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 + 2/ζ^5 + ζ^(-4) + 3/ζ^2 + 18/ζ + 18*ζ + 3*ζ^2 + ζ^4 + 2*ζ^5)
  +q(3/ζ^8 + 3*ζ^8)
  +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;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,4,5,5)
  		= 118234152
  		= η(τ)24(θ(τ,z)/η(τ))18(θ(τ,2z)/η(τ))3(θ(τ,4z)/η(τ))(θ(τ,5z)/η(τ))2

  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
  (16,4)	1
  (25,5)	2

• 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 + 2/ζ^5 + ζ^(-4) + 3/ζ^2 + 18/ζ + 18*ζ + 3*ζ^2 + ζ^4 + 2*ζ^5)
+q(762 + 3/ζ^8 + 46/ζ^7 + 95/ζ^6 + 28/ζ^5 - 256/ζ^4 - 476/ζ^3 - 223/ζ^2 + 402/ζ + 402*ζ - 223*ζ^2 - 476*ζ^3 - 256*ζ^4 + 28*ζ^5 + 95*ζ^6 + 46*ζ^7 + 3*ζ^8)
+q^2(9884 - 6/ζ^11 - 99/ζ^10 - 232/ζ^9 + 50/ζ^8 + 1198/ζ^7 + 2202/ζ^6 + 566/ζ^5 - 3968/ζ^4 - 6912/ζ^3 - 3127/ζ^2 + 5386/ζ + 5386*ζ - 3127*ζ^2 - 6912*ζ^3 - 3968*ζ^4 + 566*ζ^5 + 2202*ζ^6 + 1198*ζ^7 + 50*ζ^8 - 232*ζ^9 - 99*ζ^10 - 6*ζ^11)
+q^3(80104 + ζ^(-14) + 62/ζ^13 + 192/ζ^12 - 252/ζ^11 - 2210/ζ^10 - 4028/ζ^9 + 396/ζ^8 + 13806/ζ^7 + 23004/ζ^6 + 6226/ζ^5 - 35008/ζ^4 - 59220/ζ^3 - 26427/ζ^2 + 43406/ζ + 43406*ζ - 26427*ζ^2 - 59220*ζ^3 - 35008*ζ^4 + 6226*ζ^5 + 23004*ζ^6 + 13806*ζ^7 + 396*ζ^8 - 4028*ζ^9 - 2210*ζ^10 - 252*ζ^11 + 192*ζ^12 + 62*ζ^13 + ζ^14)
+q^4(493700 - 10/ζ^16 - 56/ζ^15 + 227/ζ^14 + 1834/ζ^13 + 3455/ζ^12 - 2690/ζ^11 - 23020/ζ^10 - 37390/ζ^9 + 1992/ζ^8 + 103632/ζ^7 + 164939/ζ^6 + 44078/ζ^5 - 227711/ζ^4 - 377798/ζ^3 - 166722/ζ^2 + 268390/ζ + 268390*ζ - 166722*ζ^2 - 377798*ζ^3 - 227711*ζ^4 + 44078*ζ^5 + 164939*ζ^6 + 103632*ζ^7 + 1992*ζ^8 - 37390*ζ^9 - 23020*ζ^10 - 2690*ζ^11 + 3455*ζ^12 + 1834*ζ^13 + 227*ζ^14 - 56*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	-1
[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	2
[162 374 864]	92	0
[6 26 128]	92	0
[26 -86 288]	92	0
[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	-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	-2
[8 12 32]	112	0
[16 -84 448]	112	0
[16 -148 1376]	112	-2
[16 -212 2816]	112	-6
[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	1
[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	-2
[16 -226 3200]	124	0
[8 34 160]	124	-2
[28 -94 320]	124	0
[10 14 32]	124	0
[14 -46 160]	124	-6
[14 -130 1216]	124	-2
[14 -158 1792]	124	-2
[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	-6
[22 -288 3776]	128	18
[18 -40 96]	128	-6
[18 -112 704]	128	4
[18 -256 3648]	128	0
[12 16 32]	128	0
[12 -64 352]	128	18
[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	-5
[72 -793 8736]	143	0
[36 -217 1312]	143	38
[42 121 352]	143	0
[6 7 32]	143	0
[6 25 128]	143	0
[18 -199 2208]	143	0
[18 -217 2624]	143	38
[12 -25 64]	143	0
[14 -89 576]	143	4
[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	22
[20 -202 2048]	156	20
[10 22 64]	156	0
[82 198 480]	156	52
[10 42 192]	156	22
[30 -102 352]	156	-14
[16 -54 192]	156	20
[16 -182 2080]	156	22
[80 -561 3936]	159	0
[40 -561 7872]	159	-79
[46 -143 448]	159	0
[28 -143 736]	159	0
[30 81 224]	159	-1
[16 -49 160]	159	-1
[12 15 32]	159	0
[14 -143 1472]	159	-1
[84 -421 2112]	167	0
[42 -379 3424]	167	0
[42 -421 4224]	167	-177
[28 -421 6336]	167	0
[36 101 288]	167	-38
[22 -69 224]	167	-38
[14 -155 1728]	167	-38
