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^(-2) + ζ^2)
  +(92 + 2/ζ^2 + 2*ζ^2)
  +q(ζ^(-6) + 114/ζ^4 + 114*ζ^4 + ζ^6)
• 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
  (20,2)	1

• From the singular part above, we compute that the valuation Ord(ψ) is -1.25, 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(1 + ζ^(-2) + ζ^2)
+(92 + 2/ζ^2 + 2*ζ^2)
+q(723120 + ζ^(-6) + 114/ζ^4 + 2048/ζ^3 - 115501/ζ^2 + 47104/ζ + 47104*ζ - 115501*ζ^2 + 2048*ζ^3 + 114*ζ^4 + ζ^6)

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	1
[6 8 24]	80	0
[14 -44 144]	80	-1
[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	-1
[4 6 32]	92	-2
[6 2 16]	92	1
[6 14 48]	92	-1
[12 -2 8]	92	3735
[8 10 24]	92	1
[12 -26 64]	92	0
[12 -38 128]	92	2
[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	0
[4 8 40]	96	2
[6 0 16]	96	0
[6 12 40]	96	2
[12 0 8]	96	-7832
[12 -24 56]	96	0
[14 -4 8]	96	-7832
[14 -32 80]	96	0
[10 12 24]	96	-2
[10 -32 112]	96	-2
[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	-88
[14 0 8]	112	219120
[8 4 16]	112	88
[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	178
[4 6 40]	124	289
[8 2 16]	124	-178
[16 -2 8]	124	1987817
[8 18 56]	124	111
[10 14 32]	124	-111
[14 -46 160]	124	-289
[4 1 32]	127	0
[8 1 16]	127	0
[16 -1 8]	127	0
[4 0 32]	128	-176
[4 8 48]	128	-400
[6 8 32]	128	-110
[22 -68 216]	128	0
[8 0 16]	128	176
[16 0 8]	128	-4421340
[8 8 24]	128	110
[8 16 48]	128	-224
[18 -4 8]	128	-4421340
[12 16 32]	128	224
[12 -40 144]	128	400
[4 5 40]	135	2048
[6 3 24]	135	0
[18 -3 8]	135	0
[8 11 32]	135	0
[10 5 16]	135	-2048
[12 -27 72]	135	0
[6 2 24]	140	0
[6 14 56]	140	312
[18 -2 8]	140	-29667528
[10 10 24]	140	-312
[4 9 56]	143	-2048
[6 1 24]	143	0
[6 7 32]	143	-2048
[18 -1 8]	143	0
[18 -55 176]	143	2048
[12 -25 64]	143	0
[14 -33 88]	143	0
[14 -47 168]	143	2048
[4 4 40]	144	-112172
[6 0 24]	144	0
[6 12 48]	144	-402
[18 0 8]	144	68380656
[20 -4 8]	144	68380656
[10 4 16]	144	112172
[10 24 72]	144	0
[12 12 24]	144	402
[38 -115 352]	151	43008
[20 -3 8]	151	0
[8 19 64]	151	2048
[10 3 16]	151	-43008
[10 13 32]	151	-2048
[4 2 40]	156	942579
[4 6 48]	156	936566
[6 6 32]	156	115797
[20 -2 8]	156	349920054
[8 10 32]	156	0
[20 -62 200]	156	-115797
[10 2 16]	156	-942579
[10 22 64]	156	-6013
[12 6 16]	156	-936566
[12 18 40]	156	6013
[4 1 40]	159	-45056
[6 9 40]	159	2048
[20 -1 8]	159	0
[8 17 56]	159	-2048
[10 1 16]	159	45056
[16 -49 160]	159	-2048
[12 15 32]	159	2048
[14 -31 80]	159	0
[4 0 40]	160	-1661396
[4 8 56]	160	-1649144
[20 0 8]	160	-840561392
[22 -4 8]	160	-840561392
[10 0 16]	160	1661396
[10 20 56]	160	12252
[14 20 40]	160	-12252
[14 -48 176]	160	1649144
[4 5 48]	167	288768
[28 -85 264]	167	-45056
[6 13 56]	167	0
