A Borcherds Product in S₁₀(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
  (20 + ζ^(-5) + ζ^(-4) + ζ^(-3) + ζ^(-2) + 10/ζ + 10*ζ + ζ^2 + ζ^3 + ζ^4 + ζ^5)
  +q(ζ^(-9) + 3/ζ^8 + 3*ζ^8 + ζ^9)
  +q^3(3/ζ^14 + 3*ζ^14)
  +q^4(-6/ζ^16 - 6*ζ^16)
• From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block

  	φ	= TB(10;1,1,1,1,1,1,1,1,1,1,2,3,4,5)
  		= 11021314151
  		= η(τ)20(θ(τ,z)/η(τ))10(θ(τ,2z)/η(τ))(θ(τ,3z)/η(τ))(θ(τ,4z)/η(τ))(θ(τ,5z)/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 5 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)	14
  (4,2)	2
  (4,14)	4
  (9,3)	1
  (16,4)	1
  (17,9)	1
  (25,5)	1

• From the singular part above, we compute that the valuation Ord(ψ) is -0.390625, 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

(20 + ζ^(-5) + ζ^(-4) + ζ^(-3) + ζ^(-2) + 10/ζ + 10*ζ + ζ^2 + ζ^3 + ζ^4 + ζ^5)
+q(490 + ζ^(-9) + 3/ζ^8 + 36/ζ^7 + 37/ζ^6 - 19/ζ^5 - 120/ζ^4 - 229/ζ^3 - 165/ζ^2 + 211/ζ + 211*ζ - 165*ζ^2 - 229*ζ^3 - 120*ζ^4 - 19*ζ^5 + 37*ζ^6 + 36*ζ^7 + 3*ζ^8 + ζ^9)
+q^2(5724 + 3/ζ^11 - 41/ζ^10 - 136/ζ^9 + 50/ζ^8 + 743/ζ^7 + 1014/ζ^6 - 20/ζ^5 - 1888/ζ^4 - 3335/ζ^3 - 1997/ζ^2 + 2745/ζ + 2745*ζ - 1997*ζ^2 - 3335*ζ^3 - 1888*ζ^4 - 20*ζ^5 + 1014*ζ^6 + 743*ζ^7 + 50*ζ^8 - 136*ζ^9 - 41*ζ^10 + 3*ζ^11)
+q^3(44072 + 3/ζ^14 + 21/ζ^13 + 56/ζ^12 - 67/ζ^11 - 1022/ζ^10 - 2235/ζ^9 + 396/ζ^8 + 7917/ζ^7 + 11004/ζ^6 + 1363/ζ^5 - 16856/ζ^4 - 28917/ζ^3 - 15617/ζ^2 + 21918/ζ + 21918*ζ - 15617*ζ^2 - 28917*ζ^3 - 16856*ζ^4 + 1363*ζ^5 + 11004*ζ^6 + 7917*ζ^7 + 396*ζ^8 - 2235*ζ^9 - 1022*ζ^10 - 67*ζ^11 + 56*ζ^12 + 21*ζ^13 + 3*ζ^14)
+q^4(263804 - 6/ζ^16 - 25/ζ^15 + 169/ζ^14 + 786/ζ^13 + 1375/ζ^12 - 923/ζ^11 - 11020/ζ^10 - 20047/ζ^9 + 1992/ζ^8 + 56842/ζ^7 + 80161/ζ^6 + 14146/ζ^5 - 110687/ζ^4 - 185617/ζ^3 - 93886/ζ^2 + 134838/ζ + 134838*ζ - 93886*ζ^2 - 185617*ζ^3 - 110687*ζ^4 + 14146*ζ^5 + 80161*ζ^6 + 56842*ζ^7 + 1992*ζ^8 - 20047*ζ^9 - 11020*ζ^10 - 923*ζ^11 + 1375*ζ^12 + 786*ζ^13 + 169*ζ^14 - 25*ζ^15 - 6*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	1
[14 -99 704]	55	1
[8 -35 160]	55	1
[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 -50 320]	60	0
[8 -114 1632]	60	0
[2 1 32]	63	0
[16 -225 3168]	63	-1
[4 33 288]	63	-1
[12 -159 2112]	63	1
[8 -97 1184]	63	-1
[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	-3
[18 -235 3072]	71	1
[120 277 640]	71	0
[10 -43 192]	71	-3
[10 -53 288]	71	1
[40 -441 4864]	79	0
[20 -121 736]	79	2
[190 441 1024]	79	0
[10 -121 1472]	79	2
[44 -397 3584]	87	0
