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^-2(1)
  +(92 + 2/ζ^2 + 2*ζ^2)
  +q(68/ζ^4 + 68*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 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
  (32,0)	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^-2(1)
+(92 + 2/ζ^2 + 2*ζ^2)
+q(17431416 + 68/ζ^4 + 94208/ζ^3 + 2128384/ζ^2 + 10555392/ζ + 10555392*ζ + 2128384*ζ^2 + 94208*ζ^3 + 68*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	-1
[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	2
[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	-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	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	2
[12 -4 8]	80	180
[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	0
[2 2 48]	92	3735
[4 2 24]	92	88
[4 6 32]	92	-1
[6 2 16]	92	88
[6 14 48]	92	1
[12 -2 8]	92	3735
[8 10 24]	92	1
[12 -26 64]	92	64
[12 -38 128]	92	-1
[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	-180
[4 8 40]	96	0
[6 0 16]	96	-180
[6 12 40]	96	-2
[12 0 8]	96	-7832
[12 -24 56]	96	-132
[14 -4 8]	96	-7832
[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	-101640
[6 6 24]	108	64
[14 -2 8]	108	-101640
[14 -30 72]	108	-1923
[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	-182
[14 0 8]	112	219120
[8 4 16]	112	-182
[16 -4 8]	112	219120
[8 12 32]	112	-4
[16 -36 88]	112	4112
[30 -91 280]	119	0
[20 -61 192]	119	94208
[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	-3823
[4 6 40]	124	24
[8 2 16]	124	-3823
[16 -2 8]	124	1987817
[8 18 56]	124	-63
[10 14 32]	124	-63
[14 -46 160]	124	24
[4 1 32]	127	0
[8 1 16]	127	0
[16 -1 8]	127	0
[4 0 32]	128	8012
[4 8 48]	128	136
[6 8 32]	128	-134
[22 -68 216]	128	2127888
[8 0 16]	128	8012
[16 0 8]	128	-4421340
[8 8 24]	128	-134
[8 16 48]	128	132
[18 -4 8]	128	-4421340
[12 16 32]	128	132
[12 -40 144]	128	136
[4 5 40]	135	-94208
[6 3 24]	135	10366976
[18 -3 8]	135	0
[8 11 32]	135	0
[10 5 16]	135	-94208
[12 -27 72]	135	94208
[6 2 24]	140	13166975
[6 14 56]	140	-25
[18 -2 8]	140	-29667528
[10 10 24]	140	-25
[4 9 56]	143	94208
[6 1 24]	143	-10461184
[6 7 32]	143	-94208
[18 -1 8]	143	0
[18 -55 176]	143	-94208
[12 -25 64]	143	-94208
[14 -33 88]	143	-94208
[14 -47 168]	143	94208
[4 4 40]	144	-2120056
[6 0 24]	144	-30589854
[6 12 48]	144	186
[18 0 8]	144	68380656
[20 -4 8]	144	68380656
[10 4 16]	144	-2120056
[10 24 72]	144	266
[12 12 24]	144	186
[38 -115 352]	151	-10366976
[20 -3 8]	151	0
[8 19 64]	151	-94208
[10 3 16]	151	-10366976
[10 13 32]	151	-94208
[4 2 40]	156	-13065336
[4 6 48]	156	-13177448
[6 6 32]	156	13173714
[20 -2 8]	156	349920054
[8 10 32]	156	-2099
[20 -62 200]	156	13173714
[10 2 16]	156	-13065336
[10 22 64]	156	1947
[12 6 16]	156	-13177448
[12 18 40]	156	1947
[4 1 40]	159	10461184
[6 9 40]	159	-94208
[20 -1 8]	159	0
[8 17 56]	159	94208
[10 1 16]	159	10461184
[16 -49 160]	159	-94208
[12 15 32]	159	94208
