A Borcherds Product in S₉(K(16))⁻

properties |

c vector |

θ vector |

ψ expansion |

Borch(ψ) coefficients

• The singular part of ψ truncated up to q^(16/4) is
  (18 + ζ^(-4) + 3/ζ^3 + 10/ζ^2 + 13/ζ + 13*ζ + 10*ζ^2 + 3*ζ^3 + ζ^4)
  +q(-ζ^(-8) - ζ^8)
  +q^3(2/ζ^14 + 2*ζ^14)
  +q^4(4/ζ^16 + 4*ζ^16)
• From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block

  	φ	= TB(9;1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4)
  		= 1132103341
  		= η(τ)18(θ(τ,z)/η(τ))13(θ(τ,2z)/η(τ))10(θ(τ,3z)/η(τ))3(θ(τ,4z)/η(τ))

  This theta block has q-order of vanishing 3. This theta block has index 3*16, 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 4 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)	27
  (4,2)	11
  (4,14)	3
  (9,3)	3
  (16,4)	1

• From the singular part above, we compute that the valuation Ord(ψ) is -0.25, 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_12,16^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^(16/4) is

(18 + ζ^(-4) + 3/ζ^3 + 10/ζ^2 + 13/ζ + 13*ζ + 10*ζ^2 + 3*ζ^3 + ζ^4)
+q(90 - ζ^(-8) - 2/ζ^7 - 14/ζ^6 - 31/ζ^5 - 44/ζ^4 - 31/ζ^3 + 14/ζ^2 + 64/ζ + 64*ζ + 14*ζ^2 - 31*ζ^3 - 44*ζ^4 - 31*ζ^5 - 14*ζ^6 - 2*ζ^7 - ζ^8)
+q^2(524 - ζ^(-11) + 2/ζ^10 + 23/ζ^9 + 42/ζ^8 + 37/ζ^7 - 56/ζ^6 - 200/ζ^5 - 304/ζ^4 - 229/ζ^3 + 54/ζ^2 + 370/ζ + 370*ζ + 54*ζ^2 - 229*ζ^3 - 304*ζ^4 - 200*ζ^5 - 56*ζ^6 + 37*ζ^7 + 42*ζ^8 + 23*ζ^9 + 2*ζ^10 - ζ^11)
+q^3(2344 + 2/ζ^14 - 5/ζ^13 - 20/ζ^12 - 25/ζ^11 + 32/ζ^10 + 202/ζ^9 + 380/ζ^8 + 339/ζ^7 - 164/ζ^6 - 976/ζ^5 - 1532/ζ^4 - 1190/ζ^3 + 130/ζ^2 + 1655/ζ + 1655*ζ + 130*ζ^2 - 1190*ζ^3 - 1532*ζ^4 - 976*ζ^5 - 164*ζ^6 + 339*ζ^7 + 380*ζ^8 + 202*ζ^9 + 32*ζ^10 - 25*ζ^11 - 20*ζ^12 - 5*ζ^13 + 2*ζ^14)
+q^4(9128 + 4/ζ^16 + 6/ζ^15 - 2/ζ^14 - 75/ζ^13 - 209/ζ^12 - 248/ζ^11 + 116/ζ^10 + 1043/ζ^9 + 1960/ζ^8 + 1773/ζ^7 - 466/ζ^6 - 3969/ζ^5 - 6319/ζ^4 - 4938/ζ^3 + 352/ζ^2 + 6408/ζ + 6408*ζ + 352*ζ^2 - 4938*ζ^3 - 6319*ζ^4 - 3969*ζ^5 - 466*ζ^6 + 1773*ζ^7 + 1960*ζ^8 + 1043*ζ^9 + 116*ζ^10 - 248*ζ^11 - 209*ζ^12 - 75*ζ^13 - 2*ζ^14 + 6*ζ^15 + 4*ζ^16)

