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 + 2/ζ^4 + 2/ζ^3 + ζ^(-2) + 10/ζ + 10*ζ + ζ^2 + 2*ζ^3 + 2*ζ^4)
  +q(ζ^(-9) + 10/ζ^8 + 10*ζ^8 + ζ^9)
  +q^3(-ζ^(-14) - ζ^14)
  +q^4(2/ζ^16 + 2*ζ^16)
• From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block

  	φ	= TB(9;1,1,1,1,1,1,1,1,1,1,2,3,3,4,4)
  		= 110213242
  		= η(τ)18(θ(τ,z)/η(τ))10(θ(τ,2z)/η(τ))(θ(τ,3z)/η(τ))2(θ(τ,4z)/η(τ))2

  This theta block has q-order of vanishing 2. 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 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)	15
  (4,2)	3
  (4,14)	1
  (9,3)	2
  (16,4)	2
  (17,9)	1

• From the singular part above, we compute that the valuation Ord(ψ) is -0.265625, 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 + 2/ζ^4 + 2/ζ^3 + ζ^(-2) + 10/ζ + 10*ζ + ζ^2 + 2*ζ^3 + 2*ζ^4)
+q(196 + ζ^(-9) + 10/ζ^8 + 17/ζ^7 - 11/ζ^6 - 62/ζ^5 - 44/ζ^4 - 34/ζ^3 - 53/ζ^2 + 78/ζ + 78*ζ - 53*ζ^2 - 34*ζ^3 - 44*ζ^4 - 62*ζ^5 - 11*ζ^6 + 17*ζ^7 + 10*ζ^8 + ζ^9)
+q^2(1416 + ζ^(-11) + 11/ζ^10 + 11/ζ^9 + 108/ζ^8 + 231/ζ^7 - 102/ζ^6 - 533/ζ^5 - 304/ζ^4 - 212/ζ^3 - 421/ζ^2 + 502/ζ + 502*ζ - 421*ζ^2 - 212*ζ^3 - 304*ζ^4 - 533*ζ^5 - 102*ζ^6 + 231*ζ^7 + 108*ζ^8 + 11*ζ^9 + 11*ζ^10 + ζ^11)
+q^3(7120 - ζ^(-14) - 21/ζ^13 - 84/ζ^12 - 10/ζ^11 + 102/ζ^10 + 47/ζ^9 + 808/ζ^8 + 1617/ζ^7 - 536/ζ^6 - 2990/ζ^5 - 1468/ζ^4 - 1043/ζ^3 - 2381/ζ^2 + 2400/ζ + 2400*ζ - 2381*ζ^2 - 1043*ζ^3 - 1468*ζ^4 - 2990*ζ^5 - 536*ζ^6 + 1617*ζ^7 + 808*ζ^8 + 47*ζ^9 + 102*ζ^10 - 10*ζ^11 - 84*ζ^12 - 21*ζ^13 - ζ^14)
+q^4(29596 + 2/ζ^16 + 19/ζ^15 + 53/ζ^14 - 254/ζ^13 - 722/ζ^12 - 62/ζ^11 + 536/ζ^10 + 97/ζ^9 + 4016/ζ^8 + 7955/ζ^7 - 2451/ζ^6 - 13220/ζ^5 - 5806/ζ^4 - 3992/ζ^3 - 10426/ζ^2 + 9457/ζ + 9457*ζ - 10426*ζ^2 - 3992*ζ^3 - 5806*ζ^4 - 13220*ζ^5 - 2451*ζ^6 + 7955*ζ^7 + 4016*ζ^8 + 97*ζ^9 + 536*ζ^10 - 62*ζ^11 - 722*ζ^12 - 254*ζ^13 + 53*ζ^14 + 19*ζ^15 + 2*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
[16 -55 192]	47	0
[2 4 32]	48	0
[26 76 224]	48	0
[6 12 32]	48	0
[14 36 96]	48	1
[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	0
[16 -50 160]	60	0
[16 -178 1984]	60	0
[10 -30 96]	60	-1
[10 -50 256]	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	-2
[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 -144 2080]	64	0
[36 -469 6112]	71	0
[18 -181 1824]	71	3
[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
[190 441 1024]	79	0
[10 -121 1472]	79	0
[44 -397 3584]	87	0
[22 -45 96]	87	1
[22 -243 2688]	87	-1
