A Borcherds Product in S₇(K(16))⁻

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c vector |

θ vector |

ψ expansion |

Borch(ψ) coefficients

• The singular part of ψ truncated up to q^(16/4) is
  (14 + ζ^(-4) + ζ^(-3) + 8/ζ^2 + 7/ζ + 7*ζ + 8*ζ^2 + ζ^3 + ζ^4)
  +q(-ζ^(-8) - ζ^8)
  +q^3(4/ζ^14 + 4*ζ^14)
  +q^4(8/ζ^16 + 8*ζ^16)
• From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block

  	φ	= TB(7;1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,4)
  		= 17283141
  		= η(τ)14(θ(τ,z)/η(τ))7(θ(τ,2z)/η(τ))8(θ(τ,3z)/η(τ))(θ(τ,4z)/η(τ))

  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 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)	17
  (4,2)	9
  (4,14)	5
  (9,3)	1
  (16,4)	1

• 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

(14 + ζ^(-4) + ζ^(-3) + 8/ζ^2 + 7/ζ + 7*ζ + 8*ζ^2 + ζ^3 + ζ^4)
+q(74 - ζ^(-8) + 2/ζ^7 - 8/ζ^6 - 21/ζ^5 - 36/ζ^4 - 29/ζ^3 + 8/ζ^2 + 48/ζ + 48*ζ + 8*ζ^2 - 29*ζ^3 - 36*ζ^4 - 21*ζ^5 - 8*ζ^6 + 2*ζ^7 - ζ^8)
+q^2(460 - 3/ζ^11 - 4/ζ^10 + 21/ζ^9 + 42/ζ^8 + 55/ζ^7 - 28/ζ^6 - 160/ζ^5 - 272/ζ^4 - 223/ζ^3 + 32/ζ^2 + 310/ζ + 310*ζ + 32*ζ^2 - 223*ζ^3 - 272*ζ^4 - 160*ζ^5 - 28*ζ^6 + 55*ζ^7 + 42*ζ^8 + 21*ζ^9 - 4*ζ^10 - 3*ζ^11)
+q^3(2152 + 4/ζ^14 - 7/ζ^13 - 28/ζ^12 - 43/ζ^11 + 4/ζ^10 + 182/ζ^9 + 380/ζ^8 + 393/ζ^7 - 68/ζ^6 - 848/ζ^5 - 1428/ζ^4 - 1162/ζ^3 + 60/ζ^2 + 1485/ζ + 1485*ζ + 60*ζ^2 - 1162*ζ^3 - 1428*ζ^4 - 848*ζ^5 - 68*ζ^6 + 393*ζ^7 + 380*ζ^8 + 182*ζ^9 + 4*ζ^10 - 43*ζ^11 - 28*ζ^12 - 7*ζ^13 + 4*ζ^14)
+q^4(8608 + 8/ζ^16 + 10/ζ^15 + 4/ζ^14 - 81/ζ^13 - 241/ζ^12 - 312/ζ^11 + 20/ζ^10 + 977/ζ^9 + 1960/ζ^8 + 1927/ζ^7 - 188/ζ^6 - 3611/ζ^5 - 6031/ζ^4 - 4862/ζ^3 + 164/ζ^2 + 5952/ζ + 5952*ζ + 164*ζ^2 - 4862*ζ^3 - 6031*ζ^4 - 3611*ζ^5 - 188*ζ^6 + 1927*ζ^7 + 1960*ζ^8 + 977*ζ^9 + 20*ζ^10 - 312*ζ^11 - 241*ζ^12 - 81*ζ^13 + 4*ζ^14 + 10*ζ^15 + 8*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	-1
[14 -42 128]	28	0
[14 -70 352]	28	0
[8 -42 224]	28	0
[8 -106 1408]	28	-1
[16 -113 800]	31	0
[8 -113 1600]	31	0
[14 -47 160]	31	1
[20 -101 512]	39	0
[10 -91 832]	39	0
[10 -101 1024]	39	0
[8 -27 96]	39	-1
[24 -73 224]	47	0
[12 -73 448]	47	0
[4 9 32]	47	0
[8 -73 672]	47	0
[16 -55 192]	47	2
[2 4 32]	48	0
[26 76 224]	48	0
[6 12 32]	48	0
[14 36 96]	48	4
[2 3 32]	55	0
[14 -29 64]	55	0
[14 -99 704]	55	0
[8 -35 160]	55	7
[2 2 32]	60	0
[30 -210 1472]	60	0
[16 -50 160]	60	6
[16 -178 1984]	60	7
[10 -30 96]	60	-2
[10 -50 256]	60	6
[8 -50 320]	60	5
[8 -114 1632]	60	0
[2 1 32]	63	0
[16 -225 3168]	63	0
[4 33 288]	63	0
[12 -159 2112]	63	-7
[8 -97 1184]	63	6
[2 0 32]	64	0
[2 8 64]	64	0
[2 16 160]	64	0
[4 8 32]	64	0