2u	det(2u)	a(u, Borch(ψ))
[4 -53 704]	7	0
[8 -89 992]	15	0
[4 -25 160]	15	0
[12 -109 992]	23	0
[6 -13 32]	23	0
[6 -19 64]	23	0
[14 -42 128]	28	0
[14 -70 352]	28	0
[8 -42 224]	28	0
[8 -106 1408]	28	0
[16 -113 800]	31	0
[8 -113 1600]	31	0
[14 -47 160]	31	0
[20 -101 512]	39	0
[10 -91 832]	39	0
[10 -101 1024]	39	0
[8 -27 96]	39	-1
[24 -73 224]	47	0
[12 -73 448]	47	0
[4 9 32]	47	0
[8 -73 672]	47	0
[16 -55 192]	47	1
[2 4 32]	48	0
[26 76 224]	48	0
[6 12 32]	48	0
[14 36 96]	48	-1
[2 3 32]	55	0
[14 -29 64]	55	0
[14 -99 704]	55	0
[8 -35 160]	55	-1
[2 2 32]	60	0
[30 -210 1472]	60	0
[16 -50 160]	60	0
[16 -178 1984]	60	0
[10 -30 96]	60	3
[10 -50 256]	60	0
[8 -50 320]	60	-2
[8 -114 1632]	60	0
[2 1 32]	63	0
[16 -225 3168]	63	0
[4 33 288]	63	0
[12 -159 2112]	63	3
[8 -97 1184]	63	-2
[2 0 32]	64	0
[2 8 64]	64	0
[2 16 160]	64	0
[4 8 32]	64	0
[20 56 160]	64	0
[10 -24 64]	64	0
[10 -144 2080]	64	0
[36 -469 6112]	71	0
[18 -181 1824]	71	0
[18 -235 3072]	71	0
[120 277 640]	71	0
[10 -43 192]	71	3
[10 -53 288]	71	0
[40 -441 4864]	79	0
[20 -121 736]	79	0
[190 441 1024]	79	0
[10 -121 1472]	79	2
[44 -397 3584]	87	0
[22 -45 96]	87	0
[22 -243 2688]	87	0
[12 -51 224]	87	1
[134 307 704]	87	0
[46 -138 416]	92	0
[46 -230 1152]	92	0
[4 6 32]	92	0
[24 -218 1984]	92	0
[26 70 192]	92	10
[162 374 864]	92	0
[6 26 128]	92	-10
[26 -86 288]	92	-13
[12 -26 64]	92	0
[12 -38 128]	92	2
[12 -122 1248]	92	10
[12 -134 1504]	92	-10
[48 -337 2368]	95	0
[24 -337 4736]	95	0
[4 17 96]	95	0
[16 -81 416]	95	0
[12 -145 1760]	95	5
[60 145 352]	95	7
[10 15 32]	95	0
[52 -261 1312]	103	0
[4 5 32]	103	0
[26 -261 2624]	103	0
[56 -169 512]	111	0
[120 297 736]	111	0
[6 9 32]	111	0
[14 -183 2400]	111	-2
[10 23 64]	111	0
[2 4 64]	112	0
[58 172 512]	112	0
[4 4 32]	112	0
[28 -84 256]	112	0
[28 -140 704]	112	0
[28 -196 1376]	112	0
[14 -28 64]	112	0
[14 -84 512]	112	16
[8 12 32]	112	0
[16 -84 448]	112	0
[16 -212 2816]	112	8
[2 3 64]	119	0
[30 -61 128]	119	0
[30 -451 6784]	119	0
[20 -61 192]	119	-26
[90 221 544]	119	17
[12 -61 320]	119	-26
[12 -125 1312]	119	-17
[2 2 64]	124	0
[62 -434 3040]	124	0
[4 2 32]	124	0
[32 -226 1600]	124	0
[16 -226 3200]	124	0
[8 34 160]	124	-6
[28 -94 320]	124	-18
[10 14 32]	124	0
[14 -46 160]	124	-15
[14 -158 1792]	124	-6
[2 1 64]	127	0
[4 1 32]	127	0
[32 -449 6304]	127	0
[2 0 64]	128	0
[2 8 96]	128	0
[2 16 192]	128	0
[4 0 32]	128	0
[4 16 96]	128	0
[4 32 288]	128	0
[6 8 32]	128	0
[22 -112 576]	128	0
[22 -200 1824]	128	44
[22 -288 3776]	128	-8
[18 -40 96]	128	44
[18 -256 3648]	128	0
[12 16 32]	128	0
[12 -64 352]	128	8
[68 -885 11520]	135	0
[34 -171 864]	135	0
[24 -267 2976]	135	-68
[8 11 32]	135	0
[72 -793 8736]	143	0
[42 121 352]	143	0
[6 7 32]	143	0
[6 25 128]	143	39
[18 -199 2208]	143	39
[12 -25 64]	143	0
