Properties

Label 349.2.a
Level 349
Weight 2
Character orbit a
Rep. character \(\chi_{349}(1,\cdot)\)
Character field \(\Q\)
Dimension 28
Newform subspaces 2
Sturm bound 58
Trace bound 1

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Defining parameters

Level: \( N \) = \( 349 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 349.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(58\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(349))\).

Total New Old
Modular forms 29 29 0
Cusp forms 28 28 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.

\(349\)Dim.
\(+\)\(11\)
\(-\)\(17\)

Trace form

\( 28q + 24q^{4} - 4q^{5} - 8q^{6} - 2q^{7} - 6q^{8} + 28q^{9} + O(q^{10}) \) \( 28q + 24q^{4} - 4q^{5} - 8q^{6} - 2q^{7} - 6q^{8} + 28q^{9} - 4q^{10} + 8q^{11} + 4q^{12} - 10q^{13} + 4q^{14} + 28q^{16} - 10q^{19} - 2q^{20} - 18q^{21} + 16q^{22} - 16q^{23} - 24q^{24} + 24q^{25} - 10q^{26} - 6q^{27} - 20q^{28} - 8q^{29} - 18q^{30} - 12q^{31} + 18q^{32} + 6q^{33} - 10q^{34} - 12q^{35} + 6q^{36} - 8q^{37} - 6q^{38} + 8q^{39} - 18q^{40} - 12q^{41} - 4q^{43} + 32q^{44} - 24q^{45} + 22q^{46} + 20q^{47} - 2q^{48} + 20q^{49} - 8q^{50} + 10q^{51} - 12q^{52} + 24q^{53} - 8q^{54} + 8q^{55} + 22q^{56} - 10q^{57} - 36q^{58} + 22q^{59} - 6q^{60} - 30q^{61} + 24q^{62} + 26q^{63} - 12q^{64} - 6q^{65} - 36q^{66} - 14q^{67} - 36q^{68} + 2q^{69} - 50q^{70} + 40q^{71} - 32q^{72} - 26q^{73} + 56q^{74} + 56q^{75} - 48q^{76} - 32q^{77} - 34q^{78} + 18q^{79} - 10q^{80} + 12q^{81} - 10q^{82} + 34q^{83} - 46q^{84} - 30q^{85} - 4q^{86} - 2q^{87} + 50q^{88} + 4q^{89} - 26q^{90} - 24q^{91} + 18q^{92} - 18q^{93} - 42q^{94} - 18q^{95} - 36q^{96} + 12q^{97} + 40q^{98} + 38q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(349))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 349
349.2.a.a \(11\) \(2.787\) \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(-5\) \(-6\) \(-9\) \(-3\) \(+\) \(q-\beta _{1}q^{2}+(-1+\beta _{2}+\beta _{4}+\beta _{6}+\beta _{7}+\cdots)q^{3}+\cdots\)
349.2.a.b \(17\) \(2.787\) \(\mathbb{Q}[x]/(x^{17} - \cdots)\) None \(5\) \(6\) \(5\) \(1\) \(-\) \(q+\beta _{1}q^{2}+\beta _{12}q^{3}+(1+\beta _{2})q^{4}+\beta _{7}q^{5}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 5 T + 21 T^{2} + 65 T^{3} + 178 T^{4} + 420 T^{5} + 906 T^{6} + 1763 T^{7} + 3196 T^{8} + 5335 T^{9} + 8367 T^{10} + 12192 T^{11} + 16734 T^{12} + 21340 T^{13} + 25568 T^{14} + 28208 T^{15} + 28992 T^{16} + 26880 T^{17} + 22784 T^{18} + 16640 T^{19} + 10752 T^{20} + 5120 T^{21} + 2048 T^{22} \))(\( 1 - 5 T + 20 T^{2} - 58 T^{3} + 150 T^{4} - 336 T^{5} + 710 T^{6} - 1393 T^{7} + 2639 T^{8} - 4759 T^{9} + 8319 T^{10} - 13988 T^{11} + 22865 T^{12} - 36276 T^{13} + 56133 T^{14} - 84661 T^{15} + 124473 T^{16} - 178392 T^{17} + 248946 T^{18} - 338644 T^{19} + 449064 T^{20} - 580416 T^{21} + 731680 T^{22} - 895232 T^{23} + 1064832 T^{24} - 1218304 T^{25} + 1351168 T^{26} - 1426432 T^{27} + 1454080 T^{28} - 1376256 T^{29} + 1228800 T^{30} - 950272 T^{31} + 655360 T^{32} - 327680 T^{33} + 131072 T^{34} \))
$3$ (\( 1 + 6 T + 33 T^{2} + 125 T^{3} + 431 T^{4} + 1245 T^{5} + 3329 T^{6} + 7963 T^{7} + 17806 T^{8} + 36586 T^{9} + 70528 T^{10} + 126077 T^{11} + 211584 T^{12} + 329274 T^{13} + 480762 T^{14} + 645003 T^{15} + 808947 T^{16} + 907605 T^{17} + 942597 T^{18} + 820125 T^{19} + 649539 T^{20} + 354294 T^{21} + 177147 T^{22} \))(\( 1 - 6 T + 31 T^{2} - 111 T^{3} + 372 T^{4} - 1031 T^{5} + 2738 T^{6} - 6372 T^{7} + 14516 T^{8} - 29903 T^{9} + 61480 T^{10} - 116559 T^{11} + 224910 T^{12} - 405155 T^{13} + 752784 T^{14} - 1312046 T^{15} + 2375014 T^{16} - 4034818 T^{17} + 7125042 T^{18} - 11808414 T^{19} + 20325168 T^{20} - 32817555 T^{21} + 54653130 T^{22} - 84971511 T^{23} + 134456760 T^{24} - 196193583 T^{25} + 285718428 T^{26} - 376260228 T^{27} + 485028486 T^{28} - 547915671 T^{29} + 593088156 T^{30} - 530909559 T^{31} + 444816117 T^{32} - 258280326 T^{33} + 129140163 T^{34} \))