[14 -197 2784]	167	0
[16 -37 96]	167	-38
[88 -265 800]	175	0
[44 -265 1600]	175	63
[4 9 64]	175	0
[8 9 32]	175	0
[20 -265 3520]	175	-123
[2 4 96]	176	0
[90 268 800]	176	0
[30 -92 288]	176	-16
[30 -332 3680]	176	0
[30 -452 6816]	176	0
[22 -44 96]	176	-32
[22 -132 800]	176	32
[10 12 32]	176	0
[18 -92 480]	176	-16
[18 -236 3104]	176	78
[2 3 96]	183	0
[46 -93 192]	183	218
[46 -323 2272]	183	0
[6 3 32]	183	0
[38 -125 416]	183	4
[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	-38
[24 -146 896]	188	-20
[24 -338 4768]	188	-24
[8 18 64]	188	0
[112 274 672]	188	56
[12 14 32]	188	0
[12 50 224]	188	-22
[32 -110 384]	188	-22
[14 34 96]	188	-38
[102 242 576]	188	56
[14 62 288]	188	14
[42 -146 512]	188	-22
[2 1 96]	191	0
[48 -673 9440]	191	-88
[4 33 320]	191	0
[6 1 32]	191	0
[32 -417 5440]	191	-153
[8 33 160]	191	31
[20 -223 2496]	191	31
[36 95 256]	191	-94
[16 -161 1632]	191	-94
[18 -95 512]	191	-153
[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	-2
[8 8 32]	192	0
[24 -72 224]	192	-8
[24 -120 608]	192	-2
[26 -56 128]	192	-4
[26 -160 992]	192	-52
[26 -368 5216]	192	76
[12 24 64]	192	0
[28 72 192]	192	-6
[14 16 32]	192	0
[14 -72 384]	192	-8
[100 -1301 16928]	199	0
[50 -501 5024]	199	-136
[50 -651 8480]	199	0
[26 -341 4480]	199	-5
[10 11 32]	199	0
[10 21 64]	199	0
[16 -213 2848]	199	-5
[104 -1145 12608]	207	0
[478 1113 2592]	207	0
[36 -327 2976]	207	71
[26 -391 5888]	207	0
[18 -39 96]	207	71
[18 -57 192]	207	-126
[108 -973 8768]	215	0
[54 -109 224]	215	-155
[54 -595 6560]	215	0
[36 -397 4384]	215	1
[6 13 64]	215	0
[24 -77 256]	215	17
[12 13 32]	215	0
[18 -109 672]	215	13
[20 -205 2112]	215	-88
[16 -179 2016]	215	17
[110 -330 992]	220	0
[110 -550 2752]	220	0
[4 6 64]	220	0
[56 -506 4576]	220	0
[28 -58 128]	220	4
[28 -282 2848]	220	-130
[28 -422 6368]	220	0
[10 10 32]	220	0
[34 86 224]	220	-76
[170 390 896]	220	0
[14 58 256]	220	8
[34 -118 416]	220	8
[16 -70 320]	220	-134
[16 -198 2464]	220	8
[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	114
[30 -453 6848]	231	0
[24 -123 640]	231	114
[40 -133 448]	231	70
[20 -123 768]	231	142
[104 251 608]	231	56
[120 -361 1088]	239	0
[248 617 1536]	239	0
[48 -151 480]	239	35
[30 -151 768]	239	35
[24 -361 5440]	239	0
[10 41 192]	239	-118
[20 -41 96]	239	-114
[198 457 1056]	239	0
[22 -137 864]	239	-63
[22 -247 2784]	239	-118
[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	288
[30 -60 128]	240	-94
[30 -180 1088]	240	144
[8 4 32]	240	0
[32 -100 320]	240	290
[32 -356 3968]	240	-68
[10 20 64]	240	0
[34 92 256]	240	-22
[12 12 32]	240	0
[20 -60 192]	240	-26
[20 -100 512]	240	290
[16 -36 96]	240	288
[16 -100 640]	240	174
[16 -164 1696]	240	-22
[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	792
[42 -210 1056]	252	-18
[8 2 32]	252	0
[32 -194 1184]	252	108
[32 -450 6336]	252	-24
[24 -126 672]	252	792
[24 -318 4224]	252	-580
[14 14 32]	252	0
[16 -66 288]	252	18
[16 -194 2368]	252	66
[18 -78 352]	252	150
[18 -222 2752]	252	18
[18 -258 3712]	252	0
[2 1 128]	255	0
[4 1 64]	255	0
[44 -575 7520]	255	184
[8 1 32]	255	0
[22 -223 2272]	255	344
[22 -289 3808]	255	88
[20 -65 224]	255	352
[14 33 96]	255	184
[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	-206
[34 -72 160]	256	120
