Properties

Label 1343.2
Level 1343
Weight 2
Dimension 73179
Nonzero newspaces 20
Sturm bound 299520
Trace bound 2

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Defining parameters

Level: \( N \) = \( 1343 = 17 \cdot 79 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 20 \)
Sturm bound: \(299520\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(1343))\).

Total New Old
Modular forms 76128 75491 637
Cusp forms 73633 73179 454
Eisenstein series 2495 2312 183

Trace form

\( 73179q - 539q^{2} - 542q^{3} - 551q^{4} - 548q^{5} - 566q^{6} - 554q^{7} - 575q^{8} - 569q^{9} + O(q^{10}) \) \( 73179q - 539q^{2} - 542q^{3} - 551q^{4} - 548q^{5} - 566q^{6} - 554q^{7} - 575q^{8} - 569q^{9} - 576q^{10} - 550q^{11} - 566q^{12} - 556q^{13} - 570q^{14} - 554q^{15} - 551q^{16} - 588q^{17} - 1191q^{18} - 574q^{19} - 600q^{20} - 578q^{21} - 606q^{22} - 586q^{23} - 630q^{24} - 567q^{25} - 584q^{26} - 602q^{27} - 602q^{28} - 580q^{29} - 618q^{30} - 562q^{31} - 607q^{32} - 594q^{33} - 546q^{34} - 1218q^{35} - 643q^{36} - 580q^{37} - 614q^{38} - 602q^{39} - 608q^{40} - 568q^{41} - 626q^{42} - 582q^{43} - 590q^{44} - 660q^{45} - 650q^{46} - 594q^{47} - 694q^{48} - 637q^{49} - 681q^{50} - 645q^{51} - 1304q^{52} - 604q^{53} - 650q^{54} - 618q^{55} - 650q^{56} - 578q^{57} - 624q^{58} - 582q^{59} - 698q^{60} - 588q^{61} - 610q^{62} - 498q^{63} - 375q^{64} - 474q^{65} - 394q^{66} - 472q^{67} - 458q^{68} - 890q^{69} - 66q^{70} - 494q^{71} + 57q^{72} - 396q^{73} - 392q^{74} - 334q^{75} + 18q^{76} - 424q^{77} - 308q^{78} - 165q^{79} - 502q^{80} - 217q^{81} - 380q^{82} - 420q^{83} - 106q^{84} - 439q^{85} - 998q^{86} - 542q^{87} + 30q^{88} - 500q^{89} - 152q^{90} - 330q^{91} - 338q^{92} - 540q^{93} - 330q^{94} - 510q^{95} - 446q^{96} - 592q^{97} - 699q^{98} - 694q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(1343))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
1343.2.a \(\chi_{1343}(1, \cdot)\) 1343.2.a.a 1 1
1343.2.a.b 15
1343.2.a.c 20
1343.2.a.d 32
1343.2.a.e 37
1343.2.b \(\chi_{1343}(475, \cdot)\) n/a 118 1
1343.2.e \(\chi_{1343}(766, \cdot)\) n/a 212 2
1343.2.f \(\chi_{1343}(633, \cdot)\) n/a 236 2
1343.2.j \(\chi_{1343}(339, \cdot)\) n/a 236 2
1343.2.k \(\chi_{1343}(712, \cdot)\) n/a 464 4
1343.2.m \(\chi_{1343}(55, \cdot)\) n/a 472 4
1343.2.o \(\chi_{1343}(18, \cdot)\) n/a 1296 12
1343.2.q \(\chi_{1343}(78, \cdot)\) n/a 944 8
1343.2.s \(\chi_{1343}(134, \cdot)\) n/a 944 8
1343.2.v \(\chi_{1343}(67, \cdot)\) n/a 1416 12
1343.2.w \(\chi_{1343}(171, \cdot)\) n/a 2544 24
1343.2.y \(\chi_{1343}(24, \cdot)\) n/a 1888 16
1343.2.ba \(\chi_{1343}(21, \cdot)\) n/a 2832 24
1343.2.bb \(\chi_{1343}(16, \cdot)\) n/a 2832 24
1343.2.bf \(\chi_{1343}(8, \cdot)\) n/a 5664 48
1343.2.bh \(\chi_{1343}(4, \cdot)\) n/a 5664 48
1343.2.bi \(\chi_{1343}(12, \cdot)\) n/a 11328 96
1343.2.bk \(\chi_{1343}(2, \cdot)\) n/a 11328 96
1343.2.bm \(\chi_{1343}(3, \cdot)\) n/a 22656 192

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(1343))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(1343)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(17))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(79))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 + 2 T + 11 T^{2} + 24 T^{3} + 73 T^{4} + 152 T^{5} + 360 T^{6} + 692 T^{7} + 1396 T^{8} + 2498 T^{9} + 4466 T^{10} + 7427 T^{11} + 12091 T^{12} + 18699 T^{13} + 28008 T^{14} + 40371 T^{15} + 56016 T^{16} + 74796 T^{17} + 96728 T^{18} + 118832 T^{19} + 142912 T^{20} + 159872 T^{21} + 178688 T^{22} + 177152 T^{23} + 184320 T^{24} + 155648 T^{25} + 149504 T^{26} + 98304 T^{27} + 90112 T^{28} + 32768 T^{29} + 32768 T^{30} \))(\( 1 + 2 T + 15 T^{2} + 28 T^{3} + 117 T^{4} + 203 T^{5} + 630 T^{6} + 1017 T^{7} + 2628 T^{8} + 3969 T^{9} + 9074 T^{10} + 12923 T^{11} + 27123 T^{12} + 36719 T^{13} + 72418 T^{14} + 93845 T^{15} + 176289 T^{16} + 219673 T^{17} + 395617 T^{18} + 474772 T^{19} + 821949 T^{20} + 949544 T^{21} + 1582468 T^{22} + 1757384 T^{23} + 2820624 T^{24} + 3003040 T^{25} + 4634752 T^{26} + 4700032 T^{27} + 6943488 T^{28} + 6616576 T^{29} + 9291776 T^{30} + 8128512 T^{31} + 10764288 T^{32} + 8331264 T^{33} + 10321920 T^{34} + 6651904 T^{35} + 7667712 T^{36} + 3670016 T^{37} + 3932160 T^{38} + 1048576 T^{39} + 1048576 T^{40} \))
$3$ (\( 1 - T + 3 T^{2} \))(\( 1 + 5 T + 35 T^{2} + 129 T^{3} + 547 T^{4} + 1645 T^{5} + 5361 T^{6} + 13857 T^{7} + 37852 T^{8} + 86608 T^{9} + 206802 T^{10} + 425985 T^{11} + 909447 T^{12} + 1701650 T^{13} + 3288118 T^{14} + 5607409 T^{15} + 9864354 T^{16} + 15314850 T^{17} + 24555069 T^{18} + 34504785 T^{19} + 50252886 T^{20} + 63137232 T^{21} + 82782324 T^{22} + 90915777 T^{23} + 105520563 T^{24} + 97135605 T^{25} + 96899409 T^{26} + 68555889 T^{27} + 55801305 T^{28} + 23914845 T^{29} + 14348907 T^{30} \))(\( 1 + 4 T + 33 T^{2} + 113 T^{3} + 549 T^{4} + 1655 T^{5} + 6127 T^{6} + 16617 T^{7} + 51525 T^{8} + 127689 T^{9} + 347489 T^{10} + 795506 T^{11} + 1951595 T^{12} + 4158459 T^{13} + 9350540 T^{14} + 18637122 T^{15} + 38820683 T^{16} + 72588146 T^{17} + 141027963 T^{18} + 247664375 T^{19} + 450699166 T^{20} + 742993125 T^{21} + 1269251667 T^{22} + 1959879942 T^{23} + 3144475323 T^{24} + 4528820646 T^{25} + 6816543660 T^{26} + 9094549833 T^{27} + 12804414795 T^{28} + 15657944598 T^{29} + 20518877961 T^{30} + 22619723283 T^{31} + 27382497525 T^{32} + 26492865291 T^{33} + 29305251063 T^{34} + 23747441085 T^{35} + 23632649829 T^{36} + 14592838419 T^{37} + 12784876137 T^{38} + 4649045868 T^{39} + 3486784401 T^{40} \))
