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
  (18 + ζ^(-3) + 7/ζ^2 - 5/ζ - 5*ζ + 7*ζ^2 + ζ^3)
  +q(2/ζ^8 + 2*ζ^8)
  +q^2(2/ζ^12 + 2*ζ^12)
  +q^3(9/ζ^14 + 9*ζ^14)
  +q^4(18/ζ^16 + 18*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,−1,2,2,2,2,2,2,2,3)
  		= 1−52731
  		= η(τ)18(θ(τ,z)/η(τ))−5(θ(τ,2z)/η(τ))7(θ(τ,3z)/η(τ))

  This theta block has q-order of vanishing 1. 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 a symmetric 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 4 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)	3
  (4,2)	7
  (4,14)	9
  (9,3)	1
  (16,12)	2

• From the singular part above, we compute that the valuation Ord(ψ) is -0.25, and hence we need to multiply ψ by Δ^1 to get a Jacobi cusp form. The definition of ψ is
      ψ = 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

(18 + ζ^(-3) + 7/ζ^2 - 5/ζ - 5*ζ + 7*ζ^2 + ζ^3)
+q(132 + 2/ζ^8 - 10/ζ^7 - 13/ζ^6 + 27/ζ^5 - 68/ζ^4 + 55/ζ^3 + 13/ζ^2 - 72/ζ - 72*ζ + 13*ζ^2 + 55*ζ^3 - 68*ζ^4 + 27*ζ^5 - 13*ζ^6 - 10*ζ^7 + 2*ζ^8)
+q^2(872 + 2/ζ^12 + 9/ζ^11 - 3/ζ^10 - 39/ζ^9 + 92/ζ^8 - 137/ζ^7 - 26/ζ^6 + 260/ζ^5 - 530/ζ^4 + 437/ζ^3 + 29/ζ^2 - 530/ζ - 530*ζ + 29*ζ^2 + 437*ζ^3 - 530*ζ^4 + 260*ζ^5 - 26*ζ^6 - 137*ζ^7 + 92*ζ^8 - 39*ζ^9 - 3*ζ^10 + 9*ζ^11 + 2*ζ^12)
+q^3(4176 + 9/ζ^14 + 17/ζ^13 - 60/ζ^12 + 113/ζ^11 - 6/ζ^10 - 334/ζ^9 + 776/ζ^8 - 867/ζ^7 - 88/ζ^6 + 1504/ζ^5 - 2804/ζ^4 + 2282/ζ^3 + 85/ζ^2 - 2715/ζ - 2715*ζ + 85*ζ^2 + 2282*ζ^3 - 2804*ζ^4 + 1504*ζ^5 - 88*ζ^6 - 867*ζ^7 + 776*ζ^8 - 334*ζ^9 - 6*ζ^10 + 113*ζ^11 - 60*ζ^12 + 17*ζ^13 + 9*ζ^14)
+q^4(16892 + 18/ζ^16 - 26/ζ^15 + 3/ζ^14 + 171/ζ^13 - 496/ζ^12 + 720/ζ^11 + 24/ζ^10 - 1855/ζ^9 + 3952/ζ^8 - 4085/ζ^7 - 181/ζ^6 + 6685/ζ^5 - 11920/ζ^4 + 9610/ζ^3 + 154/ζ^2 - 11220/ζ - 11220*ζ + 154*ζ^2 + 9610*ζ^3 - 11920*ζ^4 + 6685*ζ^5 - 181*ζ^6 - 4085*ζ^7 + 3952*ζ^8 - 1855*ζ^9 + 24*ζ^10 + 720*ζ^11 - 496*ζ^12 + 171*ζ^13 + 3*ζ^14 - 26*ζ^15 + 18*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	1
[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	5
[10 -91 832]	39	0
[10 -101 1024]	39	0
[8 -27 96]	39	0
[24 -73 224]	47	-1
[4 9 32]	47	1
[16 -55 192]	47	0
[2 4 32]	48	-8
[26 76 224]	48	0
[6 12 32]	48	0
[14 36 96]	48	0
[2 3 32]	55	1
[14 -29 64]	55	0
[14 -99 704]	55	0
[8 -35 160]	55	0
[2 2 32]	60	19
[30 -210 1472]	60	0
[16 -50 160]	60	-2
[16 -178 1984]	60	2
[10 -30 96]	60	1
[10 -50 256]	60	-2
[8 -114 1632]	60	0
[2 1 32]	63	22
[16 -225 3168]	63	0
[12 -159 2112]	63	1
[8 -97 1184]	63	0
[2 0 32]	64	0
[2 8 64]	64	12
[2 16 160]	64	0
[4 8 32]	64	12
[20 56 160]	64	0
