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 + 2/ζ^3 + 8/ζ^2 + 14/ζ + 14*ζ + 8*ζ^2 + 2*ζ^3)
  +q(15/ζ^8 + 15*ζ^8)
  +q^3(8/ζ^14 + 8*ζ^14)
  +q^4(16/ζ^16 + 16*ζ^16)
• From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block

  	φ	= TB(12;1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3)
  		= 1142832
  		= η(τ)24(θ(τ,z)/η(τ))14(θ(τ,2z)/η(τ))8(θ(τ,3z)/η(τ))2

  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)	8
  (4,14)	8
  (9,3)	2
  (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 + 2/ζ^3 + 8/ζ^2 + 14/ζ + 14*ζ + 8*ζ^2 + 2*ζ^3)
+q(33086 + 15/ζ^8 + 196/ζ^7 + 1024/ζ^6 + 3350/ζ^5 + 8092/ζ^4 + 15238/ζ^3 + 23616/ζ^2 + 30368/ζ + 30368*ζ + 23616*ζ^2 + 15238*ζ^3 + 8092*ζ^4 + 3350*ζ^5 + 1024*ζ^6 + 196*ζ^7 + 15*ζ^8)
+q^2(3081488 + 58/ζ^11 + 1008/ζ^10 + 7338/ζ^9 + 32920/ζ^8 + 108782/ζ^7 + 283464/ζ^6 + 610304/ζ^5 + 1112800/ζ^4 + 1750722/ζ^3 + 2401992/ζ^2 + 2896748/ζ + 2896748*ζ + 2401992*ζ^2 + 1750722*ζ^3 + 1112800*ζ^4 + 610304*ζ^5 + 283464*ζ^6 + 108782*ζ^7 + 32920*ζ^8 + 7338*ζ^9 + 1008*ζ^10 + 58*ζ^11)
+q^3(112732242 + 8/ζ^14 + 562/ζ^13 + 8100/ζ^12 + 58346/ζ^11 + 283800/ζ^10 + 1042348/ζ^9 + 3082152/ζ^8 + 7616850/ζ^7 + 16135880/ζ^6 + 29802016/ζ^5 + 48582508/ζ^4 + 70497708/ζ^3 + 91619224/ζ^2 + 107054362/ζ + 107054362*ζ + 91619224*ζ^2 + 70497708*ζ^3 + 48582508*ζ^4 + 29802016*ζ^5 + 16135880*ζ^6 + 7616850*ζ^7 + 3082152*ζ^8 + 1042348*ζ^9 + 283800*ζ^10 + 58346*ζ^11 + 8100*ζ^12 + 562*ζ^13 + 8*ζ^14)
+q^4(2467324704 + 16/ζ^16 + 1428/ζ^15 + 23504/ζ^14 + 197662/ζ^13 + 1112864/ζ^12 + 4727184/ζ^11 + 16134392/ζ^10 + 46039906/ζ^9 + 112727904/ζ^8 + 241236686/ζ^7 + 457190352/ζ^6 + 774921034/ζ^5 + 1183222752/ζ^4 + 1636302276/ζ^3 + 2057377640/ζ^2 + 2358050176/ζ + 2358050176*ζ + 2057377640*ζ^2 + 1636302276*ζ^3 + 1183222752*ζ^4 + 774921034*ζ^5 + 457190352*ζ^6 + 241236686*ζ^7 + 112727904*ζ^8 + 46039906*ζ^9 + 16134392*ζ^10 + 4727184*ζ^11 + 1112864*ζ^12 + 197662*ζ^13 + 23504*ζ^14 + 1428*ζ^15 + 16*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	0
[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	0
[2 1 32]	63	0
[16 -225 3168]	63	0
[4 33 288]	63	0
[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	-1
[10 -24 64]	64	1
[10 -64 416]	64	0
[10 -144 2080]	64	0
[36 -469 6112]	71	0
[18 -181 1824]	71	0
[18 -235 3072]	71	2
[120 277 640]	71	0
[10 -43 192]	71	0
[10 -53 288]	71	-2
[40 -441 4864]	79	0
[20 -121 736]	79	0
[190 441 1024]	79	0
[10 -121 1472]	79	0
[44 -397 3584]	87	0
[22 -45 96]	87	0
[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	8
[26 70 192]	92	8
[162 374 864]	92	0
[6 26 128]	92	15
[26 -86 288]	92	0
[12 -26 64]	92	-8
[12 -38 128]	92	0
[12 -122 1248]	92	-8
[12 -134 1504]	92	-15
[48 -337 2368]	95	0
[24 -337 4736]	95	0
[4 17 96]	95	14
[16 -81 416]	95	-14
[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	0
[14 -59 256]	103	0
[56 -169 512]	111	0
[28 -169 1024]	111	0
[120 297 736]	111	-2
[6 9 32]	111	0
[14 -169 2048]	111	0
[14 -183 2400]	111	0
[10 23 64]	111	2
[2 4 64]	112	0
