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(3)
  +(92 + 2/ζ^2 + 2*ζ^2)
  +q(158/ζ^4 + 158*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)	3

• 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_24,4^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(3)
+(92 + 2/ζ^2 + 2*ζ^2)
+q(198528 + 158/ζ^4 + 6144/ζ^3 + 48448/ζ^2 + 141312/ζ + 141312*ζ + 48448*ζ^2 + 6144*ζ^3 + 158*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	0
[6 6 16]	60	0
[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	0
[8 0 8]	64	-2
[8 8 16]	64	0
[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	0
[6 8 24]	80	0
[14 -44 144]	80	0
[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	3
[4 6 32]	92	0
[6 2 16]	92	-3
[6 14 48]	92	0
[12 -2 8]	92	3735
[8 10 24]	92	0
[12 -26 64]	92	0
[12 -38 128]	92	0
[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	-6
[4 8 40]	96	0
[6 0 16]	96	6
[6 12 40]	96	0
[12 0 8]	96	-7832
[12 -24 56]	96	0
[14 -4 8]	96	-7832
[14 -32 80]	96	0
[10 12 24]	96	0
[10 -32 112]	96	0
[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	0
[14 -1 8]	111	0
[8 9 24]	111	0
[12 -39 136]	111	0
[10 23 64]	111	0
[2 0 56]	112	-219120
[4 4 32]	112	-4
[14 0 8]	112	219120
[8 4 16]	112	4
[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	0
[8 3 16]	119	0
[16 -3 8]	119	0
[10 11 24]	119	0
[10 21 56]	119	0
[12 -29 80]	119	0
[4 2 32]	124	-176
[4 6 40]	124	157
[8 2 16]	124	176
[16 -2 8]	124	1987817
[8 18 56]	124	0
[10 14 32]	124	0
[14 -46 160]	124	-157
[4 1 32]	127	0
[8 1 16]	127	0
[16 -1 8]	127	0
[4 0 32]	128	360
[4 8 48]	128	-312
[6 8 32]	128	0
[22 -68 216]	128	0
[8 0 16]	128	-360
[16 0 8]	128	-4421340
[8 8 24]	128	0
[8 16 48]	128	0
[18 -4 8]	128	-4421340
[12 16 32]	128	0
[12 -40 144]	128	312
[4 5 40]	135	6144
[6 3 24]	135	0
[18 -3 8]	135	0
[8 11 32]	135	0
[10 5 16]	135	-6144
[12 -27 72]	135	0
[6 2 24]	140	0
[6 14 56]	140	0
[18 -2 8]	140	-29667528
[10 10 24]	140	0
[4 9 56]	143	-6144
[6 1 24]	143	0
[6 7 32]	143	0
[18 -1 8]	143	0
[18 -55 176]	143	0
[12 -25 64]	143	0
[14 -33 88]	143	0
[14 -47 168]	143	6144
[4 4 40]	144	48312
[6 0 24]	144	0
[6 12 48]	144	0
[18 0 8]	144	68380656
[20 -4 8]	144	68380656
[10 4 16]	144	-48312
[10 24 72]	144	0
[12 12 24]	144	0
[38 -115 352]	151	129024
[20 -3 8]	151	0
[8 19 64]	151	0
[10 3 16]	151	-129024
[10 13 32]	151	0
[4 2 40]	156	105255
[4 6 48]	156	87216
[6 6 32]	156	-466
[20 -2 8]	156	349920054
[8 10 32]	156	0
[20 -62 200]	156	466
[10 2 16]	156	-105255
[10 22 64]	156	0
[12 6 16]	156	-87216
[12 18 40]	156	0
[4 1 40]	159	-135168
[6 9 40]	159	0
[20 -1 8]	159	0
[8 17 56]	159	0
[10 1 16]	159	135168
[16 -49 160]	159	0
[12 15 32]	159	0
[14 -31 80]	159	0
[4 0 40]	160	-307448
[4 8 56]	160	-270692
[20 0 8]	160	-840561392
[22 -4 8]	160	-840561392
[10 0 16]	160	307448
