Properties

Label 1512.2.bl
Level 1512
Weight 2
Character orbit bl
Rep. character \(\chi_{1512}(593,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 64
Newform subspaces 4
Sturm bound 576
Trace bound 19

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Defining parameters

Level: \( N \) = \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1512.bl (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 4 \)
Sturm bound: \(576\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1512, [\chi])\).

Total New Old
Modular forms 624 64 560
Cusp forms 528 64 464
Eisenstein series 96 0 96

Trace form

\( 64q + 4q^{7} + O(q^{10}) \) \( 64q + 4q^{7} - 6q^{19} - 42q^{25} + 42q^{31} - 2q^{37} - 40q^{43} - 16q^{49} + 14q^{67} + 30q^{73} - 2q^{79} + 16q^{85} - 12q^{91} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1512, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1512.2.bl.a \(16\) \(12.073\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(-2\) \(q-\beta _{5}q^{5}-\beta _{2}q^{7}+(-\beta _{7}-\beta _{12})q^{11}+\cdots\)
1512.2.bl.b \(16\) \(12.073\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(-2\) \(q+(-\beta _{1}+\beta _{13}-\beta _{14})q^{5}+(-\beta _{2}-\beta _{3}+\cdots)q^{7}+\cdots\)
1512.2.bl.c \(16\) \(12.073\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(4\) \(q+\beta _{14}q^{5}+(\beta _{4}-\beta _{5}-\beta _{7}+\beta _{9})q^{7}+\cdots\)
1512.2.bl.d \(16\) \(12.073\) 16.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(4\) \(q+(\beta _{4}+\beta _{10})q^{5}-\beta _{2}q^{7}+(-\beta _{1}-\beta _{6}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1512, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1512, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(378, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(504, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(756, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( 1 - 8 T^{2} + 16 T^{4} - 208 T^{6} + 1346 T^{8} - 2360 T^{10} + 37952 T^{12} - 176152 T^{14} + 40291 T^{16} - 4403800 T^{18} + 23720000 T^{20} - 36875000 T^{22} + 525781250 T^{24} - 2031250000 T^{26} + 3906250000 T^{28} - 48828125000 T^{30} + 152587890625 T^{32} \))(\( 1 - 17 T^{2} + 157 T^{4} - 940 T^{6} + 3356 T^{8} - 908 T^{10} - 80884 T^{12} + 689507 T^{14} - 3972719 T^{16} + 17237675 T^{18} - 50552500 T^{20} - 14187500 T^{22} + 1310937500 T^{24} - 9179687500 T^{26} + 38330078125 T^{28} - 103759765625 T^{30} + 152587890625 T^{32} \))(\( 1 - 14 T^{2} + 61 T^{4} + 8 T^{6} - 580 T^{8} + 2692 T^{10} - 7864 T^{12} - 231781 T^{14} + 2288086 T^{16} - 5794525 T^{18} - 4915000 T^{20} + 42062500 T^{22} - 226562500 T^{24} + 78125000 T^{26} + 14892578125 T^{28} - 85449218750 T^{30} + 152587890625 T^{32} \))(\( 1 - 20 T^{2} + 184 T^{4} - 1096 T^{6} + 5282 T^{8} - 23132 T^{10} + 89600 T^{12} - 326908 T^{14} + 1415731 T^{16} - 8172700 T^{18} + 56000000 T^{20} - 361437500 T^{22} + 2063281250 T^{24} - 10703125000 T^{26} + 44921875000 T^{28} - 122070312500 T^{30} + 152587890625 T^{32} \))
$7$ (\( ( 1 + T - 7 T^{3} - 82 T^{4} - 49 T^{5} + 343 T^{7} + 2401 T^{8} )^{2} \))(\( ( 1 + T + 9 T^{2} + 35 T^{3} + 35 T^{4} + 245 T^{5} + 441 T^{6} + 343 T^{7} + 2401 T^{8} )^{2} \))(\( ( 1 - 2 T - 9 T^{2} - T^{3} + 116 T^{4} - 7 T^{5} - 441 T^{6} - 686 T^{7} + 2401 T^{8} )^{2} \))(\( ( 1 - 2 T + 9 T^{2} - 34 T^{3} + 92 T^{4} - 238 T^{5} + 441 T^{6} - 686 T^{7} + 2401 T^{8} )^{2} \))