[22 -45 96]	87	9
[22 -243 2688]	87	0
[12 -51 224]	87	9
[134 307 704]	87	0
[46 -138 416]	92	0
[46 -230 1152]	92	0
[4 6 32]	92	0
[24 -218 1984]	92	0
[26 70 192]	92	2
[162 374 864]	92	0
[6 26 128]	92	0
[26 -86 288]	92	1
[12 -26 64]	92	0
[12 -38 128]	92	2
[12 -122 1248]	92	2
[12 -134 1504]	92	0
[48 -337 2368]	95	0
[24 -337 4736]	95	-6
[4 17 96]	95	-1
[16 -81 416]	95	-1
[12 -145 1760]	95	-6
[60 145 352]	95	-2
[10 15 32]	95	0
[52 -261 1312]	103	0
[4 5 32]	103	0
[26 -261 2624]	103	-21
[14 -59 256]	103	-21
[56 -169 512]	111	0
[28 -169 1024]	111	13
[120 297 736]	111	9
[6 9 32]	111	0
[14 -169 2048]	111	13
[14 -183 2400]	111	0
[10 23 64]	111	9
[2 4 64]	112	0
[58 172 512]	112	0
[4 4 32]	112	0
[28 -84 256]	112	-9
[28 -140 704]	112	1
[28 -196 1376]	112	-8
[14 -28 64]	112	-8
[14 -84 512]	112	0
[8 12 32]	112	0
[16 -84 448]	112	-9
[16 -148 1376]	112	0
[16 -212 2816]	112	-1
[2 3 64]	119	0
[30 -61 128]	119	27
[30 -451 6784]	119	0
[20 -61 192]	119	7
[90 221 544]	119	0
[16 -67 288]	119	27
[12 -61 320]	119	7
[24 61 160]	119	1
[12 -125 1312]	119	0
[2 2 64]	124	0
[62 -434 3040]	124	0
[4 2 32]	124	0
[32 -226 1600]	124	0
[16 -98 608]	124	-2
[16 -226 3200]	124	0
[8 34 160]	124	-8
[28 -94 320]	124	6
[10 14 32]	124	0
[14 -46 160]	124	-9
[14 -130 1216]	124	-2
[14 -158 1792]	124	-8
[2 1 64]	127	0
[4 1 32]	127	0
[32 -449 6304]	127	-8
[16 -193 2336]	127	-8
[2 0 64]	128	0
[2 8 96]	128	0
[2 16 192]	128	0
[4 0 32]	128	0
[4 16 96]	128	9
[6 8 32]	128	0
[22 -112 576]	128	9
[22 -288 3776]	128	-9
[18 -40 96]	128	-8
[18 -112 704]	128	2
[18 -256 3648]	128	18
[12 16 32]	128	0
[12 -64 352]	128	-9
[68 -885 11520]	135	0
[34 -171 864]	135	-10
[34 -341 3424]	135	-35
[24 -267 2976]	135	-9
[8 11 32]	135	0
[12 -75 480]	135	-1
[72 -793 8736]	143	0
[36 -217 1312]	143	15
[42 121 352]	143	8
[6 7 32]	143	0
[6 25 128]	143	2
[18 -199 2208]	143	2
[12 -25 64]	143	8
[14 -89 576]	143	1
[76 -685 6176]	151	0
[38 -77 160]	151	7
[38 -115 352]	151	36
[8 19 64]	151	36
[10 13 32]	151	0
[78 -234 704]	156	0
[78 -390 1952]	156	0
[40 -202 1024]	156	44
[40 -522 6816]	156	-44
[6 6 32]	156	0
[8 10 32]	156	0
[20 -42 96]	156	10
[20 -202 2048]	156	-16
[10 22 64]	156	-44
[82 198 480]	156	-29
[10 42 192]	156	-8
[30 -102 352]	156	26
[16 -54 192]	156	2
[16 -182 2080]	156	-8
[80 -561 3936]	159	0
[40 -561 7872]	159	13
[46 -143 448]	159	-35
[28 -143 736]	159	-35
[30 81 224]	159	38
[16 -49 160]	159	20
[12 15 32]	159	0
[14 -143 1472]	159	38
[84 -421 2112]	167	0
