Properties

Label 463.2.a
Level 463
Weight 2
Character orbit a
Rep. character \(\chi_{463}(1,\cdot)\)
Character field \(\Q\)
Dimension 38
Newform subspaces 2
Sturm bound 77
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 463 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 463.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(77\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(463))\).

Total New Old
Modular forms 39 39 0
Cusp forms 38 38 0
Eisenstein series 1 1 0

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(463\)Dim.
\(+\)\(16\)
\(-\)\(22\)

Trace form

\( 38q - q^{2} - 2q^{3} + 37q^{4} - 2q^{5} - 6q^{6} - 8q^{7} - 3q^{8} + 38q^{9} + O(q^{10}) \) \( 38q - q^{2} - 2q^{3} + 37q^{4} - 2q^{5} - 6q^{6} - 8q^{7} - 3q^{8} + 38q^{9} + 6q^{10} - 4q^{11} - 4q^{12} - 4q^{13} - 10q^{15} + 39q^{16} + 10q^{17} - 11q^{18} - 4q^{19} - 8q^{20} - 10q^{21} - 4q^{22} - 16q^{23} - 16q^{24} + 34q^{25} + 16q^{26} - 26q^{27} - 24q^{28} - 2q^{29} - 8q^{30} - 7q^{32} + 10q^{33} + 26q^{34} + 4q^{35} + 37q^{36} - 2q^{38} - 10q^{39} + 34q^{40} - 8q^{41} - 4q^{42} + 8q^{43} - 8q^{44} - 20q^{45} + 2q^{46} - 4q^{47} - 16q^{48} - 2q^{49} - 47q^{50} + 14q^{51} - 40q^{52} - 10q^{53} - 24q^{54} - 34q^{55} - 34q^{56} - 32q^{57} - 34q^{58} + 14q^{59} - 66q^{60} - 6q^{61} + 2q^{62} - 20q^{63} + 33q^{64} + 26q^{65} + 34q^{66} - 8q^{67} + 12q^{68} - 10q^{69} + 40q^{70} + 18q^{71} - 11q^{72} + 10q^{73} - 2q^{74} - 4q^{75} + 6q^{76} - 14q^{77} + 10q^{78} - 14q^{79} - 4q^{80} + 30q^{81} + 46q^{82} + 8q^{83} - 10q^{84} + 24q^{85} - 64q^{86} - 40q^{87} - 28q^{88} - 24q^{89} + 32q^{90} - 38q^{91} - 10q^{92} + 10q^{93} + 2q^{94} + 70q^{95} - 10q^{96} - 20q^{97} + 85q^{98} - 28q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(463))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 463
463.2.a.a \(16\) \(3.697\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-9\) \(-6\) \(-16\) \(-10\) \(+\) \(q+(-1+\beta _{1})q^{2}-\beta _{6}q^{3}+(1-\beta _{1}-\beta _{12}+\cdots)q^{4}+\cdots\)
463.2.a.b \(22\) \(3.697\) None \(8\) \(4\) \(14\) \(2\) \(-\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 9 T + 49 T^{2} + 200 T^{3} + 674 T^{4} + 1967 T^{5} + 5123 T^{6} + 12147 T^{7} + 26591 T^{8} + 54308 T^{9} + 104285 T^{10} + 189419 T^{11} + 326932 T^{12} + 538068 T^{13} + 846573 T^{14} + 1275490 T^{15} + 1842107 T^{16} + 2550980 T^{17} + 3386292 T^{18} + 4304544 T^{19} + 5230912 T^{20} + 6061408 T^{21} + 6674240 T^{22} + 6951424 T^{23} + 6807296 T^{24} + 6219264 T^{25} + 5245952 T^{26} + 4028416 T^{27} + 2760704 T^{28} + 1638400 T^{29} + 802816 T^{30} + 294912 T^{31} + 65536 T^{32} \))