[22 -3 8]	167	0
[22 -69 224]	167	45056
[14 -43 144]	167	0
[12 5 16]	167	-288768
[16 -37 96]	167	0
[22 -2 8]	172	-3328686240
[4 9 64]	175	-331776
[22 -1 8]	175	0
[8 9 32]	175	0
[10 15 40]	175	0
[14 7 16]	175	331776
[4 4 48]	176	-26224064
[6 8 40]	176	-944396
[30 -92 288]	176	-834612
[22 0 8]	176	8397392256
[8 4 24]	176	834612
[24 -4 8]	176	8397392256
[8 12 40]	176	-109784
[10 12 32]	176	109784
[18 -56 184]	176	944396
[12 4 16]	176	26224064
[12 28 80]	176	0
[46 -139 424]	183	-198656
[6 3 32]	183	0
[8 3 24]	183	0
[24 -3 8]	183	0
[12 3 16]	183	198656
[4 2 48]	188	102968844
[4 6 56]	188	103180558
[6 2 32]	188	830808
[6 14 64]	188	1762774
[8 2 24]	188	-830808
[24 -2 8]	188	25783833735
[8 18 64]	188	931966
[12 2 16]	188	-102968844
[12 14 32]	188	-931966
[16 -50 168]	188	-1762774
[12 26 72]	188	211714
[14 6 16]	188	-103180558
[14 34 96]	188	-211714
[4 1 48]	191	-88064
[6 1 32]	191	47104
[6 7 40]	191	-288768
[8 1 24]	191	-47104
[24 -1 8]	191	0
[20 -63 208]	191	288768
[10 23 72]	191	-184320
[12 1 16]	191	88064
[12 17 40]	191	184320
[4 0 48]	192	-155361888
[6 0 32]	192	-223764
[6 12 56]	192	-1868352
[8 0 24]	192	223764
[24 0 8]	192	-69053456880
[8 8 32]	192	0
[8 16 56]	192	-1644588
[26 -4 8]	192	-69053456880
[12 0 16]	192	155361888
[14 16 32]	192	1644588
[14 -44 152]	192	1868352
[16 8 16]	192	155797792
[16 24 48]	192	435904
[4 5 56]	199	-11851776
[26 -3 8]	199	0
[8 11 40]	199	229376
[10 11 32]	199	-229376
[10 21 64]	199	186368
[14 5 16]	199	11851776
[14 19 40]	199	-186368
[6 6 40]	204	26122736
[26 -2 8]	204	-162455777264
[10 14 40]	204	0
[22 -70 232]	204	-26122736
[6 9 48]	207	104448
[26 -1 8]	207	0
[26 -79 248]	207	0
[18 -57 192]	207	-104448
[16 7 16]	207	-12095488
[4 4 56]	208	-164253887
[26 0 8]	208	469539282740
[28 -4 8]	208	469539282740
[14 4 16]	208	164253887
[14 32 88]	208	0
[54 -163 496]	215	-42846208
[36 -109 336]	215	200704
[6 13 64]	215	229376
[28 -3 8]	215	0
[8 19 72]	215	102400
[10 5 24]	215	-200704
[12 13 32]	215	-102400
[12 29 88]	215	0
[14 3 16]	215	42846208
[16 -51 176]	215	-229376
[4 2 56]	220	-1049184849
[28 -2 8]	220	819333887680
[8 10 40]	220	24844713
[28 -86 272]	220	0
[10 10 32]	220	-24844713
[14 2 16]	220	1049184849
[16 6 16]	220	1054628610
[16 -38 104]	220	5443761
[4 1 56]	223	55031808
[28 -1 8]	223	0
[8 17 64]	223	-333824
[14 1 16]	223	-55031808
[14 15 32]	223	333824
[4 0 56]	224	2223814064
[6 8 48]	224	-127915936
[38 -116 360]	224	-103694122
[28 0 8]	224	-2627636861520
[30 -4 8]	224	-2627636861520
[10 4 24]	224	103694122
[10 24 80]	224	-24221814
[12 16 40]	224	24221814
[20 -64 216]	224	127915936
[14 0 16]	224	-2223814064
[18 -8 16]	224	-2235149740
[18 -44 120]	224	-11335676
[6 3 40]	231	-200704
[30 -3 8]	231	0
