151
[8 19 64]	151	-151
[10 13 32]	151	0
[78 -234 704]	156	0
[78 -390 1952]	156	0
[40 -202 1024]	156	-132
[40 -522 6816]	156	-63
[6 6 32]	156	0
[8 10 32]	156	0
[20 -42 96]	156	-64
[20 -202 2048]	156	-131
[10 22 64]	156	-63
[82 198 480]	156	-13
[10 42 192]	156	-74
[30 -102 352]	156	50
[16 -54 192]	156	-121
[16 -182 2080]	156	-74
[80 -561 3936]	159	0
[40 -561 7872]	159	44
[46 -143 448]	159	77
[28 -143 736]	159	77
[30 81 224]	159	63
[16 -49 160]	159	53
[12 15 32]	159	0
[14 -143 1472]	159	63
[84 -421 2112]	167	0
[42 -379 3424]	167	8
[42 -421 4224]	167	58
[28 -421 6336]	167	0
[36 101 288]	167	63
[22 -69 224]	167	73
[14 -155 1728]	167	73
[14 -197 2784]	167	8
[16 -37 96]	167	63
[88 -265 800]	175	0
[4 9 64]	175	167
[8 9 32]	175	0
[20 -265 3520]	175	255
[2 4 96]	176	0
[90 268 800]	176	0
[30 -92 288]	176	42
[30 -332 3680]	176	-25
[30 -452 6816]	176	0
[22 -44 96]	176	49
[22 -132 800]	176	-60
[10 12 32]	176	0
[18 -92 480]	176	42
[18 -236 3104]	176	-70
[2 3 96]	183	0
[46 -93 192]	183	15
[46 -323 2272]	183	175
[6 3 32]	183	0
[38 -125 416]	183	138
[16 -227 3232]	183	175
[2 2 96]	188	0
[94 -658 4608]	188	0
[48 -146 448]	188	195
[48 -530 5856]	188	-13
[6 2 32]	188	0
[6 14 64]	188	-13
[142 350 864]	188	105
[24 -146 896]	188	118
[24 -338 4768]	188	64
[8 18 64]	188	195
[112 274 672]	188	-77
[12 14 32]	188	0
[12 50 224]	188	49
[32 -110 384]	188	133
[14 34 96]	188	105
[102 242 576]	188	-77
[14 62 288]	188	-77
[42 -146 512]	188	49
[2 1 96]	191	0
[48 -673 9440]	191	-53
[6 1 32]	191	0
[32 -417 5440]	191	-169
[8 33 160]	191	-110
[20 -223 2496]	191	-110
[36 95 256]	191	90
[16 -161 1632]	191	90
[18 -95 512]	191	-169
[18 -257 3680]	191	-151
[2 0 96]	192	0
[2 8 128]	192	0
[2 16 224]	192	0
[4 8 64]	192	10
[52 152 448]	192	-10
[6 0 32]	192	0
[6 24 128]	192	18
[8 8 32]	192	0
[24 -72 224]	192	14
[24 -120 608]	192	18
[26 -56 128]	192	-32
[26 -160 992]	192	76
[26 -368 5216]	192	-140
[12 24 64]	192	-10
[28 72 192]	192	10
[14 16 32]	192	0
[14 -72 384]	192	14
[100 -1301 16928]	199	0
[50 -651 8480]	199	17
[26 -341 4480]	199	-95
[10 11 32]	199	0
[10 21 64]	199	17
[16 -213 2848]	199	-95
[104 -1145 12608]	207	0
[478 1113 2592]	207	-405
[26 -391 5888]	207	0
[18 -39 96]	207	-155
[18 -57 192]	207	196
[108 -973 8768]	215	0
[54 -595 6560]	215	-228
[36 -397 4384]	215	-11
[6 13 64]	215	-228
[24 -77 256]	215	-29
[12 13 32]	215	0
[18 -109 672]	215	-94
[20 -205 2112]	215	169
[16 -179 2016]	215	-29
[110 -330 992]	220	0
[110 -550 2752]	220	0
[4 6 64]	220	652
[56 -506 4576]	220	195
[28 -58 128]	220	655
[28 -282 2848]	220	192
[28 -422 6368]	220	0
[10 10 32]	220	0
[34 86 224]	220	57
[170 390 896]	220	195
[14 58 256]	220	396
[34 -118 416]	220	-396