[10 -24 64]	64	0
[10 -144 2080]	64	0
[36 -469 6112]	71	-7
[18 -235 3072]	71	-2
[120 277 640]	71	7
[10 -53 288]	71	2
[40 -441 4864]	79	38
[190 441 1024]	79	-38
[44 -397 3584]	87	-21
[22 -243 2688]	87	0
[134 307 704]	87	-21
[46 -138 416]	92	51
[46 -230 1152]	92	-48
[4 6 32]	92	48
[24 -218 1984]	92	0
[162 374 864]	92	51
[6 26 128]	92	-2
[26 -86 288]	92	-17
[12 -26 64]	92	0
[12 -122 1248]	92	0
[12 -134 1504]	92	-2
[48 -337 2368]	95	-35
[4 17 96]	95	17
[16 -81 416]	95	-17
[10 15 32]	95	-35
[52 -261 1312]	103	-19
[4 5 32]	103	19
[56 -169 512]	111	118
[120 297 736]	111	-5
[6 9 32]	111	-118
[10 23 64]	111	5
[2 4 64]	112	-24
[58 172 512]	112	104
[4 4 32]	112	-24
[28 -84 256]	112	24
[28 -140 704]	112	48
[28 -196 1376]	112	0
[14 -28 64]	112	0
[8 12 32]	112	104
[16 -84 448]	112	24
[2 3 64]	119	-81
[30 -451 6784]	119	81
[20 -61 192]	119	21
[90 221 544]	119	-11
[12 -61 320]	119	21
[12 -125 1312]	119	11
[2 2 64]	124	-144
[62 -434 3040]	124	-98
[4 2 32]	124	-144
[16 -226 3200]	124	0
[8 34 160]	124	-38
[28 -94 320]	124	38
[10 14 32]	124	-98
[14 -158 1792]	124	-38
[2 1 64]	127	-127
[4 1 32]	127	-127
[2 0 64]	128	0
[2 8 96]	128	-160
[2 16 192]	128	0
[4 0 32]	128	0
[4 16 96]	128	44
[6 8 32]	128	-160
[22 -112 576]	128	-44
[22 -288 3776]	128	12
[18 -256 3648]	128	0
[12 16 32]	128	0
[12 -64 352]	128	-12
[68 -885 11520]	135	185
[34 -171 864]	135	139
[24 -267 2976]	135	75
[8 11 32]	135	185
[72 -793 8736]	143	-239
[42 121 352]	143	3
[6 7 32]	143	-239
[6 25 128]	143	-8
[18 -199 2208]	143	-8
[12 -25 64]	143	-3
[76 -685 6176]	151	138
[38 -115 352]	151	105
[8 19 64]	151	-105
[10 13 32]	151	-138
[78 -234 704]	156	-400
[78 -390 1952]	156	237
[40 -202 1024]	156	182
[40 -522 6816]	156	58
[6 6 32]	156	-237
[8 10 32]	156	400
[10 22 64]	156	58
[80 -561 3936]	159	28
[46 -143 448]	159	-73
[28 -143 736]	159	-73
[12 15 32]	159	28
[84 -421 2112]	167	-106
[42 -379 3424]	167	2
[28 -421 6336]	167	-106
[14 -197 2784]	167	-2
[16 -37 96]	167	33
[88 -265 800]	175	-865
[4 9 64]	175	-182
[8 9 32]	175	865
[2 4 96]	176	544
[90 268 800]	176	-680
[30 -92 288]	176	144
[30 -332 3680]	176	16
[30 -452 6816]	176	-544
[10 12 32]	176	-680
[18 -92 480]	176	144
[2 3 96]	183	693
[6 3 32]	183	693
[38 -125 416]	183	39
[16 -227 3232]	183	-72
[2 2 96]	188	518
[94 -658 4608]	188	608
[48 -146 448]	188	234
[48 -530 5856]	188	246
[6 2 32]	188	518
[6 14 64]	188	246
[142 350 864]	188	-92
[8 18 64]	188	-234
[12 14 32]	188	608
[14 34 96]	188	92
[2 1 96]	191	203
[6 1 32]	191	203
[32 -417 5440]	191	76
[8 33 160]	191	-279
[20 -223 2496]	191	-279
