A Borcherds Product in S₉(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)
  +(18 + ζ^(-3) + 3/ζ^2 + 11/ζ + 11*ζ + 3*ζ^2 + ζ^3)
  +q(19/ζ^8 + 19*ζ^8)
  +q^3(5/ζ^14 + 5*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(9;1,1,1,1,1,1,1,1,1,1,1,2,2,2,3)
  		= 1112331
  		= η(τ)18(θ(τ,z)/η(τ))11(θ(τ,2z)/η(τ))3(θ(τ,3z)/η(τ))

  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)	15
  (4,2)	3
  (4,14)	5
  (9,3)	1
  (64,0)	1

• We compute that the valuation Ord(ψ - (φ|V₂)/φ) is -0.390625, 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)
+(18 + ζ^(-3) + 3/ζ^2 + 11/ζ + 11*ζ + 3*ζ^2 + ζ^3)
+q(33070 + 19/ζ^8 + 198/ζ^7 + 1031/ζ^6 + 3355/ζ^5 + 8096/ζ^4 + 15239/ζ^3 + 23609/ζ^2 + 30360/ζ + 30360*ζ + 23609*ζ^2 + 15239*ζ^3 + 8096*ζ^4 + 3355*ζ^5 + 1031*ζ^6 + 198*ζ^7 + 19*ζ^8)
+q^2(3081440 + 57/ζ^11 + 1009/ζ^10 + 7337/ζ^9 + 32928/ζ^8 + 108791/ζ^7 + 283486/ζ^6 + 610324/ζ^5 + 1112816/ζ^4 + 1750725/ζ^3 + 2401969/ζ^2 + 2896718/ζ + 2896718*ζ + 2401969*ζ^2 + 1750725*ζ^3 + 1112816*ζ^4 + 610324*ζ^5 + 283486*ζ^6 + 108791*ζ^7 + 32928*ζ^8 + 7337*ζ^9 + 1009*ζ^10 + 57*ζ^11)
+q^3(112732114 + 5/ζ^14 + 561/ζ^13 + 8096/ζ^12 + 58337/ζ^11 + 283794/ζ^10 + 1042338/ζ^9 + 3082168/ζ^8 + 7616877/ζ^7 + 16135944/ζ^6 + 29802080/ζ^5 + 48582560/ζ^4 + 70497722/ζ^3 + 91619169/ζ^2 + 107054277/ζ + 107054277*ζ + 91619169*ζ^2 + 70497722*ζ^3 + 48582560*ζ^4 + 29802080*ζ^5 + 16135944*ζ^6 + 7616877*ζ^7 + 3082168*ζ^8 + 1042338*ζ^9 + 283794*ζ^10 + 58337*ζ^11 + 8096*ζ^12 + 561*ζ^13 + 5*ζ^14)
+q^4(2467324388 + 14/ζ^16 + 1430/ζ^15 + 23503/ζ^14 + 197659/ζ^13 + 1112848/ζ^12 + 4727152/ζ^11 + 16134360/ζ^10 + 46039873/ζ^9 + 112727936/ζ^8 + 241236763/ζ^7 + 457190519/ζ^6 + 774921213/ζ^5 + 1183222896/ζ^4 + 1636302314/ζ^3 + 2057377506/ζ^2 + 2358049948/ζ + 2358049948*ζ + 2057377506*ζ^2 + 1636302314*ζ^3 + 1183222896*ζ^4 + 774921213*ζ^5 + 457190519*ζ^6 + 241236763*ζ^7 + 112727936*ζ^8 + 46039873*ζ^9 + 16134360*ζ^10 + 4727152*ζ^11 + 1112848*ζ^12 + 197659*ζ^13 + 23503*ζ^14 + 1430*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	1
[6 -13 32]	23	-1
[6 -19 64]	23	0
[14 -42 128]	28	-1
[14 -70 352]	28	0
[8 -42 224]	28	1
[8 -106 1408]	28	0
[16 -113 800]	31	3
[8 -113 1600]	31	-3
[14 -47 160]	31	0
[20 -101 512]	39	0
[10 -91 832]	39	-5
[10 -101 1024]	39	-5
[8 -27 96]	39	0
[24 -73 224]	47	11
[12 -73 448]	47	-10
[4 9 32]	47	11
[16 -55 192]	47	0
[2 4 32]	48	0
[26 76 224]	48	8
[6 12 32]	48	-8
[14 36 96]	48	0
[2 3 32]	55	0
[14 -29 64]	55	-1
[14 -99 704]	55	-1
[8 -35 160]	55	0
[2 2 32]	60	0
[30 -210 1472]	60	-19
[16 -50 160]	60	19
[16 -178 1984]	60	0
[10 -30 96]	60	0
[10 -50 256]	60	-19
[8 -114 1632]	60	19
[2 1 32]	63	0
[16 -225 3168]	63	22
[12 -159 2112]	63	0
[8 -97 1184]	63	0
[2 0 32]	64	0
[2 8 64]	64	52