[30 -93 296]	231	0
[10 3 24]	231	200704
[10 13 40]	231	0
[12 27 80]	231	8015872
[14 21 48]	231	-8015872
[16 5 16]	231	-136972288
[6 2 40]	236	181382584
[6 14 72]	236	262956688
[30 -2 8]	236	-3154019426280
[10 2 24]	236	-181382584
[10 22 72]	236	81574104
[14 18 40]	236	-81574104
[18 -58 200]	236	-262956688
[6 1 40]	239	286720
[6 7 48]	239	274345984
[30 -1 8]	239	0
[8 9 40]	239	-262264832
[10 1 24]	239	-286720
[24 -73 232]	239	262264832
[22 -71 240]	239	-274345984
[18 -7 16]	239	93992960
[16 23 48]	239	8202240
[6 0 40]	240	-205734232
[30 0 8]	240	11871488696336
[8 4 32]	240	0
[32 -4 8]	240	11871488696336
[8 12 48]	240	-107260760
[10 0 24]	240	205734232
[12 12 32]	240	107260760
[16 4 16]	240	-15811259480
[16 20 40]	240	114923520
[16 -52 184]	240	320657752
[8 3 32]	247	0
[32 -3 8]	247	0
[16 3 16]	247	-1253093376
[6 6 48]	252	70705767
[8 2 32]	252	0
[32 -2 8]	252	7931907227145
[8 18 72]	252	241685163
[12 6 24]	252	-70705767
[12 18 48]	252	0
[14 14 32]	252	-241685163
[16 2 16]	252	33959217012
[18 -6 16]	252	33850337186
[6 9 56]	255	-1645350912
[8 1 32]	255	0
[32 -1 8]	255	0
[10 15 48]	255	254248960
[12 15 40]	255	-254248960
[20 -65 224]	255	1645350912
[14 33 96]	255	0
[16 1 16]	255	1401831424
[8 0 32]	256	0
[32 0 8]	256	-41108052986912
[8 8 40]	256	120380248
[34 -4 8]	256	-41108052986912
[10 12 40]	256	0
[26 -80 256]	256	-120380248
[16 0 16]	256	-38714894912
[16 16 32]	256	318306816
[20 -8 16]	256	-38485363904
[44 -133 408]	263	4311296000
[34 -3 8]	263	0
[8 11 48]	263	1633253376
[22 -67 216]	263	-1633253376
[12 5 24]	263	-4311296000
[18 -5 16]	263	-844242944
[18 -59 208]	263	-1370804224
[34 -2 8]	268	-2210677570160
[34 -1 8]	271	0
[34 -103 320]	271	-4530714624
[28 -87 280]	271	4530714624
[10 23 80]	271	-1625251840
[14 17 40]	271	1625251840
[20 -7 16]	271	2152323072
[16 39 112]	271	224012288
[6 8 56]	272	-2490331296
[46 -140 432]	272	-3138747106
[34 0 8]	272	92013087616560
[36 -4 8]	272	92013087616560
[12 4 24]	272	3138747106
[12 28 88]	272	648415810
[14 20 48]	272	-648415810
[22 -72 248]	272	2490331296
[18 -4 16]	272	145034798716
[6 3 48]	279	-34056996864
[36 -3 8]	279	0
[8 19 80]	279	-1637332992
[10 11 40]	279	0
[12 3 24]	279	34056996864
[20 -61 200]	279	1637332992
[18 -3 16]	279	16532129792
[16 19 40]	279	-1371234304
[18 27 56]	279	-232212480
[6 2 48]	284	-64868972488
[36 -2 8]	284	-95136385273191
[8 10 48]	284	3541470532
[36 -110 344]	284	4305653638
[10 14 48]	284	-764183106
[30 -94 304]	284	-4305653638
[12 2 24]	284	64868972488
[12 14 40]	284	764183106
[24 -74 240]	284	-3541470532
[18 -2 16]	284	-702749831405
[20 -6 16]	284	-700988187674
[16 22 48]	284	3186069548
[20 -66 232]	284	68055042036
[6 1 48]	287	29471250432