[20 56 160]	64	8
[10 -24 64]	64	-8
[10 -64 416]	64	0
[10 -144 2080]	64	0
[36 -469 6112]	71	0
[18 -181 1824]	71	0
[18 -235 3072]	71	-1
[120 277 640]	71	0
[10 -43 192]	71	5
[10 -53 288]	71	1
[40 -441 4864]	79	0
[20 -121 736]	79	0
[190 441 1024]	79	0
[10 -121 1472]	79	13
[44 -397 3584]	87	0
[22 -45 96]	87	0
[22 -243 2688]	87	-13
[12 -51 224]	87	-11
[134 307 704]	87	0
[46 -138 416]	92	0
[46 -230 1152]	92	0
[4 6 32]	92	0
[24 -218 1984]	92	27
[26 70 192]	92	-24
[162 374 864]	92	0
[6 26 128]	92	-10
[26 -86 288]	92	-14
[12 -26 64]	92	27
[12 -38 128]	92	-17
[12 -122 1248]	92	-24
[12 -134 1504]	92	-10
[48 -337 2368]	95	0
[24 -337 4736]	95	0
[4 17 96]	95	14
[16 -81 416]	95	-14
[12 -145 1760]	95	-4
[60 145 352]	95	-19
[10 15 32]	95	0
[52 -261 1312]	103	0
[4 5 32]	103	0
[26 -261 2624]	103	0
[14 -59 256]	103	3
[56 -169 512]	111	0
[28 -169 1024]	111	0
[120 297 736]	111	22
[6 9 32]	111	0
[14 -169 2048]	111	-13
[14 -183 2400]	111	-4
[10 23 64]	111	-22
[2 4 64]	112	0
[58 172 512]	112	0
[4 4 32]	112	0
[28 -84 256]	112	-28
[28 -140 704]	112	20
[28 -196 1376]	112	48
[14 -28 64]	112	-48
[14 -84 512]	112	-16
[8 12 32]	112	0
[16 -84 448]	112	-28
[16 -148 1376]	112	-16
[16 -212 2816]	112	-20
[2 3 64]	119	0
[30 -61 128]	119	0
[30 -451 6784]	119	0
[20 -61 192]	119	9
[90 221 544]	119	-13
[16 -67 288]	119	-31
[12 -61 320]	119	9
[24 61 160]	119	-21
[12 -125 1312]	119	13
[2 2 64]	124	0
[62 -434 3040]	124	0
[4 2 32]	124	0
[32 -226 1600]	124	-42
[16 -98 608]	124	-8
[16 -226 3200]	124	-42
[8 34 160]	124	18
[28 -94 320]	124	24
[10 14 32]	124	0
[14 -46 160]	124	-26
[14 -130 1216]	124	-8
[14 -158 1792]	124	18
[2 1 64]	127	0
[4 1 32]	127	0
[32 -449 6304]	127	0
[16 -193 2336]	127	-39
[2 0 64]	128	0
[2 8 96]	128	0
[2 16 192]	128	0
[4 0 32]	128	0
[4 16 96]	128	-16
[6 8 32]	128	0
[22 -112 576]	128	16
[22 -288 3776]	128	32
[18 -40 96]	128	-48
[18 -112 704]	128	0
[18 -256 3648]	128	0
[12 16 32]	128	0
[12 -64 352]	128	-32
[68 -885 11520]	135	0
[34 -171 864]	135	-99
[34 -341 3424]	135	0
[24 -267 2976]	135	-51
[8 11 32]	135	0
[12 -75 480]	135	6
[72 -793 8736]	143	0
[36 -217 1312]	143	0
[42 121 352]	143	86
[6 7 32]	143	0
[6 25 128]	143	53
[18 -199 2208]	143	53
[12 -25 64]	143	-86
[14 -89 576]	143	-42
[76 -685 6176]	151	0
[38 -77 160]	151	0
[38 -115 352]	151	67
[8 19 64]	151	-67
[10 13 32]	151	0
[78 -234 704]	156	0
[78 -390 1952]	156	0
[40 -202 1024]	156	57
[40 -522 6816]	156	-38
[6 6 32]	156	0
[8 10 32]	156	0
[20 -42 96]	156	8
[20 -202 2048]	156	27
[10 22 64]	156	-38
[82 198 480]	156	38
[10 42 192]	156	-12
[30 -102 352]	156	28
[16 -54 192]	156	-7
[16 -182 2080]	156	-12