[16 -70 320]	220	451
[16 -198 2464]	220	396
[112 -785 5504]	223	0
[4 17 128]	223	85
[8 17 64]	223	85
[14 15 32]	223	0
[116 -581 2912]	231	0
[4 5 64]	231	-322
[40 -123 384]	231	-173
[30 -453 6848]	231	0
[24 -123 640]	231	-173
[40 -133 448]	231	-266
[20 -123 768]	231	-195
[104 251 608]	231	-231
[120 -361 1088]	239	0
[248 617 1536]	239	397
[48 -151 480]	239	304
[30 -151 768]	239	304
[24 -361 5440]	239	0
[10 41 192]	239	357
[20 -41 96]	239	323
[198 457 1056]	239	397
[22 -247 2784]	239	357
[2 4 128]	240	0
[122 364 1088]	240	0
[4 4 64]	240	-453
[60 -180 544]	240	308
[60 -300 1504]	240	384
[60 -420 2944]	240	-455
[6 12 64]	240	384
[46 132 384]	240	-240
[30 -60 128]	240	-636
[30 -180 1088]	240	564
[8 4 32]	240	0
[32 -100 320]	240	-343
[32 -356 3968]	240	-389
[10 20 64]	240	308
[34 92 256]	240	-268
[12 12 32]	240	0
[20 -60 192]	240	-437
[20 -100 512]	240	-343
[16 -36 96]	240	-240
[16 -164 1696]	240	-268
[16 -228 3264]	240	-455
[2 3 128]	247	0
[62 -931 13984]	247	925
[8 3 32]	247	0
[2 2 128]	252	0
[126 -882 6176]	252	0
[4 2 64]	252	-520
[42 -126 384]	252	471
[42 -210 1056]	252	-123
[8 2 32]	252	0
[32 -194 1184]	252	-128
[32 -450 6336]	252	-523
[24 -126 672]	252	471
[24 -318 4224]	252	745
[14 14 32]	252	0
[16 -194 2368]	252	-266
[18 -222 2752]	252	-385
[18 -258 3712]	252	-131
[2 1 128]	255	0
[4 1 64]	255	-418
[44 -575 7520]	255	107
[8 1 32]	255	0
[22 -223 2272]	255	-213
[22 -289 3808]	255	-225
[20 -65 224]	255	-303
[14 33 96]	255	107
[2 0 128]	256	0
[2 8 160]	256	0
[2 16 256]	256	0
[4 0 64]	256	908
[4 16 128]	256	-672
[8 0 32]	256	0
[8 16 64]	256	-672
[8 32 160]	256	576
[34 -72 160]	256	-128
[34 -480 6784]	256	1280
[26 -80 256]	256	-320
[26 -288 3200]	256	576
[26 -392 5920]	256	0
[16 16 32]	256	0
[16 -48 160]	256	-320
[20 -48 128]	256	-544
[20 -288 4160]	256	820
[132 -1717 22336]	263	0
[66 -331 1664]	263	-10
[408 949 2208]	263	-10
[22 -309 4352]	263	-165
[22 -331 4992]	263	0
[24 -53 128]	263	-33
[12 53 256]	263	2
[18 -203 2304]	263	2
[136 -1497 16480]	271	0
[74 217 640]	271	-426
[34 -103 320]	271	749
[28 -423 6400]	271	0
[134 327 800]	271	-410
[20 -103 544]	271	749
[212 487 1120]	271	-426
[124 295 704]	271	-410
[140 -1261 11360]	279	0
[70 -211 640]	279	-670
[28 -365 4768]	279	-20
[268 621 1440]	279	-670
[24 -339 4800]	279	-390
[42 -141 480]	279	-477
[20 -301 4544]	279	0
[98 237 576]	279	-1073
[18 -237 3136]	279	-20
[20 -109 608]	279	787
[142 -426 1280]	284	0
[142 -710 3552]	284	0
[72 -362 1824]	284	-624
[72 -938 12224]	284	-963
[450 1046 2432]	284	-624
[6 26 160]	284	258
[58 -182 576]	284	-698
[36 -74 160]	284	-320
[36 -182 928]	284	-698
[36 -362 3648]	284	-1267
[36 -470 6144]	284	-569
[30 -154 800]	284	258