$5$ (\( 1 - T + 5 T^{2} \))(\( 1 + 4 T + 55 T^{2} + 186 T^{3} + 1440 T^{4} + 4260 T^{5} + 24203 T^{6} + 63953 T^{7} + 294690 T^{8} + 703916 T^{9} + 2765884 T^{10} + 6008397 T^{11} + 20718707 T^{12} + 40992967 T^{13} + 126257090 T^{14} + 226985071 T^{15} + 631285450 T^{16} + 1024824175 T^{17} + 2589838375 T^{18} + 3755248125 T^{19} + 8643387500 T^{20} + 10998687500 T^{21} + 23022656250 T^{22} + 24981640625 T^{23} + 47271484375 T^{24} + 41601562500 T^{25} + 70312500000 T^{26} + 45410156250 T^{27} + 67138671875 T^{28} + 24414062500 T^{29} + 30517578125 T^{30} \))(\( 1 + 7 T + 85 T^{2} + 468 T^{3} + 3330 T^{4} + 15392 T^{5} + 82361 T^{6} + 331390 T^{7} + 1462511 T^{8} + 5238319 T^{9} + 19967234 T^{10} + 64597865 T^{11} + 218403648 T^{12} + 644452757 T^{13} + 1964895213 T^{14} + 5322073849 T^{15} + 14790071877 T^{16} + 36916520405 T^{17} + 94150012075 T^{18} + 216964861451 T^{19} + 509887366191 T^{20} + 1084824307255 T^{21} + 2353750301875 T^{22} + 4614565050625 T^{23} + 9243794923125 T^{24} + 16631480778125 T^{25} + 30701487703125 T^{26} + 50347871640625 T^{27} + 85313925000000 T^{28} + 126167705078125 T^{29} + 194992519531250 T^{30} + 255777294921875 T^{31} + 357058349609375 T^{32} + 404528808593750 T^{33} + 502691650390625 T^{34} + 469726562500000 T^{35} + 508117675781250 T^{36} + 357055664062500 T^{37} + 324249267578125 T^{38} + 133514404296875 T^{39} + 95367431640625 T^{40} \))
$7$ (\( 1 - T + 7 T^{2} \))(\( 1 + 14 T + 161 T^{2} + 1332 T^{3} + 9554 T^{4} + 58351 T^{5} + 319877 T^{6} + 1572682 T^{7} + 7083308 T^{8} + 29277544 T^{9} + 112219389 T^{10} + 399469391 T^{11} + 1328175572 T^{12} + 4127326236 T^{13} + 12025166682 T^{14} + 32837995971 T^{15} + 84176166774 T^{16} + 202238985564 T^{17} + 455564221196 T^{18} + 959126007791 T^{19} + 1886071270923 T^{20} + 3444473774056 T^{21} + 5833408720244 T^{22} + 9066198766282 T^{23} + 12908190746339 T^{24} + 16482713254399 T^{25} + 18891379702622 T^{26} + 18436594551732 T^{27} + 15599130675527 T^{28} + 9495123019886 T^{29} + 4747561509943 T^{30} \))(\( 1 + 11 T + 134 T^{2} + 985 T^{3} + 7339 T^{4} + 42303 T^{5} + 242924 T^{6} + 1180148 T^{7} + 5686702 T^{8} + 24210927 T^{9} + 102138851 T^{10} + 390137173 T^{11} + 1476820653 T^{12} + 5137762132 T^{13} + 17722299284 T^{14} + 56715766917 T^{15} + 180063186750 T^{16} + 533490044589 T^{17} + 1568753267395 T^{18} + 4318773801675 T^{19} + 11803133484430 T^{20} + 30231416611725 T^{21} + 76868910102355 T^{22} + 182987085294027 T^{23} + 432331711386750 T^{24} + 953221894574019 T^{25} + 2085010788463316 T^{26} + 4231168039473676 T^{27} + 8513577177235053 T^{28} + 15743442155333011 T^{29} + 28851697368798899 T^{30} + 47872913429920761 T^{31} + 78711275608501102 T^{32} + 114343371853800236 T^{33} + 164756661748770476 T^{34} + 200836094555118729 T^{35} + 243896477450301739 T^{36} + 229141056277398895 T^{37} + 218207422120000166 T^{38} + 125387847039104573 T^{39} + 79792266297612001 T^{40} \))
$11$ (\( 1 + 2 T + 11 T^{2} \))(\( 1 + T + 105 T^{2} + 65 T^{3} + 5498 T^{4} + 1935 T^{5} + 190848 T^{6} + 31708 T^{7} + 4903952 T^{8} + 207184 T^{9} + 98596899 T^{10} - 3235204 T^{11} + 1599361061 T^{12} - 110551113 T^{13} + 21284402821 T^{14} - 1594312863 T^{15} + 234128431031 T^{16} - 13376684673 T^{17} + 2128749572191 T^{18} - 47366621764 T^{19} + 15879129180849 T^{20} + 367039094224 T^{21} + 95564151199792 T^{22} + 6796891398748 T^{23} + 450009600931968 T^{24} + 50188916602935 T^{25} + 1568643565019278 T^{26} + 203997844486865 T^{27} + 3624884775112755 T^{28} + 379749833583241 T^{29} + 4177248169415651 T^{30} \))(\( 1 + 7 T + 155 T^{2} + 985 T^{3} + 11774 T^{4} + 67785 T^{5} + 580366 T^{6} + 3034176 T^{7} + 20796848 T^{8} + 99189758 T^{9} + 576683922 T^{10} + 2523258379 T^{11} + 12884528182 T^{12} + 52020154304 T^{13} + 238754471919 T^{14} + 894367796446 T^{15} + 3750118806985 T^{16} + 13092711379645 T^{17} + 50735768131969 T^{18} + 165532483367234 T^{19} + 597253821596124 T^{20} + 1820857317039574 T^{21} + 6139027943968249 T^{22} + 17426398846307495 T^{23} + 54905489453067385 T^{24} + 144038827985424746 T^{25} + 422968111027295559 T^{26} + 1013725642368433984 T^{27} + 2761913043306484342 T^{28} + 5949711268559452889 T^{29} + 14957695745483965122 T^{30} + 28299995562480802138 T^{31} + 65269417909553375408 T^{32} + \)\(10\!\cdots\!56\)\( T^{33} + \)\(22\!\cdots\!06\)\( T^{34} + \)\(28\!\cdots\!35\)\( T^{35} + \)\(54\!\cdots\!14\)\( T^{36} + \)\(49\!\cdots\!35\)\( T^{37} + \)\(86\!\cdots\!55\)\( T^{38} + \)\(42\!\cdots\!37\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \))
$13$ (\( 1 + 5 T + 13 T^{2} \))(\( 1 + 12 T + 178 T^{2} + 1500 T^{3} + 13264 T^{4} + 87993 T^{5} + 587598 T^{6} + 3254077 T^{7} + 17937276 T^{8} + 86048020 T^{9} + 410970269 T^{10} + 1755407187 T^{11} + 7504398131 T^{12} + 29169946154 T^{13} + 114161263890 T^{14} + 409674176869 T^{15} + 1484096430570 T^{16} + 4929720900026 T^{17} + 16487162693807 T^{18} + 50136184667907 T^{19} + 152590384087817 T^{20} + 415337357368180 T^{21} + 1125537468019692 T^{22} + 2654450577399517 T^{23} + 6231182622576054 T^{24} + 12130582273269057 T^{25} + 23771215466506768 T^{26} + 34947127683721500 T^{27} + 53911768973421034 T^{28} + 47248516628391468 T^{29} + 51185893014090757 T^{30} \))(\( 1 + 19 T + 344 T^{2} + 4151 T^{3} + 46404 T^{4} + 426792 T^{5} + 3662233 T^{6} + 27742880 T^{7} + 197785690 T^{8} + 1286343173 T^{9} + 7928354664 T^{10} + 45387184734 T^{11} + 247516357278 T^{12} + 1267290750938 T^{13} + 6204355877921 T^{14} + 28709931547664 T^{15} + 127361927405287 T^{16} + 536225793820673 T^{17} + 2167875239494297 T^{18} + 8336708655861664 T^{19} + 30808786041802401 T^{20} + 108377212526201632 T^{21} + 366370915474536193 T^{22} + 1178088069024018581 T^{23} + 3637584008622402007 T^{24} + 10659796614126809552 T^{25} + 29947240790751984089 T^{26} + 79520615229175858946 T^{27} + \)\(20\!