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(1)
  +(92 + ζ^(-4) + ζ^(-2) + 12/ζ + 12*ζ + ζ^2 + ζ^4)
  +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;1,1,1,1,1,1,1,1,1,1,1,1,2,4)
  		= 1122141
  		= η(τ)92(θ(τ,z)/η(τ))12(θ(τ,2z)/η(τ))(θ(τ,4z)/η(τ))

  This theta block has q-order of vanishing 5. 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 an antisymmetric 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 0 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)	14
  (4,2)	2
  (16,0)	1
  (16,4)	1

• From the singular part above, we compute that the valuation Ord(ψ) is -1, 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(1)
+(92 + ζ^(-4) + ζ^(-2) + 12/ζ + 12*ζ + ζ^2 + ζ^4)
+q(132108 + 68/ζ^4 - 12/ζ^3 + 32320/ζ^2 + 12/ζ + 12*ζ + 32320*ζ^2 - 12*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	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	0
[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	0
[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	0
[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	0
[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	0
[12 2 16]	188	0
[12 14 32]	188	0
[16 -50 168]	188	0
[12 26 72]	188	0
[14 6 16]	188	0
[14 34 96]	188	0
[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	-1
[12 1 16]	191	0
[16 -33 80]	191	0
[12 17 40]	191	1
[18 -41 104]	191	0
[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	0
[26 -4 8]	192	0
[12 0 16]	192	0
[12 24 64]	192	0
[14 16 32]	192	0
[14 -44 152]	192	0
[16 8 16]	192	0
[16 24 48]	192	0
[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	0
[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	0
[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	0
[54 -163 496]	215	0
[36 -109 336]	215	0
[6 13 64]	215	0
[28 -3 8]	215	0
[8 19 72]	215	1
[10 5 24]	215	0
[12 13 32]	215	-1
[18 -37 88]	215	0
[20 -45 112]	215	0
[12 29 88]	215	0
[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	0
[16 6 16]	220	0
[16 -38 104]	220	0
[4 1 56]	223	0
[28 -1 8]	223	0
[8 17 64]	223	-1
[14 1 16]	223	0
[14 15 32]	223	1
[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	12
[12 16 40]	224	-12
[20 -64 216]	224	0
[14 0 16]	224	0
[14 28 72]	224	0
[18 -8 16]	224	0
[18 -44 120]	224	0
[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	0
[20 -43 104]	231	0
[12 27 80]	231	66
[14 21 48]	231	-66
[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	0
[14 18 40]	236	0
[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	-1
[10 1 24]	239	0
[24 -73 232]	239	1
[12 25 72]	239	-67
[22 -49 120]	239	0
[22 -71 240]	239	0
[18 -7 16]	239	0
[16 23 48]	239	67
[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	-12
[10 0 24]	240	0
[10 20 64]	240	-24
[12 12 32]	240	12
[16 4 16]	240	0
[16 20 40]	240	24
[16 -36 96]	240	0
[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	0
[12 6 24]	252	0
[12 18 48]	252	0
[14 14 32]	252	0
[16 2 16]	252	0
[16 -34 88]	252	0
[18 -6 16]	252	0
[18 -42 112]	252	0
[4 1 64]	255	0
[6 9 56]	255	0
[8 1 32]	255	0
[32 -1 8]	255	0
[10 15 48]	255	-66
[12 15 40]	255	66
[20 -65 224]	255	0
[14 33 96]	255	0
[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	12
[8 16 64]	256	24
