A Borcherds Product in S₁₂(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
  (24 + ζ^(-5) + ζ^(-4) + 3/ζ^3 + 3/ζ^2 + 16/ζ + 16*ζ + 3*ζ^2 + 3*ζ^3 + ζ^4 + ζ^5)
  +q(ζ^(-9) + 3/ζ^8 + 3*ζ^8 + ζ^9)
  +q^3(ζ^(-14) + ζ^14)
  +q^4(-10/ζ^16 - 10*ζ^16)
• From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block

  	φ	= TB(12;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5)
  		= 11623334151
  		= η(τ)24(θ(τ,z)/η(τ))16(θ(τ,2z)/η(τ))3(θ(τ,3z)/η(τ))3(θ(τ,4z)/η(τ))(θ(τ,5z)/η(τ))

  This theta block has q-order of vanishing 3. This theta block has index 3*16, and so Borch(ψ) begins with Jacobi coefficient #3.
• 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 6 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)	24
  (4,2)	4
  (4,14)	2
  (9,3)	3
  (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

(24 + ζ^(-5) + ζ^(-4) + 3/ζ^3 + 3/ζ^2 + 16/ζ + 16*ζ + 3*ζ^2 + 3*ζ^3 + ζ^4 + ζ^5)
+q(506 + ζ^(-9) + 3/ζ^8 + 32/ζ^7 + 31/ζ^6 - 29/ζ^5 - 128/ζ^4 - 231/ζ^3 - 159/ζ^2 + 227/ζ + 227*ζ - 159*ζ^2 - 231*ζ^3 - 128*ζ^4 - 29*ζ^5 + 31*ζ^6 + 32*ζ^7 + 3*ζ^8 + ζ^9)
+q^2(5788 + 5/ζ^11 - 35/ζ^10 - 134/ζ^9 + 50/ζ^8 + 725/ζ^7 + 986/ζ^6 - 60/ζ^5 - 1920/ζ^4 - 3341/ζ^3 - 1975/ζ^2 + 2805/ζ + 2805*ζ - 1975*ζ^2 - 3341*ζ^3 - 1920*ζ^4 - 60*ζ^5 + 986*ζ^6 + 725*ζ^7 + 50*ζ^8 - 134*ζ^9 - 35*ζ^10 + 5*ζ^11)
+q^3(44264 + ζ^(-14) + 23/ζ^13 + 64/ζ^12 - 49/ζ^11 - 994/ζ^10 - 2215/ζ^9 + 396/ζ^8 + 7863/ζ^7 + 10908/ζ^6 + 1235/ζ^5 - 16960/ζ^4 - 28945/ζ^3 - 15547/ζ^2 + 22088/ζ + 22088*ζ - 15547*ζ^2 - 28945*ζ^3 - 16960*ζ^4 + 1235*ζ^5 + 10908*ζ^6 + 7863*ζ^7 + 396*ζ^8 - 2215*ζ^9 - 994*ζ^10 - 49*ζ^11 + 64*ζ^12 + 23*ζ^13 + ζ^14)
+q^4(264324 - 10/ζ^16 - 29/ζ^15 + 163/ζ^14 + 792/ζ^13 + 1407/ζ^12 - 859/ζ^11 - 10924/ζ^10 - 19981/ζ^9 + 1992/ζ^8 + 56688/ζ^7 + 79883/ζ^6 + 13788/ζ^5 - 110975/ζ^4 - 185693/ζ^3 - 93698/ζ^2 + 135294/ζ + 135294*ζ - 93698*ζ^2 - 185693*ζ^3 - 110975*ζ^4 + 13788*ζ^5 + 79883*ζ^6 + 56688*ζ^7 + 1992*ζ^8 - 19981*ζ^9 - 10924*ζ^10 - 859*ζ^11 + 1407*ζ^12 + 792*ζ^13 + 163*ζ^14 - 29*ζ^15 - 10*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	0
[14 -99 704]	55	0
[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	0
[10 -50 256]	60	0
[8 -50 320]	60	0
[8 -114 1632]	60	0
[2 1 32]	63	0
[16 -225 3168]	63	0
[4 33 288]	63	0
[12 -159 2112]	63	0
[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	0
[10 -24 64]	64	0
[10 -64 416]	64	0
[10 -144 2080]	64	0
[36 -469 6112]	71	0
[18 -181 1824]	71	1
[18 -235 3072]	71	0
