A Borcherds Product in S₁₁(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
  q^-1(1)
  +(22 + 2/ζ^3 + ζ^(-2) + 10/ζ + 10*ζ + ζ^2 + 2*ζ^3)
  +q(ζ^(-9) + 19/ζ^8 + 19*ζ^8 + ζ^9)
  +q^3(7/ζ^14 + 7*ζ^14)
  +q^4(10/ζ^16 + 10*ζ^16)
• 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 1*16, and so Borch(ψ) begins with Jacobi coefficient #1.
• From the truncated singular part of ψ, we compute that Borch(ψ) is an antisymmetric 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 2 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
  (4,14)	7
  (9,3)	2
  (17,9)	1
  (64,0)	1

• We compute that the valuation Ord(ψ - (φ|V₂)/φ) is -0.5625, and hence we need to multiply (ψ - (φ|V₂)/φ) by Δ^1 to get a Jacobi cusp form. The definition of ψ is
      ψ = (φ|V₂)/φ + 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

q^-1(1)
+(22 + 2/ζ^3 + ζ^(-2) + 10/ζ + 10*ζ + ζ^2 + 2*ζ^3)
+q(33150 + ζ^(-9) + 19/ζ^8 + 209/ζ^7 + 1021/ζ^6 + 3330/ζ^5 + 8120/ζ^4 + 15262/ζ^3 + 23555/ζ^2 + 30350/ζ + 30350*ζ + 23555*ζ^2 + 15262*ζ^3 + 8120*ζ^4 + 3330*ζ^5 + 1021*ζ^6 + 209*ζ^7 + 19*ζ^8 + ζ^9)
+q^2(3082016 + 65/ζ^11 + 1019/ζ^10 + 7307/ζ^9 + 32928/ζ^8 + 108903/ζ^7 + 283402/ζ^6 + 610091/ζ^5 + 1113040/ζ^4 + 1750956/ζ^3 + 2401531/ζ^2 + 2896630/ζ + 2896630*ζ + 2401531*ζ^2 + 1750956*ζ^3 + 1113040*ζ^4 + 610091*ζ^5 + 283402*ζ^6 + 108903*ζ^7 + 32928*ζ^8 + 7307*ζ^9 + 1019*ζ^10 + 65*ζ^11)
+q^3(112735122 + 7/ζ^14 + 555/ζ^13 + 8072/ζ^12 + 58422/ζ^11 + 283878/ζ^10 + 1042031/ζ^9 + 3082168/ζ^8 + 7617681/ζ^7 + 16135448/ζ^6 + 29800722/ζ^5 + 48583896/ζ^4 + 70498989/ζ^3 + 91616763/ζ^2 + 107053792/ζ + 107053792*ζ + 91616763*ζ^2 + 70498989*ζ^3 + 48583896*ζ^4 + 29800722*ζ^5 + 16135448*ζ^6 + 7617681*ζ^7 + 3082168*ζ^8 + 1042031*ζ^9 + 283878*ζ^10 + 58422*ζ^11 + 8072*ζ^12 + 555*ζ^13 + 7*ζ^14)
+q^4(2467337196 + 10/ζ^16 + 1427/ζ^15 + 23557/ζ^14 + 197570/ζ^13 + 1112624/ζ^12 + 4727746/ζ^11 + 16134856/ζ^10 + 46038049/ζ^9 + 112727936/ζ^8 + 241240787/ζ^7 + 457188221/ζ^6 + 774915036/ζ^5 + 1183229008/ζ^4 + 1636308008/ζ^3 + 2057366966/ζ^2 + 2358047729/ζ + 2358047729*ζ + 2057366966*ζ^2 + 1636308008*ζ^3 + 1183229008*ζ^4 + 774915036*ζ^5 + 457188221*ζ^6 + 241240787*ζ^7 + 112727936*ζ^8 + 46038049*ζ^9 + 16134856*ζ^10 + 4727746*ζ^11 + 1112624*ζ^12 + 197570*ζ^13 + 23557*ζ^14 + 1427*ζ^15 + 10*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	0
[16 -113 800]	31	1
[8 -113 1600]	31	-1
[14 -47 160]	31	0
[20 -101 512]	39	0
[10 -91 832]	39	0
[10 -101 1024]	39	0
[8 -27 96]	39	0
[24 -73 224]	47	-1
[12 -73 448]	47	0
[4 9 32]	47	-1
[8 -73 672]	47	0
[16 -55 192]	47	0
[2 4 32]	48	0
[26 76 224]	48	-2
[6 12 32]	48	2
[14 36 96]	48	0
[2 3 32]	55	0