[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	2
[162 374 864]	92	0
[6 26 128]	92	-2
[26 -86 288]	92	-1
[12 -26 64]	92	-8
[12 -38 128]	92	10
[12 -122 1248]	92	2
[12 -134 1504]	92	-2
[48 -337 2368]	95	0
[24 -337 4736]	95	-5
[4 17 96]	95	2
[16 -81 416]	95	-2
[60 145 352]	95	5
[10 15 32]	95	0
[52 -261 1312]	103	0
[4 5 32]	103	0
[56 -169 512]	111	0
[120 297 736]	111	17
[6 9 32]	111	0
[14 -183 2400]	111	19
[10 23 64]	111	-17
[2 4 64]	112	0
[58 172 512]	112	0
[4 4 32]	112	0
[28 -84 256]	112	-6
[28 -140 704]	112	-10
[28 -196 1376]	112	-4
[14 -28 64]	112	4
[8 12 32]	112	0
[16 -84 448]	112	-6
[16 -212 2816]	112	-6
[2 3 64]	119	0
[30 -451 6784]	119	0
[20 -61 192]	119	-4
[90 221 544]	119	1
[12 -61 320]	119	-4
[12 -125 1312]	119	-1
[2 2 64]	124	0
[62 -434 3040]	124	0
[4 2 32]	124	0
[32 -226 1600]	124	-24
[16 -226 3200]	124	-24
[8 34 160]	124	18
[28 -94 320]	124	6
[10 14 32]	124	0
[14 -158 1792]	124	18
[2 1 64]	127	0
[4 1 32]	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	20
[18 -256 3648]	128	0
[12 16 32]	128	0
[12 -64 352]	128	2
[68 -885 11520]	135	0
[34 -171 864]	135	44
[24 -267 2976]	135	-4
[8 11 32]	135	0
[72 -793 8736]	143	0
[42 121 352]	143	-44
[6 7 32]	143	0
[6 25 128]	143	0
[18 -199 2208]	143	0
[12 -25 64]	143	44
[76 -685 6176]	151	0
[38 -115 352]	151	7
[8 19 64]	151	-7
[10 13 32]	151	0
[78 -234 704]	156	0
[78 -390 1952]	156	0
[40 -202 1024]	156	-112
[40 -522 6816]	156	72
[6 6 32]	156	0
[8 10 32]	156	0
[20 -42 96]	156	-22
[10 22 64]	156	72
[10 42 192]	156	-66
[30 -102 352]	156	-4
[16 -182 2080]	156	-66
[80 -561 3936]	159	0
[46 -143 448]	159	-69
[28 -143 736]	159	-69
[12 15 32]	159	0
[84 -421 2112]	167	0
[42 -379 3424]	167	-12
[28 -421 6336]	167	0
[36 101 288]	167	24
[22 -69 224]	167	-6
[14 -155 1728]	167	6
[14 -197 2784]	167	12
[16 -37 96]	167	-24
[88 -265 800]	175	0
[4 9 64]	175	-183
[8 9 32]	175	0
[2 4 96]	176	0
[90 268 800]	176	0
[30 -92 288]	176	42
[30 -332 3680]	176	-7
[30 -452 6816]	176	0
[22 -44 96]	176	-37
[10 12 32]	176	0
[18 -92 480]	176	42
[2 3 96]	183	0
[46 -323 2272]	183	113
[6 3 32]	183	0
[38 -125 416]	183	-47
[16 -227 3232]	183	113
[2 2 96]	188	0
[94 -658 4608]	188	0
[48 -146 448]	188	56
[48 -530 5856]	188	-136
[6 2 32]	188	0
[6 14 64]	188	-136
[142 350 864]	188	39
[24 -338 4768]	188	-76
[8 18 64]	188	-56
[12 14 32]	188	0
[14 34 96]	188	-39
[102 242 576]	188	58
[2 1 96]	191	0
[6 1 32]	191	0
[32 -417 5440]	191	73
[8 33 160]	191	-71
[20 -223 2496]	191	-71