\cdots\!38\)\( T^{28} + \)\(48\!\cdots\!82\)\( T^{29} + \)\(10\!\cdots\!36\)\( T^{30} + \)\(23\!\cdots\!01\)\( T^{31} + \)\(46\!\cdots\!90\)\( T^{32} + \)\(84\!\cdots\!40\)\( T^{33} + \)\(14\!\cdots\!37\)\( T^{34} + \)\(21\!\cdots\!44\)\( T^{35} + \)\(30\!\cdots\!64\)\( T^{36} + \)\(35\!\cdots\!83\)\( T^{37} + \)\(38\!\cdots\!76\)\( T^{38} + \)\(27\!\cdots\!63\)\( T^{39} + \)\(19\!\cdots\!01\)\( T^{40} \))
$17$ (\( 1 - T \))(\( ( 1 - T )^{15} \))(\( ( 1 + T )^{20} \))
$19$ (\( 1 - 8 T + 19 T^{2} \))(\( 1 + 13 T + 245 T^{2} + 2395 T^{3} + 26914 T^{4} + 215423 T^{5} + 1839040 T^{6} + 12604329 T^{7} + 89238320 T^{8} + 537235333 T^{9} + 3289855311 T^{10} + 17655633407 T^{11} + 95638781351 T^{12} + 461268814408 T^{13} + 2237464937702 T^{14} + 9730130200883 T^{15} + 42511833816338 T^{16} + 166518042001288 T^{17} + 655986401286509 T^{18} + 2300899801233647 T^{19} + 8146007445711789 T^{20} + 25274709545313373 T^{21} + 79767612283838480 T^{22} + 214066416161004489 T^{23} + 593435583723492160 T^{24} + 1320772686454264823 T^{25} + 3135218827986666166 T^{26} + 5300889231163455595 T^{27} + 10302980948252979455 T^{28} + 10387086915177493573 T^{29} + 15181127029874798299 T^{30} \))(\( 1 + 9 T + 273 T^{2} + 2155 T^{3} + 35806 T^{4} + 253211 T^{5} + 3024420 T^{6} + 19468405 T^{7} + 185798896 T^{8} + 1101787129 T^{9} + 8877582418 T^{10} + 48919511102 T^{11} + 344059600186 T^{12} + 1771761319057 T^{13} + 11120156313488 T^{14} + 53663181597520 T^{15} + 305297418579062 T^{16} + 1380823773009847 T^{17} + 7203394305123965 T^{18} + 30466057799879314 T^{19} + 147010155336358062 T^{20} + 578855098197706966 T^{21} + 2600425344149751365 T^{22} + 9471070259074540573 T^{23} + 39786664886641938902 T^{24} + \)\(13\!\cdots\!80\)\( T^{25} + \)\(52\!\cdots\!28\)\( T^{26} + \)\(15\!\cdots\!23\)\( T^{27} + \)\(58\!\cdots\!26\)\( T^{28} + \)\(15\!\cdots\!58\)\( T^{29} + \)\(54\!\cdots\!18\)\( T^{30} + \)\(12\!\cdots\!51\)\( T^{31} + \)\(41\!\cdots\!56\)\( T^{32} + \)\(81\!\cdots\!95\)\( T^{33} + \)\(24\!\cdots\!20\)\( T^{34} + \)\(38\!\cdots\!89\)\( T^{35} + \)\(10\!\cdots\!86\)\( T^{36} + \)\(11\!\cdots\!45\)\( T^{37} + \)\(28\!\cdots\!93\)\( T^{38} + \)\(17\!\cdots\!11\)\( T^{39} + \)\(37\!\cdots\!01\)\( T^{40} \))
$23$ (\( 1 + 6 T + 23 T^{2} \))(\( 1 + 3 T + 212 T^{2} + 462 T^{3} + 21522 T^{4} + 31413 T^{5} + 1412664 T^{6} + 1177555 T^{7} + 68173668 T^{8} + 21179658 T^{9} + 2598507888 T^{10} - 224574122 T^{11} + 81761908337 T^{12} - 26730297370 T^{13} + 2183748825843 T^{14} - 847652696641 T^{15} + 50226222994389 T^{16} - 14140327308730 T^{17} + 994797138736279 T^{18} - 62845046874602 T^{19} + 16724888055373584 T^{20} + 3135349500745962 T^{21} + 232119439621729596 T^{22} + 92215492272567955 T^{23} + 2544423523352967432 T^{24} + 1301330996754356037 T^{25} + 20506371609823536894 T^{26} + 10124556487593388302 T^{27} + \)\(10\!\cdots\!96\)\( T^{28} + 34778508973616249427 T^{29} + \)\(26\!\cdots\!07\)\( T^{30} \))(\( 1 + 15 T + 382 T^{2} + 4400 T^{3} + 65680 T^{4} + 625343 T^{5} + 7009800 T^{6} + 57461341 T^{7} + 530559168 T^{8} + 3841889360 T^{9} + 30641703873 T^{10} + 199544946139 T^{11} + 1415302637067 T^{12} + 8402069264544 T^{13} + 54063751355993 T^{14} + 295785832799534 T^{15} + 1752008289643105 T^{16} + 8910668742471937 T^{17} + 49093789538255991 T^{18} + 233526659246157046 T^{19} + 1204255481030904814 T^{20} + 5371113162661612058 T^{21} + 25970614665737419239 T^{22} + \)\(10\!\cdots\!79\)\( T^{23} + \)\(49\!\cdots\!05\)\( T^{24} + \)\(19\!\cdots\!62\)\( T^{25} + \)\(80\!\cdots\!77\)\( T^{26} + \)\(28\!\cdots\!68\)\( T^{27} + \)\(11\!\cdots\!27\)\( T^{28} + \)\(35\!\cdots\!57\)\( T^{29} + \)\(12\!\cdots\!77\)\( T^{30} + \)\(36\!\cdots\!20\)\( T^{31} + \)\(11\!\cdots\!28\)\( T^{32} + \)\(28\!\cdots\!03\)\( T^{33} + \)\(81\!\cdots\!00\)\( T^{34} + \)\(16\!\cdots\!01\)\( T^{35} + \)\(40\!\cdots\!80\)\( T^{36} + \)\(62\!\cdots\!00\)\( T^{37} + \)\(12\!\cdots\!58\)\( T^{38} + \)\(11\!\cdots\!05\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \))
$29$ (\( 1 + 2 T + 29 T^{2} \))(\( 1 + 5 T + 296 T^{2} + 1253 T^{3} + 41697 T^{4} + 149782 T^{5} + 3753794 T^{6} + 11481236 T^{7} + 244725239 T^{8} + 641898457 T^{9} + 12400172925 T^{10} + 28270002583 T^{11} + 510348586626 T^{12} + 1032081593047 T^{13} + 17520597520862 T^{14} + 32206464175955 T^{15} + 508097328104998 T^{16} + 867980619752527 T^{17} + 12446891679221514 T^{18} + 19994835696906823 T^{19} + 254341794490440825 T^{20} + 381816171937515697 T^{21} + 4221480102440462851 T^{22} + 5743447125358699796 T^{23} + 54456837521341196986 T^{24} + 63014370818170706182 T^{25} + \)\(50\!\cdots\!13\)\( T^{26} + \)\(44\!\cdots\!73\)\( T^{27} + \)\(30\!\cdots\!44\)\( T^{28} + \)\(14\!\cdots\!05\)\( T^{29} + \)\(86\!\cdots\!49\)\( T^{30} \))(\( 1 + 11 T + 337 T^{2} + 3304 T^{3} + 56063 T^{4} + 499449 T^{5} + 6188771 T^{6} + 50738790 T^{7} + 512029928 T^{8} + 3894234212 T^{9} + 33880286175 T^{10} + 240193481814 T^{11} + 1863203604676 T^{12} + 12348780430631 T^{13} + 87222519332195 T^{14} + 541411957537868 T^{15} + 3528840663427007 T^{16} + 20536792467770813 T^{17} + 124545197938173418 T^{18} + 679770750705093252 T^{19} + 3854103929129307626 T^{20} + 19713351770447704308 T^{21} + \)\(10\!\cdots\!38\)\( T^{22} + \)\(50\!\cdots\!57\)\( T^{23} + \)\(24\!\cdots\!67\)\( T^{24} + \)\(11\!\cdots\!32\)\( T^{25} + \)\(51\!\cdots\!95\)\( T^{26} + \)\(21\!\cdots\!79\)\( T^{27} + \)\(93\!\cdots\!36\)\( T^{28} + \)\(34\!\cdots\!66\)\( T^{29} + \)\(14\!\cdots\!75\)\( T^{30} + \)\(47\!\cdots\!48\)\( T^{31} + \)\(18\!\cdots\!48\)\( T^{32} + \)\(52\!\cdots\!10\)\( T^{33} + \)\(18\!\cdots\!51\)\( T^{34} + \)\(43\!\cdots\!01\)\( T^{35} + \)\(14\!\cdots\!23\)\( T^{36} + \)\(23\!\cdots\!36\)\( T^{37} + \)\(70\!\cdots\!57\)\( T^{38} + \)\(67\!\cdots\!59\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \))