$3$ (\( 1 + 6 T + 38 T^{2} + 157 T^{3} + 622 T^{4} + 2019 T^{5} + 6236 T^{6} + 17033 T^{7} + 44461 T^{8} + 106447 T^{9} + 244932 T^{10} + 527645 T^{11} + 1097885 T^{12} + 2164934 T^{13} + 4138331 T^{14} + 7547384 T^{15} + 13368629 T^{16} + 22642152 T^{17} + 37244979 T^{18} + 58453218 T^{19} + 88928685 T^{20} + 128217735 T^{21} + 178555428 T^{22} + 232799589 T^{23} + 291708621 T^{24} + 335260539 T^{25} + 368229564 T^{26} + 357659793 T^{27} + 330556302 T^{28} + 250308711 T^{29} + 181752822 T^{30} + 86093442 T^{31} + 43046721 T^{32} \))
$5$ (\( 1 + 16 T + 164 T^{2} + 1239 T^{3} + 7670 T^{4} + 40397 T^{5} + 187090 T^{6} + 775240 T^{7} + 2920138 T^{8} + 10101040 T^{9} + 32397429 T^{10} + 97014264 T^{11} + 273046335 T^{12} + 725849048 T^{13} + 1830809103 T^{14} + 4394362365 T^{15} + 10059427855 T^{16} + 21971811825 T^{17} + 45770227575 T^{18} + 90731131000 T^{19} + 170653959375 T^{20} + 303169575000 T^{21} + 506209828125 T^{22} + 789143750000 T^{23} + 1140678906250 T^{24} + 1514140625000 T^{25} + 1827050781250 T^{26} + 1972509765625 T^{27} + 1872558593750 T^{28} + 1512451171875 T^{29} + 1000976562500 T^{30} + 488281250000 T^{31} + 152587890625 T^{32} \))
$7$ (\( 1 + 10 T + 108 T^{2} + 745 T^{3} + 4963 T^{4} + 26786 T^{5} + 137741 T^{6} + 620747 T^{7} + 2670428 T^{8} + 10436232 T^{9} + 39087274 T^{10} + 135539122 T^{11} + 451920564 T^{12} + 1409599289 T^{13} + 4236678129 T^{14} + 11974387100 T^{15} + 32644350937 T^{16} + 83820709700 T^{17} + 207597228321 T^{18} + 483492556127 T^{19} + 1085061274164 T^{20} + 2278006023454 T^{21} + 4598578698826 T^{22} + 8594685809976 T^{23} + 15394486004828 T^{24} + 25049380484429 T^{25} + 38908423272509 T^{26} + 52964674137998 T^{27} + 68694308378563 T^{28} + 72182312753215 T^{29} + 73248091867692 T^{30} + 47475615099430 T^{31} + 33232930569601 T^{32} \))
$11$ (\( 1 + 7 T + 107 T^{2} + 684 T^{3} + 6021 T^{4} + 34156 T^{5} + 227346 T^{6} + 1149645 T^{7} + 6360149 T^{8} + 28929830 T^{9} + 139124283 T^{10} + 573439464 T^{11} + 2455528474 T^{12} + 9218924625 T^{13} + 35632659197 T^{14} + 122148896422 T^{15} + 429428989519 T^{16} + 1343637860642 T^{17} + 4311551762837 T^{18} + 12270388675875 T^{19} + 35951392387834 T^{20} + 92352999116664 T^{21} + 246467153915763 T^{22} + 563760544210930 T^{23} + 1363354422633269 T^{24} + 2710802773219695 T^{25} + 5896769733338946 T^{26} + 9745105421389316 T^{27} + 18896477256237141 T^{28} + 23613535106448804 T^{29} + 40633232193406787 T^{30} + 29240737185909557 T^{31} + 45949729863572161 T^{32} \))
