A Borcherds Product in S₉(K(8))⁻

properties |

c vector |

θ vector |

ψ expansion |

Borch(ψ) coefficients

• The singular part of ψ truncated up to q^(8/4) is
  (18 + ζ^(-3) + 3/ζ^2 + 11/ζ + 11*ζ + 3*ζ^2 + ζ^3)
  +q(ζ^(-6) + ζ^6)
  +q^2(-2/ζ^8 - 2*ζ^8)
• From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block

  	φ	= TB(9;1,1,1,1,1,1,1,1,1,1,1,2,2,2,3)
  		= 1112331
  		= η(τ)18(θ(τ,z)/η(τ))11(θ(τ,2z)/η(τ))3(θ(τ,3z)/η(τ))

  This theta block has q-order of vanishing 2. This theta block has index 2*8, and so Borch(ψ) begins with Jacobi coefficient #2.
• 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 -1 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)	15
  (4,2)	3
  (4,6)	1
  (9,3)	1

• From the singular part above, we compute that the valuation Ord(ψ) is -0.28125, and hence we need to multiply ψ by Δ^1 to get a Jacobi cusp form. The definition of ψ is
      ψ = c · θ/ Δ^1.
  Here c is a vector of coefficients, θ is a basis of of J_12,8^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^(8/4) is

(18 + ζ^(-3) + 3/ζ^2 + 11/ζ + 11*ζ + 3*ζ^2 + ζ^3)
+q(160 + ζ^(-6) + 9/ζ^5 - 16/ζ^4 - 63/ζ^3 - 65/ζ^2 + 54/ζ + 54*ζ - 65*ζ^2 - 63*ζ^3 - 16*ζ^4 + 9*ζ^5 + ζ^6)
+q^2(1156 - 2/ζ^8 - 10/ζ^7 + 61/ζ^6 + 171/ζ^5 - 64/ζ^4 - 585/ζ^3 - 573/ζ^2 + 424/ζ + 424*ζ - 573*ζ^2 - 585*ζ^3 - 64*ζ^4 + 171*ζ^5 + 61*ζ^6 - 10*ζ^7 - 2*ζ^8)

