A Borcherds Product in S₁₄(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)
  +(28 + 2/ζ^3 + 2/ζ^2 + 6/ζ + 6*ζ + 2*ζ^2 + 2*ζ^3)
  +q(2/ζ^9 + 15/ζ^8 + 15*ζ^8 + 2*ζ^9)
  +q^3(14/ζ^14 + 14*ζ^14)
  +q^4(12/ζ^16 + 12*ζ^16)
• 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 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)	10
  (4,2)	2
  (4,14)	14
  (9,3)	2
  (17,9)	2
  (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)
+(28 + 2/ζ^3 + 2/ζ^2 + 6/ζ + 6*ζ + 2*ζ^2 + 2*ζ^3)
+q(33230 + 2/ζ^9 + 15/ζ^8 + 222/ζ^7 + 1010/ζ^6 + 3310/ζ^5 + 8148/ζ^4 + 15286/ζ^3 + 23502/ζ^2 + 30332/ζ + 30332*ζ + 23502*ζ^2 + 15286*ζ^3 + 8148*ζ^4 + 3310*ζ^5 + 1010*ζ^6 + 222*ζ^7 + 15*ζ^8 + 2*ζ^9)
+q^2(3082576 + 72/ζ^11 + 1022/ζ^10 + 7276/ζ^9 + 32920/ζ^8 + 109024/ζ^7 + 283324/ζ^6 + 609878/ζ^5 + 1113280/ζ^4 + 1751190/ζ^3 + 2401094/ζ^2 + 2896512/ζ + 2896512*ζ + 2401094*ζ^2 + 1751190*ζ^3 + 1113280*ζ^4 + 609878*ζ^5 + 283324*ζ^6 + 109024*ζ^7 + 32920*ζ^8 + 7276*ζ^9 + 1022*ζ^10 + 72*ζ^11)
+q^3(112738066 + 14/ζ^14 + 548/ζ^13 + 8044/ζ^12 + 58498/ζ^11 + 283940/ζ^10 + 1041714/ζ^9 + 3082152/ζ^8 + 7618512/ζ^7 + 16134984/ζ^6 + 29799428/ζ^5 + 48585284/ζ^4 + 70500270/ζ^3 + 91614342/ζ^2 + 107053222/ζ + 107053222*ζ + 91614342*ζ^2 + 70500270*ζ^3 + 48585284*ζ^4 + 29799428*ζ^5 + 16134984*ζ^6 + 7618512*ζ^7 + 3082152*ζ^8 + 1041714*ζ^9 + 283940*ζ^10 + 58498*ζ^11 + 8044*ζ^12 + 548*ζ^13 + 14*ζ^14)
+q^4(2467349800 + 12/ζ^16 + 1426/ζ^15 + 23618/ζ^14 + 197478/ζ^13 + 1112384/ζ^12 + 4728308/ζ^11 + 16135288/ζ^10 + 46036192/ζ^9 + 112727904/ζ^8 + 241244888/ζ^7 + 457186034/ζ^6 + 774909038/ζ^5 + 1183235264/ζ^4 + 1636313740/ζ^3 + 2057356372/ζ^2 + 2358045282/ζ + 2358045282*ζ + 2057356372*ζ^2 + 1636313740*ζ^3 + 1183235264*ζ^4 + 774909038*ζ^5 + 457186034*ζ^6 + 241244888*ζ^7 + 112727904*ζ^8 + 46036192*ζ^9 + 16135288*ζ^10 + 4728308*ζ^11 + 1112384*ζ^12 + 197478*ζ^13 + 23618*ζ^14 + 1426*ζ^15 + 12*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	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	0
[12 -73 448]	47	0
[4 9 32]	47	0
[8 -73 672]	47	0
[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	-1
[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	1
[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	-1
[2 16 160]	64	2
[4 8 32]	64	1
[20 56 160]	64	0
[10 -24 64]	64	0
[10 -64 416]	64	0
[10 -144 2080]	64	-2
[36 -469 6112]	71	2
[18 -181 1824]	71	0
[18 -235 3072]	71	0
[120 277 640]	71	-2
[10 -43 192]	71	0
[10 -53 288]	71	0
[40 -441 4864]	79	6
[20 -121 736]	79	0