[58 172 512]	112	0
[4 4 32]	112	0
[28 -84 256]	112	-20
[28 -140 704]	112	0
[28 -196 1376]	112	-28
[14 -28 64]	112	28
[14 -84 512]	112	28
[8 12 32]	112	0
[16 -84 448]	112	20
[16 -148 1376]	112	-28
[16 -212 2816]	112	0
[2 3 64]	119	0
[30 -61 128]	119	0
[30 -451 6784]	119	0
[20 -61 192]	119	28
[90 221 544]	119	42
[16 -67 288]	119	0
[12 -61 320]	119	-28
[24 61 160]	119	0
[12 -125 1312]	119	-42
[2 2 64]	124	0
[62 -434 3040]	124	0
[4 2 32]	124	0
[32 -226 1600]	124	56
[16 -98 608]	124	-56
[16 -226 3200]	124	-56
[8 34 160]	124	-89
[28 -94 320]	124	0
[10 14 32]	124	0
[14 -46 160]	124	0
[14 -130 1216]	124	56
[14 -158 1792]	124	89
[2 1 64]	127	0
[4 1 32]	127	0
[32 -449 6304]	127	0
[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	-83
[6 8 32]	128	0
[22 -112 576]	128	83
[22 -288 3776]	128	-93
[18 -40 96]	128	48
[18 -112 704]	128	70
[18 -256 3648]	128	70
[12 16 32]	128	0
[12 -64 352]	128	93
[68 -885 11520]	135	0
[34 -171 864]	135	0
[34 -341 3424]	135	0
[24 -267 2976]	135	0
[8 11 32]	135	0
[12 -75 480]	135	0
[72 -793 8736]	143	0
[36 -217 1312]	143	0
[42 121 352]	143	-42
[6 7 32]	143	0
[6 25 128]	143	-14
[18 -199 2208]	143	14
[12 -25 64]	143	42
[14 -89 576]	143	0
[76 -685 6176]	151	0
[38 -77 160]	151	0
[38 -115 352]	151	68
[8 19 64]	151	-68
[10 13 32]	151	0
[78 -234 704]	156	0
[78 -390 1952]	156	0
[40 -202 1024]	156	0
[40 -522 6816]	156	85
[6 6 32]	156	0
[8 10 32]	156	0
[20 -42 96]	156	-104
[20 -202 2048]	156	-104
[10 22 64]	156	-85
[82 198 480]	156	0
[10 42 192]	156	230
[30 -102 352]	156	0
[16 -54 192]	156	0
[16 -182 2080]	156	-230
[80 -561 3936]	159	0
[40 -561 7872]	159	0
[46 -143 448]	159	254
[28 -143 736]	159	-254
[30 81 224]	159	-212
[16 -49 160]	159	0
[12 15 32]	159	0
[14 -143 1472]	159	212
[84 -421 2112]	167	0
[42 -379 3424]	167	170
[42 -421 4224]	167	0
[28 -421 6336]	167	0
[36 101 288]	167	82
[22 -69 224]	167	-130
[14 -155 1728]	167	130
[14 -197 2784]	167	-170
[16 -37 96]	167	-82
[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	104
[30 -332 3680]	176	0
[30 -452 6816]	176	0
[22 -44 96]	176	64
[22 -132 800]	176	64
[10 12 32]	176	0
[18 -92 480]	176	-104
[18 -236 3104]	176	0
[2 3 96]	183	0
[46 -93 192]	183	0
[46 -323 2272]	183	-268
[6 3 32]	183	0
[38 -125 416]	183	0
[16 -227 3232]	183	268
[2 2 96]	188	0
[94 -658 4608]	188	0
[48 -146 448]	188	-16
[48 -530 5856]	188	-329
[6 2 32]	188	0
[6 14 64]	188	329
[142 350 864]	188	56
[24 -146 896]	188	112
[24 -338 4768]	188	112
[8 18 64]	188	16
[112 274 672]	188	-257
[12 14 32]	188	0
[12 50 224]	188	-7
[32 -110 384]	188	0
[14 34 96]	188	-56
[102 242 576]	188	257
[14 62 288]	188	0
[42 -146 512]	188	7
[2 1 96]	191	0
[48 -673 9440]	191	0
[6 1 32]	191	0
[32 -417 5440]	191	-180
[8 33 160]	191	-266
[20 -223 2496]	191	266
[36 95 256]	191	140
[16 -161 1632]	191	-140
[18 -95 512]	191	180
[18 -257 3680]	191	-140
[2 0 96]	192	0
[2 8 128]	192	0
[2 16 224]	192	0
[4 8 64]	192	0
[52 152 448]	192	-16
[6 0 32]	192	0
[6 24 128]	192	-478
[8 8 32]	192	0
[24 -72 224]	192	-238