[18 -95 512]	191	-76
[18 -257 3680]	191	182
[2 0 96]	192	0
[2 8 128]	192	832
[2 16 224]	192	0
[4 8 64]	192	-192
[52 152 448]	192	0
[6 0 32]	192	0
[6 24 128]	192	8
[8 8 32]	192	832
[24 -120 608]	192	-8
[12 24 64]	192	0
[14 16 32]	192	0
[100 -1301 16928]	199	-1247
[50 -651 8480]	199	210
[10 11 32]	199	-1247
[10 21 64]	199	210
[104 -1145 12608]	207	319
[478 1113 2592]	207	182
[26 -391 5888]	207	-319
[108 -973 8768]	215	-159
[54 -595 6560]	215	579
[36 -397 4384]	215	-542
[6 13 64]	215	579
[12 13 32]	215	159
[110 -330 992]	220	1065
[110 -550 2752]	220	80
[4 6 64]	220	-78
[56 -506 4576]	220	126
[28 -422 6368]	220	80
[10 10 32]	220	-1065
[170 390 896]	220	126
[112 -785 5504]	223	1632
[4 17 128]	223	-405
[8 17 64]	223	-405
[14 15 32]	223	1632
[116 -581 2912]	231	967
[4 5 64]	231	187
[40 -123 384]	231	282
[30 -453 6848]	231	967
[24 -123 640]	231	282
[120 -361 1088]	239	1895
[248 617 1536]	239	20
[48 -151 480]	239	-159
[30 -151 768]	239	-159
[24 -361 5440]	239	1895
[198 457 1056]	239	20
[2 4 128]	240	-1976
[122 364 1088]	240	456
[4 4 64]	240	528
[60 -180 544]	240	-504
[60 -300 1504]	240	-960
[6 12 64]	240	960
[8 4 32]	240	-1976
[32 -356 3968]	240	448
[10 20 64]	240	504
[12 12 32]	240	456
[16 -36 96]	240	384
[16 -228 3264]	240	72
[2 3 128]	247	-2339
[62 -931 13984]	247	-894
[8 3 32]	247	-2339
[2 2 128]	252	-1760
[126 -882 6176]	252	151
[4 2 64]	252	1130
[42 -126 384]	252	-382
[42 -210 1056]	252	1466
[8 2 32]	252	-1760
[24 -126 672]	252	-382
[14 14 32]	252	151
[18 -258 3712]	252	74
[2 1 128]	255	-474
[4 1 64]	255	824
[44 -575 7520]	255	187
[8 1 32]	255	-474
[14 33 96]	255	187
[2 0 128]	256	0
[2 8 160]	256	-1920
[4 0 64]	256	0
[4 16 128]	256	-928
[8 0 32]	256	0
[8 16 64]	256	-928
[8 32 160]	256	-304
[26 -288 3200]	256	-304
[26 -392 5920]	256	1920
[16 16 32]	256	0
[20 -288 4160]	256	0
[132 -1717 22336]	263	2916
[66 -331 1664]	263	-1434
[408 949 2208]	263	-1434
[22 -331 4992]	263	-2916
[136 -1497 16480]	271	500
[74 217 640]	271	-490
[28 -423 6400]	271	-500
[212 487 1120]	271	490
[140 -1261 11360]	279	-841
[70 -211 640]	279	-1065
[268 621 1440]	279	-1065
[20 -301 4544]	279	-841
[142 -426 1280]	284	-880
[142 -710 3552]	284	-2203
[72 -362 1824]	284	-1440
[72 -938 12224]	284	-336
[450 1046 2432]	284	-1440
[6 26 160]	284	-111
[58 -182 576]	284	-652
[36 -182 928]	284	-652
[30 -154 800]	284	111
[30 -454 6880]	284	-2203
[24 -362 5472]	284	-880
[240 554 1280]	284	336
[78 -239 736]	287	1446
[48 -241 1216]	287	1768
[338 785 1824]	287	1446
[18 -271 4096]	287	7152
[148 -741 3712]	295	-3320
[38 -421 4672]	295	788