[6 7 56]	287	84229795840
[36 -1 8]	287	0
[12 1 24]	287	-29471250432
[12 17 48]	287	0
[14 7 24]	287	-84229795840
[18 -1 16]	287	-15769495552
[18 -55 184]	287	-266299392
[6 0 48]	288	136126334784
[36 0 8]	288	-4413649854240
[38 -4 8]	288	-4413649854240
[12 0 24]	288	-136126334784
[18 0 16]	288	1052405827256
[22 -8 16]	288	1048641685360
[18 24 48]	288	-5028726972
[18 -60 216]	288	-141155061756
[38 -3 8]	295	0
[38 -117 368]	295	29471354880
[10 5 32]	295	-29471354880
[20 -5 16]	295	44190197760
[6 6 56]	300	408556106496
[38 -2 8]	300	528629708975400
[10 10 40]	300	0
[14 6 24]	300	-408556106496
[14 34 104]	300	0
[38 -1 8]	303	0
[8 9 48]	303	-84366665728
[26 -81 264]	303	84366665728
[22 -7 16]	303	-62069323776
[22 -73 256]	303	529901398016
[38 0 8]	304	-1067427147614080
[8 4 40]	304	66824884232
[40 -4 8]	304	-1067427147614080
[8 12 56]	304	66561841516
[10 4 32]	304	-66824884232
[10 24 88]	304	-263042716
[14 16 40]	304	263042716
[22 -68 224]	304	-66561841516
[20 -4 16]	304	884535985984
[52 -157 480]	311	1116731277312
[8 3 40]	311	219416576
[40 -3 8]	311	0
[10 3 32]	311	-219416576
[10 13 48]	311	-45751738368
[12 13 40]	311	45751738368
[12 29 96]	311	50506418176
[14 5 24]	311	-1116731277312
[14 19 48]	311	-50506418176
[20 -3 16]	311	-106120534016
[20 -67 240]	311	-441360459776
[8 2 40]	316	-68182989960
[40 -2 8]	316	-1492793971437868
[10 2 32]	316	68182989960
[20 -2 16]	316	3786297193418
[16 18 40]	316	71654169098
[20 -62 208]	316	139837159058
[22 -6 16]	316	3762625151933
[8 1 40]	319	-24898023424
[40 -1 8]	319	0
[10 1 32]	319	24898023424
[32 -97 304]	319	0
[20 -1 16]	319	59672211456
[54 -164 504]	320	-2054578286240
[8 0 40]	320	-6185326396
[40 0 8]	320	5391275648088600
[8 8 48]	320	-424504293952
[42 -4 8]	320	5391275648088600
[10 0 32]	320	6185326396
[12 16 48]	320	0
[28 -88 288]	320	424504293952
[14 4 24]	320	2054578286240
[20 0 16]	320	-7864880960640
[16 8 24]	320	2050907667048
[24 -8 16]	320	-7813543640448
[16 24 56]	320	-3670619192
[18 20 40]	320	-145410591056
[18 -56 192]	320	-139225264660
[6 3 56]	327	-6199277322240
[42 -3 8]	327	0
[8 11 56]	327	576001376256
[14 3 24]	327	6199277322240
[24 -75 248]	327	-576001376256
[22 -5 16]	327	-621850126336
[16 21 48]	327	157182826496
[6 2 56]	332	-4802057520968
[42 -2 8]	332	1194664150575720
[14 2 24]	332	4802057520968
[18 26 56]	332	-14684767104
[22 -74 264]	332	4787372753864
[6 1 56]	335	4999961370624
[42 -1 8]	335	0
[42 -127 392]	335	-1116351117312
[10 15 56]	335	241317480448
[30 -95 312]	335	1116351117312
[14 1 24]	335	-4999961370624
[14 15 40]	335	-241317480448
[16 7 24]	335	-9682012921856
[24 -7 16]	335	742295767040
[18 23 48]	335	-106930759680
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