$5$ (\( 1 + 9 T + 63 T^{2} + 321 T^{3} + 1440 T^{4} + 5535 T^{5} + 19431 T^{6} + 61267 T^{7} + 179102 T^{8} + 479850 T^{9} + 1201056 T^{10} + 2775966 T^{11} + 6005280 T^{12} + 11996250 T^{13} + 22387750 T^{14} + 38291875 T^{15} + 60721875 T^{16} + 86484375 T^{17} + 112500000 T^{18} + 125390625 T^{19} + 123046875 T^{20} + 87890625 T^{21} + 48828125 T^{22} \))(\( 1 - 5 T + 48 T^{2} - 186 T^{3} + 1044 T^{4} - 3339 T^{5} + 14253 T^{6} - 39282 T^{7} + 142051 T^{8} - 349952 T^{9} + 1141627 T^{10} - 2597236 T^{11} + 7929703 T^{12} - 17031398 T^{13} + 49308966 T^{14} - 100628638 T^{15} + 275804904 T^{16} - 532910478 T^{17} + 1379024520 T^{18} - 2515715950 T^{19} + 6163620750 T^{20} - 10644623750 T^{21} + 24780321875 T^{22} - 40581812500 T^{23} + 89189609375 T^{24} - 136700000000 T^{25} + 277443359375 T^{26} - 383613281250 T^{27} + 695947265625 T^{28} - 815185546875 T^{29} + 1274414062500 T^{30} - 1135253906250 T^{31} + 1464843750000 T^{32} - 762939453125 T^{33} + 762939453125 T^{34} \))
$7$ (\( 1 + 3 T + 36 T^{2} + 73 T^{3} + 599 T^{4} + 886 T^{5} + 6706 T^{6} + 7260 T^{7} + 58319 T^{8} + 44843 T^{9} + 431862 T^{10} + 269606 T^{11} + 3023034 T^{12} + 2197307 T^{13} + 20003417 T^{14} + 17431260 T^{15} + 112707742 T^{16} + 104237014 T^{17} + 493302257 T^{18} + 420830473 T^{19} + 1452729852 T^{20} + 847425747 T^{21} + 1977326743 T^{22} \))(\( 1 - T + 57 T^{2} - 4 T^{3} + 1600 T^{4} + 1112 T^{5} + 31168 T^{6} + 38728 T^{7} + 481569 T^{8} + 753299 T^{9} + 6192483 T^{10} + 10621139 T^{11} + 67559581 T^{12} + 118360004 T^{13} + 632316767 T^{14} + 1082756247 T^{15} + 5109754045 T^{16} + 8273159638 T^{17} + 35768278315 T^{18} + 53055056103 T^{19} + 216884651081 T^{20} + 284182369604 T^{21} + 1135473877867 T^{22} + 1249566382211 T^{23} + 5099776027269 T^{24} + 4342618828499 T^{25} + 19433046169383 T^{26} + 10939701443272 T^{27} + 61629319925824 T^{28} + 15391511367512 T^{29} + 155022416651200 T^{30} - 2712892291396 T^{31} + 270611006066751 T^{32} - 33232930569601 T^{33} + 232630513987207 T^{34} \))
$11$ (\( 1 + 31 T + 531 T^{2} + 6413 T^{3} + 60422 T^{4} + 467688 T^{5} + 3069703 T^{6} + 17433416 T^{7} + 86846022 T^{8} + 382944129 T^{9} + 1503494100 T^{10} + 5273070684 T^{11} + 16538435100 T^{12} + 46336239609 T^{13} + 115592055282 T^{14} + 255242643656 T^{15} + 494378737853 T^{16} + 828537820968 T^{17} + 1177453846162 T^{18} + 1374683503853 T^{19} + 1252070223921 T^{20} + 804060162631 T^{21} + 285311670611 T^{22} \))(\( 1 - 39 T + 824 T^{2} - 12326 T^{3} + 145216 T^{4} - 1425628 T^{5} + 12073177 T^{6} - 90258106 T^{7} + 605453969 T^{8} - 3688064149 T^{9} + 20584700049 T^{10} - 105999554279 T^{11} + 506259989221 T^{12} - 2251727597066 T^{13} + 9355439535330 T^{14} - 36391023577503 T^{15} + 132731023630338 T^{16} - 454346180348312 T^{17} + 1460041259933718 T^{18} - 4403313852877863 T^{19} + 12452090021524230 T^{20} - 32967543748643306 T^{21} + 81533677524031271 T^{22} - 187784676378059519 T^{23} + 401137569838571379 T^{24} - 790569304035857269 T^{25} + 1427628788210335579 T^{26} - 2341062819004065706 T^{27} + 3444618299452301147 T^{28} - 4474231369848005788 T^{29} + 5013250166693084096 T^{30} - 4680796448747028566 T^{31} + 3442052491598496424 T^{32} - 1792039464679314279 T^{33} + 505447028499293771 T^{34} \))
$13$ (\( 1 + 4 T + 70 T^{2} + 205 T^{3} + 2502 T^{4} + 6277 T^{5} + 64760 T^{6} + 145370 T^{7} + 1294441 T^{8} + 2599706 T^{9} + 20644875 T^{10} + 37409596 T^{11} + 268383375 T^{12} + 439350314 T^{13} + 2843886877 T^{14} + 4151912570 T^{15} + 24044934680 T^{16} + 30297880093 T^{17} + 156996789534 T^{18} + 167224797805 T^{19} + 742314956110 T^{20} + 551433967396 T^{21} + 1792160394037 T^{22} \))(\( 1 + 6 T + 134 T^{2} + 719 T^{3} + 8529 T^{4} + 41841 T^{5} + 346292 T^{6} + 1584845 T^{7} + 10165280 T^{8} + 44248787 T^{9} + 232349055 T^{10} + 978858157 T^{11} + 4364141841 T^{12} + 18004351591 T^{13} + 70379139094 T^{14} + 284510622359 T^{15} + 1008816511090 T^{16} + 3938739150750 T^{17} + 13114614644170 T^{18} + 48082295178671 T^{19} + 154622968589518 T^{20} + 514222285790551 T^{21} + 1620375316570413 T^{22} + 4724761361931013 T^{23} + 14579558627601435 T^{24} + 36095094922885427 T^{25} + 107797705386369440 T^{26} + 218484341514428405 T^{27} + 620610807171860804 T^{28} + 974815179609727521 T^{29} + 2583221784125325837 T^{30} + 2830973621317788791 T^{31} + 6858909663888161438 T^{32} + 3992499655099079046 T^{33} + 8650415919381337933 T^{34} \))
