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^(-1) + ζ)
  +(92 + 2/ζ^2 + 2*ζ^2)
  +q(18/ζ^4 + 18*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
  (17,1)	1

• From the singular part above, we compute that the valuation Ord(ψ) is -1.0625, 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^(-1) + ζ)
+(92 + 2/ζ^2 + 2*ζ^2)
+q(267120 + 18/ζ^4 - 1773/ζ^3 + 16064/ζ^2 + 147457/ζ + 147457*ζ + 16064*ζ^2 - 1773*ζ^3 + 18*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	1
[6 3 16]	87	-1
[12 -3 8]	87	0
[8 19 56]	87	0
[2 2 48]	92	-3735
[4 2 24]	92	1
[4 6 32]	92	0
[6 2 16]	92	-1
[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	-1
[6 1 16]	95	1
[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	0
[6 0 16]	96	2
[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	-2
[14 -3 8]	103	0
[14 -45 152]	103	2
[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	1
[6 9 32]	111	0
[14 -1 8]	111	0
[8 9 24]	111	0
[12 -39 136]	111	-1
[10 23 64]	111	0
[2 0 56]	112	-219120
[4 4 32]	112	0
[14 0 8]	112	219120
[8 4 16]	112	0
[16 -4 8]	112	219120
[8 12 32]	112	0
[16 -36 88]	112	0
[30 -91 280]	119	-90
[20 -61 192]	119	0
[6 13 48]	119	0
[8 3 16]	119	90
[16 -3 8]	119	0
[10 11 24]	119	0
[10 21 56]	119	0
[12 -29 80]	119	0
[4 2 32]	124	0
[4 6 40]	124	15
[8 2 16]	124	0
[16 -2 8]	124	1987817
[8 18 56]	124	0
[10 14 32]	124	0
[14 -46 160]	124	-15
[4 1 32]	127	92
[8 1 16]	127	-92
[16 -1 8]	127	0
[4 0 32]	128	0
[4 8 48]	128	-32
[6 8 32]	128	0
[22 -68 216]	128	0
[8 0 16]	128	0
[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	32
[4 5 40]	135	-1592
[6 3 24]	135	0
[18 -3 8]	135	0
[8 11 32]	135	0
[10 5 16]	135	1592
[12 -27 72]	135	0
[6 2 24]	140	0
[6 14 56]	140	-1
[18 -2 8]	140	-29667528
[10 10 24]	140	1
[4 9 56]	143	1683
[6 1 24]	143	0
[6 7 32]	143	-15
[18 -1 8]	143	0
[18 -55 176]	143	15
[12 -25 64]	143	0
[14 -33 88]	143	0
[14 -47 168]	143	-1683
[4 4 40]	144	15848
[6 0 24]	144	0
[6 12 48]	144	2
[18 0 8]	144	68380656
[20 -4 8]	144	68380656
[10 4 16]	144	-15848
[10 24 72]	144	0
[12 12 24]	144	-2
[38 -115 352]	151	154918
[20 -3 8]	151	0
[8 19 64]	151	0
[10 3 16]	151	-154918
[10 13 32]	151	0
[4 2 40]	156	231005
[4 6 48]	156	-168064
[6 6 32]	156	1757
[20 -2 8]	156	349920054
[8 10 32]	156	0
[20 -62 200]	156	-1757
[10 2 16]	156	-231005
[10 22 64]	156	0
[12 6 16]	156	168064
[12 18 40]	156	0
[4 1 40]	159	-153327
[6 9 40]	159	32
[20 -1 8]	159	0
[8 17 56]	159	0
[10 1 16]	159	153327
[16 -49 160]	159	-32
[12 15 32]	159	0
[14 -31 80]	159	0
[4 0 40]	160	-493736
[4 8 56]	160	304436
[20 0 8]	160	-840561392
[22 -4 8]	160	-840561392
[10 0 16]	160	493736
[10 20 56]	160	0
[14 20 40]	160	0
[14 -48 176]	160	-304436
[4 5 48]	167	-893918
[28 -85 264]	167	-14090