[2 16 160]	64	0
[4 8 32]	64	-52
[20 56 160]	64	52
[10 -24 64]	64	52
[10 -144 2080]	64	0
[36 -469 6112]	71	-30
[18 -235 3072]	71	-38
[120 277 640]	71	-30
[10 -53 288]	71	-38
[40 -441 4864]	79	-133
[190 441 1024]	79	-133
[10 -121 1472]	79	0
[44 -397 3584]	87	-52
[22 -243 2688]	87	0
[134 307 704]	87	52
[46 -138 416]	92	-80
[46 -230 1152]	92	-179
[4 6 32]	92	-179
[24 -218 1984]	92	-336
[162 374 864]	92	80
[6 26 128]	92	157
[26 -86 288]	92	0
[12 -26 64]	92	336
[12 -38 128]	92	0
[12 -122 1248]	92	77
[12 -134 1504]	92	-157
[48 -337 2368]	95	71
[4 17 96]	95	11
[16 -81 416]	95	11
[10 15 32]	95	-71
[52 -261 1312]	103	52
[4 5 32]	103	52
[56 -169 512]	111	165
[120 297 736]	111	10
[6 9 32]	111	165
[10 23 64]	111	10
[2 4 64]	112	-232
[58 172 512]	112	-104
[4 4 32]	112	232
[28 -84 256]	112	336
[28 -140 704]	112	0
[28 -196 1376]	112	-104
[14 -28 64]	112	-104
[8 12 32]	112	104
[16 -84 448]	112	-336
[2 3 64]	119	404
[30 -451 6784]	119	404
[20 -61 192]	119	-365
[90 221 544]	119	595
[12 -61 320]	119	365
[12 -125 1312]	119	595
[2 2 64]	124	-286
[62 -434 3040]	124	-240
[4 2 32]	124	286
[16 -226 3200]	124	1008
[8 34 160]	124	-722
[28 -94 320]	124	0
[10 14 32]	124	240
[14 -158 1792]	124	722
[2 1 64]	127	78
[4 1 32]	127	-78
[2 0 64]	128	0
[2 8 96]	128	160
[2 16 192]	128	0
[4 0 32]	128	0
[4 16 96]	128	160
[6 8 32]	128	-160
[22 -112 576]	128	160
[22 -288 3776]	128	160
[18 -40 96]	128	160
[18 -256 3648]	128	0
[12 16 32]	128	0
[12 -64 352]	128	160
[68 -885 11520]	135	211
[34 -171 864]	135	0
[24 -267 2976]	135	0
[8 11 32]	135	-211
[72 -793 8736]	143	273
[42 121 352]	143	595
[6 7 32]	143	-273
[6 25 128]	143	605
[18 -199 2208]	143	-605
[12 -25 64]	143	595
[76 -685 6176]	151	-415
[38 -115 352]	151	-815
[8 19 64]	151	-815
[10 13 32]	151	-415
[78 -234 704]	156	403
[78 -390 1952]	156	1040
[40 -202 1024]	156	0
[40 -522 6816]	156	-1443
[6 6 32]	156	1040
[8 10 32]	156	403
[10 22 64]	156	1443
[30 -102 352]	156	0
[80 -561 3936]	159	52
[46 -143 448]	159	-209
[28 -143 736]	159	209
[12 15 32]	159	-52
[84 -421 2112]	167	-940
[42 -379 3424]	167	-227
[28 -421 6336]	167	940
[22 -69 224]	167	923
[14 -155 1728]	167	923
[14 -197 2784]	167	-227
[16 -37 96]	167	-1480
[88 -265 800]	175	-858
[4 9 64]	175	0
[8 9 32]	175	-858
[2 4 96]	176	680
[90 268 800]	176	-544
[30 -92 288]	176	-136
[30 -332 3680]	176	0
[30 -452 6816]	176	680
[10 12 32]	176	544
[18 -92 480]	176	136
[2 3 96]	183	-1831
[6 3 32]	183	1831
[38 -125 416]	183	0
[16 -227 3232]	183	1027
[2 2 96]	188	672
[94 -658 4608]	188	762
[48 -146 448]	188	-2416
[48 -530 5856]	188	-1834
[6 2 32]	188	-672
[6 14 64]	188	1834
[142 350 864]	188	400
[8 18 64]	188	-2416
[12 14 32]	188	-762
[14 34 96]	188	400
[2 1 96]	191	710
[6 1 32]	191	-710
[32 -417 5440]	191	1016