$17$ (\( 1 + T + 84 T^{2} + 63 T^{3} + 3698 T^{4} + 1926 T^{5} + 110912 T^{6} + 30427 T^{7} + 2571395 T^{8} + 181737 T^{9} + 49922464 T^{10} - 452741 T^{11} + 848681888 T^{12} + 52521993 T^{13} + 12633263635 T^{14} + 2541293467 T^{15} + 157479179584 T^{16} + 46488957894 T^{17} + 1517432412754 T^{18} + 439472718783 T^{19} + 9961381625748 T^{20} + 2015993900449 T^{21} + 34271896307633 T^{22} \))(\( 1 - T + 181 T^{2} - 74 T^{3} + 15981 T^{4} + 1750 T^{5} + 925095 T^{6} + 506473 T^{7} + 39650408 T^{8} + 35557136 T^{9} + 1342145378 T^{10} + 1543199155 T^{11} + 37267915520 T^{12} + 48471134510 T^{13} + 868880331098 T^{14} + 1170434743288 T^{15} + 17243063174476 T^{16} + 22309383674893 T^{17} + 293132073966092 T^{18} + 338255640810232 T^{19} + 4268809066684474 T^{20} + 4048357625409710 T^{21} + 52915110726480640 T^{22} + 37249076084554195 T^{23} + 550734153381603394 T^{24} + 248037956032648976 T^{25} + 4702057686959660776 T^{26} + 1021046478742106377 T^{27} + 31704759914709750135 T^{28} + 1019588915152081750 T^{29} + \)\(15\!\cdots\!97\)\( T^{30} - 12459959165395668746 T^{31} + \)\(51\!\cdots\!33\)\( T^{32} - 48661191875666868481 T^{33} + \)\(82\!\cdots\!77\)\( T^{34} \))
$19$ (\( 1 + 17 T + 226 T^{2} + 2068 T^{3} + 16001 T^{4} + 99620 T^{5} + 537641 T^{6} + 2402987 T^{7} + 9332226 T^{8} + 29918227 T^{9} + 91775769 T^{10} + 313894131 T^{11} + 1743739611 T^{12} + 10800479947 T^{13} + 64009738134 T^{14} + 313159668827 T^{15} + 1331252342459 T^{16} + 4686710665220 T^{17} + 14302841695739 T^{18} + 35122008368788 T^{19} + 72927419698054 T^{20} + 104228126382617 T^{21} + 116490258898219 T^{22} \))(\( 1 - 7 T + 200 T^{2} - 1146 T^{3} + 18872 T^{4} - 90709 T^{5} + 1136805 T^{6} - 4635185 T^{7} + 49627502 T^{8} - 172083168 T^{9} + 1687498764 T^{10} - 4973821868 T^{11} + 47013959799 T^{12} - 118494834509 T^{13} + 1118825958150 T^{14} - 2473678070857 T^{15} + 23541332034957 T^{16} - 48055653048750 T^{17} + 447285308664183 T^{18} - 892997783579377 T^{19} + 7674027246950850 T^{20} - 15442365328047389 T^{21} + 116411218844344101 T^{22} - 233997831717125708 T^{23} + 1508407454737030596 T^{24} - 2922585332022993888 T^{25} + 16014184366902718058 T^{26} - 28418626352165328185 T^{27} + \)\(13\!\cdots\!95\)\( T^{28} - \)\(20\!\cdots\!49\)\( T^{29} + \)\(79\!\cdots\!48\)\( T^{30} - \)\(91\!\cdots\!66\)\( T^{31} + \)\(30\!\cdots\!00\)\( T^{32} - \)\(20\!\cdots\!67\)\( T^{33} + \)\(54\!\cdots\!39\)\( T^{34} \))
$23$ (\( 1 + 24 T + 456 T^{2} + 6060 T^{3} + 69390 T^{4} + 658653 T^{5} + 5584079 T^{6} + 41423274 T^{7} + 279404880 T^{8} + 1686913894 T^{9} + 9341224951 T^{10} + 46717149029 T^{11} + 214848173873 T^{12} + 892377449926 T^{13} + 3399519174960 T^{14} + 11591930419434 T^{15} + 35941047783097 T^{16} + 97504282397517 T^{17} + 236260837767330 T^{18} + 474564570802860 T^{19} + 821325613627128 T^{20} + 994236269127576 T^{21} + 952809757913927 T^{22} \))(\( 1 - 8 T + 214 T^{2} - 1356 T^{3} + 21853 T^{4} - 117935 T^{5} + 1489111 T^{6} - 7132216 T^{7} + 77294283 T^{8} - 335817683 T^{9} + 3255641878 T^{10} - 12996669057 T^{11} + 115230456614 T^{12} - 426173466055 T^{13} + 3498529197768 T^{14} - 12050936012779 T^{15} + 92199776371748 T^{16} - 296636825810398 T^{17} + 2120594856550204 T^{18} - 6374945150760091 T^{19} + 42566604749243256 T^{20} - 119260808914297255 T^{21} + 741662742814322602 T^{22} - 1923973457891786673 T^{23} + 11084892312533269466 T^{24} - 26298213630512523923 T^{25} + \)\(13\!\cdots\!29\)\( T^{26} - \)\(29\!\cdots\!84\)\( T^{27} + \)\(14\!\cdots\!97\)\( T^{28} - \)\(25\!\cdots\!35\)\( T^{29} + \)\(11\!\cdots\!99\)\( T^{30} - \)\(15\!\cdots\!04\)\( T^{31} + \)\(57\!\cdots\!98\)\( T^{32} - \)\(49\!\cdots\!88\)\( T^{33} + \)\(14\!\cdots\!03\)\( T^{34} \))
$29$ (\( 1 + 17 T + 312 T^{2} + 3279 T^{3} + 35609 T^{4} + 278714 T^{5} + 2291527 T^{6} + 14722036 T^{7} + 102293462 T^{8} + 575540002 T^{9} + 3582194714 T^{10} + 18273808685 T^{11} + 103883646706 T^{12} + 484029141682 T^{13} + 2494835244718 T^{14} + 10412616344116 T^{15} + 47001851734523 T^{16} + 165785587089194 T^{17} + 614250845487181 T^{18} + 1640307988099119 T^{19} + 4526229544471128 T^{20} + 7152022966103417 T^{21} + 12200509765705829 T^{22} \))(\( 1 - 9 T + 229 T^{2} - 1922 T^{3} + 28036 T^{4} - 216572 T^{5} + 2382347 T^{6} - 17006184 T^{7} + 156138921 T^{8} - 1035931284 T^{9} + 8342507847 T^{10} - 51641810851 T^{11} + 374856739205 T^{12} - 2169918937795 T^{13} + 14427576553368 T^{14} - 78145530035404 T^{15} + 480787887338763 T^{16} - 2433554823998601 T^{17} + 13942848732824127 T^{18} - 65720390759774764 T^{19} + 351874164560092152 T^{20} - 1534742436242585395 T^{21} + 7688742431487896545 T^{22} - 30717753432845656171 T^{23} + \)\(14\!\cdots\!23\)\( T^{24} - \)\(51\!\cdots\!24\)\( T^{25} + \)\(22\!\cdots\!49\)\( T^{26} - \)\(71\!\cdots\!84\)\( T^{27} + \)\(29\!\cdots\!63\)\( T^{28} - \)\(76\!\cdots\!52\)\( T^{29} + \)\(28\!\cdots\!04\)\( T^{30} - \)\(57\!\cdots\!82\)\( T^{31} + \)\(19\!\cdots\!21\)\( T^{32} - \)\(22\!\cdots\!89\)\( T^{33} + \)\(72\!\cdots\!09\)\( T^{34} \))