[6 13 56]	167	-16
[22 -3 8]	167	0
[22 -69 224]	167	14090
[14 -43 144]	167	16
[12 5 16]	167	893918
[16 -37 96]	167	0
[22 -2 8]	172	-3328686240
[4 9 64]	175	738909
[22 -1 8]	175	0
[8 9 32]	175	0
[10 15 40]	175	0
[14 7 16]	175	-738909
[4 4 48]	176	565632
[6 8 40]	176	-1772
[30 -92 288]	176	-165258
[22 0 8]	176	8397392256
[8 4 24]	176	165258
[24 -4 8]	176	8397392256
[8 12 40]	176	1
[10 12 32]	176	-1
[18 -56 184]	176	1772
[12 4 16]	176	-565632
[12 28 80]	176	0
[46 -139 424]	183	4622878
[6 3 32]	183	-397856
[8 3 24]	183	397856
[24 -3 8]	183	0
[12 3 16]	183	-4622878
[4 2 48]	188	3154048
[4 6 56]	188	-5250270
[6 2 32]	188	-236513
[6 14 64]	188	-1696
[8 2 24]	188	236513
[24 -2 8]	188	25783833735
[8 18 64]	188	16
[12 2 16]	188	-3154048
[12 14 32]	188	-16
[16 -50 168]	188	1696
[12 26 72]	188	0
[14 6 16]	188	5250270
[14 34 96]	188	0
[4 1 48]	191	-3730644
[6 1 32]	191	411961
[6 7 40]	191	168096
[8 1 24]	191	-411961
[24 -1 8]	191	0
[20 -63 208]	191	-168096
[10 23 72]	191	1
[12 1 16]	191	3730644
[12 17 40]	191	-1
[4 0 48]	192	-7103168
[6 0 32]	192	800028
[6 12 56]	192	3424
[8 0 24]	192	-800028
[24 0 8]	192	-69053456880
[8 8 32]	192	0
[8 16 56]	192	-34
[26 -4 8]	192	-69053456880
[12 0 16]	192	7103168
[14 16 32]	192	34
[14 -44 152]	192	-3424
[16 8 16]	192	-8907264
[16 24 48]	192	0
[4 5 56]	199	-24591
[26 -3 8]	199	0
[8 11 40]	199	12
[10 11 32]	199	-12
[10 21 64]	199	-1
[14 5 16]	199	24591
[14 19 40]	199	1
[6 6 40]	204	884315
[26 -2 8]	204	-162455777264
[10 14 40]	204	0
[22 -70 232]	204	-884315
[6 9 48]	207	-126943
[26 -1 8]	207	0
[26 -79 248]	207	0
[18 -57 192]	207	126943
[16 7 16]	207	4445053
[4 4 56]	208	-40250592
[26 0 8]	208	469539282740
[28 -4 8]	208	469539282740
[14 4 16]	208	40250592
[14 32 88]	208	0
[54 -163 496]	215	-246314823
[36 -109 336]	215	-397568
[6 13 64]	215	-27064
[28 -3 8]	215	0
[8 19 72]	215	-1832
[10 5 24]	215	397568
[12 13 32]	215	1832
[12 29 88]	215	0
[14 3 16]	215	246314823
[16 -51 176]	215	27064
[4 2 56]	220	-277519631
[28 -2 8]	220	819333887680
[8 10 40]	220	-177479
[28 -86 272]	220	0
[10 10 32]	220	177479
[14 2 16]	220	277519631
[16 6 16]	220	-288054656
[16 -38 104]	220	1
[4 1 56]	223	245598824
[28 -1 8]	223	0
[8 17 64]	223	1803
[14 1 16]	223	-245598824
[14 15 32]	223	-1803
[4 0 56]	224	645432084
[6 8 48]	224	-69470
[38 -116 360]	224	-3838520
[28 0 8]	224	-2627636861520
[30 -4 8]	224	-2627636861520
[10 4 24]	224	3838520
[10 24 80]	224	-16
[12 16 40]	224	16
[20 -64 216]	224	69470
[14 0 16]	224	-645432084
[18 -8 16]	224	500928752
[18 -44 120]	224	-2
[6 3 40]	231	-3719680
[30 -3 8]	231	0
[30 -93 296]	231	0
[10 3 24]	231	3719680
[10 13 40]	231	0
[12 27 80]	231	-17
[14 21 48]	231	17
[16 5 16]	231	-755609248