[18 -95 512]	191	-73
[18 -257 3680]	191	73
[2 0 96]	192	0
[2 8 128]	192	0
[2 16 224]	192	0
[4 8 64]	192	200
[52 152 448]	192	168
[6 0 32]	192	0
[6 24 128]	192	68
[8 8 32]	192	0
[24 -72 224]	192	92
[24 -120 608]	192	-68
[26 -368 5216]	192	0
[12 24 64]	192	168
[14 16 32]	192	0
[14 -72 384]	192	92
[100 -1301 16928]	199	0
[50 -651 8480]	199	-177
[26 -341 4480]	199	117
[10 11 32]	199	0
[10 21 64]	199	-177
[16 -213 2848]	199	117
[104 -1145 12608]	207	0
[478 1113 2592]	207	133
[26 -391 5888]	207	0
[18 -39 96]	207	-32
[108 -973 8768]	215	0
[54 -595 6560]	215	104
[36 -397 4384]	215	43
[6 13 64]	215	104
[12 13 32]	215	0
[110 -330 992]	220	0
[110 -550 2752]	220	0
[4 6 64]	220	-16
[56 -506 4576]	220	8
[28 -422 6368]	220	0
[10 10 32]	220	0
[170 390 896]	220	8
[112 -785 5504]	223	0
[4 17 128]	223	299
[8 17 64]	223	299
[14 15 32]	223	0
[116 -581 2912]	231	0
[4 5 64]	231	226
[40 -123 384]	231	-38
[30 -453 6848]	231	0
[24 -123 640]	231	-38
[40 -133 448]	231	77
[120 -361 1088]	239	0
[248 617 1536]	239	423
[48 -151 480]	239	222
[30 -151 768]	239	222
[24 -361 5440]	239	0
[10 41 192]	239	298
[20 -41 96]	239	32
[198 457 1056]	239	423
[22 -247 2784]	239	298
[2 4 128]	240	0
[122 364 1088]	240	0
[4 4 64]	240	-430
[60 -180 544]	240	-200
[60 -300 1504]	240	-124
[60 -420 2944]	240	-354
[6 12 64]	240	124
[46 132 384]	240	-132
[8 4 32]	240	0
[32 -100 320]	240	-226
[32 -356 3968]	240	-130
[10 20 64]	240	200
[12 12 32]	240	0
[20 -100 512]	240	-226
[16 -36 96]	240	132
[16 -228 3264]	240	-354
[2 3 128]	247	0
[62 -931 13984]	247	-541
[8 3 32]	247	0
[2 2 128]	252	0
[126 -882 6176]	252	0
[4 2 64]	252	-512
[42 -126 384]	252	270
[42 -210 1056]	252	9
[8 2 32]	252	0
[24 -126 672]	252	270
[14 14 32]	252	0
[18 -258 3712]	252	-336
[2 1 128]	255	0
[4 1 64]	255	322
[44 -575 7520]	255	247
[8 1 32]	255	0
[14 33 96]	255	247
[2 0 128]	256	0
[2 8 160]	256	0
[2 16 256]	256	0
[4 0 64]	256	0
[4 16 128]	256	-688
[8 0 32]	256	0
[8 16 64]	256	-688
[8 32 160]	256	-72
[26 -288 3200]	256	-72
[26 -392 5920]	256	0
[16 16 32]	256	0
[20 -288 4160]	256	0
[132 -1717 22336]	263	0
[66 -331 1664]	263	378
[408 949 2208]	263	378
[22 -309 4352]	263	-350
[22 -331 4992]	263	0
[136 -1497 16480]	271	0
[74 217 640]	271	-70
[34 -103 320]	271	-92
[28 -423 6400]	271	0
[20 -103 544]	271	-92
[212 487 1120]	271	70
[140 -1261 11360]	279	0
[70 -211 640]	279	-158
[268 621 1440]	279	-158
[24 -339 4800]	279	289
[20 -301 4544]	279	0
[20 -109 608]	279	-222