$31$ (\( 1 + 10 T + 209 T^{2} + 1511 T^{3} + 18966 T^{4} + 113255 T^{5} + 1109705 T^{6} + 5858249 T^{7} + 49248292 T^{8} + 238704003 T^{9} + 1803457161 T^{10} + 8093669384 T^{11} + 55907171991 T^{12} + 229394546883 T^{13} + 1467155866972 T^{14} + 5410215974729 T^{15} + 31769912010455 T^{16} + 100514229391655 T^{17} + 521804239229226 T^{18} + 1288718357573351 T^{19} + 5525881031580239 T^{20} + 8196282869808010 T^{21} + 25408476896404831 T^{22} \))(\( 1 + 2 T + 287 T^{2} + 403 T^{3} + 40895 T^{4} + 31579 T^{5} + 3847612 T^{6} + 383772 T^{7} + 268562331 T^{8} - 160101624 T^{9} + 14850629033 T^{10} - 18874433910 T^{11} + 680551134050 T^{12} - 1255068400403 T^{13} + 26763039995205 T^{14} - 59024724946809 T^{15} + 927423118377644 T^{16} - 2091461908060580 T^{17} + 28750116669706964 T^{18} - 56722760673883449 T^{19} + 797297724497152155 T^{20} - 1159082024208578963 T^{21} + 19483601179938691550 T^{22} - 16751129571916222710 T^{23} + \)\(40\!\cdots\!63\)\( T^{24} - \)\(13\!\cdots\!84\)\( T^{25} + \)\(71\!\cdots\!01\)\( T^{26} + \)\(31\!\cdots\!72\)\( T^{27} + \)\(97\!\cdots\!72\)\( T^{28} + \)\(24\!\cdots\!19\)\( T^{29} + \)\(99\!\cdots\!45\)\( T^{30} + \)\(30\!\cdots\!63\)\( T^{31} + \)\(67\!\cdots\!37\)\( T^{32} + \)\(14\!\cdots\!62\)\( T^{33} + \)\(22\!\cdots\!11\)\( T^{34} \))
$37$ (\( 1 + T + 148 T^{2} + 671 T^{3} + 11897 T^{4} + 84685 T^{5} + 812968 T^{6} + 5833528 T^{7} + 47735175 T^{8} + 297515886 T^{9} + 2211783831 T^{10} + 12276997715 T^{11} + 81836001747 T^{12} + 407299247934 T^{13} + 2417929819275 T^{14} + 10932970670008 T^{15} + 56374418034376 T^{16} + 217278540946165 T^{17} + 1129404542251301 T^{18} + 2356873713580991 T^{19} + 19234337489671396 T^{20} + 4808584372417849 T^{21} + 177917621779460413 T^{22} \))(\( 1 + 7 T + 305 T^{2} + 1934 T^{3} + 46856 T^{4} + 280627 T^{5} + 4937326 T^{6} + 28342799 T^{7} + 402762781 T^{8} + 2212475869 T^{9} + 26937621598 T^{10} + 140823919532 T^{11} + 1523045258623 T^{12} + 7545771331120 T^{13} + 74109107202793 T^{14} + 346646303087377 T^{15} + 3136352355428545 T^{16} + 13773533176896479 T^{17} + 116045037150856165 T^{18} + 474558788926619113 T^{19} + 3753848607143073829 T^{20} + 14141990343703190320 T^{21} + \)\(10\!\cdots\!11\)\( T^{22} + \)\(36\!\cdots\!88\)\( T^{23} + \)\(25\!\cdots\!34\)\( T^{24} + \)\(77\!\cdots\!49\)\( T^{25} + \)\(52\!\cdots\!37\)\( T^{26} + \)\(13\!\cdots\!51\)\( T^{27} + \)\(87\!\cdots\!38\)\( T^{28} + \)\(18\!\cdots\!87\)\( T^{29} + \)\(11\!\cdots\!32\)\( T^{30} + \)\(17\!\cdots\!26\)\( T^{31} + \)\(10\!\cdots\!65\)\( T^{32} + \)\(86\!\cdots\!87\)\( T^{33} + \)\(45\!\cdots\!17\)\( T^{34} \))
$41$ (\( 1 + 15 T + 464 T^{2} + 5462 T^{3} + 94566 T^{4} + 913668 T^{5} + 11425824 T^{6} + 92868482 T^{7} + 920707511 T^{8} + 6370944600 T^{9} + 52310068897 T^{10} + 308742520868 T^{11} + 2144712824777 T^{12} + 10709557872600 T^{13} + 63456082365631 T^{14} + 262424134564802 T^{15} + 1323752561934624 T^{16} + 4340018241665988 T^{17} + 18417132663830646 T^{18} + 43613661601458902 T^{19} + 151905217558797904 T^{20} + 201339889652286015 T^{21} + 550329031716248441 T^{22} \))(\( 1 - 3 T + 377 T^{2} - 697 T^{3} + 70387 T^{4} - 68829 T^{5} + 8813051 T^{6} - 2865581 T^{7} + 836114500 T^{8} + 122049873 T^{9} + 64017892219 T^{10} + 29326561973 T^{11} + 4101630769056 T^{12} + 2600570289327 T^{13} + 224489668362810 T^{14} + 157563661771252 T^{15} + 10609886000694774 T^{16} + 7296132373049024 T^{17} + 435005326028485734 T^{18} + 264864515437474612 T^{19} + 15472052433233228010 T^{20} + 7348590101338952847 T^{21} + \)\(47\!\cdots\!56\)\( T^{22} + \)\(13\!\cdots\!93\)\( T^{23} + \)\(12\!\cdots\!39\)\( T^{24} + \)\(97\!\cdots\!33\)\( T^{25} + \)\(27\!\cdots\!00\)\( T^{26} - \)\(38\!\cdots\!81\)\( T^{27} + \)\(48\!\cdots\!91\)\( T^{28} - \)\(15\!\cdots\!49\)\( T^{29} + \)\(65\!\cdots\!27\)\( T^{30} - \)\(26\!\cdots\!17\)\( T^{31} + \)\(58\!\cdots\!77\)\( T^{32} - \)\(19\!\cdots\!23\)\( T^{33} + \)\(26\!\cdots\!81\)\( T^{34} \))
