A Borcherds Product in S₁₁(K(16))⁻

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c vector |

θ vector |

ψ expansion |

Borch(ψ) coefficients

• The singular part of ψ truncated up to q^(16/4) is
  (22 + ζ^(-4) + 12/ζ^2 + 12*ζ^2 + ζ^4)
  +q(-2/ζ^8 - 2*ζ^8)
  +q^2(ζ^(-12) + ζ^12)
  +q^3(12/ζ^14 + 12*ζ^14)
  +q^4(22/ζ^16 + 22*ζ^16)
• From the truncated singular part of ψ, the leading Jacobi coefficient of Borch(ψ) is the theta block

  	φ	= TB(11;2,2,2,2,2,2,2,2,2,2,2,2,4)
  		= 21241
  		= η(τ)22(θ(τ,2z)/η(τ))12(θ(τ,4z)/η(τ))

  This theta block has q-order of vanishing 2. This theta block has index 2*16, 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 5 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)	13
  (4,2)	13
  (4,14)	13
  (16,4)	1
  (16,12)	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

(22 + ζ^(-4) + 12/ζ^2 + 12*ζ^2 + ζ^4)
+q(132 - 2/ζ^8 - 12/ζ^6 - 64/ζ^4 + 12/ζ^2 + 12*ζ^2 - 64*ζ^4 - 12*ζ^6 - 2*ζ^8)
+q^2(856 + ζ^(-12) - 12/ζ^10 + 84/ζ^8 - 24/ζ^6 - 513/ζ^4 + 36/ζ^2 + 36*ζ^2 - 513*ζ^4 - 24*ζ^6 + 84*ζ^8 - 12*ζ^10 + ζ^12)
+q^3(4112 + 12/ζ^14 - 64/ζ^12 - 24/ζ^10 + 760/ζ^8 - 48/ζ^6 - 2752/ζ^4 + 60/ζ^2 + 60*ζ^2 - 2752*ζ^4 - 48*ζ^6 + 760*ζ^8 - 24*ζ^10 - 64*ζ^12 + 12*ζ^14)
+q^4(16692 + 22/ζ^16 + 12/ζ^14 - 513/ζ^12 - 48/ζ^10 + 3920/ζ^8 - 84/ζ^6 - 11775/ζ^4 + 120/ζ^2 + 120*ζ^2 - 11775*ζ^4 - 84*ζ^6 + 3920*ζ^8 - 48*ζ^10 - 513*ζ^12 + 12*ζ^14 + 22*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	0
[24 -73 224]	47	0
[12 -73 448]	47	0
[4 9 32]	47	0
[8 -73 672]	47	0
[16 -55 192]	47	0
[2 4 32]	48	0
[26 76 224]	48	0
[6 12 32]	48	0
[14 36 96]	48	0
[2 3 32]	55	0
[14 -29 64]	55	0
[14 -99 704]	55	0
[8 -35 160]	55	0
[2 2 32]	60	0
[30 -210 1472]	60	0
[16 -50 160]	60	-1
[16 -178 1984]	60	-1
[10 -30 96]	60	1
[10 -50 256]	60	-1
[8 -50 320]	60	0
[8 -114 1632]	60	0
[2 1 32]	63	0
[16 -225 3168]	63	0
[4 33 288]	63	0
[12 -159 2112]	63	0
[8 -97 1184]	63	0
[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 -64 416]	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	0
[10 -53 288]	71	0
[40 -441 4864]	79	0
[20 -121 736]	79	0
[190 441 1024]	79	0
[10 -121 1472]	79	0
[44 -397 3584]	87	0
[22 -45 96]	87	0
[22 -243 2688]	87	0
[12 -51 224]	87	0
[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	0
[162 374 864]	92	0
[6 26 128]	92	3
[26 -86 288]	92	3
[12 -26 64]	92	0
[12 -38 128]	92	3
[12 -122 1248]	92	0
[12 -134 1504]	92	3
[48 -337 2368]	95	0
[24 -337 4736]	95	0