[142 -426 1280]	284	0
[142 -710 3552]	284	0
[72 -362 1824]	284	-344
[72 -938 12224]	284	640
[450 1046 2432]	284	-344
[6 26 160]	284	368
[58 -182 576]	284	-84
[36 -182 928]	284	-84
[36 -470 6144]	284	-556
[30 -154 800]	284	-368
[30 -454 6880]	284	0
[24 -362 5472]	284	0
[240 554 1280]	284	-640
[18 -38 96]	284	-118
[20 -106 576]	284	556
[144 -1009 7072]	287	0
[78 -239 736]	287	-762
[48 -241 1216]	287	43
[338 785 1824]	287	-762
[26 -369 5248]	287	598
[18 -271 4096]	287	0
[148 -741 3712]	295	0
[74 -667 6016]	295	-728
[38 -421 4672]	295	295
[10 5 32]	295	0
[226 517 1184]	295	-728
[152 -457 1376]	303	0
[4 9 96]	303	-87
[6 9 64]	303	-87
[26 -393 5952]	303	0
[2 4 160]	304	0
[154 460 1376]	304	0
[10 4 32]	304	0
[50 -164 544]	304	-238
[22 -332 5024]	304	0
[2 3 160]	311	0
[52 -157 480]	311	-392
[218 541 1344]	311	-181
[8 35 192]	311	-685
[10 3 32]	311	0
[26 -131 672]	311	685
[12 29 96]	311	392
[16 35 96]	311	181
[240 547 1248]	311	-802
[2 2 160]	316	0
[158 -1106 7744]	316	0
[80 -242 736]	316	848
[80 -882 9728]	316	-88
[380 882 2048]	316	88
[10 2 32]	316	0
[310 718 1664]	316	848
[20 -302 4576]	316	0
[2 1 160]	319	0
[50 -159 512]	319	-415
[10 1 32]	319	0
[32 -353 3904]	319	415
[20 -289 4192]	319	259
[2 0 160]	320	0
[2 8 192]	320	0
[4 8 96]	320	-60
[84 248 736]	320	140
[6 8 64]	320	-60
[54 -272 1376]	320	552
[54 -704 9184]	320	-832
[10 0 32]	320	0
[10 40 192]	320	140
[28 -312 3488]	320	140
[28 -424 6432]	320	0
[14 32 96]	320	-832
[254 584 1344]	320	-140
[20 -40 96]	320	-164
[18 -272 4128]	320	0
[164 -2133 27744]	327	0
[82 -1067 13888]	327	-318
[56 -619 6848]	327	728
[42 -213 1088]	327	-728
[282 651 1504]	327	318
[24 -363 5504]	327	0
[168 -1849 20352]	335	0
[766 1785 4160]	335	-153
[6 7 64]	335	153
[6 25 160]	335	-1451
[36 -185 960]	335	1451
[30 -455 6912]	335	0
[172 -1549 13952]	343	0
[86 -947 10432]	343	-631
[422 979 2272]	343	631
[22 -333 5056]	343	0
[174 -522 1568]	348	0
[174 -870 4352]	348	0
[4 6 96]	348	-328
[88 -794 7168]	348	912
[6 6 64]	348	-328
[44 -134 416]	348	-128
[44 -486 5376]	348	-1024
[12 6 32]	348	0
[26 -134 704]	348	-128
[26 -394 5984]	348	0
[22 -310 4384]	348	694
[268 614 1408]	348	912
[4 17 160]	351	769
[60 -303 1536]	351	-1928
[352 815 1888]	351	-769
[20 -303 4608]	351	0
[180 -901 4512]	359	0
[4 5 96]	359	330
[60 -901 13536]	359	-330
[12 5 32]	359	0
[18 37 96]	359	-645
[184 -553 1664]	367	0
[376 937 2336]	367	-290
[46 -599 7808]	367	1076
[28 -425 6464]	367	0
