A Borcherds Product in S₁₃(K(8))⁻

properties |

c vector |

θ vector |

ψ expansion |

Borch(ψ) coefficients

• The singular part of ψ truncated up to q^(8/4) is
  (26 + 2/ζ^3 + 3/ζ^2 + 18/ζ + 18*ζ + 3*ζ^2 + 2*ζ^3)
  +q(ζ^(-6) + ζ^6)
  +q^2(-10/ζ^8 - 10*ζ^8)
• From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block

  	φ	= TB(13;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3)
  		= 1182332
  		= η(τ)26(θ(τ,z)/η(τ))18(θ(τ,2z)/η(τ))3(θ(τ,3z)/η(τ))2

  This theta block has q-order of vanishing 3. This theta block has index 3*8, and so Borch(ψ) begins with Jacobi coefficient #3.
• 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)	23
  (4,2)	3
  (4,6)	1
  (9,3)	2

• 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

(26 + 2/ζ^3 + 3/ζ^2 + 18/ζ + 18*ζ + 3*ζ^2 + 2*ζ^3)
+q(288 + ζ^(-6) + 22/ζ^5 - 16/ζ^4 - 118/ζ^3 - 129/ζ^2 + 96/ζ + 96*ζ - 129*ζ^2 - 118*ζ^3 - 16*ζ^4 + 22*ζ^5 + ζ^6)
+q^2(2196 - 10/ζ^8 - 24/ζ^7 + 125/ζ^6 + 358/ζ^5 - 64/ζ^4 - 1142/ζ^3 - 1149/ζ^2 + 808/ζ + 808*ζ - 1149*ζ^2 - 1142*ζ^3 - 64*ζ^4 + 358*ζ^5 + 125*ζ^6 - 24*ζ^7 - 10*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	0
[6 -19 64]	23	0
[2 2 16]	28	0
[14 -42 128]	28	0
[4 6 16]	28	0
[8 -42 224]	28	0
[2 1 16]	31	0
[8 -49 304]	31	0
[4 17 80]	31	0
[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
[10 -21 48]	39	1
[8 -27 96]	39	-1
[24 -73 224]	47	0
[12 -73 448]	47	3
[4 9 32]	47	0
[6 7 16]	47	0
[6 25 112]	47	-3
[2 4 32]	48	0
[4 4 16]	48	0
[12 -36 112]	48	0
[6 12 32]	48	0
[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	0
[6 6 16]	60	0
[10 -30 96]	60	2
[8 -18 48]	60	-2
[8 -50 320]	60	0
[2 1 32]	63	0
[4 1 16]	63	0
[12 -63 336]	63	1
[2 0 32]	64	0
[2 8 64]	64	0
[4 0 16]	64	0
[4 8 32]	64	0
[4 16 80]	64	0
[8 8 16]	64	0
[8 -24 80]	64	0
[10 -24 64]	64	0
[10 -64 416]	64	0
[36 -181 912]	71	0
[18 -91 464]	71	0
[12 -85 608]	71	0
[20 53 144]	71	0
[10 -53 288]	71	3
[40 -121 368]	79	0
[26 73 208]	79	0
[10 -71 512]	79	0
[2 4 48]	80	0
[14 -44 144]	80	12
[14 -100 720]	80	0
[10 -20 48]	80	-12
[2 3 48]	87	0
[22 -67 208]	87	0
[6 3 16]	87	0
[26 67 176]	87	0
[2 2 48]	92	0
[46 -138 416]	92	0
[4 6 32]	92	0
[24 -122 624]	92	0
[6 2 16]	92	0
[6 14 48]	92	18
[26 70 192]	92	0
[6 26 128]	92	30
[12 -74 464]	92	-30
[12 -86 624]	92	0
[2 1 48]	95	0
[4 17 96]	95	0
[6 1 16]	95	0
[16 -81 416]	95	33