$31$ (\( 1 + 2 T + 31 T^{2} \))(\( 1 - 3 T + 278 T^{2} - 1113 T^{3} + 39811 T^{4} - 181712 T^{5} + 3873275 T^{6} - 18449631 T^{7} + 282784266 T^{8} - 1333386151 T^{9} + 16225324671 T^{10} - 73255285100 T^{11} + 749908791335 T^{12} - 3165737580326 T^{13} + 28299326386568 T^{14} - 109442121702655 T^{15} + 877279117983608 T^{16} - 3042273814693286 T^{17} + 22340532802660985 T^{18} - 67652794150837100 T^{19} + 464517270030084321 T^{20} - 1183385117206921831 T^{21} + 7780134387120377526 T^{22} - 15735524923993634271 T^{23} + \)\(10\!\cdots\!25\)\( T^{24} - \)\(14\!\cdots\!12\)\( T^{25} + \)\(10\!\cdots\!41\)\( T^{26} - \)\(87\!\cdots\!93\)\( T^{27} + \)\(67\!\cdots\!98\)\( T^{28} - \)\(22\!\cdots\!63\)\( T^{29} + \)\(23\!\cdots\!51\)\( T^{30} \))(\( 1 + 13 T + 402 T^{2} + 4717 T^{3} + 81079 T^{4} + 850396 T^{5} + 10780931 T^{6} + 101525761 T^{7} + 1056613894 T^{8} + 9012084121 T^{9} + 81206386480 T^{10} + 632870695777 T^{11} + 5089064267949 T^{12} + 36512403922907 T^{13} + 266880570020541 T^{14} + 1772912603563763 T^{15} + 11915704620123491 T^{16} + 73574456829398433 T^{17} + 457933291091217011 T^{18} + 2633362510544333781 T^{19} + 15240223114663089536 T^{20} + 81634237826874347211 T^{21} + \)\(44\!\cdots\!71\)\( T^{22} + \)\(21\!\cdots\!03\)\( T^{23} + \)\(11\!\cdots\!11\)\( T^{24} + \)\(50\!\cdots\!13\)\( T^{25} + \)\(23\!\cdots\!21\)\( T^{26} + \)\(10\!\cdots\!77\)\( T^{27} + \)\(43\!\cdots\!09\)\( T^{28} + \)\(16\!\cdots\!67\)\( T^{29} + \)\(66\!\cdots\!80\)\( T^{30} + \)\(22\!\cdots\!51\)\( T^{31} + \)\(83\!\cdots\!34\)\( T^{32} + \)\(24\!\cdots\!51\)\( T^{33} + \)\(81\!\cdots\!51\)\( T^{34} + \)\(19\!\cdots\!96\)\( T^{35} + \)\(58\!\cdots\!99\)\( T^{36} + \)\(10\!\cdots\!87\)\( T^{37} + \)\(28\!\cdots\!82\)\( T^{38} + \)\(28\!\cdots\!23\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \))
$37$ (\( 1 + 6 T + 37 T^{2} \))(\( 1 + 28 T + 737 T^{2} + 13028 T^{3} + 211132 T^{4} + 2821766 T^{5} + 34895842 T^{6} + 380541287 T^{7} + 3876683907 T^{8} + 35887706768 T^{9} + 312504743316 T^{10} + 2511326994472 T^{11} + 19068516372841 T^{12} + 134704353903580 T^{13} + 901355950418349 T^{14} + 5631166084053363 T^{15} + 33350170165478913 T^{16} + 184410260494001020 T^{17} + 965877559833515173 T^{18} + 4706631111286637992 T^{19} + 21670315482800741412 T^{20} + 92078037013105636112 T^{21} + \)\(36\!\cdots\!31\)\( T^{22} + \)\(13\!\cdots\!27\)\( T^{23} + \)\(45\!\cdots\!34\)\( T^{24} + \)\(13\!\cdots\!34\)\( T^{25} + \)\(37\!\cdots\!16\)\( T^{26} + \)\(85\!\cdots\!68\)\( T^{27} + \)\(17\!\cdots\!89\)\( T^{28} + \)\(25\!\cdots\!92\)\( T^{29} + \)\(33\!\cdots\!93\)\( T^{30} \))(\( 1 + 26 T + 680 T^{2} + 11208 T^{3} + 178373 T^{4} + 2236904 T^{5} + 27165432 T^{6} + 280870617 T^{7} + 2834784352 T^{8} + 25253208585 T^{9} + 221483944967 T^{10} + 1747457405286 T^{11} + 13701695835246 T^{12} + 97711761844620 T^{13} + 700662377957548 T^{14} + 4604861608682748 T^{15} + 30898367863168743 T^{16} + 191409632644113634 T^{17} + 1231898923130297821 T^{18} + 7371494589048669624 T^{19} + 46426106772411204074 T^{20} + \)\(27\!\cdots\!88\)\( T^{21} + \)\(16\!\cdots\!49\)\( T^{22} + \)\(96\!\cdots\!02\)\( T^{23} + \)\(57\!\cdots\!23\)\( T^{24} + \)\(31\!\cdots\!36\)\( T^{25} + \)\(17\!\cdots\!32\)\( T^{26} + \)\(92\!\cdots\!60\)\( T^{27} + \)\(48\!\cdots\!66\)\( T^{28} + \)\(22\!\cdots\!22\)\( T^{29} + \)\(10\!\cdots\!83\)\( T^{30} + \)\(44\!\cdots\!05\)\( T^{31} + \)\(18\!\cdots\!12\)\( T^{32} + \)\(68\!\cdots\!49\)\( T^{33} + \)\(24\!\cdots\!48\)\( T^{34} + \)\(74\!\cdots\!72\)\( T^{35} + \)\(22\!\cdots\!93\)\( T^{36} + \)\(51\!\cdots\!36\)\( T^{37} + \)\(11\!\cdots\!20\)\( T^{38} + \)\(16\!\cdots\!98\)\( T^{39} + \)\(23\!\cdots\!01\)\( T^{40} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 + 10 T + 501 T^{2} + 4291 T^{3} + 118494 T^{4} + 884991 T^{5} + 17675217 T^{6} + 116505267 T^{7} + 1871081273 T^{8} + 10980373078 T^{9} + 149775216545 T^{10} + 787758732055 T^{11} + 9416547989787 T^{12} + 44595639328835 T^{13} + 475554021741895 T^{14} + 2031714096117781 T^{15} + 19497714891417695 T^{16} + 74965269711771635 T^{17} + 648997904004109827 T^{18} + 2226017902450468855 T^{19} + 17352387592856045545 T^{20} + 52157916725570023798 T^{21} + \)\(36\!\cdots\!13\)\( T^{22} + \)\(93\!\cdots\!07\)\( T^{23} + \)\(57\!\cdots\!37\)\( T^{24} + \)\(11\!\cdots\!91\)\( T^{25} + \)\(65\!\cdots\!54\)\( T^{26} + \)\(96\!\cdots\!71\)\( T^{27} + \)\(46\!\cdots\!21\)\( T^{28} + \)\(37\!\cdots\!10\)\( T^{29} + \)\(15\!