[80 -561 3936]	159	0
[40 -561 7872]	159	0
[46 -143 448]	159	10
[28 -143 736]	159	10
[30 81 224]	159	-62
[16 -49 160]	159	-95
[12 15 32]	159	0
[14 -143 1472]	159	-62
[84 -421 2112]	167	0
[42 -379 3424]	167	113
[42 -421 4224]	167	0
[28 -421 6336]	167	0
[36 101 288]	167	77
[22 -69 224]	167	46
[14 -155 1728]	167	-46
[14 -197 2784]	167	-113
[16 -37 96]	167	-77
[88 -265 800]	175	0
[4 9 64]	175	-225
[8 9 32]	175	0
[20 -265 3520]	175	15
[2 4 96]	176	0
[90 268 800]	176	0
[30 -92 288]	176	24
[30 -332 3680]	176	84
[30 -452 6816]	176	0
[22 -44 96]	176	108
[22 -132 800]	176	48
[10 12 32]	176	0
[18 -92 480]	176	24
[18 -236 3104]	176	-72
[2 3 96]	183	0
[46 -93 192]	183	0
[46 -323 2272]	183	-15
[6 3 32]	183	0
[38 -125 416]	183	61
[16 -227 3232]	183	-15
[2 2 96]	188	0
[94 -658 4608]	188	0
[48 -146 448]	188	-4
[48 -530 5856]	188	118
[6 2 32]	188	0
[6 14 64]	188	118
[142 350 864]	188	170
[24 -146 896]	188	42
[24 -338 4768]	188	144
[8 18 64]	188	4
[112 274 672]	188	-46
[12 14 32]	188	0
[12 50 224]	188	-58
[32 -110 384]	188	-56
[14 34 96]	188	-170
[102 242 576]	188	-46
[14 62 288]	188	-130
[42 -146 512]	188	58
[2 1 96]	191	0
[48 -673 9440]	191	0
[6 1 32]	191	0
[32 -417 5440]	191	1
[8 33 160]	191	59
[20 -223 2496]	191	59
[36 95 256]	191	142
[16 -161 1632]	191	142
[18 -95 512]	191	-1
[18 -257 3680]	191	204
[2 0 96]	192	0
[2 8 128]	192	0
[2 16 224]	192	0
[4 8 64]	192	336
[52 152 448]	192	16
[6 0 32]	192	0
[6 24 128]	192	-80
[8 8 32]	192	0
[24 -72 224]	192	16
[24 -120 608]	192	80
[26 -56 128]	192	192
[26 -160 992]	192	0
[26 -368 5216]	192	0
[12 24 64]	192	16
[28 72 192]	192	208
[14 16 32]	192	0
[14 -72 384]	192	16
[100 -1301 16928]	199	0
[50 -651 8480]	199	165
[26 -341 4480]	199	38
[10 11 32]	199	0
[10 21 64]	199	165
[16 -213 2848]	199	38
[104 -1145 12608]	207	0
[478 1113 2592]	207	-147
[26 -391 5888]	207	0
[18 -39 96]	207	279
[18 -57 192]	207	-207
[108 -973 8768]	215	0
[54 -595 6560]	215	-229
[36 -397 4384]	215	88
[6 13 64]	215	-229
[24 -77 256]	215	183
[12 13 32]	215	0
[18 -109 672]	215	139
[20 -205 2112]	215	25
[16 -179 2016]	215	-183
[110 -330 992]	220	0
[110 -550 2752]	220	0
[4 6 64]	220	-581
[56 -506 4576]	220	110
[28 -58 128]	220	-319
[28 -282 2848]	220	104
[28 -422 6368]	220	0
[10 10 32]	220	0
[34 86 224]	220	-158
[170 390 896]	220	110
[14 58 256]	220	200
[34 -118 416]	220	56
[16 -70 320]	220	271
[16 -198 2464]	220	200
[112 -785 5504]	223	0
[4 17 128]	223	203
[8 17 64]	223	203
[14 15 32]	223	0
[116 -581 2912]	231	0
[4 5 64]	231	167
[40 -123 384]	231	-27
[30 -453 6848]	231	0
[24 -123 640]	231	-27
[40 -133 448]	231	87
[20 -123 768]	231	57