[14 -29 64]	55	-7
[14 -99 704]	55	-7
[8 -35 160]	55	0
[2 2 32]	60	0
[30 -210 1472]	60	-8
[16 -50 160]	60	1
[16 -178 1984]	60	0
[10 -30 96]	60	0
[10 -50 256]	60	-1
[8 -50 320]	60	0
[8 -114 1632]	60	8
[2 1 32]	63	0
[16 -225 3168]	63	3
[12 -159 2112]	63	0
[8 -97 1184]	63	0
[2 0 32]	64	0
[2 8 64]	64	-10
[2 16 160]	64	0
[4 8 32]	64	10
[20 56 160]	64	-1
[10 -24 64]	64	-1
[10 -64 416]	64	0
[10 -144 2080]	64	0
[36 -469 6112]	71	19
[18 -181 1824]	71	-21
[18 -235 3072]	71	2
[120 277 640]	71	19
[10 -43 192]	71	0
[10 -53 288]	71	2
[40 -441 4864]	79	44
[20 -121 736]	79	-35
[190 441 1024]	79	44
[10 -121 1472]	79	0
[44 -397 3584]	87	-44
[22 -45 96]	87	48
[22 -243 2688]	87	0
[12 -51 224]	87	0
[134 307 704]	87	44
[46 -138 416]	92	88
[46 -230 1152]	92	112
[4 6 32]	92	112
[24 -218 1984]	92	50
[26 70 192]	92	6
[162 374 864]	92	-88
[6 26 128]	92	19
[26 -86 288]	92	0
[12 -26 64]	92	-50
[12 -38 128]	92	0
[12 -122 1248]	92	-6
[12 -134 1504]	92	-19
[48 -337 2368]	95	18
[24 -337 4736]	95	-3
[4 17 96]	95	-9
[16 -81 416]	95	-9
[12 -145 1760]	95	0
[60 145 352]	95	0
[10 15 32]	95	-18
[52 -261 1312]	103	-185
[4 5 32]	103	-185
[26 -261 2624]	103	168
[14 -59 256]	103	0
[56 -169 512]	111	-258
[28 -169 1024]	111	294
[120 297 736]	111	-112
[6 9 32]	111	-258
[14 -169 2048]	111	0
[14 -183 2400]	111	0
[10 23 64]	111	-112
[2 4 64]	112	-220
[58 172 512]	112	-172
[4 4 32]	112	220
[28 -84 256]	112	46
[28 -140 704]	112	0
[28 -196 1376]	112	-414
[14 -28 64]	112	-414
[14 -84 512]	112	-14
[8 12 32]	112	172
[16 -84 448]	112	-46
[16 -148 1376]	112	14
[16 -212 2816]	112	0
[2 3 64]	119	221
[30 -61 128]	119	-102
[30 -451 6784]	119	221
[20 -61 192]	119	-27
[90 221 544]	119	-53
[12 -61 320]	119	27
[24 61 160]	119	0
[12 -125 1312]	119	-53
[2 2 64]	124	-208
[62 -434 3040]	124	-88
[4 2 32]	124	208
[16 -98 608]	124	14
[16 -226 3200]	124	778
[8 34 160]	124	-161
[28 -94 320]	124	0
[10 14 32]	124	88
[14 -46 160]	124	0
[14 -158 1792]	124	161
[2 1 64]	127	143
[4 1 32]	127	-143
[32 -449 6304]	127	81
[2 0 64]	128	0
[2 8 96]	128	-488
[2 16 192]	128	0
[4 0 32]	128	0
[4 16 96]	128	-63
[6 8 32]	128	488
[22 -112 576]	128	-63
[22 -288 3776]	128	-105
[18 -40 96]	128	208
[18 -112 704]	128	0
[18 -256 3648]	128	0
[12 16 32]	128	0
[12 -64 352]	128	-105
[68 -885 11520]	135	407
[34 -171 864]	135	0
[34 -341 3424]	135	-378
[24 -267 2976]	135	0
[8 11 32]	135	-407
[12 -75 480]	135	0
[72 -793 8736]	143	506
[36 -217 1312]	143	-672
[42 121 352]	143	487
[6 7 32]	143	-506
[6 25 128]	143	180
[18 -199 2208]	143	-180
[12 -25 64]	143	487
[14 -89 576]	143	0
[76 -685 6176]	151	-198
[38 -77 160]	151	109
[38 -115 352]	151	224
[8 19 64]	151	224
[10 13 32]	151	-198