[76 -685 6176]	151	0
[38 -115 352]	151	0
[8 19 64]	151	0
[10 13 32]	151	0
[78 -234 704]	156	0
[78 -390 1952]	156	0
[40 -202 1024]	156	0
[40 -522 6816]	156	0
[6 6 32]	156	0
[8 10 32]	156	0
[20 -42 96]	156	-110
[20 -202 2048]	156	50
[10 22 64]	156	0
[10 42 192]	156	22
[30 -102 352]	156	28
[16 -182 2080]	156	22
[80 -561 3936]	159	0
[46 -143 448]	159	0
[28 -143 736]	159	0
[16 -49 160]	159	30
[12 15 32]	159	0
[14 -143 1472]	159	27
[84 -421 2112]	167	0
[42 -379 3424]	167	0
[28 -421 6336]	167	0
[36 101 288]	167	15
[22 -69 224]	167	-7
[14 -155 1728]	167	7
[14 -197 2784]	167	0
[16 -37 96]	167	-15
[88 -265 800]	175	0
[4 9 64]	175	0
[8 9 32]	175	0
[20 -265 3520]	175	-15
[2 4 96]	176	0
[90 268 800]	176	0
[30 -92 288]	176	78
[30 -332 3680]	176	159
[30 -452 6816]	176	0
[22 -44 96]	176	165
[10 12 32]	176	0
[18 -92 480]	176	78
[18 -236 3104]	176	30
[2 3 96]	183	0
[46 -323 2272]	183	0
[6 3 32]	183	0
[38 -125 416]	183	75
[16 -227 3232]	183	0
[2 2 96]	188	0
[94 -658 4608]	188	0
[48 -146 448]	188	0
[48 -530 5856]	188	0
[6 2 32]	188	0
[6 14 64]	188	0
[142 350 864]	188	-141
[24 -338 4768]	188	132
[8 18 64]	188	0
[112 274 672]	188	-38
[12 14 32]	188	0
[12 50 224]	188	-76
[32 -110 384]	188	-4
[14 34 96]	188	141
[102 242 576]	188	-38
[42 -146 512]	188	76
[2 1 96]	191	0
[4 33 320]	191	0
[6 1 32]	191	0
[32 -417 5440]	191	-86
[8 33 160]	191	31
[20 -223 2496]	191	31
[18 -95 512]	191	86
[18 -257 3680]	191	0
[2 0 96]	192	0
[2 8 128]	192	0
[2 16 224]	192	0
[4 8 64]	192	0
[52 152 448]	192	0
[6 0 32]	192	0
[6 24 128]	192	-72
[8 8 32]	192	0
[24 -72 224]	192	-56
[24 -120 608]	192	72
[26 -56 128]	192	64
[26 -368 5216]	192	0
[12 24 64]	192	0
[14 16 32]	192	0
[14 -72 384]	192	-56
[100 -1301 16928]	199	0
[50 -651 8480]	199	0
[26 -341 4480]	199	-104
[10 11 32]	199	0
[10 21 64]	199	0
[16 -213 2848]	199	-104
[104 -1145 12608]	207	0
[478 1113 2592]	207	0
[36 -327 2976]	207	-115
[26 -391 5888]	207	0
[18 -39 96]	207	-115
[108 -973 8768]	215	0
[54 -595 6560]	215	0
[36 -397 4384]	215	-18
[6 13 64]	215	0
[24 -77 256]	215	119
[12 13 32]	215	0
[20 -205 2112]	215	44
[16 -179 2016]	215	-119
[110 -330 992]	220	0
[110 -550 2752]	220	0
[4 6 64]	220	0
[56 -506 4576]	220	0
[28 -58 128]	220	22
[28 -422 6368]	220	0
[10 10 32]	220	0
[170 390 896]	220	0
[112 -785 5504]	223	0
[4 17 128]	223	0
[8 17 64]	223	0
[14 15 32]	223	0
[116 -581 2912]	231	0
[4 5 64]	231	0
[40 -123 384]	231	-51
[30 -453 6848]	231	0
[24 -123 640]	231	-51
[40 -133 448]	231	-81
[120 -361 1088]	239	0
[248 617 1536]	239	0
[48 -151 480]	239	-70
[30 -151 768]	239	-70
[24 -361 5440]	239	0
[10 41 192]	239	11
[20 -41 96]	239	393
[198 457 1056]	239	0
[22 -247 2784]	239	11
[2 4 128]	240	0
[122 364 1088]	240	0
[4 4 64]	240	0
[60 -180 544]	240	0
[60 -300 1504]	240	0