[8 17 48]	95	-33
[10 15 32]	95	0
[2 0 48]	96	0
[2 8 80]	96	0
[6 0 16]	96	0
[6 24 112]	96	0
[14 -88 560]	96	0
[10 -72 528]	96	0
[52 -261 1312]	103	0
[4 5 32]	103	0
[14 -101 736]	103	0
[56 -169 512]	111	0
[4 9 48]	111	0
[6 9 32]	111	0
[14 -71 368]	111	42
[12 -87 640]	111	0
[10 23 64]	111	-42
[2 4 64]	112	0
[4 4 32]	112	0
[28 -84 256]	112	0
[8 4 16]	112	0
[8 12 32]	112	0
[16 -84 448]	112	-64
[2 3 64]	119	0
[30 -211 1488]	119	0
[20 -61 192]	119	-81
[6 13 48]	119	-150
[8 3 16]	119	0
[24 61 160]	119	81
[2 2 64]	124	0
[62 -186 560]	124	0
[4 2 32]	124	0
[32 -98 304]	124	0
[8 2 16]	124	0
[10 14 32]	124	0
[14 -102 752]	124	0
[2 1 64]	127	0
[4 1 32]	127	0
[8 1 16]	127	0
[2 0 64]	128	0
[2 8 96]	128	0
[4 0 32]	128	0
[4 8 48]	128	0
[4 16 96]	128	0
[6 8 32]	128	0
[22 -112 576]	128	-264
[8 0 16]	128	0
[8 16 48]	128	264
[12 16 32]	128	0
[12 -88 656]	128	0
[68 -341 1712]	135	0
[34 -171 864]	135	0
[24 -75 240]	135	149
[8 11 32]	135	0
[10 5 16]	135	0
[12 -75 480]	135	149
[72 -217 656]	143	0
[42 121 352]	143	0
[6 7 32]	143	0
[6 25 128]	143	-198
[14 -89 576]	143	198
[14 -103 768]	143	0
[2 4 80]	144	0
[6 12 48]	144	764
[10 4 16]	144	0
[2 3 80]	151	0
[38 -115 352]	151	0
[8 19 64]	151	-727
[10 3 16]	151	0
[10 13 32]	151	0
[2 2 80]	156	0
[78 -234 704]	156	0
[4 6 48]	156	0
[40 -202 1024]	156	0
[6 6 32]	156	0
[26 -78 240]	156	462
[8 10 32]	156	0
[20 -102 528]	156	-794
[10 2 16]	156	0
[10 22 64]	156	794
[30 78 208]	156	-462
[12 6 16]	156	0
[2 1 80]	159	0
[4 17 112]	159	0
[28 -143 736]	159	1376
[30 81 224]	159	1376
[10 1 16]	159	0
[12 15 32]	159	0
[2 0 80]	160	0
[2 8 112]	160	0
[10 0 16]	160	0
[14 -104 784]	160	0
[84 -421 2112]	167	0
[4 5 48]	167	0
[28 -197 1392]	167	0
[36 101 288]	167	2640
[12 5 16]	167	0
[88 -265 800]	175	0
[4 9 64]	175	0
[8 9 32]	175	0
[14 7 16]	175	0
[2 4 96]	176	0
[4 4 48]	176	0
[44 -132 400]	176	0
[30 -92 288]	176	-528
[30 -212 1504]	176	0
[10 12 32]	176	0
[12 4 16]	176	0
[36 92 240]	176	528
[2 3 96]	183	0
[46 -323 2272]	183	0
[6 3 32]	183	0
[12 3 16]	183	0
[2 2 96]	188	0
[94 -282 848]	188	0
[4 2 48]	188	0
[48 -146 448]	188	0
[6 2 32]	188	0
[6 14 64]	188	5238
[42 118 336]	188	-6474
[6 26 144]	188	1476
[8 18 64]	188	5208
[36 98 272]	188	-5238
[12 2 16]	188	0
[12 14 32]	188	0
[14 6 16]	188	0
[42 106 272]	188	1476
[2 1 96]	191	0
[4 1 48]	191	0
[6 1 32]	191	0