[42 -379 3424]	167	27
[42 -421 4224]	167	72
[28 -421 6336]	167	0
[36 101 288]	167	5
[22 -69 224]	167	23
[14 -155 1728]	167	23
[14 -197 2784]	167	27
[16 -37 96]	167	5
[88 -265 800]	175	0
[4 9 64]	175	-111
[8 9 32]	175	0
[20 -265 3520]	175	98
[2 4 96]	176	0
[90 268 800]	176	0
[30 -92 288]	176	-50
[30 -332 3680]	176	34
[30 -452 6816]	176	0
[22 -44 96]	176	0
[22 -132 800]	176	64
[10 12 32]	176	0
[18 -92 480]	176	-50
[18 -236 3104]	176	-78
[2 3 96]	183	0
[46 -93 192]	183	-139
[46 -323 2272]	183	18
[6 3 32]	183	0
[38 -125 416]	183	-44
[16 -227 3232]	183	18
[2 2 96]	188	0
[94 -658 4608]	188	0
[48 -146 448]	188	-76
[48 -530 5856]	188	76
[6 2 32]	188	0
[6 14 64]	188	76
[142 350 864]	188	-71
[24 -146 896]	188	16
[24 -338 4768]	188	-12
[8 18 64]	188	-76
[112 274 672]	188	-86
[12 14 32]	188	0
[12 50 224]	188	72
[32 -110 384]	188	-68
[14 34 96]	188	-71
[102 242 576]	188	-86
[14 62 288]	188	65
[42 -146 512]	188	72
[2 1 96]	191	0
[48 -673 9440]	191	42
[6 1 32]	191	0
[32 -417 5440]	191	124
[8 33 160]	191	18
[20 -223 2496]	191	18
[36 95 256]	191	-49
[16 -161 1632]	191	-49
[18 -95 512]	191	124
[18 -257 3680]	191	-63
[2 0 96]	192	0
[2 8 128]	192	0
[2 16 224]	192	0
[4 8 64]	192	174
[52 152 448]	192	-54
[6 0 32]	192	0
[6 24 128]	192	16
[8 8 32]	192	0
[24 -72 224]	192	-100
[24 -120 608]	192	16
[26 -56 128]	192	72
[26 -160 992]	192	-112
[26 -368 5216]	192	16
[12 24 64]	192	-54
[28 72 192]	192	18
[14 16 32]	192	0
[14 -72 384]	192	-100
[100 -1301 16928]	199	0
[50 -651 8480]	199	138
[26 -341 4480]	199	134
[10 11 32]	199	0
[10 21 64]	199	138
[16 -213 2848]	199	134
[104 -1145 12608]	207	0
[478 1113 2592]	207	-167
[26 -391 5888]	207	0
[18 -39 96]	207	70
[18 -57 192]	207	-12
[108 -973 8768]	215	0
[54 -595 6560]	215	-107
[36 -397 4384]	215	-99
[6 13 64]	215	-107
[24 -77 256]	215	15
[12 13 32]	215	0
[18 -109 672]	215	-47
[20 -205 2112]	215	-13
[16 -179 2016]	215	15
[110 -330 992]	220	0
[110 -550 2752]	220	0
[4 6 64]	220	76
[56 -506 4576]	220	-76
[28 -58 128]	220	176
[28 -282 2848]	220	-206
[28 -422 6368]	220	0
[10 10 32]	220	0
[34 86 224]	220	61
[170 390 896]	220	-76
[14 58 256]	220	-128
[34 -118 416]	220	8
[16 -70 320]	220	98
[16 -198 2464]	220	-128
[112 -785 5504]	223	0
[4 17 128]	223	71
[8 17 64]	223	71
[14 15 32]	223	0
[116 -581 2912]	231	0
[4 5 64]	231	34
[40 -123 384]	231	223
[30 -453 6848]	231	0
[24 -123 640]	231	223
[40 -133 448]	231	-159
[20 -123 768]	231	34
[104 251 608]	231	117
[120 -361 1088]	239	0
[248 617 1536]	239	162
