A Borcherds Product in S₁₃(K(16))⁺

properties |

c vector |

θ vector |

ψ expansion |

Borch(ψ) coefficients

• The singular part of ψ truncated up to q^(16/4) is
  q^-1(1)
  +(26 + 7/ζ^2 + 4/ζ + 4*ζ + 7*ζ^2)
  +q(19/ζ^8 + 19*ζ^8)
  +q^3(ζ^(-15) + ζ^(-14) + ζ^14 + ζ^15)
  +q^4(ζ^(-17) + 6/ζ^16 + 6*ζ^16 + ζ^17)
• From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block

  	φ	= TB(13;1,1,1,1,2,2,2,2,2,2,2)
  		= 1427
  		= η(τ)26(θ(τ,z)/η(τ))4(θ(τ,2z)/η(τ))7

  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 3 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)	11
  (4,2)	7
  (4,14)	1
  (33,15)	1
  (64,0)	1

• We compute that the valuation Ord(ψ - (φ|V₂)/φ) is -0.515625, 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)
+(26 + 7/ζ^2 + 4/ζ + 4*ζ + 7*ζ^2)
+q(34382 + 19/ζ^8 + 185/ζ^7 + 1083/ζ^6 + 3199/ζ^5 + 8464/ζ^4 + 14585/ζ^3 + 24581/ζ^2 + 29135/ζ + 29135*ζ + 24581*ζ^2 + 14585*ζ^3 + 8464*ζ^4 + 3199*ζ^5 + 1083*ζ^6 + 185*ζ^7 + 19*ζ^8)
+q^2(3110240 + 70/ζ^11 + 957/ζ^10 + 7415/ζ^9 + 32928/ζ^8 + 108219/ζ^7 + 285670/ζ^6 + 605026/ζ^5 + 1122992/ζ^4 + 1734468/ζ^3 + 2424413/ζ^2 + 2869602/ζ + 2869602*ζ + 2424413*ζ^2 + 1734468*ζ^3 + 1122992*ζ^4 + 605026*ζ^5 + 285670*ζ^6 + 108219*ζ^7 + 32928*ζ^8 + 7415*ζ^9 + 957*ζ^10 + 70*ζ^11)
+q^3(113075538 + ζ^(-15) + ζ^(-14) + 645/ζ^13 + 7728/ζ^12 + 59391/ζ^11 + 281610/ζ^10 + 1045016/ζ^9 + 3082168/ζ^8 + 7606386/ζ^7 + 16170376/ζ^6 + 29726001/ζ^5 + 48718416/ζ^4 + 70291211/ζ^3 + 91894125/ζ^2 + 106729141/ζ + 106729141*ζ + 91894125*ζ^2 + 70291211*ζ^3 + 48718416*ζ^4 + 29726001*ζ^5 + 16170376*ζ^6 + 7606386*ζ^7 + 3082168*ζ^8 + 1045016*ζ^9 + 281610*ζ^10 + 59391*ζ^11 + 7728*ζ^12 + 645*ζ^13 + ζ^14 + ζ^15)
+q^4(2470241780 + ζ^(-17) + 6/ζ^16 + 1609/ζ^15 + 22531/ζ^14 + 201326/ζ^13 + 1102672/ζ^12 + 4748625/ζ^11 + 16099928/ζ^10 + 46075845/ζ^9 + 112727936/ζ^8 + 241123461/ζ^7 + 457537163/ζ^6 + 774195945/ζ^5 + 1184461360/ζ^4 + 1634478748/ζ^3 + 2059753242/ζ^2 + 2355276840/ζ + 2355276840*ζ + 2059753242*ζ^2 + 1634478748*ζ^3 + 1184461360*ζ^4 + 774195945*ζ^5 + 457537163*ζ^6 + 241123461*ζ^7 + 112727936*ζ^8 + 46075845*ζ^9 + 16099928*ζ^10 + 4748625*ζ^11 + 1102672*ζ^12 + 201326*ζ^13 + 22531*ζ^14 + 1609*ζ^15 + 6*ζ^16 + ζ^17)