[104 251 608]	231	98
[120 -361 1088]	239	0
[248 617 1536]	239	247
[48 -151 480]	239	119
[30 -151 768]	239	119
[24 -361 5440]	239	0
[10 41 192]	239	126
[20 -41 96]	239	-245
[198 457 1056]	239	247
[22 -247 2784]	239	126
[2 4 128]	240	0
[122 364 1088]	240	0
[4 4 64]	240	412
[60 -180 544]	240	48
[60 -300 1504]	240	32
[60 -420 2944]	240	396
[6 12 64]	240	-32
[46 132 384]	240	224
[30 -60 128]	240	176
[30 -180 1088]	240	16
[8 4 32]	240	0
[32 -100 320]	240	172
[32 -356 3968]	240	324
[10 20 64]	240	-48
[34 92 256]	240	48
[12 12 32]	240	0
[20 -60 192]	240	188
[20 -100 512]	240	172
[16 -36 96]	240	-224
[16 -164 1696]	240	48
[16 -228 3264]	240	396
[2 3 128]	247	0
[62 -931 13984]	247	265
[8 3 32]	247	0
[2 2 128]	252	0
[126 -882 6176]	252	0
[4 2 64]	252	-69
[42 -126 384]	252	-62
[42 -210 1056]	252	466
[8 2 32]	252	0
[32 -194 1184]	252	-240
[32 -450 6336]	252	-33
[24 -126 672]	252	-62
[24 -318 4224]	252	-138
[14 14 32]	252	0
[16 -194 2368]	252	-159
[18 -222 2752]	252	234
[18 -258 3712]	252	-462
[2 1 128]	255	0
[4 1 64]	255	51
[44 -575 7520]	255	111
[8 1 32]	255	0
[22 -223 2272]	255	-406
[22 -289 3808]	255	-8
[20 -65 224]	255	-216
[14 33 96]	255	111
[2 0 128]	256	0
[2 8 160]	256	0
[2 16 256]	256	0
[4 0 64]	256	0
[4 16 128]	256	-320
[8 0 32]	256	0
[8 16 64]	256	-320
[8 32 160]	256	-64
[34 -72 160]	256	64
[34 -480 6784]	256	0
[26 -80 256]	256	128
[26 -288 3200]	256	-64
[26 -392 5920]	256	0
[16 16 32]	256	0
[16 -48 160]	256	128
[20 -48 128]	256	64
[20 -288 4160]	256	0
[132 -1717 22336]	263	0
[66 -331 1664]	263	-568
[408 949 2208]	263	-568
[22 -309 4352]	263	169
[22 -331 4992]	263	0
[24 -53 128]	263	-25
[12 53 256]	263	-413
[18 -203 2304]	263	-413
[136 -1497 16480]	271	0
[74 217 640]	271	237
[34 -103 320]	271	-214
[28 -423 6400]	271	0
[134 327 800]	271	17
[20 -103 544]	271	-214
[212 487 1120]	271	-237
[124 295 704]	271	17
[140 -1261 11360]	279	0
[70 -211 640]	279	348
[28 -365 4768]	279	-339
[268 621 1440]	279	348
[24 -339 4800]	279	-343
[42 -141 480]	279	256
[20 -301 4544]	279	0
[98 237 576]	279	-363
[18 -237 3136]	279	-339
[20 -109 608]	279	99
[142 -426 1280]	284	0
[142 -710 3552]	284	0
[72 -362 1824]	284	408
[72 -938 12224]	284	75
[450 1046 2432]	284	408
[6 26 160]	284	252
[58 -182 576]	284	-228
[36 -74 160]	284	120
[36 -182 928]	284	-228
[36 -362 3648]	284	195
[36 -470 6144]	284	33
[30 -154 800]	284	-252
[30 -394 5184]	284	468
[30 -454 6880]	284	0
[24 -362 5472]	284	0
[240 554 1280]	284	-75
[18 -38 96]	284	-48
[18 -182 1856]	284	-360
[20 -106 576]	284	-33
[20 -266 3552]	284	468
[144 -1009 7072]	287	0
[78 -239 736]	287	7
[48 -241 1216]	287	-145
