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(ζ^(-4) + ζ^4)
  +(92 + ζ^(-3) + 2/ζ^2 - ζ^(-1) - ζ + 2*ζ^2 + ζ^3)
  +q(ζ^(-5) + 68/ζ^4 + 68*ζ^4 + ζ^5)
• From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block

  	φ	= TB(46;−1,2,2,3)
  		= 1−12231
  		= η(τ)92(θ(τ,z)/η(τ))−1(θ(τ,2z)/η(τ))2(θ(τ,3z)/η(τ))

  This theta block has q-order of vanishing 4. This theta block has index 2*4, and so Borch(ψ) begins with Jacobi coefficient #2.
• From the truncated singular part of ψ, we compute that Borch(ψ) is a symmetric 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 11 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
  (9,3)	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(ζ^(-4) + ζ^4)
+(92 + ζ^(-3) + 2/ζ^2 - ζ^(-1) - ζ + 2*ζ^2 + ζ^3)
+q(17440160 + ζ^(-5) + 68/ζ^4 - 93931/ζ^3 + 2128384/ζ^2 - 10557718/ζ - 10557718*ζ + 2128384*ζ^2 - 93931*ζ^3 + 68*ζ^4 + ζ^5)

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	0
[4 2 16]	60	0
[4 6 24]	60	0
[6 6 16]	60	0
[8 2 8]	60	0
[8 -18 48]	60	0
[2 1 32]	63	0
[4 1 16]	63	0
[16 -33 72]	63	0
[6 9 24]	63	0
[8 1 8]	63	0
[2 0 32]	64	0
[2 4 40]	64	0
[4 0 16]	64	0
[4 8 32]	64	0
[8 0 8]	64	0
[8 8 16]	64	0
[8 -16 40]	64	0
[10 -4 8]	64	0
[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	0
[10 -2 8]	76	0
[10 -22 56]	76	1
[2 1 40]	79	0
[4 9 40]	79	0
[10 -1 8]	79	0
[10 -31 104]	79	0
[2 0 40]	80	0
[2 4 48]	80	0
[4 4 24]	80	0
[6 8 24]	80	0
[14 -44 144]	80	0
[10 0 8]	80	0
[10 -20 48]	80	-2
[12 -4 8]	80	0
[12 -28 72]	80	-2
[2 3 48]	87	0
[22 -67 208]	87	0
[6 3 16]	87	0
[12 -3 8]	87	0
[8 19 56]	87	-1
[2 2 48]	92	0
[4 2 24]	92	0
[4 6 32]	92	0
[6 2 16]	92	0
[6 14 48]	92	1
[12 -2 8]	92	0
[8 10 24]	92	1
[12 -26 64]	92	-64
[12 -38 128]	92	0
[2 1 48]	95	0
[4 1 24]	95	0
[6 1 16]	95	0
[6 7 24]	95	-1
[12 -1 8]	95	0
[8 17 48]	95	1
[10 15 32]	95	1
[2 0 48]	96	0
[2 4 56]	96	0
[4 0 24]	96	0
[4 8 40]	96	0
[6 0 16]	96	0
[6 12 40]	96	-2
[12 0 8]	96	0
[12 -24 56]	96	132
[14 -4 8]	96	0
[14 -32 80]	96	132
[10 12 24]	96	-2
[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	0
[6 6 24]	108	-1
[14 -2 8]	108	0
[14 -30 72]	108	1923
[2 1 56]	111	0
[4 9 48]	111	0
[6 9 32]	111	3
[14 -1 8]	111	0
[8 9 24]	111	3
[12 -39 136]	111	0
[10 23 64]	111	69
[2 0 56]	112	0
[2 4 64]	112	0
[4 4 32]	112	0
[14 0 8]	112	0
[14 -28 64]	112	-4112
[8 4 16]	112	0
[16 -4 8]	112	0
[8 12 32]	112	-4
[16 -36 88]	112	-4112
[2 3 64]	119	0
[30 -91 280]	119	1
[20 -61 192]	119	1
[6 13 48]	119	-2
[8 3 16]	119	1
[16 -3 8]	119	0
[10 11 24]	119	-2
[10 21 56]	119	-70
[12 -29 80]	119	-70
[2 2 64]	124	0
[4 2 32]	124	1
[4 6 40]	124	-1
[8 2 16]	124	1
[16 -2 8]	124	0
[16 -34 80]	124	-35776
[8 18 56]	124	-65
[10 14 32]	124	-65
[14 -46 160]	124	-1
[2 1 64]	127	0
[4 1 32]	127	-1
[8 1 16]	127	-1
[16 -1 8]	127	0
[2 0 64]	128	0
[4 0 32]	128	-2
[4 8 48]	128	2
[6 8 32]	128	0
[22 -68 216]	128	2
[8 0 16]	128	-2
[16 0 8]	128	0
[8 8 24]	128	0
[8 16 48]	128	136
[18 -4 8]	128	0
[18 -40 96]	128	79904
[12 16 32]	128	136