[10 5 32]	295	3320
[226 517 1184]	295	-484
[152 -457 1376]	303	359
[4 9 96]	303	906
[6 9 64]	303	906
[26 -393 5952]	303	359
[2 4 160]	304	3016
[154 460 1376]	304	6304
[10 4 32]	304	3016
[22 -332 5024]	304	-6304
[2 3 160]	311	3373
[52 -157 480]	311	-594
[10 3 32]	311	3373
[12 29 96]	311	594
[16 35 96]	311	-1076
[240 547 1248]	311	-665
[2 2 160]	316	5130
[80 -242 736]	316	-2098
[80 -882 9728]	316	-2462
[380 882 2048]	316	2462
[10 2 32]	316	5130
[310 718 1664]	316	-2098
[20 -302 4576]	316	9424
[2 1 160]	319	2872
[50 -159 512]	319	-1081
[10 1 32]	319	2872
[32 -353 3904]	319	1081
[20 -289 4192]	319	-2005
[2 0 160]	320	0
[2 8 192]	320	1200
[4 8 96]	320	280
[84 248 736]	320	-920
[6 8 64]	320	280
[54 -272 1376]	320	2112
[54 -704 9184]	320	128
[10 0 32]	320	0
[28 -424 6432]	320	-1200
[14 32 96]	320	128
[254 584 1344]	320	920
[18 -272 4128]	320	0
[164 -2133 27744]	327	1060
[82 -1067 13888]	327	-934
[56 -619 6848]	327	1796
[42 -213 1088]	327	-1796
[282 651 1504]	327	934
[24 -363 5504]	327	-1060
[168 -1849 20352]	335	-295
[766 1785 4160]	335	782
[6 7 64]	335	-782
[6 25 160]	335	-341
[36 -185 960]	335	341
[30 -455 6912]	335	295
[86 -947 10432]	343	-4735
[422 979 2272]	343	4735
[22 -333 5056]	343	2884
[174 -522 1568]	348	-703
[174 -870 4352]	348	816
[4 6 96]	348	-1680
[6 6 64]	348	-1680
[44 -486 5376]	348	-1232
[12 6 32]	348	-816
[26 -394 5984]	348	-703
[268 614 1408]	348	-1824
[4 17 160]	351	2305
[60 -303 1536]	351	2724
[352 815 1888]	351	-2305
[20 -303 4608]	351	-5272
[180 -901 4512]	359	5126
[4 5 96]	359	-2356
[60 -901 13536]	359	2356
[12 5 32]	359	-5126
[184 -553 1664]	367	-3695
[376 937 2336]	367	1985
[28 -425 6464]	367	-3695
[296 681 1568]	367	1985
[2 4 192]	368	-3768
[4 4 96]	368	-3616
[92 -276 832]	368	1680
[92 -460 2304]	368	5448
[62 -188 576]	368	-320
[62 -684 7552]	368	4992
[62 -932 14016]	368	3616
[8 12 64]	368	-5448
[48 -244 1248]	368	-4992
[12 4 32]	368	-3768
[12 28 96]	368	320
[324 748 1728]	368	1680
[24 -364 5536]	368	16440
[282 644 1472]	368	-152
[2 3 192]	375	-227
[94 -1411 21184]	375	4547
[6 3 64]	375	-4547
[12 3 32]	375	-227
[2 2 192]	380	-6208
[4 2 96]	380	-3900
[6 2 64]	380	-3900
[6 14 96]	380	-2777
[60 -190 608]	380	-2254
[38 -190 960]	380	-2254
[12 2 32]	380	-6208
[16 34 96]	380	-2470
[22 -334 5088]	380	-23357
[20 30 64]	380	-2172
[2 1 192]	383	-3551
[4 1 96]	383	-1944
[6 1 64]	383	-1944
[64 -833 10848]	383	1064
[12 1 32]	383	-3551
[14 31 96]	383	1064
[2 0 192]	384	0
[2 8 224]	384	192
[4 0 96]	384	0
[4 16 160]	384	4352
[6 0 64]	384	0
[6 24 160]	384	-2464
[10 24 96]	384	-2464