$13$ (\( 1 + 17 T + 237 T^{2} + 2367 T^{3} + 20729 T^{4} + 154510 T^{5} + 1049037 T^{6} + 6413808 T^{7} + 36512877 T^{8} + 192264502 T^{9} + 955580662 T^{10} + 4460191066 T^{11} + 19824977553 T^{12} + 83518590378 T^{13} + 336953487467 T^{14} + 1295062269759 T^{15} + 4780082457321 T^{16} + 16835809506867 T^{17} + 56945139381923 T^{18} + 183490343060466 T^{19} + 566221183891233 T^{20} + 1656037721468338 T^{21} + 4612405339567558 T^{22} + 12064312372243534 T^{23} + 29784675480994317 T^{24} + 68015222914542384 T^{25} + 144618658713799413 T^{26} + 276906702482656870 T^{27} + 482946006503908649 T^{28} + 716905377303862851 T^{29} + 933158203410731493 T^{30} + 870160181239542869 T^{31} + 665416609183179841 T^{32} \))
$17$ (\( 1 + 46 T + 1151 T^{2} + 20397 T^{3} + 284658 T^{4} + 3309258 T^{5} + 33172431 T^{6} + 293391976 T^{7} + 2326951570 T^{8} + 16747024957 T^{9} + 110343784205 T^{10} + 670086643491 T^{11} + 3769622619104 T^{12} + 19719983088147 T^{13} + 96197488990675 T^{14} + 438419761649477 T^{15} + 1868801178296941 T^{16} + 7453135948041109 T^{17} + 27801074318305075 T^{18} + 96884276912066211 T^{19} + 314842650770185184 T^{20} + 951427211367200787 T^{21} + 2663430704969297645 T^{22} + 6871951997553262061 T^{23} + 16232249729274132370 T^{24} + 34792731415098788072 T^{25} + 66875418559065321519 T^{26} + \)\(11\!\cdots\!14\)\( T^{27} + \)\(16\!\cdots\!38\)\( T^{28} + \)\(20\!\cdots\!89\)\( T^{29} + \)\(19\!\cdots\!79\)\( T^{30} + \)\(13\!\cdots\!78\)\( T^{31} + 48661191875666868481 T^{32} \))
$19$ (\( 1 + 7 T + 149 T^{2} + 833 T^{3} + 11013 T^{4} + 54362 T^{5} + 565199 T^{6} + 2549279 T^{7} + 22493928 T^{8} + 93983219 T^{9} + 732594627 T^{10} + 2855938059 T^{11} + 20119079204 T^{12} + 73337692764 T^{13} + 473935514548 T^{14} + 1615169043294 T^{15} + 9666346397599 T^{16} + 30688211822586 T^{17} + 171090720751828 T^{18} + 503023234668276 T^{19} + 2621938520944484 T^{20} + 7071585371951841 T^{21} + 34465559643081387 T^{22} + 84008943404347841 T^{23} + 382027044227715048 T^{24} + 822620971506351341 T^{25} + 3465272517842867399 T^{26} + 6332643454224981278 T^{27} + 24375237203675631093 T^{28} + 35030135224060130147 T^{29} + \)\(11\!\cdots\!29\)\( T^{30} + \)\(10\!\cdots\!93\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( 1 + 16 T + 303 T^{2} + 3595 T^{3} + 42304 T^{4} + 405596 T^{5} + 3723831 T^{6} + 30275593 T^{7} + 234448264 T^{8} + 1664439512 T^{9} + 11272177113 T^{10} + 71171919755 T^{11} + 429776245453 T^{12} + 2440758842489 T^{13} + 13283573765680 T^{14} + 68284348271540 T^{15} + 336739221398203 T^{16} + 1570540010245420 T^{17} + 7027010522044720 T^{18} + 29696712836563663 T^{19} + 120269014303812973 T^{20} + 458086887511655965 T^{21} + 1668686759888408457 T^{22} + 5667126005449861864 T^{23} + 18359874551260002184 T^{24} + 54530964909320572559 T^{25} + \)\(15\!\cdots\!19\)\( T^{26} + \)\(38\!\cdots\!92\)\( T^{27} + \)\(92\!\cdots\!84\)\( T^{28} + \)\(18\!