A Borcherds Product in S₁₃(K(16))⁻

properties |

c vector |

θ vector |

ψ expansion |

Borch(ψ) coefficients

• The singular part of ψ truncated up to q^(16/4) is
  (26 + 2/ζ^5 + 2/ζ^3 + 3/ζ^2 + 16/ζ + 16*ζ + 3*ζ^2 + 2*ζ^3 + 2*ζ^5)
  +q(3/ζ^8 + 3*ζ^8)
  +q^2(ζ^(-12) + ζ^12)
  +q^3(ζ^(-14) + ζ^14)
  +q^4(-12/ζ^16 - 12*ζ^16)
• From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block

  	φ	= TB(13;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,5,5)
  		= 116233252
  		= η(τ)26(θ(τ,z)/η(τ))16(θ(τ,2z)/η(τ))3(θ(τ,3z)/η(τ))2(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)	23
  (4,2)	3
  (4,14)	1
  (9,3)	2
  (16,12)	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

(26 + 2/ζ^5 + 2/ζ^3 + 3/ζ^2 + 16/ζ + 16*ζ + 3*ζ^2 + 2*ζ^3 + 2*ζ^5)
+q(778 + 3/ζ^8 + 34/ζ^7 + 87/ζ^6 + 50/ζ^5 - 264/ζ^4 - 422/ζ^3 - 215/ζ^2 + 338/ζ + 338*ζ - 215*ζ^2 - 422*ζ^3 - 264*ζ^4 + 50*ζ^5 + 87*ζ^6 + 34*ζ^7 + 3*ζ^8)
+q^2(9948 + ζ^(-12) + 4/ζ^11 - 91/ζ^10 - 270/ζ^9 + 50/ζ^8 + 1052/ζ^7 + 2178/ζ^6 + 806/ζ^5 - 4001/ζ^4 - 6478/ζ^3 - 3111/ζ^2 + 4886/ζ + 4886*ζ - 3111*ζ^2 - 6478*ζ^3 - 4001*ζ^4 + 806*ζ^5 + 2178*ζ^6 + 1052*ζ^7 + 50*ζ^8 - 270*ζ^9 - 91*ζ^10 + 4*ζ^11 + ζ^12)
+q^3(80296 + ζ^(-14) + 80/ζ^13 + 200/ζ^12 - 130/ζ^11 - 2186/ζ^10 - 4352/ζ^9 + 396/ζ^8 + 12912/ζ^7 + 22900/ζ^6 + 7666/ζ^5 - 35112/ζ^4 - 56952/ζ^3 - 26347/ζ^2 + 40776/ζ + 40776*ζ - 26347*ζ^2 - 56952*ζ^3 - 35112*ζ^4 + 7666*ζ^5 + 22900*ζ^6 + 12912*ζ^7 + 396*ζ^8 - 4352*ζ^9 - 2186*ζ^10 - 130*ζ^11 + 200*ζ^12 + 80*ζ^13 + ζ^14)
+q^4(494216 - 12/ζ^16 - 84/ζ^15 + 219/ζ^14 + 2008/ζ^13 + 3488/ζ^12 - 1938/ζ^11 - 22916/ζ^10 - 39212/ζ^9 + 1992/ζ^8 + 99470/ζ^7 + 164675/ζ^6 + 50584/ζ^5 - 228000/ζ^4 - 368226/ζ^3 - 166554/ζ^2 + 257398/ζ + 257398*ζ - 166554*ζ^2 - 368226*ζ^3 - 228000*ζ^4 + 50584*ζ^5 + 164675*ζ^6 + 99470*ζ^7 + 1992*ζ^8 - 39212*ζ^9 - 22916*ζ^10 - 1938*ζ^11 + 3488*ζ^12 + 2008*ζ^13 + 219*ζ^14 - 84*ζ^15 - 12*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 -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	0
[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	1
[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	0
[26 -86 288]	92	0
[12 -26 64]	92	0
[12 -38 128]	92	0
[12 -122 1248]	92	0
[12 -134 1504]	92	0
[48 -337 2368]	95	0
[24 -337 4736]	95	3
[4 17 96]	95	0
[16 -81 416]	95	0
[12 -145 1760]	95	3
[60 145 352]	95	-1
[10 15 32]	95	0
[52 -261 1312]	103	0
[4 5 32]	103	0
[26 -261 2624]	103	-2
[56 -169 512]	111	0
[120 297 736]	111	0
[6 9 32]	111	0
[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	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 -212 2816]	112	-2
[2 3 64]	119	0
[30 -61 128]	119	12