[48 -151 480]	239	-213
[30 -151 768]	239	-213
[24 -361 5440]	239	0
[10 41 192]	239	98
[20 -41 96]	239	-29
[198 457 1056]	239	162
[22 -247 2784]	239	98
[2 4 128]	240	0
[122 364 1088]	240	0
[4 4 64]	240	-97
[60 -180 544]	240	-260
[60 -300 1504]	240	140
[60 -420 2944]	240	-23
[6 12 64]	240	140
[46 132 384]	240	-92
[30 -60 128]	240	-256
[30 -180 1088]	240	136
[8 4 32]	240	0
[32 -100 320]	240	-163
[32 -356 3968]	240	269
[10 20 64]	240	-260
[34 92 256]	240	-36
[12 12 32]	240	0
[20 -60 192]	240	-213
[20 -100 512]	240	-163
[16 -36 96]	240	-92
[16 -164 1696]	240	-36
[16 -228 3264]	240	-23
[2 3 128]	247	0
[62 -931 13984]	247	112
[8 3 32]	247	0
[2 2 128]	252	0
[126 -882 6176]	252	0
[4 2 64]	252	-120
[42 -126 384]	252	-648
[42 -210 1056]	252	369
[8 2 32]	252	0
[32 -194 1184]	252	196
[32 -450 6336]	252	-160
[24 -126 672]	252	-648
[24 -318 4224]	252	-470
[14 14 32]	252	0
[16 -194 2368]	252	28
[18 -222 2752]	252	8
[18 -258 3712]	252	120
[2 1 128]	255	0
[4 1 64]	255	142
[44 -575 7520]	255	421
[8 1 32]	255	0
[22 -223 2272]	255	179
[22 -289 3808]	255	550
[20 -65 224]	255	52
[14 33 96]	255	421
[2 0 128]	256	0
[2 8 160]	256	0
[2 16 256]	256	0
[4 0 64]	256	-156
[4 16 128]	256	8
[8 0 32]	256	0
[8 16 64]	256	8
[8 32 160]	256	-12
[34 -72 160]	256	-32
[34 -480 6784]	256	368
[26 -80 256]	256	148
[26 -288 3200]	256	-12
[26 -392 5920]	256	0
[16 16 32]	256	0
[16 -48 160]	256	148
[20 -48 128]	256	-156
[20 -288 4160]	256	172
[132 -1717 22336]	263	0
[66 -331 1664]	263	-222
[408 949 2208]	263	-222
[22 -309 4352]	263	-306
[22 -331 4992]	263	0
[24 -53 128]	263	-148
[12 53 256]	263	-148
[18 -203 2304]	263	-148
[136 -1497 16480]	271	0
[74 217 640]	271	158
[34 -103 320]	271	382
[28 -423 6400]	271	0
[134 327 800]	271	431
[20 -103 544]	271	382
[212 487 1120]	271	158
[124 295 704]	271	431
[140 -1261 11360]	279	0
[70 -211 640]	279	195
[28 -365 4768]	279	485
[268 621 1440]	279	195
[24 -339 4800]	279	289
[42 -141 480]	279	212
[20 -301 4544]	279	0
[98 237 576]	279	-62
[18 -237 3136]	279	485
[20 -109 608]	279	602
[142 -426 1280]	284	0
[142 -710 3552]	284	0
[72 -362 1824]	284	212
[72 -938 12224]	284	-212
[450 1046 2432]	284	212
[6 26 160]	284	-554
[58 -182 576]	284	826
[36 -74 160]	284	950
[36 -182 928]	284	826
[36 -362 3648]	284	-848
[36 -470 6144]	284	-662
[30 -154 800]	284	-554
[30 -394 5184]	284	-986
[30 -454 6880]	284	0
[24 -362 5472]	284	0
[240 554 1280]	284	-212
[18 -38 96]	284	-78
[18 -182 1856]	284	-152
[20 -106 576]	284	-662