[10 20 56]	160	0
[14 20 40]	160	0
[14 -48 176]	160	270692
[4 5 48]	167	866304
[28 -85 264]	167	-18432
[6 13 56]	167	0
[22 -3 8]	167	0
[22 -69 224]	167	18432
[14 -43 144]	167	0
[12 5 16]	167	-866304
[16 -37 96]	167	0
[22 -2 8]	172	-3328686240
[4 9 64]	175	-995328
[22 -1 8]	175	0
[8 9 32]	175	0
[10 15 40]	175	0
[14 7 16]	175	995328
[4 4 48]	176	1648800
[6 8 40]	176	624
[30 -92 288]	176	-144036
[22 0 8]	176	8397392256
[8 4 24]	176	144036
[24 -4 8]	176	8397392256
[8 12 40]	176	0
[10 12 32]	176	0
[18 -56 184]	176	-624
[12 4 16]	176	-1648800
[12 28 80]	176	0
[46 -139 424]	183	-595968
[6 3 32]	183	-387072
[8 3 24]	183	387072
[24 -3 8]	183	0
[12 3 16]	183	595968
[4 2 48]	188	-3501360
[4 6 56]	188	-2866218
[6 2 32]	188	-297090
[6 14 64]	188	-12159
[8 2 24]	188	297090
[24 -2 8]	188	25783833735
[8 18 64]	188	0
[12 2 16]	188	3501360
[12 14 32]	188	0
[16 -50 168]	188	12159
[12 26 72]	188	0
[14 6 16]	188	2866218
[14 34 96]	188	0
[4 1 48]	191	-264192
[6 1 32]	191	405504
[6 7 40]	191	12288
[8 1 24]	191	-405504
[24 -1 8]	191	0
[20 -63 208]	191	-12288
[10 23 72]	191	0
[12 1 16]	191	264192
[12 17 40]	191	0
[4 0 48]	192	3531312
[6 0 32]	192	883184
[6 12 56]	192	24008
[8 0 24]	192	-883184
[24 0 8]	192	-69053456880
[8 8 32]	192	0
[8 16 56]	192	0
[26 -4 8]	192	-69053456880
[12 0 16]	192	-3531312
[14 16 32]	192	0
[14 -44 152]	192	-24008
[16 8 16]	192	-2223600
[16 24 48]	192	0
[4 5 56]	199	-35555328
[26 -3 8]	199	0
[8 11 40]	199	0
[10 11 32]	199	0
[10 21 64]	199	0
[14 5 16]	199	35555328
[14 19 40]	199	0
[6 6 40]	204	-174696
[26 -2 8]	204	-162455777264
[10 14 40]	204	0
[22 -70 232]	204	174696
[6 9 48]	207	-811008
[26 -1 8]	207	0
[26 -79 248]	207	0
[18 -57 192]	207	811008
[16 7 16]	207	-36286464
[4 4 56]	208	-119073440
[26 0 8]	208	469539282740
[28 -4 8]	208	469539282740
[14 4 16]	208	119073440
[14 32 88]	208	0
[54 -163 496]	215	-128538624
[36 -109 336]	215	-1732608
[6 13 64]	215	817152
[28 -3 8]	215	0
[8 19 72]	215	0
[10 5 24]	215	1732608
[12 13 32]	215	0
[12 29 88]	215	0
[14 3 16]	215	128538624
[16 -51 176]	215	-817152
[4 2 56]	220	24113139
[28 -2 8]	220	819333887680
[8 10 40]	220	35692
[28 -86 272]	220	0
[10 10 32]	220	-35692
[14 2 16]	220	-24113139
[16 6 16]	220	-7781856
[16 -38 104]	220	0
[4 1 56]	223	165095424
[28 -1 8]	223	0
[8 17 64]	223	0
[14 1 16]	223	-165095424
[14 15 32]	223	0
[4 0 56]	224	196194108
[6 8 48]	224	-19554798
[38 -116 360]	224	-3321096
[28 0 8]	224	-2627636861520
[30 -4 8]	224	-2627636861520
[10 4 24]	224	3321096
[10 24 80]	224	0
[12 16 40]	224	0
[20 -64 216]	224	19554798
[14 0 16]	224	-196194108
[18 -8 16]	224	-230201136
[18 -44 120]	224	0
[6 3 40]	231	1191936
[30 -3 8]	231	0
[30 -93 296]	231	0
[10 3 24]	231	-1191936
[10 13 40]	231	0
[12 27 80]	231	0
[14 21 48]	231	0