[14 -31 80]	159	4149248
[4 0 40]	160	30370736
[4 8 56]	160	30603016
[20 0 8]	160	-840561392
[22 -4 8]	160	-840561392
[10 0 16]	160	30370736
[10 20 56]	160	-4296
[14 20 40]	160	-4296
[14 -48 176]	160	30603016
[4 5 48]	167	-390770688
[28 -85 264]	167	390770688
[6 13 56]	167	94208
[22 -3 8]	167	0
[22 -69 224]	167	390770688
[14 -43 144]	167	94208
[12 5 16]	167	-390770688
[16 -37 96]	167	-4055040
[22 -2 8]	172	-3328686240
[4 9 64]	175	401137664
[22 -1 8]	175	0
[8 9 32]	175	4149248
[10 15 40]	175	188416
[14 7 16]	175	401137664
[4 4 48]	176	-2880842704
[6 8 40]	176	-30603062
[30 -92 288]	176	2880623766
[22 0 8]	176	8397392256
[8 4 24]	176	2880623766
[24 -4 8]	176	8397392256
[8 12 40]	176	-2123594
[10 12 32]	176	-2123594
[18 -56 184]	176	-30603062
[12 4 16]	176	-2880842704
[12 28 80]	176	2119664
[46 -139 424]	183	-7918096384
[6 3 32]	183	7918096384
[8 3 24]	183	7918096384
[24 -3 8]	183	0
[12 3 16]	183	-7918096384
[4 2 48]	188	-6546781803
[4 6 56]	188	-6544097632
[6 2 32]	188	6544797809
[6 14 64]	188	13179546
[8 2 24]	188	6544797809
[24 -2 8]	188	25783833735
[8 18 64]	188	-13175660
[12 2 16]	188	-6546781803
[12 14 32]	188	-13175660
[16 -50 168]	188	13179546
[12 26 72]	188	13139860
[14 6 16]	188	-6544097632
[14 34 96]	188	13139860
[4 1 48]	191	8308772864
[6 1 32]	191	-8308772864
[6 7 40]	191	-401043456
[8 1 24]	191	-8308772864
[24 -1 8]	191	0
[20 -63 208]	191	-401043456
[10 23 72]	191	-4055040
[12 1 16]	191	8308772864
[12 17 40]	191	-4055040
[4 0 48]	192	18881603640
[6 0 32]	192	-18877190312
[6 12 56]	192	4243652
[8 0 24]	192	-18877190312
[24 0 8]	192	-69053456880
[8 8 32]	192	-1505838704
[8 16 56]	192	30602976
[26 -4 8]	192	-69053456880
[12 0 16]	192	18881603640
[14 16 32]	192	30602976
[14 -44 152]	192	4243652
[16 8 16]	192	18875995144
[16 24 48]	192	-30523208
[4 5 56]	199	-101888110592
[26 -3 8]	199	0
[8 11 40]	199	-14422016
[10 11 32]	199	-14422016
[10 21 64]	199	3960832
[14 5 16]	199	-101888110592
[14 19 40]	199	3960832
[6 6 40]	204	6531025558
[26 -2 8]	204	-162455777264
[10 14 40]	204	-26311651
[22 -70 232]	204	6531025558
[6 9 48]	207	-405286912
[26 -1 8]	207	0
[26 -79 248]	207	-46034071552
[18 -57 192]	207	-405286912
[16 7 16]	207	109795745792
[4 4 56]	208	-464118135806
[26 0 8]	208	469539282740
[28 -4 8]	208	469539282740
[14 4 16]	208	-464118135806
[14 32 88]	208	-1505679256
[54 -163 496]	215	-899091402752
[36 -109 336]	215	101497339904
[6 13 64]	215	405192704
[28 -3 8]	215	0
[8 19 72]	215	-394731520
[10 5 24]	215	101497339904
[12 13 32]	215	-394731520
[12 29 88]	215	18477056
[14 3 16]	215	-899091402752
[16 -51 176]	215	405192704
[4 2 56]	220	-489082775816
[28 -2 8]	220	819333887680
[8 10 40]	220	6847274450
[28 -86 272]	220	-611188244559
[10 10 32]	220	6847274450
[14 2 16]	220	-489082775816
[16 6 16]	220	-489143933162
[16 -38 104]	220	6860857997
[4 1 56]	223	1000578281472
[28 -1 8]	223	0
[8 17 64]	223	405098496