[12 -40 144]	128	2
[4 5 40]	135	-3
[6 3 24]	135	1
[18 -3 8]	135	0
[8 11 32]	135	64
[10 5 16]	135	-3
[12 -27 72]	135	91676
[6 2 24]	140	-1
[6 14 56]	140	-87
[18 -2 8]	140	0
[10 10 24]	140	-87
[4 9 56]	143	2
[6 1 24]	143	-1
[6 7 32]	143	87
[18 -1 8]	143	0
[18 -55 176]	143	87
[12 -25 64]	143	-91607
[14 -33 88]	143	-91607
[14 -47 168]	143	2
[4 4 40]	144	0
[6 0 24]	144	0
[6 12 48]	144	176
[18 0 8]	144	0
[20 -4 8]	144	0
[10 4 16]	144	0
[10 24 72]	144	-93662
[12 12 24]	144	176
[38 -115 352]	151	-87
[20 -3 8]	151	0
[8 19 64]	151	93818
[10 3 16]	151	-87
[10 13 32]	151	93818
[4 2 40]	156	-87
[4 6 48]	156	87
[6 6 32]	156	86
[20 -2 8]	156	0
[8 10 32]	156	-94040
[20 -62 200]	156	86
[10 2 16]	156	-87
[10 22 64]	156	1901
[12 6 16]	156	87
[12 18 40]	156	1901
[4 1 40]	159	90
[6 9 40]	159	-266
[20 -1 8]	159	0
[8 17 56]	159	-93885
[10 1 16]	159	90
[16 -49 160]	159	-266
[12 15 32]	159	-93885
[14 -31 80]	159	4216595
[4 0 40]	160	176
[4 8 56]	160	-176
[20 0 8]	160	0
[22 -4 8]	160	0
[10 0 16]	160	176
[10 20 56]	160	183656
[14 20 40]	160	183656
[14 -48 176]	160	-176
[4 5 48]	167	267
[28 -85 264]	167	-93
[6 13 56]	167	178
[22 -3 8]	167	0
[22 -69 224]	167	-93
[14 -29 72]	167	-4124917
[14 -43 144]	167	178
[12 5 16]	167	267
[16 -37 96]	167	-4124917
[22 -2 8]	172	0
[4 9 64]	175	-179
[22 -1 8]	175	0
[8 9 32]	175	2033847
[10 15 40]	175	1938420
[14 7 16]	175	-179
[4 4 48]	176	0
[6 8 40]	176	4
[30 -92 288]	176	-176
[22 0 8]	176	0
[8 4 24]	176	-176
[24 -4 8]	176	0
[8 12 40]	176	-1939984
[10 12 32]	176	-1939984
[18 -56 184]	176	4
[12 4 16]	176	0
[12 28 80]	176	-6025200
[46 -139 424]	183	3651
[6 3 32]	183	-177
[8 3 24]	183	-177
[24 -3 8]	183	0
[12 3 16]	183	3651
[16 -35 88]	183	-92810913
[4 2 48]	188	3649
[4 6 56]	188	-3649
[6 2 32]	188	1
[6 14 64]	188	3714
[8 2 24]	188	1
[24 -2 8]	188	0
[8 18 64]	188	-13071017
[12 2 16]	188	3649
[12 14 32]	188	-13071017
[16 -50 168]	188	3714
[12 26 72]	188	-13202798
[14 6 16]	188	-3649
[14 34 96]	188	-13202798
[4 1 48]	191	-3920
[6 1 32]	191	180
[6 7 40]	191	-3715
[8 1 24]	191	180
[24 -1 8]	191	0
[20 -63 208]	191	-3715
[10 23 72]	191	19187708
[12 1 16]	191	-3920
[12 17 40]	191	19187708
[18 -41 104]	191	97027368
[4 0 48]	192	-7476
[4 8 64]	192	7476
[6 0 32]	192	176
[6 12 56]	192	-7612
[8 0 24]	192	176
[24 0 8]	192	0
[8 8 32]	192	-8335532
[8 16 56]	192	30210082
[26 -4 8]	192	0
[12 0 16]	192	-7476
[12 24 64]	192	38451920
[14 16 32]	192	30210082
[14 -44 152]	192	-7612
[16 8 16]	192	7476
[16 24 48]	192	38451920
[4 5 56]	199	-11483
[26 -3 8]	199	0
[8 11 40]	199	21500530
[10 11 32]	199	21500530
[10 21 64]	199	-21219944
[14 5 16]	199	-11483
[14 19 40]	199	-21219944
[6 6 40]	204	90391
[26 -2 8]	204	0
[10 14 40]	204	-40871964
[22 -70 232]	204	90391
[4 9 72]	207	7743
[6 9 48]	207	-82335
[26 -1 8]	207	0
[26 -79 248]	207	4934289
[18 -57 192]	207	-82335
[16 7 16]	207	7743
[4 4 56]	208	0
[26 0 8]	208	0
[28 -4 8]	208	0
[14 4 16]	208	0
[14 32 88]	208	1584302280
[54 -163 496]	215	-98257
[36 -109 336]	215	-2029927