$11$ (\( 1 + 40 T^{2} + 592 T^{4} + 6800 T^{6} + 131138 T^{8} + 1759960 T^{10} + 13053248 T^{12} + 168275960 T^{14} + 2611238371 T^{16} + 20361391160 T^{18} + 191112603968 T^{20} + 3117876497560 T^{22} + 28110594936578 T^{24} + 176374487286800 T^{26} + 1857949599018832 T^{28} + 15189993343329640 T^{30} + 45949729863572161 T^{32} \))(\( 1 + 58 T^{2} + 1801 T^{4} + 37304 T^{6} + 562052 T^{8} + 6203728 T^{10} + 48440420 T^{12} + 243538427 T^{14} + 1239374062 T^{16} + 29468149667 T^{18} + 709216189220 T^{20} + 10990282579408 T^{22} + 120480837783812 T^{24} + 967569687315704 T^{26} + 5652309506474521 T^{28} + 22025490347827978 T^{30} + 45949729863572161 T^{32} \))(\( 1 + 19 T^{2} - 23 T^{4} - 4456 T^{6} - 28228 T^{8} + 533860 T^{10} + 7671380 T^{12} - 29265493 T^{14} - 1170886031 T^{16} - 3541124653 T^{18} + 112316674580 T^{20} + 945765555460 T^{22} - 6050922492868 T^{24} - 115577164022056 T^{26} - 72183852664583 T^{28} + 7215246838081579 T^{30} + 45949729863572161 T^{32} \))(\( 1 + 28 T^{2} + 316 T^{4} - 472 T^{6} - 56470 T^{8} - 785948 T^{10} - 3198064 T^{12} + 52576484 T^{14} + 1079693059 T^{16} + 6361754564 T^{18} - 46822855024 T^{20} - 1392354824828 T^{22} - 12104846010070 T^{24} - 12242464411672 T^{26} + 991743367043836 T^{28} + 10632995340330748 T^{30} + 45949729863572161 T^{32} \))
$13$ (\( ( 1 - 43 T^{2} + 981 T^{4} - 14582 T^{6} + 191918 T^{8} - 2464358 T^{10} + 28018341 T^{12} - 207552787 T^{14} + 815730721 T^{16} )^{2} \))(\( ( 1 - 82 T^{2} + 3057 T^{4} - 69371 T^{6} + 1073570 T^{8} - 11723699 T^{10} + 87310977 T^{12} - 395798338 T^{14} + 815730721 T^{16} )^{2} \))(\( ( 1 + 2 T^{2} + 81 T^{4} - 2123 T^{6} - 31690 T^{8} - 358787 T^{10} + 2313441 T^{12} + 9653618 T^{14} + 815730721 T^{16} )^{2} \))(\( ( 1 - 82 T^{2} + 3129 T^{4} - 73034 T^{6} + 1144484 T^{8} - 12342746 T^{10} + 89367369 T^{12} - 395798338 T^{14} + 815730721 T^{16} )^{2} \))
$17$ (\( 1 - 60 T^{2} + 1608 T^{4} - 25848 T^{6} + 271234 T^{8} - 432516 T^{10} - 52024320 T^{12} + 967086156 T^{14} - 12371794029 T^{16} + 279487899084 T^{18} - 4345123230720 T^{20} - 10439884793604 T^{22} + 1892062593752194 T^{24} - 52109410338805752 T^{26} + 936856557465455688 T^{28} - 10102669593564055740 T^{30} + 48661191875666868481 T^{32} \))(\( 1 - 18 T^{2} + 255 T^{4} + 10500 T^{6} - 283019 T^{8} + 4453854 T^{10} + 2888559 T^{12} - 1373182944 T^{14} + 31467444945 T^{16} - 396849870816 T^{18} + 241255336239 T^{20} + 107505208240926 T^{22} - 1974271895194379 T^{24} + 21167935954714500 T^{26} + 148568670493589055 T^{28} - 3030800878069216722 T^{30} + 48661191875666868481 T^{32} \))(\( 1 - 102 T^{2} + 5415 T^{4} - 206640 T^{6} + 6384925 T^{8} - 167030250 T^{10} + 3794350455 T^{12} - 76308529980 T^{14} + 1371191367105 T^{16} - 22053165164220 T^{18} + 316907944352055 T^{20} - 4031704184462250 T^{22} + 44539688078976925 T^{24} - 416584979588781360 T^{26} + 3154899414599155815 T^{28} - 17174538309058894758 T^{30} + 48661191875666868481 T^{32} \))(\( 1 - 84 T^{2} + 3672 T^{4} - 107400 T^{6} + 2312962 T^{8} - 37424988 T^{10} + 449629056 T^{12} - 4067785692 T^{14} + 43331761299 T^{16} - 1175590064988 T^{18} + 37553468386176 T^{20} - 903348230174172 T^{22} + 16134661882250242 T^{24} - 216517744908222600 T^{26} + 2139388855107682392 T^{28} - 14143737430989678036 T^{30} + 48661191875666868481 T^{32} \))