[60 -420 2944]	240	0
[6 12 64]	240	0
[46 132 384]	240	-432
[30 -60 128]	240	-176
[8 4 32]	240	0
[32 -100 320]	240	64
[32 -356 3968]	240	216
[10 20 64]	240	0
[12 12 32]	240	0
[20 -100 512]	240	64
[16 -36 96]	240	432
[16 -228 3264]	240	0
[2 3 128]	247	0
[62 -931 13984]	247	0
[8 3 32]	247	0
[2 2 128]	252	0
[126 -882 6176]	252	0
[4 2 64]	252	0
[64 -450 3168]	252	0
[42 -126 384]	252	-514
[42 -210 1056]	252	797
[8 2 32]	252	0
[32 -450 6336]	252	-198
[24 -126 672]	252	-514
[14 14 32]	252	0
[18 -258 3712]	252	0
[2 1 128]	255	0
[4 1 64]	255	0
[44 -575 7520]	255	-218
[8 1 32]	255	0
[14 33 96]	255	-218
[2 0 128]	256	0
[2 8 160]	256	0
[2 16 256]	256	0
[4 0 64]	256	0
[4 16 128]	256	0
[8 0 32]	256	0
[8 16 64]	256	0
[8 32 160]	256	-144
[34 -480 6784]	256	0
[26 -80 256]	256	-208
[26 -288 3200]	256	-144
[26 -392 5920]	256	0
[16 16 32]	256	0
[16 -48 160]	256	-208
[20 -48 128]	256	416
[20 -288 4160]	256	0
[132 -1717 22336]	263	0
[66 -331 1664]	263	0
[408 949 2208]	263	0
[22 -309 4352]	263	451
[22 -331 4992]	263	0
[24 -53 128]	263	420
[136 -1497 16480]	271	0
[74 217 640]	271	0
[34 -103 320]	271	205
[28 -423 6400]	271	0
[20 -103 544]	271	205
[212 487 1120]	271	0
[140 -1261 11360]	279	0
[70 -211 640]	279	0
[164 403 992]	279	-337
[28 -365 4768]	279	-167
[268 621 1440]	279	0
[24 -339 4800]	279	99
[42 -141 480]	279	348
[20 -301 4544]	279	0
[18 -237 3136]	279	-167
[20 -109 608]	279	-337
[142 -426 1280]	284	0
[142 -710 3552]	284	0
[72 -362 1824]	284	0
[72 -938 12224]	284	0
[58 166 480]	284	470
[450 1046 2432]	284	0
[6 26 160]	284	208
[58 -182 576]	284	76
[36 -182 928]	284	76
[36 -470 6144]	284	204
[30 -154 800]	284	-208
[30 -454 6880]	284	0
[24 -362 5472]	284	0
[240 554 1280]	284	0
[18 -38 96]	284	-470
[20 -106 576]	284	-204
[20 -266 3552]	284	88
[144 -1009 7072]	287	0
[78 -239 736]	287	0
[48 -241 1216]	287	-1641
[338 785 1824]	287	0
[12 49 224]	287	677
[26 -369 5248]	287	-825
[24 -271 3072]	287	677
[18 -271 4096]	287	0
[148 -741 3712]	295	0
[74 -667 6016]	295	0
[38 -421 4672]	295	-425
[10 5 32]	295	0
[226 517 1184]	295	0
[152 -457 1376]	303	0
[4 9 96]	303	0
[6 9 64]	303	0
[26 -55 128]	303	-880
[26 -393 5952]	303	0
[2 4 160]	304	0
[154 460 1376]	304	0
[10 4 32]	304	0
[10 44 224]	304	86
[50 -164 544]	304	-90
[22 -244 2720]	304	86
[22 -332 5024]	304	0
[2 3 160]	311	0
[78 -547 3840]	311	0
[52 -157 480]	311	935
[218 541 1344]	311	-1226
[8 35 192]	311	465
[10 3 32]	311	0
[26 -131 672]	311	-465
[12 29 96]	311	-935
[16 35 96]	311	1226
[240 547 1248]	311	0
[2 2 160]	316	0
[158 -1106 7744]	316	0
[80 -242 736]	316	0
[80 -882 9728]	316	0
[380 882 2048]	316	0
[10 2 32]	316	0
[310 718 1664]	316	0
[20 -302 4576]	316	0
[22 -50 128]	316	-986
[2 1 160]	319	0
[50 -159 512]	319	-206
[10 1 32]	319	0