[6 13 64]	215	86142
[28 -3 8]	215	0
[8 19 72]	215	345176832
[10 5 24]	215	-2029927
[12 13 32]	215	345176832
[12 29 88]	215	-1453109423
[14 3 16]	215	-98257
[16 -51 176]	215	86142
[4 2 56]	220	-98078
[4 6 64]	220	98078
[28 -2 8]	220	0
[8 10 40]	220	-380414845
[28 -86 272]	220	13269998
[10 10 32]	220	-380414845
[14 2 16]	220	-98078
[14 30 80]	220	-6860693807
[16 6 16]	220	98078
[16 -38 104]	220	-6860693807
[4 1 56]	223	109917
[28 -1 8]	223	0
[8 17 64]	223	-368711296
[14 1 16]	223	109917
[14 15 32]	223	-368711296
[4 0 56]	224	203808
[6 8 48]	224	1939984
[38 -116 360]	224	8343010
[28 0 8]	224	0
[30 -4 8]	224	0
[10 4 24]	224	8343010
[10 24 80]	224	1121223282
[12 16 40]	224	1121223282
[20 -64 216]	224	1939984
[14 0 16]	224	203808
[18 -8 16]	224	-203808
[18 -44 120]	224	10578821468
[4 5 64]	231	317376
[6 3 40]	231	-4930367
[30 -3 8]	231	0
[30 -93 296]	231	-6967662
[10 3 24]	231	-4930367
[10 13 40]	231	-719586680
[12 27 80]	231	5622018533
[14 21 48]	231	5622018533
[16 5 16]	231	317376
[6 2 40]	236	-13273560
[6 14 72]	236	12972938
[30 -2 8]	236	0
[10 2 24]	236	-13273560
[10 22 72]	236	-6833478913
[14 18 40]	236	-6833478913
[18 -58 200]	236	12972938
[6 1 40]	239	6964277
[6 7 48]	239	-21402274
[30 -1 8]	239	0
[8 9 40]	239	2481221632
[10 1 24]	239	6964277
[24 -73 232]	239	2481221632
[12 25 72]	239	-4188002931
[22 -71 240]	239	-21402274
[18 -7 16]	239	-215290
[16 23 48]	239	-4188002931
[4 4 64]	240	0
[6 0 40]	240	9680488
[6 12 64]	240	-30006272
[30 0 8]	240	0
[8 4 32]	240	-9680492
[32 -4 8]	240	0
[8 12 48]	240	-2112709202
[10 0 24]	240	9680488
[10 20 64]	240	11510135332
[12 12 32]	240	-2112709202
[16 4 16]	240	0
[16 20 40]	240	11510135332
[16 -36 96]	240	148633598212
[16 -52 184]	240	-30006272
[62 -187 568]	247	1901129
[8 3 32]	247	-6960829
[32 -3 8]	247	0
[16 3 16]	247	1901129
[4 2 64]	252	1893387
[6 6 48]	252	380512988
[8 2 32]	252	13183082
[32 -2 8]	252	0
[8 18 72]	252	-6177727257
[12 6 24]	252	380512988
[12 18 48]	252	31085280270
[14 14 32]	252	-6177727257
[16 2 16]	252	1893387
[18 -6 16]	252	-1893387
[18 -42 112]	252	-650126853694
[4 1 64]	255	-2226066
[6 9 56]	255	-345494277
[8 1 32]	255	6960292
[32 -1 8]	255	0
[10 15 48]	255	-34633050949
[12 15 40]	255	-34633050949
[20 -65 224]	255	-345494277
[14 33 96]	255	-179058913662
[16 1 16]	255	-2226066
[4 0 64]	256	-3998056
[8 0 32]	256	-16686024
[32 0 8]	256	0
[8 8 40]	256	-5412212800
[8 16 64]	256	17358373500
[34 -4 8]	256	0
[10 12 40]	256	39006531532
[26 -80 256]	256	-5412212800
[16 0 16]	256	-3998056
[16 16 32]	256	17358373500
[20 -8 16]	256	3998056
[20 -48 128]	256	1016630990944
[44 -133 408]	263	-2481425367
[6 13 72]	263	368926656
[34 -3 8]	263	0
[8 11 48]	263	11967558996
[22 -67 216]	263	11967558996
[12 5 24]	263	-2481425367
[18 -5 16]	263	-6322384
[18 -59 208]	263	368926656
[34 -2 8]	268	0
[34 -1 8]	271	0
[34 -103 320]	271	1499921646
[28 -87 280]	271	1499921646
[10 23 80]	271	183527100309
[14 17 40]	271	183527100309
