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^(-1) + ζ)
  +(92 + 2/ζ^2 + 24/ζ + 24*ζ + 2*ζ^2)
  +q(-52/ζ^4 - 52*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
  (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(ζ^(-1) + ζ)
+(92 + 2/ζ^2 + 24/ζ + 24*ζ + 2*ζ^2)
+q(201064 - 52/ζ^4 - 3845/ζ^3 - 128/ζ^2 + 100377/ζ + 100377*ζ - 128*ζ^2 - 3845*ζ^3 - 52*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	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	0
[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	0
[14 -4 8]	96	0
[14 -32 80]	96	0
[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	0
[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	0
[8 4 16]	112	0
[16 -4 8]	112	0
[8 12 32]	112	0
[16 -36 88]	112	0
[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	0
[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	0
[8 16 48]	128	0
[18 -4 8]	128	0
[18 -40 96]	128	0
[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	0
[2 2 72]	140	0
[6 2 24]	140	0
[6 14 56]	140	0
[18 -2 8]	140	0
[18 -38 88]	140	0
[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	0
[14 -33 88]	143	0
[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	0
[20 -4 8]	144	0
[20 -44 104]	144	0
[10 4 16]	144	0
[10 24 72]	144	0
[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	1
[20 -62 200]	156	0
[10 2 16]	156	0
[10 22 64]	156	0
[12 6 16]	156	0
[12 18 40]	156	0
[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	0
[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	-2
[10 0 16]	160	0
[10 20 56]	160	0
[14 20 40]	160	0
[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	0
[14 -43 144]	167	0
[12 5 16]	167	0
[16 -37 96]	167	0
[22 -2 8]	172	0
[4 9 64]	175	0
[22 -1 8]	175	0
[8 9 32]	175	0
[10 15 40]	175	-1
[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	0
[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	-1
[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	1
[14 6 16]	188	0
[14 34 96]	188	1
[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	2
[12 1 16]	191	0
[16 -33 80]	191	1
[12 17 40]	191	2
[18 -41 104]	191	1
[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	-2
[14 16 32]	192	-2
[14 -44 152]	192	0
[16 8 16]	192	0
[16 24 48]	192	-2
[4 5 56]	199	0
[26 -3 8]	199	0
[8 11 40]	199	0
[10 11 32]	199	0
[10 21 64]	199	-1
[14 5 16]	199	0
[14 19 40]	199	-1
[6 6 40]	204	0
[26 -2 8]	204	0
[10 14 40]	204	24
[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	-52
[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	-3
[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	51
[20 -45 112]	215	51
[12 29 88]	215	-56
[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	55
[16 6 16]	220	0
[16 -38 104]	220	55
[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	-26
[12 16 40]	224	-26
[20 -64 216]	224	0
[14 0 16]	224	0
[14 28 72]	224	-106
[18 -8 16]	224	0
[18 -44 120]	224	-106
[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	-275
[20 -43 104]	231	-1311
[12 27 80]	231	83
[14 21 48]	231	83
[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	32
[14 18 40]	236	32
[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	-29
[22 -49 120]	239	1259
[22 -71 240]	239	0
[18 -7 16]	239	0
[16 23 48]	239	-29
[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	-60
[12 12 32]	240	274
[16 4 16]	240	0
[16 20 40]	240	-60
[16 -36 96]	240	-136
[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	-2595
[14 14 32]	252	-64
[16 2 16]	252	0
[16 -34 88]	252	1450
[18 -6 16]	252	0
[18 -42 112]	252	1450
[4 1 64]	255	0
[6 9 56]	255	0
[8 1 32]	255	0
[32 -1 8]	255	0
[10 15 48]	255	90
[12 15 40]	255	90
[20 -65 224]	255	0
[14 33 96]	255	-1475
[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	2000
[26 -80 256]	256	0
[16 0 16]	256	0
[16 16 32]	256	-420
[16 -32 80]	256	-2734
[20 -8 16]	256	0
[20 -48 128]	256	-2734
[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	3016
[14 17 40]	271	3016
[14 31 88]	271	5381
[20 -7 16]	271	0
[16 39 112]	271	5381
[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	2812
[14 20 48]	272	2812
[22 -72 248]	272	0
[18 -4 16]	272	0
[18 -40 104]	272	-6728
[70 -211 640]	279	0
[6 3 48]	279	0
[36 -3 8]	279	0
[8 19 80]	279	-440
[10 11 40]	279	-10349
[10 21 72]	279	-2831
[12 3 24]	279	0
[20 -61 200]	279	-440
[14 29 80]	279	-3988
[18 -3 16]	279	0
[16 19 40]	279	-2831
[18 27 56]	279	-3988
[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	352
[30 -94 304]	284	0
[12 2 24]	284	0
[12 14 40]	284	352
[24 -74 240]	284	10075
[12 26 80]	284	4034
[18 -2 16]	284	0
[18 -38 96]	284	24365
[20 -6 16]	284	0
[16 22 48]	284	4034
[20 -46 120]	284	24365
[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	78160
[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	-8452
[18 0 16]	288	0
[22 -8 16]	288	0
[22 -52 136]	288	-37964
[18 24 48]	288	-8452
[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	-28410
[6 6 56]	300	0
[38 -2 8]	300	0
[10 10 40]	300	40456
[14 6 24]	300	0
[14 34 104]	300	-68468
[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	-80608
[14 16 40]	304	-80608
[22 -68 224]	304	2640
[20 -4 16]	304	0
[20 -44 112]	304	-129430
[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	-3016
[12 13 40]	311	-3016
[12 29 96]	311	-201483
[14 5 24]	311	0
[14 19 48]	311	-201483
[20 -3 16]	311	0
[16 35 96]	311	107653
[20 -67 240]	311	0
[18 43 120]	311	107653
[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	196256
[20 -2 16]	316	0
[16 18 40]	316	196256
[20 -62 208]	316	-8175
[22 -6 16]	316	0
[22 -50 128]	316	292544
[4 1 80]	319	0
[8 1 40]	319	0
[40 -1 8]	319	0
[10 1 32]	319	0
[32 -97 304]	319	-123695
[20 -1 16]	319	0
[16 33 88]	319	-84594
[20 -49 136]	319	-84594
[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	-236000
[12 16 48]	320	-851022
[28 -88 288]	320	113620
[14 4 24]	320	0
[14 32 96]	320	-17982
[20 0 16]	320	0
[16 8 24]	320	0
[24 -8 16]	320	0
[16 24 56]	320	-17982
[24 -56 144]	320	-369600
[18 20 40]	320	-236000
[18 -56 192]	320	-9080
[6 3 56]	327	0
[42 -3 8]	327	0
[8 11 56]	327	10424
[12 27 88]	327	304937
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