[394 912 2112]	384	-4352
[12 0 32]	384	0
[30 -456 6944]	384	-192
[20 -304 4640]	384	0
[22 32 64]	384	0
[98 -491 2464]	391	5043
[8 11 64]	391	-5043
[26 -395 6016]	391	14000
[200 -2201 24224]	399	-1765
[50 -551 6080]	399	-4962
[14 7 32]	399	-1765
[310 711 1632]	399	-4925
[102 -307 928]	407	1531
[68 -749 8256]	407	8466
[6 13 96]	407	-5822
[8 19 96]	407	8466
[366 845 1952]	407	1531
[24 -365 5568]	407	-3326
[206 -1030 5152]	412	14641
[104 -522 2624]	412	3904
[104 -1354 17632]	412	-2320
[8 10 64]	412	-3904
[14 6 32]	412	-14641
[338 778 1792]	412	2320
[28 -426 6496]	412	4016
[110 -335 1024]	415	-8038
[10 15 64]	415	8038
[22 -335 5120]	415	-25514
[212 -1061 5312]	423	5230
[72 -219 672]	423	-120
[12 27 96]	423	120
[14 5 32]	423	-5230
[18 27 64]	423	-1824
[4 9 128]	431	8
[80 -247 768]	431	3985
[8 9 64]	431	8
[10 23 96]	431	-3985
[30 -457 6976]	431	-4537
[2 4 224]	432	2656
[6 12 96]	432	-9744
[14 4 32]	432	2656
[26 -396 6048]	432	-4864
[2 3 224]	439	-9748
[70 -221 704]	439	-1317
[44 -221 1120]	439	-1317
[14 3 32]	439	-9748
[20 29 64]	439	-4017
[2 2 224]	444	-5435
[112 -338 1024]	444	4874
[112 -1234 13600]	444	8710
[74 -222 672]	444	-1249
[74 -370 1856]	444	-11686
[8 18 96]	444	11686
[10 14 64]	444	8710
[12 18 64]	444	-4874
[14 2 32]	444	-5435
[14 30 96]	444	1249
[24 -366 5600]	444	1520
[2 1 224]	447	-11704
[14 1 32]	447	-11704
[16 33 96]	447	-5744
[22 31 64]	447	5824
[2 0 224]	448	0
[4 8 128]	448	3904
[8 8 64]	448	3904
[56 -280 1408]	448	11392
[14 0 32]	448	0
[16 8 32]	448	13312
[16 24 64]	448	6976
[22 -336 5152]	448	0
[114 -1483 19296]	455	-3373
[696 1621 3776]	455	8594
[14 21 64]	455	-3373
[28 -427 6528]	455	-8857
[1054 2457 5728]	463	-8026
[58 -871 13088]	463	-8026
[16 7 32]	463	-4141
[118 -1299 14304]	471	13535
[10 13 64]	471	13535
[26 -397 6080]	471	-2896
[4 6 128]	476	10320
[738 1718 4000]	476	4240
[6 26 192]	476	1632
[90 -278 864]	476	9302
[60 -902 13568]	476	-10320
[10 22 96]	476	-9302
[12 26 96]	476	1632
[16 6 32]	476	25872
[18 26 64]	476	-5184
[30 -458 7008]	476	-11238
[4 17 192]	479	-4367
[80 -401 2016]	479	-14503
[8 17 96]	479	14503
[12 17 64]	479	-4367
[24 -367 5632]	479	47393
[4 5 128]	487	10261
[62 -933 14048]	487	-10261
[16 5 32]	487	37195
[6 9 96]	495	1426
[62 -311 1568]	495	17112
[16 23 64]	495	14058
[18 9 32]	495	1685
[4 4 128]	496	12368
[124 -372 1120]	496	3936
[124 -620 3104]	496	-13392
[8 4 64]	496	12368
[64 -708 7840]	496	5584
[10 12 64]	496	13392
[50 -252 1280]	496	-5584
[14 20 64]	496	-3936
[16 4 32]	496	23216
[20 28 64]	496	-4960
[28 -428 6560]	496	-17328