A Borcherds Product in S₁₂(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)
  +(24 + ζ^(-4) + 2/ζ^3 + 3/ζ^2 + 18/ζ + 18*ζ + 3*ζ^2 + 2*ζ^3 + ζ^4)
  +q(26/ζ^8 + 26*ζ^8)
  +q^3(ζ^(-14) + ζ^14)
  +q^4(14/ζ^16 + 14*ζ^16)
• From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block

  	φ	= TB(12;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4)
  		= 118233241
  		= η(τ)24(θ(τ,z)/η(τ))18(θ(τ,2z)/η(τ))3(θ(τ,3z)/η(τ))2(θ(τ,4z)/η(τ))

  This theta block has q-order of vanishing 3. 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 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)	24
  (4,2)	4
  (4,14)	2
  (9,3)	2
  (16,4)	1
  (64,0)	1

• From the singular part above, we compute that the valuation Ord(ψ) is -1, and hence we need to multiply ψ by Δ^2 to get a Jacobi cusp form. The definition of ψ is
      ψ = c · θ/ Δ^2.
  Here c is a vector of coefficients, θ is a basis of of J_24,16^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)
+(24 + ζ^(-4) + 2/ζ^3 + 3/ζ^2 + 18/ζ + 18*ζ + 3*ζ^2 + 2*ζ^3 + ζ^4)
+q(33128 + 26/ζ^8 + 200/ζ^7 + 1031/ζ^6 + 3334/ζ^5 + 8060/ζ^4 + 15210/ζ^3 + 23609/ζ^2 + 30408/ζ + 30408*ζ + 23609*ζ^2 + 15210*ζ^3 + 8060*ζ^4 + 3334*ζ^5 + 1031*ζ^6 + 200*ζ^7 + 26*ζ^8)
+q^2(3081868 + 54/ζ^11 + 1013/ζ^10 + 7358/ζ^9 + 32986/ζ^8 + 108846/ζ^7 + 283474/ζ^6 + 610164/ζ^5 + 1112544/ζ^4 + 1750502/ζ^3 + 2401977/ζ^2 + 2897028/ζ + 2897028*ζ + 2401977*ζ^2 + 1750502*ζ^3 + 1112544*ζ^4 + 610164*ζ^5 + 283474*ζ^6 + 108846*ζ^7 + 32986*ζ^8 + 7358*ζ^9 + 1013*ζ^10 + 54*ζ^11)
+q^3(112734202 + ζ^(-14) + 554/ζ^13 + 8068/ζ^12 + 58294/ζ^11 + 283814/ζ^10 + 1042520/ζ^9 + 3082580/ζ^8 + 7617270/ζ^7 + 16135908/ζ^6 + 29801232/ζ^5 + 48581132/ζ^4 + 70496560/ζ^3 + 91619189/ζ^2 + 107055762/ζ + 107055762*ζ + 91619189*ζ^2 + 70496560*ζ^3 + 48581132*ζ^4 + 29801232*ζ^5 + 16135908*ζ^6 + 7617270*ζ^7 + 3082580*ζ^8 + 1042520*ζ^9 + 283814*ζ^10 + 58294*ζ^11 + 8068*ζ^12 + 554*ζ^13 + ζ^14)
+q^4(2467332884 + 14/ζ^16 + 1440/ζ^15 + 23499/ζ^14 + 197578/ζ^13 + 1112607/ζ^12 + 4726840/ζ^11 + 16134412/ζ^10 + 46040850/ζ^9 + 112729960/ζ^8 + 241238690/ζ^7 + 457190387/ζ^6 + 774917602/ζ^5 + 1183216865/ζ^4 + 1636297452/ζ^3 + 2057377590/ζ^2 + 2358055900/ζ + 2358055900*ζ + 2057377590*ζ^2 + 1636297452*ζ^3 + 1183216865*ζ^4 + 774917602*ζ^5 + 457190387*ζ^6 + 241238690*ζ^7 + 112729960*ζ^8 + 46040850*ζ^9 + 16134412*ζ^10 + 4726840*ζ^11 + 1112607*ζ^12 + 197578*ζ^13 + 23499*ζ^14 + 1440*ζ^15 + 14*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	1
[10 -101 1024]	39	-1
[8 -27 96]	39	0
[24 -73 224]	47	0
[12 -73 448]	47	1
[4 9 32]	47	0