\cdots\!85\)\( T^{29} + \)\(35\!\cdots\!27\)\( T^{30} + \)\(42\!\cdots\!12\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 + 10 T + 276 T^{2} + 2506 T^{3} + 38483 T^{4} + 318119 T^{5} + 3583236 T^{6} + 27074292 T^{7} + 248515668 T^{8} + 1724640721 T^{9} + 13601257526 T^{10} + 87045543356 T^{11} + 608135828802 T^{12} + 3596744512050 T^{13} + 22690195762264 T^{14} + 123967572880569 T^{15} + 714780732953961 T^{16} + 3595059613536501 T^{17} + 19082454636064024 T^{18} + 87721001904387450 T^{19} + 430122917130907362 T^{20} + 1785404109560876044 T^{21} + 8090345171391563846 T^{22} + 29749839114714578789 T^{23} + \)\(12\!\cdots\!48\)\( T^{24} + \)\(39\!\cdots\!48\)\( T^{25} + \)\(15\!\cdots\!36\)\( T^{26} + \)\(38\!\cdots\!51\)\( T^{27} + \)\(13\!\cdots\!03\)\( T^{28} + \)\(25\!\cdots\!34\)\( T^{29} + \)\(82\!\cdots\!56\)\( T^{30} + \)\(86\!\cdots\!90\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( 1 - 4 T + 208 T^{2} - 402 T^{3} + 21035 T^{4} - 5002 T^{5} + 1464898 T^{6} + 1859786 T^{7} + 80577673 T^{8} + 207963817 T^{9} + 3740504823 T^{10} + 13505828628 T^{11} + 151820063282 T^{12} + 647956692455 T^{13} + 5497468604771 T^{14} + 24658600065472 T^{15} + 179363028516845 T^{16} + 764416602029632 T^{17} + 5283067329184931 T^{18} + 19303277824926905 T^{19} + 140209016662255922 T^{20} + 386660407171134828 T^{21} + 3319711799210753463 T^{22} + 5721628246171621687 T^{23} + 68723975119551654793 T^{24} + 49172039139705676406 T^{25} + \)\(12\!\cdots\!98\)\( T^{26} - \)\(12\!\cdots\!62\)\( T^{27} + \)\(16\!\cdots\!35\)\( T^{28} - \)\(98\!\cdots\!82\)\( T^{29} + \)\(15\!\cdots\!68\)\( T^{30} - \)\(93\!\cdots\!04\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( 1 - T + 315 T^{2} - 266 T^{3} + 49695 T^{4} - 39922 T^{5} + 5253723 T^{6} - 4454833 T^{7} + 419263532 T^{8} - 392426463 T^{9} + 26912926240 T^{10} - 27526754943 T^{11} + 1443038514749 T^{12} - 1547428463923 T^{13} + 66136785770301 T^{14} - 70347593281825 T^{15} + 2624391452680307 T^{16} - 2602860951427525 T^{17} + 90541259719542069 T^{18} - 78381893983091719 T^{19} + 2704486505840500589 T^{20} - 1908814111116929451 T^{21} + 69051205597437072160 T^{22} - 37253780769253770579 T^{23} + \)\(14\!\cdots\!72\)\( T^{24} - \)\(57\!\cdots\!41\)\( T^{25} + \)\(25\!\cdots\!27\)\( T^{26} - \)\(71\!\cdots\!86\)\( T^{27} + \)\(32\!\cdots\!95\)\( T^{28} - \)\(64\!\cdots\!02\)\( T^{29} + \)\(28\!\cdots\!35\)\( T^{30} - \)\(33\!\cdots\!93\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( 1 + 56 T + 1821 T^{2} + 42599 T^{3} + 792881 T^{4} + 12353483 T^{5} + 166738992 T^{6} + 1995626570 T^{7} + 21563575796 T^{8} + 213268347833 T^{9} + 1952126518936 T^{10} + 16682396648893 T^{11} + 134040309819760 T^{12} + 1018182003769421 T^{13} + 7343170706327340 T^{14} + 50434843594345640 T^{15} + 330510603282251459 T^{16} + 2067828587368171240 T^{17} + 12343869957336258540 T^{18} + 70174121881792264741 T^{19} + \)\(37\!