[32 -161 816]	191	2298
[36 95 256]	191	2298
[12 1 16]	191	0
[2 0 96]	192	0
[2 8 128]	192	0
[4 0 48]	192	0
[4 8 64]	192	0
[4 16 112]	192	0
[6 0 32]	192	0
[6 24 128]	192	0
[8 8 32]	192	0
[12 0 16]	192	0
[14 16 32]	192	0
[16 8 16]	192	0
[16 24 48]	192	0
[100 -501 2512]	199	0
[50 -251 1264]	199	0
[26 -133 688]	199	6538
[10 11 32]	199	0
[10 21 64]	199	-6538
[14 5 16]	199	0
[104 -313 944]	207	0
[58 169 496]	207	0
[6 9 48]	207	-11367
[26 -183 1296]	207	0
[16 7 16]	207	0
[2 4 112]	208	0
[14 4 16]	208	0
[2 3 112]	215	0
[54 -163 496]	215	0
[36 -109 336]	215	1341
[6 13 64]	215	-15201
[42 115 320]	215	15201
[12 13 32]	215	0
[42 109 288]	215	-1341
[14 3 16]	215	0
[2 2 112]	220	0
[110 -330 992]	220	0
[4 6 64]	220	0
[56 -282 1424]	220	0
[28 -198 1408]	220	0
[10 10 32]	220	0
[14 2 16]	220	0
[16 6 16]	220	0
[2 1 112]	223	0
[4 17 128]	223	0
[8 17 64]	223	-24851
[14 1 16]	223	0
[14 15 32]	223	0
[2 0 112]	224	0
[2 8 144]	224	0
[6 8 48]	224	13776
[38 -192 976]	224	-9072
[42 112 304]	224	-9072
[14 0 16]	224	0
[116 -581 2912]	231	0
[4 5 64]	231	0
[40 -123 384]	231	6503
[30 -213 1520]	231	0
[14 21 48]	231	6503
[16 5 16]	231	0
[120 -361 1088]	239	0
[4 9 80]	239	0
[6 7 48]	239	-9591
[6 25 144]	239	-3906
[24 -169 1200]	239	0
[16 23 48]	239	3906
[2 4 128]	240	0
[4 4 64]	240	0
[60 -180 544]	240	0
[6 12 64]	240	33616
[8 4 32]	240	0
[8 12 48]	240	33616
[32 -164 848]	240	-32144
[10 20 64]	240	32144
[12 12 32]	240	0
[16 4 16]	240	0
[2 3 128]	247	0
[62 -435 3056]	247	0
[8 3 32]	247	0
[16 3 16]	247	0
[2 2 128]	252	0
[126 -378 1136]	252	0
[4 2 64]	252	0
[64 -194 592]	252	0
[6 6 48]	252	-1518
[42 -126 384]	252	-1518
[8 2 32]	252	0
[12 18 48]	252	-1518
[14 14 32]	252	0
[16 2 16]	252	0
[2 1 128]	255	0
[4 1 64]	255	0
[44 -223 1136]	255	27083
[8 1 32]	255	0
[10 15 48]	255	-27083
[16 1 16]	255	0
[2 0 128]	256	0
[4 0 64]	256	0
[4 8 80]	256	0
[4 16 128]	256	0
[8 0 32]	256	0
[8 16 64]	256	81728
[26 -184 1312]	256	0
[16 0 16]	256	0
[16 16 32]	256	0
[132 -661 3312]	263	0
[66 -331 1664]	263	0
[44 -309 2176]	263	-15180
[52 149 432]	263	54681
[8 11 48]	263	-54681
[22 -155 1104]	263	0
[74 217 640]	271	0
[28 -199 1424]	271	0
[2 4 144]	272	0
[46 -140 432]	272	-20964
[46 -324 2288]	272	26772
[14 20 48]	272	-20964
[2 3 144]	279	0
[70 -211 640]	279	0
[6 3 48]	279	34086
[8 19 80]	279	-108843
[40 109 304]	279	108843