[190 441 1024]	79	-6
[10 -121 1472]	79	0
[44 -397 3584]	87	2
[22 -45 96]	87	0
[22 -243 2688]	87	0
[12 -51 224]	87	0
[134 307 704]	87	-2
[46 -138 416]	92	-10
[46 -230 1152]	92	-13
[4 6 32]	92	13
[24 -218 1984]	92	0
[26 70 192]	92	0
[162 374 864]	92	10
[6 26 128]	92	-1
[26 -86 288]	92	0
[12 -26 64]	92	0
[12 -38 128]	92	0
[12 -122 1248]	92	0
[12 -134 1504]	92	1
[48 -337 2368]	95	-4
[24 -337 4736]	95	0
[4 17 96]	95	2
[16 -81 416]	95	-2
[12 -145 1760]	95	0
[60 145 352]	95	0
[10 15 32]	95	4
[52 -261 1312]	103	10
[4 5 32]	103	-10
[26 -261 2624]	103	0
[14 -59 256]	103	0
[56 -169 512]	111	20
[28 -169 1024]	111	0
[120 297 736]	111	2
[6 9 32]	111	-20
[14 -169 2048]	111	0
[14 -183 2400]	111	0
[10 23 64]	111	-2
[2 4 64]	112	2
[58 172 512]	112	-2
[4 4 32]	112	-2
[28 -84 256]	112	-4
[28 -140 704]	112	0
[28 -196 1376]	112	14
[14 -28 64]	112	-14
[14 -84 512]	112	0
[8 12 32]	112	2
[16 -84 448]	112	4
[16 -148 1376]	112	0
[16 -212 2816]	112	0
[2 3 64]	119	-2
[30 -61 128]	119	0
[30 -451 6784]	119	2
[20 -61 192]	119	2
[90 221 544]	119	0
[12 -61 320]	119	-2
[24 61 160]	119	0
[12 -125 1312]	119	0
[2 2 64]	124	-3
[62 -434 3040]	124	26
[4 2 32]	124	3
[16 -98 608]	124	0
[16 -226 3200]	124	16
[8 34 160]	124	15
[28 -94 320]	124	0
[10 14 32]	124	-26
[14 -46 160]	124	0
[14 -158 1792]	124	-15
[2 1 64]	127	-14
[4 1 32]	127	14
[32 -449 6304]	127	0
[2 0 64]	128	30
[2 8 96]	128	-2
[2 16 192]	128	-26
[4 0 32]	128	-30
[4 16 96]	128	3
[6 8 32]	128	2
[22 -112 576]	128	-3
[22 -288 3776]	128	-1
[18 -40 96]	128	0
[18 -112 704]	128	0
[18 -256 3648]	128	-4
[12 16 32]	128	26
[12 -64 352]	128	1
[68 -885 11520]	135	-30
[34 -171 864]	135	0
[34 -341 3424]	135	0
[24 -267 2976]	135	0
[8 11 32]	135	30
[12 -75 480]	135	0
[72 -793 8736]	143	-108
[36 -217 1312]	143	0
[42 121 352]	143	40
[6 7 32]	143	108
[6 25 128]	143	-24
[18 -199 2208]	143	24
[12 -25 64]	143	-40
[14 -89 576]	143	0
[76 -685 6176]	151	-58
[38 -77 160]	151	0
[38 -115 352]	151	62
[8 19 64]	151	-62
[10 13 32]	151	58
[78 -234 704]	156	188
[78 -390 1952]	156	270
[40 -202 1024]	156	0
[40 -522 6816]	156	-119
[6 6 32]	156	-270
[8 10 32]	156	-188
[20 -42 96]	156	91
[20 -202 2048]	156	0
[10 22 64]	156	119
[82 198 480]	156	0
[10 42 192]	156	-93
[30 -102 352]	156	0
[16 -54 192]	156	0
[16 -182 2080]	156	93
[80 -561 3936]	159	62
[40 -561 7872]	159	0
[46 -143 448]	159	158
[28 -143 736]	159	-158
[30 81 224]	159	0
[16 -49 160]	159	0
[12 15 32]	159	-62
[14 -143 1472]	159	0
[84 -421 2112]	167	-214
[42 -379 3424]	167	354
