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 + 2/ζ^2 + 24/ζ + 24*ζ + 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;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2)
  		= 12422
  		= η(τ)92(θ(τ,z)/η(τ))24(θ(τ,2z)/η(τ))2

  This theta block has q-order of vanishing 6. This theta block has index 4*4, and so Borch(ψ) begins with Jacobi coefficient #4.
• 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)	26
  (4,2)	2
  (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 + 2/ζ^2 + 24/ζ + 24*ζ + 2*ζ^2)
+q(17436064 + 68/ζ^4 - 94232/ζ^3 + 2128384/ζ^2 - 10555368/ζ - 10555368*ζ + 2128384*ζ^2 - 94232*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	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
[18 -37 80]	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
[20 -41 88]	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	0
[2 2 48]	92	0
[4 2 24]	92	0
[4 6 32]	92	0
[6 2 16]	92	0
[6 14 48]	92	0
[12 -2 8]	92	0
[8 10 24]	92	0
[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	0
[12 -1 8]	95	0
[8 17 48]	95	0
[10 15 32]	95	0
[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	0
[12 0 8]	96	0
[12 -24 56]	96	-132
[14 -4 8]	96	0
[14 -32 80]	96	-132
[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	0
[6 6 24]	108	0
[14 -2 8]	108	0
[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	24
[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	0
[16 -36 88]	112	4112
[2 3 64]	119	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	-24
[12 -29 80]	119	-24
[2 2 64]	124	0
[4 2 32]	124	0
[4 6 40]	124	0
[8 2 16]	124	0
[16 -2 8]	124	0
[16 -34 80]	124	35776
[8 18 56]	124	0
[10 14 32]	124	0
[14 -46 160]	124	0
[2 1 64]	127	0
[4 1 32]	127	0
[8 1 16]	127	0
[16 -1 8]	127	0
[2 0 64]	128	0
[2 4 72]	128	0
[4 0 32]	128	0
[4 8 48]	128	0
[6 8 32]	128	0
[22 -68 216]	128	0
[8 0 16]	128	0
[16 0 8]	128	0
[8 8 24]	128	0
[16 -32 72]	128	-79904
[8 16 48]	128	0
[18 -4 8]	128	0
[18 -40 96]	128	-79904
[12 16 32]	128	0
[12 -40 144]	128	0
[2 3 72]	135	0
[4 5 40]	135	0
[6 3 24]	135	0
[18 -3 8]	135	0
[8 11 32]	135	0
[10 5 16]	135	0
[12 -27 72]	135	-95864
[2 2 72]	140	0
[6 2 24]	140	0
[6 14 56]	140	0
[18 -2 8]	140	0
[18 -38 88]	140	-456171
[10 10 24]	140	0
[2 1 72]	143	0
[4 9 56]	143	0
[6 1 24]	143	0
[6 7 32]	143	0
[18 -1 8]	143	0
[18 -55 176]	143	0
[12 -25 64]	143	95888
[14 -33 88]	143	95888
[14 -47 168]	143	0
[2 0 72]	144	0
[2 4 80]	144	0
[4 4 40]	144	0
[6 0 24]	144	0
[6 12 48]	144	0
[18 0 8]	144	0
[18 -36 80]	144	1075992
[20 -4 8]	144	0
[20 -44 104]	144	1075992
[10 4 16]	144	0
[10 24 72]	144	-274
