A Borcherds Product in S₄₆(K(4))⁻

properties |

c vector |

θ vector |

ψ expansion |

Borch(ψ) coefficients

• The singular part of ψ truncated up to q^(4/4) is
  q^-1(1 + ζ^(-4) + ζ^4)
  +(92 + 2/ζ^2 + 2*ζ^2)
  +q(138/ζ^4 + 138*ζ^4)
• From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block

  	φ	= TB(46;2,2)
  		= 22
  		= η(τ)92(θ(τ,2z)/η(τ))2

  This theta block has q-order of vanishing 4. This theta block has index 1*4, and so Borch(ψ) begins with Jacobi coefficient #1.
• 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 0 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)	2
  (4,2)	2
  (16,0)	1
  (32,4)	1

• From the singular part above, we compute that the valuation Ord(ψ) is -2, and hence we need to multiply ψ by Δ^3 to get a Jacobi cusp form. The definition of ψ is
      ψ = c · θ/ Δ^3.
  Here c is a vector of coefficients, θ is a basis of of J_36,4^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^(4/4) is

q^-1(1 + ζ^(-4) + ζ^4)
+(92 + 2/ζ^2 + 2*ζ^2)
+q(17502120 + 138/ζ^4 - 92160/ζ^3 + 2144576/ζ^2 - 10508288/ζ - 10508288*ζ + 2144576*ζ^2 - 92160*ζ^3 + 138*ζ^4)