\cdots\!60\)\( T^{20} + \)\(19\!\cdots\!93\)\( T^{21} + \)\(92\!\cdots\!76\)\( T^{22} + \)\(41\!\cdots\!73\)\( T^{23} + \)\(17\!\cdots\!16\)\( T^{24} + \)\(65\!\cdots\!70\)\( T^{25} + \)\(22\!\cdots\!92\)\( T^{26} + \)\(67\!\cdots\!03\)\( T^{27} + \)\(17\!\cdots\!61\)\( T^{28} + \)\(39\!\cdots\!79\)\( T^{29} + \)\(69\!\cdots\!81\)\( T^{30} + \)\(87\!\cdots\!56\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( 1 - 10 T + 317 T^{2} - 3392 T^{3} + 58797 T^{4} - 587059 T^{5} + 7740970 T^{6} - 70756949 T^{7} + 777598776 T^{8} - 6535550864 T^{9} + 62639050236 T^{10} - 484186860828 T^{11} + 4153498450862 T^{12} - 29582233798419 T^{13} + 230251967712969 T^{14} - 1511284667503638 T^{15} + 10767173032188079 T^{16} - 64985240702656434 T^{17} + 425735888301279681 T^{18} - 2351994662610899433 T^{19} + 14199984657305456462 T^{20} - 71179556531381970804 T^{21} + \)\(39\!\cdots\!64\)\( T^{22} - \)\(17\!\cdots\!48\)\( T^{23} + \)\(90\!\cdots\!76\)\( T^{24} - \)\(35\!\cdots\!07\)\( T^{25} + \)\(16\!\cdots\!30\)\( T^{26} - \)\(54\!\cdots\!13\)\( T^{27} + \)\(23\!\cdots\!97\)\( T^{28} - \)\(58\!\cdots\!56\)\( T^{29} + \)\(23\!\cdots\!33\)\( T^{30} - \)\(31\!\cdots\!70\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 + 28 T + 781 T^{2} + 14394 T^{3} + 247532 T^{4} + 3512208 T^{5} + 46557360 T^{6} + 547656096 T^{7} + 6072112428 T^{8} + 61727302831 T^{9} + 596125491007 T^{10} + 5368886829679 T^{11} + 46188564582229 T^{12} + 374027742363357 T^{13} + 2902692501347872 T^{14} + 21304403348597243 T^{15} + 150077864520214339 T^{16} + 1001306957384070421 T^{17} + 6412047735477449248 T^{18} + 38832682295390813811 T^{19} + \)\(22\!\cdots\!49\)\( T^{20} + \)\(12\!\cdots\!53\)\( T^{21} + \)\(64\!\cdots\!03\)\( T^{22} + \)\(31\!\cdots\!53\)\( T^{23} + \)\(14\!\cdots\!08\)\( T^{24} + \)\(61\!\cdots\!32\)\( T^{25} + \)\(24\!\cdots\!40\)\( T^{26} + \)\(86\!\cdots\!24\)\( T^{27} + \)\(28\!\cdots\!12\)\( T^{28} + \)\(78\!\cdots\!38\)\( T^{29} + \)\(20\!\cdots\!89\)\( T^{30} + \)\(33\!\cdots\!04\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 + 36 T + 1104 T^{2} + 24333 T^{3} + 470904 T^{4} + 7781117 T^{5} + 116471852 T^{6} + 1569181791 T^{7} + 19528251952 T^{8} + 223998573096 T^{9} + 2401699591191 T^{10} + 24038856196335 T^{11} + 226540120222210 T^{12} + 2007260871152187 T^{13} + 16816241090176206 T^{14} + 132949422822654628 T^{15} + 995770895571066497 T^{16} + 7046319409600695284 T^{17} + 47236821222304962654 T^{18} + \)\(29\!\cdots\!99\)\( T^{19} + \)\(17\!