[14 1 16]	223	1000578281472
[14 15 32]	223	405098496
[4 0 56]	224	1919428812424
[6 8 48]	224	-17370148608
[38 -116 360]	224	461241933382
[28 0 8]	224	-2627636861520
[30 -4 8]	224	-2627636861520
[10 4 24]	224	461241933382
[10 24 80]	224	1536362224
[12 16 40]	224	1536362224
[20 -64 216]	224	-17370148608
[14 0 16]	224	1919428812424
[18 -8 16]	224	1919556967712
[18 -44 120]	224	-10736628484
[6 3 40]	231	891173306368
[30 -3 8]	231	0
[30 -93 296]	231	-3779892563968
[10 3 24]	231	891173306368
[10 13 40]	231	-872914944
[12 27 80]	231	486199296
[14 21 48]	231	486199296
[16 5 16]	231	-5603389698048
[6 2 40]	236	482567645623
[6 14 72]	236	-303176705
[30 -2 8]	236	-3154019426280
[10 2 24]	236	482567645623
[10 22 72]	236	-6873581284
[14 18 40]	236	-6873581284
[18 -58 200]	236	-303176705
[6 1 40]	239	-992269508608
[6 7 48]	239	-63761674240
[30 -1 8]	239	0
[8 9 40]	239	186299711488
[10 1 24]	239	-992269508608
[24 -73 232]	239	186299711488
[22 -71 240]	239	-63761674240
[18 -7 16]	239	6494161866752
[16 23 48]	239	-500715520
[6 0 40]	240	-1900620002948
[30 0 8]	240	11871488696336
[8 4 32]	240	-12348095218764
[32 -4 8]	240	11871488696336
[8 12 48]	240	-13614052090
[10 0 24]	240	-1900620002948
[12 12 32]	240	-13614052090
[16 4 16]	240	-17650869417034
[16 20 40]	240	10731308660
[16 -52 184]	240	16497018388
[8 3 32]	247	-20464154939392
[32 -3 8]	247	0
[16 3 16]	247	-24248668102656
[6 6 48]	252	1100330189880
[8 2 32]	252	-9282712727678
[32 -2 8]	252	7931907227145
[8 18 72]	252	342553691
[12 6 24]	252	1100330189880
[12 18 48]	252	-13708164400
[14 14 32]	252	342553691
[16 2 16]	252	-6103921355949
[18 -6 16]	252	-6102546002560
[6 9 56]	255	-296095363072
[8 1 32]	255	24290077425664
[32 -1 8]	255	0
[10 15 48]	255	46530637824
[12 15 40]	255	46530637824
[20 -65 224]	255	-296095363072
[14 33 96]	255	-46225117184
[16 1 16]	255	29741861011456
[8 0 32]	256	44487004063616
[32 0 8]	256	-41108052986912
[8 8 40]	256	4231586368008
[34 -4 8]	256	-41108052986912
[10 12 40]	256	-155640261112
[26 -80 256]	256	4231586368008
[16 0 16]	256	48450143777264
[16 16 32]	256	15833778240
[20 -8 16]	256	48447259314576
[44 -133 408]	263	9383282356224
[34 -3 8]	263	0
[8 11 48]	263	-148549529600
[22 -67 216]	263	-148549529600
[12 5 24]	263	9383282356224
[18 -5 16]	263	-64204136628224
[18 -59 208]	263	250452062208
[34 -2 8]	268	-2210677570160
[34 -1 8]	271	0
[34 -103 320]	271	40511943491584
[28 -87 280]	271	40511943491584
[10 23 80]	271	-185788440576
[14 17 40]	271	-185788440576
[20 -7 16]	271	87443928043520
[16 39 112]	271	186146791424
[6 8 56]	272	-6151112951778
[46 -140 432]	272	29998898375198
[34 0 8]	272	92013087616560
[36 -4 8]	272	92013087616560
[12 4 24]	272	29998898375198
[12 28 88]	272	163496261718
[14 20 48]	272	163496261718
[22 -72 248]	272	-6151112951778
[18 -4 16]	272	-105428010142616
[6 3 48]	279	44712833409024
[36 -3 8]	279	0