[30 -451 6784]	119	0
[20 -61 192]	119	1
[90 221 544]	119	1
[12 -61 320]	119	1
[12 -125 1312]	119	-1
[2 2 64]	124	0
[62 -434 3040]	124	0
[4 2 32]	124	0
[32 -226 1600]	124	0
[16 -226 3200]	124	0
[8 34 160]	124	2
[28 -94 320]	124	0
[10 14 32]	124	0
[14 -46 160]	124	2
[14 -158 1792]	124	2
[2 1 64]	127	0
[4 1 32]	127	0
[32 -449 6304]	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	0
[4 32 288]	128	0
[6 8 32]	128	0
[22 -112 576]	128	0
[22 -200 1824]	128	-2
[22 -288 3776]	128	2
[18 -40 96]	128	-2
[18 -256 3648]	128	0
[12 16 32]	128	0
[12 -64 352]	128	-2
[68 -885 11520]	135	0
[34 -171 864]	135	0
[24 -267 2976]	135	0
[8 11 32]	135	0
[72 -793 8736]	143	0
[42 121 352]	143	0
[6 7 32]	143	0
[6 25 128]	143	0
[18 -199 2208]	143	0
[12 -25 64]	143	0
[76 -685 6176]	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	0
[40 -522 6816]	156	0
[6 6 32]	156	0
[8 10 32]	156	0
[20 -42 96]	156	4
[20 -202 2048]	156	2
[10 22 64]	156	0
[10 42 192]	156	-4
[30 -102 352]	156	-16
[16 -182 2080]	156	-4
[80 -561 3936]	159	0
[46 -143 448]	159	0
[28 -143 736]	159	0
[16 -49 160]	159	-2
[12 15 32]	159	0
[14 -143 1472]	159	2
[84 -421 2112]	167	0
[42 -379 3424]	167	0
[28 -421 6336]	167	0
[36 101 288]	167	-1
[22 -69 224]	167	1
[14 -155 1728]	167	-1
[14 -197 2784]	167	0
[16 -37 96]	167	1
[88 -265 800]	175	0
[4 9 64]	175	0
[8 9 32]	175	0
[20 -265 3520]	175	5
[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	-16
[10 12 32]	176	0
[18 -92 480]	176	-16
[18 -236 3104]	176	-2
[2 3 96]	183	0
[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	-12
[24 -338 4768]	188	-28
[8 18 64]	188	0
[112 274 672]	188	10
[12 14 32]	188	0
[12 50 224]	188	12
[32 -110 384]	188	18
[14 34 96]	188	12
[102 242 576]	188	10
[42 -146 512]	188	-12
[2 1 96]	191	0
[4 33 320]	191	0
[6 1 32]	191	0
[32 -417 5440]	191	-29
[8 33 160]	191	-32
[20 -223 2496]	191	-32
[18 -95 512]	191	29
[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	16
[24 -120 608]	192	-2
[26 -56 128]	192	-16
[26 -368 5216]	192	0
[12 24 64]	192	0
[14 16 32]	192	0
[14 -72 384]	192	16
[100 -1301 16928]	199	0
[50 -651 8480]	199	0
[26 -341 4480]	199	25
[10 11 32]	199	0
[10 21 64]	199	0
[16 -213 2848]	199	25
[104 -1145 12608]	207	0
[478 1113 2592]	207	0
[36 -327 2976]	207	24
[26 -391 5888]	207	0
[18 -39 96]	207	24
[108 -973 8768]	215	0
[54 -595 6560]	215	0
[36 -397 4384]	215	-1
[6 13 64]	215	0
[24 -77 256]	215	-20
[12 13 32]	215	0
[20 -205 2112]	215	5
[16 -179 2016]	215	20
[110 -330 992]	220	0
[110 -550 2752]	220	0
[4 6 64]	220	0
[56 -506 4576]	220	0
[28 -58 128]	220	-30
[28 -422 6368]	220	0
[10 10 32]	220	0
[170 390 896]	220	0
[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	121
