A Borcherds Product in S₁₁(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
  (22 + ζ^(-4) + 12/ζ^2 + 12*ζ^2 + ζ^4)
  +q(-ζ^(-8) - ζ^8)
  +q^3(ζ^(-15) + ζ^15)
  +q^4(ζ^(-17) + ζ^17)
• From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block

  	φ	= TB(11;2,2,2,2,2,2,2,2,2,2,2,2,4)
  		= 21241
  		= η(τ)22(θ(τ,2z)/η(τ))12(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 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)	13
  (4,2)	13
  (4,14)	1
  (16,4)	1
  (33,15)	1

• From the singular part above, we compute that the valuation Ord(ψ) is -0.515625, 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

(22 + ζ^(-4) + 12/ζ^2 + 12*ζ^2 + ζ^4)
+q(1386 - ζ^(-8) - 11/ζ^7 + 44/ζ^6 - 177/ζ^5 + 332/ζ^4 - 683/ζ^3 + 980/ζ^2 - 1177/ζ - 1177*ζ + 980*ζ^2 - 683*ζ^3 + 332*ζ^4 - 177*ζ^5 + 44*ζ^6 - 11*ζ^7 - ζ^8)
+q^2(29260 + 10/ζ^11 - 56/ζ^10 + 99/ζ^9 + 42/ζ^8 - 517/ζ^7 + 2156/ζ^6 - 5458/ζ^5 + 9904/ζ^4 - 16480/ζ^3 + 22476/ζ^2 - 26806/ζ - 26806*ζ + 22476*ζ^2 - 16480*ζ^3 + 9904*ζ^4 - 5458*ζ^5 + 2156*ζ^6 - 517*ζ^7 + 42*ζ^8 + 99*ζ^9 - 56*ζ^10 + 10*ζ^11)
+q^3(345576 + ζ^(-15) + 77/ζ^13 - 396/ζ^12 + 1011/ζ^11 - 2180/ζ^10 + 2860/ζ^9 + 380/ζ^8 - 10098/ζ^7 + 34364/ζ^6 - 76927/ζ^5 + 134428/ζ^4 - 207673/ζ^3 + 275016/ζ^2 - 323651/ζ - 323651*ζ + 275016*ζ^2 - 207673*ζ^3 + 134428*ζ^4 - 76927*ζ^5 + 34364*ζ^6 - 10098*ζ^7 + 380*ζ^8 + 2860*ζ^9 - 2180*ζ^10 + 1011*ζ^11 - 396*ζ^12 + 77*ζ^13 + ζ^15)
+q^4(2926000 + ζ^(-17) + 189/ζ^15 - 968/ζ^14 + 3586/ζ^13 - 10417/ζ^12 + 21161/ζ^11 - 34412/ζ^10 + 36949/ζ^9 + 1960/ζ^8 - 111375/ζ^7 + 346456/ζ^6 - 728879/ζ^5 + 1232433/ζ^4 - 1828428/ζ^3 + 2375900/ζ^2 - 2767156/ζ - 2767156*ζ + 2375900*ζ^2 - 1828428*ζ^3 + 1232433*ζ^4 - 728879*ζ^5 + 346456*ζ^6 - 111375*ζ^7 + 1960*ζ^8 + 36949*ζ^9 - 34412*ζ^10 + 21161*ζ^11 - 10417*ζ^12 + 3586*ζ^13 - 968*ζ^14 + 189*ζ^15 + ζ^17)

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	1
[2 2 32]	60	0
[30 -210 1472]	60	0
[16 -50 160]	60	-1
[16 -178 1984]	60	-1
[10 -30 96]	60	0
[10 -50 256]	60	-1
[8 -50 320]	60	-1
[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	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	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	0
[18 -235 3072]	71	0
[120 277 640]	71	0
[10 -43 192]	71	1
[10 -53 288]	71	0
[40 -441 4864]	79	0
[20 -121 736]	79	0
[190 441 1024]	79	0
[10 -121 1472]	79	2
[44 -397 3584]	87	0
[22 -45 96]	87	0
[22 -243 2688]	87	0
[12 -51 224]	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	0
[26 70 192]	92	0
[162 374 864]	92	0
[6 26 128]	92	1
[26 -86 288]	92	2
[12 -26 64]	92	0
[12 -38 128]	92	0
[12 -122 1248]	92	0
[12 -134 1504]	92	1
[48 -337 2368]	95	0
[24 -337 4736]	95	0
[4 17 96]	95	0
[16 -81 416]	95	0
[12 -145 1760]	95	2
[60 145 352]	95	0
[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	0
[6 9 32]	111	0
[14 -169 2048]	111	3
[14 -183 2400]	111	-1
[10 23 64]	111	0
[2 4 64]	112	0
[58 172 512]	112	0
[4 4 32]	112	0
[28 -84 256]	112	12
[28 -140 704]	112	-12
[28 -196 1376]	112	0
