A Borcherds Product in S₁₃(K(8))⁺

properties |

c vector |

θ vector |

ψ expansion |

Borch(ψ) coefficients

• The singular part of ψ truncated up to q^(8/4) is
  q^-1(1)
  +(26 + 3/ζ^2 + 20/ζ + 20*ζ + 3*ζ^2)
  +q(ζ^(-6) + ζ^6)
  +q^2(14/ζ^8 + 14*ζ^8)
• From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block

  	φ	= TB(13;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2)
  		= 12023
  		= η(τ)26(θ(τ,z)/η(τ))20(θ(τ,2z)/η(τ))3

  This theta block has q-order of vanishing 3. This theta block has index 2*8, and so Borch(ψ) begins with Jacobi coefficient #2.
• From the truncated singular part of ψ, we compute that Borch(ψ) is an antisymmetric 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 -1 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
  (4,6)	1
  (32,0)	1

• From the singular part above, we compute that the valuation Ord(ψ) is -1, and hence we need to multiply ψ by Δ^2 to get a Jacobi cusp form. The definition of ψ is
      ψ = c · θ/ Δ^2.
  Here c is a vector of coefficients, θ is a basis of of J_24,8^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^(8/4) is

q^-1(1)
+(26 + 3/ζ^2 + 20/ζ + 20*ζ + 3*ζ^2)
+q(46780 + ζ^(-6) + 236/ζ^5 + 2284/ζ^4 + 9432/ζ^3 + 23615/ζ^2 + 39484/ζ + 39484*ζ + 23615*ζ^2 + 9432*ζ^3 + 2284*ζ^4 + 236*ζ^5 + ζ^6)
+q^2(4357924 + 14/ζ^8 + 1812/ζ^7 + 23485/ζ^6 + 142512/ζ^5 + 532064/ζ^4 + 1379956/ζ^3 + 2638403/ζ^2 + 3849672/ζ + 3849672*ζ + 2638403*ζ^2 + 1379956*ζ^3 + 532064*ζ^4 + 142512*ζ^5 + 23485*ζ^6 + 1812*ζ^7 + 14*ζ^8)