2u	det(2u)	a(u, Borch(ψ))
[2 3 8]	7	0
[2 2 8]	12	0
[2 1 8]	15	0
[4 -9 24]	15	0
[2 0 8]	16	0
[2 4 16]	16	0
[4 4 8]	16	0
[2 3 16]	23	0
[6 -13 32]	23	0
[6 -19 64]	23	0
[2 2 16]	28	0
[4 2 8]	28	0
[4 6 16]	28	0
[2 1 16]	31	0
[4 1 8]	31	0
[8 -17 40]	31	0
[2 0 16]	32	0
[2 4 24]	32	0
[4 0 8]	32	0
[4 8 24]	32	0
[6 8 16]	32	0
[6 -20 72]	32	0
[2 3 24]	39	0
[4 5 16]	39	0
[10 -21 48]	39	0
[6 3 8]	39	0
[2 2 24]	44	0
[6 2 8]	44	0
[6 14 40]	44	0
[2 1 24]	47	0
[12 -25 56]	47	0
[4 9 32]	47	0
[6 1 8]	47	0
[6 7 16]	47	0
[2 0 24]	48	0
[2 4 32]	48	0
[4 4 16]	48	0
[6 0 8]	48	0
[6 12 32]	48	0
[8 4 8]	48	0
[8 12 24]	48	0
[2 3 32]	55	0
[14 -29 64]	55	0
[14 -43 136]	55	0
[8 3 8]	55	0
[2 2 32]	60	-1
[4 2 16]	60	0
[4 6 24]	60	1
[6 6 16]	60	-1
[8 2 8]	60	1
[8 -18 48]	60	0
[2 1 32]	63	0
[4 1 16]	63	0
[6 9 24]	63	0
[8 1 8]	63	0
[2 0 32]	64	2
[2 4 40]	64	2
[4 0 16]	64	0
[4 8 32]	64	-2
[8 0 8]	64	-2
[8 8 16]	64	2
[8 -16 40]	64	0
[10 -4 8]	64	-2
[10 -24 64]	64	0
[2 3 40]	71	0
[4 5 24]	71	0
[12 -37 120]	71	0
[6 13 40]	71	0
[10 -3 8]	71	0
[8 11 24]	71	0
[2 2 40]	76	88
[10 -2 8]	76	-88
[10 -22 56]	76	0
[2 1 40]	79	0
[4 9 40]	79	0
[10 -1 8]	79	0
[10 -31 104]	79	0
[2 0 40]	80	-180
[2 4 48]	80	-180
[4 4 24]	80	-2
[6 8 24]	80	0
[14 -44 144]	80	2
[10 0 8]	80	180
[10 -20 48]	80	0
[12 -4 8]	80	180
[12 -28 72]	80	0
[2 3 48]	87	0
[22 -67 208]	87	0
[6 3 16]	87	0
[12 -3 8]	87	0
[8 19 56]	87	0
[2 2 48]	92	-3735
[4 2 24]	92	2
[4 6 32]	92	-88
[6 2 16]	92	-2
[6 14 48]	92	-135
[12 -2 8]	92	3735
[8 10 24]	92	135
[12 -26 64]	92	0
[12 -38 128]	92	88
[2 1 48]	95	0
[4 1 24]	95	0
[6 1 16]	95	0
[6 7 24]	95	0
[12 -1 8]	95	0
[8 17 48]	95	0
[10 15 32]	95	0
[2 0 48]	96	7832
[2 4 56]	96	7832
[4 0 24]	96	-2
[4 8 40]	96	180
[6 0 16]	96	2
[6 12 40]	96	272
[12 0 8]	96	-7832
[12 -24 56]	96	0
[14 -4 8]	96	-7832
[14 -32 80]	96	0
[10 12 24]	96	-272
[10 -32 112]	96	-180
[2 3 56]	103	0
[4 5 32]	103	0
[14 -3 8]	103	0
[14 -45 152]	103	0
[2 2 56]	108	101640
[6 6 24]	108	0
[14 -2 8]	108	-101640
[14 -30 72]	108	0
[2 1 56]	111	0
[4 9 48]	111	0
[6 9 32]	111	92160
[14 -1 8]	111	0
[8 9 24]	111	-92160
[12 -39 136]	111	0
[10 23 64]	111	0
[2 0 56]	112	-219120
[4 4 32]	112	180
[14 0 8]	112	219120
[8 4 16]	112	-180
[16 -4 8]	112	219120
[8 12 32]	112	0
[16 -36 88]	112	0
[30 -91 280]	119	0
[20 -61 192]	119	0
[6 13 48]	119	-92160
[8 3 16]	119	0
[16 -3 8]	119	0
[10 11 24]	119	92160
[10 21 56]	119	0
[12 -29 80]	119	0
[4 2 32]	124	-88
[4 6 40]	124	3870
[8 2 16]	124	88
[16 -2 8]	124	1987817
[8 18 56]	124	8953
[10 14 32]	124	-8953
[14 -46 160]	124	-3870
[4 1 32]	127	0
[8 1 16]	127	0
[16 -1 8]	127	0
[4 0 32]	128	-4
[4 8 48]	128	-8104
[6 8 32]	128	-2144120
[22 -68 216]	128	0
[8 0 16]	128	4
[16 0 8]	128	-4421340
[8 8 24]	128	2144120
[8 16 48]	128	-18180
[18 -4 8]	128	-4421340
[12 16 32]	128	18180
[12 -40 144]	128	8104
[4 5 40]	135	-92160
[6 3 24]	135	0
[18 -3 8]	135	0
[8 11 32]	135	0
[10 5 16]	135	92160
[12 -27 72]	135	0
[6 2 24]	140	0
[6 14 56]	140	-13196352
[18 -2 8]	140	-29667528
[10 10 24]	140	13196352
[4 9 56]	143	92160
[6 1 24]	143	0
[6 7 32]	143	10323968
[18 -1 8]	143	0
[18 -55 176]	143	-10323968