$19$ (\( ( 1 + 3 T + 53 T^{2} + 150 T^{3} + 1437 T^{4} + 3069 T^{5} + 23386 T^{6} + 40311 T^{7} + 338342 T^{8} + 765909 T^{9} + 8442346 T^{10} + 21050271 T^{11} + 187271277 T^{12} + 371414850 T^{13} + 2493431693 T^{14} + 2681615217 T^{15} + 16983563041 T^{16} )^{2} \))(\( ( 1 - 3 T + 29 T^{2} - 78 T^{3} + 252 T^{4} + 714 T^{5} - 4994 T^{6} + 81579 T^{7} - 245935 T^{8} + 1550001 T^{9} - 1802834 T^{10} + 4897326 T^{11} + 32840892 T^{12} - 193135722 T^{13} + 1364330549 T^{14} - 2681615217 T^{15} + 16983563041 T^{16} )^{2} \))(\( ( 1 + 9 T + 77 T^{2} + 450 T^{3} + 2244 T^{4} + 11718 T^{5} + 59482 T^{6} + 301239 T^{7} + 1433825 T^{8} + 5723541 T^{9} + 21473002 T^{10} + 80373762 T^{11} + 292440324 T^{12} + 1114244550 T^{13} + 3622532837 T^{14} + 8044845651 T^{15} + 16983563041 T^{16} )^{2} \))(\( ( 1 - 6 T + 23 T^{2} - 66 T^{3} + 9 T^{4} + 336 T^{5} - 6794 T^{6} + 56748 T^{7} - 289030 T^{8} + 1078212 T^{9} - 2452634 T^{10} + 2304624 T^{11} + 1172889 T^{12} - 163422534 T^{13} + 1082055263 T^{14} - 5363230434 T^{15} + 16983563041 T^{16} )^{2} \))
$23$ (\( ( 1 + 42 T^{2} + 1235 T^{4} + 22218 T^{6} + 279841 T^{8} )^{4} \))(\( 1 + 90 T^{2} + 3623 T^{4} + 86400 T^{6} + 1618033 T^{8} + 39578310 T^{10} + 1297280747 T^{12} + 38239323840 T^{14} + 946740877921 T^{16} + 20228602311360 T^{18} + 363032341521227 T^{20} + 5859010305967590 T^{22} + 126709758447172273 T^{24} + 3579250568859273600 T^{26} + 79396684317209622983 T^{28} + \)\(10\!\cdots\!10\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 + 102 T^{2} + 6023 T^{4} + 223644 T^{6} + 5387737 T^{8} + 54053766 T^{10} - 1721404285 T^{12} - 106929141708 T^{14} - 3095815766783 T^{16} - 56565515963532 T^{18} - 481719496518685 T^{20} + 8001897303607974 T^{22} + 421918992904899097 T^{24} + 9264790673865316956 T^{26} + \)\(13\!\cdots\!83\)\( T^{28} + \)\(11\!\cdots\!18\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 + 48 T^{2} + 32 T^{4} - 22848 T^{6} + 227938 T^{8} + 14969952 T^{10} - 71080576 T^{12} - 4283552784 T^{14} - 14608125629 T^{16} - 2265999422736 T^{18} - 19891259468416 T^{20} + 2216090152607328 T^{22} + 17850049362980578 T^{24} - 946512928209452352 T^{26} + 701267981824650272 T^{28} + \)\(55\!\cdots\!32\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( ( 1 + 20 T^{2} + 72 T^{4} - 21860 T^{6} - 211090 T^{8} - 18384260 T^{10} + 50924232 T^{12} + 11896466420 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 - 202 T^{2} + 18483 T^{4} - 1005083 T^{6} + 35645051 T^{8} - 845274803 T^{10} + 13072674723 T^{12} - 120154310842 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 - 154 T^{2} + 12171 T^{4} - 609827 T^{6} + 21113243 T^{8} - 512864507 T^{10} + 8608317051 T^{12} - 91602791434 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 - 172 T^{2} + 13800 T^{4} - 688724 T^{6} + 23709710 T^{8} - 579216884 T^{10} + 9760477800 T^{12} - 102309611212 T^{14} + 500246412961 T^{16} )^{2} \))