[14 -28 64]	112	0
[14 -84 512]	112	12
[8 12 32]	112	0
[16 -84 448]	112	12
[16 -148 1376]	112	12
[16 -212 2816]	112	10
[2 3 64]	119	0
[30 -61 128]	119	0
[30 -451 6784]	119	0
[20 -61 192]	119	9
[90 221 544]	119	-11
[16 -67 288]	119	-17
[12 -61 320]	119	9
[24 61 160]	119	-9
[12 -125 1312]	119	11
[2 2 64]	124	0
[62 -434 3040]	124	0
[4 2 32]	124	0
[32 -226 1600]	124	0
[16 -98 608]	124	0
[16 -226 3200]	124	0
[8 34 160]	124	1
[28 -94 320]	124	2
[10 14 32]	124	0
[14 -46 160]	124	-8
[14 -130 1216]	124	0
[14 -158 1792]	124	1
[2 1 64]	127	0
[4 1 32]	127	0
[32 -449 6304]	127	0
[16 -193 2336]	127	-16
[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
[6 8 32]	128	0
[22 -112 576]	128	0
[22 -288 3776]	128	0
[18 -40 96]	128	0
[18 -112 704]	128	0
[18 -256 3648]	128	0
[12 16 32]	128	0
[12 -64 352]	128	0
[68 -885 11520]	135	0
[34 -171 864]	135	0
[34 -341 3424]	135	0
[24 -267 2976]	135	-11
[8 11 32]	135	0
[12 -75 480]	135	-11
[72 -793 8736]	143	0
[36 -217 1312]	143	0
[42 121 352]	143	0
[6 7 32]	143	0
[6 25 128]	143	21
[18 -199 2208]	143	21
[12 -25 64]	143	0
[14 -89 576]	143	-21
[76 -685 6176]	151	0
[38 -77 160]	151	0
[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	65
[40 -522 6816]	156	65
[6 6 32]	156	0
[8 10 32]	156	0
[20 -42 96]	156	0
[20 -202 2048]	156	65
[10 22 64]	156	65
[82 198 480]	156	10
[10 42 192]	156	-18
[30 -102 352]	156	-8
[16 -54 192]	156	-83
[16 -182 2080]	156	-18
[80 -561 3936]	159	0
[40 -561 7872]	159	0
[46 -143 448]	159	0
[28 -143 736]	159	0
[30 81 224]	159	0
[16 -49 160]	159	1
[12 15 32]	159	0
[14 -143 1472]	159	0
[84 -421 2112]	167	0
[42 -379 3424]	167	0
[42 -421 4224]	167	0
[28 -421 6336]	167	0
[36 101 288]	167	-53
[22 -69 224]	167	73
[14 -155 1728]	167	-73
[14 -197 2784]	167	0
[16 -37 96]	167	53
[88 -265 800]	175	0
[4 9 64]	175	0
[8 9 32]	175	0
[20 -265 3520]	175	-101
[2 4 96]	176	0
[90 268 800]	176	0
[30 -92 288]	176	-44
[30 -332 3680]	176	44
[30 -452 6816]	176	0
[22 -44 96]	176	0
[22 -132 800]	176	0
[10 12 32]	176	0
[18 -92 480]	176	-44
[18 -236 3104]	176	64
[2 3 96]	183	0
[46 -93 192]	183	0
[46 -323 2272]	183	0
[6 3 32]	183	0
[38 -125 416]	183	-135
[16 -227 3232]	183	0
[2 2 96]	188	0
[94 -658 4608]	188	0
[48 -146 448]	188	-57
[48 -530 5856]	188	-57
[6 2 32]	188	0
[6 14 64]	188	-57
[142 350 864]	188	0
[24 -146 896]	188	-57
[24 -338 4768]	188	0
[8 18 64]	188	57
[112 274 672]	188	35
[12 14 32]	188	0
[12 50 224]	188	65
[32 -110 384]	188	131
[14 34 96]	188	0
[102 242 576]	188	35
[14 62 288]	188	-94
[42 -146 512]	188	-65
[2 1 96]	191	0
[48 -673 9440]	191	0
[6 1 32]	191	0
[32 -417 5440]	191	44
[8 33 160]	191	-82
[20 -223 2496]	191	-82
[36 95 256]	191	44
[16 -161 1632]	191	44
[18 -95 512]	191	-44
[18 -257 3680]	191	0
[2 0 96]	192	0
[2 8 128]	192	0
[2 16 224]	192	0
[4 8 64]	192	208
[52 152 448]	192	-208
[6 0 32]	192	0
[6 24 128]	192	-88
[8 8 32]	192	0
[24 -72 224]	192	-80
[24 -120 608]	192	88
[26 -56 128]	192	208
[26 -160 992]	192	0
[26 -368 5216]	192	0
[12 24 64]	192	-208
[28 72 192]	192	80