[8 33 160]	191	-3311
[20 -223 2496]	191	3311
[18 -95 512]	191	1016
[18 -257 3680]	191	1711
[2 0 96]	192	0
[2 8 128]	192	-1856
[2 16 224]	192	0
[4 8 64]	192	0
[52 152 448]	192	-3712
[6 0 32]	192	0
[6 24 128]	192	-1856
[8 8 32]	192	1856
[24 -120 608]	192	-1856
[12 24 64]	192	3712
[14 16 32]	192	0
[100 -1301 16928]	199	377
[50 -651 8480]	199	1835
[10 11 32]	199	-377
[10 21 64]	199	-1835
[104 -1145 12608]	207	2012
[478 1113 2592]	207	0
[26 -391 5888]	207	2012
[18 -39 96]	207	573
[108 -973 8768]	215	2483
[54 -595 6560]	215	1914
[36 -397 4384]	215	0
[6 13 64]	215	-1914
[12 13 32]	215	2483
[110 -330 992]	220	-208
[110 -550 2752]	220	-1193
[4 6 64]	220	0
[56 -506 4576]	220	-1401
[28 -422 6368]	220	1193
[10 10 32]	220	-208
[170 390 896]	220	1401
[112 -785 5504]	223	-2327
[4 17 128]	223	-4619
[8 17 64]	223	4619
[14 15 32]	223	2327
[116 -581 2912]	231	4703
[4 5 64]	231	0
[40 -123 384]	231	3967
[30 -453 6848]	231	-4703
[24 -123 640]	231	-3967
[40 -133 448]	231	0
[120 -361 1088]	239	418
[248 617 1536]	239	-122
[48 -151 480]	239	-141
[30 -151 768]	239	141
[24 -361 5440]	239	-418
[198 457 1056]	239	122
[2 4 128]	240	1976
[122 364 1088]	240	4408
[4 4 64]	240	0
[60 -180 544]	240	-6008
[60 -300 1504]	240	376
[6 12 64]	240	376
[8 4 32]	240	-1976
[32 -100 320]	240	-6384
[32 -356 3968]	240	0
[10 20 64]	240	-6008
[12 12 32]	240	-4408
[20 -100 512]	240	6384
[16 -36 96]	240	6008
[16 -228 3264]	240	-6384
[2 3 128]	247	2002
[62 -931 13984]	247	0
[8 3 32]	247	-2002
[2 2 128]	252	2665
[126 -882 6176]	252	4576
[4 2 64]	252	0
[42 -126 384]	252	2760
[42 -210 1056]	252	0
[8 2 32]	252	-2665
[24 -126 672]	252	-2760
[14 14 32]	252	-4576
[18 -258 3712]	252	-7241
[2 1 128]	255	-7339
[4 1 64]	255	0
[44 -575 7520]	255	1033
[8 1 32]	255	7339
[14 33 96]	255	-1033
[2 0 128]	256	0
[2 8 160]	256	1920
[4 0 64]	256	0
[4 16 128]	256	-2624
[8 0 32]	256	0
[8 16 64]	256	2624
[8 32 160]	256	1920
[26 -288 3200]	256	-1920
[26 -392 5920]	256	1920
[16 16 32]	256	0
[20 -288 4160]	256	0
[132 -1717 22336]	263	-5423
[66 -331 1664]	263	-4705
[408 949 2208]	263	4705
[22 -331 4992]	263	-5423
[136 -1497 16480]	271	-6310
[74 217 640]	271	-5739
[28 -423 6400]	271	-6310
[212 487 1120]	271	-5739
[140 -1261 11360]	279	2214
[70 -211 640]	279	3680
[268 621 1440]	279	-3680
[20 -301 4544]	279	-2214
[142 -426 1280]	284	-2533
[142 -710 3552]	284	-3856
[72 -362 1824]	284	9749
[72 -938 12224]	284	4320
[450 1046 2432]	284	-9749
[6 26 160]	284	-800
[58 -182 576]	284	4629
[36 -182 928]	284	-4629
[30 -154 800]	284	-800
[30 -454 6880]	284	3856
[24 -362 5472]	284	2533
[240 554 1280]	284	4320
[18 -38 96]	284	-3360
[78 -239 736]	287	7860
[48 -241 1216]	287	0
[338 785 1824]	287	-7860
[18 -271 4096]	287	2983