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	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	1
[4 9 32]	47	-1
[8 -73 672]	47	-1
[16 -55 192]	47	0
[2 4 32]	48	0
[26 76 224]	48	0
[6 12 32]	48	0
[14 36 96]	48	0
[2 3 32]	55	0
[14 -29 64]	55	0
[14 -99 704]	55	0
[8 -35 160]	55	0
[2 2 32]	60	0
[30 -210 1472]	60	0
[16 -50 160]	60	0
[16 -178 1984]	60	0
[10 -30 96]	60	0
[10 -50 256]	60	0
[8 -50 320]	60	0
[8 -114 1632]	60	0
[2 1 32]	63	0
[16 -225 3168]	63	0
[12 -159 2112]	63	0
[8 -97 1184]	63	0
[2 0 32]	64	0
[2 8 64]	64	-4
[2 16 160]	64	0
[4 8 32]	64	4
[20 56 160]	64	-1
[10 -24 64]	64	-1
[10 -64 416]	64	0
[10 -144 2080]	64	0
[36 -469 6112]	71	-7
[18 -181 1824]	71	7
[18 -235 3072]	71	0
[120 277 640]	71	-7
[10 -43 192]	71	0
[10 -53 288]	71	0
[40 -441 4864]	79	-1
[20 -121 736]	79	1
[190 441 1024]	79	-1
[10 -121 1472]	79	0
[44 -397 3584]	87	-21
[22 -45 96]	87	21
[22 -243 2688]	87	0
[12 -51 224]	87	0
[134 307 704]	87	21
[46 -138 416]	92	-24
[46 -230 1152]	92	-24
[4 6 32]	92	-24
[24 -218 1984]	92	-6
[26 70 192]	92	1
[162 374 864]	92	24
[6 26 128]	92	-1
[26 -86 288]	92	0
[12 -26 64]	92	6
[12 -38 128]	92	0
[12 -122 1248]	92	-1
[12 -134 1504]	92	1
[48 -337 2368]	95	-35
[24 -337 4736]	95	35
[4 17 96]	95	-19
[16 -81 416]	95	-19
[12 -145 1760]	95	0
[60 145 352]	95	0
[10 15 32]	95	35
[52 -261 1312]	103	20
[4 5 32]	103	20
[26 -261 2624]	103	-20
[14 -59 256]	103	0
[56 -169 512]	111	3
[28 -169 1024]	111	-3
[120 297 736]	111	-3
[6 9 32]	111	3
[14 -169 2048]	111	0
[14 -183 2400]	111	0
[10 23 64]	111	-3
[2 4 64]	112	-56
[58 172 512]	112	-56
[4 4 32]	112	56
[28 -84 256]	112	70
[28 -140 704]	112	0
[28 -196 1376]	112	-14
[14 -28 64]	112	-14
[14 -84 512]	112	-70
[8 12 32]	112	56
[16 -84 448]	112	-70
[16 -148 1376]	112	70
[16 -212 2816]	112	0
[2 3 64]	119	84
[30 -61 128]	119	-84
[30 -451 6784]	119	84
[20 -61 192]	119	0
[90 221 544]	119	-70
[12 -61 320]	119	0
[24 61 160]	119	0
[12 -125 1312]	119	-70
[2 2 64]	124	56
[62 -434 3040]	124	56
[4 2 32]	124	-56
[16 -98 608]	124	-115
[16 -226 3200]	124	-14
[8 34 160]	124	19
[28 -94 320]	124	0
[10 14 32]	124	-56
[14 -46 160]	124	0
[14 -158 1792]	124	-19
[2 1 64]	127	-154
[4 1 32]	127	154
[32 -449 6304]	127	-154
[2 0 64]	128	0
[2 8 96]	128	-64
[2 16 192]	128	0
[4 0 32]	128	0
[4 16 96]	128	-109
[6 8 32]	128	64
[22 -112 576]	128	-109
[22 -288 3776]	128	19
[18 -40 96]	128	-4
[18 -112 704]	128	0
[18 -256 3648]	128	0
[12 16 32]	128	0
[12 -64 352]	128	19
[68 -885 11520]	135	48
[34 -171 864]	135	0
[34 -341 3424]	135	-48
[24 -267 2976]	135	0
[8 11 32]	135	-48
[12 -75 480]	135	0
[72 -793 8736]	143	79
[36 -217 1312]	143	-79
[42 121 352]	143	-19
[6 7 32]	143	-79
[6 25 128]	143	169
[18 -199 2208]	143	-169