[8 19 80]	279	-241661050880
[10 11 40]	279	649649008640
[12 3 24]	279	44712833409024
[20 -61 200]	279	-241661050880
[18 -3 16]	279	-27224748789760
[16 19 40]	279	140145233920
[18 27 56]	279	-140411928576
[6 2 48]	284	15386297228909
[36 -2 8]	284	-95136385273191
[8 10 48]	284	-9199923262769
[36 -110 344]	284	227678644578059
[10 14 48]	284	1254811216693
[30 -94 304]	284	227678644578059
[12 2 24]	284	15386297228909
[12 14 40]	284	1254811216693
[24 -74 240]	284	-9199923262769
[18 -2 16]	284	86684278134856
[20 -6 16]	284	86655474388681
[16 22 48]	284	-650483830283
[20 -66 232]	284	9682219917747
[6 1 48]	287	-54031948898304
[6 7 56]	287	-47005704314880
[36 -1 8]	287	0
[12 1 24]	287	-54031948898304
[12 17 48]	287	-372542025728
[14 7 24]	287	-47005704314880
[18 -1 16]	287	84825318576128
[18 -55 184]	287	249554169856
[6 0 48]	288	-92936337650224
[36 0 8]	288	-4413649854240
[38 -4 8]	288	-4413649854240
[12 0 24]	288	-92936337650224
[18 0 16]	288	45866534520416
[22 -8 16]	288	45927154398640
[18 24 48]	288	998345092432
[18 -60 216]	288	-10198386594548
[38 -3 8]	295	0
[38 -117 368]	295	803147685498880
[10 5 32]	295	803147685498880
[20 -5 16]	295	618923180011520
[6 6 56]	300	-221582599933530
[38 -2 8]	300	528629708975400
[10 10 40]	300	13634371034203
[14 6 24]	300	-221582599933530
[14 34 104]	300	-1911492099257
[38 -1 8]	303	0
[8 9 48]	303	-89630609432576
[26 -81 264]	303	-89630609432576
[22 -7 16]	303	-622432952324096
[22 -73 256]	303	82950147854336
[38 0 8]	304	-1067427147614080
[8 4 40]	304	1770519930281106
[40 -4 8]	304	-1067427147614080
[8 12 56]	304	10818144991282
[10 4 32]	304	1770519930281106
[10 24 88]	304	-5228395191330
[14 16 40]	304	-5228395191330
[22 -68 224]	304	10818144991282
[20 -4 16]	304	2194785235992592
[52 -157 480]	311	-739045046210560
[8 3 40]	311	2174093749153792
[40 -3 8]	311	0
[10 3 32]	311	2174093749153792
[10 13 48]	311	932560396288
[12 13 40]	311	932560396288
[12 29 96]	311	-517063802880
[14 5 24]	311	-739045046210560
[14 19 48]	311	-517063802880
[20 -3 16]	311	3064424844046336
[20 -67 240]	311	-46210579968000
[8 2 40]	316	453251469881784
[40 -2 8]	316	-1492793971437868
[10 2 32]	316	453251469881784
[20 -2 16]	316	696174589189626
[16 18 40]	316	9843849155383
[20 -62 208]	316	-10930471176977
[22 -6 16]	316	696704667737768
[8 1 40]	319	-3017939663339520
[40 -1 8]	319	0
[10 1 32]	319	-3017939663339520
[32 -97 304]	319	-63206927609856
[20 -1 16]	319	-3777184215490560
[54 -164 504]	320	-1665552321510660
[8 0 40]	320	-4911376952519500
[40 0 8]	320	5391275648088600
[8 8 48]	320	-576604967299904
[42 -4 8]	320	5391275648088600
[10 0 32]	320	-4911376952519500
[12 16 48]	320	-14822175830808
[28 -88 288]	320	-576604967299904
[14 4 24]	320	-1665552321510660
[20 0 16]	320	-6051141068211576
[16 8 24]	320	523926126038524
[24 -8 16]	320	-6052264724047968
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