[78 -234 704]	156	440
[78 -390 1952]	156	40
[40 -202 1024]	156	0
[40 -522 6816]	156	217
[6 6 32]	156	40
[8 10 32]	156	440
[20 -42 96]	156	650
[20 -202 2048]	156	86
[10 22 64]	156	-217
[82 198 480]	156	0
[10 42 192]	156	798
[30 -102 352]	156	0
[16 -54 192]	156	0
[16 -182 2080]	156	-798
[80 -561 3936]	159	-1
[40 -561 7872]	159	-240
[46 -143 448]	159	-348
[28 -143 736]	159	348
[30 81 224]	159	-154
[16 -49 160]	159	0
[12 15 32]	159	1
[14 -143 1472]	159	154
[84 -421 2112]	167	579
[42 -379 3424]	167	272
[42 -421 4224]	167	-294
[28 -421 6336]	167	-579
[22 -69 224]	167	358
[14 -155 1728]	167	358
[14 -197 2784]	167	272
[16 -37 96]	167	1145
[88 -265 800]	175	465
[4 9 64]	175	0
[8 9 32]	175	465
[20 -265 3520]	175	0
[2 4 96]	176	726
[90 268 800]	176	216
[30 -92 288]	176	-670
[30 -332 3680]	176	0
[30 -452 6816]	176	726
[22 -44 96]	176	-1176
[22 -132 800]	176	-666
[10 12 32]	176	-216
[18 -92 480]	176	670
[18 -236 3104]	176	0
[2 3 96]	183	-686
[6 3 32]	183	686
[38 -125 416]	183	0
[16 -227 3232]	183	2436
[2 2 96]	188	1072
[94 -658 4608]	188	376
[48 -146 448]	188	494
[48 -530 5856]	188	-401
[6 2 32]	188	-1072
[6 14 64]	188	401
[142 350 864]	188	1968
[24 -146 896]	188	1020
[24 -338 4768]	188	3484
[8 18 64]	188	494
[112 274 672]	188	835
[12 14 32]	188	-376
[12 50 224]	188	-2551
[32 -110 384]	188	0
[14 34 96]	188	1968
[102 242 576]	188	-835
[14 62 288]	188	0
[42 -146 512]	188	-2551
[2 1 96]	191	-1120
[6 1 32]	191	1120
[32 -417 5440]	191	-256
[8 33 160]	191	-1732
[20 -223 2496]	191	1732
[16 -161 1632]	191	166
[18 -95 512]	191	-256
[18 -257 3680]	191	1819
[2 0 96]	192	0
[2 8 128]	192	2672
[2 16 224]	192	0
[4 8 64]	192	0
[52 152 448]	192	3064
[6 0 32]	192	0
[6 24 128]	192	698
[8 8 32]	192	-2672
[24 -72 224]	192	-14
[24 -120 608]	192	698
[26 -56 128]	192	2744
[26 -368 5216]	192	0
[12 24 64]	192	-3064
[28 72 192]	192	0
[14 16 32]	192	0
[14 -72 384]	192	14
[100 -1301 16928]	199	-2433
[50 -651 8480]	199	-1870
[26 -341 4480]	199	1508
[10 11 32]	199	2433
[10 21 64]	199	1870
[16 -213 2848]	199	-1508
[104 -1145 12608]	207	-4565
[478 1113 2592]	207	0
[26 -391 5888]	207	-4565
[18 -39 96]	207	6144
[18 -57 192]	207	0
[108 -973 8768]	215	2442
[54 -595 6560]	215	502
[36 -397 4384]	215	0
[6 13 64]	215	-502
[24 -77 256]	215	-1654
[12 13 32]	215	2442
[18 -109 672]	215	875
[20 -205 2112]	215	2412
[16 -179 2016]	215	-1654
[110 -330 992]	220	-6248
[110 -550 2752]	220	-3560
[4 6 64]	220	0
[56 -506 4576]	220	-8465
[28 -58 128]	220	10510
[28 -282 2848]	220	5922
[28 -422 6368]	220	3560
[10 10 32]	220	-6248
[34 86 224]	220	0
[170 390 896]	220	8465
[34 -118 416]	220	0
[16 -198 2464]	220	-5472
[112 -785 5504]	223	-587