\cdots\!01\)\( T^{30} \))(\( 1 + 6 T + 416 T^{2} + 2261 T^{3} + 82281 T^{4} + 389942 T^{5} + 10301955 T^{6} + 40435688 T^{7} + 923112319 T^{8} + 2781624891 T^{9} + 64302590698 T^{10} + 133171949946 T^{11} + 3768952862589 T^{12} + 4770900886360 T^{13} + 201152115528337 T^{14} + 165268043132596 T^{15} + 10243297187662476 T^{16} + 7807698222759897 T^{17} + 491793542040637722 T^{18} + 403709797415297645 T^{19} + 21390164161825361132 T^{20} + 16552101694027203445 T^{21} + \)\(82\!\cdots\!82\)\( T^{22} + \)\(53\!\cdots\!37\)\( T^{23} + \)\(28\!\cdots\!36\)\( T^{24} + \)\(19\!\cdots\!96\)\( T^{25} + \)\(95\!\cdots\!17\)\( T^{26} + \)\(92\!\cdots\!60\)\( T^{27} + \)\(30\!\cdots\!69\)\( T^{28} + \)\(43\!\cdots\!06\)\( T^{29} + \)\(86\!\cdots\!98\)\( T^{30} + \)\(15\!\cdots\!31\)\( T^{31} + \)\(20\!\cdots\!39\)\( T^{32} + \)\(37\!\cdots\!48\)\( T^{33} + \)\(39\!\cdots\!55\)\( T^{34} + \)\(60\!\cdots\!42\)\( T^{35} + \)\(52\!\cdots\!21\)\( T^{36} + \)\(59\!\cdots\!41\)\( T^{37} + \)\(44\!\cdots\!36\)\( T^{38} + \)\(26\!\cdots\!66\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 + 34 T + 871 T^{2} + 16100 T^{3} + 255594 T^{4} + 3451819 T^{5} + 42110548 T^{6} + 462033352 T^{7} + 4699115510 T^{8} + 44143045624 T^{9} + 390119874034 T^{10} + 3231163495914 T^{11} + 25396455042940 T^{12} + 188570697014796 T^{13} + 1334835190620896 T^{14} + 8958199265977165 T^{15} + 57397913196698528 T^{16} + 348667218780357804 T^{17} + 2019195951099030580 T^{18} + 11046704990994279114 T^{19} + 57350915265094469062 T^{20} + \)\(27\!\cdots\!76\)\( T^{21} + \)\(12\!\cdots\!70\)\( T^{22} + \)\(54\!\cdots\!52\)\( T^{23} + \)\(21\!\cdots\!64\)\( T^{24} + \)\(74\!\cdots\!31\)\( T^{25} + \)\(23\!\cdots\!58\)\( T^{26} + \)\(64\!\cdots\!00\)\( T^{27} + \)\(14\!\cdots\!53\)\( T^{28} + \)\(25\!\cdots\!66\)\( T^{29} + \)\(31\!\cdots\!07\)\( T^{30} \))(\( 1 + 32 T + 920 T^{2} + 18372 T^{3} + 329191 T^{4} + 4965093 T^{5} + 68600286 T^{6} + 848878809 T^{7} + 9784116543 T^{8} + 103999022040 T^{9} + 1042537586678 T^{10} + 9805098185968 T^{11} + 87811742796571 T^{12} + 746505414548173 T^{13} + 6090755606708942 T^{14} + 47587160987240004 T^{15} + 359171633758391490 T^{16} + 2613250836352446103 T^{17} + 18459704443881645318 T^{18} + \)\(12\!\cdots\!12\)\( T^{19} + \)\(84\!\cdots\!08\)\( T^{20} + \)\(54\!\cdots\!16\)\( T^{21} + \)\(34\!\cdots\!82\)\( T^{22} + \)\(20\!\cdots\!21\)\( T^{23} + \)\(12\!\cdots\!90\)\( T^{24} + \)\(69\!\cdots\!72\)\( T^{25} + \)\(38\!\cdots\!58\)\( T^{26} + \)\(20\!\cdots\!11\)\( T^{27} + \)\(10\!\cdots\!71\)\( T^{28} + \)\(49\!\cdots\!24\)\( T^{29} + \)\(22\!\cdots\!22\)\( T^{30} + \)\(96\!\cdots\!80\)\( T^{31} + \)\(39\!\cdots\!43\)\( T^{32} + \)\(14\!\cdots\!87\)\( T^{33} + \)\(50\!\cdots\!14\)\( T^{34} + \)\(15\!\cdots\!51\)\( T^{35} + \)\(44\!\cdots\!91\)\( T^{36} + \)\(10\!\cdots\!96\)\( T^{37} + \)\(23\!\cdots\!80\)\( T^{38} + \)\(34\!\cdots\!24\)\( T^{39} + \)\(46\!\cdots\!01\)\( T^{40} \))
$47$ (\( 1 + 3 T + 47 T^{2} \))(\( 1 + 14 T + 500 T^{2} + 5908 T^{3} + 118891 T^{4} + 1233191 T^{5} + 18210703 T^{6} + 169555757 T^{7} + 2031443894 T^{8} + 17180136777 T^{9} + 175740287866 T^{10} + 1357307270872 T^{11} + 12211486276317 T^{12} + 86221244688572 T^{13} + 694853982506021 T^{14} + 4473315096067911 T^{15} + 32658137177782987 T^{16} + 190462729517055548 T^{17} + 1267833139666059891 T^{18} + 6623226500835951832 T^{19} + 40305157550809785062 T^{20} + \)\(18\!\cdots\!33\)\( T^{21} + \)\(10\!\cdots\!22\)\( T^{22} + \)\(40\!\cdots\!77\)\( T^{23} + \)\(20\!\cdots\!01\)\( T^{24} + \)\(64\!\cdots\!59\)\( T^{25} + \)\(29\!\cdots\!73\)\( T^{26} + \)\(68\!\cdots\!28\)\( T^{27} + \)\(27\!\cdots\!00\)\( T^{28} + \)\(35\!\cdots\!66\)\( T^{29} + \)\(12\!\cdots\!43\)\( T^{30} \))(\( 1 - 3 T + 599 T^{2} - 1614 T^{3} + 174865 T^{4} - 440722 T^{5} + 33370749 T^{6} - 81692171 T^{7} + 4702316329 T^{8} - 11474763237 T^{9} + 522874940266 T^{10} - 1285674027589 T^{11} + 47791593145124 T^{12} - 118121454381605 T^{13} + 3687015042111761 T^{14} - 9054319077083189 T^{15} + 244286843455104677 T^{16} - 585900492543014734 T^{17} + 14052573680955717749 T^{18} - 32250593794691098066 T^{19} + \)\(70\!\cdots\!08\)\( T^{20} - \)\(15\!\cdots\!02\)\( T^{21} + \)\(31\!\cdots\!41\)\( T^{22} - \)\(60\!\cdots\!82\)\( T^{23} + \)\(11\!\cdots\!37\)\( T^{24} - \)\(20\!\cdots\!23\)\( T^{25} + \)\(39\!\cdots\!69\)\( T^{26} - \)\(59\!\cdots\!15\)\( T^{27} + \)\(11\!\cdots\!64\)\( T^{28} - \)\(14\!\cdots\!63\)\( T^{29} + \)\(27\!\cdots\!34\)\( T^{30} - \)\(28\!\cdots\!11\)\( T^{31} + \)\(54\!\cdots\!89\)\( T^{32} - \)\(44\!\cdots\!17\)\( T^{33} + \)\(85\!\cdots\!81\)\( T^{34} - \)\(53\!\cdots\!46\)\( T^{35} + \)\(99\!\cdots\!65\)\( T^{36} - \)\(43\!\cdots\!18\)\( T^{37} + \)\(75\!\cdots\!11\)\( T^{38} - \)\(17\!\cdots\!49\)\( T^{39} + \)\(27\!\cdots\!01\)\( T^{40} \))
$53$ (\( 1 - 8 T + 53 T^{2} \))(\( 1 + 26 T + 716 T^{2} + 12120 T^{3} + 202185 T^{4} + 2670172 T^{5} + 34283759 T^{6} + 384016749 T^{7} + 4150159669 T^{8} + 41128713159 T^{9} + 390341553630 T^{10} + 3495779922277 T^{11} + 29789497851247 T^{12} + 243497287580375 T^{13} + 1885714233357263 T^{14} + 14114423017001699 T^{15} + 99942854367934939 T^{16} + 683983880813273375 T^{17} + 4434971071600099619 T^{18} + 27583385056908145237 T^{19} + \)\(16\!\cdots\!90\)\( T^{20} + \)\(91\!\cdots\!11\)\( T^{21} + \)\(48\!\cdots\!53\)\( T^{22} + \)\(23\!\cdots\!89\)\( T^{23} + \)\(11\!\cdots\!47\)\( T^{24} + \)\(46\!\cdots\!28\)\( T^{25} + \)\(18\!\cdots\!45\)\( T^{26} + \)\(59\!\cdots\!20\)\( T^{27} + \)\(18\!\cdots\!68\)\( T^{28} + \)\(35\!\cdots\!94\)\( T^{29} + \)\(73\!