2u	det(2u)	a(u, Borch(ψ))
[4 -21 112]	7	0
[8 -25 80]	15	0
[4 -25 160]	15	0
[2 4 16]	16	0
[2 3 16]	23	0
[6 -13 32]	23	1
[6 -19 64]	23	1
[2 2 16]	28	0
[14 -42 128]	28	0
[4 6 16]	28	0
[8 -42 224]	28	0
[2 1 16]	31	0
[8 -49 304]	31	3
[4 17 80]	31	3
[2 0 16]	32	0
[2 8 48]	32	0
[6 8 16]	32	0
[6 -32 176]	32	0
[20 -101 512]	39	0
[4 5 16]	39	0
[10 -21 48]	39	-11
[8 -27 96]	39	-11
[24 -73 224]	47	0
[4 9 32]	47	0
[6 7 16]	47	0
[2 4 32]	48	0
[4 4 16]	48	0
[12 -36 112]	48	-20
[6 12 32]	48	20
[2 3 32]	55	0
[14 -99 704]	55	0
[2 2 32]	60	0
[30 -90 272]	60	0
[4 2 16]	60	0
[16 -50 160]	60	40
[6 6 16]	60	0
[10 -30 96]	60	-216
[8 -18 48]	60	-216
[8 -50 320]	60	-40
[2 1 32]	63	0
[4 1 16]	63	0
[12 -63 336]	63	0
[2 0 32]	64	0
[2 8 64]	64	0
[4 0 16]	64	0
[4 8 32]	64	0
[4 16 80]	64	0
[8 8 16]	64	0
[10 -64 416]	64	0
[36 -181 912]	71	0
[18 -91 464]	71	-187
[12 -85 608]	71	0
[20 53 144]	71	187
[10 -53 288]	71	0
[40 -121 368]	79	0
[26 73 208]	79	0
[10 -71 512]	79	0
[2 4 48]	80	0
[14 -44 144]	80	-864
[14 -100 720]	80	0
[10 -20 48]	80	-864
[2 3 48]	87	0
[22 -67 208]	87	-187
[6 3 16]	87	0
[26 67 176]	87	-187
[2 2 48]	92	0
[46 -138 416]	92	0
[4 6 32]	92	0
[24 -122 624]	92	1080
[6 2 16]	92	0
[6 14 48]	92	0
[26 70 192]	92	-1080
[12 -86 624]	92	0
[2 1 48]	95	0
[4 17 96]	95	374
[6 1 16]	95	0
[16 -81 416]	95	-2255
[8 17 48]	95	-2255
[10 15 32]	95	374
[2 0 48]	96	0
[2 8 80]	96	0
[6 0 16]	96	0
[10 -72 528]	96	0
[52 -261 1312]	103	0
[4 5 32]	103	0
[14 -101 736]	103	0
[56 -169 512]	111	0
[4 9 48]	111	4278
[6 9 32]	111	-4278
[12 -87 640]	111	0
[2 4 64]	112	0
[4 4 32]	112	0
[28 -84 256]	112	0
[8 4 16]	112	0
[8 12 32]	112	0
[16 -84 448]	112	0
[2 3 64]	119	0
[30 -211 1488]	119	0
[20 -61 192]	119	7728
[6 13 48]	119	0
[8 3 16]	119	0
[24 61 160]	119	7728
[2 2 64]	124	0
[62 -186 560]	124	0
[4 2 32]	124	0
[32 -98 304]	124	3240
[8 2 16]	124	0
[10 14 32]	124	-3240
[14 -102 752]	124	0
[2 1 64]	127	0
[4 1 32]	127	0
[8 1 16]	127	0
[2 0 64]	128	0
[2 8 96]	128	0
[4 0 32]	128	0
[4 8 48]	128	-12144
[4 16 96]	128	0
[6 8 32]	128	12144
[22 -112 576]	128	6864
[8 0 16]	128	0
[8 16 48]	128	6864
[12 16 32]	128	0
[12 -88 656]	128	0
[68 -341 1712]	135	0
[34 -171 864]	135	-4301
[8 11 32]	135	-4301
[10 5 16]	135	0
[72 -217 656]	143	0
[42 121 352]	143	-24794
[6 7 32]	143	-24794
[14 -103 768]	143	0
[2 4 80]	144	0
[6 12 48]	144	0
[10 4 16]	144	0
[2 3 80]	151	0
[38 -115 352]	151	-12903
[8 19 64]	151	0
[10 3 16]	151	0
[10 13 32]	151	12903
[2 2 80]	156	0
[78 -234 704]	156	0
[4 6 48]	156	-34408
[40 -202 1024]	156	24728
[6 6 32]	156	34408
[26 -78 240]	156	-15488
[8 10 32]	156	24728
[10 2 16]	156	0
[30 78 208]	156	-15488
[12 6 16]	156	0
[2 1 80]	159	0
[4 17 112]	159	12742
[28 -143 736]	159	11268
[30 81 224]	159	-11268
[10 1 16]	159	0
[12 15 32]	159	12742
[2 0 80]	160	0
[2 8 112]	160	0
[10 0 16]	160	0
[14 -104 784]	160	0