[4 17 128]	223	3841
[8 17 64]	223	-3841
[14 15 32]	223	587
[116 -581 2912]	231	1423
[4 5 64]	231	0
[40 -123 384]	231	-796
[30 -453 6848]	231	-1423
[24 -123 640]	231	796
[40 -133 448]	231	0
[20 -123 768]	231	-528
[104 251 608]	231	0
[120 -361 1088]	239	7582
[248 617 1536]	239	6578
[48 -151 480]	239	482
[30 -151 768]	239	-482
[24 -361 5440]	239	-7582
[10 41 192]	239	8413
[20 -41 96]	239	10640
[198 457 1056]	239	-6578
[22 -247 2784]	239	-8413
[2 4 128]	240	2980
[122 364 1088]	240	6068
[4 4 64]	240	0
[60 -180 544]	240	2908
[60 -300 1504]	240	1338
[6 12 64]	240	1338
[30 -60 128]	240	10682
[30 -180 1088]	240	-1814
[8 4 32]	240	-2980
[32 -100 320]	240	2214
[32 -356 3968]	240	0
[10 20 64]	240	2908
[12 12 32]	240	-6068
[20 -60 192]	240	0
[20 -100 512]	240	-2214
[16 -36 96]	240	-5462
[16 -164 1696]	240	-1036
[16 -228 3264]	240	-10778
[2 3 128]	247	-5042
[62 -931 13984]	247	0
[8 3 32]	247	5042
[2 2 128]	252	536
[126 -882 6176]	252	2192
[4 2 64]	252	0
[42 -126 384]	252	-5277
[42 -210 1056]	252	0
[8 2 32]	252	-536
[32 -450 6336]	252	-3460
[24 -126 672]	252	5277
[24 -318 4224]	252	0
[14 14 32]	252	-2192
[18 -258 3712]	252	-6807
[2 1 128]	255	4165
[4 1 64]	255	0
[44 -575 7520]	255	-8152
[8 1 32]	255	-4165
[22 -223 2272]	255	5468
[22 -289 3808]	255	0
[20 -65 224]	255	0
[14 33 96]	255	8152
[2 0 128]	256	0
[2 8 160]	256	1888
[4 0 64]	256	0
[4 16 128]	256	-1456
[8 0 32]	256	0
[8 16 64]	256	1456
[8 32 160]	256	-5164
[34 -72 160]	256	-6464
[34 -480 6784]	256	0
[26 -288 3200]	256	5164
[26 -392 5920]	256	1888
[16 16 32]	256	0
[16 -48 160]	256	2828
[20 -48 128]	256	-9808
[20 -288 4160]	256	0
[132 -1717 22336]	263	-3699
[66 -331 1664]	263	1358
[408 949 2208]	263	-1358
[22 -309 4352]	263	868
[22 -331 4992]	263	-3699
[24 -53 128]	263	-5884
[18 -203 2304]	263	5423
[136 -1497 16480]	271	6160
[74 217 640]	271	-8538
[34 -103 320]	271	6224
[28 -423 6400]	271	6160
[20 -103 544]	271	-6224
[212 487 1120]	271	-8538
[124 295 704]	271	5904
[140 -1261 11360]	279	-726
[70 -211 640]	279	-12299
[268 621 1440]	279	12299
[24 -339 4800]	279	20466
[42 -141 480]	279	0
[20 -301 4544]	279	726
[98 237 576]	279	0
[18 -237 3136]	279	6276
[20 -109 608]	279	11523
[142 -426 1280]	284	11792
[142 -710 3552]	284	3512
[72 -362 1824]	284	-8386
[72 -938 12224]	284	-12500
[450 1046 2432]	284	8386
[6 26 160]	284	-1256
[58 -182 576]	284	1876
[36 -74 160]	284	-1090
[36 -182 928]	284	-1876
[36 -362 3648]	284	18042
[30 -154 800]	284	-1256
[30 -454 6880]	284	-3512
[24 -362 5472]	284	-11792
[240 554 1280]	284	-12500
[18 -38 96]	284	-536
[18 -182 1856]	284	-4894
[20 -106 576]	284	10988
[20 -266 3552]	284	5792
[78 -239 736]	287	411