\cdots\!57\)\( T^{30} \))(\( 1 + 20 T + 805 T^{2} + 13400 T^{3} + 305717 T^{4} + 4371504 T^{5} + 73671126 T^{6} + 925488469 T^{7} + 12747293665 T^{8} + 143097086366 T^{9} + 1696883169074 T^{10} + 17252326006690 T^{11} + 181699444931188 T^{12} + 1691548276964420 T^{13} + 16147504316318064 T^{14} + 138867618948762805 T^{15} + 1218324557762187704 T^{16} + 9743251460111187102 T^{17} + 79297819914734031867 T^{18} + \)\(59\!\cdots\!52\)\( T^{19} + \)\(44\!\cdots\!58\)\( T^{20} + \)\(31\!\cdots\!56\)\( T^{21} + \)\(22\!\cdots\!03\)\( T^{22} + \)\(14\!\cdots\!54\)\( T^{23} + \)\(96\!\cdots\!24\)\( T^{24} + \)\(58\!\cdots\!65\)\( T^{25} + \)\(35\!\cdots\!56\)\( T^{26} + \)\(19\!\cdots\!40\)\( T^{27} + \)\(11\!\cdots\!68\)\( T^{28} + \)\(56\!\cdots\!70\)\( T^{29} + \)\(29\!\cdots\!26\)\( T^{30} + \)\(13\!\cdots\!02\)\( T^{31} + \)\(62\!\cdots\!65\)\( T^{32} + \)\(24\!\cdots\!37\)\( T^{33} + \)\(10\!\cdots\!94\)\( T^{34} + \)\(31\!\cdots\!28\)\( T^{35} + \)\(11\!\cdots\!57\)\( T^{36} + \)\(27\!\cdots\!00\)\( T^{37} + \)\(87\!\cdots\!45\)\( T^{38} + \)\(11\!\cdots\!40\)\( T^{39} + \)\(30\!\cdots\!01\)\( T^{40} \))
$59$ (\( 1 + T + 59 T^{2} \))(\( 1 + T + 409 T^{2} + 755 T^{3} + 81345 T^{4} + 148229 T^{5} + 10374453 T^{6} + 9103985 T^{7} + 920562181 T^{8} - 1102729978 T^{9} + 58006830984 T^{10} - 301912806882 T^{11} + 2626178332696 T^{12} - 34338349850952 T^{13} + 100085590946124 T^{14} - 2470353747085387 T^{15} + 5905049865821316 T^{16} - 119531795831163912 T^{17} + 539361879790771784 T^{18} - 3658386471512478402 T^{19} + 41470492978447680216 T^{20} - 46513738933968189898 T^{21} + \)\(22\!\cdots\!39\)\( T^{22} + \)\(13\!\cdots\!85\)\( T^{23} + \)\(89\!\cdots\!67\)\( T^{24} + \)\(75\!\cdots\!29\)\( T^{25} + \)\(24\!\cdots\!55\)\( T^{26} + \)\(13\!\cdots\!55\)\( T^{27} + \)\(42\!\cdots\!11\)\( T^{28} + \)\(61\!\cdots\!61\)\( T^{29} + \)\(36\!\cdots\!99\)\( T^{30} \))(\( 1 - 26 T + 1105 T^{2} - 21841 T^{3} + 543139 T^{4} - 8828109 T^{5} + 164731203 T^{6} - 2300920909 T^{7} + 35348516048 T^{8} - 435980492806 T^{9} + 5777983968772 T^{10} - 64045481184303 T^{11} + 752276922465829 T^{12} - 7581553118044085 T^{13} + 80268255806004173 T^{14} - 741133355905276646 T^{15} + 7149656460497707844 T^{16} - 60763016488155835949 T^{17} + \)\(53\!\cdots\!51\)\( T^{18} - \)\(42\!\cdots\!60\)\( T^{19} + \)\(34\!\cdots\!46\)\( T^{20} - \)\(24\!\cdots\!40\)\( T^{21} + \)\(18\!\cdots\!31\)\( T^{22} - \)\(12\!\cdots\!71\)\( T^{23} + \)\(86\!\cdots\!84\)\( T^{24} - \)\(52\!\cdots\!54\)\( T^{25} + \)\(33\!\cdots\!93\)\( T^{26} - \)\(18\!\cdots\!15\)\( T^{27} + \)\(11\!\cdots\!09\)\( T^{28} - \)\(55\!\cdots\!17\)\( T^{29} + \)\(29\!\cdots\!72\)\( T^{30} - \)\(13\!\cdots\!54\)\( T^{31} + \)\(62\!\cdots\!88\)\( T^{32} - \)\(24\!\cdots\!11\)\( T^{33} + \)\(10\!\cdots\!83\)\( T^{34} - \)\(32\!\cdots\!91\)\( T^{35} + \)\(11\!\cdots\!99\)\( T^{36} - \)\(27\!\cdots\!79\)\( T^{37} + \)\(82\!\cdots\!05\)\( T^{38} - \)\(11\!\cdots\!14\)\( T^{39} + \)\(26\!\cdots\!01\)\( T^{40} \))
$61$ (\( 1 - 12 T + 61 T^{2} \))(\( 1 + 15 T + 742 T^{2} + 9755 T^{3} + 263240 T^{4} + 3077021 T^{5} + 59419472 T^{6} + 623236524 T^{7} + 9563512055 T^{8} + 90491662796 T^{9} + 1164255571997 T^{10} + 9959835813655 T^{11} + 110861479783462 T^{12} + 856772226327094 T^{13} + 8413236975648457 T^{14} + 58513223777527365 T^{15} + 513207455514555877 T^{16} + 3188049454163116774 T^{17} + 25163449542729988222 T^{18} + \)\(13\!\cdots\!55\)\( T^{19} + \)\(98\!\cdots\!97\)\( T^{20} + \)\(46\!\cdots\!56\)\( T^{21} + \)\(30\!\cdots\!55\)\( T^{22} + \)\(11\!\cdots\!44\)\( T^{23} + \)\(69\!\cdots\!52\)\( T^{24} + \)\(21\!\cdots\!21\)\( T^{25} + \)\(11\!\cdots\!40\)\( T^{26} + \)\(25\!\cdots\!55\)\( T^{27} + \)\(12\!\cdots\!02\)\( T^{28} + \)\(14\!\cdots\!15\)\( T^{29} + \)\(60\!\cdots\!01\)\( T^{30} \))(\( 1 + 33 T + 1267 T^{2} + 29294 T^{3} + 682226 T^{4} + 12472134 T^{5} + 222203044 T^{6} + 3412433277 T^{7} + 50623265086 T^{8} + 676518910917 T^{9} + 8718593025943 T^{10} + 103670110278113 T^{11} + 1189420303345133 T^{12} + 12769387474133203 T^{13} + 132425616440396833 T^{14} + 1296132077751296232 T^{15} + 12267915425643753345 T^{16} + \)\(11\!\cdots\!63\)\( T^{17} + \)\(95\!\cdots\!53\)\( T^{18} + \)\(79\!\cdots\!34\)\( T^{19} + \)\(63\!\cdots\!50\)\( T^{20} + \)\(48\!\cdots\!74\)\( T^{21} + \)\(35\!\cdots\!13\)\( T^{22} + \)\(25\!\cdots\!03\)\( T^{23} + \)\(16\!\cdots\!45\)\( T^{24} + \)\(10\!\cdots\!32\)\( T^{25} + \)\(68\!\cdots\!13\)\( T^{26} + \)\(40\!\cdots\!63\)\( T^{27} + \)\(22\!\cdots\!73\)\( T^{28} + \)\(12\!\cdots\!33\)\( T^{29} + \)\(62\!\cdots\!43\)\( T^{30} + \)\(29\!\cdots\!37\)\( T^{31} + \)\(13\!\cdots\!06\)\( T^{32} + \)\(55\!\cdots\!37\)\( T^{33} + \)\(21\!\cdots\!04\)\( T^{34} + \)\(75\!\cdots\!34\)\( T^{35} + \)\(25\!\cdots\!86\)\( T^{36} + \)\(65\!\cdots\!74\)\( T^{37} + \)\(17\!\cdots\!27\)\( T^{38} + \)\(27\!\cdots\!53\)\( T^{39} + \)\(50\!\cdots\!01\)\( T^{40} \))