$43$ (\( 1 + 5 T + 397 T^{2} + 1835 T^{3} + 74045 T^{4} + 314401 T^{5} + 8578800 T^{6} + 33083766 T^{7} + 686577253 T^{8} + 2365470392 T^{9} + 39868889363 T^{10} + 120003944224 T^{11} + 1714362242609 T^{12} + 4373754754808 T^{13} + 54587697654271 T^{14} + 113106812284566 T^{15} + 1261156030808400 T^{16} + 1987442863968649 T^{17} + 20126809059417815 T^{18} + 21447847509397835 T^{19} + 199529266938926671 T^{20} + 108057411566421245 T^{21} + 929293739471222707 T^{22} \))(\( 1 - T + 274 T^{2} - 506 T^{3} + 41073 T^{4} - 105403 T^{5} + 4428027 T^{6} - 13622481 T^{7} + 382361662 T^{8} - 1291092373 T^{9} + 27825938921 T^{10} - 97315131261 T^{11} + 1749574466768 T^{12} - 6106641140091 T^{13} + 96301713459316 T^{14} - 326974036157220 T^{15} + 4675783501084611 T^{16} - 15109533829277132 T^{17} + 201058690546638273 T^{18} - 604574992854699780 T^{19} + 7656660332009837212 T^{20} - 20877390836384250891 T^{21} + \)\(25\!\cdots\!24\)\( T^{22} - \)\(61\!\cdots\!89\)\( T^{23} + \)\(75\!\cdots\!47\)\( T^{24} - \)\(15\!\cdots\!73\)\( T^{25} + \)\(19\!\cdots\!66\)\( T^{26} - \)\(29\!\cdots\!69\)\( T^{27} + \)\(41\!\cdots\!89\)\( T^{28} - \)\(42\!\cdots\!03\)\( T^{29} + \)\(70\!\cdots\!39\)\( T^{30} - \)\(37\!\cdots\!94\)\( T^{31} + \)\(87\!\cdots\!18\)\( T^{32} - \)\(13\!\cdots\!01\)\( T^{33} + \)\(58\!\cdots\!43\)\( T^{34} \))
$47$ (\( 1 - 4 T + 133 T^{2} - 1165 T^{3} + 15951 T^{4} - 122352 T^{5} + 1476135 T^{6} - 10679138 T^{7} + 99342939 T^{8} - 714139231 T^{9} + 5858599654 T^{10} - 35823804252 T^{11} + 275354183738 T^{12} - 1577533561279 T^{13} + 10314081955797 T^{14} - 52110786794978 T^{15} + 338544191907945 T^{16} - 1318858553933808 T^{17} + 8081145394505313 T^{18} - 27740148960951565 T^{19} + 148844352922668011 T^{20} - 210396528943320196 T^{21} + 2472159215084012303 T^{22} \))(\( 1 - 16 T + 478 T^{2} - 6401 T^{3} + 108759 T^{4} - 1244375 T^{5} + 15666296 T^{6} - 155988945 T^{7} + 1603678559 T^{8} - 14141466804 T^{9} + 124640758901 T^{10} - 991844899290 T^{11} + 7750878282645 T^{12} - 57017435210467 T^{13} + 409119465021707 T^{14} - 2874074187472665 T^{15} + 19742513247228737 T^{16} - 136648596986997038 T^{17} + 927898122619750639 T^{18} - 6348829880127116985 T^{19} + 42476010216948685861 T^{20} - \)\(27\!\cdots\!27\)\( T^{21} + \)\(17\!\cdots\!15\)\( T^{22} - \)\(10\!\cdots\!10\)\( T^{23} + \)\(63\!\cdots\!63\)\( T^{24} - \)\(33\!\cdots\!44\)\( T^{25} + \)\(17\!\cdots\!53\)\( T^{26} - \)\(82\!\cdots\!05\)\( T^{27} + \)\(38\!\cdots\!88\)\( T^{28} - \)\(14\!\cdots\!75\)\( T^{29} + \)\(59\!\cdots\!93\)\( T^{30} - \)\(16\!\cdots\!69\)\( T^{31} + \)\(57\!\cdots\!54\)\( T^{32} - \)\(90\!\cdots\!36\)\( T^{33} + \)\(26\!\cdots\!87\)\( T^{34} \))
$53$ (\( 1 + 3 T + 131 T^{2} - 113 T^{3} + 11900 T^{4} - 35620 T^{5} + 973068 T^{6} - 2905877 T^{7} + 73723127 T^{8} - 217817831 T^{9} + 4472676520 T^{10} - 12277722876 T^{11} + 237051855560 T^{12} - 611850287279 T^{13} + 10975677978379 T^{14} - 22928767256837 T^{15} + 406932651982524 T^{16} - 789494543414980 T^{17} + 13979062564060300 T^{18} - 7035345016483793 T^{19} + 432269030526079423 T^{20} + 524662411096539147 T^{21} + 9269035929372191597 T^{22} \))(\( 1 - 27 T + 839 T^{2} - 14937 T^{3} + 274361 T^{4} - 3729565 T^{5} + 51590897 T^{6} - 581068266 T^{7} + 6709416320 T^{8} - 66254874956 T^{9} + 676970234685 T^{10} - 6069922084934 T^{11} + 56487811317102 T^{12} - 466722036331991 T^{13} + 4001081004882664 T^{14} - 30646165151498655 T^{15} + 243830697121406879 T^{16} - 1740631728849083882 T^{17} + 12923026947434564587 T^{18} - 86085077910559721895 T^{19} + \)\(59\!\cdots\!28\)\( T^{20} - \)\(36\!\cdots\!71\)\( T^{21} + \)\(23\!\cdots\!86\)\( T^{22} - \)\(13\!\cdots\!86\)\( T^{23} + \)\(79\!\cdots\!45\)\( T^{24} - \)\(41\!\cdots\!16\)\( T^{25} + \)\(22\!\cdots\!60\)\( T^{26} - \)\(10\!\cdots\!34\)\( T^{27} + \)\(47\!\cdots\!09\)\( T^{28} - \)\(18\!\cdots\!65\)\( T^{29} + \)\(71\!\cdots\!53\)\( T^{30} - \)\(20\!\cdots\!53\)\( T^{31} + \)\(61\!\cdots\!23\)\( T^{32} - \)\(10\!\cdots\!67\)\( T^{33} + \)\(20\!\cdots\!13\)\( T^{34} \))