[48 -241 1216]	287	0
[338 785 1824]	287	-411
[24 -241 2432]	287	-10398
[12 49 224]	287	-25471
[26 -369 5248]	287	13818
[24 -271 3072]	287	25471
[18 -271 4096]	287	329
[148 -741 3712]	295	3975
[38 -421 4672]	295	0
[10 5 32]	295	3975
[226 517 1184]	295	171
[152 -457 1376]	303	-28451
[4 9 96]	303	-11104
[6 9 64]	303	11104
[26 -55 128]	303	2549
[26 -393 5952]	303	28451
[2 4 160]	304	-9544
[154 460 1376]	304	-20734
[38 -76 160]	304	146
[10 4 32]	304	9544
[10 44 224]	304	9642
[50 -164 544]	304	0
[22 -68 224]	304	0
[22 -244 2720]	304	-9642
[22 -332 5024]	304	-20734
[2 3 160]	311	28816
[52 -157 480]	311	-2867
[8 35 192]	311	-6316
[10 3 32]	311	-28816
[26 -131 672]	311	-6316
[28 -285 2912]	311	32320
[12 29 96]	311	-2867
[16 35 96]	311	27205
[20 -227 2592]	311	-16528
[240 547 1248]	311	6381
[2 2 160]	316	-9848
[80 -242 736]	316	-6115
[80 -882 9728]	316	-16290
[40 -562 7904]	316	-4382
[380 882 2048]	316	-16290
[10 2 32]	316	9848
[310 718 1664]	316	6115
[20 -302 4576]	316	-9368
[22 -50 128]	316	-33051
[22 -226 2336]	316	-10096
[2 1 160]	319	-17519
[50 -159 512]	319	8361
[10 1 32]	319	17519
[32 -353 3904]	319	8361
[20 -289 4192]	319	-6667
[2 0 160]	320	0
[2 8 192]	320	-25048
[4 8 96]	320	15206
[84 248 736]	320	-28270
[6 8 64]	320	-15206
[54 -272 1376]	320	0
[54 -704 9184]	320	17136
[42 -88 192]	320	-27388
[42 -592 8352]	320	0
[10 0 32]	320	0
[10 40 192]	320	29954
[28 -312 3488]	320	-29954
[28 -424 6432]	320	-25048
[14 32 96]	320	-17136
[126 304 736]	320	0
[254 584 1344]	320	-28270
[20 -40 96]	320	11562
[20 -120 736]	320	-2306
[18 -56 192]	320	-7126
[18 -272 4128]	320	0
[164 -2133 27744]	327	42335
[82 -1067 13888]	327	14581
[56 -619 6848]	327	-9082
[42 -213 1088]	327	-9082
[282 651 1504]	327	14581
[24 -363 5504]	327	42335
[44 -149 512]	327	0
[168 -1849 20352]	335	10563
[766 1785 4160]	335	27106
[6 7 64]	335	27106
[6 25 160]	335	-14162
[36 -185 960]	335	-14162
[28 -57 128]	335	7610
[30 -455 6912]	335	10563
[24 -313 4096]	335	0
[54 -185 640]	335	56082
[118 281 672]	335	912
[86 -947 10432]	343	8905
[422 979 2272]	343	8905
[28 -371 4928]	343	0
[22 -333 5056]	343	28457
[174 -522 1568]	348	24904
[174 -870 4352]	348	28944
[4 6 96]	348	29730
[6 6 64]	348	-29730
[44 -90 192]	348	-51054
[44 -134 416]	348	15558
[44 -486 5376]	348	0
[12 6 32]	348	28944
[26 -134 704]	348	-15558
[26 -342 4512]	348	0
[26 -394 5984]	348	-24904
[22 -310 4384]	348	296
[268 614 1408]	348	-45280
[4 17 160]	351	-8812
[60 -303 1536]	351	0
[352 815 1888]	351	-8812
[30 -303 3072]	351	-14010
[22 -111 576]	351	23116
[20 -303 4608]	351	5395
[20 -47 128]	351	11934
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