$67$ (\( 1 + 12 T + 67 T^{2} \))(\( 1 + 3 T + 698 T^{2} + 1740 T^{3} + 228863 T^{4} + 440874 T^{5} + 46876731 T^{6} + 60660049 T^{7} + 6756876295 T^{8} + 4120043843 T^{9} + 737901948417 T^{10} - 47829575031 T^{11} + 64959824490208 T^{12} - 38191060182672 T^{13} + 4887697883899138 T^{14} - 3742571722634919 T^{15} + 327475758221242246 T^{16} - 171439669160014608 T^{17} + 19537511693148428704 T^{18} - 963819553828259751 T^{19} + \)\(99\!\cdots\!19\)\( T^{20} + \)\(37\!\cdots\!67\)\( T^{21} + \)\(40\!\cdots\!85\)\( T^{22} + \)\(24\!\cdots\!09\)\( T^{23} + \)\(12\!\cdots\!57\)\( T^{24} + \)\(80\!\cdots\!26\)\( T^{25} + \)\(27\!\cdots\!29\)\( T^{26} + \)\(14\!\cdots\!40\)\( T^{27} + \)\(38\!\cdots\!26\)\( T^{28} + \)\(11\!\cdots\!87\)\( T^{29} + \)\(24\!\cdots\!43\)\( T^{30} \))(\( 1 + 31 T + 1170 T^{2} + 26344 T^{3} + 609579 T^{4} + 11064054 T^{5} + 197705967 T^{6} + 3051968525 T^{7} + 45770971367 T^{8} + 620342669191 T^{9} + 8136633731384 T^{10} + 98815335168230 T^{11} + 1160800965520938 T^{12} + 12805609846296624 T^{13} + 136735672352665635 T^{14} + 1382894977525655735 T^{15} + 13549732629531415755 T^{16} + \)\(12\!\cdots\!39\)\( T^{17} + \)\(11\!\cdots\!53\)\( T^{18} + \)\(98\!\cdots\!70\)\( T^{19} + \)\(82\!\cdots\!70\)\( T^{20} + \)\(66\!\cdots\!90\)\( T^{21} + \)\(51\!\cdots\!17\)\( T^{22} + \)\(38\!\cdots\!57\)\( T^{23} + \)\(27\!\cdots\!55\)\( T^{24} + \)\(18\!\cdots\!45\)\( T^{25} + \)\(12\!\cdots\!15\)\( T^{26} + \)\(77\!\cdots\!52\)\( T^{27} + \)\(47\!\cdots\!58\)\( T^{28} + \)\(26\!\cdots\!10\)\( T^{29} + \)\(14\!\cdots\!16\)\( T^{30} + \)\(75\!\cdots\!53\)\( T^{31} + \)\(37\!\cdots\!87\)\( T^{32} + \)\(16\!\cdots\!75\)\( T^{33} + \)\(72\!\cdots\!43\)\( T^{34} + \)\(27\!\cdots\!22\)\( T^{35} + \)\(10\!\cdots\!99\)\( T^{36} + \)\(29\!\cdots\!88\)\( T^{37} + \)\(86\!\cdots\!30\)\( T^{38} + \)\(15\!\cdots\!93\)\( T^{39} + \)\(33\!\cdots\!01\)\( T^{40} \))
$71$ (\( 1 + 7 T + 71 T^{2} \))(\( 1 - 3 T + 601 T^{2} - 1012 T^{3} + 178541 T^{4} - 85093 T^{5} + 35180743 T^{6} + 20272160 T^{7} + 5192138179 T^{8} + 7396213449 T^{9} + 612139307534 T^{10} + 1235099866004 T^{11} + 59848219863744 T^{12} + 138426659434371 T^{13} + 4959923920563372 T^{14} + 11366050712183617 T^{15} + 352154598359999412 T^{16} + 697808790208664211 T^{17} + 21420336219652478784 T^{18} + 31385963798036392724 T^{19} + \)\(11\!\cdots\!34\)\( T^{20} + \)\(94\!\cdots\!29\)\( T^{21} + \)\(47\!\cdots\!89\)\( T^{22} + \)\(13\!\cdots\!60\)\( T^{23} + \)\(16\!\cdots\!33\)\( T^{24} - \)\(27\!\cdots\!93\)\( T^{25} + \)\(41\!\cdots\!11\)\( T^{26} - \)\(16\!\cdots\!92\)\( T^{27} + \)\(70\!\cdots\!11\)\( T^{28} - \)\(24\!\cdots\!43\)\( T^{29} + \)\(58\!\cdots\!51\)\( T^{30} \))(\( 1 - 2 T + 701 T^{2} - 534 T^{3} + 250050 T^{4} + 48684 T^{5} + 60553691 T^{6} + 57813628 T^{7} + 11155983417 T^{8} + 17483422816 T^{9} + 1660923601567 T^{10} + 3414409124637 T^{11} + 207479982167340 T^{12} + 503694435191397 T^{13} + 22311948642034294 T^{14} + 59790361697638972 T^{15} + 2103772568196114445 T^{16} + 5897422641416472420 T^{17} + \)\(17\!\cdots\!71\)\( T^{18} + \)\(49\!\cdots\!68\)\( T^{19} + \)\(13\!\cdots\!22\)\( T^{20} + \)\(34\!\cdots\!28\)\( T^{21} + \)\(88\!\cdots\!11\)\( T^{22} + \)\(21\!\cdots\!20\)\( T^{23} + \)\(53\!\cdots\!45\)\( T^{24} + \)\(10\!\cdots\!72\)\( T^{25} + \)\(28\!\cdots\!74\)\( T^{26} + \)\(45\!\cdots\!27\)\( T^{27} + \)\(13\!\cdots\!40\)\( T^{28} + \)\(15\!\cdots\!47\)\( T^{29} + \)\(54\!\cdots\!67\)\( T^{30} + \)\(40\!\cdots\!36\)\( T^{31} + \)\(18\!\cdots\!97\)\( T^{32} + \)\(67\!\cdots\!08\)\( T^{33} + \)\(50\!\cdots\!71\)\( T^{34} + \)\(28\!\cdots\!84\)\( T^{35} + \)\(10\!\cdots\!50\)\( T^{36} - \)\(15\!\cdots\!94\)\( T^{37} + \)\(14\!\cdots\!61\)\( T^{38} - \)\(29\!\cdots\!62\)\( T^{39} + \)\(10\!\cdots\!01\)\( T^{40} \))
$73$ (\( 1 - 10 T + 73 T^{2} \))(\( 1 + 65 T + 2568 T^{2} + 73498 T^{3} + 1695448 T^{4} + 32949928 T^{5} + 558469479 T^{6} + 8420580841 T^{7} + 114888099440 T^{8} + 1435046042800 T^{9} + 16589414461501 T^{10} + 178986233611711 T^{11} + 1816955133781779 T^{12} + 17463237858844293 T^{13} + 159782551729754693 T^{14} + 1396044677171789299 T^{15} + 11664126276272092589 T^{16} + 93061594549781237397 T^{17} + \)\(70\!\cdots\!43\)\( T^{18} + \)\(50\!\cdots\!51\)\( T^{19} + \)\(34\!\cdots\!93\)\( T^{20} + \)\(21\!\cdots\!00\)\( T^{21} + \)\(12\!\cdots\!80\)\( T^{22} + \)\(67\!\cdots\!21\)\( T^{23} + \)\(32\!\cdots\!27\)\( T^{24} + \)\(14\!\cdots\!72\)\( T^{25} + \)\(53\!\cdots\!96\)\( T^{26} + \)\(16\!\cdots\!58\)\( T^{27} + \)\(42\!\cdots\!44\)\( T^{28} + \)\(79\!\cdots\!85\)\( T^{29} + \)\(89\!\cdots\!57\)\( T^{30} \))(\( 1 + 71 T + 3138 T^{2} + 103488 T^{3} + 2801036 T^{4} + 65026576 T^{5} + 1334037765 T^{6} + 24649846753 T^{7} + 416097576592 T^{8} + 6483383508536 T^{9} + 94010953865954 T^{10} + 1276707687575920 T^{11} + 16322987910017499 T^{12} + 197301236998452807 T^{13} + 2262537017060156937 T^{14} + 24684871894707172757 T^{15} + \)\(25\!\cdots\!76\)\( T^{16} + \)\(25\!\cdots\!44\)\( T^{17} + \)\(24\!\cdots\!59\)\( T^{18} + \)\(22\!\cdots\!55\)\( T^{19} + \)\(19\!\cdots\!84\)\( T^{20} + \)\(16\!\cdots\!15\)\( T^{21} + \)\(12\!\cdots\!11\)\( T^{22} + \)\(99\!\cdots\!48\)\( T^{23} + \)\(72\!\cdots\!16\)\( T^{24} + \)\(51\!\cdots\!01\)\( T^{25} + \)\(34\!\cdots\!93\)\( T^{26} + \)\(21\!\cdots\!79\)\( T^{27} + \)\(13\!\cdots\!19\)\( T^{28} + \)\(75\!\cdots\!60\)\( T^{29} + \)\(40\!\cdots\!46\)\( T^{30} + \)\(20\!\cdots\!72\)\( T^{31} + \)\(95\!\cdots\!32\)\( T^{32} + \)\(41\!\cdots\!49\)\( T^{33} + \)\(16\!\cdots\!85\)\( T^{34} + \)\(57\!\cdots\!32\)\( T^{35} + \)\(18\!\cdots\!96\)\( T^{36} + \)\(49\!\cdots\!64\)\( T^{37} + \)\(10\!\cdots\!22\)\( T^{38} + \)\(17\!\cdots\!27\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \))
$79$ (\( 1 - T \))(\( ( 1 - T )^{15} \))(\( ( 1 + T )^{20} \))