$59$ (\( 1 + 52 T + 1591 T^{2} + 35655 T^{3} + 646035 T^{4} + 9906888 T^{5} + 132455588 T^{6} + 1571731651 T^{7} + 16762500494 T^{8} + 161934343201 T^{9} + 1424678579860 T^{10} + 11445471606214 T^{11} + 84056036211740 T^{12} + 563693448682681 T^{13} + 3442665588957226 T^{14} + 19045239810293011 T^{15} + 94695718399532812 T^{16} + 417877822561619208 T^{17} + 1607755961995042665 T^{18} + 5235239252782065255 T^{19} + 13782826347480007949 T^{20} + 26578071171633352852 T^{21} + 30155888444737842659 T^{22} \))(\( 1 - 74 T + 3278 T^{2} - 105185 T^{3} + 2707019 T^{4} - 58538343 T^{5} + 1099455345 T^{6} - 18308964734 T^{7} + 274712321571 T^{8} - 3756066692537 T^{9} + 47227583970761 T^{10} - 549773449142659 T^{11} + 5957583227005193 T^{12} - 60338789248142446 T^{13} + 572999171911577244 T^{14} - 5113312798421402983 T^{15} + 42948622142212805403 T^{16} - \)\(33\!\cdots\!80\)\( T^{17} + \)\(25\!\cdots\!77\)\( T^{18} - \)\(17\!\cdots\!23\)\( T^{19} + \)\(11\!\cdots\!76\)\( T^{20} - \)\(73\!\cdots\!06\)\( T^{21} + \)\(42\!\cdots\!07\)\( T^{22} - \)\(23\!\cdots\!19\)\( T^{23} + \)\(11\!\cdots\!59\)\( T^{24} - \)\(55\!\cdots\!77\)\( T^{25} + \)\(23\!\cdots\!69\)\( T^{26} - \)\(93\!\cdots\!34\)\( T^{27} + \)\(33\!\cdots\!55\)\( T^{28} - \)\(10\!\cdots\!83\)\( T^{29} + \)\(28\!\cdots\!01\)\( T^{30} - \)\(65\!\cdots\!85\)\( T^{31} + \)\(11\!\cdots\!22\)\( T^{32} - \)\(15\!\cdots\!34\)\( T^{33} + \)\(12\!\cdots\!19\)\( T^{34} \))
$61$ (\( 1 + 305 T^{2} - 1022 T^{3} + 46959 T^{4} - 238047 T^{5} + 5591294 T^{6} - 27036499 T^{7} + 525969946 T^{8} - 2368610576 T^{9} + 38193372810 T^{10} - 167452201806 T^{11} + 2329795741410 T^{12} - 8813599953296 T^{13} + 119385184313026 T^{14} - 374343066350659 T^{15} + 4722386230203494 T^{16} - 12264270555512967 T^{17} + 147580060836710139 T^{18} - 195924873883221182 T^{19} + 3566714558314413005 T^{20} + 43513917611435838661 T^{22} \))(\( 1 + 30 T + 987 T^{2} + 19590 T^{3} + 384216 T^{4} + 5814631 T^{5} + 85281087 T^{6} + 1051015089 T^{7} + 12526179632 T^{8} + 130454460599 T^{9} + 1319655959857 T^{10} + 11913884249779 T^{11} + 105669259627098 T^{12} + 850172531474345 T^{13} + 6887787888345491 T^{14} + 51800172134294910 T^{15} + 408570998958673159 T^{16} + 3076437485416293174 T^{17} + 24922830936479062699 T^{18} + \)\(19\!\cdots\!10\)\( T^{19} + \)\(15\!\cdots\!71\)\( T^{20} + \)\(11\!\cdots\!45\)\( T^{21} + \)\(89\!\cdots\!98\)\( T^{22} + \)\(61\!\cdots\!19\)\( T^{23} + \)\(41\!\cdots\!97\)\( T^{24} + \)\(25\!\cdots\!19\)\( T^{25} + \)\(14\!\cdots\!12\)\( T^{26} + \)\(74\!\cdots\!89\)\( T^{27} + \)\(37\!\cdots\!07\)\( T^{28} + \)\(15\!\cdots\!51\)\( T^{29} + \)\(62\!\cdots\!96\)\( T^{30} + \)\(19\!\cdots\!90\)\( T^{31} + \)\(59\!\cdots\!87\)\( T^{32} + \)\(11\!\cdots\!30\)\( T^{33} + \)\(22\!\cdots\!21\)\( T^{34} \))
$67$ (\( 1 + 23 T + 810 T^{2} + 13243 T^{3} + 269718 T^{4} + 3466456 T^{5} + 52096518 T^{6} + 553620189 T^{7} + 6706250203 T^{8} + 60455149013 T^{9} + 614420394552 T^{10} + 4744114263579 T^{11} + 41166166434984 T^{12} + 271383163919357 T^{13} + 2016991929804889 T^{14} + 11156067416581869 T^{15} + 70336816939077426 T^{16} + 313570001620023064 T^{17} + 1634683012764508914 T^{18} + 5377554253882596763 T^{19} + 22037292860998907070 T^{20} + 41925269504690513327 T^{21} + \)\(12\!\cdots\!83\)\( T^{22} \))(\( 1 - 9 T + 524 T^{2} - 4135 T^{3} + 130677 T^{4} - 926965 T^{5} + 21015536 T^{6} - 138573578 T^{7} + 2514501640 T^{8} - 16059520836 T^{9} + 247230885612 T^{10} - 1588804246137 T^{11} + 21527250878740 T^{12} - 141223694409929 T^{13} + 1725886193997133 T^{14} - 11345250222512998 T^{15} + 127372418975845143 T^{16} - 810122938319325226 T^{17} + 8533952071381624581 T^{18} - 50928828248860848022 T^{19} + \)\(51\!\cdots\!79\)\( T^{20} - \)\(28\!\cdots\!09\)\( T^{21} + \)\(29\!\cdots\!80\)\( T^{22} - \)\(14\!\cdots\!53\)\( T^{23} + \)\(14\!\cdots\!76\)\( T^{24} - \)\(65\!\cdots\!76\)\( T^{25} + \)\(68\!\cdots\!80\)\( T^{26} - \)\(25\!\cdots\!22\)\( T^{27} + \)\(25\!\cdots\!88\)\( T^{28} - \)\(75\!\cdots\!65\)\( T^{29} + \)\(71\!\cdots\!99\)\( T^{30} - \)\(15\!\cdots\!15\)\( T^{31} + \)\(12\!\cdots\!32\)\( T^{32} - \)\(14\!\cdots\!29\)\( T^{33} + \)\(11\!\cdots\!27\)\( T^{34} \))