[12 -25 64]	143	0
[14 -33 88]	143	0
[14 -47 168]	143	-92160
[4 4 40]	144	2136288
[6 0 24]	144	0
[6 12 48]	144	30680940
[18 0 8]	144	68380656
[20 -4 8]	144	68380656
[10 4 16]	144	-2136288
[10 24 72]	144	0
[12 12 24]	144	-30680940
[38 -115 352]	151	-10323968
[20 -3 8]	151	0
[8 19 64]	151	-2117632
[10 3 16]	151	10323968
[10 13 32]	151	2117632
[4 2 40]	156	13212837
[4 6 48]	156	13094712
[6 6 32]	156	-13209237
[20 -2 8]	156	349920054
[8 10 32]	156	0
[20 -62 200]	156	13209237
[10 2 16]	156	-13212837
[10 22 64]	156	-388710
[12 6 16]	156	-13094712
[12 18 40]	156	388710
[4 1 40]	159	10416128
[6 9 40]	159	390481920
[20 -1 8]	159	0
[8 17 56]	159	2209792
[10 1 16]	159	-10416128
[16 -49 160]	159	-390481920
[12 15 32]	159	-2209792
[14 -31 80]	159	0
[4 0 40]	160	-30706352
[4 8 56]	160	-30461820
[20 0 8]	160	-840561392
[22 -4 8]	160	-840561392
[10 0 16]	160	30706352
[10 20 56]	160	795872
[14 20 40]	160	-795872
[14 -48 176]	160	30461820
[4 5 48]	167	-390481920
[28 -85 264]	167	-10323968
[6 13 56]	167	-400805888
[22 -3 8]	167	0
[22 -69 224]	167	10323968
[14 -43 144]	167	400805888
[12 5 16]	167	390481920
[16 -37 96]	167	0
[22 -2 8]	172	-3328686240
[4 9 64]	175	400805888
[22 -1 8]	175	0
[8 9 32]	175	0
[10 15 40]	175	0
[14 7 16]	175	-400805888
[4 4 48]	176	2881384288
[6 8 40]	176	-2881165168
[30 -92 288]	176	28562412
[22 0 8]	176	8397392256
[8 4 24]	176	-28562412
[24 -4 8]	176	8397392256
[8 12 40]	176	1433546960
[10 12 32]	176	-1433546960
[18 -56 184]	176	2881165168
[12 4 16]	176	-2881384288
[12 28 80]	176	0
[46 -139 424]	183	-7918295040
[6 3 32]	183	-92160
[8 3 24]	183	92160
[24 -3 8]	183	0
[12 3 16]	183	7918295040
[4 2 48]	188	6542923936
[4 6 56]	188	6545819821
[6 2 32]	188	-26422072
[6 14 64]	188	-6543840957
[8 2 24]	188	26422072
[24 -2 8]	188	25783833735
[8 18 64]	188	-5923320920
[12 2 16]	188	-6542923936
[12 14 32]	188	5923320920
[16 -50 168]	188	6543840957
[12 26 72]	188	12423015
[14 6 16]	188	-6545819821
[14 34 96]	188	-12423015
[4 1 48]	191	8308684800
[6 1 32]	191	0
[6 7 40]	191	7918295040
[8 1 24]	191	0
[24 -1 8]	191	0
[20 -63 208]	191	-7918295040
[10 23 72]	191	-169758720
[12 1 16]	191	-8308684800
[12 17 40]	191	169758720
[4 0 48]	192	-18874789488
[6 0 32]	192	26426304
[6 12 56]	192	18876430728
[8 0 24]	192	-26426304
[24 0 8]	192	-69053456880
[8 8 32]	192	0
[8 16 56]	192	8975259504
[26 -4 8]	192	-69053456880
[12 0 16]	192	18874789488
[14 16 32]	192	-8975259504
[14 -44 152]	192	-18876430728
[16 8 16]	192	18880833888
[16 24 48]	192	25660080
[4 5 56]	199	-101899962368
[26 -3 8]	199	0
[8 11 40]	199	-43516815360
[10 11 32]	199	43516815360
[10 21 64]	199	167548928
[14 5 16]	199	101899962368
[14 19 40]	199	-167548928
[6 6 40]	204	-6543025576
[26 -2 8]	204	-162455777264
[10 14 40]	204	0
[22 -70 232]	204	6543025576
[6 9 48]	207	101902080000
[26 -1 8]	207	0
[26 -79 248]	207	0
[18 -57 192]	207	-101902080000
[16 7 16]	207	-109807841280
[4 4 56]	208	464078724382
[26 0 8]	208	469539282740
[28 -4 8]	208	469539282740
[14 4 16]	208	-464078724382
[14 32 88]	208	0
[54 -163 496]	215	-899134248960
[36 -109 336]	215	-8308684800