$83$ (\( 1 + 10 T + 83 T^{2} \))(\( 1 + 5 T + 890 T^{2} + 3026 T^{3} + 376000 T^{4} + 766699 T^{5} + 101592298 T^{6} + 95368841 T^{7} + 19920338615 T^{8} + 2669547495 T^{9} + 3033815383589 T^{10} - 1110110824024 T^{11} + 372282981481910 T^{12} - 222233402373595 T^{13} + 37453640001803056 T^{14} - 23262068538180107 T^{15} + 3108652120149653648 T^{16} - 1530965908951695955 T^{17} + \)\(21\!\cdots\!70\)\( T^{18} - 52683995832105503704 T^{19} + \)\(11\!\cdots\!27\)\( T^{20} + \)\(87\!\cdots\!55\)\( T^{21} + \)\(54\!\cdots\!05\)\( T^{22} + \)\(21\!\cdots\!81\)\( T^{23} + \)\(18\!\cdots\!94\)\( T^{24} + \)\(11\!\cdots\!51\)\( T^{25} + \)\(48\!\cdots\!00\)\( T^{26} + \)\(32\!\cdots\!86\)\( T^{27} + \)\(78\!\cdots\!70\)\( T^{28} + \)\(36\!\cdots\!45\)\( T^{29} + \)\(61\!\cdots\!07\)\( T^{30} \))(\( 1 - 21 T + 1131 T^{2} - 18395 T^{3} + 571432 T^{4} - 7619985 T^{5} + 178678738 T^{6} - 2024116136 T^{7} + 39954302752 T^{8} - 393433266059 T^{9} + 6941993556763 T^{10} - 60293925419540 T^{11} + 988178746237337 T^{12} - 7648108022670594 T^{13} + 119618896160340440 T^{14} - 833887010797679656 T^{15} + 12680389966434130455 T^{16} - 80870915508044437622 T^{17} + \)\(12\!\cdots\!84\)\( T^{18} - \)\(71\!\cdots\!38\)\( T^{19} + \)\(10\!\cdots\!26\)\( T^{20} - \)\(59\!\cdots\!54\)\( T^{21} + \)\(83\!\cdots\!76\)\( T^{22} - \)\(46\!\cdots\!14\)\( T^{23} + \)\(60\!\cdots\!55\)\( T^{24} - \)\(32\!\cdots\!08\)\( T^{25} + \)\(39\!\cdots\!60\)\( T^{26} - \)\(20\!\cdots\!38\)\( T^{27} + \)\(22\!\cdots\!17\)\( T^{28} - \)\(11\!\cdots\!20\)\( T^{29} + \)\(10\!\cdots\!87\)\( T^{30} - \)\(50\!\cdots\!53\)\( T^{31} + \)\(42\!\cdots\!72\)\( T^{32} - \)\(17\!\cdots\!68\)\( T^{33} + \)\(13\!\cdots\!02\)\( T^{34} - \)\(46\!\cdots\!95\)\( T^{35} + \)\(28\!\cdots\!92\)\( T^{36} - \)\(77\!\cdots\!85\)\( T^{37} + \)\(39\!\cdots\!79\)\( T^{38} - \)\(60\!\cdots\!87\)\( T^{39} + \)\(24\!\cdots\!01\)\( T^{40} \))
$89$ (\( 1 + 7 T + 89 T^{2} \))(\( 1 + 26 T + 1118 T^{2} + 22769 T^{3} + 570036 T^{4} + 9600260 T^{5} + 179709180 T^{6} + 2593747713 T^{7} + 39837526604 T^{8} + 504981400197 T^{9} + 6673242302064 T^{10} + 75588909782524 T^{11} + 884000658927887 T^{12} + 9051698506779760 T^{13} + 95300321830678317 T^{14} + 887125551740376499 T^{15} + 8481728642930370213 T^{16} + 71698503872202478960 T^{17} + \)\(62\!\cdots\!03\)\( T^{18} + \)\(47\!\cdots\!84\)\( T^{19} + \)\(37\!\cdots\!36\)\( T^{20} + \)\(25\!\cdots\!17\)\( T^{21} + \)\(17\!\cdots\!16\)\( T^{22} + \)\(10\!\cdots\!53\)\( T^{23} + \)\(62\!\cdots\!20\)\( T^{24} + \)\(29\!\cdots\!60\)\( T^{25} + \)\(15\!\cdots\!04\)\( T^{26} + \)\(56\!\cdots\!49\)\( T^{27} + \)\(24\!\cdots\!42\)\( T^{28} + \)\(50\!\cdots\!66\)\( T^{29} + \)\(17\!\cdots\!49\)\( T^{30} \))(\( 1 + 15 T + 1250 T^{2} + 17422 T^{3} + 769183 T^{4} + 9994207 T^{5} + 309854565 T^{6} + 3762025538 T^{7} + 91669733232 T^{8} + 1041623868834 T^{9} + 21180582347874 T^{10} + 225418731342232 T^{11} + 3967685064296720 T^{12} + 39550665061845988 T^{13} + 617369400284294161 T^{14} + 5758884931249964979 T^{15} + 81077100494320905446 T^{16} + \)\(70\!\cdots\!22\)\( T^{17} + \)\(90\!\cdots\!87\)\( T^{18} + \)\(73\!\cdots\!67\)\( T^{19} + \)\(87\!\cdots\!05\)\( T^{20} + \)\(65\!\cdots\!63\)\( T^{21} + \)\(71\!\cdots\!27\)\( T^{22} + \)\(49\!\cdots\!18\)\( T^{23} + \)\(50\!\cdots\!86\)\( T^{24} + \)\(32\!\cdots\!71\)\( T^{25} + \)\(30\!\cdots\!21\)\( T^{26} + \)\(17\!\cdots\!52\)\( T^{27} + \)\(15\!\cdots\!20\)\( T^{28} + \)\(78\!\cdots\!88\)\( T^{29} + \)\(66\!\cdots\!74\)\( T^{30} + \)\(28\!\cdots\!26\)\( T^{31} + \)\(22\!\cdots\!72\)\( T^{32} + \)\(82\!\cdots\!22\)\( T^{33} + \)\(60\!\cdots\!65\)\( T^{34} + \)\(17\!\cdots\!43\)\( T^{35} + \)\(11\!\cdots\!63\)\( T^{36} + \)\(24\!\cdots\!38\)\( T^{37} + \)\(15\!\cdots\!50\)\( T^{38} + \)\(16\!\cdots\!35\)\( T^{39} + \)\(97\!\cdots\!01\)\( T^{40} \))
$97$ (\( 1 + 7 T + 97 T^{2} \))(\( 1 + 6 T + 627 T^{2} + 2794 T^{3} + 187541 T^{4} + 617687 T^{5} + 36232727 T^{6} + 85232118 T^{7} + 5099669503 T^{8} + 7471197037 T^{9} + 558828113097 T^{10} + 269576246798 T^{11} + 51033527097079 T^{12} - 31057010685037 T^{13} + 4411201406575250 T^{14} - 5627383992143573 T^{15} + 427886536437799250 T^{16} - 292215413535513133 T^{17} + 46576922276272382167 T^{18} + 23865391303705492238 T^{19} + \)\(47\!\cdots\!29\)\( T^{20} + \)\(62\!\cdots\!73\)\( T^{21} + \)\(41\!\cdots\!39\)\( T^{22} + \)\(66\!\cdots\!98\)\( T^{23} + \)\(27\!\cdots\!59\)\( T^{24} + \)\(45\!\cdots\!63\)\( T^{25} + \)\(13\!\cdots\!73\)\( T^{26} + \)\(19\!\cdots\!54\)\( T^{27} + \)\(42\!\cdots\!79\)\( T^{28} + \)\(39\!\cdots\!14\)\( T^{29} + \)\(63\!\cdots\!93\)\( T^{30} \))(\( 1 + 47 T + 2033 T^{2} + 58572 T^{3} + 1560829 T^{4} + 34106652 T^{5} + 700865910 T^{6} + 12705410076 T^{7} + 219323513178 T^{8} + 3461620579552 T^{9} + 52466012298494 T^{10} + 741981502391271 T^{11} + 10131918166842398 T^{12} + 130709795625996212 T^{13} + 1633986938906440772 T^{14} + 19450993452866578418 T^{15} + \)\(22\!\cdots\!16\)\( T^{16} + \)\(24\!\cdots\!56\)\( T^{17} + \)\(26\!\cdots\!52\)\( T^{18} + \)\(27\!\cdots\!28\)\( T^{19} + \)\(27\!\cdots\!23\)\( T^{20} + \)\(26\!\cdots\!16\)\( T^{21} + \)\(25\!\cdots\!68\)\( T^{22} + \)\(22\!\cdots\!88\)\( T^{23} + \)\(19\!\cdots\!96\)\( T^{24} + \)\(16\!\cdots\!26\)\( T^{25} + \)\(13\!\cdots\!88\)\( T^{26} + \)\(10\!\cdots\!56\)\( T^{27} + \)\(79\!\cdots\!78\)\( T^{28} + \)\(56\!\cdots\!07\)\( T^{29} + \)\(38\!\cdots\!06\)\( T^{30} + \)\(24\!\cdots\!56\)\( T^{31} + \)\(15\!\cdots\!98\)\( T^{32} + \)\(85\!\cdots\!52\)\( T^{33} + \)\(45\!\cdots\!90\)\( T^{34} + \)\(21\!\cdots\!36\)\( T^{35} + \)\(95\!\cdots\!09\)\( T^{36} + \)\(34\!\cdots\!64\)\( T^{37} + \)\(11\!\cdots\!37\)\( T^{38} + \)\(26\!\cdots\!51\)\( T^{39} + \)\(54\!\cdots\!01\)\( T^{40} \))
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