[32 -353 3904]	319	206
[20 -289 4192]	319	0
[2 0 160]	320	0
[2 8 192]	320	0
[2 16 288]	320	0
[4 8 96]	320	0
[84 248 736]	320	0
[6 8 64]	320	0
[54 -272 1376]	320	24
[54 -704 9184]	320	-640
[10 0 32]	320	0
[10 40 192]	320	-192
[28 -312 3488]	320	-192
[28 -424 6432]	320	0
[14 32 96]	320	-640
[254 584 1344]	320	0
[20 -40 96]	320	112
[18 -272 4128]	320	0
[164 -2133 27744]	327	0
[82 -1067 13888]	327	0
[56 -619 6848]	327	-72
[42 -213 1088]	327	72
[282 651 1504]	327	0
[24 -363 5504]	327	0
[168 -1849 20352]	335	0
[766 1785 4160]	335	0
[6 7 64]	335	0
[6 25 160]	335	-431
[36 -185 960]	335	431
[28 -57 128]	335	600
[30 -455 6912]	335	0
[172 -1549 13952]	343	0
[86 -947 10432]	343	0
[422 979 2272]	343	0
[22 -333 5056]	343	0
[174 -522 1568]	348	0
[174 -870 4352]	348	0
[4 6 96]	348	0
[88 -794 7168]	348	0
[6 6 64]	348	0
[44 -134 416]	348	-96
[44 -486 5376]	348	72
[12 6 32]	348	0
[26 -134 704]	348	-96
[26 -394 5984]	348	0
[22 -310 4384]	348	630
[268 614 1408]	348	0
[176 -1233 8640]	351	0
[4 17 160]	351	0
[60 -303 1536]	351	4373
[36 -111 352]	351	-155
[352 815 1888]	351	0
[22 -111 576]	351	-155
[20 -303 4608]	351	0
[20 -47 128]	351	-475
[180 -901 4512]	359	0
[4 5 96]	359	0
[60 -901 13536]	359	0
[68 197 576]	359	-1569
[30 -421 5920]	359	-436
[12 5 32]	359	0
[18 37 96]	359	-1569
[184 -553 1664]	367	0
[376 937 2336]	367	0
[46 -599 7808]	367	355
[28 -425 6464]	367	0
[296 681 1568]	367	0
[26 -137 736]	367	-355
[2 4 192]	368	0
[186 556 1664]	368	0
[4 4 96]	368	0
[92 -276 832]	368	0
[92 -460 2304]	368	0
[92 -644 4512]	368	0
[62 -188 576]	368	192
[62 -684 7552]	368	1200
[62 -932 14016]	368	0
[8 12 64]	368	0
[48 -244 1248]	368	-1200
[48 -628 8224]	368	-1664
[12 4 32]	368	0
[12 28 96]	368	-192
[324 748 1728]	368	0
[52 -172 576]	368	-1024
[24 -52 128]	368	488
[24 -340 4832]	368	-2544
[24 -364 5536]	368	0
[18 44 128]	368	-1664
[282 644 1472]	368	0
[2 3 192]	375	0
[94 -1411 21184]	375	0
[6 3 64]	375	0
[40 -445 4960]	375	-642
[12 3 32]	375	0
[32 -451 6368]	375	924
[2 2 192]	380	0
[190 -1330 9312]	380	0
[4 2 96]	380	0
[6 2 64]	380	0
[6 14 96]	380	5001
[270 670 1664]	380	344
[8 34 192]	380	-1046
[60 -190 608]	380	-710
[38 -114 352]	380	715
[38 -190 960]	380	-710
[12 2 32]	380	0
[32 -162 832]	380	1046
[16 34 96]	380	-344
[26 -370 5280]	380	2992
[22 -114 608]	380	715
[22 -334 5088]	380	0
[20 30 64]	380	0
[2 1 192]	383	0
[4 1 96]	383	0
[6 1 64]	383	0
[64 -833 10848]	383	2159
[12 1 32]	383	0
[34 -481 6816]	383	-550
[14 31 96]	383	2159
[2 0 192]	384	0
[2 8 224]	384	0
[4 0 96]	384	0
[4 16 160]	384	0
[6 0 64]	384	0
[6 24 160]	384	132
[10 24 96]	384	132
[394 912 2112]	384	0
[12 0 32]	384	0
[12 48 224]	384	-1264
[30 -336 3776]	384	-1264
[30 -456 6944]	384	0
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