[12 12 24]	144	0
[2 3 80]	151	0
[38 -115 352]	151	0
[20 -3 8]	151	0
[8 19 64]	151	0
[10 3 16]	151	0
[10 13 32]	151	0
[2 2 80]	156	0
[4 2 40]	156	0
[4 6 48]	156	0
[6 6 32]	156	0
[20 -2 8]	156	0
[8 10 32]	156	0
[20 -42 96]	156	4140864
[20 -62 200]	156	0
[10 2 16]	156	0
[10 22 64]	156	64
[12 6 16]	156	0
[12 18 40]	156	64
[2 1 80]	159	0
[4 1 40]	159	0
[6 9 40]	159	0
[20 -1 8]	159	0
[8 17 56]	159	0
[10 1 16]	159	0
[16 -49 160]	159	0
[12 15 32]	159	0
[14 -31 80]	159	-4095104
[2 0 80]	160	0
[4 0 40]	160	0
[4 8 56]	160	0
[20 0 8]	160	0
[22 -4 8]	160	0
[22 -48 112]	160	-10505000
[10 0 16]	160	0
[10 20 56]	160	420
[14 20 40]	160	420
[14 -48 176]	160	0
[4 5 48]	167	0
[28 -85 264]	167	0
[6 13 56]	167	0
[22 -3 8]	167	0
[22 -69 224]	167	0
[14 -29 72]	167	3999240
[14 -43 144]	167	0
[12 5 16]	167	0
[16 -37 96]	167	3999240
[22 -2 8]	172	0
[4 9 64]	175	0
[22 -1 8]	175	0
[8 9 32]	175	0
[10 15 40]	175	1976
[14 7 16]	175	0
[4 4 48]	176	0
[6 8 40]	176	0
[30 -92 288]	176	0
[22 0 8]	176	0
[8 4 24]	176	0
[24 -4 8]	176	0
[8 12 40]	176	0
[10 12 32]	176	0
[18 -56 184]	176	0
[12 4 16]	176	0
[12 28 80]	176	4408848
[46 -139 424]	183	0
[6 3 32]	183	0
[8 3 24]	183	0
[24 -3 8]	183	0
[12 3 16]	183	0
[16 -35 88]	183	90263448
[4 2 48]	188	0
[4 6 56]	188	0
[6 2 32]	188	0
[6 14 64]	188	0
[8 2 24]	188	0
[24 -2 8]	188	0
[8 18 64]	188	1
[12 2 16]	188	0
[12 14 32]	188	1
[16 -50 168]	188	0
[12 26 72]	188	13170624
[14 6 16]	188	0
[14 34 96]	188	13170624
[4 1 48]	191	0
[6 1 32]	191	0
[6 7 40]	191	0
[8 1 24]	191	0
[24 -1 8]	191	0
[20 -63 208]	191	0
[10 23 72]	191	440
[12 1 16]	191	0
[16 -33 80]	191	-94358600
[12 17 40]	191	440
[18 -41 104]	191	-94358600
[4 0 48]	192	0
[4 8 64]	192	0
[6 0 32]	192	0
[6 12 56]	192	0
[8 0 24]	192	0
[24 0 8]	192	0
[8 8 32]	192	0
[8 16 56]	192	-2
[26 -4 8]	192	0
[12 0 16]	192	0
[12 24 64]	192	-35159072
[14 16 32]	192	-2
[14 -44 152]	192	0
[16 8 16]	192	0
[16 24 48]	192	-35159072
[4 5 56]	199	0
[26 -3 8]	199	0
[8 11 40]	199	0
[10 11 32]	199	0
[10 21 64]	199	-2416
[14 5 16]	199	0
[14 19 40]	199	-2416
[6 6 40]	204	0
[26 -2 8]	204	0
[10 14 40]	204	-10075
[22 -70 232]	204	0
[4 9 72]	207	0
[6 9 48]	207	0
[26 -1 8]	207	0
[26 -79 248]	207	0
[18 -39 96]	207	-31043864
[18 -57 192]	207	0
[16 7 16]	207	0
[4 4 56]	208	0
[26 0 8]	208	0
[28 -4 8]	208	0
[14 4 16]	208	0
[14 32 88]	208	-1413803400
[54 -163 496]	215	0
[36 -109 336]	215	0
[6 13 64]	215	0
[28 -3 8]	215	0
[8 19 72]	215	-24
[10 5 24]	215	0
[12 13 32]	215	-24
[18 -37 88]	215	121499088
[20 -45 112]	215	121499088
[12 29 88]	215	-87590776
[14 3 16]	215	0
[16 -51 176]	215	0