[14 16 32]	192	0
[14 -72 384]	192	-80
[100 -1301 16928]	199	0
[50 -651 8480]	199	0
[26 -341 4480]	199	-34
[10 11 32]	199	0
[10 21 64]	199	0
[16 -213 2848]	199	-34
[104 -1145 12608]	207	0
[478 1113 2592]	207	0
[26 -391 5888]	207	0
[18 -39 96]	207	-143
[18 -57 192]	207	62
[108 -973 8768]	215	0
[54 -595 6560]	215	0
[36 -397 4384]	215	-45
[6 13 64]	215	0
[24 -77 256]	215	-48
[12 13 32]	215	0
[18 -109 672]	215	-45
[20 -205 2112]	215	0
[16 -179 2016]	215	48
[110 -330 992]	220	0
[110 -550 2752]	220	0
[4 6 64]	220	-429
[56 -506 4576]	220	-429
[28 -58 128]	220	-429
[28 -282 2848]	220	0
[28 -422 6368]	220	0
[10 10 32]	220	0
[34 86 224]	220	-278
[170 390 896]	220	-429
[14 58 256]	220	266
[34 -118 416]	220	246
[16 -70 320]	220	737
[16 -198 2464]	220	266
[112 -785 5504]	223	0
[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	166
[30 -453 6848]	231	0
[24 -123 640]	231	166
[40 -133 448]	231	-237
[20 -123 768]	231	166
[104 251 608]	231	-236
[120 -361 1088]	239	0
[248 617 1536]	239	0
[48 -151 480]	239	-156
[30 -151 768]	239	-156
[24 -361 5440]	239	0
[10 41 192]	239	-193
[20 -41 96]	239	221
[198 457 1056]	239	0
[22 -247 2784]	239	-193
[2 4 128]	240	0
[122 364 1088]	240	0
[4 4 64]	240	572
[60 -180 544]	240	320
[60 -300 1504]	240	-320
[60 -420 2944]	240	-572
[6 12 64]	240	320
[46 132 384]	240	0
[30 -60 128]	240	572
[30 -180 1088]	240	320
[8 4 32]	240	0
[32 -100 320]	240	-812
[32 -356 3968]	240	-834
[10 20 64]	240	-320
[34 92 256]	240	-12
[12 12 32]	240	0
[20 -60 192]	240	12
[20 -100 512]	240	-812
[16 -36 96]	240	0
[16 -164 1696]	240	-12
[16 -228 3264]	240	-572
[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	-429
[42 -126 384]	252	211
[42 -210 1056]	252	226
[8 2 32]	252	0
[32 -194 1184]	252	0
[32 -450 6336]	252	429
[24 -126 672]	252	211
[24 -318 4224]	252	-507
[14 14 32]	252	0
[16 -194 2368]	252	487
[18 -222 2752]	252	-81
[18 -258 3712]	252	429
[2 1 128]	255	0
[4 1 64]	255	0
[44 -575 7520]	255	28
[8 1 32]	255	0
[22 -223 2272]	255	28
[22 -289 3808]	255	164
[20 -65 224]	255	60
[14 33 96]	255	28
[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	32
[34 -72 160]	256	0
[34 -480 6784]	256	0
[26 -80 256]	256	48
[26 -288 3200]	256	32
[26 -392 5920]	256	0
[16 16 32]	256	0
[16 -48 160]	256	48
[20 -48 128]	256	128
[20 -288 4160]	256	0
[132 -1717 22336]	263	0
[66 -331 1664]	263	0
[408 949 2208]	263	0
[22 -309 4352]	263	143
[22 -331 4992]	263	0
[24 -53 128]	263	0
[12 53 256]	263	-78
[18 -203 2304]	263	-78
[136 -1497 16480]	271	0
[74 217 640]	271	0
[34 -103 320]	271	590
[28 -423 6400]	271	0
[134 327 800]	271	-424
[20 -103 544]	271	590
[212 487 1120]	271	0
[124 295 704]	271	-424
[140 -1261 11360]	279	0
[70 -211 640]	279	0
[28 -365 4768]	279	-294
[268 621 1440]	279	0
[24 -339 4800]	279	143
[42 -141 480]	279	-380
[20 -301 4544]	279	0
[98 237 576]	279	315
[18 -237 3136]	279	-294
[20 -109 608]	279	-212
[142 -426 1280]	284	0
[142 -710 3552]	284	0
[72 -362 1824]	284	780
[72 -938 12224]	284	780
[450 1046 2432]	284	780