\cdots\!10\)\( T^{20} + \)\(10\!\cdots\!55\)\( T^{21} + \)\(53\!\cdots\!39\)\( T^{22} + \)\(26\!\cdots\!52\)\( T^{23} + \)\(12\!\cdots\!72\)\( T^{24} + \)\(51\!\cdots\!03\)\( T^{25} + \)\(20\!\cdots\!48\)\( T^{26} + \)\(72\!\cdots\!49\)\( T^{27} + \)\(23\!\cdots\!64\)\( T^{28} + \)\(63\!\cdots\!09\)\( T^{29} + \)\(15\!\cdots\!76\)\( T^{30} + \)\(26\!\cdots\!52\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 + 7 T + 544 T^{2} + 2542 T^{3} + 132785 T^{4} + 296016 T^{5} + 19584685 T^{6} - 10767304 T^{7} + 2031780748 T^{8} - 7472565097 T^{9} + 170534528036 T^{10} - 1108040414148 T^{11} + 13136085226539 T^{12} - 102290485541641 T^{13} + 953567413858504 T^{14} - 7193355393370478 T^{15} + 61091551379316177 T^{16} - 424407968208858202 T^{17} + 3319368167641452424 T^{18} - 21008317630056686939 T^{19} + \)\(15\!\cdots\!79\)\( T^{20} - \)\(79\!\cdots\!52\)\( T^{21} + \)\(71\!\cdots\!76\)\( T^{22} - \)\(18\!\cdots\!43\)\( T^{23} + \)\(29\!\cdots\!08\)\( T^{24} - \)\(93\!\cdots\!56\)\( T^{25} + \)\(10\!\cdots\!85\)\( T^{26} + \)\(89\!\cdots\!44\)\( T^{27} + \)\(23\!\cdots\!85\)\( T^{28} + \)\(26\!\cdots\!18\)\( T^{29} + \)\(33\!\cdots\!84\)\( T^{30} + \)\(25\!\cdots\!93\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 - 4 T + 627 T^{2} - 2623 T^{3} + 195248 T^{4} - 850174 T^{5} + 40104585 T^{6} - 179631513 T^{7} + 6082906119 T^{8} - 27566901240 T^{9} + 722612323528 T^{10} - 3247228912999 T^{11} + 69574713810719 T^{12} - 302864858532674 T^{13} + 5540759986565961 T^{14} - 22755711267014294 T^{15} + 368973778826370119 T^{16} - 1388098387287871934 T^{17} + 20617167910011940881 T^{18} - 68744568454604877194 T^{19} + \)\(96\!\cdots\!79\)\( T^{20} - \)\(27\!\cdots\!99\)\( T^{21} + \)\(37\!\cdots\!08\)\( T^{22} - \)\(86\!\cdots\!40\)\( T^{23} + \)\(11\!\cdots\!39\)\( T^{24} - \)\(21\!\cdots\!33\)\( T^{25} + \)\(28\!\cdots\!85\)\( T^{26} - \)\(36\!\cdots\!14\)\( T^{27} + \)\(51\!\cdots\!08\)\( T^{28} - \)\(42\!\cdots\!63\)\( T^{29} + \)\(61\!\cdots\!07\)\( T^{30} - \)\(24\!\cdots\!04\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( 1 + 774 T^{2} - 585 T^{3} + 292119 T^{4} - 387016 T^{5} + 71599029 T^{6} - 123414785 T^{7} + 12774184641 T^{8} - 25161450889 T^{9} + 1759776888375 T^{10} - 3662398506453 T^{11} + 193614208939517 T^{12} - 402107730720888 T^{13} + 17349345699357120 T^{14} - 34271812954815351 T^{15} + 1279323145242601319 T^{16} - 2296211467972628517 T^{17} + 77881212844414111680 T^{18} - \)\(12\!\cdots\!44\)\( T^{19} + \)\(39\!\cdots\!57\)\( T^{20} - \)\(49\!\cdots\!71\)\( T^{21} + \)\(15\!\cdots\!75\)\( T^{22} - \)\(15\!\cdots\!47\)\( T^{23} + \)\(51\!\cdots\!81\)\( T^{24} - \)\(33\!\cdots\!95\)\( T^{25} + \)\(13\!