[4 2 56]	220	0
[4 6 64]	220	0
[28 -2 8]	220	0
[8 10 40]	220	0
[28 -86 272]	220	0
[10 10 32]	220	0
[14 2 16]	220	0
[14 30 80]	220	6860777792
[16 6 16]	220	0
[16 -38 104]	220	6860777792
[4 1 56]	223	0
[28 -1 8]	223	0
[8 17 64]	223	24
[14 1 16]	223	0
[14 15 32]	223	24
[4 0 56]	224	0
[4 8 72]	224	0
[6 8 48]	224	0
[38 -116 360]	224	0
[28 0 8]	224	0
[30 -4 8]	224	0
[10 4 24]	224	0
[10 24 80]	224	-2642
[12 16 40]	224	-2642
[20 -64 216]	224	0
[14 0 16]	224	0
[14 28 72]	224	-10920290872
[18 -8 16]	224	0
[18 -44 120]	224	-10920290872
[4 5 64]	231	0
[6 3 40]	231	0
[30 -3 8]	231	0
[30 -93 296]	231	0
[10 3 24]	231	0
[10 13 40]	231	38480
[20 -43 104]	231	-10882227288
[12 27 80]	231	-786346728
[14 21 48]	231	-786346728
[16 5 16]	231	0
[6 2 40]	236	0
[6 14 72]	236	0
[30 -2 8]	236	0
[10 2 24]	236	0
[10 22 72]	236	8111
[14 18 40]	236	8111
[18 -58 200]	236	0
[4 9 80]	239	0
[6 1 40]	239	0
[6 7 48]	239	0
[30 -1 8]	239	0
[8 9 40]	239	0
[10 1 24]	239	0
[24 -73 232]	239	0
[12 25 72]	239	873937064
[22 -49 120]	239	10859181928
[22 -71 240]	239	0
[18 -7 16]	239	0
[16 23 48]	239	873937064
[4 4 64]	240	0
[6 0 40]	240	0
[6 12 64]	240	0
[30 0 8]	240	0
[8 4 32]	240	0
[32 -4 8]	240	0
[8 12 48]	240	274
[10 0 24]	240	0
[10 20 64]	240	9212
[12 12 32]	240	274
[16 4 16]	240	0
[16 20 40]	240	9212
[16 -36 96]	240	-152339302144
[16 -52 184]	240	0
[62 -187 568]	247	0
[8 3 32]	247	0
[32 -3 8]	247	0
[16 3 16]	247	0
[4 2 64]	252	0
[4 6 72]	252	0
[6 6 48]	252	0
[8 2 32]	252	0
[32 -2 8]	252	0
[8 18 72]	252	-64
[12 6 24]	252	0
[12 18 48]	252	1040819050
[14 14 32]	252	-64
[16 2 16]	252	0
[16 -34 88]	252	650138834560
[18 -6 16]	252	0
[18 -42 112]	252	650138834560
[4 1 64]	255	0
[6 9 56]	255	0
[8 1 32]	255	0
[32 -1 8]	255	0
[10 15 48]	255	-10400
[12 15 40]	255	-10400
[20 -65 224]	255	0
[14 33 96]	255	80563543384
[16 1 16]	255	0
[4 0 64]	256	0
[4 8 80]	256	0
[8 0 32]	256	0
[32 0 8]	256	0
[8 8 40]	256	0
[8 16 64]	256	-420
[34 -4 8]	256	0
[10 12 40]	256	-113620
[26 -80 256]	256	0
[16 0 16]	256	0
[16 16 32]	256	-420
[16 -32 80]	256	-1009250302400
[20 -8 16]	256	0
[20 -48 128]	256	-1009250302400
[4 5 72]	263	0
[44 -133 408]	263	0
[6 13 72]	263	0
[34 -3 8]	263	0
[8 11 48]	263	-1976
[22 -67 216]	263	-1976
[12 5 24]	263	0
[18 -5 16]	263	0
[18 -59 208]	263	0
[34 -2 8]	268	0
[4 9 88]	271	0
[34 -1 8]	271	0
[34 -103 320]	271	0
[28 -87 280]	271	0
[10 23 80]	271	54128
[14 17 40]	271	54128
[14 31 88]	271	-315218640824
[20 -7 16]	271	0
[16 39 112]	271	-315218640824
[4 4 72]	272	0
[6 8 56]	272	0
[46 -140 432]	272	0
[34 0 8]	272	0
[36 -4 8]	272	0
[12 4 24]	272	0