[6 26 160]	284	82
[58 -182 576]	284	-412
[36 -74 160]	284	0
[36 -182 928]	284	-412
[36 -362 3648]	284	780
[36 -470 6144]	284	-350
[30 -154 800]	284	-82
[30 -394 5184]	284	454
[30 -454 6880]	284	0
[24 -362 5472]	284	0
[240 554 1280]	284	-780
[18 -38 96]	284	0
[18 -182 1856]	284	-350
[20 -106 576]	284	350
[20 -266 3552]	284	454
[144 -1009 7072]	287	0
[78 -239 736]	287	0
[48 -241 1216]	287	-726
[338 785 1824]	287	0
[24 -241 2432]	287	-726
[12 49 224]	287	36
[26 -369 5248]	287	-429
[24 -271 3072]	287	36
[18 -271 4096]	287	0
[148 -741 3712]	295	0
[74 -667 6016]	295	0
[38 -421 4672]	295	1315
[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	0
[26 -393 5952]	303	0
[22 -73 256]	303	-628
[2 4 160]	304	0
[154 460 1376]	304	0
[38 -76 160]	304	0
[10 4 32]	304	0
[10 44 224]	304	-196
[50 -164 544]	304	296
[22 -68 224]	304	-364
[22 -244 2720]	304	-196
[22 -332 5024]	304	0
[2 3 160]	311	0
[52 -157 480]	311	-556
[218 541 1344]	311	583
[8 35 192]	311	-1921
[10 3 32]	311	0
[26 -131 672]	311	1921
[26 -157 960]	311	-556
[28 -285 2912]	311	-583
[12 29 96]	311	556
[16 35 96]	311	-583
[20 -227 2592]	311	468
[240 547 1248]	311	0
[2 2 160]	316	0
[158 -1106 7744]	316	0
[80 -242 736]	316	-345
[80 -882 9728]	316	-345
[40 -562 7904]	316	0
[380 882 2048]	316	345
[10 2 32]	316	0
[310 718 1664]	316	-345
[20 -302 4576]	316	0
[22 -50 128]	316	-875
[22 -226 2336]	316	470
[2 1 160]	319	0
[50 -159 512]	319	-1074
[10 1 32]	319	0
[32 -353 3904]	319	1074
[38 97 256]	319	-1001
[20 -289 4192]	319	0
[2 0 160]	320	0
[2 8 192]	320	0
[4 8 96]	320	768
[84 248 736]	320	-768
[6 8 64]	320	768
[54 -272 1376]	320	-416
[54 -704 9184]	320	-352
[42 -88 192]	320	768
[42 -592 8352]	320	0
[10 0 32]	320	0
[10 40 192]	320	400
[28 -312 3488]	320	400
[28 -424 6432]	320	0
[14 32 96]	320	-352
[126 304 736]	320	-1408
[254 584 1344]	320	768
[20 -40 96]	320	0
[20 -120 736]	320	128
[18 -56 192]	320	368
[18 -272 4128]	320	0
[164 -2133 27744]	327	0
[82 -1067 13888]	327	0
[56 -619 6848]	327	-999
[42 -213 1088]	327	999
[282 651 1504]	327	0
[24 -363 5504]	327	0
[44 -149 512]	327	83
[168 -1849 20352]	335	0
[766 1785 4160]	335	0
[6 7 64]	335	0
[6 25 160]	335	-258
[36 -185 960]	335	258
[28 -57 128]	335	0
[30 -455 6912]	335	0
[24 -313 4096]	335	159
[14 57 256]	335	-335
[54 -185 640]	335	335
[118 281 672]	335	97
[172 -1549 13952]	343	0
[86 -947 10432]	343	0
[422 979 2272]	343	0
[28 -371 4928]	343	-380
[22 -333 5056]	343	0
[174 -522 1568]	348	0
[174 -870 4352]	348	0
[4 6 96]	348	572
[88 -794 7168]	348	572
[6 6 64]	348	572
[44 -90 192]	348	572
[44 -134 416]	348	-776
[44 -486 5376]	348	-496
[12 6 32]	348	0
[26 -134 704]	348	-776
[26 -342 4512]	348	-246
[26 -394 5984]	348	0
[22 -310 4384]	348	0
[268 614 1408]	348	572
[4 17 160]	351	0
[60 -303 1536]	351	2198
[36 -111 352]	351	-533
[352 815 1888]	351	0
[30 -303 3072]	351	2198
[22 -111 576]	351	-533
[20 -303 4608]	351	0
[20 -47 128]	351	163
[180 -901 4512]	359	0
[4 5 96]	359	0
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