$71$ (\( 1 + 30 T + 879 T^{2} + 16593 T^{3} + 295747 T^{4} + 4228076 T^{5} + 57278317 T^{6} + 671008140 T^{7} + 7489732329 T^{8} + 74677531487 T^{9} + 711427669644 T^{10} + 6130590425082 T^{11} + 50511364544724 T^{12} + 376449436225967 T^{13} + 2680657587604719 T^{14} + 17051444802083340 T^{15} + 103343220707282267 T^{16} + 541617736039565996 T^{17} + 2689854501483663077 T^{18} + 10714988343960912273 T^{19} + 40300832131516698249 T^{20} + 97657306530296436030 T^{21} + \)\(23\!\cdots\!71\)\( T^{22} \))(\( 1 - 70 T + 3028 T^{2} - 96081 T^{3} + 2476957 T^{4} - 54154321 T^{5} + 1037295504 T^{6} - 17744331291 T^{7} + 275375201797 T^{8} - 3918770484624 T^{9} + 51603019810981 T^{10} - 632914403419616 T^{11} + 7270749123649303 T^{12} - 78543817320779319 T^{13} + 800541429573002823 T^{14} - 7715155108302091123 T^{15} + 70425336524578166553 T^{16} - \)\(60\!\cdots\!68\)\( T^{17} + \)\(50\!\cdots\!63\)\( T^{18} - \)\(38\!\cdots\!43\)\( T^{19} + \)\(28\!\cdots\!53\)\( T^{20} - \)\(19\!\cdots\!39\)\( T^{21} + \)\(13\!\cdots\!53\)\( T^{22} - \)\(81\!\cdots\!36\)\( T^{23} + \)\(46\!\cdots\!71\)\( T^{24} - \)\(25\!\cdots\!64\)\( T^{25} + \)\(12\!\cdots\!07\)\( T^{26} - \)\(57\!\cdots\!91\)\( T^{27} + \)\(23\!\cdots\!84\)\( T^{28} - \)\(88\!\cdots\!61\)\( T^{29} + \)\(28\!\cdots\!27\)\( T^{30} - \)\(79\!\cdots\!61\)\( T^{31} + \)\(17\!\cdots\!28\)\( T^{32} - \)\(29\!\cdots\!70\)\( T^{33} + \)\(29\!\cdots\!91\)\( T^{34} \))
$73$ (\( 1 - 12 T + 379 T^{2} - 2866 T^{3} + 64346 T^{4} - 309225 T^{5} + 6811163 T^{6} - 12873028 T^{7} + 506136815 T^{8} + 687792516 T^{9} + 32137115719 T^{10} + 114959140581 T^{11} + 2346009447487 T^{12} + 3665246317764 T^{13} + 196895825360855 T^{14} - 365571351543748 T^{15} + 14120028530592659 T^{16} - 46796326124216025 T^{17} + 710855905109815562 T^{18} - 2311314623368436146 T^{19} + 22312331362433539027 T^{20} - 51571509956442691788 T^{21} + \)\(31\!\cdots\!77\)\( T^{22} \))(\( 1 + 38 T + 1458 T^{2} + 35260 T^{3} + 817804 T^{4} + 14982695 T^{5} + 262885705 T^{6} + 3941266949 T^{7} + 56912490172 T^{8} + 729453582581 T^{9} + 9068872756649 T^{10} + 102315434018594 T^{11} + 1127973827704328 T^{12} + 11448919855388484 T^{13} + 114352984457995388 T^{14} + 1061688680299556166 T^{15} + 9751909910938630702 T^{16} + 83682487854376099853 T^{17} + \)\(71\!\cdots\!46\)\( T^{18} + \)\(56\!\cdots\!14\)\( T^{19} + \)\(44\!\cdots\!96\)\( T^{20} + \)\(32\!\cdots\!44\)\( T^{21} + \)\(23\!\cdots\!04\)\( T^{22} + \)\(15\!\cdots\!66\)\( T^{23} + \)\(10\!\cdots\!53\)\( T^{24} + \)\(58\!\cdots\!61\)\( T^{25} + \)\(33\!\cdots\!36\)\( T^{26} + \)\(16\!\cdots\!01\)\( T^{27} + \)\(82\!\cdots\!85\)\( T^{28} + \)\(34\!\cdots\!95\)\( T^{29} + \)\(13\!\cdots\!32\)\( T^{30} + \)\(43\!\cdots\!40\)\( T^{31} + \)\(12\!\cdots\!06\)\( T^{32} + \)\(24\!\cdots\!18\)\( T^{33} + \)\(47\!\cdots\!53\)\( T^{34} \))
$79$ (\( 1 - 11 T + 561 T^{2} - 4855 T^{3} + 148594 T^{4} - 1083574 T^{5} + 25875251 T^{6} - 165665964 T^{7} + 3338743854 T^{8} - 19013446671 T^{9} + 333875716292 T^{10} - 1694281754028 T^{11} + 26376181587068 T^{12} - 118662920673711 T^{13} + 1646130931032306 T^{14} - 6452702716743084 T^{15} + 79619606665281149 T^{16} - 263403246528712054 T^{17} + 2853585651889310446 T^{18} - 7365563272096353655 T^{19} + 67236745346248876959 T^{20} - \)\(10\!\cdots\!11\)\( T^{21} + \)\(74\!\cdots\!79\)\( T^{22} \))(\( 1 - 7 T + 870 T^{2} - 6104 T^{3} + 356114 T^{4} - 2508516 T^{5} + 91824131 T^{6} - 649332772 T^{7} + 16941292845 T^{8} - 119756651477 T^{9} + 2422941373127 T^{10} - 16945206226005 T^{11} + 285069863498099 T^{12} - 1941697470143002 T^{13} + 28797184714272512 T^{14} - 187982498955122177 T^{15} + 2560256763682626504 T^{16} - 15834890903507516172 T^{17} + \)\(20\!\cdots\!16\)\( T^{18} - \)\(11\!\cdots\!57\)\( T^{19} + \)\(14\!\cdots\!68\)\( T^{20} - \)\(75\!\cdots\!62\)\( T^{21} + \)\(87\!\cdots\!01\)\( T^{22} - \)\(41\!\cdots\!05\)\( T^{23} + \)\(46\!\cdots\!93\)\( T^{24} - \)\(18\!\cdots\!97\)\( T^{25} + \)\(20\!\cdots\!55\)\( T^{26} - \)\(61\!\cdots\!72\)\( T^{27} + \)\(68\!\cdots\!49\)\( T^{28} - \)\(14\!\cdots\!56\)\( T^{29} + \)\(16\!\cdots\!46\)\( T^{30} - \)\(22\!\cdots\!24\)\( T^{31} + \)\(25\!\cdots\!30\)\( T^{32} - \)\(16\!\cdots\!47\)\( T^{33} + \)\(18\!\cdots\!59\)\( T^{34} \))