[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	3
[14 -99 704]	55	-3
[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	-2
[4 33 288]	63	2
[12 -159 2112]	63	0
[8 -97 1184]	63	0
[2 0 32]	64	0
[2 8 64]	64	0
[2 16 160]	64	0
[4 8 32]	64	0
[20 56 160]	64	-2
[10 -24 64]	64	2
[10 -64 416]	64	0
[10 -144 2080]	64	0
[36 -469 6112]	71	0
[18 -181 1824]	71	11
[18 -235 3072]	71	1
[120 277 640]	71	0
[10 -43 192]	71	0
[10 -53 288]	71	-1
[40 -441 4864]	79	0
[20 -121 736]	79	-14
[190 441 1024]	79	0
[10 -121 1472]	79	0
[44 -397 3584]	87	0
[22 -45 96]	87	-39
[22 -243 2688]	87	0
[12 -51 224]	87	0
[134 307 704]	87	0
[46 -138 416]	92	0
[46 -230 1152]	92	0
[4 6 32]	92	0
[24 -218 1984]	92	-12
[26 70 192]	92	-52
[162 374 864]	92	0
[6 26 128]	92	0
[26 -86 288]	92	0
[12 -26 64]	92	12
[12 -38 128]	92	0
[12 -122 1248]	92	52
[12 -134 1504]	92	0
[48 -337 2368]	95	0
[24 -337 4736]	95	27
[4 17 96]	95	-1
[16 -81 416]	95	1
[12 -145 1760]	95	0
[60 145 352]	95	0
[10 15 32]	95	0
[52 -261 1312]	103	0
[4 5 32]	103	0
[26 -261 2624]	103	-33
[14 -59 256]	103	0
[56 -169 512]	111	0
[28 -169 1024]	111	74
[120 297 736]	111	32
[6 9 32]	111	0
[14 -169 2048]	111	0
[14 -183 2400]	111	0
[10 23 64]	111	-32
[2 4 64]	112	0
[58 172 512]	112	0
[4 4 32]	112	0
[28 -84 256]	112	-16
[28 -140 704]	112	0
[28 -196 1376]	112	32
[14 -28 64]	112	-32
[14 -84 512]	112	144
[8 12 32]	112	0
[16 -84 448]	112	16
[16 -148 1376]	112	-144
[16 -212 2816]	112	0
[2 3 64]	119	0
[30 -61 128]	119	173
[30 -451 6784]	119	0
[20 -61 192]	119	-26
[90 221 544]	119	81
[16 -67 288]	119	0
[12 -61 320]	119	26
[24 61 160]	119	0
[12 -125 1312]	119	-81
[2 2 64]	124	0
[62 -434 3040]	124	0
[4 2 32]	124	0
[32 -226 1600]	124	12
[16 -98 608]	124	-52
[16 -226 3200]	124	-12
[8 34 160]	124	0
[28 -94 320]	124	0
[10 14 32]	124	0
[14 -46 160]	124	0
[14 -130 1216]	124	52
[14 -158 1792]	124	0
[2 1 64]	127	0
[4 1 32]	127	0
[32 -449 6304]	127	-127
[16 -193 2336]	127	0
[2 0 64]	128	0
[2 8 96]	128	0
[2 16 192]	128	0
[4 0 32]	128	0
[4 16 96]	128	18
[6 8 32]	128	0
[22 -112 576]	128	-18
[22 -288 3776]	128	-2
[18 -40 96]	128	352
[18 -112 704]	128	-292
[18 -256 3648]	128	60
[12 16 32]	128	0
[12 -64 352]	128	2
[68 -885 11520]	135	0
[34 -171 864]	135	0
[34 -341 3424]	135	-61
[24 -267 2976]	135	0
[8 11 32]	135	0
[12 -75 480]	135	0
[72 -793 8736]	143	0
[36 -217 1312]	143	-137
[42 121 352]	143	83
[6 7 32]	143	0
[6 25 128]	143	25
[18 -199 2208]	143	-25
[12 -25 64]	143	-83
[14 -89 576]	143	0
[76 -685 6176]	151	0
[38 -77 160]	151	-106
[38 -115 352]	151	117
[8 19 64]	151	-117
[10 13 32]	151	0
[78 -234 704]	156	0