[84 -421 2112]	167	0
[4 5 48]	167	24035
[28 -197 1392]	167	24035
[36 101 288]	167	0
[12 5 16]	167	0
[88 -265 800]	175	0
[4 9 64]	175	78309
[8 9 32]	175	-78309
[14 7 16]	175	0
[2 4 96]	176	0
[4 4 48]	176	19228
[44 -132 400]	176	25168
[30 -212 1504]	176	19228
[10 12 32]	176	-25168
[12 4 16]	176	0
[2 3 96]	183	0
[46 -323 2272]	183	81719
[6 3 32]	183	81719
[12 3 16]	183	0
[2 2 96]	188	0
[94 -282 848]	188	0
[4 2 48]	188	118864
[48 -146 448]	188	72424
[6 2 32]	188	-118864
[6 14 64]	188	137808
[42 118 336]	188	0
[8 18 64]	188	0
[36 98 272]	188	137808
[12 2 16]	188	0
[12 14 32]	188	-72424
[14 6 16]	188	0
[2 1 96]	191	0
[4 1 48]	191	-89148
[6 1 32]	191	89148
[32 -161 816]	191	59409
[36 95 256]	191	-59409
[12 1 16]	191	0
[2 0 96]	192	0
[4 0 48]	192	0
[4 8 64]	192	-159232
[6 0 32]	192	0
[8 8 32]	192	159232
[12 0 16]	192	0
[14 16 32]	192	0
[16 8 16]	192	0
[100 -501 2512]	199	0
[50 -251 1264]	199	3927
[10 11 32]	199	3927
[14 5 16]	199	0
[58 169 496]	207	-201457
[6 9 48]	207	0
[26 -183 1296]	207	201457
[16 7 16]	207	0
[2 4 112]	208	0
[14 4 16]	208	0
[2 3 112]	215	0
[54 -163 496]	215	-202279
[6 13 64]	215	-412236
[42 115 320]	215	-412236
[12 13 32]	215	202279
[14 3 16]	215	0
[2 2 112]	220	0
[4 6 64]	220	-103224
[56 -282 1424]	220	103224
[28 -198 1408]	220	-103224
[10 10 32]	220	103224
[14 2 16]	220	0
[16 6 16]	220	0
[2 1 112]	223	0
[8 17 64]	223	0
[14 1 16]	223	0
[14 15 32]	223	144837
[2 0 112]	224	0
[6 8 48]	224	0
[38 -192 976]	224	-338688
[42 112 304]	224	338688
[14 0 16]	224	0
[18 -8 16]	224	0
[4 5 64]	231	-150535
[30 -213 1520]	231	-150535
[16 5 16]	231	0
[4 9 80]	239	249777
[6 7 48]	239	0
[24 -169 1200]	239	249777
[18 -7 16]	239	0
[4 4 64]	240	400384
[60 -180 544]	240	318464
[6 12 64]	240	553248
[8 4 32]	240	-400384
[8 12 48]	240	-553248
[12 12 32]	240	-318464
[16 4 16]	240	0
[62 -435 3056]	247	441807
[8 3 32]	247	441807
[16 3 16]	247	0
[4 2 64]	252	269192
[6 6 48]	252	0
[8 2 32]	252	-269192
[14 14 32]	252	-481712
[16 2 16]	252	0
[18 -6 16]	252	0
[4 1 64]	255	-81719
[44 -223 1136]	255	525943
[8 1 32]	255	81719
[10 15 48]	255	525943
[16 1 16]	255	0
[4 0 64]	256	0
[4 8 80]	256	-211008
[8 0 32]	256	0
[8 16 64]	256	0
[26 -184 1312]	256	-211008
[16 0 16]	256	0
[16 16 32]	256	0
[66 -331 1664]	263	283492
[44 -309 2176]	263	0
[52 149 432]	263	202785
[8 11 48]	263	202785
[22 -155 1104]	263	-283492
[18 -5 16]	263	0
[74 217 640]	271	167739
[28 -199 1424]	271	-167739
[46 -324 2288]	272	0
[18 -4 16]	272	0
[6 3 48]	279	0
[20 -141 1008]	279	-690072
[18 -3 16]	279	0
[4 6 80]	284	567528
[72 -362 1824]	284	-266208
[6 2 48]	284	0
[6 14 80]	284	-159616
[58 166 480]	284	-80792
[8 10 48]	284	-80792
[10 14 48]	284	159616
[30 -214 1536]	284	567528
[24 -170 1216]	284	266208
[18 -2 16]	284	0
[6 1 48]	287	0
[46 129 368]	287	0
[18 -1 16]	287	0
[18 -127 912]	287	-338261
[6 0 48]	288	0