[12 -85 608]	71	1
[120 277 640]	71	0
[10 -43 192]	71	1
[10 -53 288]	71	0
[40 -441 4864]	79	0
[20 -121 736]	79	-1
[190 441 1024]	79	0
[10 -121 1472]	79	-1
[44 -397 3584]	87	0
[22 -45 96]	87	-3
[22 -243 2688]	87	0
[12 -51 224]	87	-3
[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	-1
[162 374 864]	92	0
[6 26 128]	92	0
[26 -86 288]	92	0
[12 -26 64]	92	0
[12 -38 128]	92	-1
[12 -122 1248]	92	-1
[12 -134 1504]	92	0
[48 -337 2368]	95	0
[24 -337 4736]	95	2
[4 17 96]	95	0
[16 -81 416]	95	0
[12 -145 1760]	95	2
[60 145 352]	95	1
[10 15 32]	95	0
[52 -261 1312]	103	0
[4 5 32]	103	0
[26 -261 2624]	103	13
[14 -59 256]	103	13
[56 -169 512]	111	0
[28 -169 1024]	111	-10
[120 297 736]	111	0
[6 9 32]	111	0
[14 -169 2048]	111	-10
[14 -183 2400]	111	3
[10 23 64]	111	0
[2 4 64]	112	0
[58 172 512]	112	0
[4 4 32]	112	0
[28 -84 256]	112	0
[28 -140 704]	112	0
[28 -196 1376]	112	0
[14 -28 64]	112	0
[14 -84 512]	112	0
[8 12 32]	112	0
[16 -84 448]	112	0
[16 -148 1376]	112	0
[16 -212 2816]	112	-3
[2 3 64]	119	0
[30 -61 128]	119	-33
[30 -451 6784]	119	0
[20 -61 192]	119	1
[90 221 544]	119	3
[16 -67 288]	119	-33
[12 -61 320]	119	1
[24 61 160]	119	0
[12 -125 1312]	119	3
[2 2 64]	124	0
[62 -434 3040]	124	0
[4 2 32]	124	0
[32 -226 1600]	124	0
[16 -98 608]	124	1
[16 -226 3200]	124	0
[8 34 160]	124	0
[28 -94 320]	124	1
[10 14 32]	124	0
[14 -46 160]	124	0
[14 -130 1216]	124	1
[14 -158 1792]	124	0
[2 1 64]	127	0
[4 1 32]	127	0
[32 -449 6304]	127	21
[16 -193 2336]	127	21
[2 0 64]	128	0
[2 8 96]	128	0
[2 16 192]	128	0
[4 0 32]	128	0
[4 16 96]	128	0
[4 32 288]	128	0
[6 8 32]	128	0
[22 -112 576]	128	0
[22 -200 1824]	128	0
[22 -288 3776]	128	-3
[18 -40 96]	128	0
[18 -112 704]	128	0
[18 -256 3648]	128	0
[12 16 32]	128	0
[12 -64 352]	128	-3
[68 -885 11520]	135	0
[34 -171 864]	135	0
[34 -341 3424]	135	69
[24 -267 2976]	135	0
[8 11 32]	135	0
[18 -75 320]	135	69
[12 -75 480]	135	0
[72 -793 8736]	143	0
[36 -217 1312]	143	-38
[42 121 352]	143	0
[6 7 32]	143	0
[6 25 128]	143	-1
[18 -199 2208]	143	-1
[18 -217 2624]	143	-38
[12 -25 64]	143	0
[14 -89 576]	143	-1
[76 -685 6176]	151	0
[38 -77 160]	151	-137
[38 -115 352]	151	0
[8 19 64]	151	0
[10 13 32]	151	0
[78 -234 704]	156	0
[78 -390 1952]	156	0
[40 -202 1024]	156	0
[40 -522 6816]	156	0
[6 6 32]	156	0
[8 10 32]	156	0
[20 -42 96]	156	-9
[20 -202 2048]	156	-12
[10 22 64]	156	0
[82 198 480]	156	-5
[10 42 192]	156	-12
[30 -102 352]	156	12
[16 -54 192]	156	-9
[16 -182 2080]	156	-12