[6 2 40]	236	-782586
[6 14 72]	236	25224089
[30 -2 8]	236	-3154019426280
[10 2 24]	236	782586
[10 22 72]	236	-30
[14 18 40]	236	30
[18 -58 200]	236	-25224089
[6 1 40]	239	3949120
[6 7 48]	239	-20962825
[30 -1 8]	239	0
[8 9 40]	239	-843199
[10 1 24]	239	-3949120
[24 -73 232]	239	843199
[22 -71 240]	239	20962825
[18 -7 16]	239	505410364
[16 23 48]	239	-18
[6 0 40]	240	7477128
[30 0 8]	240	11871488696336
[8 4 32]	240	0
[32 -4 8]	240	11871488696336
[8 12 48]	240	195144
[10 0 24]	240	-7477128
[12 12 32]	240	-195144
[16 4 16]	240	-798515200
[16 20 40]	240	-92
[16 -52 184]	240	51747356
[8 3 32]	247	0
[32 -3 8]	247	0
[16 3 16]	247	-2965050783
[6 6 48]	252	-530631403
[8 2 32]	252	0
[32 -2 8]	252	7931907227145
[8 18 72]	252	10275
[12 6 24]	252	530631403
[12 18 48]	252	0
[14 14 32]	252	-10275
[16 2 16]	252	-4441387776
[18 -6 16]	252	3769282894
[6 9 56]	255	532734536
[8 1 32]	255	0
[32 -1 8]	255	0
[10 15 48]	255	925
[12 15 40]	255	-925
[20 -65 224]	255	-532734536
[14 33 96]	255	0
[16 1 16]	255	3716952296
[8 0 32]	256	0
[32 0 8]	256	-41108052986912
[8 8 40]	256	25368198
[34 -4 8]	256	-41108052986912
[10 12 40]	256	0
[26 -80 256]	256	-25368198
[16 0 16]	256	11073393664
[16 16 32]	256	52368
[20 -8 16]	256	-6510318976
[44 -133 408]	263	-3511216223
[34 -3 8]	263	0
[8 11 48]	263	-25964651
[22 -67 216]	263	25964651
[12 5 24]	263	3511216223
[18 -5 16]	263	15358730289
[18 -59 208]	263	510976288
[34 -2 8]	268	-2210677570160
[34 -1 8]	271	0
[34 -103 320]	271	556116384
[28 -87 280]	271	-556116384
[10 23 80]	271	179310
[14 17 40]	271	-179310
[20 -7 16]	271	-12151656410
[16 39 112]	271	-44
[6 8 56]	272	4401779760
[46 -140 432]	272	-11466720322
[34 0 8]	272	92013087616560
[36 -4 8]	272	92013087616560
[12 4 24]	272	11466720322
[12 28 88]	272	1804
[14 20 48]	272	-1804
[22 -72 248]	272	-4401779760
[18 -4 16]	272	6456760888
[6 3 48]	279	-18778391923
[36 -3 8]	279	0
[8 19 80]	279	49459843
[10 11 40]	279	0
[12 3 24]	279	18778391923
[20 -61 200]	279	-49459843
[18 -3 16]	279	-5997977753
[16 19 40]	279	180190
[18 27 56]	279	62
[6 2 48]	284	-8220293929
[36 -2 8]	284	-95136385273191
[8 10 48]	284	-541446672
[36 -110 344]	284	4107515412
[10 14 48]	284	3182323
[30 -94 304]	284	-4107515412
[12 2 24]	284	8220293929
[12 14 40]	284	-3182323
[24 -74 240]	284	541446672
[18 -2 16]	284	18347462445
[20 -6 16]	284	82373248
[16 22 48]	284	864
[20 -66 232]	284	-5566935661
[6 1 48]	287	22310697930
[6 7 56]	287	5440901992
[36 -1 8]	287	0
[12 1 24]	287	-22310697930
[12 17 48]	287	0
[14 7 24]	287	-5440901992
[18 -1 16]	287	-9871501961
[18 -55 184]	287	22666173
[6 0 48]	288	40435433636
[36 0 8]	288	-4413649854240
[38 -4 8]	288	-4413649854240
[12 0 24]	288	-40435433636
[18 0 16]	288	-58159338800