[4 17 96]	95	0
[16 -81 416]	95	0
[12 -145 1760]	95	0
[60 145 352]	95	0
[10 15 32]	95	0
[52 -261 1312]	103	0
[4 5 32]	103	0
[26 -261 2624]	103	0
[14 -59 256]	103	0
[56 -169 512]	111	0
[28 -169 1024]	111	0
[120 297 736]	111	0
[6 9 32]	111	0
[14 -169 2048]	111	0
[14 -183 2400]	111	0
[10 23 64]	111	0
[2 4 64]	112	0
[58 172 512]	112	0
[4 4 32]	112	0
[28 -84 256]	112	12
[28 -140 704]	112	-12
[28 -196 1376]	112	0
[14 -28 64]	112	0
[14 -84 512]	112	0
[8 12 32]	112	0
[16 -84 448]	112	12
[16 -148 1376]	112	0
[16 -212 2816]	112	12
[2 3 64]	119	0
[30 -61 128]	119	0
[30 -451 6784]	119	0
[20 -61 192]	119	0
[90 221 544]	119	0
[16 -67 288]	119	0
[12 -61 320]	119	0
[24 61 160]	119	0
[12 -125 1312]	119	0
[2 2 64]	124	0
[62 -434 3040]	124	0
[4 2 32]	124	0
[32 -226 1600]	124	0
[16 -98 608]	124	0
[16 -226 3200]	124	0
[8 34 160]	124	-15
[28 -94 320]	124	-15
[10 14 32]	124	0
[14 -46 160]	124	-15
[14 -130 1216]	124	0
[14 -158 1792]	124	-15
[2 1 64]	127	0
[4 1 32]	127	0
[32 -449 6304]	127	0
[16 -193 2336]	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
[6 8 32]	128	0
[22 -112 576]	128	0
[22 -288 3776]	128	0
[18 -40 96]	128	0
[18 -112 704]	128	0
[18 -256 3648]	128	0
[12 16 32]	128	0
[12 -64 352]	128	0
[68 -885 11520]	135	0
[34 -171 864]	135	0
[34 -341 3424]	135	0
[24 -267 2976]	135	0
[8 11 32]	135	0
[12 -75 480]	135	0
[72 -793 8736]	143	0
[36 -217 1312]	143	0
[42 121 352]	143	0
[6 7 32]	143	0
[6 25 128]	143	0
[18 -199 2208]	143	0
[12 -25 64]	143	0
[14 -89 576]	143	0
[76 -685 6176]	151	0
[38 -77 160]	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	65
[40 -522 6816]	156	65
[6 6 32]	156	0
[8 10 32]	156	0
[20 -42 96]	156	0
[20 -202 2048]	156	0
[10 22 64]	156	65
[82 198 480]	156	65
[10 42 192]	156	-50
[30 -102 352]	156	-50
[16 -54 192]	156	-50
[16 -182 2080]	156	-50
[80 -561 3936]	159	0
[40 -561 7872]	159	0
[46 -143 448]	159	0
[28 -143 736]	159	0
[30 81 224]	159	0
[16 -49 160]	159	0
[12 15 32]	159	0
[14 -143 1472]	159	0
[84 -421 2112]	167	0
[42 -379 3424]	167	0
[42 -421 4224]	167	0
[28 -421 6336]	167	0
[36 101 288]	167	0
[22 -69 224]	167	0
[14 -155 1728]	167	0
[14 -197 2784]	167	0
[16 -37 96]	167	0
[88 -265 800]	175	0
[4 9 64]	175	0
[8 9 32]	175	0
[20 -265 3520]	175	0
[2 4 96]	176	0
[90 268 800]	176	0
[30 -92 288]	176	0
[30 -332 3680]	176	0
[30 -452 6816]	176	0
[22 -44 96]	176	0
[22 -132 800]	176	0
[10 12 32]	176	0
[18 -92 480]	176	0
[18 -236 3104]	176	0
[2 3 96]	183	0