$31$ (\( ( 1 - 12 T + 128 T^{2} - 960 T^{3} + 6003 T^{4} - 31164 T^{5} + 137488 T^{6} - 590424 T^{7} + 2895416 T^{8} - 18303144 T^{9} + 132125968 T^{10} - 928406724 T^{11} + 5543896563 T^{12} - 27483984960 T^{13} + 113600471168 T^{14} - 330151369332 T^{15} + 852891037441 T^{16} )^{2} \))(\( ( 1 - 3 T + 56 T^{2} - 159 T^{3} + 429 T^{4} - 4290 T^{5} + 41239 T^{6} - 259953 T^{7} + 3221834 T^{8} - 8058543 T^{9} + 39630679 T^{10} - 127803390 T^{11} + 396190509 T^{12} - 4552035009 T^{13} + 49700206136 T^{14} - 82537842333 T^{15} + 852891037441 T^{16} )^{2} \))(\( ( 1 - 12 T + 143 T^{2} - 1140 T^{3} + 8619 T^{4} - 60864 T^{5} + 400903 T^{6} - 2550678 T^{7} + 14613095 T^{8} - 79071018 T^{9} + 385267783 T^{10} - 1813199424 T^{11} + 7959827499 T^{12} - 32637232140 T^{13} + 126913026383 T^{14} - 330151369332 T^{15} + 852891037441 T^{16} )^{2} \))(\( ( 1 + 6 T + 92 T^{2} + 480 T^{3} + 4695 T^{4} + 32340 T^{5} + 192520 T^{6} + 1356474 T^{7} + 5862920 T^{8} + 42050694 T^{9} + 185011720 T^{10} + 963440940 T^{11} + 4335931095 T^{12} + 13741992480 T^{13} + 81650338652 T^{14} + 165075684666 T^{15} + 852891037441 T^{16} )^{2} \))
$37$ (\( ( 1 + 13 T + 30 T^{2} - 433 T^{3} - 2773 T^{4} + 1236 T^{5} + 49036 T^{6} + 50032 T^{7} - 855540 T^{8} + 1851184 T^{9} + 67130284 T^{10} + 62607108 T^{11} - 5197048453 T^{12} - 30025933381 T^{13} + 76971792270 T^{14} + 1234114402729 T^{15} + 3512479453921 T^{16} )^{2} \))(\( ( 1 + 4 T - 81 T^{2} - 658 T^{3} + 2861 T^{4} + 34434 T^{5} + 29479 T^{6} - 689822 T^{7} - 3398823 T^{8} - 25523414 T^{9} + 40356751 T^{10} + 1744185402 T^{11} + 5361974621 T^{12} - 45628323706 T^{13} - 207823839129 T^{14} + 379727508532 T^{15} + 3512479453921 T^{16} )^{2} \))(\( ( 1 - 8 T - 57 T^{2} + 338 T^{3} + 3173 T^{4} - 3270 T^{5} - 191105 T^{6} + 39214 T^{7} + 7776297 T^{8} + 1450918 T^{9} - 261622745 T^{10} - 165635310 T^{11} + 5946712853 T^{12} + 23438257466 T^{13} - 146246405313 T^{14} - 759455017064 T^{15} + 3512479453921 T^{16} )^{2} \))(\( ( 1 - 8 T - 75 T^{2} + 608 T^{3} + 4901 T^{4} - 30720 T^{5} - 182438 T^{6} + 434008 T^{7} + 7645374 T^{8} + 16058296 T^{9} - 249757622 T^{10} - 1556060160 T^{11} + 9185263061 T^{12} + 42161125856 T^{13} - 192429480675 T^{14} - 759455017064 T^{15} + 3512479453921 T^{16} )^{2} \))
$41$ (\( ( 1 + 156 T^{2} + 13128 T^{4} + 762324 T^{6} + 34666958 T^{8} + 1281466644 T^{10} + 37096590408 T^{12} + 741016261596 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 + 198 T^{2} + 18957 T^{4} + 1188399 T^{6} + 55361954 T^{8} + 1997698719 T^{10} + 53567951277 T^{12} + 940520639718 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 + 222 T^{2} + 24117 T^{4} + 1673415 T^{6} + 81144722 T^{8} + 2813010615 T^{10} + 68148878037 T^{12} + 1054523141502 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 - 12 T^{2} + 3384 T^{4} - 4212 T^{6} + 5494286 T^{8} - 7080372 T^{10} + 9562375224 T^{12} - 57001250892 T^{14} + 7984925229121 T^{16} )^{2} \))
$43$ (\( ( 1 + T + 43 T^{2} )^{16} \))(\( ( 1 - 11 T + 136 T^{2} - 1220 T^{3} + 8482 T^{4} - 52460 T^{5} + 251464 T^{6} - 874577 T^{7} + 3418801 T^{8} )^{4} \))(\( ( 1 + 13 T + 112 T^{2} + 604 T^{3} + 3358 T^{4} + 25972 T^{5} + 207088 T^{6} + 1033591 T^{7} + 3418801 T^{8} )^{4} \))(\( ( 1 + 4 T + 94 T^{2} + 496 T^{3} + 4735 T^{4} + 21328 T^{5} + 173806 T^{6} + 318028 T^{7} + 3418801 T^{8} )^{4} \))