[20 -141 1008]	279	0
[2 2 144]	284	0
[4 6 80]	284	0
[72 -362 1824]	284	0
[6 2 48]	284	-34086
[6 14 80]	284	59634
[58 166 480]	284	-59664
[6 26 160]	284	20580
[8 10 48]	284	59664
[10 14 48]	284	59634
[30 -214 1536]	284	0
[24 -170 1216]	284	0
[16 22 48]	284	-20580
[2 1 144]	287	0
[4 17 144]	287	0
[6 1 48]	287	22287
[48 -241 1216]	287	3228
[46 129 368]	287	188733
[12 17 48]	287	-3228
[18 -127 912]	287	0
[2 0 144]	288	0
[6 0 48]	288	0
[6 24 144]	288	0
[4 5 80]	295	0
[10 5 32]	295	0
[4 9 96]	303	0
[6 9 64]	303	-30757
[8 9 48]	303	-30757
[26 -185 1328]	303	0
[4 4 80]	304	0
[76 -228 688]	304	0
[10 4 32]	304	0
[22 -156 1120]	304	0
[78 -547 3840]	311	0
[52 -157 480]	311	50400
[6 13 80]	311	-90756
[10 3 32]	311	0
[10 13 48]	311	-90756
[14 19 48]	311	50400
[4 2 80]	316	0
[80 -242 736]	316	0
[8 18 80]	316	276210
[52 146 416]	316	-306216
[10 2 32]	316	0
[46 126 352]	316	-276210
[20 -142 1024]	316	0
[4 1 80]	319	0
[10 1 32]	319	0
[4 0 80]	320	0
[4 8 96]	320	0
[4 16 144]	320	0
[6 8 64]	320	-20880
[54 -272 1376]	320	-14208
[8 8 48]	320	-20880
[10 0 32]	320	0
[12 16 48]	320	14208
[28 -200 1440]	320	0
[82 -411 2064]	327	0
[56 -171 528]	327	63745
[24 -171 1232]	327	0
[16 21 48]	327	63745
[90 265 784]	335	0
[6 7 64]	335	57879
[6 25 160]	335	-24639
[42 -295 2080]	335	-57879
[30 -215 1552]	335	0
[6 12 80]	336	72820
[10 12 48]	336	72820
[86 -259 784]	343	0
[58 163 464]	343	339813
[22 -157 1136]	343	0
[4 6 96]	348	0
[88 -442 2224]	348	0
[6 6 64]	348	-67584
[58 -174 528]	348	-92422
[44 -310 2192]	348	67584
[12 6 32]	348	0
[14 18 48]	348	-92422
[26 -186 1344]	348	0
[4 17 160]	351	0
[60 -303 1536]	351	4904
[8 17 80]	351	-545202
[52 143 400]	351	545202
[12 15 48]	351	-4904
[4 5 96]	359	0
[60 -421 2960]	359	-81675
[68 197 576]	359	-38412
[46 -325 2304]	359	-81675
[10 11 48]	359	38412
[12 5 32]	359	0
[4 9 112]	367	0
[28 -201 1456]	367	0
[4 4 96]	368	0
[92 -276 832]	368	0
[62 -188 576]	368	-129264
[62 -436 3072]	368	109296
[8 4 48]	368	-109296
[8 12 64]	368	230208
[12 4 32]	368	0
[16 20 48]	368	-129264
[24 -172 1248]	368	0
[94 -659 4624]	375	0
[6 3 64]	375	99843
[8 3 48]	375	99843
[12 3 32]	375	0
[4 2 96]	380	0
[96 -290 880]	380	0
[6 2 64]	380	-33948
[6 14 96]	380	-99834
[74 214 624]	380	194808
[6 26 176]	380	66222
[8 2 48]	380	-33948
[10 10 48]	380	-194808
[12 2 32]	380	0
[12 14 48]	380	-99834