[22 -8 16]	288	-2898288760
[18 24 48]	288	1876
[18 -60 216]	288	18868017968
[38 -3 8]	295	0
[38 -117 368]	295	15801879985
[10 5 32]	295	-15801879985
[20 -5 16]	295	-105700023081
[6 6 56]	300	-93160506693
[38 -2 8]	300	528629708975400
[10 10 40]	300	0
[14 6 24]	300	93160506693
[14 34 104]	300	0
[38 -1 8]	303	0
[8 9 48]	303	-4423397573
[26 -81 264]	303	4423397573
[22 -7 16]	303	108231268648
[22 -73 256]	303	-129091391967
[38 0 8]	304	-1067427147614080
[8 4 40]	304	34327899047
[40 -4 8]	304	-1067427147614080
[8 12 56]	304	808116786
[10 4 32]	304	-34327899047
[10 24 88]	304	-2712864
[14 16 40]	304	2712864
[22 -68 224]	304	-808116786
[20 -4 16]	304	15397093248
[52 -157 480]	311	-564866263736
[8 3 40]	311	38781706319
[40 -3 8]	311	0
[10 3 32]	311	-38781706319
[10 13 48]	311	-288706102
[12 13 40]	311	288706102
[12 29 96]	311	-19957
[14 5 24]	311	564866263736
[14 19 48]	311	19957
[20 -3 16]	311	516397521760
[20 -67 240]	311	131017770016
[8 2 40]	316	4358827657
[40 -2 8]	316	-1492793971437868
[10 2 32]	316	-4358827657
[20 -2 16]	316	294232634112
[16 18 40]	316	-22757311
[20 -62 208]	316	665327408
[22 -6 16]	316	-543058275885
[8 1 40]	319	-55138859501
[40 -1 8]	319	0
[10 1 32]	319	55138859501
[32 -97 304]	319	0
[20 -1 16]	319	-399051122307
[54 -164 504]	320	-1469925872128
[8 0 40]	320	-85638865672
[40 0 8]	320	5391275648088600
[8 8 48]	320	-6270430384
[42 -4 8]	320	5391275648088600
[10 0 32]	320	85638865672
[12 16 48]	320	0
[28 -88 288]	320	6270430384
[14 4 24]	320	1469925872128
[20 0 16]	320	-605826322688
[16 8 24]	320	-706507747076
[24 -8 16]	320	995926745088
[16 24 56]	320	178524
[18 20 40]	320	46812008
[18 -56 192]	320	-746247680
[6 3 56]	327	-1867560933376
[42 -3 8]	327	0
[8 11 56]	327	10604498447
[14 3 24]	327	1867560933376
[24 -75 248]	327	-10604498447
[22 -5 16]	327	-478011306805
[16 21 48]	327	3532131
[6 2 56]	332	-401550309587
[42 -2 8]	332	1194664150575720
[14 2 24]	332	401550309587
[18 26 56]	332	-4899
[22 -74 264]	332	-584271135133
[6 1 56]	335	2426453587664
[42 -1 8]	335	0
[42 -127 392]	335	88002301029
[10 15 56]	335	1175917390
[30 -95 312]	335	-88002301029
[14 1 24]	335	-2426453587664
[14 15 40]	335	-1175917390
[16 7 24]	335	-990185543228
[24 -7 16]	335	-32478874030
[18 23 48]	335	-3551166
[6 0 56]	336	3920419362272
[42 0 8]	336	-14406718985411544
[44 -4 8]	336	-14406718985411544
[10 12 48]	336	-16455692682
[34 -104 328]	336	0
[12 12 40]	336	16455692682
[14 0 24]	336	-3920419362272
[22 -4 16]	336	-923838492704
[20 28 56]	336	-350728
[20 -68 248]	336	2372303345272
[44 -3 8]	343	0
[14 21 56]	343	0
[22 -3 16]	343	-6037760109570
[22 -69 232]	343	-7141143553
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