[80 -561 3936]	159	0
[40 -561 7872]	159	79
[46 -143 448]	159	0
[28 -143 736]	159	0
[30 81 224]	159	2
[16 -49 160]	159	-1
[12 15 32]	159	0
[14 -143 1472]	159	2
[84 -421 2112]	167	0
[42 -379 3424]	167	0
[42 -421 4224]	167	177
[28 -421 6336]	167	0
[36 101 288]	167	10
[22 -69 224]	167	13
[14 -155 1728]	167	13
[14 -197 2784]	167	0
[16 -37 96]	167	10
[88 -265 800]	175	0
[44 -265 1600]	175	-63
[4 9 64]	175	0
[8 9 32]	175	0
[20 -265 3520]	175	12
[2 4 96]	176	0
[90 268 800]	176	0
[30 -92 288]	176	-13
[30 -332 3680]	176	1
[30 -452 6816]	176	0
[22 -44 96]	176	24
[22 -132 800]	176	-24
[10 12 32]	176	0
[18 -92 480]	176	-13
[18 -236 3104]	176	-7
[2 3 96]	183	0
[46 -93 192]	183	-218
[46 -323 2272]	183	0
[6 3 32]	183	0
[38 -125 416]	183	-15
[16 -227 3232]	183	0
[2 2 96]	188	0
[94 -658 4608]	188	0
[48 -146 448]	188	0
[48 -530 5856]	188	0
[6 2 32]	188	0
[6 14 64]	188	0
[142 350 864]	188	-39
[24 -146 896]	188	12
[24 -338 4768]	188	10
[8 18 64]	188	0
[112 274 672]	188	-19
[12 14 32]	188	0
[12 50 224]	188	12
[32 -110 384]	188	10
[14 34 96]	188	-39
[102 242 576]	188	-19
[14 62 288]	188	-12
[42 -146 512]	188	12
[2 1 96]	191	0
[48 -673 9440]	191	88
[4 33 320]	191	0
[6 1 32]	191	0
[32 -417 5440]	191	42
[8 33 160]	191	0
[20 -223 2496]	191	0
[36 95 256]	191	39
[16 -161 1632]	191	39
[18 -95 512]	191	42
[18 -257 3680]	191	0
[2 0 96]	192	0
[2 8 128]	192	0
[2 16 224]	192	0
[4 8 64]	192	0
[52 152 448]	192	0
[6 0 32]	192	0
[6 24 128]	192	15
[8 8 32]	192	0
[24 -72 224]	192	-41
[24 -120 608]	192	15
[26 -56 128]	192	0
[26 -160 992]	192	48
[26 -368 5216]	192	-48
[12 24 64]	192	0
[28 72 192]	192	-26
[14 16 32]	192	0
[14 -72 384]	192	-41
[100 -1301 16928]	199	0
[50 -501 5024]	199	136
[50 -651 8480]	199	0
[26 -341 4480]	199	45
[10 11 32]	199	0
[10 21 64]	199	0
[16 -213 2848]	199	45
[104 -1145 12608]	207	0
[478 1113 2592]	207	0
[36 -327 2976]	207	12
[26 -391 5888]	207	0
[18 -39 96]	207	12
[18 -57 192]	207	51
[108 -973 8768]	215	0
[54 -109 224]	215	155
[54 -595 6560]	215	0
[36 -397 4384]	215	-16
[6 13 64]	215	0
[24 -77 256]	215	36
[12 13 32]	215	0
[18 -109 672]	215	21
[20 -205 2112]	215	49
[16 -179 2016]	215	36
[110 -330 992]	220	0
[110 -550 2752]	220	0
[4 6 64]	220	0
[56 -506 4576]	220	0
[28 -58 128]	220	-28
[28 -282 2848]	220	91
[28 -422 6368]	220	0
[10 10 32]	220	0
[34 86 224]	220	-49
[170 390 896]	220	0
[14 58 256]	220	-28
[34 -118 416]	220	28
[16 -70 320]	220	91
[16 -198 2464]	220	-28
[112 -785 5504]	223	0
[56 -785 11008]	223	-203