[16 5 16]	231	-410916864
[6 2 40]	236	6697800
[6 14 72]	236	-9365688
[30 -2 8]	236	-3154019426280
[10 2 24]	236	-6697800
[10 22 72]	236	0
[14 18 40]	236	0
[18 -58 200]	236	9365688
[6 1 40]	239	528384
[6 7 48]	239	-223647744
[30 -1 8]	239	0
[8 9 40]	239	2463744
[10 1 24]	239	-528384
[24 -73 232]	239	-2463744
[22 -71 240]	239	223647744
[18 -7 16]	239	281978880
[16 23 48]	239	0
[6 0 40]	240	-6405264
[30 0 8]	240	11871488696336
[8 4 32]	240	0
[32 -4 8]	240	11871488696336
[8 12 48]	240	-48004
[10 0 24]	240	6405264
[12 12 32]	240	48004
[16 4 16]	240	-2356384260
[16 20 40]	240	0
[16 -52 184]	240	-58478424
[8 3 32]	247	0
[32 -3 8]	247	0
[16 3 16]	247	-3759280128
[6 6 48]	252	-1390587516
[8 2 32]	252	0
[32 -2 8]	252	7931907227145
[8 18 72]	252	619201
[12 6 24]	252	1390587516
[12 18 48]	252	0
[14 14 32]	252	-619201
[16 2 16]	252	-858293616
[18 -6 16]	252	-1184933094
[6 9 56]	255	-703236096
[8 1 32]	255	0
[32 -1 8]	255	0
[10 15 48]	255	0
[12 15 40]	255	0
[20 -65 224]	255	703236096
[14 33 96]	255	0
[16 1 16]	255	4205494272
[8 0 32]	256	0
[32 0 8]	256	-41108052986912
[8 8 40]	256	60017840
[34 -4 8]	256	-41108052986912
[10 12 40]	256	0
[26 -80 256]	256	-60017840
[16 0 16]	256	6449541088
[16 16 32]	256	1214720
[20 -8 16]	256	7138134112
[44 -133 408]	263	-5048985600
[34 -3 8]	263	0
[8 11 48]	263	-1652736
[22 -67 216]	263	1652736
[12 5 24]	263	5048985600
[18 -5 16]	263	-2532728832
[18 -59 208]	263	-929003520
[34 -2 8]	268	-2210677570160
[34 -1 8]	271	0
[34 -103 320]	271	675919872
[28 -87 280]	271	-675919872
[10 23 80]	271	0
[14 17 40]	271	0
[20 -7 16]	271	6456969216
[16 39 112]	271	0
[6 8 56]	272	-10159424016
[46 -140 432]	272	-10982075328
[34 0 8]	272	92013087616560
[36 -4 8]	272	92013087616560
[12 4 24]	272	10982075328
[12 28 88]	272	0
[14 20 48]	272	0
[22 -72 248]	272	10159424016
[18 -4 16]	272	18702272424
[6 3 48]	279	-12939982848
[36 -3 8]	279	0
[8 19 80]	279	52568064
[10 11 40]	279	0
[12 3 24]	279	12939982848
[20 -61 200]	279	-52568064
[18 -3 16]	279	49596389376
[16 19 40]	279	0
[18 27 56]	279	0
[6 2 48]	284	-2111544033
[36 -2 8]	284	-95136385273191
[8 10 48]	284	18469728
[36 -110 344]	284	4186119783
[10 14 48]	284	-1796829
[30 -94 304]	284	-4186119783
[12 2 24]	284	2111544033
[12 14 40]	284	1796829
[24 -74 240]	284	-18469728
[18 -2 16]	284	25420847559
[20 -6 16]	284	30705778752
[16 22 48]	284	0
[20 -66 232]	284	1773054687
[6 1 48]	287	18213427200
[6 7 56]	287	-70872705024
[36 -1 8]	287	0
[12 1 24]	287	-18213427200
[12 17 48]	287	0
[14 7 24]	287	70872705024
[18 -1 16]	287	-47308486656
[18 -55 184]	287	53397504
[6 0 48]	288	29007547668
[36 0 8]	288	-4413649854240
[38 -4 8]	288	-4413649854240
[12 0 24]	288	-29007547668
[18 0 16]	288	-85421607312
[22 -8 16]	288	-96714033000
[18 24 48]	288	0
[18 -60 216]	288	-26653368816