$83$ (\( 1 + 13 T + 601 T^{2} + 7481 T^{3} + 184200 T^{4} + 2103509 T^{5} + 36953980 T^{6} + 380316811 T^{7} + 5333867435 T^{8} + 48916596181 T^{9} + 578558064924 T^{10} + 4678349738759 T^{11} + 48020319388692 T^{12} + 336986431090909 T^{13} + 3049836059056345 T^{14} + 18049197298134331 T^{15} + 145563229140609140 T^{16} + 687722017845051821 T^{17} + 4998460592289293400 T^{18} + 16849398188632165721 T^{19} + \)\(11\!\cdots\!03\)\( T^{20} + \)\(20\!\cdots\!37\)\( T^{21} + \)\(12\!\cdots\!67\)\( T^{22} \))(\( 1 - 47 T + 1911 T^{2} - 52485 T^{3} + 1293469 T^{4} - 26129798 T^{5} + 486787331 T^{6} - 7952973120 T^{7} + 122126971698 T^{8} - 1701495305079 T^{9} + 22588813266459 T^{10} - 277553764934125 T^{11} + 3280505101331932 T^{12} - 36312725260292464 T^{13} + 388826963320656326 T^{14} - 3923786302587828670 T^{15} + 38403627283756914763 T^{16} - \)\(35\!\cdots\!24\)\( T^{17} + \)\(31\!\cdots\!29\)\( T^{18} - \)\(27\!\cdots\!30\)\( T^{19} + \)\(22\!\cdots\!62\)\( T^{20} - \)\(17\!\cdots\!44\)\( T^{21} + \)\(12\!\cdots\!76\)\( T^{22} - \)\(90\!\cdots\!25\)\( T^{23} + \)\(61\!\cdots\!93\)\( T^{24} - \)\(38\!\cdots\!39\)\( T^{25} + \)\(22\!\cdots\!94\)\( T^{26} - \)\(12\!\cdots\!80\)\( T^{27} + \)\(62\!\cdots\!77\)\( T^{28} - \)\(27\!\cdots\!78\)\( T^{29} + \)\(11\!\cdots\!47\)\( T^{30} - \)\(38\!\cdots\!65\)\( T^{31} + \)\(11\!\cdots\!77\)\( T^{32} - \)\(23\!\cdots\!07\)\( T^{33} + \)\(42\!\cdots\!23\)\( T^{34} \))
$89$ (\( 1 + 19 T + 466 T^{2} + 5953 T^{3} + 93394 T^{4} + 896288 T^{5} + 10921543 T^{6} + 77200192 T^{7} + 815530713 T^{8} + 3671172923 T^{9} + 44897516730 T^{10} + 145517283008 T^{11} + 3995878988970 T^{12} + 29079360723083 T^{13} + 574923871212897 T^{14} + 4843713051710272 T^{15} + 60986545386809807 T^{16} + 445438367312852768 T^{17} + 4130941291233035426 T^{18} + 23434513160344488193 T^{19} + \)\(16\!\cdots\!94\)\( T^{20} + \)\(59\!\cdots\!19\)\( T^{21} + \)\(27\!\cdots\!89\)\( T^{22} \))(\( 1 - 23 T + 1134 T^{2} - 21743 T^{3} + 608451 T^{4} - 10014955 T^{5} + 207797823 T^{6} - 3005884069 T^{7} + 51156014686 T^{8} - 662033156512 T^{9} + 9719318180981 T^{10} - 113964453081701 T^{11} + 1485165050888844 T^{12} - 15909287246357393 T^{13} + 187219420722253150 T^{14} - 1840691161327135335 T^{15} + 19761871087393564594 T^{16} - \)\(17\!\cdots\!90\)\( T^{17} + \)\(17\!\cdots\!66\)\( T^{18} - \)\(14\!\cdots\!35\)\( T^{19} + \)\(13\!\cdots\!50\)\( T^{20} - \)\(99\!\cdots\!13\)\( T^{21} + \)\(82\!\cdots\!56\)\( T^{22} - \)\(56\!\cdots\!61\)\( T^{23} + \)\(42\!\cdots\!49\)\( T^{24} - \)\(26\!\cdots\!72\)\( T^{25} + \)\(17\!\cdots\!74\)\( T^{26} - \)\(93\!\cdots\!69\)\( T^{27} + \)\(57\!\cdots\!47\)\( T^{28} - \)\(24\!\cdots\!55\)\( T^{29} + \)\(13\!\cdots\!19\)\( T^{30} - \)\(42\!\cdots\!63\)\( T^{31} + \)\(19\!\cdots\!66\)\( T^{32} - \)\(35\!\cdots\!03\)\( T^{33} + \)\(13\!\cdots\!29\)\( T^{34} \))
$97$ (\( 1 - 26 T + 836 T^{2} - 14101 T^{3} + 278601 T^{4} - 3720808 T^{5} + 58309271 T^{6} - 670846910 T^{7} + 9018538123 T^{8} - 91787845546 T^{9} + 1089314495233 T^{10} - 9927024051904 T^{11} + 105663506037601 T^{12} - 863631838742314 T^{13} + 8230976244332779 T^{14} - 59389594603371710 T^{15} + 500721550214622647 T^{16} - 3099328899715862632 T^{17} + 22510482853886759913 T^{18} - \)\(11\!\cdots\!61\)\( T^{19} + \)\(63\!\cdots\!12\)\( T^{20} - \)\(19\!\cdots\!74\)\( T^{21} + \)\(71\!\cdots\!53\)\( T^{22} \))(\( 1 + 14 T + 978 T^{2} + 12057 T^{3} + 466690 T^{4} + 5172956 T^{5} + 146327151 T^{6} + 1478856591 T^{7} + 34063734517 T^{8} + 316852322736 T^{9} + 6288008605578 T^{10} + 54193394649113 T^{11} + 958626958505064 T^{12} + 7692144498678444 T^{13} + 123957579198078067 T^{14} + 928735303312769343 T^{15} + 13825209937501111010 T^{16} + 96754109374678229620 T^{17} + \)\(13\!\cdots\!70\)\( T^{18} + \)\(87\!\cdots\!87\)\( T^{19} + \)\(11\!\cdots\!91\)\( T^{20} + \)\(68\!\cdots\!64\)\( T^{21} + \)\(82\!\cdots\!48\)\( T^{22} + \)\(45\!\cdots\!77\)\( T^{23} + \)\(50\!\cdots\!14\)\( T^{24} + \)\(24\!\cdots\!96\)\( T^{25} + \)\(25\!\cdots\!89\)\( T^{26} + \)\(10\!\cdots\!59\)\( T^{27} + \)\(10\!\cdots\!03\)\( T^{28} + \)\(35\!\cdots\!96\)\( T^{29} + \)\(31\!\cdots\!30\)\( T^{30} + \)\(78\!\cdots\!33\)\( T^{31} + \)\(61\!\cdots\!54\)\( T^{32} + \)\(85\!\cdots\!94\)\( T^{33} + \)\(59\!\cdots\!37\)\( T^{34} \))
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