[6 13 64]	215	-109810051072
[28 -3 8]	215	0
[8 19 72]	215	186455164928
[10 5 24]	215	8308684800
[12 13 32]	215	-186455164928
[12 29 88]	215	0
[14 3 16]	215	899134248960
[16 -51 176]	215	109810051072
[4 2 56]	220	489151995035
[28 -2 8]	220	819333887680
[8 10 40]	220	627469518667
[28 -86 272]	220	0
[10 10 32]	220	-627469518667
[14 2 16]	220	-489151995035
[16 6 16]	220	-489085393928
[16 -38 104]	220	311570352
[4 1 56]	223	1000633313280
[28 -1 8]	223	0
[8 17 64]	223	-142928025600
[14 1 16]	223	-1000633313280
[14 15 32]	223	142928025600
[4 0 56]	224	-1919492162572
[6 8 48]	224	-465516692682
[38 -116 360]	224	18874781928
[28 0 8]	224	-2627636861520
[30 -4 8]	224	-2627636861520
[10 4 24]	224	-18874781928
[10 24 80]	224	31154311380
[12 16 40]	224	-31154311380
[20 -64 216]	224	465516692682
[14 0 16]	224	1919492162572
[18 -8 16]	224	1919352671608
[18 -44 120]	224	-649596384
[6 3 40]	231	-8308776960
[30 -3 8]	231	0
[30 -93 296]	231	0
[10 3 24]	231	8308776960
[10 13 40]	231	0
[12 27 80]	231	11617318912
[14 21 48]	231	-11617318912
[16 5 16]	231	5603252725760
[6 2 40]	236	-6556120376
[6 14 72]	236	-483191740536
[30 -2 8]	236	-3154019426280
[10 2 24]	236	6556120376
[10 22 72]	236	-195837232960
[14 18 40]	236	195837232960
[18 -58 200]	236	483191740536
[6 1 40]	239	8308776960
[6 7 48]	239	942651064320
[30 -1 8]	239	0
[8 9 40]	239	-3928277839872
[10 1 24]	239	-8308776960
[24 -73 232]	239	3928277839872
[22 -71 240]	239	-942651064320
[18 -7 16]	239	-6494067873792
[16 23 48]	239	11784960000
[6 0 40]	240	-5762549456
[30 0 8]	240	11871488696336
[8 4 32]	240	0
[32 -4 8]	240	11871488696336
[8 12 48]	240	-4575549122252
[10 0 24]	240	5762549456
[12 12 32]	240	4575549122252
[16 4 16]	240	-17651648351564
[16 20 40]	240	-332206544736
[16 -52 184]	240	-1910445792760
[8 3 32]	247	0
[32 -3 8]	247	0
[16 3 16]	247	24247415009280
[6 6 48]	252	-1116619516932
[8 2 32]	252	0
[32 -2 8]	252	7931907227145
[8 18 72]	252	10022972959263
[12 6 24]	252	1116619516932
[12 18 48]	252	0
[14 14 32]	252	-10022972959263
[16 2 16]	252	-6102834343104
[18 -6 16]	252	-6104318576319
[6 9 56]	255	5416797560832
[8 1 32]	255	0
[32 -1 8]	255	0
[10 15 48]	255	867510190080
[12 15 40]	255	-867510190080
[20 -65 224]	255	-5416797560832
[14 33 96]	255	0
[16 1 16]	255	-29740459180032
[8 0 32]	256	0
[32 0 8]	256	-41108052986912
[8 8 40]	256	12704128369424
[34 -4 8]	256	-41108052986912
[10 12 40]	256	0
[26 -80 256]	256	-12704128369424
[16 0 16]	256	48449400699200
[16 16 32]	256	12149813129472
[20 -8 16]	256	48452514692896
[44 -133 408]	263	2927642408960
[34 -3 8]	263	0
[8 11 48]	263	40655689515008
[22 -67 216]	263	-40655689515008
[12 5 24]	263	-2927642408960
[18 -5 16]	263	64203292385280
[18 -59 208]	263	6351139848192
[34 -2 8]	268	-2210677570160
[34 -1 8]	271	0
[34 -103 320]	271	-21312322498560
[28 -87 280]	271	21312322498560
[10 23 80]	271	-5327408486400
[14 17 40]	271	5327408486400
[20 -7 16]	271	-87441775720448
[16 39 112]	271	355022594048
[6 8 56]	272	-13076030848656
[46 -140 432]	272	-10783202440772
[34 0 8]	272	92013087616560
[36 -4 8]	272	92013087616560
[12 4 24]	272	10783202440772