[42 -421 4224]	167	0
[28 -421 6336]	167	214
[22 -69 224]	167	42
[14 -155 1728]	167	-42
[14 -197 2784]	167	-354
[16 -37 96]	167	26
[88 -265 800]	175	-414
[4 9 64]	175	0
[8 9 32]	175	414
[20 -265 3520]	175	0
[2 4 96]	176	-56
[90 268 800]	176	56
[30 -92 288]	176	-94
[30 -332 3680]	176	0
[30 -452 6816]	176	56
[22 -44 96]	176	-16
[22 -132 800]	176	0
[10 12 32]	176	-56
[18 -92 480]	176	94
[18 -236 3104]	176	0
[2 3 96]	183	36
[6 3 32]	183	-36
[38 -125 416]	183	0
[16 -227 3232]	183	-406
[2 2 96]	188	124
[94 -658 4608]	188	-259
[48 -146 448]	188	256
[48 -530 5856]	188	-150
[6 2 32]	188	-124
[6 14 64]	188	150
[142 350 864]	188	-90
[24 -146 896]	188	0
[24 -338 4768]	188	-7
[8 18 64]	188	-256
[112 274 672]	188	11
[12 14 32]	188	259
[12 50 224]	188	290
[32 -110 384]	188	0
[14 34 96]	188	90
[102 242 576]	188	-11
[14 62 288]	188	0
[42 -146 512]	188	-290
[2 1 96]	191	264
[6 1 32]	191	-264
[32 -417 5440]	191	234
[8 33 160]	191	318
[20 -223 2496]	191	-318
[16 -161 1632]	191	0
[18 -95 512]	191	-234
[18 -257 3680]	191	802
[2 0 96]	192	-652
[2 8 128]	192	296
[2 16 224]	192	60
[4 8 64]	192	0
[52 152 448]	192	-88
[6 0 32]	192	652
[6 24 128]	192	-370
[8 8 32]	192	-296
[24 -72 224]	192	-118
[24 -120 608]	192	370
[26 -56 128]	192	-364
[26 -368 5216]	192	-136
[12 24 64]	192	88
[28 72 192]	192	0
[14 16 32]	192	-60
[14 -72 384]	192	118
[100 -1301 16928]	199	120
[50 -651 8480]	199	-664
[26 -341 4480]	199	50
[10 11 32]	199	-120
[10 21 64]	199	664
[16 -213 2848]	199	-50
[104 -1145 12608]	207	762
[478 1113 2592]	207	0
[26 -391 5888]	207	-762
[18 -39 96]	207	576
[18 -57 192]	207	0
[108 -973 8768]	215	702
[54 -595 6560]	215	-4
[36 -397 4384]	215	0
[6 13 64]	215	4
[24 -77 256]	215	-346
[12 13 32]	215	-702
[18 -109 672]	215	0
[20 -205 2112]	215	-154
[16 -179 2016]	215	346
[110 -330 992]	220	-1386
[110 -550 2752]	220	-2390
[4 6 64]	220	0
[56 -506 4576]	220	2525
[28 -58 128]	220	-336
[28 -282 2848]	220	1001
[28 -422 6368]	220	2390
[10 10 32]	220	1386
[34 86 224]	220	0
[170 390 896]	220	-2525
[34 -118 416]	220	0
[16 -198 2464]	220	369
[112 -785 5504]	223	-310
[4 17 128]	223	1986
[8 17 64]	223	-1986
[14 15 32]	223	310
[116 -581 2912]	231	1966
[4 5 64]	231	0
[40 -123 384]	231	-918
[30 -453 6848]	231	-1966
[24 -123 640]	231	918
[40 -133 448]	231	0
[20 -123 768]	231	0
[104 251 608]	231	0
[120 -361 1088]	239	3630
[248 617 1536]	239	-4594
[48 -151 480]	239	-1076
[30 -151 768]	239	1076
[24 -361 5440]	239	-3630
[10 41 192]	239	-1926
[20 -41 96]	239	3146