[78 -390 1952]	156	0
[40 -202 1024]	156	0
[40 -522 6816]	156	-252
[6 6 32]	156	0
[8 10 32]	156	0
[20 -42 96]	156	-252
[20 -202 2048]	156	1596
[10 22 64]	156	252
[82 198 480]	156	0
[10 42 192]	156	0
[30 -102 352]	156	0
[16 -54 192]	156	0
[16 -182 2080]	156	0
[80 -561 3936]	159	0
[40 -561 7872]	159	108
[46 -143 448]	159	-150
[28 -143 736]	159	150
[30 81 224]	159	735
[16 -49 160]	159	0
[12 15 32]	159	0
[14 -143 1472]	159	-735
[84 -421 2112]	167	0
[42 -379 3424]	167	-182
[42 -421 4224]	167	490
[28 -421 6336]	167	0
[36 101 288]	167	775
[22 -69 224]	167	311
[14 -155 1728]	167	-311
[14 -197 2784]	167	182
[16 -37 96]	167	-775
[88 -265 800]	175	0
[4 9 64]	175	0
[8 9 32]	175	0
[20 -265 3520]	175	0
[2 4 96]	176	0
[90 268 800]	176	0
[30 -92 288]	176	272
[30 -332 3680]	176	0
[30 -452 6816]	176	0
[22 -44 96]	176	-1216
[22 -132 800]	176	1472
[10 12 32]	176	0
[18 -92 480]	176	-272
[18 -236 3104]	176	0
[2 3 96]	183	0
[46 -93 192]	183	-1553
[46 -323 2272]	183	-287
[6 3 32]	183	0
[38 -125 416]	183	0
[16 -227 3232]	183	287
[2 2 96]	188	0
[94 -658 4608]	188	0
[48 -146 448]	188	-500
[48 -530 5856]	188	764
[6 2 32]	188	0
[6 14 64]	188	-764
[142 350 864]	188	-128
[24 -146 896]	188	-1608
[24 -338 4768]	188	200
[8 18 64]	188	500
[112 274 672]	188	-184
[12 14 32]	188	0
[12 50 224]	188	0
[32 -110 384]	188	0
[14 34 96]	188	128
[102 242 576]	188	184
[14 62 288]	188	0
[42 -146 512]	188	0
[2 1 96]	191	0
[48 -673 9440]	191	1115
[6 1 32]	191	0
[32 -417 5440]	191	-581
[8 33 160]	191	-285
[20 -223 2496]	191	285
[36 95 256]	191	2248
[16 -161 1632]	191	-2248
[18 -95 512]	191	581
[18 -257 3680]	191	-353
[2 0 96]	192	0
[2 8 128]	192	0
[2 16 224]	192	0
[4 8 64]	192	0
[52 152 448]	192	-480
[6 0 32]	192	0
[6 24 128]	192	-252
[8 8 32]	192	0
[24 -72 224]	192	228
[24 -120 608]	192	252
[26 -56 128]	192	480
[26 -160 992]	192	-3648
[26 -368 5216]	192	1728
[12 24 64]	192	480
[28 72 192]	192	0
[14 16 32]	192	0
[14 -72 384]	192	-228
[100 -1301 16928]	199	0
[50 -651 8480]	199	1298
[26 -341 4480]	199	-945
[10 11 32]	199	0
[10 21 64]	199	-1298
[16 -213 2848]	199	945
[104 -1145 12608]	207	0
[478 1113 2592]	207	0
[26 -391 5888]	207	0
[18 -39 96]	207	-2187
[18 -57 192]	207	0
[108 -973 8768]	215	0
[54 -595 6560]	215	-2639
[36 -397 4384]	215	0
[6 13 64]	215	2639
[24 -77 256]	215	-2261
[12 13 32]	215	0
[18 -109 672]	215	-3469
[20 -205 2112]	215	-2754
[16 -179 2016]	215	2261
[110 -330 992]	220	0
[110 -550 2752]	220	0
[4 6 64]	220	0
[56 -506 4576]	220	756
[28 -58 128]	220	-4788
[28 -282 2848]	220	756
[28 -422 6368]	220	0
[10 10 32]	220	0