[24 -120 608]	192	478
[26 -56 128]	192	-16
[26 -160 992]	192	-224
[26 -368 5216]	192	-224
[12 24 64]	192	16
[28 72 192]	192	0
[14 16 32]	192	0
[14 -72 384]	192	238
[100 -1301 16928]	199	0
[50 -651 8480]	199	-448
[26 -341 4480]	199	954
[10 11 32]	199	0
[10 21 64]	199	448
[16 -213 2848]	199	-954
[104 -1145 12608]	207	0
[478 1113 2592]	207	0
[26 -391 5888]	207	0
[18 -39 96]	207	1092
[18 -57 192]	207	0
[108 -973 8768]	215	0
[54 -595 6560]	215	-352
[36 -397 4384]	215	0
[6 13 64]	215	352
[24 -77 256]	215	-378
[12 13 32]	215	0
[18 -109 672]	215	-870
[20 -205 2112]	215	-254
[16 -179 2016]	215	378
[110 -330 992]	220	0
[110 -550 2752]	220	0
[4 6 64]	220	0
[56 -506 4576]	220	-313
[28 -58 128]	220	632
[28 -282 2848]	220	632
[28 -422 6368]	220	0
[10 10 32]	220	0
[34 86 224]	220	0
[170 390 896]	220	313
[14 58 256]	220	-1472
[34 -118 416]	220	0
[16 -70 320]	220	0
[16 -198 2464]	220	1472
[112 -785 5504]	223	0
[4 17 128]	223	-534
[8 17 64]	223	534
[14 15 32]	223	0
[116 -581 2912]	231	0
[4 5 64]	231	0
[40 -123 384]	231	-1106
[30 -453 6848]	231	0
[24 -123 640]	231	1106
[40 -133 448]	231	0
[20 -123 768]	231	2002
[104 251 608]	231	0
[120 -361 1088]	239	0
[248 617 1536]	239	518
[48 -151 480]	239	1498
[30 -151 768]	239	-1498
[24 -361 5440]	239	0
[10 41 192]	239	142
[20 -41 96]	239	-1820
[198 457 1056]	239	-518
[22 -247 2784]	239	-142
[2 4 128]	240	0
[122 364 1088]	240	0
[4 4 64]	240	0
[60 -180 544]	240	996
[60 -300 1504]	240	2008
[60 -420 2944]	240	244
[6 12 64]	240	-2008
[46 132 384]	240	808
[30 -60 128]	240	-1212
[30 -180 1088]	240	-1212
[8 4 32]	240	0
[32 -100 320]	240	2572
[32 -356 3968]	240	0
[10 20 64]	240	-996
[34 92 256]	240	1820
[12 12 32]	240	0
[20 -60 192]	240	0
[20 -100 512]	240	-2572
[16 -36 96]	240	-808
[16 -100 640]	240	0
[16 -164 1696]	240	-1820
[16 -228 3264]	240	-244
[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	2577
[42 -210 1056]	252	0
[8 2 32]	252	0
[32 -194 1184]	252	1008
[32 -450 6336]	252	1008
[24 -126 672]	252	-2577
[24 -318 4224]	252	0
[14 14 32]	252	0
[16 -194 2368]	252	0
[18 -222 2752]	252	-3303
[18 -258 3712]	252	-741
[2 1 128]	255	0
[4 1 64]	255	0
[44 -575 7520]	255	-814
[8 1 32]	255	0
[22 -223 2272]	255	2574
[22 -289 3808]	255	0
[20 -65 224]	255	0
[14 33 96]	255	814
[2 0 128]	256	0
[2 8 160]	256	0
[2 16 256]	256	0
[4 0 64]	256	0
[4 16 128]	256	1584
[8 0 32]	256	0
[8 16 64]	256	-1584
[8 32 160]	256	2736
[34 -72 160]	256	-1728
[34 -480 6784]	256	-672
[26 -80 256]	256	1360
[26 -288 3200]	256	-2736
[26 -392 5920]	256	0
[16 16 32]	256	0
[16 -48 160]	256	-1360
[20 -48 128]	256	-1072
[20 -288 4160]	256	1440
[132 -1717 22336]	263	0
[66 -331 1664]	263	-2560
[408 949 2208]	263	2560
[22 -309 4352]	263	-1470
[22 -331 4992]	263	0
[24 -53 128]	263	-2350
[12 53 256]	263	1678
[18 -203 2304]	263	-1678
[136 -1497 16480]	271	0
[74 217 640]	271	266
[34 -103 320]	271	-1288
[28 -423 6400]	271	0
[134 327 800]	271	-2126
[20 -103 544]	271	1288
[212 487 1120]	271	-266
[124 295 704]	271	2126
[140 -1261 11360]	279	0