\cdots\!21\)\( T^{26} - \)\(47\!\cdots\!28\)\( T^{27} + \)\(23\!\cdots\!59\)\( T^{28} - \)\(32\!\cdots\!95\)\( T^{29} + \)\(28\!\cdots\!46\)\( T^{30} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( 1 - 11 T + 571 T^{2} - 6455 T^{3} + 175229 T^{4} - 1941133 T^{5} + 37395936 T^{6} - 395768757 T^{7} + 6107127995 T^{8} - 60987869635 T^{9} + 801200207454 T^{10} - 7497626301149 T^{11} + 86825896929286 T^{12} - 757521163788183 T^{13} + 7899548066381555 T^{14} - 63930096603886802 T^{15} + 608641644755527409 T^{16} - 4539036858875962942 T^{17} + 39821621802629418755 T^{18} - \)\(27\!\cdots\!13\)\( T^{19} + \)\(22\!\cdots\!66\)\( T^{20} - \)\(13\!\cdots\!99\)\( T^{21} + \)\(10\!\cdots\!34\)\( T^{22} - \)\(55\!\cdots\!85\)\( T^{23} + \)\(39\!\cdots\!95\)\( T^{24} - \)\(18\!\cdots\!67\)\( T^{25} + \)\(12\!\cdots\!36\)\( T^{26} - \)\(44\!\cdots\!43\)\( T^{27} + \)\(28\!\cdots\!89\)\( T^{28} - \)\(75\!\cdots\!05\)\( T^{29} + \)\(47\!\cdots\!51\)\( T^{30} - \)\(64\!\cdots\!61\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \))
$73$ (\( 1 + 20 T + 630 T^{2} + 10055 T^{3} + 193974 T^{4} + 2657239 T^{5} + 40038495 T^{6} + 489163896 T^{7} + 6282825098 T^{8} + 69913877541 T^{9} + 797216239082 T^{10} + 8178439452445 T^{11} + 84689201091229 T^{12} + 806617204342058 T^{13} + 7688541042638728 T^{14} + 68195336738049393 T^{15} + 602942485839269345 T^{16} + 4978259581877605689 T^{17} + 40972235216221781512 T^{18} + \)\(31\!\cdots\!86\)\( T^{19} + \)\(24\!\cdots\!89\)\( T^{20} + \)\(16\!\cdots\!85\)\( T^{21} + \)\(12\!\cdots\!98\)\( T^{22} + \)\(77\!\cdots\!77\)\( T^{23} + \)\(50\!\cdots\!38\)\( T^{24} + \)\(28\!\cdots\!48\)\( T^{25} + \)\(17\!\cdots\!55\)\( T^{26} + \)\(83\!\cdots\!03\)\( T^{27} + \)\(44\!\cdots\!54\)\( T^{28} + \)\(16\!\cdots\!15\)\( T^{29} + \)\(76\!\cdots\!70\)\( T^{30} + \)\(17\!\cdots\!40\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 - 15 T + 785 T^{2} - 9282 T^{3} + 281001 T^{4} - 2746948 T^{5} + 63749536 T^{6} - 536086902 T^{7} + 10687053025 T^{8} - 80054319933 T^{9} + 1443818057444 T^{10} - 9878750625776 T^{11} + 164300464812096 T^{12} - 1040313748157692 T^{13} + 16057714863236720 T^{14} - 94619498305386588 T^{15} + 1360338039721430693 T^{16} - 7474940366125540452 T^{17} + \)\(10\!\cdots\!20\)\( T^{18} - \)\(51\!\cdots\!88\)\( T^{19} + \)\(63\!\cdots\!76\)\( T^{20} - \)\(30\!\cdots\!24\)\( T^{21} + \)\(35\!\cdots\!24\)\( T^{22} - \)\(15\!\cdots\!47\)\( T^{23} + \)\(16\!\cdots\!25\)\( T^{24} - \)\(64\!\cdots\!38\)\( T^{25} + \)\(60\!\cdots\!36\)\( T^{26} - \)\(20\!\cdots\!92\)\( T^{27} + \)\(16\!\cdots\!41\)\( T^{28} - \)\(43\!\cdots\!98\)\( T^{29} + \)\(28\!\cdots\!85\)\( T^{30} - \)\(43\!