[12 28 88]	272	17805910908
[14 20 48]	272	17805910908
[22 -72 248]	272	0
[18 -4 16]	272	0
[18 -40 104]	272	-6351251143082
[70 -211 640]	279	0
[6 3 48]	279	0
[36 -3 8]	279	0
[8 19 80]	279	-440
[10 11 40]	279	263120
[10 21 72]	279	-82208
[12 3 24]	279	0
[20 -61 200]	279	-440
[14 29 80]	279	235441446584
[18 -3 16]	279	0
[16 19 40]	279	-82208
[18 27 56]	279	235441446584
[4 2 72]	284	0
[4 6 80]	284	0
[6 2 48]	284	0
[6 14 80]	284	0
[36 -2 8]	284	0
[8 10 48]	284	10075
[36 -110 344]	284	0
[10 14 48]	284	183680
[30 -94 304]	284	0
[12 2 24]	284	0
[12 14 40]	284	183680
[24 -74 240]	284	10075
[12 26 80]	284	5833138221
[18 -2 16]	284	0
[18 -38 96]	284	19840248712512
[20 -6 16]	284	0
[16 22 48]	284	5833138221
[20 -46 120]	284	19840248712512
[20 -66 232]	284	0
[4 1 72]	287	0
[6 1 48]	287	0
[6 7 56]	287	0
[36 -1 8]	287	0
[8 17 72]	287	2416
[12 1 24]	287	0
[12 17 48]	287	-8478840272
[14 7 24]	287	0
[18 -1 16]	287	0
[18 -55 184]	287	2416
[4 0 72]	288	0
[6 0 48]	288	0
[6 12 72]	288	0
[36 0 8]	288	0
[38 -4 8]	288	0
[12 0 24]	288	0
[12 24 72]	288	-49359731076
[18 0 16]	288	0
[22 -8 16]	288	0
[22 -52 136]	288	-28256458566936
[18 24 48]	288	-49359731076
[18 -60 216]	288	0
[4 5 80]	295	0
[38 -3 8]	295	0
[38 -117 368]	295	0
[10 5 32]	295	0
[20 -5 16]	295	0
[16 37 104]	295	2588242230288
[6 6 56]	300	0
[38 -2 8]	300	0
[10 10 40]	300	-476906
[14 6 24]	300	0
[14 34 104]	300	-2090160213629
[6 9 64]	303	0
[38 -1 8]	303	0
[8 9 48]	303	-38480
[26 -81 264]	303	-38480
[22 -7 16]	303	0
[22 -73 256]	303	0
[4 4 80]	304	0
[38 0 8]	304	0
[8 4 40]	304	0
[40 -4 8]	304	0
[8 12 56]	304	2640
[10 4 32]	304	0
[10 24 88]	304	-4106912
[14 16 40]	304	-4106912
[22 -68 224]	304	2640
[20 -4 16]	304	0
[20 -44 112]	304	-99036122333008
[78 -235 712]	311	0
[52 -157 480]	311	0
[6 13 80]	311	0
[8 3 40]	311	0
[40 -3 8]	311	0
[10 3 32]	311	0
[10 13 48]	311	-1102112
[12 13 40]	311	-1102112
[12 29 96]	311	-235636611984
[14 5 24]	311	0
[14 19 48]	311	-235636611984
[20 -3 16]	311	0
[16 35 96]	311	-8860530792160
[20 -67 240]	311	0
[18 43 120]	311	-8860530792160
[4 2 80]	316	0
[8 2 40]	316	0
[40 -2 8]	316	0
[8 18 80]	316	-8175
[10 2 32]	316	0
[10 22 80]	316	-13320064
[20 -2 16]	316	0
[16 18 40]	316	-13320064
[20 -62 208]	316	-8175
[22 -6 16]	316	0
[22 -50 128]	316	223950689495424
[4 1 80]	319	0
[8 1 40]	319	0
[40 -1 8]	319	0
[10 1 32]	319	0
[32 -97 304]	319	657824
[20 -1 16]	319	0
[16 33 88]	319	6586633268048
[20 -49 136]	319	6586633268048
[4 0 80]	320	0
[6 8 64]	320	0
[54 -164 504]	320	0
[8 0 40]	320	0
[40 0 8]	320	0
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