[4 17 128]	223	0
[8 17 64]	223	0
[14 15 32]	223	0
[116 -581 2912]	231	0
[4 5 64]	231	0
[40 -123 384]	231	75
[30 -453 6848]	231	0
[24 -123 640]	231	75
[40 -133 448]	231	-89
[20 -123 768]	231	-14
[104 251 608]	231	155
[120 -361 1088]	239	0
[248 617 1536]	239	0
[48 -151 480]	239	-104
[30 -151 768]	239	-104
[24 -361 5440]	239	0
[10 41 192]	239	53
[20 -41 96]	239	23
[198 457 1056]	239	0
[22 -137 864]	239	-51
[22 -247 2784]	239	53
[2 4 128]	240	0
[122 364 1088]	240	0
[4 4 64]	240	0
[60 -180 544]	240	0
[60 -300 1504]	240	0
[60 -420 2944]	240	0
[6 12 64]	240	0
[46 132 384]	240	-72
[30 -60 128]	240	128
[30 -180 1088]	240	-128
[8 4 32]	240	0
[32 -100 320]	240	-102
[32 -356 3968]	240	105
[10 20 64]	240	0
[34 92 256]	240	-176
[12 12 32]	240	0
[20 -60 192]	240	-146
[20 -100 512]	240	-102
[16 -36 96]	240	-72
[16 -100 640]	240	3
[16 -164 1696]	240	-176
[16 -228 3264]	240	0
[2 3 128]	247	0
[62 -931 13984]	247	0
[8 3 32]	247	0
[2 2 128]	252	0
[126 -882 6176]	252	0
[4 2 64]	252	0
[64 -450 3168]	252	0
[42 -126 384]	252	-282
[42 -210 1056]	252	117
[8 2 32]	252	0
[32 -194 1184]	252	-82
[32 -450 6336]	252	40
[24 -126 672]	252	-282
[24 -318 4224]	252	-315
[14 14 32]	252	0
[16 -66 288]	252	40
[16 -194 2368]	252	-82
[18 -78 352]	252	-40
[18 -222 2752]	252	40
[18 -258 3712]	252	0
[2 1 128]	255	0
[4 1 64]	255	0
[44 -575 7520]	255	264
[8 1 32]	255	0
[22 -223 2272]	255	4
[22 -289 3808]	255	293
[20 -65 224]	255	-26
[14 33 96]	255	264
[2 0 128]	256	0
[2 8 160]	256	0
[2 16 256]	256	0
[4 0 64]	256	0
[4 16 128]	256	0
[8 0 32]	256	0
[8 16 64]	256	0
[8 32 160]	256	26
[34 -72 160]	256	0
[34 -480 6784]	256	-256
[26 -80 256]	256	-158
[26 -288 3200]	256	26
[26 -392 5920]	256	0
[16 16 32]	256	0
[16 -48 160]	256	-158
[20 -48 128]	256	-158
[20 -288 4160]	256	0
[132 -1717 22336]	263	0
[66 -331 1664]	263	0
[408 949 2208]	263	0
[22 -309 4352]	263	-107
[22 -331 4992]	263	0
[24 -53 128]	263	121
[12 53 256]	263	121
[18 -203 2304]	263	121
[136 -1497 16480]	271	0
[74 217 640]	271	0
[34 -103 320]	271	389
[28 -423 6400]	271	0
[134 327 800]	271	416
[20 -103 544]	271	389
[212 487 1120]	271	0
[124 295 704]	271	416
[140 -1261 11360]	279	0
[70 -211 640]	279	0
[164 403 992]	279	411
[28 -365 4768]	279	446
[268 621 1440]	279	0
[24 -339 4800]	279	-150
[42 -141 480]	279	-255
[20 -301 4544]	279	0
[98 237 576]	279	466
[18 -237 3136]	279	446
[20 -109 608]	279	411
[142 -426 1280]	284	0
[142 -710 3552]	284	0
[72 -362 1824]	284	0