[46 -93 192]	183	0
[46 -323 2272]	183	0
[6 3 32]	183	0
[38 -125 416]	183	0
[16 -227 3232]	183	0
[2 2 96]	188	0
[94 -658 4608]	188	0
[48 -146 448]	188	-57
[48 -530 5856]	188	-57
[6 2 32]	188	0
[6 14 64]	188	-57
[142 350 864]	188	-57
[24 -146 896]	188	0
[24 -338 4768]	188	0
[8 18 64]	188	57
[112 274 672]	188	57
[12 14 32]	188	0
[12 50 224]	188	87
[32 -110 384]	188	87
[14 34 96]	188	57
[102 242 576]	188	57
[14 62 288]	188	-87
[42 -146 512]	188	-87
[2 1 96]	191	0
[48 -673 9440]	191	0
[6 1 32]	191	0
[32 -417 5440]	191	0
[8 33 160]	191	0
[20 -223 2496]	191	0
[36 95 256]	191	0
[16 -161 1632]	191	0
[18 -95 512]	191	0
[18 -257 3680]	191	0
[2 0 96]	192	0
[2 8 128]	192	0
[2 16 224]	192	0
[4 8 64]	192	208
[52 152 448]	192	-208
[6 0 32]	192	0
[6 24 128]	192	0
[8 8 32]	192	0
[24 -72 224]	192	0
[24 -120 608]	192	0
[26 -56 128]	192	0
[26 -160 992]	192	0
[26 -368 5216]	192	0
[12 24 64]	192	-208
[28 72 192]	192	208
[14 16 32]	192	0
[14 -72 384]	192	0
[100 -1301 16928]	199	0
[50 -651 8480]	199	0
[26 -341 4480]	199	0
[10 11 32]	199	0
[10 21 64]	199	0
[16 -213 2848]	199	0
[104 -1145 12608]	207	0
[478 1113 2592]	207	0
[26 -391 5888]	207	0
[18 -39 96]	207	0
[18 -57 192]	207	0
[108 -973 8768]	215	0
[54 -595 6560]	215	0
[36 -397 4384]	215	0
[6 13 64]	215	0
[24 -77 256]	215	0
[12 13 32]	215	0
[18 -109 672]	215	0
[20 -205 2112]	215	0
[16 -179 2016]	215	0
[110 -330 992]	220	0
[110 -550 2752]	220	0
[4 6 64]	220	-429
[56 -506 4576]	220	-429
[28 -58 128]	220	0
[28 -282 2848]	220	0
[28 -422 6368]	220	0
[10 10 32]	220	0
[34 86 224]	220	-429
[170 390 896]	220	-429
[14 58 256]	220	336
[34 -118 416]	220	336
[16 -70 320]	220	336
[16 -198 2464]	220	336
[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	0
[30 -453 6848]	231	0
[24 -123 640]	231	0
[40 -133 448]	231	0
[20 -123 768]	231	0
[104 251 608]	231	0
[120 -361 1088]	239	0
[248 617 1536]	239	0
[48 -151 480]	239	0
[30 -151 768]	239	0
[24 -361 5440]	239	0
[10 41 192]	239	0
[20 -41 96]	239	0
[198 457 1056]	239	0
[22 -247 2784]	239	0
[2 4 128]	240	0
[122 364 1088]	240	0
[4 4 64]	240	572
[60 -180 544]	240	320
[60 -300 1504]	240	-320
[60 -420 2944]	240	-572
[6 12 64]	240	320
[46 132 384]	240	-320
[30 -60 128]	240	0
[30 -180 1088]	240	0
[8 4 32]	240	0
[32 -100 320]	240	-252
[32 -356 3968]	240	-252
[10 20 64]	240	-320
[34 92 256]	240	320
[12 12 32]	240	0
[20 -60 192]	240	252
[20 -100 512]	240	-252
[16 -36 96]	240	320
[16 -164 1696]	240	320
[16 -228 3264]	240	-572