$47$ (\( 1 - 164 T^{2} + 11848 T^{4} - 531784 T^{6} + 18114050 T^{8} - 446890844 T^{10} - 12645926656 T^{12} + 2562333918356 T^{14} - 161926607596397 T^{16} + 5660195625648404 T^{18} - 61708088030676736 T^{20} - 4817132636034547676 T^{22} + \)\(43\!\cdots\!50\)\( T^{24} - \)\(27\!\cdots\!16\)\( T^{26} + \)\(13\!\cdots\!68\)\( T^{28} - \)\(42\!\cdots\!16\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 - 125 T^{2} + 6121 T^{4} - 237712 T^{6} + 8683520 T^{8} - 2987672 T^{10} - 24143549800 T^{12} + 2234488097219 T^{14} - 131390741463215 T^{16} + 4935984206756771 T^{18} - 117812821231613800 T^{20} - 32204759820424088 T^{22} + \)\(20\!\cdots\!20\)\( T^{24} - \)\(12\!\cdots\!88\)\( T^{26} + \)\(71\!\cdots\!61\)\( T^{28} - \)\(32\!\cdots\!25\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 - 173 T^{2} + 16633 T^{4} - 1089952 T^{6} + 47343872 T^{8} - 818110040 T^{10} - 64424097160 T^{12} + 7478687522819 T^{14} - 437436851957231 T^{16} + 16520420737907171 T^{18} - 314369042853805960 T^{20} - 8818584283976803160 T^{22} + \)\(11\!\cdots\!92\)\( T^{24} - \)\(57\!\cdots\!48\)\( T^{26} + \)\(19\!\cdots\!53\)\( T^{28} - \)\(44\!\cdots\!37\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 - 212 T^{2} + 22024 T^{4} - 1457032 T^{6} + 71730530 T^{8} - 3319728188 T^{10} + 189334539584 T^{12} - 11915633567836 T^{14} + 635975079098131 T^{16} - 26321634551349724 T^{18} + 923892155451792704 T^{20} - 35784064972202993852 T^{22} + \)\(17\!\cdots\!30\)\( T^{24} - \)\(76\!\cdots\!68\)\( T^{26} + \)\(25\!\cdots\!84\)\( T^{28} - \)\(54\!\cdots\!28\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 + 136 T^{2} + 9700 T^{4} + 6128 T^{6} - 41933782 T^{8} - 3532199048 T^{10} - 71962032112 T^{12} + 5883147483704 T^{14} + 625107841095283 T^{16} + 16525761281724536 T^{18} - 567815047101125872 T^{20} - 78288935279382005192 T^{22} - \)\(26\!\cdots\!02\)\( T^{24} + \)\(10\!\cdots\!72\)\( T^{26} + \)\(47\!\cdots\!00\)\( T^{28} + \)\(18\!\cdots\!84\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 + 106 T^{2} + 1615 T^{4} - 125848 T^{6} - 2732311 T^{8} - 209584850 T^{10} - 23241931957 T^{12} - 759974533696 T^{14} - 20858808219215 T^{16} - 2134768465152064 T^{18} - 183390022510001317 T^{20} - 4645314302567295650 T^{22} - \)\(17\!\cdots\!71\)\( T^{24} - \)\(22\!\cdots\!52\)\( T^{26} + \)\(79\!\cdots\!15\)\( T^{28} + \)\(14\!\cdots\!14\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 + 133 T^{2} + 8944 T^{4} + 599099 T^{6} + 29717867 T^{8} + 797091448 T^{10} + 31263307613 T^{12} + 880034743289 T^{14} - 19324329468344 T^{16} + 2472017593898801 T^{18} + 246682534717531853 T^{20} + 17667022706309524792 T^{22} + \)\(18\!\cdots\!87\)\( T^{24} + \)\(10\!\cdots\!51\)\( T^{26} + \)\(43\!\cdots\!04\)\( T^{28} + \)\(18\!\cdots\!77\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 + 304 T^{2} + 48352 T^{4} + 5330432 T^{6} + 461119874 T^{8} + 34199329984 T^{10} + 2292888431744 T^{12} + 140988863024816 T^{14} + 7874149661572771 T^{16} + 396037716236708144 T^{18} + 18091992605795828864 T^{20} + \)\(75\!\cdots\!36\)\( T^{22} + \)\(28\!\cdots\!14\)\( T^{24} + \)\(93\!\cdots\!68\)\( T^{26} + \)\(23\!\cdots\!32\)\( T^{28} + \)\(41\!\cdots\!76\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 - 328 T^{2} + 54000 T^{4} - 6551312 T^{6} + 674381378 T^{8} - 59438391192 T^{10} + 4550306378176 T^{12} - 314097266893720 T^{14} + 19606433379307875 T^{16} - 1093372586057039320 T^{18} + 55137705044961113536 T^{20} - \)\(25\!\cdots\!72\)\( T^{22} + \)\(99\!\cdots\!38\)\( T^{24} - \)\(33\!\cdots\!12\)\( T^{26} + \)\(96\!\cdots\!00\)\( T^{28} - \)\(20\!