[4 1 96]	383	0
[6 1 64]	383	-15939
[64 -321 1616]	383	-152772
[8 1 48]	383	-15939
[12 1 32]	383	0
[14 17 48]	383	152772
[4 0 96]	384	0
[4 8 112]	384	0
[6 0 64]	384	0
[8 0 48]	384	0
[8 16 80]	384	816928
[58 160 448]	384	-816928
[12 0 32]	384	0
[30 -216 1568]	384	0
[98 -491 2464]	391	0
[8 11 64]	391	-15961
[26 -187 1360]	391	0
[106 313 928]	399	0
[6 9 80]	399	250419
[40 -281 1984]	399	-250419
[14 7 32]	399	0
[102 -307 928]	407	0
[68 -205 624]	407	156807
[6 13 96]	407	368751
[12 13 48]	407	368751
[16 19 48]	407	156807
[4 6 112]	412	0
[104 -522 2624]	412	0
[8 10 64]	412	-261728
[14 6 32]	412	0
[28 -202 1472]	412	0
[62 177 512]	415	830138
[10 15 64]	415	-830138
[6 8 80]	416	-159264
[70 -352 1776]	416	294240
[42 -296 2096]	416	159264
[14 16 48]	416	-294240
[4 5 112]	423	0
[72 -219 672]	423	28227
[14 5 32]	423	0
[4 9 128]	431	0
[6 7 80]	431	95943
[8 9 64]	431	617820
[44 -311 2208]	431	-95943
[30 -217 1584]	431	0
[4 4 112]	432	0
[108 -324 976]	432	0
[6 12 96]	432	-702896
[12 12 48]	432	-702896
[14 4 32]	432	0
[110 -771 5408]	439	0
[14 3 32]	439	0
[4 2 112]	444	0
[6 6 80]	444	-141448
[74 -222 672]	444	-85786
[68 194 560]	444	-363814
[10 14 64]	444	363814
[46 -326 2320]	444	141448
[14 2 32]	444	0
[16 18 48]	444	-85786
[4 1 112]	447	0
[76 -383 1936]	447	-604385
[14 1 32]	447	0
[14 15 48]	447	604385
[4 0 112]	448	0
[4 8 128]	448	0
[8 8 64]	448	-940672
[14 0 32]	448	0
[16 8 32]	448	0
[114 -571 2864]	455	0
[76 -533 3744]	455	-20817
[84 245 720]	455	-778695
[10 5 48]	455	20817
[38 -267 1888]	455	-778695
[122 361 1072]	463	0
[58 -407 2864]	463	-930244
[16 7 32]	463	0
[78 -236 720]	464	325932
[78 -548 3856]	464	-395496
[10 4 48]	464	395496
[6 3 80]	471	-626991
[74 211 608]	471	-421111
[10 3 48]	471	-626991
[10 13 64]	471	421111
[4 6 128]	476	0
[120 -602 3024]	476	0
[6 2 80]	476	313236
[6 14 112]	476	-985950
[90 262 768]	476	437514
[60 -422 2976]	476	472224
[10 2 48]	476	313236
[40 -282 2000]	476	437514
[14 14 48]	476	-985950
[16 6 32]	476	0
[6 1 80]	479	55683
[80 -401 2016]	479	-408
[10 1 48]	479	55683
[16 17 48]	479	408
[6 0 80]	480	0
[10 0 48]	480	0
[4 5 128]	487	0
[62 -437 3088]	487	138726
[16 5 32]	487	0
[4 9 144]	495	0
[6 9 96]	495	125589
[42 -297 2112]	495	-125589
[4 4 128]	496	0
[8 4 64]	496	601344
[8 12 80]	496	-1264016
[10 12 64]	496	-1264016
[16 4 32]	496	0