[18 0 16]	288	0
[4 5 80]	295	-420555
[10 5 32]	295	420555
[4 9 96]	303	-761464
[6 9 64]	303	-1768041
[8 9 48]	303	1768041
[26 -185 1328]	303	-761464
[4 4 80]	304	-206448
[10 4 32]	304	206448
[22 -156 1120]	304	345972
[78 -547 3840]	311	-496869
[6 13 80]	311	1234052
[10 3 32]	311	-496869
[10 13 48]	311	-1234052
[4 2 80]	316	-103224
[52 146 416]	316	0
[10 2 32]	316	103224
[20 -142 1024]	316	-245784
[4 1 80]	319	-245157
[10 1 32]	319	245157
[4 0 80]	320	0
[4 8 96]	320	1601536
[6 8 64]	320	4239776
[8 8 48]	320	-4239776
[10 0 32]	320	0
[28 -200 1440]	320	1601536
[18 -128 928]	320	0
[24 -171 1232]	327	-64141
[90 265 784]	335	1472552
[6 7 64]	335	-2930877
[42 -295 2080]	335	-2930877
[30 -215 1552]	335	-1472552
[6 12 80]	336	-465696
[10 12 48]	336	465696
[58 163 464]	343	0
[22 -157 1136]	343	2619309
[4 6 96]	348	-197472
[6 6 64]	348	-2872584
[44 -310 2192]	348	-2872584
[12 6 32]	348	197472
[26 -186 1344]	348	1096568
[20 -143 1040]	351	2020909
[4 5 96]	359	1787060
[60 -421 2960]	359	-5831595
[68 197 576]	359	1000285
[46 -325 2304]	359	5831595
[10 11 48]	359	1000285
[12 5 32]	359	-1787060
[28 -201 1456]	367	-2595987
[4 4 96]	368	-2192384
[62 -436 3072]	368	2378592
[8 4 48]	368	2378592
[8 12 64]	368	0
[12 4 32]	368	2192384
[24 -172 1248]	368	-4404224
[94 -659 4624]	375	-3122802
[6 3 64]	375	1453761
[8 3 48]	375	-1453761
[12 3 32]	375	-3122802
[4 2 96]	380	-5177392
[6 2 64]	380	-8215416
[74 214 624]	380	2134528
[8 2 48]	380	8215416
[10 10 48]	380	2134528
[12 2 32]	380	5177392
[22 -158 1152]	380	-6959392
[4 1 96]	383	4790164
[6 1 64]	383	10629657
[8 1 48]	383	-10629657
[12 1 32]	383	-4790164
[4 0 96]	384	0
[6 0 64]	384	0
[8 0 48]	384	0
[12 0 32]	384	0
[30 -216 1568]	384	2189088
[8 11 64]	391	0
[26 -187 1360]	391	2696727
[6 9 80]	399	6242572
[40 -281 1984]	399	6242572
[14 7 32]	399	829311
[24 -173 1264]	407	8369999
[8 10 64]	412	0
[14 6 32]	412	-853944
[28 -202 1472]	412	-412896
[6 8 80]	416	1114624
[42 -296 2096]	416	1114624
[14 5 32]	423	-1366295
[6 7 80]	431	-12951356
[8 9 64]	431	0
[44 -311 2208]	431	-12951356
[30 -217 1584]	431	937244
[14 4 32]	432	8867072
[26 -188 1376]	432	-3517712
[14 3 32]	439	-12466641
[6 6 80]	444	6818304
[46 -326 2320]	444	6818304
[14 2 32]	444	4485736
[14 1 32]	447	3433463
[8 8 64]	448	0
[14 0 32]	448	0
[16 8 32]	448	0
[76 -533 3744]	455	-6090966
[10 5 48]	455	-6090966
[28 -203 1488]	455	2504355
[58 -407 2864]	463	0
[16 7 32]	463	2116092
[78 -548 3856]	464	-3060288
[10 4 48]	464	-3060288
[6 3 80]	471	-10142852
[10 3 48]	471	10142852
[6 2 80]	476	-19011328
[60 -422 2976]	476	0
[10 2 48]	476	19011328
[16 6 32]	476	3225376
[30 -218 1600]	476	-7466536
[6 1 80]	479	38163026
[10 1 48]	479	-38163026
[6 0 80]	480	0
[10 0 48]	480	0
[62 -437 3088]	487	0
[16 5 32]	487	-5862450
[18 9 32]	495	-7064398
[8 4 64]	496	0
[16 4 32]	496	-13212672