[148 -741 3712]	295	-8121
[38 -421 4672]	295	0
[10 5 32]	295	-8121
[226 517 1184]	295	3726
[152 -457 1376]	303	308
[4 9 96]	303	4655
[6 9 64]	303	-4655
[26 -393 5952]	303	-308
[2 4 160]	304	-6304
[154 460 1376]	304	-3016
[10 4 32]	304	6304
[50 -164 544]	304	0
[22 -332 5024]	304	-3016
[2 3 160]	311	-7553
[52 -157 480]	311	3705
[8 35 192]	311	7305
[10 3 32]	311	7553
[26 -131 672]	311	7305
[12 29 96]	311	3705
[16 35 96]	311	-839
[240 547 1248]	311	-2654
[2 2 160]	316	-2736
[80 -242 736]	316	21546
[80 -882 9728]	316	32528
[380 882 2048]	316	32528
[10 2 32]	316	2736
[310 718 1664]	316	-21546
[20 -302 4576]	316	-17290
[2 1 160]	319	19430
[50 -159 512]	319	-13603
[10 1 32]	319	-19430
[32 -353 3904]	319	-13603
[20 -289 4192]	319	2538
[2 0 160]	320	0
[2 8 192]	320	5200
[4 8 96]	320	-2960
[84 248 736]	320	7440
[6 8 64]	320	2960
[54 -272 1376]	320	0
[54 -704 9184]	320	4480
[10 0 32]	320	0
[28 -424 6432]	320	5200
[14 32 96]	320	-4480
[254 584 1344]	320	7440
[18 -272 4128]	320	0
[164 -2133 27744]	327	10210
[82 -1067 13888]	327	9739
[56 -619 6848]	327	11059
[42 -213 1088]	327	11059
[282 651 1504]	327	9739
[24 -363 5504]	327	10210
[168 -1849 20352]	335	-1258
[766 1785 4160]	335	-2205
[6 7 64]	335	-2205
[6 25 160]	335	8292
[36 -185 960]	335	8292
[30 -455 6912]	335	-1258
[86 -947 10432]	343	-8702
[422 979 2272]	343	-8702
[22 -333 5056]	343	22316
[174 -522 1568]	348	8528
[174 -870 4352]	348	10047
[4 6 96]	348	17103
[6 6 64]	348	-17103
[44 -486 5376]	348	0
[12 6 32]	348	10047
[26 -394 5984]	348	-8528
[268 614 1408]	348	-14784
[4 17 160]	351	4090
[60 -303 1536]	351	0
[352 815 1888]	351	4090
[20 -303 4608]	351	15314
[180 -901 4512]	359	97
[4 5 96]	359	-22129
[60 -901 13536]	359	-22129
[12 5 32]	359	97
[18 37 96]	359	13806
[184 -553 1664]	367	7659
[376 937 2336]	367	3892
[28 -425 6464]	367	-7659
[296 681 1568]	367	-3892
[2 4 192]	368	-3656
[4 4 96]	368	2824
[92 -276 832]	368	20736
[92 -460 2304]	368	752
[62 -188 576]	368	17912
[62 -684 7552]	368	-3576
[62 -932 14016]	368	2824
[8 12 64]	368	752
[48 -244 1248]	368	-3576
[12 4 32]	368	3656
[12 28 96]	368	17912
[324 748 1728]	368	-20736
[24 -364 5536]	368	-16328
[282 644 1472]	368	-17160
[2 3 192]	375	42378
[94 -1411 21184]	375	8232
[6 3 64]	375	8232
[40 -445 4960]	375	0
[12 3 32]	375	-42378
[2 2 192]	380	-27965
[4 2 96]	380	17613
[6 2 64]	380	-17613
[6 14 96]	380	0
[8 34 192]	380	3984
[60 -190 608]	380	-16115
[38 -190 960]	380	16115
[12 2 32]	380	27965
[32 -162 832]	380	3984
[16 34 96]	380	-20099
[22 -334 5088]	380	1600
[20 30 64]	380	26928
[2 1 192]	383	-22731
[4 1 96]	383	-1948
[6 1 64]	383	1948
[64 -833 10848]	383	-1930
[12 1 32]	383	22731
[14 31 96]	383	1930
[2 0 192]	384	0
[2 8 224]	384	-192
[4 0 96]	384	0
[4 16 160]	384	28800