[12 -25 64]	143	-19
[14 -89 576]	143	0
[76 -685 6176]	151	168
[38 -77 160]	151	-168
[38 -115 352]	151	544
[8 19 64]	151	544
[10 13 32]	151	168
[78 -234 704]	156	120
[78 -390 1952]	156	120
[40 -202 1024]	156	0
[40 -522 6816]	156	-150
[6 6 32]	156	120
[8 10 32]	156	120
[20 -42 96]	156	3
[20 -202 2048]	156	150
[10 22 64]	156	150
[82 198 480]	156	0
[10 42 192]	156	-165
[30 -102 352]	156	0
[16 -54 192]	156	0
[16 -182 2080]	156	165
[80 -561 3936]	159	294
[40 -561 7872]	159	-294
[46 -143 448]	159	324
[28 -143 736]	159	-324
[30 81 224]	159	96
[16 -49 160]	159	0
[12 15 32]	159	-294
[14 -143 1472]	159	-96
[84 -421 2112]	167	-67
[42 -379 3424]	167	-52
[42 -421 4224]	167	67
[28 -421 6336]	167	67
[22 -69 224]	167	-70
[14 -155 1728]	167	-70
[14 -197 2784]	167	-52
[16 -37 96]	167	221
[88 -265 800]	175	77
[4 9 64]	175	0
[8 9 32]	175	77
[20 -265 3520]	175	0
[2 4 96]	176	640
[90 268 800]	176	640
[30 -92 288]	176	-692
[30 -332 3680]	176	0
[30 -452 6816]	176	640
[22 -44 96]	176	8
[22 -132 800]	176	-1336
[10 12 32]	176	-640
[18 -92 480]	176	692
[18 -236 3104]	176	0
[2 3 96]	183	-966
[6 3 32]	183	966
[38 -125 416]	183	0
[16 -227 3232]	183	84
[2 2 96]	188	-784
[94 -658 4608]	188	-784
[48 -146 448]	188	2678
[48 -530 5856]	188	862
[6 2 32]	188	784
[6 14 64]	188	-862
[142 350 864]	188	-448
[24 -146 896]	188	-3036
[24 -338 4768]	188	14
[8 18 64]	188	2678
[112 274 672]	188	1477
[12 14 32]	188	784
[12 50 224]	188	874
[32 -110 384]	188	0
[14 34 96]	188	-448
[102 242 576]	188	-1477
[14 62 288]	188	0
[42 -146 512]	188	874
[2 1 96]	191	2114
[6 1 32]	191	-2114
[32 -417 5440]	191	1586
[8 33 160]	191	-765
[20 -223 2496]	191	765
[16 -161 1632]	191	-1477
[18 -95 512]	191	1586
[18 -257 3680]	191	-98
[2 0 96]	192	0
[2 8 128]	192	1152
[2 16 224]	192	0
[4 8 64]	192	0
[52 152 448]	192	1440
[6 0 32]	192	0
[6 24 128]	192	1248
[8 8 32]	192	-1152
[24 -72 224]	192	168
[24 -120 608]	192	1248
[26 -56 128]	192	1440
[26 -368 5216]	192	0
[12 24 64]	192	-1440
[28 72 192]	192	0
[14 16 32]	192	0
[14 -72 384]	192	-168
[100 -1301 16928]	199	-54
[50 -651 8480]	199	-551
[26 -341 4480]	199	-691
[10 11 32]	199	54
[10 21 64]	199	551
[16 -213 2848]	199	691
[104 -1145 12608]	207	-1047
[478 1113 2592]	207	0
[26 -391 5888]	207	-1047
[18 -39 96]	207	-1287
[18 -57 192]	207	0
[108 -973 8768]	215	-78
[54 -595 6560]	215	1383
[36 -397 4384]	215	0
[6 13 64]	215	-1383
[24 -77 256]	215	685
[12 13 32]	215	-78
[18 -109 672]	215	2941
[20 -205 2112]	215	-900
[16 -179 2016]	215	685
[110 -330 992]	220	1256
[110 -550 2752]	220	1256
[4 6 64]	220	0
[56 -506 4576]	220	1570
[28 -58 128]	220	-1570
[28 -282 2848]	220	-3481
[28 -422 6368]	220	-1256
[10 10 32]	220	1256
[34 86 224]	220	0
[170 390 896]	220	-1570
