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
  (92 + ζ^(-4) + 5/ζ^2 - 4/ζ - 4*ζ + 5*ζ^2 + ζ^4)
  +q(78/ζ^4 + 78*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,−1,2,2,2,2,2,4)
  		= 1−42541
  		= η(τ)92(θ(τ,z)/η(τ))−4(θ(τ,2z)/η(τ))5(θ(τ,4z)/η(τ))

  This theta block has q-order of vanishing 4. 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)	2
  (4,2)	6
  (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

(92 + ζ^(-4) + 5/ζ^2 - 4/ζ - 4*ζ + 5*ζ^2 + ζ^4)
+q(66404 + 78/ζ^4 - 2044/ζ^3 + 15872/ζ^2 - 47108/ζ - 47108*ζ + 15872*ζ^2 - 2044*ζ^3 + 78*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	0
[12 1 16]	191	0
[16 -33 80]	191	0
[12 17 40]	191	0
[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	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	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	0
[10 5 24]	215	0
[12 13 32]	215	0
[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	0
[14 1 16]	223	0
[14 15 32]	223	0
[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	0
[12 16 40]	224	0
[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	1
[10 3 24]	231	0
[10 13 40]	231	1
[20 -43 104]	231	0
[12 27 80]	231	0
[14 21 48]	231	0
[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	0
[22 -49 120]	239	0
[22 -71 240]	239	0
[18 -7 16]	239	0
[16 23 48]	239	0
[4 4 64]	240	0
[6 0 40]	240	0
[6 12 64]	240	0
[30 0 8]	240	0
[8 4 32]	240	4
[32 -4 8]	240	0
[8 12 48]	240	0
[10 0 24]	240	0
[10 20 64]	240	0
[12 12 32]	240	0
[16 4 16]	240	0
[16 20 40]	240	0
[16 -36 96]	240	0
[16 -52 184]	240	0
[62 -187 568]	247	0
[8 3 32]	247	5
[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	-6
[32 -1 8]	255	0
[10 15 48]	255	-5
[12 15 40]	255	-5
[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	-8
[32 0 8]	256	0
[8 8 40]	256	-4
[8 16 64]	256	0
[34 -4 8]	256	0
[10 12 40]	256	4
[26 -80 256]	256	-4
[16 0 16]	256	0
[16 16 32]	256	0
[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	5
[22 -67 216]	263	5
[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	-10
[28 -87 280]	271	-10
[10 23 80]	271	9
[14 17 40]	271	9
[14 31 88]	271	0
[20 -7 16]	271	0
[16 39 112]	271	0
[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	0
[14 20 48]	272	0
[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	-9
[10 11 40]	279	15
[10 21 72]	279	-5
[12 3 24]	279	0
[20 -61 200]	279	-9
[14 29 80]	279	0
[18 -3 16]	279	0
[16 19 40]	279	-5
[18 27 56]	279	0
[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	16
[36 -110 344]	284	-16
[10 14 48]	284	-16
[30 -94 304]	284	-16
[12 2 24]	284	0
[12 14 40]	284	-16
[24 -74 240]	284	16
[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	5
[12 1 24]	287	0
[12 17 48]	287	-53
[14 7 24]	287	0
[18 -1 16]	287	0
[18 -55 184]	287	5
[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	0
[18 0 16]	288	0
[22 -8 16]	288	0
[22 -52 136]	288	0
[18 24 48]	288	0
[18 -60 216]	288	0
[4 5 80]	295	0
[38 -3 8]	295	0
[38 -117 368]	295	-95
[10 5 32]	295	-95
[20 -5 16]	295	0
[16 37 104]	295	0
[6 6 56]	300	0
[38 -2 8]	300	0
[10 10 40]	300	32
[14 6 24]	300	0
[14 34 104]	300	0
[6 9 64]	303	0
[38 -1 8]	303	0
[8 9 48]	303	110
[26 -81 264]	303	110
[22 -7 16]	303	0
[22 -73 256]	303	0
[4 4 80]	304	0
[38 0 8]	304	0
[8 4 40]	304	-336
[40 -4 8]	304	0
[8 12 56]	304	-16
[10 4 32]	304	-336
[10 24 88]	304	16
[14 16 40]	304	16
[22 -68 224]	304	-16
[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	-413
[40 -3 8]	311	0
[10 3 32]	311	-413
[10 13 48]	311	-57
[12 13 40]	311	-57
[12 29 96]	311	340
[14 5 24]	311	0
[14 19 48]	311	340
[20 -3 16]	311	0
[16 35 96]	311	0
[20 -67 240]	311	0
[18 43 120]	311	0
[4 2 80]	316	0
[8 2 40]	316	16
[40 -2 8]	316	0
[8 18 80]	316	-16
[10 2 32]	316	16
[10 22 80]	316	16
[20 -2 16]	316	0
[16 18 40]	316	16
[20 -62 208]	316	-16
[22 -6 16]	316	0
[22 -50 128]	316	0
[4 1 80]	319	0
[8 1 40]	319	519
[40 -1 8]	319	0
[10 1 32]	319	519
[32 -97 304]	319	143
[20 -1 16]	319	0
[16 33 88]	319	0
[20 -49 136]	319	0
[4 0 80]	320	0
[6 8 64]	320	0
[54 -164 504]	320	0
[8 0 40]	320	680
[40 0 8]	320	0
[8 8 48]	320	360
[8 16 72]	320	40
[42 -4 8]	320	0
[10 0 32]	320	680
[10 20 72]	320	-40
[12 16 48]	320	1792
[28 -88 288]	320	360
[14 4 24]	320	0
[14 32 96]	320	0
[20 0 16]	320	0
[16 8 24]	320	0
[24 -8 16]	320	0
[16 24 56]	320	0
[24 -56 144]	320	0
[18 20 40]	320	-40
[18 -56 192]	320	40
[6 3 56]	327	0
[42 -3 8]	327	0
[8 11 56]	327	-435
[12 27 88]	327	-652
[14 3 24]	327	0
[24 -75 248]	327	-435
[22 -5 16]	327	0
[16 21 48]	327	-652
[6 2 56]	332	0
[6 14 88]	332	0
[42 -2 8]	332	0
[14 2 24]	332	0
[14 30 88]	332	0
[18 26 56]	332	0
[22 -74 264]	332	0
[6 1 56]	335	0
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