[30 -394 5184]	284	-634
[30 -454 6880]	284	0
[24 -362 5472]	284	0
[240 554 1280]	284	-963
[18 -38 96]	284	160
[18 -182 1856]	284	-240
[20 -106 576]	284	-569
[20 -266 3552]	284	-634
[144 -1009 7072]	287	0
[78 -239 736]	287	682
[48 -241 1216]	287	480
[338 785 1824]	287	682
[24 -241 2432]	287	-91
[12 49 224]	287	-162
[26 -369 5248]	287	347
[24 -271 3072]	287	-162
[18 -271 4096]	287	0
[148 -741 3712]	295	0
[74 -667 6016]	295	60
[38 -421 4672]	295	-36
[10 5 32]	295	0
[226 517 1184]	295	60
[152 -457 1376]	303	0
[4 9 96]	303	503
[6 9 64]	303	503
[26 -55 128]	303	-184
[26 -393 5952]	303	0
[22 -73 256]	303	-528
[2 4 160]	304	0
[154 460 1376]	304	0
[38 -76 160]	304	508
[10 4 32]	304	0
[10 44 224]	304	-298
[50 -164 544]	304	134
[22 -68 224]	304	683
[22 -244 2720]	304	-298
[22 -332 5024]	304	0
[2 3 160]	311	0
[52 -157 480]	311	-379
[218 541 1344]	311	139
[8 35 192]	311	58
[10 3 32]	311	0
[26 -131 672]	311	58
[26 -157 960]	311	799
[28 -285 2912]	311	13
[12 29 96]	311	-379
[16 35 96]	311	139
[20 -227 2592]	311	308
[240 547 1248]	311	1042
[2 2 160]	316	0
[158 -1106 7744]	316	0
[80 -242 736]	316	127
[80 -882 9728]	316	627
[40 -562 7904]	316	128
[380 882 2048]	316	627
[10 2 32]	316	0
[310 718 1664]	316	127
[20 -302 4576]	316	0
[22 -50 128]	316	-1
[22 -226 2336]	316	669
[2 1 160]	319	0
[50 -159 512]	319	-799
[10 1 32]	319	0
[32 -353 3904]	319	-799
[38 97 256]	319	1145
[20 -289 4192]	319	-949
[2 0 160]	320	0
[2 8 192]	320	0
[4 8 96]	320	436
[84 248 736]	320	-184
[6 8 64]	320	436
[54 -272 1376]	320	-278
[54 -704 9184]	320	-834
[42 -88 192]	320	380
[42 -592 8352]	320	-644
[10 0 32]	320	0
[10 40 192]	320	-58
[28 -312 3488]	320	-58
[28 -424 6432]	320	0
[14 32 96]	320	-834
[126 304 736]	320	498
[254 584 1344]	320	-184
[20 -40 96]	320	-236
[20 -120 736]	320	488
[18 -56 192]	320	182
[18 -272 4128]	320	0
[164 -2133 27744]	327	0
[82 -1067 13888]	327	730
[56 -619 6848]	327	200
[42 -213 1088]	327	200
[282 651 1504]	327	730
[24 -363 5504]	327	0
[44 -149 512]	327	944
[168 -1849 20352]	335	0
[766 1785 4160]	335	-783
[6 7 64]	335	-783
[6 25 160]	335	-244
[36 -185 960]	335	-244
[28 -57 128]	335	-1022
[30 -455 6912]	335	0
[24 -313 4096]	335	-1446
[14 57 256]	335	-790
[54 -185 640]	335	-790
[118 281 672]	335	161
[172 -1549 13952]	343	0
[86 -947 10432]	343	-1865
[422 979 2272]	343	-1865
[28 -371 4928]	343	854
[22 -333 5056]	343	0
[174 -522 1568]	348	0
[174 -870 4352]	348	0
[4 6 96]	348	-88
[88 -794 7168]	348	1271
[6 6 64]	348	-88
[44 -90 192]	348	607
[44 -134 416]	348	-1048
[44 -486 5376]	348	1655
[12 6 32]	348	0
[26 -134 704]	348	-1048
[26 -342 4512]	348	840
[26 -394 5984]	348	0
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