[34 -4 8]	256	0
[10 12 40]	256	0
[26 -80 256]	256	-12
[16 0 16]	256	0
[16 16 32]	256	-24
[16 -32 80]	256	0
[20 -8 16]	256	0
[20 -48 128]	256	0
[4 5 72]	263	0
[44 -133 408]	263	0
[6 13 72]	263	0
[34 -3 8]	263	0
[8 11 48]	263	66
[22 -67 216]	263	-66
[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	-65
[28 -87 280]	271	65
[10 23 80]	271	-54
[14 17 40]	271	54
[14 31 88]	271	-2054
[20 -7 16]	271	0
[16 39 112]	271	2054
[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	-816
[14 20 48]	272	816
[22 -72 248]	272	0
[18 -4 16]	272	0
[18 -40 104]	272	0
[70 -211 640]	279	0
[6 3 48]	279	0
[36 -3 8]	279	0
[8 19 80]	279	-12
[10 11 40]	279	0
[10 21 72]	279	120
[12 3 24]	279	0
[20 -61 200]	279	12
[14 29 80]	279	2121
[18 -3 16]	279	0
[16 19 40]	279	-120
[18 27 56]	279	-2121
[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	-208
[36 -110 344]	284	208
[10 14 48]	284	208
[30 -94 304]	284	-208
[12 2 24]	284	0
[12 14 40]	284	-208
[24 -74 240]	284	208
[12 26 80]	284	0
[18 -2 16]	284	0
[18 -38 96]	284	0
[20 -6 16]	284	0
[16 22 48]	284	0
[20 -46 120]	284	0
[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	-53
[12 1 24]	287	0
[12 17 48]	287	0
[14 7 24]	287	0
[18 -1 16]	287	0
[18 -55 184]	287	53
[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	1656
[18 0 16]	288	0
[22 -8 16]	288	0
[22 -52 136]	288	0
[18 24 48]	288	-1656
[18 -60 216]	288	0
[4 5 80]	295	0
[38 -3 8]	295	0
[38 -117 368]	295	-428
[10 5 32]	295	428
[20 -5 16]	295	0
[16 37 104]	295	0
[6 6 56]	300	0
[38 -2 8]	300	0
[10 10 40]	300	0
[14 6 24]	300	0
[14 34 104]	300	0
[6 9 64]	303	0
[38 -1 8]	303	0
[8 9 48]	303	441
[26 -81 264]	303	-441
[22 -7 16]	303	0
[22 -73 256]	303	0
[4 4 80]	304	0
[38 0 8]	304	0
[8 4 40]	304	560
[40 -4 8]	304	0
[8 12 56]	304	496
[10 4 32]	304	-560
[10 24 88]	304	320
[14 16 40]	304	-320
[22 -68 224]	304	-496
[20 -4 16]	304	0
[20 -44 112]	304	0
[78 -235 712]	311	0
[52 -157 480]	311	0
[6 13 80]	311	0
[8 3 40]	311	-364
[40 -3 8]	311	0
[10 3 32]	311	364
[10 13 48]	311	-375
[12 13 40]	311	375
[12 29 96]	311	36794
[14 5 24]	311	0
[14 19 48]	311	-36794
[20 -3 16]	311	0
[16 35 96]	311	39821
[20 -67 240]	311	0
[18 43 120]	311	-39821
[4 2 80]	316	0
[8 2 40]	316	-208
[40 -2 8]	316	0
[8 18 80]	316	208
[10 2 32]	316	208
[10 22 80]	316	-208
[20 -2 16]	316	0
[16 18 40]	316	208
[20 -62 208]	316	-208
[22 -6 16]	316	0
[22 -50 128]	316	0
[4 1 80]	319	0
[8 1 40]	319	858
[40 -1 8]	319	0
[10 1 32]	319	-858
[32 -97 304]	319	0
[20 -1 16]	319	0
[16 33 88]	319	-41941
[20 -49 136]	319	41941
[4 0 80]	320	0
[6 8 64]	320	0
[54 -164 504]	320	0
[8 0 40]	320	-1144
[40 0 8]	320	0
[8 8 48]	320	-1080
[8 16 72]	320	-1016
[42 -4 8]	320	0
[10 0 32]	320	1144
[10 20 72]	320	-640
[12 16 48]	320	0
[28 -88 288]	320	1080
[14 4 24]	320	0
[14 32 96]	320	26256
[20 0 16]	320	0
[16 8 24]	320	0
[24 -8 16]	320	0
[16 24 56]	320	-26256
[24 -56 144]	320	0
[18 20 40]	320	640
[18 -56 192]	320	1016
[6 3 56]	327	0
[42 -3 8]	327	0
[8 11 56]	327	-3717
[12 27 88]	327	101277
[14 3 24]	327	0
[24 -75 248]	327	3717
[22 -5 16]	327	0
[16 21 48]	327	-101277
[6 2 56]	332	0
[6 14 88]	332	0
[42 -2 8]	332	0
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