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(2)
  +(92 + 2/ζ^2 + 24/ζ + 24*ζ + 2*ζ^2)
  +q(88/ζ^4 + 88*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
  (16,0)	2

• 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(2)
+(92 + 2/ζ^2 + 24/ζ + 24*ζ + 2*ζ^2)
+q(132472 + 88/ζ^4 + 4072/ζ^3 + 32256/ζ^2 + 94232/ζ + 94232*ζ + 32256*ζ^2 + 4072*ζ^3 + 88*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	1
[12 2 16]	188	0
[12 14 32]	188	1
[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	-2
[26 -4 8]	192	0
[12 0 16]	192	0
[12 24 64]	192	0
[14 16 32]	192	-2
[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	-2
[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	-24
[10 5 24]	215	0
[12 13 32]	215	-24
[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	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	2
[12 16 40]	224	2
[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	48
[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	-86
[14 18 40]	236	-86
[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	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	0
[32 -4 8]	240	0
[8 12 48]	240	274
[10 0 24]	240	0
[10 20 64]	240	172
[12 12 32]	240	274
[16 4 16]	240	0
[16 20 40]	240	172
[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	-64
[12 6 24]	252	0
[12 18 48]	252	170
[14 14 32]	252	-64
[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	-24
[12 15 40]	255	-24
[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	0
[8 16 64]	256	-420
[34 -4 8]	256	0
[10 12 40]	256	-548
[26 -80 256]	256	0
[16 0 16]	256	0
[16 16 32]	256	-420
[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	-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	-2008
[14 17 40]	271	-2008
[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	-172
[14 20 48]	272	-172
[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	-440
[10 11 40]	279	3952
[10 21 72]	279	1984
[12 3 24]	279	0
[20 -61 200]	279	-440
[14 29 80]	279	0
[18 -3 16]	279	0
[16 19 40]	279	1984
[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	10075
[36 -110 344]	284	0
[10 14 48]	284	64
[30 -94 304]	284	0
[12 2 24]	284	0
[12 14 40]	284	64
[24 -74 240]	284	10075
[12 26 80]	284	3565
[18 -2 16]	284	0
[18 -38 96]	284	0
[20 -6 16]	284	0
[16 22 48]	284	3565
[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	2416
[12 1 24]	287	0
[12 17 48]	287	3968
[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	-7132
[18 0 16]	288	0
[22 -8 16]	288	0
[22 -52 136]	288	0
[18 24 48]	288	-7132
[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	0
[6 6 56]	300	0
[38 -2 8]	300	0
[10 10 40]	300	-20150
[14 6 24]	300	0
[14 34 104]	300	-6958
[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	41936
[14 16 40]	304	41936
[22 -68 224]	304	2640
[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	0
[40 -3 8]	311	0
[10 3 32]	311	0
[10 13 48]	311	440
[12 13 40]	311	440
[12 29 96]	311	-1960
[14 5 24]	311	0
[14 19 48]	311	-1960
[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	0
[40 -2 8]	316	0
[8 18 80]	316	-8175
[10 2 32]	316	0
[10 22 80]	316	-62016
[20 -2 16]	316	0
[16 18 40]	316	-62016
[20 -62 208]	316	-8175
[22 -6 16]	316	0
[22 -50 128]	316	0
[4 1 80]	319	0
[8 1 40]	319	0
[40 -1 8]	319	0
[10 1 32]	319	0
[32 -97 304]	319	76960
[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	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	41128
[12 16 48]	320	-83032
[28 -88 288]	320	113620
[14 4 24]	320	0
[14 32 96]	320	7130
[20 0 16]	320	0
[16 8 24]	320	0
[24 -8 16]	320	0
[16 24 56]	320	7130
[24 -56 144]	320	0
[18 20 40]	320	41128
[18 -56 192]	320	-9080
[6 3 56]	327	0
[42 -3 8]	327	0
[8 11 56]	327	10424
[12 27 88]	327	170448
[14 3 24]	327	0
[24 -75 248]	327	10424
[22 -5 16]	327	0
[16 21 48]	327	170448
[6 2 56]	332	0
[6 14 88]	332	0
[42 -2 8]	332	0
[14 2 24]	332	0
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