[14 31 88]	271	676720633728
[20 -7 16]	271	4315345
[16 39 112]	271	676720633728
[6 8 56]	272	2112802864
[46 -140 432]	272	5413948976
[34 0 8]	272	0
[36 -4 8]	272	0
[12 4 24]	272	5413948976
[12 28 88]	272	8574273236
[14 20 48]	272	8574273236
[22 -72 248]	272	2112802864
[18 -4 16]	272	0
[6 3 48]	279	-1521532093
[36 -3 8]	279	0
[8 19 80]	279	74714511552
[10 11 40]	279	-554501863827
[10 21 72]	279	-148195962944
[12 3 24]	279	-1521532093
[20 -61 200]	279	74714511552
[18 -3 16]	279	-28087332
[16 19 40]	279	-148195962944
[18 27 56]	279	-503262424773
[6 2 48]	284	-6937507103
[6 14 80]	284	6179618743
[36 -2 8]	284	0
[8 10 48]	284	-93908209179
[36 -110 344]	284	7318020025
[10 14 48]	284	364859683357
[30 -94 304]	284	7318020025
[12 2 24]	284	-6937507103
[12 14 40]	284	364859683357
[24 -74 240]	284	-93908209179
[12 26 80]	284	-694613720317
[18 -2 16]	284	-27872132
[20 -6 16]	284	27872132
[16 22 48]	284	-694613720317
[20 -66 232]	284	6179618743
[6 1 48]	287	4024441893
[6 7 56]	287	-11971398545
[36 -1 8]	287	0
[8 17 72]	287	-89168226377
[12 1 24]	287	4024441893
[12 17 48]	287	-381202250041
[14 7 24]	287	-11971398545
[18 -1 16]	287	34616995
[18 -55 184]	287	-89168226377
[6 0 48]	288	2282202528
[36 0 8]	288	0
[38 -4 8]	288	0
[12 0 24]	288	2282202528
[18 0 16]	288	59945952
[22 -8 16]	288	-59945952
[18 24 48]	288	1307692028740
[18 -60 216]	288	-17362555212
[38 -3 8]	295	0
[38 -117 368]	295	-6505565266
[10 5 32]	295	-6505565266
[20 -5 16]	295	96464520
[16 37 104]	295	-4698395884541
[6 6 56]	300	93947187756
[38 -2 8]	300	0
[10 10 40]	300	3220949209015
[14 6 24]	300	93947187756
[14 34 104]	300	-1354768114021
[6 9 64]	303	-74727376876
[38 -1 8]	303	0
[8 9 48]	303	356747647555
[26 -81 264]	303	356747647555
[22 -7 16]	303	-66264687
[22 -73 256]	303	-74727376876
[38 0 8]	304	0
[8 4 40]	304	3130010096
[40 -4 8]	304	0
[8 12 56]	304	-268734253824
[10 4 32]	304	3130010096
[10 24 88]	304	-4165816343988
[14 16 40]	304	-4165816343988
[22 -68 224]	304	-268734253824
[20 -4 16]	304	0
[52 -157 480]	311	-356025834810
[8 3 40]	311	-5524352234
[40 -3 8]	311	0
[10 3 32]	311	-5524352234
[10 13 48]	311	1098386373666
[12 13 40]	311	1098386373666
[12 29 96]	311	4758553301906
[14 5 24]	311	-356025834810
[14 19 48]	311	4758553301906
[20 -3 16]	311	328249086
[20 -67 240]	311	89185130976
[18 43 120]	311	10678475673543
[8 2 40]	316	-380512987
[40 -2 8]	316	0
[8 18 80]	316	-394881913152
[10 2 32]	316	-380512987
[10 22 80]	316	9318116245121
[20 -2 16]	316	323937661
[16 18 40]	316	9318116245121
[20 -62 208]	316	-394881913152
[22 -6 16]	316	-323937661
[8 1 40]	319	8027179873
[40 -1 8]	319	0
[10 1 32]	319	8027179873
[32 -97 304]	319	-9246617279532
[20 -1 16]	319	-428802091
[20 -49 136]	319	-6652591293672
[6 8 64]	320	267613030544
[54 -164 504]	320	528839053440
[8 0 40]	320	-8545898880
[40 0 8]	320	0
[8 8 48]	320	-567841587052
[42 -4 8]	320	0
[10 0 32]	320	-8545898880
[12 16 48]	320	-5064202315672
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