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^(-2) + ζ^2)
  +(92 + 2/ζ^2 + 24/ζ + 24*ζ + 2*ζ^2)
  +q(ζ^(-6) + 44/ζ^4 + 44*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;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
  (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(ζ^(-2) + ζ^2)
+(92 + 2/ζ^2 + 24/ζ + 24*ζ + 2*ζ^2)
+q(657064 + ζ^(-6) + 44/ζ^4 - 24/ζ^3 - 131693/ζ^2 + 24/ζ + 24*ζ - 131693*ζ^2 - 24*ζ^3 + 44*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	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	0
[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	0
[12 -4 8]	80	0
[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	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	1
[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	-2
[14 -4 8]	96	0
[14 -32 80]	96	-2
[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	-40
[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	0
[2 4 64]	112	0
[4 4 32]	112	0
[14 0 8]	112	0
[14 -28 64]	112	84
[8 4 16]	112	0
[16 -4 8]	112	0
[8 12 32]	112	0
[16 -36 88]	112	84
[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	0
[12 -29 80]	119	0
[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	687
[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	-1544
[8 16 48]	128	0
[18 -4 8]	128	0
[18 -40 96]	128	-1544
[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	-24
[2 2 72]	140	0
[6 2 24]	140	0
[6 14 56]	140	0
[18 -2 8]	140	0
[18 -38 88]	140	-6200
[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	24
[14 -33 88]	143	24
[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	15568
[20 -4 8]	144	0
[20 -44 104]	144	15568
[10 4 16]	144	0
[10 24 72]	144	-1
[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	24637
[20 -62 200]	156	0
[10 2 16]	156	0
[10 22 64]	156	2
[12 6 16]	156	0
[12 18 40]	156	2
[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	1080
[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	-81780
[10 0 16]	160	0
[10 20 56]	160	-2
[14 20 40]	160	-2
[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	-1104
[14 -43 144]	167	0
[12 5 16]	167	0
[16 -37 96]	167	-1104
[22 -2 8]	172	0
[4 9 64]	175	0
[22 -1 8]	175	0
[8 9 32]	175	0
[10 15 40]	175	24
[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	314
[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	-21600
[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	-146
[14 6 16]	188	0
[14 34 96]	188	-146
[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	-24
[12 1 16]	191	0
[16 -33 80]	191	22680
[12 17 40]	191	-24
[18 -41 104]	191	22680
[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	-340
[14 16 32]	192	-2
[14 -44 152]	192	0
[16 8 16]	192	0
[16 24 48]	192	-340
[4 5 56]	199	0
[26 -3 8]	199	0
[8 11 40]	199	0
[10 11 32]	199	0
[10 21 64]	199	0
[14 5 16]	199	0
[14 19 40]	199	0
[6 6 40]	204	0
[26 -2 8]	204	0
[10 14 40]	204	-274
[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	244488
[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	118108
[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	-266040
[20 -45 112]	215	-266040
[12 29 88]	215	-3056
[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	-916454
[16 6 16]	220	0
[16 -38 104]	220	-916454
[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	62
[12 16 40]	224	62
[20 -64 216]	224	0
[14 0 16]	224	0
[14 28 72]	224	1596988
[18 -8 16]	224	0
[18 -44 120]	224	1596988
[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	1976
[20 -43 104]	231	-1582200
[12 27 80]	231	640
[14 21 48]	231	640
[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	106
[14 18 40]	236	106
[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	2440
[22 -49 120]	239	1824480
[22 -71 240]	239	0
[18 -7 16]	239	0
[16 23 48]	239	2440
[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	212
[12 12 32]	240	274
[16 4 16]	240	0
[16 20 40]	240	212
[16 -36 96]	240	33297568
[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	-108719
[14 14 32]	252	-64
[16 2 16]	252	0
[16 -34 88]	252	-147932956
[18 -6 16]	252	0
[18 -42 112]	252	-147932956
[4 1 64]	255	0
[6 9 56]	255	0
[8 1 32]	255	0
[32 -1 8]	255	0
[10 15 48]	255	464
[12 15 40]	255	464
[20 -65 224]	255	0
[14 33 96]	255	-3045328
[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	-10076
[26 -80 256]	256	0
[16 0 16]	256	0
[16 16 32]	256	-420
[16 -32 80]	256	231104360
[20 -8 16]	256	0
[20 -48 128]	256	231104360
[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	640
[14 17 40]	271	640
[14 31 88]	271	18937776
[20 -7 16]	271	0
[16 39 112]	271	18937776
[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	920700
[14 20 48]	272	920700
[22 -72 248]	272	0
[18 -4 16]	272	0
[18 -40 104]	272	1558413075
[70 -211 640]	279	0
[6 3 48]	279	0
[36 -3 8]	279	0
[8 19 80]	279	-440
[10 11 40]	279	38504
[10 21 72]	279	-3080
[12 3 24]	279	0
[20 -61 200]	279	-440
[14 29 80]	279	-15893088
[18 -3 16]	279	0
[16 19 40]	279	-3080
[18 27 56]	279	-15893088
[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	-2702
[30 -94 304]	284	0
[12 2 24]	284	0
[12 14 40]	284	-2702
[24 -74 240]	284	10075
[12 26 80]	284	-1724646
[18 -2 16]	284	0
[18 -38 96]	284	-5012701046
[20 -6 16]	284	0
[16 22 48]	284	-1724646
[20 -46 120]	284	-5012701046
[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	3028424
[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	1825204
[18 0 16]	288	0
[22 -8 16]	288	0
[22 -52 136]	288	7201248172
[18 24 48]	288	1825204
[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	-747322440
[6 6 56]	300	0
[38 -2 8]	300	0
[10 10 40]	300	-113894
[14 6 24]	300	0
[14 34 104]	300	1656698
[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	116896
[14 16 40]	304	116896
[22 -68 224]	304	2640
[20 -4 16]	304	0
[20 -44 112]	304	25006405182
[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	-10888
[12 13 40]	311	-10888
[12 29 96]	311	-18949568
[14 5 24]	311	0
[14 19 48]	311	-18949568
[20 -3 16]	311	0
[16 35 96]	311	2699212904
[20 -67 240]	311	0
[18 43 120]	311	2699212904
[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	-911474
[20 -2 16]	316	0
[16 18 40]	316	-911474
[20 -62 208]	316	-8175
[22 -6 16]	316	0
[22 -50 128]	316	-55356418404
[4 1 80]	319	0
[8 1 40]	319	0
[40 -1 8]	319	0
[10 1 32]	319	0
[32 -97 304]	319	265096
[20 -1 16]	319	0
[16 33 88]	319	-1970830680
[20 -49 136]	319	-1970830680
[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
[8 8 48]	320	113620
[8 16 72]	320	-9080
[42 -4 8]	320	0
[10 0 32]	320	0
[10 20 72]	320	1614712
[12 16 48]	320	-34834172
[28 -88 288]	320	113620
[14 4 24]	320	0
[14 32 96]	320	-2830894
[20 0 16]	320	0
[16 8 24]	320	0
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