[296 681 1568]	367	-290
[26 -137 736]	367	-1076
[2 4 192]	368	0
[186 556 1664]	368	0
[4 4 96]	368	-180
[92 -276 832]	368	562
[92 -460 2304]	368	-66
[62 -188 576]	368	-228
[62 -684 7552]	368	-1180
[62 -932 14016]	368	180
[8 12 64]	368	66
[48 -244 1248]	368	1180
[12 4 32]	368	0
[12 28 96]	368	228
[324 748 1728]	368	562
[24 -340 4832]	368	76
[24 -364 5536]	368	0
[18 44 128]	368	136
[282 644 1472]	368	808
[2 3 192]	375	0
[94 -1411 21184]	375	-76
[6 3 64]	375	76
[40 -445 4960]	375	-245
[12 3 32]	375	0
[2 2 192]	380	0
[4 2 96]	380	216
[6 2 64]	380	216
[6 14 96]	380	-2235
[270 670 1664]	380	-208
[8 34 192]	380	610
[60 -190 608]	380	-870
[38 -190 960]	380	-870
[12 2 32]	380	0
[32 -162 832]	380	-610
[16 34 96]	380	208
[26 -370 5280]	380	-2408
[22 -334 5088]	380	0
[20 30 64]	380	-72
[2 1 192]	383	0
[4 1 96]	383	-429
[6 1 64]	383	-429
[64 -833 10848]	383	1808
[12 1 32]	383	0
[14 31 96]	383	1808
[2 0 192]	384	0
[2 8 224]	384	0
[4 0 96]	384	0
[4 16 160]	384	1200
[6 0 64]	384	0
[6 24 160]	384	2364
[10 24 96]	384	2364
[394 912 2112]	384	-1200
[12 0 32]	384	0
[30 -456 6944]	384	0
[28 -400 5728]	384	0
[20 -304 4640]	384	0
[22 32 64]	384	0
[196 -2549 33152]	391	0
[98 -491 2464]	391	-1265
[8 11 64]	391	1265
[26 -395 6016]	391	0
[200 -2201 24224]	399	0
[106 313 928]	399	928
[50 -551 6080]	399	1972
[52 -167 544]	399	1699
[34 -377 4192]	399	-1699
[14 7 32]	399	0
[310 711 1632]	399	-928
[20 -281 3968]	399	-829
[102 -307 928]	407	-143
[68 -749 8256]	407	-329
[6 13 96]	407	294
[8 19 96]	407	-329
[366 845 1952]	407	-143
[24 -365 5568]	407	0
[206 -618 1856]	412	0
[206 -1030 5152]	412	0
[104 -522 2624]	412	1360
[104 -1354 17632]	412	-1624
[8 10 64]	412	-1360
[14 6 32]	412	0
[338 778 1792]	412	1624
[28 -426 6496]	412	0
[110 -335 1024]	415	3192
[10 15 64]	415	-3192
[22 -335 5120]	415	0
[212 -1061 5312]	423	0
[72 -219 672]	423	1909
[54 -165 512]	423	-258
[12 27 96]	423	-1909
[14 5 32]	423	0
[32 -165 864]	423	-258
[18 27 64]	423	-958
[216 -649 1952]	431	0
[4 9 128]	431	715
[80 -247 768]	431	1336
[8 9 64]	431	715
[10 23 96]	431	-1336
[30 -457 6976]	431	0
[22 -311 4416]	431	-589
[2 4 224]	432	0
[6 12 96]	432	2051
[14 4 32]	432	0
[18 36 96]	432	2150
[26 -396 6048]	432	0
[2 3 224]	439	0
[70 -221 704]	439	4264
[44 -221 1120]	439	4264
[14 3 32]	439	0
[20 29 64]	439	-2465
[2 2 224]	444	0
[112 -338 1024]	444	-3656
[112 -1234 13600]	444	1392
[74 -222 672]	444	2819
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