[30 -453 6848]	231	0
[24 -123 640]	231	121
[40 -133 448]	231	113
[120 -361 1088]	239	0
[248 617 1536]	239	0
[48 -151 480]	239	32
[30 -151 768]	239	32
[24 -361 5440]	239	0
[10 41 192]	239	4
[20 -41 96]	239	-24
[198 457 1056]	239	0
[22 -247 2784]	239	4
[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	32
[30 -60 128]	240	90
[8 4 32]	240	0
[32 -100 320]	240	-42
[32 -356 3968]	240	64
[10 20 64]	240	0
[12 12 32]	240	0
[20 -100 512]	240	-42
[16 -36 96]	240	-32
[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	-204
[42 -210 1056]	252	-16
[8 2 32]	252	0
[32 -450 6336]	252	28
[24 -126 672]	252	-204
[14 14 32]	252	0
[18 -258 3712]	252	0
[2 1 128]	255	0
[4 1 64]	255	0
[44 -575 7520]	255	-88
[8 1 32]	255	0
[14 33 96]	255	-88
[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	238
[34 -480 6784]	256	0
[26 -80 256]	256	-46
[26 -288 3200]	256	238
[26 -392 5920]	256	0
[16 16 32]	256	0
[16 -48 160]	256	-46
[20 -48 128]	256	-58
[20 -288 4160]	256	0
[132 -1717 22336]	263	0
[66 -331 1664]	263	0
[408 949 2208]	263	0
[22 -309 4352]	263	-56
[22 -331 4992]	263	0
[24 -53 128]	263	152
[136 -1497 16480]	271	0
[74 217 640]	271	0
[34 -103 320]	271	39
[28 -423 6400]	271	0
[20 -103 544]	271	39
[212 487 1120]	271	0
[140 -1261 11360]	279	0
[70 -211 640]	279	0
[164 403 992]	279	195
[28 -365 4768]	279	-66
[268 621 1440]	279	0
[24 -339 4800]	279	62
[42 -141 480]	279	98
[20 -301 4544]	279	0
[18 -237 3136]	279	-66
[20 -109 608]	279	195
[142 -426 1280]	284	0
[142 -710 3552]	284	0
[72 -362 1824]	284	0
[72 -938 12224]	284	0
[58 166 480]	284	128
[450 1046 2432]	284	0
[6 26 160]	284	572
[58 -182 576]	284	-236
[36 -182 928]	284	-236
[36 -470 6144]	284	158
[30 -154 800]	284	-572
[30 -454 6880]	284	0
[24 -362 5472]	284	0
[240 554 1280]	284	0
[18 -38 96]	284	-128
[20 -106 576]	284	-158
[20 -266 3552]	284	-78
[144 -1009 7072]	287	0
[78 -239 736]	287	0
[48 -241 1216]	287	117
[338 785 1824]	287	0
[12 49 224]	287	-125
[26 -369 5248]	287	166
[24 -271 3072]	287	-125
[18 -271 4096]	287	0
[148 -741 3712]	295	0
[74 -667 6016]	295	0
[38 -421 4672]	295	-473
[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	-70
[26 -393 5952]	303	0
[2 4 160]	304	0
[154 460 1376]	304	0
[10 4 32]	304	0
[10 44 224]	304	96
[50 -164 544]	304	-570
[22 -244 2720]	304	96
[22 -332 5024]	304	0
[2 3 160]	311	0
[78 -547 3840]	311	0
[52 -157 480]	311	931
[218 541 1344]	311	290
[8 35 192]	311	-482
[10 3 32]	311	0
[26 -131 672]	311	482
[12 29 96]	311	-931
[16 35 96]	311	-290
[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
[380 882 2048]	316	0
[10 2 32]	316	0
[310 718 1664]	316	0
[20 -302 4576]	316	0