\cdots\!08\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 - 166 T^{2} + 9693 T^{4} - 138704 T^{6} - 8826568 T^{8} + 813721572 T^{10} - 53184751892 T^{12} + 953209874183 T^{14} + 65187402782238 T^{16} + 3318123572031023 T^{18} - 644458838370797012 T^{20} + 34323210142153403652 T^{22} - \)\(12\!\cdots\!28\)\( T^{24} - \)\(70\!\cdots\!04\)\( T^{26} + \)\(17\!\cdots\!33\)\( T^{28} - \)\(10\!\cdots\!26\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 - 190 T^{2} + 11541 T^{4} - 443000 T^{6} + 56915720 T^{8} - 4100356020 T^{10} + 95822008324 T^{12} - 9747744998665 T^{14} + 1100644208561934 T^{16} - 33931900340352865 T^{18} + 1161109866606912964 T^{20} - \)\(17\!\cdots\!20\)\( T^{22} + \)\(83\!\cdots\!20\)\( T^{24} - \)\(22\!\cdots\!00\)\( T^{26} + \)\(20\!\cdots\!21\)\( T^{28} - \)\(11\!\cdots\!90\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 + 44 T^{2} - 2292 T^{4} + 465352 T^{6} + 27444554 T^{8} - 775368300 T^{10} + 157834889296 T^{12} + 8699229607316 T^{14} - 256293216247293 T^{16} + 30282018263066996 T^{18} + 1912542331994667856 T^{20} - 32705448662314980300 T^{22} + \)\(40\!\cdots\!34\)\( T^{24} + \)\(23\!\cdots\!52\)\( T^{26} - \)\(40\!\cdots\!52\)\( T^{28} + \)\(27\!\cdots\!84\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( ( 1 - 30 T + 526 T^{2} - 6780 T^{3} + 71293 T^{4} - 689040 T^{5} + 6452458 T^{6} - 57925758 T^{7} + 478605124 T^{8} - 3533471238 T^{9} + 24009596218 T^{10} - 156398988240 T^{11} + 987111542413 T^{12} - 5726362920780 T^{13} + 27099716913886 T^{14} - 94282285080630 T^{15} + 191707312997281 T^{16} )^{2} \))(\( ( 1 + 3 T + 148 T^{2} + 435 T^{3} + 10381 T^{4} + 51408 T^{5} + 640546 T^{6} + 4589094 T^{7} + 40871512 T^{8} + 279934734 T^{9} + 2383471666 T^{10} + 11668639248 T^{11} + 143733675421 T^{12} + 367399390935 T^{13} + 7625015405428 T^{14} + 9428228508063 T^{15} + 191707312997281 T^{16} )^{2} \))(\( ( 1 - 3 T + 118 T^{2} - 345 T^{3} + 7303 T^{4} - 40824 T^{5} + 56536 T^{6} - 3083880 T^{7} - 8201036 T^{8} - 188116680 T^{9} + 210370456 T^{10} - 9266272344 T^{11} + 101116176823 T^{12} - 291385723845 T^{13} + 6079404174598 T^{14} - 9428228508063 T^{15} + 191707312997281 T^{16} )^{2} \))(\( ( 1 + 30 T + 568 T^{2} + 8040 T^{3} + 93811 T^{4} + 983664 T^{5} + 9476836 T^{6} + 84606546 T^{7} + 693799792 T^{8} + 5160999306 T^{9} + 35263306756 T^{10} + 223273038384 T^{11} + 1298892190051 T^{12} + 6790554260040 T^{13} + 29263572637048 T^{14} + 94282285080630 T^{15} + 191707312997281 T^{16} )^{2} \))
$67$ (\( ( 1 - T - 218 T^{2} - 35 T^{3} + 27473 T^{4} + 14480 T^{5} - 2498242 T^{6} - 460454 T^{7} + 180583204 T^{8} - 30850418 T^{9} - 11214608338 T^{10} + 4355048240 T^{11} + 553611747233 T^{12} - 47254378745 T^{13} - 19719927312842 T^{14} - 6060711605323 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 - 10 T - 47 T^{2} + 40 T^{3} + 3785 T^{4} + 43910 T^{5} - 126337 T^{6} - 2491340 T^{7} + 10362079 T^{8} - 166919780 T^{9} - 567126793 T^{10} + 13206503330 T^{11} + 76271992985 T^{12} + 54005004280 T^{13} - 4251543961943 T^{14} - 60607116053230 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 + 2 T - 179 T^{2} - 908 T^{3} + 16841 T^{4} + 97754 T^{5} - 850057 T^{6} - 3582776 T^{7} + 41096215 T^{8} - 240045992 T^{9} - 3815905873 T^{10} + 29400786302 T^{11} + 339365028761 T^{12} - 1225913597156 T^{13} - 16192050408251 T^{14} + 12121423210646 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 + 2 T - 65 T^{2} + 838 T^{3} + 2213 T^{4} - 60592 T^{5} + 564770 T^{6} + 3320308 T^{7} - 36775142 T^{8} + 222460636 T^{9} + 2535252530 T^{10} - 18223831696 T^{11} + 44594430773 T^{12} + 1131404839666 T^{13} - 5879794840985 T^{14} + 12121423210646 T^{15} + 406067677556641 T^{16} )^{2} \))