[20 -266 3552]	284	-986
[144 -1009 7072]	287	0
[78 -239 736]	287	6
[48 -241 1216]	287	-1180
[338 785 1824]	287	6
[24 -241 2432]	287	-107
[12 49 224]	287	-181
[26 -369 5248]	287	-30
[24 -271 3072]	287	-181
[18 -271 4096]	287	0
[148 -741 3712]	295	0
[74 -667 6016]	295	-59
[38 -421 4672]	295	-908
[10 5 32]	295	0
[226 517 1184]	295	-59
[152 -457 1376]	303	0
[4 9 96]	303	203
[6 9 64]	303	203
[26 -55 128]	303	-853
[26 -393 5952]	303	0
[22 -73 256]	303	-885
[2 4 160]	304	0
[154 460 1376]	304	0
[38 -76 160]	304	-1088
[10 4 32]	304	0
[10 44 224]	304	-118
[50 -164 544]	304	438
[22 -68 224]	304	518
[22 -244 2720]	304	-118
[22 -332 5024]	304	0
[2 3 160]	311	0
[52 -157 480]	311	1526
[218 541 1344]	311	501
[8 35 192]	311	682
[10 3 32]	311	0
[26 -131 672]	311	682
[26 -157 960]	311	-335
[28 -285 2912]	311	-627
[12 29 96]	311	1526
[16 35 96]	311	501
[20 -227 2592]	311	-568
[240 547 1248]	311	-359
[2 2 160]	316	0
[158 -1106 7744]	316	0
[80 -242 736]	316	248
[80 -882 9728]	316	-248
[40 -562 7904]	316	-742
[380 882 2048]	316	-248
[10 2 32]	316	0
[310 718 1664]	316	248
[20 -302 4576]	316	0
[22 -50 128]	316	614
[22 -226 2336]	316	299
[2 1 160]	319	0
[50 -159 512]	319	-111
[10 1 32]	319	0
[32 -353 3904]	319	-111
[38 97 256]	319	187
[20 -289 4192]	319	-75
[2 0 160]	320	0
[2 8 192]	320	0
[4 8 96]	320	-860
[84 248 736]	320	-40
[6 8 64]	320	-860
[54 -272 1376]	320	2236
[54 -704 9184]	320	-1180
[42 -88 192]	320	-580
[42 -592 8352]	320	2112
[10 0 32]	320	0
[10 40 192]	320	-326
[28 -312 3488]	320	-326
[28 -424 6432]	320	0
[14 32 96]	320	-1180
[126 304 736]	320	-1220
[254 584 1344]	320	-40
[20 -40 96]	320	-132
[20 -120 736]	320	272
[18 -56 192]	320	146
[18 -272 4128]	320	0
[164 -2133 27744]	327	0
[82 -1067 13888]	327	-233
[56 -619 6848]	327	-1584
[42 -213 1088]	327	-1584
[282 651 1504]	327	-233
[24 -363 5504]	327	0
[44 -149 512]	327	186
[168 -1849 20352]	335	0
[766 1785 4160]	335	987
[6 7 64]	335	987
[6 25 160]	335	919
[36 -185 960]	335	919
[28 -57 128]	335	682
[30 -455 6912]	335	0
[24 -313 4096]	335	-997
[14 57 256]	335	473
[54 -185 640]	335	473
[118 281 672]	335	708
[172 -1549 13952]	343	0
[86 -947 10432]	343	346
[422 979 2272]	343	346
[28 -371 4928]	343	2151
[22 -333 5056]	343	0
[174 -522 1568]	348	0
[174 -870 4352]	348	0
[4 6 96]	348	-324
[88 -794 7168]	348	324
[6 6 64]	348	-324
[44 -90 192]	348	2448
[44 -134 416]	348	-866
[44 -486 5376]	348	1850
[12 6 32]	348	0
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