[72 -938 12224]	284	0
[58 166 480]	284	-97
[450 1046 2432]	284	0
[6 26 160]	284	-238
[58 -182 576]	284	443
[36 -74 160]	284	-551
[36 -182 928]	284	443
[36 -362 3648]	284	836
[36 -470 6144]	284	-577
[30 -154 800]	284	-238
[30 -394 5184]	284	-591
[30 -454 6880]	284	0
[24 -362 5472]	284	0
[40 106 288]	284	-306
[240 554 1280]	284	0
[18 -38 96]	284	-97
[18 -182 1856]	284	-306
[18 -218 2656]	284	188
[20 -106 576]	284	-577
[20 -246 3040]	284	-188
[20 -266 3552]	284	-591
[144 -1009 7072]	287	0
[78 -239 736]	287	0
[48 -241 1216]	287	-515
[338 785 1824]	287	0
[24 -241 2432]	287	320
[12 49 224]	287	16
[26 -369 5248]	287	209
[24 -271 3072]	287	16
[18 -271 4096]	287	0
[148 -741 3712]	295	0
[74 -667 6016]	295	0
[38 -421 4672]	295	-482
[10 5 32]	295	0
[226 517 1184]	295	0
[152 -457 1376]	303	0
[4 9 96]	303	0
[6 9 64]	303	0
[26 -55 128]	303	-40
[26 -393 5952]	303	0
[22 -73 256]	303	254
[2 4 160]	304	0
[154 460 1376]	304	0
[38 -76 160]	304	760
[38 -228 1376]	304	-760
[10 4 32]	304	0
[10 44 224]	304	-407
[50 -164 544]	304	571
[22 -68 224]	304	-365
[22 -244 2720]	304	-407
[22 -332 5024]	304	0
[2 3 160]	311	0
[78 -547 3840]	311	0
[52 -157 480]	311	1192
[218 541 1344]	311	427
[8 35 192]	311	537
[10 3 32]	311	0
[26 -131 672]	311	537
[26 -157 960]	311	45
[28 -285 2912]	311	78
[12 29 96]	311	1192
[14 61 288]	311	-65
[16 35 96]	311	427
[20 -227 2592]	311	-65
[240 547 1248]	311	0
[2 2 160]	316	0
[158 -1106 7744]	316	0
[80 -242 736]	316	0
[80 -882 9728]	316	0
[40 -562 7904]	316	459
[380 882 2048]	316	0
[10 2 32]	316	0
[310 718 1664]	316	0
[20 -302 4576]	316	0
[22 -50 128]	316	-479
[22 -226 2336]	316	-350
[2 1 160]	319	0
[50 -159 512]	319	-386
[10 1 32]	319	0
[32 -353 3904]	319	-386
[38 97 256]	319	251
[20 -289 4192]	319	0
[2 0 160]	320	0
[2 8 192]	320	0
[2 16 288]	320	0
[4 8 96]	320	0
[84 248 736]	320	0
[6 8 64]	320	0
[54 -272 1376]	320	1510
[54 -704 9184]	320	-1230
[42 -88 192]	320	0
[42 -592 8352]	320	-1472
[10 0 32]	320	0
[10 40 192]	320	-12
[28 -88 288]	320	-364
[28 -312 3488]	320	-12
[28 -424 6432]	320	0
[14 32 96]	320	-1230
[126 304 736]	320	-1562
[254 584 1344]	320	0
[20 -40 96]	320	-4
[20 -120 736]	320	-372
[18 -56 192]	320	-364
[18 -272 4128]	320	0
[164 -2133 27744]	327	0
[82 -1067 13888]	327	0
[56 -619 6848]	327	-1314
[42 -213 1088]	327	-1314
[28 -171 1056]	327	-196
[282 651 1504]	327	0
[24 -363 5504]	327	0
[44 -149 512]	327	284
[168 -1849 20352]	335	0
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