2u	det(2u)	a(u, Borch(ψ))
[4 -21 112]	7	0
[8 -25 80]	15	0
[4 -25 160]	15	0
[2 4 16]	16	0
[2 3 16]	23	0
[6 -13 32]	23	-1
[6 -19 64]	23	1
[2 2 16]	28	0
[14 -42 128]	28	0
[4 6 16]	28	0
[8 -42 224]	28	1
[2 1 16]	31	0
[8 -49 304]	31	-3
[4 17 80]	31	3
[2 0 16]	32	0
[2 8 48]	32	0
[6 8 16]	32	0
[6 -32 176]	32	0
[20 -101 512]	39	0
[4 5 16]	39	0
[24 -73 224]	47	0
[4 9 32]	47	11
[6 7 16]	47	0
[2 4 32]	48	0
[4 4 16]	48	0
[12 -36 112]	48	-8
[6 12 32]	48	-8
[2 3 32]	55	0
[14 -99 704]	55	0
[2 2 32]	60	0
[30 -90 272]	60	0
[4 2 16]	60	0
[16 -50 160]	60	19
[6 6 16]	60	0
[10 -30 96]	60	0
[8 -18 48]	60	0
[8 -50 320]	60	19
[2 1 32]	63	0
[4 1 16]	63	0
[12 -63 336]	63	-3
[2 0 32]	64	0
[2 8 64]	64	0
[4 0 16]	64	0
[4 8 32]	64	-52
[4 16 80]	64	0
[8 8 16]	64	0
[10 -64 416]	64	0
[36 -181 912]	71	0
[18 -91 464]	71	-30
[12 -85 608]	71	0
[20 53 144]	71	-30
[40 -121 368]	79	0
[26 73 208]	79	-133
[10 -71 512]	79	0
[2 4 48]	80	0
[14 -100 720]	80	0
[2 3 48]	87	0
[22 -67 208]	87	-52
[6 3 16]	87	0
[26 67 176]	87	52
[2 2 48]	92	0
[46 -138 416]	92	0
[4 6 32]	92	-179
[24 -122 624]	92	80
[6 2 16]	92	0
[6 14 48]	92	-80
[26 70 192]	92	80
[12 -86 624]	92	0
[2 1 48]	95	0
[4 17 96]	95	71
[6 1 16]	95	0
[16 -81 416]	95	-60
[8 17 48]	95	60
[10 15 32]	95	-71
[2 0 48]	96	0
[2 8 80]	96	0
[6 0 16]	96	0
[10 -72 528]	96	0
[52 -261 1312]	103	0
[4 5 32]	103	52
[14 -101 736]	103	0
[56 -169 512]	111	0
[4 9 48]	111	165
[6 9 32]	111	165
[12 -87 640]	111	0
[2 4 64]	112	0
[4 4 32]	112	232
[28 -84 256]	112	104
[8 4 16]	112	0
[8 12 32]	112	104
[2 3 64]	119	0
[30 -211 1488]	119	404
[6 13 48]	119	508
[8 3 16]	119	0
[2 2 64]	124	0
[62 -186 560]	124	0
[4 2 32]	124	286
[32 -98 304]	124	240
[8 2 16]	124	0
[10 14 32]	124	240
[14 -102 752]	124	0
[2 1 64]	127	0
[4 1 32]	127	-78
[8 1 16]	127	0
[2 0 64]	128	0
[2 8 96]	128	0
[4 0 32]	128	0
[4 8 48]	128	-160
[4 16 96]	128	0
[6 8 32]	128	-160
[22 -112 576]	128	160
[8 0 16]	128	0
[8 16 48]	128	-160
[12 16 32]	128	0
[12 -88 656]	128	0
[68 -341 1712]	135	0
[34 -171 864]	135	211
[8 11 32]	135	-211
[10 5 16]	135	0
[72 -217 656]	143	0
[42 121 352]	143	273
[6 7 32]	143	-273
[14 -103 768]	143	0
[2 4 80]	144	0
[6 12 48]	144	-1344
[10 4 16]	144	0
[2 3 80]	151	0
[38 -115 352]	151	-415
[10 3 16]	151	0
[10 13 32]	151	-415
[2 2 80]	156	0
[78 -234 704]	156	0
[4 6 48]	156	1040
[40 -202 1024]	156	-403
[6 6 32]	156	1040
[8 10 32]	156	403
[10 2 16]	156	0
[12 6 16]	156	0
[2 1 80]	159	0
[4 17 112]	159	52
[28 -143 736]	159	261
[30 81 224]	159	261
[10 1 16]	159	0
[12 15 32]	159	-52
[2 0 80]	160	0
[2 8 112]	160	0
[10 0 16]	160	0
[84 -421 2112]	167	0
[4 5 48]	167	-940
[28 -197 1392]	167	940
[36 101 288]	167	-1270
[12 5 16]	167	0
[88 -265 800]	175	0
[4 9 64]	175	-858
[8 9 32]	175	-858
[2 4 96]	176	0
[4 4 48]	176	-680
[44 -132 400]	176	544
[30 -212 1504]	176	680
[10 12 32]	176	544
[12 4 16]	176	0
[2 3 96]	183	0
[46 -323 2272]	183	-1831
[6 3 32]	183	1831
[12 3 16]	183	0
[2 2 96]	188	0
[94 -282 848]	188	0
[4 2 48]	188	-672
[48 -146 448]	188	-762
[6 2 32]	188	-672
[6 14 64]	188	1072
[42 118 336]	188	-1072
[36 98 272]	188	-1072
[12 2 16]	188	0
[12 14 32]	188	-762
[2 1 96]	191	0
[4 1 48]	191	-710
[6 1 32]	191	-710
[12 1 16]	191	0
[2 0 96]	192	0