[34 86 224]	220	0
[170 390 896]	220	-756
[14 58 256]	220	0
[34 -118 416]	220	0
[16 -70 320]	220	0
[16 -198 2464]	220	0
[112 -785 5504]	223	0
[4 17 128]	223	1261
[8 17 64]	223	-1261
[14 15 32]	223	0
[116 -581 2912]	231	0
[4 5 64]	231	0
[40 -123 384]	231	-861
[30 -453 6848]	231	0
[24 -123 640]	231	861
[40 -133 448]	231	0
[20 -123 768]	231	-249
[104 251 608]	231	0
[120 -361 1088]	239	0
[248 617 1536]	239	2393
[48 -151 480]	239	1211
[30 -151 768]	239	-1211
[24 -361 5440]	239	0
[10 41 192]	239	1951
[20 -41 96]	239	601
[198 457 1056]	239	-2393
[22 -247 2784]	239	-1951
[2 4 128]	240	0
[122 364 1088]	240	0
[4 4 64]	240	0
[60 -180 544]	240	-4824
[60 -300 1504]	240	6456
[60 -420 2944]	240	2064
[6 12 64]	240	-6456
[46 132 384]	240	-2472
[30 -60 128]	240	-3216
[30 -180 1088]	240	-1632
[8 4 32]	240	0
[32 -100 320]	240	-2064
[32 -356 3968]	240	0
[10 20 64]	240	4824
[34 92 256]	240	-744
[12 12 32]	240	0
[20 -60 192]	240	0
[20 -100 512]	240	2064
[16 -36 96]	240	2472
[16 -100 640]	240	0
[16 -164 1696]	240	744
[16 -228 3264]	240	-2064
[2 3 128]	247	0
[62 -931 13984]	247	0
[8 3 32]	247	0
[2 2 128]	252	0
[126 -882 6176]	252	0
[4 2 64]	252	0
[42 -126 384]	252	5292
[42 -210 1056]	252	0
[8 2 32]	252	0
[32 -194 1184]	252	-504
[32 -450 6336]	252	3192
[24 -126 672]	252	-5292
[24 -318 4224]	252	0
[14 14 32]	252	0
[16 -194 2368]	252	0
[18 -222 2752]	252	0
[18 -258 3712]	252	504
[2 1 128]	255	0
[4 1 64]	255	0
[44 -575 7520]	255	-4185
[8 1 32]	255	0
[22 -223 2272]	255	-184
[22 -289 3808]	255	0
[20 -65 224]	255	0
[14 33 96]	255	4185
[2 0 128]	256	0
[2 8 160]	256	0
[2 16 256]	256	0
[4 0 64]	256	0
[4 16 128]	256	-1184
[8 0 32]	256	0
[8 16 64]	256	1184
[8 32 160]	256	1568
[34 -72 160]	256	-8576
[34 -480 6784]	256	5184
[26 -80 256]	256	-1184
[26 -288 3200]	256	-1568
[26 -392 5920]	256	0
[16 16 32]	256	0
[16 -48 160]	256	1184
[20 -48 128]	256	160
[20 -288 4160]	256	3136
[132 -1717 22336]	263	0
[66 -331 1664]	263	-11217
[408 949 2208]	263	11217
[22 -309 4352]	263	9027
[22 -331 4992]	263	0
[24 -53 128]	263	-7583
[12 53 256]	263	11119
[18 -203 2304]	263	-11119
[136 -1497 16480]	271	0
[74 217 640]	271	-2270
[34 -103 320]	271	1946
[28 -423 6400]	271	0
[134 327 800]	271	-10164
[20 -103 544]	271	-1946
[212 487 1120]	271	2270
[124 295 704]	271	10164
[140 -1261 11360]	279	0
[70 -211 640]	279	13255
[28 -365 4768]	279	4456
[268 621 1440]	279	-13255
[24 -339 4800]	279	4509
[42 -141 480]	279	0
[20 -301 4544]	279	0
[98 237 576]	279	0
[18 -237 3136]	279	-4456
[20 -109 608]	279	4665
[142 -426 1280]	284	0
[142 -710 3552]	284	0
[72 -362 1824]	284	13012