\cdots\!85\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( 1 + 41 T + 1613 T^{2} + 42290 T^{3} + 1025255 T^{4} + 20483117 T^{5} + 380303363 T^{6} + 6245017551 T^{7} + 96075029593 T^{8} + 1350488764246 T^{9} + 17903876948450 T^{10} + 220685000981518 T^{11} + 2578004136191155 T^{12} + 28276424426216196 T^{13} + 294859290249995375 T^{14} + 2901744972509969845 T^{15} + 27190892974329355177 T^{16} + \)\(24\!\cdots\!35\)\( T^{17} + \)\(20\!\cdots\!75\)\( T^{18} + \)\(16\!\cdots\!52\)\( T^{19} + \)\(12\!\cdots\!55\)\( T^{20} + \)\(86\!\cdots\!74\)\( T^{21} + \)\(58\!\cdots\!50\)\( T^{22} + \)\(36\!\cdots\!42\)\( T^{23} + \)\(21\!\cdots\!13\)\( T^{24} + \)\(11\!\cdots\!53\)\( T^{25} + \)\(59\!\cdots\!87\)\( T^{26} + \)\(26\!\cdots\!39\)\( T^{27} + \)\(10\!\cdots\!55\)\( T^{28} + \)\(37\!\cdots\!70\)\( T^{29} + \)\(11\!\cdots\!77\)\( T^{30} + \)\(25\!\cdots\!87\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( 1 + 73 T + 3290 T^{2} + 107721 T^{3} + 2848502 T^{4} + 63472129 T^{5} + 1234024197 T^{6} + 21361297789 T^{7} + 335181490762 T^{8} + 4826433661288 T^{9} + 64494947891656 T^{10} + 806278826706712 T^{11} + 9497892222602545 T^{12} + 105948935547091924 T^{13} + 1123818761160252427 T^{14} + 11362000424219221134 T^{15} + \)\(10\!\cdots\!61\)\( T^{16} + \)\(10\!\cdots\!26\)\( T^{17} + \)\(89\!\cdots\!67\)\( T^{18} + \)\(74\!\cdots\!56\)\( T^{19} + \)\(59\!\cdots\!45\)\( T^{20} + \)\(45\!\cdots\!88\)\( T^{21} + \)\(32\!\cdots\!16\)\( T^{22} + \)\(21\!\cdots\!52\)\( T^{23} + \)\(13\!\cdots\!22\)\( T^{24} + \)\(74\!\cdots\!01\)\( T^{25} + \)\(38\!\cdots\!97\)\( T^{26} + \)\(17\!\cdots\!81\)\( T^{27} + \)\(70\!\cdots\!42\)\( T^{28} + \)\(23\!\cdots\!49\)\( T^{29} + \)\(64\!\cdots\!90\)\( T^{30} + \)\(12\!\cdots\!77\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( 1 + 45 T + 1645 T^{2} + 43033 T^{3} + 978031 T^{4} + 18808529 T^{5} + 325694752 T^{6} + 5042387216 T^{7} + 71762457887 T^{8} + 938639537904 T^{9} + 11456948370021 T^{10} + 131120631238020 T^{11} + 1423105699407328 T^{12} + 14809600679966425 T^{13} + 149453406616929061 T^{14} + 1485348566606872334 T^{15} + 14635205209769473043 T^{16} + \)\(14\!\cdots\!98\)\( T^{17} + \)\(14\!\cdots\!49\)\( T^{18} + \)\(13\!\cdots\!25\)\( T^{19} + \)\(12\!\cdots\!68\)\( T^{20} + \)\(11\!\cdots\!40\)\( T^{21} + \)\(95\!\cdots\!09\)\( T^{22} + \)\(75\!\cdots\!52\)\( T^{23} + \)\(56\!\cdots\!07\)\( T^{24} + \)\(38\!\cdots\!72\)\( T^{25} + \)\(24\!\cdots\!48\)\( T^{26} + \)\(13\!\cdots\!37\)\( T^{27} + \)\(67\!\cdots\!71\)\( T^{28} + \)\(28\!\cdots\!41\)\( T^{29} + \)\(10\!\cdots\!05\)\( T^{30} + \)\(28\!\cdots\!85\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))
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