$71$ (\( ( 1 - 44 T^{2} + 15496 T^{4} - 542276 T^{6} + 106966222 T^{8} - 2733613316 T^{10} + 393779408776 T^{12} - 5636412492524 T^{14} + 645753531245761 T^{16} )^{2} \))(\( ( 1 - 539 T^{2} + 129076 T^{4} - 17929268 T^{6} + 1579617304 T^{8} - 90381439988 T^{10} + 3280038136756 T^{12} - 69046053033419 T^{14} + 645753531245761 T^{16} )^{2} \))(\( ( 1 - 71 T^{2} + 9904 T^{4} - 812624 T^{6} + 56885380 T^{8} - 4096437584 T^{10} + 251677288624 T^{12} - 9095120158391 T^{14} + 645753531245761 T^{16} )^{2} \))(\( ( 1 - 428 T^{2} + 86452 T^{4} - 10845908 T^{6} + 924059350 T^{8} - 54674222228 T^{10} + 2196890645812 T^{12} - 54826921518188 T^{14} + 645753531245761 T^{16} )^{2} \))
$73$ (\( ( 1 - 15 T + 219 T^{2} - 2160 T^{3} + 16119 T^{4} - 15975 T^{5} - 773154 T^{6} + 16053945 T^{7} - 163050496 T^{8} + 1171937985 T^{9} - 4120137666 T^{10} - 6214546575 T^{11} + 457751246679 T^{12} - 4477834640880 T^{13} + 33142195557291 T^{14} - 165710977786455 T^{15} + 806460091894081 T^{16} )^{2} \))(\( ( 1 + 273 T^{2} + 45342 T^{4} + 12600 T^{5} + 5059740 T^{6} + 1838781 T^{7} + 426414266 T^{8} + 134231013 T^{9} + 26963354460 T^{10} + 4901614200 T^{11} + 1287633043422 T^{12} + 41314243776897 T^{14} + 806460091894081 T^{16} )^{2} \))(\( ( 1 + 6 T + 159 T^{2} + 882 T^{3} + 12318 T^{4} + 98346 T^{5} + 584490 T^{6} + 8845941 T^{7} + 32868122 T^{8} + 645753693 T^{9} + 3114747210 T^{10} + 38258265882 T^{11} + 349809532638 T^{12} + 1828449145026 T^{13} + 24062141979951 T^{14} + 66284391114582 T^{15} + 806460091894081 T^{16} )^{2} \))(\( ( 1 - 6 T + 255 T^{2} - 1458 T^{3} + 37305 T^{4} - 256020 T^{5} + 3961950 T^{6} - 27178752 T^{7} + 315641642 T^{8} - 1984048896 T^{9} + 21113231550 T^{10} - 99596132340 T^{11} + 1059396380505 T^{12} - 3022538382594 T^{13} + 38590227703695 T^{14} - 66284391114582 T^{15} + 806460091894081 T^{16} )^{2} \))
$79$ (\( ( 1 - 5 T - 44 T^{2} - 733 T^{3} + 6395 T^{4} + 22072 T^{5} + 1021340 T^{6} - 6862144 T^{7} - 46543880 T^{8} - 542109376 T^{9} + 6374182940 T^{10} + 10882356808 T^{11} + 249085767995 T^{12} - 2255482340467 T^{13} - 10695848042924 T^{14} - 96019544930795 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 + 7 T - 158 T^{2} - 889 T^{3} + 11435 T^{4} + 9100 T^{5} - 1504939 T^{6} + 1418207 T^{7} + 173668174 T^{8} + 112038353 T^{9} - 9392324299 T^{10} + 4486654900 T^{11} + 445394176235 T^{12} - 2735503138711 T^{13} - 38407817972318 T^{14} + 134427362903113 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 - 17 T - 2 T^{2} + 611 T^{3} + 7367 T^{4} - 5816 T^{5} - 1127755 T^{6} + 4949663 T^{7} + 11692942 T^{8} + 391023377 T^{9} - 7038318955 T^{10} - 2867514824 T^{11} + 286945246727 T^{12} + 1880081459789 T^{13} - 486174911042 T^{14} - 326466452764703 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 + 16 T + T^{2} - 736 T^{3} - 271 T^{4} - 52304 T^{5} - 888178 T^{6} + 790928 T^{7} + 66112030 T^{8} + 62483312 T^{9} - 5543118898 T^{10} - 25787911856 T^{11} - 10555471951 