[72 -938 12224]	284	-9592
[450 1046 2432]	284	-13012
[6 26 160]	284	192
[58 -182 576]	284	-4540
[36 -74 160]	284	-36
[36 -182 928]	284	4540
[36 -362 3648]	284	-18204
[36 -470 6144]	284	4056
[30 -154 800]	284	-192
[30 -394 5184]	284	7596
[30 -454 6880]	284	0
[24 -362 5472]	284	0
[240 554 1280]	284	9592
[18 -38 96]	284	5184
[18 -182 1856]	284	4580
[20 -106 576]	284	-4056
[20 -266 3552]	284	-7596
[144 -1009 7072]	287	0
[78 -239 736]	287	-4251
[48 -241 1216]	287	0
[338 785 1824]	287	4251
[24 -241 2432]	287	-2208
[12 49 224]	287	-8883
[26 -369 5248]	287	-11094
[24 -271 3072]	287	8883
[18 -271 4096]	287	0
[148 -741 3712]	295	0
[74 -667 6016]	295	-9545
[38 -421 4672]	295	0
[10 5 32]	295	0
[226 517 1184]	295	9545
[152 -457 1376]	303	0
[4 9 96]	303	-6951
[6 9 64]	303	6951
[26 -55 128]	303	-13503
[26 -393 5952]	303	0
[22 -73 256]	303	0
[2 4 160]	304	0
[154 460 1376]	304	0
[38 -76 160]	304	17216
[38 -228 1376]	304	-12352
[10 4 32]	304	0
[10 44 224]	304	9552
[50 -164 544]	304	0
[22 -68 224]	304	0
[22 -244 2720]	304	-9552
[22 -332 5024]	304	0
[2 3 160]	311	0
[52 -157 480]	311	-18264
[218 541 1344]	311	-6800
[8 35 192]	311	1706
[10 3 32]	311	0
[26 -131 672]	311	-1706
[26 -157 960]	311	-18136
[28 -285 2912]	311	20614
[12 29 96]	311	18264
[16 35 96]	311	6800
[20 -227 2592]	311	38974
[240 547 1248]	311	3574
[2 2 160]	316	0
[158 -1106 7744]	316	0
[80 -242 736]	316	-26056
[80 -882 9728]	316	21868
[40 -562 7904]	316	-668
[380 882 2048]	316	-21868
[10 2 32]	316	0
[310 718 1664]	316	26056
[20 -302 4576]	316	0
[22 -50 128]	316	17116
[22 -226 2336]	316	-15936
[2 1 160]	319	0
[50 -159 512]	319	-1379
[10 1 32]	319	0
[32 -353 3904]	319	1379
[38 97 256]	319	0
[20 -289 4192]	319	-11582
[2 0 160]	320	0
[2 8 192]	320	0
[4 8 96]	320	-7992
[84 248 736]	320	3840
[6 8 64]	320	7992
[54 -272 1376]	320	0
[54 -704 9184]	320	28256
[42 -88 192]	320	-3016
[42 -592 8352]	320	-20224
[10 0 32]	320	0
[10 40 192]	320	-6532
[28 -312 3488]	320	6532
[28 -424 6432]	320	0
[14 32 96]	320	-28256
[126 304 736]	320	0
[254 584 1344]	320	-3840
[20 -40 96]	320	2024
[20 -120 736]	320	10464
[18 -56 192]	320	1212
[18 -272 4128]	320	0
[164 -2133 27744]	327	0
[82 -1067 13888]	327	29675
[56 -619 6848]	327	16866
[42 -213 1088]	327	-16866
[282 651 1504]	327	-29675
[24 -363 5504]	327	0
[44 -149 512]	327	0
[168 -1849 20352]	335	0
[766 1785 4160]	335	26547
[6 7 64]	335	-26547
[6 25 160]	335	2170
[36 -185 960]	335	-2170
[28 -57 128]	335	21039
[30 -455 6912]	335	0
[24 -313 4096]	335	0
[14 57 256]	335	28141
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