[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	-429
[42 -126 384]	252	-15
[42 -210 1056]	252	15
[8 2 32]	252	0
[32 -194 1184]	252	0
[32 -450 6336]	252	0
[24 -126 672]	252	-15
[24 -318 4224]	252	-15
[14 14 32]	252	0
[16 -194 2368]	252	207
[18 -222 2752]	252	-207
[18 -258 3712]	252	429
[2 1 128]	255	0
[4 1 64]	255	0
[44 -575 7520]	255	0
[8 1 32]	255	0
[22 -223 2272]	255	0
[22 -289 3808]	255	0
[20 -65 224]	255	0
[14 33 96]	255	0
[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	0
[34 -72 160]	256	0
[34 -480 6784]	256	0
[26 -80 256]	256	0
[26 -288 3200]	256	0
[26 -392 5920]	256	0
[16 16 32]	256	0
[16 -48 160]	256	0
[20 -48 128]	256	0
[20 -288 4160]	256	0
[132 -1717 22336]	263	0
[66 -331 1664]	263	0
[408 949 2208]	263	0
[22 -309 4352]	263	0
[22 -331 4992]	263	0
[24 -53 128]	263	0
[12 53 256]	263	0
[18 -203 2304]	263	0
[136 -1497 16480]	271	0
[74 217 640]	271	0
[34 -103 320]	271	0
[28 -423 6400]	271	0
[134 327 800]	271	0
[20 -103 544]	271	0
[212 487 1120]	271	0
[124 295 704]	271	0
[140 -1261 11360]	279	0
[70 -211 640]	279	0
[28 -365 4768]	279	0
[268 621 1440]	279	0
[24 -339 4800]	279	0
[42 -141 480]	279	0
[20 -301 4544]	279	0
[98 237 576]	279	0
[18 -237 3136]	279	0
[20 -109 608]	279	0
[142 -426 1280]	284	0
[142 -710 3552]	284	0
[72 -362 1824]	284	780
[72 -938 12224]	284	780
[450 1046 2432]	284	780
[6 26 160]	284	-330
[58 -182 576]	284	-330
[36 -74 160]	284	0
[36 -182 928]	284	-330
[36 -362 3648]	284	0
[36 -470 6144]	284	-330
[30 -154 800]	284	330
[30 -394 5184]	284	330
[30 -454 6880]	284	0
[24 -362 5472]	284	0
[240 554 1280]	284	-780
[18 -38 96]	284	-780
[18 -182 1856]	284	-780
[20 -106 576]	284	330
[20 -266 3552]	284	330
[144 -1009 7072]	287	0
[78 -239 736]	287	0
[48 -241 1216]	287	0
[338 785 1824]	287	0
[24 -241 2432]	287	0
[12 49 224]	287	0
[26 -369 5248]	287	0
[24 -271 3072]	287	0
[18 -271 4096]	287	0
[148 -741 3712]	295	0
[74 -667 6016]	295	0
[38 -421 4672]	295	0
[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	0
[26 -393 5952]	303	0
[22 -73 256]	303	0
[2 4 160]	304	0
[154 460 1376]	304	0
[38 -76 160]	304	0
[10 4 32]	304	0
[10 44 224]	304	0
[50 -164 544]	304	0
[22 -68 224]	304	0
[22 -244 2720]	304	0
[22 -332 5024]	304	0
[2 3 160]	311	0
[52 -157 480]	311	0
[218 541 1344]	311	0
[8 35 192]	311	0
[10 3 32]	311	0
[26 -131 672]	311	0
[26 -157 960]	311	0
[28 -285 2912]	311	0
[12 29 96]	311	0
[16 35 96]	311	0
[20 -227 2592]	311	0
[240 547 1248]	311	0