[70 -211 640]	279	-714
[28 -365 4768]	279	-348
[268 621 1440]	279	714
[24 -339 4800]	279	6522
[42 -141 480]	279	0
[20 -301 4544]	279	0
[98 237 576]	279	0
[18 -237 3136]	279	348
[20 -109 608]	279	264
[142 -426 1280]	284	0
[142 -710 3552]	284	0
[72 -362 1824]	284	-1028
[72 -938 12224]	284	-1224
[450 1046 2432]	284	1028
[6 26 160]	284	-184
[58 -182 576]	284	1246
[36 -74 160]	284	1000
[36 -182 928]	284	-1246
[36 -362 3648]	284	1000
[36 -470 6144]	284	-2184
[30 -154 800]	284	184
[30 -394 5184]	284	-754
[30 -454 6880]	284	0
[24 -362 5472]	284	0
[240 554 1280]	284	1224
[18 -38 96]	284	1240
[18 -182 1856]	284	1044
[20 -106 576]	284	2184
[20 -266 3552]	284	754
[144 -1009 7072]	287	0
[78 -239 736]	287	-1638
[48 -241 1216]	287	0
[338 785 1824]	287	1638
[24 -241 2432]	287	-4438
[12 49 224]	287	2394
[26 -369 5248]	287	-4200
[24 -271 3072]	287	-2394
[18 -271 4096]	287	0
[148 -741 3712]	295	0
[74 -667 6016]	295	56
[38 -421 4672]	295	0
[10 5 32]	295	0
[226 517 1184]	295	-56
[152 -457 1376]	303	0
[4 9 96]	303	6424
[6 9 64]	303	-6424
[26 -55 128]	303	6688
[26 -393 5952]	303	0
[22 -73 256]	303	0
[2 4 160]	304	0
[154 460 1376]	304	0
[38 -76 160]	304	1216
[38 -228 1376]	304	1216
[10 4 32]	304	0
[10 44 224]	304	-5112
[50 -164 544]	304	0
[22 -68 224]	304	0
[22 -244 2720]	304	5112
[22 -332 5024]	304	0
[2 3 160]	311	0
[52 -157 480]	311	-1694
[218 541 1344]	311	-362
[8 35 192]	311	-2014
[10 3 32]	311	0
[26 -131 672]	311	2014
[26 -157 960]	311	-2042
[28 -285 2912]	311	-5626
[12 29 96]	311	1694
[16 35 96]	311	362
[20 -227 2592]	311	-1988
[240 547 1248]	311	2044
[2 2 160]	316	0
[158 -1106 7744]	316	0
[80 -242 736]	316	-1281
[80 -882 9728]	316	-1040
[40 -562 7904]	316	-1576
[380 882 2048]	316	1040
[10 2 32]	316	0
[310 718 1664]	316	1281
[20 -302 4576]	316	0
[22 -50 128]	316	-71
[22 -226 2336]	316	-312
[2 1 160]	319	0
[50 -159 512]	319	-3766
[10 1 32]	319	0
[32 -353 3904]	319	3766
[38 97 256]	319	0
[20 -289 4192]	319	-1570
[2 0 160]	320	0
[2 8 192]	320	0
[4 8 96]	320	-5268
[84 248 736]	320	-1288
[6 8 64]	320	5268
[54 -272 1376]	320	0
[54 -704 9184]	320	240
[42 -88 192]	320	3612
[42 -592 8352]	320	896
[10 0 32]	320	0
[10 40 192]	320	-1090
[28 -312 3488]	320	1090
[28 -424 6432]	320	0
[14 32 96]	320	-240
[126 304 736]	320	0
[254 584 1344]	320	1288
[20 -40 96]	320	-212
[20 -120 736]	320	3768
[18 -56 192]	320	-1570
[18 -272 4128]	320	0
[164 -2133 27744]	327	0
[82 -1067 13888]	327	1290
[56 -619 6848]	327	-10164
[42 -213 1088]	327	10164
[282 651 1504]	327	-1290
[24 -363 5504]	327	0
[44 -149 512]	327	0
[168 -1849 20352]	335	0
[766 1785 4160]	335	-3528
[6 7 64]	335	3528
[6 25 160]	335	12238
[36 -185 960]	335	-12238
[28 -57 128]	335	-4200
[30 -455 6912]	335	0
[24 -313 4096]	335	0
[14 57 256]	335	-102
[54 -185 640]	335	102
[118 281 672]	335	5476
[172 -1549 13952]	343	0
[86 -947 10432]	343	5612
[422 979 2272]	343	-5612
[28 -371 4928]	343	0
[22 -333 5056]	343	0
[174 -522 1568]	348	0
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