[198 457 1056]	239	4594
[22 -247 2784]	239	1926
[2 4 128]	240	706
[122 364 1088]	240	-706
[4 4 64]	240	0
[60 -180 544]	240	-178
[60 -300 1504]	240	-858
[6 12 64]	240	858
[30 -60 128]	240	-32
[30 -180 1088]	240	-2002
[8 4 32]	240	-706
[32 -100 320]	240	988
[32 -356 3968]	240	0
[10 20 64]	240	178
[12 12 32]	240	706
[20 -60 192]	240	0
[20 -100 512]	240	-988
[16 -36 96]	240	972
[16 -164 1696]	240	50
[16 -228 3264]	240	-5124
[2 3 128]	247	-290
[62 -931 13984]	247	0
[8 3 32]	247	290
[2 2 128]	252	-1893
[126 -882 6176]	252	1116
[4 2 64]	252	0
[42 -126 384]	252	549
[42 -210 1056]	252	0
[8 2 32]	252	1893
[32 -450 6336]	252	336
[24 -126 672]	252	-549
[24 -318 4224]	252	0
[14 14 32]	252	-1116
[18 -258 3712]	252	2406
[2 1 128]	255	-1996
[4 1 64]	255	0
[44 -575 7520]	255	2378
[8 1 32]	255	1996
[22 -223 2272]	255	546
[22 -289 3808]	255	0
[20 -65 224]	255	0
[14 33 96]	255	-2378
[2 0 128]	256	6192
[2 8 160]	256	-3504
[4 0 64]	256	0
[4 16 128]	256	1112
[8 0 32]	256	-6192
[8 16 64]	256	-1112
[8 32 160]	256	2852
[34 -72 160]	256	2184
[34 -480 6784]	256	792
[26 -288 3200]	256	-2852
[26 -392 5920]	256	3504
[16 16 32]	256	-816
[16 -48 160]	256	740
[20 -48 128]	256	-1112
[20 -288 4160]	256	10064
[132 -1717 22336]	263	546
[66 -331 1664]	263	-1082
[408 949 2208]	263	1082
[22 -309 4352]	263	2128
[22 -331 4992]	263	-546
[24 -53 128]	263	-2938
[18 -203 2304]	263	-1206
[136 -1497 16480]	271	-2226
[74 217 640]	271	50
[34 -103 320]	271	872
[28 -423 6400]	271	2226
[20 -103 544]	271	-872
[212 487 1120]	271	-50
[124 295 704]	271	474
[140 -1261 11360]	279	-4554
[70 -211 640]	279	5646
[268 621 1440]	279	-5646
[24 -339 4800]	279	-4626
[42 -141 480]	279	0
[20 -301 4544]	279	4554
[98 237 576]	279	0
[18 -237 3136]	279	-2316
[20 -109 608]	279	-1116
[142 -426 1280]	284	4101
[142 -710 3552]	284	11234
[72 -362 1824]	284	3287
[72 -938 12224]	284	-11104
[450 1046 2432]	284	-3287
[6 26 160]	284	-5592
[58 -182 576]	284	-5959
[36 -74 160]	284	1044
[36 -182 928]	284	5959
[36 -362 3648]	284	-7280
[30 -154 800]	284	5592
[30 -454 6880]	284	-11234
[24 -362 5472]	284	-4101
[240 554 1280]	284	11104
[18 -38 96]	284	7392
[18 -182 1856]	284	-105
[20 -106 576]	284	4560
[20 -266 3552]	284	613
[78 -239 736]	287	-6134
[48 -241 1216]	287	0
[338 785 1824]	287	6134
[24 -241 2432]	287	-1274
[12 49 224]	287	5528
[26 -369 5248]	287	-1250
[24 -271 3072]	287	-5528
[18 -271 4096]	287	168
[148 -741 3712]	295	-9676
[38 -421 4672]	295	0
[10 5 32]	295	9676
[226 517 1184]	295	-17068