[22 -50 128]	316	-270
[2 1 160]	319	0
[50 -159 512]	319	1144
[10 1 32]	319	0
[32 -353 3904]	319	-1144
[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	-514
[54 -704 9184]	320	510
[10 0 32]	320	0
[10 40 192]	320	364
[28 -312 3488]	320	364
[28 -424 6432]	320	0
[14 32 96]	320	510
[254 584 1344]	320	0
[20 -40 96]	320	4
[18 -272 4128]	320	0
[164 -2133 27744]	327	0
[82 -1067 13888]	327	0
[56 -619 6848]	327	-1057
[42 -213 1088]	327	1057
[282 651 1504]	327	0
[24 -363 5504]	327	0
[168 -1849 20352]	335	0
[766 1785 4160]	335	0
[6 7 64]	335	0
[6 25 160]	335	-1900
[36 -185 960]	335	1900
[28 -57 128]	335	547
[30 -455 6912]	335	0
[172 -1549 13952]	343	0
[86 -947 10432]	343	0
[422 979 2272]	343	0
[22 -333 5056]	343	0
[174 -522 1568]	348	0
[174 -870 4352]	348	0
[4 6 96]	348	0
[88 -794 7168]	348	0
[6 6 64]	348	0
[44 -134 416]	348	-1474
[44 -486 5376]	348	2158
[12 6 32]	348	0
[26 -134 704]	348	-1474
[26 -394 5984]	348	0
[22 -310 4384]	348	-16
[268 614 1408]	348	0
[176 -1233 8640]	351	0
[4 17 160]	351	0
[60 -303 1536]	351	1495
[36 -111 352]	351	880
[352 815 1888]	351	0
[22 -111 576]	351	880
[20 -303 4608]	351	0
[20 -47 128]	351	768
[180 -901 4512]	359	0
[4 5 96]	359	0
[60 -901 13536]	359	0
[68 197 576]	359	-523
[30 -421 5920]	359	1531
[12 5 32]	359	0
[18 37 96]	359	-523
[184 -553 1664]	367	0
[376 937 2336]	367	0
[46 -599 7808]	367	-3064
[28 -425 6464]	367	0
[296 681 1568]	367	0
[26 -137 736]	367	3064
[2 4 192]	368	0
[186 556 1664]	368	0
[4 4 96]	368	0
[92 -276 832]	368	0
[92 -460 2304]	368	0
[92 -644 4512]	368	0
[62 -188 576]	368	-3096
[62 -684 7552]	368	3192
[62 -932 14016]	368	0
[8 12 64]	368	0
[48 -244 1248]	368	-3192
[48 -628 8224]	368	2474
[12 4 32]	368	0
[12 28 96]	368	3096
[324 748 1728]	368	0
[52 -172 576]	368	-1426
[24 -52 128]	368	-1118
[24 -340 4832]	368	-96
[24 -364 5536]	368	0
[18 44 128]	368	2474
[282 644 1472]	368	0
[2 3 192]	375	0
[94 -1411 21184]	375	0
[6 3 64]	375	0
[40 -445 4960]	375	-2688
[12 3 32]	375	0
[32 -451 6368]	375	-660
[2 2 192]	380	0
[190 -1330 9312]	380	0
[4 2 96]	380	0
[6 2 64]	380	0
[6 14 96]	380	2972
[270 670 1664]	380	-1440
[8 34 192]	380	3044
[60 -190 608]	380	-4204
[38 -114 352]	380	-1124
[38 -190 960]	380	-4204
[12 2 32]	380	0
[32 -162 832]	380	-3044
[16 34 96]	380	1440
[26 -370 5280]	380	-676
[22 -114 608]	380	-1124
[22 -334 5088]	380	0
[20 30 64]	380	0
[2 1 192]	383	0
[4 1 96]	383	0
[6 1 64]	383	0
[64 -833 10848]	383	-2257
[12 1 32]	383	0
[34 -481 6816]	383	-916
[14 31 96]	383	-2257
[2 0 192]	384	0
[2 8 224]	384	0
[4 0 96]	384	0
[4 16 160]	384	0
[6 0 64]	384	0
[6 24 160]	384	4798
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