[34 -118 416]	220	0
[16 -198 2464]	220	3201
[112 -785 5504]	223	84
[4 17 128]	223	-793
[8 17 64]	223	793
[14 15 32]	223	-84
[116 -581 2912]	231	-1890
[4 5 64]	231	0
[40 -123 384]	231	-4683
[30 -453 6848]	231	1890
[24 -123 640]	231	4683
[40 -133 448]	231	0
[20 -123 768]	231	2898
[104 251 608]	231	0
[120 -361 1088]	239	-979
[248 617 1536]	239	-7058
[48 -151 480]	239	-7250
[30 -151 768]	239	7250
[24 -361 5440]	239	979
[10 41 192]	239	1954
[20 -41 96]	239	83
[198 457 1056]	239	7058
[22 -247 2784]	239	-1954
[2 4 128]	240	-2136
[122 364 1088]	240	-2136
[4 4 64]	240	0
[60 -180 544]	240	1512
[60 -300 1504]	240	-1158
[6 12 64]	240	-1158
[30 -60 128]	240	-2670
[30 -180 1088]	240	-2838
[8 4 32]	240	2136
[32 -100 320]	240	1950
[32 -356 3968]	240	0
[10 20 64]	240	1512
[12 12 32]	240	2136
[20 -60 192]	240	0
[20 -100 512]	240	-1950
[16 -36 96]	240	-258
[16 -164 1696]	240	2364
[16 -228 3264]	240	2670
[2 3 128]	247	2967
[62 -931 13984]	247	0
[8 3 32]	247	-2967
[2 2 128]	252	4368
[126 -882 6176]	252	4368
[4 2 64]	252	0
[42 -126 384]	252	-9639
[42 -210 1056]	252	0
[8 2 32]	252	-4368
[32 -450 6336]	252	-5460
[24 -126 672]	252	9639
[24 -318 4224]	252	0
[14 14 32]	252	-4368
[18 -258 3712]	252	-5460
[2 1 128]	255	-11130
[4 1 64]	255	0
[44 -575 7520]	255	6201
[8 1 32]	255	11130
[22 -223 2272]	255	-6450
[22 -289 3808]	255	0
[20 -65 224]	255	0
[14 33 96]	255	-6201
[2 0 128]	256	0
[2 8 160]	256	-4864
[4 0 64]	256	0
[4 16 128]	256	7512
[8 0 32]	256	0
[8 16 64]	256	-7512
[8 32 160]	256	-3568
[34 -72 160]	256	5328
[34 -480 6784]	256	0
[26 -288 3200]	256	3568
[26 -392 5920]	256	-4864
[16 16 32]	256	0
[16 -48 160]	256	-712
[20 -48 128]	256	-5000
[20 -288 4160]	256	0
[132 -1717 22336]	263	969
[66 -331 1664]	263	4070
[408 949 2208]	263	-4070
[22 -309 4352]	263	-4411
[22 -331 4992]	263	969
[24 -53 128]	263	-2682
[18 -203 2304]	263	-2494
[136 -1497 16480]	271	5827
[74 217 640]	271	-3304
[34 -103 320]	271	8377
[28 -423 6400]	271	5827
[20 -103 544]	271	-8377
[212 487 1120]	271	-3304
[124 295 704]	271	14192
[140 -1261 11360]	279	-1530
[70 -211 640]	279	5130
[268 621 1440]	279	-5130
[24 -339 4800]	279	3132
[42 -141 480]	279	0
[20 -301 4544]	279	1530
[98 237 576]	279	0
[18 -237 3136]	279	-2046
[20 -109 608]	279	-6951
[142 -426 1280]	284	-12664
[142 -710 3552]	284	-12664
[72 -362 1824]	284	11354
[72 -938 12224]	284	22396
[450 1046 2432]	284	-11354
[6 26 160]	284	-8016
[58 -182 576]	284	-2750
[36 -74 160]	284	-15763
[36 -182 928]	284	2750
[36 -362 3648]	284	-31366
[30 -154 800]	284	-8016
[30 -454 6880]	284	12664
[24 -362 5472]	284	12664
[240 554 1280]	284	22396
[18 -38 96]	284	-2976
[18 -182 1856]	284	16992
[20 -106 576]	284	-18876
[20 -266 3552]	284	-7060