T^{12} - 2264713509664 T^{13} + 243087455521 T^{14} + 307262543778544 T^{15} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( ( 1 + 420 T^{2} + 87492 T^{4} + 11760060 T^{6} + 1135300886 T^{8} + 81015053340 T^{10} + 4152223420932 T^{12} + 137314956814980 T^{14} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 + 366 T^{2} + 69987 T^{4} + 9026481 T^{6} + 861047891 T^{8} + 62183427609 T^{10} + 3321465511827 T^{12} + 119660176653054 T^{14} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 + 429 T^{2} + 94512 T^{4} + 13346472 T^{6} + 1314527120 T^{8} + 91943845608 T^{10} + 4485380834352 T^{12} + 140257420175301 T^{14} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 + 384 T^{2} + 77916 T^{4} + 10524288 T^{6} + 1018130534 T^{8} + 72501820032 T^{10} + 3697762539036 T^{12} + 125545103373696 T^{14} + 2252292232139041 T^{16} )^{2} \))
$89$ (\( 1 - 336 T^{2} + 57888 T^{4} - 4687200 T^{6} + 3621442 T^{8} + 39434248560 T^{10} - 2636071502976 T^{12} - 175948648527792 T^{14} + 38039673771388995 T^{16} - 1393689244988640432 T^{18} - \)\(16\!\cdots\!16\)\( T^{20} + \)\(19\!\cdots\!60\)\( T^{22} + \)\(14\!\cdots\!02\)\( T^{24} - \)\(14\!\cdots\!00\)\( T^{26} + \)\(14\!\cdots\!48\)\( T^{28} - \)\(65\!\cdots\!76\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( 1 + 117 T^{2} - 8940 T^{4} - 1399587 T^{6} + 32843593 T^{8} + 5083158288 T^{10} - 427680145674 T^{12} - 10515396835650 T^{14} + 3896148919948920 T^{16} - 83292458335183650 T^{18} - 26833610770793215434 T^{20} + \)\(25\!\cdots\!68\)\( T^{22} + \)\(12\!\cdots\!33\)\( T^{24} - \)\(43\!\cdots\!87\)\( T^{26} - \)\(22\!\cdots\!40\)\( T^{28} + \)\(22\!\cdots\!97\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( 1 - 459 T^{2} + 117876 T^{4} - 19489923 T^{6} + 2201471017 T^{8} - 146522371824 T^{10} + 290930466582 T^{12} + 1282936732245342 T^{14} - 164379910475968584 T^{16} + 10162141856115353982 T^{18} + 18253629448530290262 T^{20} - \)\(72\!\cdots\!64\)\( T^{22} + \)\(86\!\cdots\!77\)\( T^{24} - \)\(60\!\cdots\!23\)\( T^{26} + \)\(29\!\cdots\!96\)\( T^{28} - \)\(89\!\cdots\!19\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( 1 - 372 T^{2} + 60216 T^{4} - 6828360 T^{6} + 821869570 T^{8} - 102264008700 T^{10} + 10913950851840 T^{12} - 1024801936202364 T^{14} + 92017663830511443 T^{16} - 8117456136658925244 T^{18} + \)\(68\!\cdots\!40\)\( T^{20} - \)\(50\!\cdots\!00\)\( T^{22} + \)\(32\!\cdots\!70\)\( T^{24} - \)\(21\!\cdots\!60\)\( T^{26} + \)\(14\!\cdots\!36\)\( T^{28} - \)\(72\!\cdots\!52\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( ( 1 - 636 T^{2} + 187434 T^{4} - 33404592 T^{6} + 3937386611 T^{8} - 314303806128 T^{10} + 16593397254954 T^{12} - 529770195134844 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 - 225 T^{2} + 22662 T^{4} - 2838687 T^{6} + 365425586 T^{8} - 26709205983 T^{10} + 2006250566022 T^{12} - 187418701109025 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 - 252 T^{2} + 22644 T^{4} + 38076 T^{6} - 140127082 T^{8} + 358257084 T^{10} + 2004657038964 T^{12} - 209908945242108 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 - 372 T^{2} + 87042 T^{4} - 13239552 T^{6} + 1512714731 T^{8} - 124570944768 T^{10} + 7705765676802 T^{12} - 309865585833588 T^{14} + 7837433594376961 T^{16} )^{2} \))
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