[12 28 88]	272	-329917077244
[14 20 48]	272	329917077244
[22 -72 248]	272	13076030848656
[18 -4 16]	272	-105421887386848
[6 3 48]	279	20268172369920
[36 -3 8]	279	0
[8 19 80]	279	-87922830809088
[10 11 40]	279	0
[12 3 24]	279	-20268172369920
[20 -61 200]	279	87922830809088
[18 -3 16]	279	27241280919552
[16 19 40]	279	-4416771962880
[18 27 56]	279	-366809763840
[6 2 48]	284	-10764623689108
[36 -2 8]	284	-95136385273191
[8 10 48]	284	-214551339900528
[36 -110 344]	284	10902941216340
[10 14 48]	284	2393041342662
[30 -94 304]	284	-10902941216340
[12 2 24]	284	10764623689108
[12 14 40]	284	-2393041342662
[24 -74 240]	284	214551339900528
[18 -2 16]	284	86664017764470
[20 -6 16]	284	86694583154376
[16 22 48]	284	-10025770802800
[20 -66 232]	284	16126941226818
[6 1 48]	287	-24239967025152
[6 7 56]	287	-16408274505728
[36 -1 8]	287	0
[12 1 24]	287	24239967025152
[12 17 48]	287	0
[14 7 24]	287	16408274505728
[18 -1 16]	287	-84841088071680
[18 -55 184]	287	-51195408809984
[6 0 48]	288	46272950999292
[36 0 8]	288	-4413649854240
[38 -4 8]	288	-4413649854240
[12 0 24]	288	-46272950999292
[18 0 16]	288	45898766966400
[22 -8 16]	288	45834382946280
[18 24 48]	288	19341184108488
[18 -60 216]	288	-60601486465104
[38 -3 8]	295	0
[38 -117 368]	295	21210823342080
[10 5 32]	295	-21210823342080
[20 -5 16]	295	-618878989813760
[6 6 56]	300	208448475889896
[38 -2 8]	300	528629708975400
[10 10 40]	300	0
[14 6 24]	300	-208448475889896
[14 34 104]	300	0
[38 -1 8]	303	0
[8 9 48]	303	786733541992448
[26 -81 264]	303	-786733541992448
[22 -7 16]	303	622370883000320
[22 -73 256]	303	-152126292953088
[38 0 8]	304	-1067427147614080
[8 4 40]	304	-34032870673312
[40 -4 8]	304	-1067427147614080
[8 12 56]	304	489951114625260
[10 4 32]	304	34032870673312
[10 24 88]	304	165589934991952
[14 16 40]	304	-165589934991952
[22 -68 224]	304	-489951114625260
[20 -4 16]	304	2194805015909088
[52 -157 480]	311	-816474001172480
[8 3 40]	311	3029143683072
[40 -3 8]	311	0
[10 3 32]	311	-3029143683072
[10 13 48]	311	-306947134869504
[12 13 40]	311	306947134869504
[12 29 96]	311	-183462798346240
[14 5 24]	311	816474001172480
[14 19 48]	311	183462798346240
[20 -3 16]	311	-3064530964580352
[20 -67 240]	311	138637352079360
[8 2 40]	316	21654475244670
[40 -2 8]	316	-1492793971437868
[10 2 32]	316	-21654475244670
[20 -2 16]	316	696588472253160
[16 18 40]	316	942641433066355
[20 -62 208]	316	198660227060200
[22 -6 16]	316	696034721663533
[8 1 40]	319	1044152246272
[40 -1 8]	319	0
[10 1 32]	319	-1044152246272
[32 -97 304]	319	0
[20 -1 16]	319	3777243887702016
[54 -164 504]	320	1871579200696088
[8 0 40]	320	-23715154463404
[40 0 8]	320	5391275648088600
[8 8 48]	320	-1823129804436408
[42 -4 8]	320	5391275648088600
[10 0 32]	320	23715154463404
[12 16 48]	320	0
[28 -88 288]	320	1823129804436408
[14 4 24]	320	-1871579200696088
[20 0 16]	320	-6052097238462784
[16 8 24]	320	595404996884880
[24 -8 16]	320	-6050922245306200
[16 24 56]	320	64996564562048
[18 20 40]	320	-1550566090178224
[18 -56 192]	320	178887294943260
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