[6 0 64]	384	0
[6 24 160]	384	28800
[10 24 96]	384	-28800
[394 912 2112]	384	28800
[12 0 32]	384	0
[30 -456 6944]	384	-192
[20 -304 4640]	384	0
[22 32 64]	384	0
[98 -491 2464]	391	46853
[8 11 64]	391	46853
[26 -395 6016]	391	5473
[200 -2201 24224]	399	21081
[50 -551 6080]	399	0
[14 7 32]	399	-21081
[310 711 1632]	399	10970
[102 -307 928]	407	4478
[68 -749 8256]	407	17897
[6 13 96]	407	0
[8 19 96]	407	-17897
[366 845 1952]	407	-4478
[24 -365 5568]	407	-16986
[206 -1030 5152]	412	208
[104 -522 2624]	412	-13248
[104 -1354 17632]	412	10145
[8 10 64]	412	-13248
[14 6 32]	412	208
[338 778 1792]	412	10145
[28 -426 6496]	412	10417
[110 -335 1024]	415	-47468
[10 15 64]	415	-47468
[22 -335 5120]	415	-36153
[212 -1061 5312]	423	5606
[72 -219 672]	423	-52230
[12 27 96]	423	-52230
[14 5 32]	423	5606
[18 27 64]	423	6077
[4 9 128]	431	12933
[80 -247 768]	431	43090
[8 9 64]	431	-12933
[10 23 96]	431	43090
[30 -457 6976]	431	12260
[2 4 224]	432	-4864
[6 12 96]	432	0
[14 4 32]	432	4864
[18 36 96]	432	4584
[26 -396 6048]	432	-2656
[2 3 224]	439	-46202
[70 -221 704]	439	38426
[44 -221 1120]	439	-38426
[14 3 32]	439	46202
[20 29 64]	439	5032
[2 2 224]	444	37744
[112 -338 1024]	444	-39056
[112 -1234 13600]	444	-82587
[74 -222 672]	444	19040
[74 -370 1856]	444	24491
[8 18 96]	444	24491
[10 14 64]	444	82587
[46 -510 5664]	444	0
[12 18 64]	444	-39056
[14 2 32]	444	-37744
[14 30 96]	444	19040
[24 -366 5600]	444	41659
[2 1 224]	447	41249
[8 33 192]	447	-10496
[38 -193 992]	447	-10496
[14 1 32]	447	-41249
[16 33 96]	447	3367
[22 31 64]	447	-10162
[2 0 224]	448	0
[4 8 128]	448	48256
[8 8 64]	448	-48256
[56 -280 1408]	448	0
[14 0 32]	448	0
[16 8 32]	448	29696
[16 24 64]	448	11136
[22 -336 5152]	448	0
[114 -1483 19296]	455	-41441
[696 1621 3776]	455	0
[14 21 64]	455	41441
[28 -427 6528]	455	-7904
[1054 2457 5728]	463	-23881
[58 -871 13088]	463	23881
[16 7 32]	463	24688
[118 -1299 14304]	471	12729
[10 13 64]	471	-12729
[26 -397 6080]	471	-1612
[4 6 128]	476	-94400
[738 1718 4000]	476	0
[6 26 192]	476	64262
[90 -278 864]	476	-880
[60 -902 13568]	476	-94400
[10 22 96]	476	-880
[12 26 96]	476	-64262
[16 6 32]	476	-30106
[18 26 64]	476	-54602
[30 -458 7008]	476	15472
[4 17 192]	479	17722
[80 -401 2016]	479	-2435
[8 17 96]	479	-2435
[12 17 64]	479	-17722
[24 -367 5632]	479	-2106
[4 5 128]	487	39653
[62 -933 14048]	487	39653
[16 5 32]	487	8949
[6 9 96]	495	0
[62 -311 1568]	495	0
[16 23 64]	495	-11063
[18 9 32]	495	10180
[4 4 128]	496	-82080
[124 -372 1120]	496	-52496
[124 -620 3104]	496	91792
[8 4 64]	496	82080
[64 -708 7840]	496	-9712
[10 12 64]	496	91792
[50 -252 1280]	496	-9712
[14 20 64]	496	-52496
[16 4 32]	496	-75088
[20 28 64]	496	-62208
[28 -428 6560]	496	69200