[2 2 160]	316	0
[158 -1106 7744]	316	0
[80 -242 736]	316	-345
[80 -882 9728]	316	-345
[40 -562 7904]	316	0
[380 882 2048]	316	345
[10 2 32]	316	0
[310 718 1664]	316	-345
[20 -302 4576]	316	0
[22 -50 128]	316	-345
[22 -226 2336]	316	-345
[2 1 160]	319	0
[50 -159 512]	319	0
[10 1 32]	319	0
[32 -353 3904]	319	0
[38 97 256]	319	0
[20 -289 4192]	319	0
[2 0 160]	320	0
[2 8 192]	320	0
[4 8 96]	320	768
[84 248 736]	320	-768
[6 8 64]	320	768
[54 -272 1376]	320	-768
[54 -704 9184]	320	-768
[42 -88 192]	320	0
[42 -592 8352]	320	0
[10 0 32]	320	0
[10 40 192]	320	0
[28 -312 3488]	320	0
[28 -424 6432]	320	0
[14 32 96]	320	-768
[126 304 736]	320	-768
[254 584 1344]	320	768
[20 -40 96]	320	768
[20 -120 736]	320	-768
[18 -56 192]	320	0
[18 -272 4128]	320	0
[164 -2133 27744]	327	0
[82 -1067 13888]	327	0
[56 -619 6848]	327	0
[42 -213 1088]	327	0
[282 651 1504]	327	0
[24 -363 5504]	327	0
[44 -149 512]	327	0
[168 -1849 20352]	335	0
[766 1785 4160]	335	0
[6 7 64]	335	0
[6 25 160]	335	0
[36 -185 960]	335	0
[28 -57 128]	335	0
[30 -455 6912]	335	0
[24 -313 4096]	335	0
[14 57 256]	335	0
[54 -185 640]	335	0
[118 281 672]	335	0
[172 -1549 13952]	343	0
[86 -947 10432]	343	0
[422 979 2272]	343	0
[28 -371 4928]	343	0
[22 -333 5056]	343	0
[174 -522 1568]	348	0
[174 -870 4352]	348	0
[4 6 96]	348	572
[88 -794 7168]	348	572
[6 6 64]	348	572
[44 -90 192]	348	0
[44 -134 416]	348	324
[44 -486 5376]	348	324
[12 6 32]	348	0
[26 -134 704]	348	324
[26 -342 4512]	348	324
[26 -394 5984]	348	0
[22 -310 4384]	348	-572
[268 614 1408]	348	572
[4 17 160]	351	0
[60 -303 1536]	351	0
[36 -111 352]	351	0
[352 815 1888]	351	0
[30 -303 3072]	351	0
[22 -111 576]	351	0
[20 -303 4608]	351	0
[20 -47 128]	351	0
[180 -901 4512]	359	0
[4 5 96]	359	0
[60 -901 13536]	359	0
[30 -421 5920]	359	0
[12 5 32]	359	0
[32 -69 160]	359	0
[20 -261 3424]	359	0
[18 37 96]	359	0
[22 -229 2400]	359	0
[22 -251 2880]	359	0
[184 -553 1664]	367	0
[376 937 2336]	367	0
[46 -599 7808]	367	0
[28 -425 6464]	367	0
[296 681 1568]	367	0
[26 -137 736]	367	0
[2 4 192]	368	0
[186 556 1664]	368	0
[4 4 96]	368	-2496
[92 -276 832]	368	-228
[92 -460 2304]	368	228
[62 -188 576]	368	-2496
[62 -684 7552]	368	-2496
[62 -932 14016]	368	2496
[46 -92 192]	368	0
[8 12 64]	368	-228
[48 -244 1248]	368	2496
[12 4 32]	368	0
[12 28 96]	368	2496
[324 748 1728]	368	-228
[12 52 256]	368	-2268
[52 -172 576]	368	-2268
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