[78 -239 736]	287	-21182
[48 -241 1216]	287	0
[338 785 1824]	287	21182
[24 -241 2432]	287	-15211
[12 49 224]	287	-4886
[26 -369 5248]	287	4998
[24 -271 3072]	287	4886
[18 -271 4096]	287	-8694
[148 -741 3712]	295	19258
[38 -421 4672]	295	0
[10 5 32]	295	19258
[226 517 1184]	295	-18821
[152 -457 1376]	303	7788
[4 9 96]	303	6528
[6 9 64]	303	-6528
[26 -55 128]	303	-1722
[26 -393 5952]	303	-7788
[2 4 160]	304	-2432
[154 460 1376]	304	-2432
[38 -76 160]	304	-6648
[10 4 32]	304	2432
[10 44 224]	304	-3756
[50 -164 544]	304	0
[22 -68 224]	304	0
[22 -244 2720]	304	3756
[22 -332 5024]	304	-2432
[2 3 160]	311	8500
[52 -157 480]	311	-18316
[8 35 192]	311	-10809
[10 3 32]	311	-8500
[26 -131 672]	311	-10809
[28 -285 2912]	311	2225
[12 29 96]	311	-18316
[16 35 96]	311	3909
[20 -227 2592]	311	951
[240 547 1248]	311	11090
[2 2 160]	316	-11624
[80 -242 736]	316	-308
[80 -882 9728]	316	17398
[40 -562 7904]	316	-30689
[380 882 2048]	316	17398
[10 2 32]	316	11624
[310 718 1664]	316	308
[20 -302 4576]	316	-11624
[22 -50 128]	316	-1009
[22 -226 2336]	316	8952
[2 1 160]	319	24493
[50 -159 512]	319	25129
[10 1 32]	319	-24493
[32 -353 3904]	319	25129
[20 -289 4192]	319	-33859
[2 0 160]	320	0
[2 8 192]	320	-1872
[4 8 96]	320	13274
[84 248 736]	320	10934
[6 8 64]	320	-13274
[54 -272 1376]	320	0
[54 -704 9184]	320	9960
[42 -88 192]	320	-6100
[42 -592 8352]	320	0
[10 0 32]	320	0
[10 40 192]	320	-11954
[28 -312 3488]	320	11954
[28 -424 6432]	320	-1872
[14 32 96]	320	-9960
[126 304 736]	320	0
[254 584 1344]	320	10934
[20 -40 96]	320	1658
[20 -120 736]	320	-10850
[18 -56 192]	320	5326
[18 -272 4128]	320	0
[164 -2133 27744]	327	-14319
[82 -1067 13888]	327	14067
[56 -619 6848]	327	33714
[42 -213 1088]	327	33714
[282 651 1504]	327	14067
[24 -363 5504]	327	-14319
[44 -149 512]	327	0
[168 -1849 20352]	335	-13210
[766 1785 4160]	335	-8498
[6 7 64]	335	-8498
[6 25 160]	335	7063
[36 -185 960]	335	7063
[28 -57 128]	335	7090
[30 -455 6912]	335	-13210
[24 -313 4096]	335	0
[54 -185 640]	335	23747
[118 281 672]	335	6363
[86 -947 10432]	343	-37947
[422 979 2272]	343	-37947
[28 -371 4928]	343	0
[22 -333 5056]	343	11333
[174 -522 1568]	348	36312
[174 -870 4352]	348	36312
[4 6 96]	348	12798
[6 6 64]	348	-12798
[44 -90 192]	348	-80622
[44 -134 416]	348	22572
[44 -486 5376]	348	0
[12 6 32]	348	36312
[26 -134 704]	348	-22572
[26 -342 4512]	348	0
[26 -394 5984]	348	-36312
[22 -310 4384]	348	-5760
[268 614 1408]	348	-86352
[4 17 160]	351	26526
[60 -303 1536]	351	0
[352 815 1888]	351	26526
[30 -303 3072]	351	-21492
[22 -111 576]	351	-7704
[20 -303 4608]	351	33747
[20 -47 128]	351	-3156
[180 -901 4512]	359	-77815
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