[34 -480 6784]	256	200
[26 -80 256]	256	-30
[26 -288 3200]	256	-206
[26 -392 5920]	256	0
[16 16 32]	256	0
[16 -48 160]	256	-30
[20 -48 128]	256	-34
[20 -288 4160]	256	0
[132 -1717 22336]	263	0
[66 -331 1664]	263	0
[408 949 2208]	263	0
[22 -309 4352]	263	3
[22 -331 4992]	263	0
[24 -53 128]	263	54
[12 53 256]	263	46
[18 -203 2304]	263	46
[136 -1497 16480]	271	0
[74 217 640]	271	0
[34 -103 320]	271	-198
[28 -423 6400]	271	0
[134 327 800]	271	-173
[20 -103 544]	271	-198
[212 487 1120]	271	0
[124 295 704]	271	-173
[140 -1261 11360]	279	0
[70 -211 640]	279	0
[164 403 992]	279	-658
[28 -365 4768]	279	-591
[268 621 1440]	279	0
[24 -339 4800]	279	500
[42 -141 480]	279	487
[20 -301 4544]	279	0
[98 237 576]	279	-596
[18 -237 3136]	279	-591
[20 -109 608]	279	-658
[142 -426 1280]	284	0
[142 -710 3552]	284	0
[72 -362 1824]	284	0
[72 -938 12224]	284	0
[58 166 480]	284	-246
[450 1046 2432]	284	0
[6 26 160]	284	-456
[58 -182 576]	284	-282
[36 -74 160]	284	354
[36 -182 928]	284	-282
[36 -362 3648]	284	-924
[36 -470 6144]	284	636
[30 -154 800]	284	-456
[30 -394 5184]	284	872
[30 -454 6880]	284	0
[24 -362 5472]	284	0
[40 106 288]	284	54
[240 554 1280]	284	0
[18 -38 96]	284	-246
[18 -182 1856]	284	54
[18 -218 2656]	284	-220
[20 -106 576]	284	636
[20 -246 3040]	284	60
[20 -266 3552]	284	872
[144 -1009 7072]	287	0
[78 -239 736]	287	0
[48 -241 1216]	287	150
[338 785 1824]	287	0
[24 -241 2432]	287	-310
[12 49 224]	287	-251
[26 -369 5248]	287	-424
[24 -271 3072]	287	-251
[18 -271 4096]	287	0
[148 -741 3712]	295	0
[74 -667 6016]	295	0
[38 -421 4672]	295	526
[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	-423
[26 -393 5952]	303	0
[22 -73 256]	303	-519
[2 4 160]	304	0
[154 460 1376]	304	0
[38 -76 160]	304	-672
[38 -228 1376]	304	672
[10 4 32]	304	0
[10 44 224]	304	492
[50 -164 544]	304	-202
[22 -68 224]	304	92
[22 -244 2720]	304	492
[22 -332 5024]	304	0
[2 3 160]	311	0
[78 -547 3840]	311	0
[52 -157 480]	311	-2677
[218 541 1344]	311	-1132
[8 35 192]	311	-1141
[10 3 32]	311	0
[26 -131 672]	311	-1141
[26 -157 960]	311	-814
[28 -285 2912]	311	-668
[12 29 96]	311	-2677
[14 61 288]	311	-629
[16 35 96]	311	-1132
[20 -227 2592]	311	-629
[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	-222
[380 882 2048]	316	0
[10 2 32]	316	0
[310 718 1664]	316	0
[20 -302 4576]	316	0
[22 -50 128]	316	1204
[22 -226 2336]	316	506
[2 1 160]	319	0
[50 -159 512]	319	782
[10 1 32]	319	0
[32 -353 3904]	319	782
[38 97 256]	319	-587
[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	-762
[54 -704 9184]	320	-206
[42 -88 192]	320	80
[42 -592 8352]	320	1040
[10 0 32]	320	0
[10 40 192]	320	-116
[28 -88 288]	320	1108
[28 -312 3488]	320	-116
[28 -424 6432]	320	0
[14 32 96]	320	-206
[126 304 736]	320	2810
[254 584 1344]	320	0
[20 -40 96]	320	-136
[20 -120 736]	320	1208
[18 -56 192]	320	1108
[18 -272 4128]	320	0
[164 -2133 27744]	327	0
[82 -1067 13888]	327	0
[56 -619 6848]	327	1371
[42 -213 1088]	327	1371
[28 -171 1056]	327	136
[282 651 1504]	327	0
[24 -363 5504]	327	0
[44 -149 512]	327	-114
[168 -1849 20352]	335	0
[766 1785 4160]	335	0
[6 7 64]	335	0
[6 25 160]	335	1010
[36 -185 960]	335	1010
[28 -57 128]	335	857
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