[2 8 128]	192	0
[4 0 48]	192	0
[4 8 64]	192	1856
[4 16 112]	192	0
[6 0 32]	192	0
[8 8 32]	192	1856
[12 0 16]	192	0
[100 -501 2512]	199	0
[50 -251 1264]	199	377
[10 11 32]	199	-377
[104 -313 944]	207	0
[58 169 496]	207	2012
[6 9 48]	207	-3015
[26 -183 1296]	207	2012
[2 4 112]	208	0
[2 3 112]	215	0
[54 -163 496]	215	2483
[6 13 64]	215	569
[42 115 320]	215	-569
[12 13 32]	215	2483
[2 2 112]	220	0
[110 -330 992]	220	0
[4 6 64]	220	-1193
[56 -282 1424]	220	208
[28 -198 1408]	220	1193
[10 10 32]	220	-208
[2 1 112]	223	0
[4 17 128]	223	-2327
[2 0 112]	224	0
[6 8 48]	224	1280
[116 -581 2912]	231	0
[4 5 64]	231	4703
[30 -213 1520]	231	-4703
[4 9 80]	239	418
[6 7 48]	239	-799
[24 -169 1200]	239	-418
[2 4 128]	240	0
[4 4 64]	240	-1976
[60 -180 544]	240	-4408
[6 12 64]	240	-4032
[8 4 32]	240	-1976
[8 12 48]	240	-4032
[12 12 32]	240	-4408
[2 3 128]	247	0
[62 -435 3056]	247	2002
[8 3 32]	247	-2002
[2 2 128]	252	0
[4 2 64]	252	-2665
[64 -194 592]	252	-4576
[6 6 48]	252	6000
[8 2 32]	252	-2665
[2 1 128]	255	0
[4 1 64]	255	7339
[8 1 32]	255	7339
[2 0 128]	256	0
[4 0 64]	256	0
[4 8 80]	256	-1920
[4 16 128]	256	0
[8 0 32]	256	0
[26 -184 1312]	256	1920
[66 -331 1664]	263	-5423
[44 -309 2176]	263	6078
[52 149 432]	263	-718
[8 11 48]	263	718
[22 -155 1104]	263	-5423
[74 217 640]	271	-6310
[28 -199 1424]	271	-6310
[46 -324 2288]	272	6720
[70 -211 640]	279	2214
[6 3 48]	279	15528
[4 6 80]	284	-3856
[72 -362 1824]	284	2533
[6 2 48]	284	-6416
[58 166 480]	284	-7216
[8 10 48]	284	7216
[30 -214 1536]	284	3856
[24 -170 1216]	284	2533
[4 17 144]	287	2983
[6 1 48]	287	-4085
[6 0 48]	288	0
[4 5 80]	295	-8121
[10 5 32]	295	-8121
[4 9 96]	303	308
[6 9 64]	303	-4347
[8 9 48]	303	-4347
[26 -185 1328]	303	-308
[4 4 80]	304	6304
[76 -228 688]	304	3016
[10 4 32]	304	6304
[78 -547 3840]	311	-7553
[10 3 32]	311	7553
[4 2 80]	316	2736
[80 -242 736]	316	17290
[10 2 32]	316	2736
[4 1 80]	319	-19430
[10 1 32]	319	-19430
[4 0 80]	320	0
[4 8 96]	320	-5200
[6 8 64]	320	-2240
[8 8 48]	320	-2240
[10 0 32]	320	0
[28 -200 1440]	320	5200
[82 -411 2064]	327	10210
[90 265 784]	335	-1258
[6 7 64]	335	-947
[42 -295 2080]	335	947
[30 -215 1552]	335	-1258
[86 -259 784]	343	-22316
[4 6 96]	348	10047
[88 -442 2224]	348	-8528
[6 6 64]	348	-7056
[44 -310 2192]	348	7056
[12 6 32]	348	10047
[4 5 96]	359	97
[60 -421 2960]	359	-22226
[46 -325 2304]	359	-22226
[12 5 32]	359	97
[4 9 112]	367	7659
[4 4 96]	368	3656
[92 -276 832]	368	16328
[62 -436 3072]	368	-832
[8 4 48]	368	832
[12 4 32]	368	3656
[94 -659 4624]	375	42378
[6 3 64]	375	-34146
[8 3 48]	375	-34146
[12 3 32]	375	-42378
[4 2 96]	380	27965
[6 2 64]	380	10352
[8 2 48]	380	10352
[12 2 32]	380	27965
[4 1 96]	383	22731
[6 1 64]	383	24679
[8 1 48]	383	24679
[12 1 32]	383	22731
[4 0 96]	384	0
[4 8 112]	384	192
[6 0 64]	384	0
[8 0 48]	384	0
[12 0 32]	384	0
[98 -491 2464]	391	5473
[106 313 928]	399	21081
[4 6 112]	412	208
[104 -522 2624]	412	10417
[4 5 112]	423	5606
[4 9 128]	431	-12260
[4 4 112]	432	4864
[110 -771 5408]	439	-46202
[4 2 112]	444	-37744
[4 1 112]	447	-41249
[4 0 112]	448	0
[4 8 128]	448	29696
[122 361 1072]	463	-24688
[4 6 128]	476	-30106
[4 5 128]	487	8949
[4 4 128]	496	-75088