[38 -3 8]	295	0
[38 -117 368]	295	15324733440
[10 5 32]	295	-15324733440
[20 -5 16]	295	132570593280
[6 6 56]	300	-286285772760
[38 -2 8]	300	528629708975400
[10 10 40]	300	0
[14 6 24]	300	286285772760
[14 34 104]	300	0
[38 -1 8]	303	0
[8 9 48]	303	1409458176
[26 -81 264]	303	-1409458176
[22 -7 16]	303	-186207971328
[22 -73 256]	303	130105755648
[38 0 8]	304	-1067427147614080
[8 4 40]	304	33543127024
[40 -4 8]	304	-1067427147614080
[8 12 56]	304	1849134236
[10 4 32]	304	-33543127024
[10 24 88]	304	2427568
[14 16 40]	304	-2427568
[22 -68 224]	304	-1849134236
[20 -4 16]	304	54956926240
[52 -157 480]	311	-720933365760
[8 3 40]	311	39462641664
[40 -3 8]	311	0
[10 3 32]	311	-39462641664
[10 13 48]	311	-161789952
[12 13 40]	311	161789952
[12 29 96]	311	0
[14 5 24]	311	720933365760
[14 19 48]	311	0
[20 -3 16]	311	-318361602048
[20 -67 240]	311	-206025848832
[8 2 40]	316	6227981123
[40 -2 8]	316	-1492793971437868
[10 2 32]	316	-6227981123
[20 -2 16]	316	-344967769360
[16 18 40]	316	23321136
[20 -62 208]	316	-446671856
[22 -6 16]	316	-415983893815
[8 1 40]	319	-55465758720
[40 -1 8]	319	0
[10 1 32]	319	55465758720
[32 -97 304]	319	0
[20 -1 16]	319	179016634368
[54 -164 504]	320	-1131342481704
[8 0 40]	320	-88034562924
[40 0 8]	320	5391275648088600
[8 8 48]	320	20312177952
[42 -4 8]	320	5391275648088600
[10 0 32]	320	88034562924
[12 16 48]	320	0
[28 -88 288]	320	-20312177952
[14 4 24]	320	1131342481704
[20 0 16]	320	504352721472
[16 8 24]	320	1157015890896
[24 -8 16]	320	658364682048
[16 24 56]	320	0
[18 20 40]	320	-45453744
[18 -56 192]	320	4748875740
[6 3 56]	327	-929174999040
[42 -3 8]	327	0
[8 11 56]	327	35132033024
[14 3 24]	327	929174999040
[24 -75 248]	327	-35132033024
[22 -5 16]	327	-1865550379008
[16 21 48]	327	0
[6 2 56]	332	216174900240
[42 -2 8]	332	1194664150575720
[14 2 24]	332	-216174900240
[18 26 56]	332	0
[22 -74 264]	332	-215207880720
[6 1 56]	335	1721683488768
[42 -1 8]	335	0
[42 -127 392]	335	141636550656
[10 15 56]	335	109228032
[30 -95 312]	335	-141636550656
[14 1 24]	335	-1721683488768
[14 15 40]	335	-109228032
[16 7 24]	335	5200612442112
[24 -7 16]	335	2226887301120
[18 23 48]	335	0
[6 0 56]	336	2423244239840
[42 0 8]	336	-14406718985411544
[44 -4 8]	336	-14406718985411544
[10 12 48]	336	-5840133012
[34 -104 328]	336	0
[12 12 40]	336	5840133012
[14 0 24]	336	-2423244239840
[22 -4 16]	336	-2862794148960
[20 28 56]	336	0
[20 -68 248]	336	-2478136554560
[44 -3 8]	343	0
[14 21 56]	343	0
[22 -3 16]	343	-70276663296
[22 -69 232]	343	-71479144448
[44 -2 8]	348	9568647859470645
[8 10 56]	348	422151936821
[44 -134 416]	348	572547793392
[12 6 32]	348	-572547793392
[12 18 56]	348	66870665
[14 18 48]	348	-66870665
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