[152 -457 1376]	303	-16564
[4 9 96]	303	2372
[6 9 64]	303	-2372
[26 -55 128]	303	-9882
[26 -393 5952]	303	16564
[2 4 160]	304	-5208
[154 460 1376]	304	5208
[38 -76 160]	304	704
[10 4 32]	304	5208
[10 44 224]	304	-4038
[50 -164 544]	304	0
[22 -68 224]	304	0
[22 -244 2720]	304	4038
[22 -332 5024]	304	-5208
[2 3 160]	311	1518
[52 -157 480]	311	-7124
[8 35 192]	311	6440
[10 3 32]	311	-1518
[26 -131 672]	311	-6440
[28 -285 2912]	311	7744
[12 29 96]	311	7124
[16 35 96]	311	-2612
[20 -227 2592]	311	3544
[240 547 1248]	311	-9950
[2 2 160]	316	15214
[80 -242 736]	316	15689
[80 -882 9728]	316	-10384
[40 -562 7904]	316	-3687
[380 882 2048]	316	10384
[10 2 32]	316	-15214
[310 718 1664]	316	-15689
[20 -302 4576]	316	66
[22 -50 128]	316	-13610
[22 -226 2336]	316	4714
[2 1 160]	319	6696
[50 -159 512]	319	16712
[10 1 32]	319	-6696
[32 -353 3904]	319	-16712
[20 -289 4192]	319	21874
[2 0 160]	320	-32868
[2 8 192]	320	19724
[4 8 96]	320	136
[84 248 736]	320	-22868
[6 8 64]	320	-136
[54 -272 1376]	320	0
[54 -704 9184]	320	21704
[42 -88 192]	320	-6384
[42 -592 8352]	320	-2492
[10 0 32]	320	32868
[10 40 192]	320	-15910
[28 -312 3488]	320	15910
[28 -424 6432]	320	-19724
[14 32 96]	320	-21704
[126 304 736]	320	0
[254 584 1344]	320	22868
[20 -40 96]	320	22644
[20 -120 736]	320	0
[18 -56 192]	320	-4510
[18 -272 4128]	320	6580
[164 -2133 27744]	327	-7054
[82 -1067 13888]	327	284
[56 -619 6848]	327	-3446
[42 -213 1088]	327	3446
[282 651 1504]	327	-284
[24 -363 5504]	327	7054
[44 -149 512]	327	0
[168 -1849 20352]	335	-1160
[766 1785 4160]	335	-5228
[6 7 64]	335	5228
[6 25 160]	335	-5254
[36 -185 960]	335	5254
[28 -57 128]	335	1774
[30 -455 6912]	335	1160
[24 -313 4096]	335	0
[54 -185 640]	335	7408
[118 281 672]	335	-1198
[86 -947 10432]	343	-12564
[422 979 2272]	343	12564
[28 -371 4928]	343	0
[22 -333 5056]	343	-15888
[174 -522 1568]	348	4970
[174 -870 4352]	348	-26399
[4 6 96]	348	-5845
[6 6 64]	348	5845
[44 -90 192]	348	-7040
[44 -134 416]	348	19691
[44 -486 5376]	348	0
[12 6 32]	348	26399
[26 -134 704]	348	-19691
[26 -342 4512]	348	0
[26 -394 5984]	348	-4970
[22 -310 4384]	348	14284
[268 614 1408]	348	4624
[4 17 160]	351	-12540
[60 -303 1536]	351	0
[352 815 1888]	351	12540
[30 -303 3072]	351	-10152
[22 -111 576]	351	16962
[20 -303 4608]	351	-7968
[20 -47 128]	351	-7908
[180 -901 4512]	359	23870
[4 5 96]	359	734
[60 -901 13536]	359	-734
[30 -421 5920]	359	10986
[12 5 32]	359	-23870
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