[338 785 1824]	287	7
[24 -241 2432]	287	442
[12 49 224]	287	398
[26 -369 5248]	287	69
[24 -271 3072]	287	398
[18 -271 4096]	287	0
[148 -741 3712]	295	0
[74 -667 6016]	295	79
[38 -421 4672]	295	-201
[10 5 32]	295	0
[226 517 1184]	295	79
[152 -457 1376]	303	0
[4 9 96]	303	-324
[6 9 64]	303	-324
[26 -55 128]	303	-201
[26 -393 5952]	303	0
[22 -73 256]	303	-424
[2 4 160]	304	0
[154 460 1376]	304	0
[38 -76 160]	304	-272
[10 4 32]	304	0
[10 44 224]	304	184
[50 -164 544]	304	88
[22 -68 224]	304	204
[22 -244 2720]	304	184
[22 -332 5024]	304	0
[2 3 160]	311	0
[52 -157 480]	311	477
[218 541 1344]	311	323
[8 35 192]	311	422
[10 3 32]	311	0
[26 -131 672]	311	-422
[26 -157 960]	311	-245
[28 -285 2912]	311	549
[12 29 96]	311	-477
[16 35 96]	311	-323
[20 -227 2592]	311	-421
[240 547 1248]	311	247
[2 2 160]	316	0
[158 -1106 7744]	316	0
[80 -242 736]	316	-458
[80 -882 9728]	316	608
[40 -562 7904]	316	-216
[380 882 2048]	316	-608
[10 2 32]	316	0
[310 718 1664]	316	-458
[20 -302 4576]	316	0
[22 -50 128]	316	-402
[22 -226 2336]	316	130
[2 1 160]	319	0
[50 -159 512]	319	306
[10 1 32]	319	0
[32 -353 3904]	319	-306
[38 97 256]	319	-661
[20 -289 4192]	319	563
[2 0 160]	320	0
[2 8 192]	320	0
[4 8 96]	320	288
[84 248 736]	320	384
[6 8 64]	320	288
[54 -272 1376]	320	-864
[54 -704 9184]	320	384
[42 -88 192]	320	-96
[42 -592 8352]	320	0
[10 0 32]	320	0
[10 40 192]	320	176
[28 -312 3488]	320	176
[28 -424 6432]	320	0
[14 32 96]	320	384
[126 304 736]	320	96
[254 584 1344]	320	-384
[20 -40 96]	320	192
[20 -120 736]	320	96
[18 -56 192]	320	-272
[18 -272 4128]	320	0
[164 -2133 27744]	327	0
[82 -1067 13888]	327	215
[56 -619 6848]	327	373
[42 -213 1088]	327	-373
[282 651 1504]	327	-215
[24 -363 5504]	327	0
[44 -149 512]	327	358
[168 -1849 20352]	335	0
[766 1785 4160]	335	68
[6 7 64]	335	-68
[6 25 160]	335	0
[36 -185 960]	335	0
[28 -57 128]	335	227
[30 -455 6912]	335	0
[24 -313 4096]	335	256
[14 57 256]	335	398
[54 -185 640]	335	-398
[118 281 672]	335	20
[172 -1549 13952]	343	0
[86 -947 10432]	343	-301
[422 979 2272]	343	301
[28 -371 4928]	343	-490
[22 -333 5056]	343	0
[174 -522 1568]	348	0
[174 -870 4352]	348	0
[4 6 96]	348	568
[88 -794 7168]	348	897
[6 6 64]	348	568
[44 -90 192]	348	793
[44 -134 416]	348	-232
[44 -486 5376]	348	-561
[12 6 32]	348	0
[26 -134 704]	348	-232
[26 -342 4512]	348	208
[26 -394 5984]	348	0
[22 -310 4384]	348	32
[268 614 1408]	348	897
[4 17 160]	351	21
[60 -303 1536]	351	81
[36 -111 352]	351	-438
[352 815 1888]	351	-21
[30 -303 3072]	351	267
[22 -111 576]	351	-438
[20 -303 4608]	351	0
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