Properties

Label 2002.2.g
Level 2002
Weight 2
Character orbit g
Rep. character \(\chi_{2002}(155,\cdot)\)
Character field \(\Q\)
Dimension 68
Newform subspaces 4
Sturm bound 672
Trace bound 3

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Defining parameters

Level: \( N \) = \( 2002 = 2 \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2002.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 13 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(672\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(2002, [\chi])\).

Total New Old
Modular forms 344 68 276
Cusp forms 328 68 260
Eisenstein series 16 0 16

Trace form

\( 68q - 68q^{4} + 60q^{9} + O(q^{10}) \) \( 68q - 68q^{4} + 60q^{9} + 16q^{10} + 8q^{13} - 8q^{14} + 68q^{16} - 4q^{22} - 16q^{23} - 84q^{25} - 4q^{26} + 24q^{29} + 16q^{30} + 8q^{35} - 60q^{36} - 8q^{38} + 40q^{39} - 16q^{40} - 8q^{42} - 24q^{43} - 68q^{49} - 8q^{52} - 16q^{53} + 8q^{56} + 32q^{61} - 48q^{62} - 68q^{64} + 16q^{65} - 16q^{69} - 16q^{74} + 80q^{75} + 8q^{77} + 32q^{78} - 8q^{79} - 28q^{81} - 80q^{82} - 64q^{87} + 4q^{88} + 48q^{90} + 8q^{91} + 16q^{92} + 16q^{94} - 64q^{95} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(2002, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
2002.2.g.a \(12\) \(15.986\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(-2\) \(0\) \(0\) \(q+\beta _{7}q^{2}+\beta _{10}q^{3}-q^{4}+(-\beta _{3}-\beta _{7}+\cdots)q^{5}+\cdots\)
2002.2.g.b \(18\) \(15.986\) \(\mathbb{Q}[x]/(x^{18} + \cdots)\) None \(0\) \(-2\) \(0\) \(0\) \(q+\beta _{5}q^{2}-\beta _{2}q^{3}-q^{4}-\beta _{9}q^{5}+\beta _{1}q^{6}+\cdots\)
2002.2.g.c \(18\) \(15.986\) \(\mathbb{Q}[x]/(x^{18} + \cdots)\) None \(0\) \(2\) \(0\) \(0\) \(q-\beta _{8}q^{2}-\beta _{3}q^{3}-q^{4}+(\beta _{6}+\beta _{8})q^{5}+\cdots\)
2002.2.g.d \(20\) \(15.986\) \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(2\) \(0\) \(0\) \(q+\beta _{11}q^{2}+\beta _{6}q^{3}-q^{4}+\beta _{14}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(2002, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(2002, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(91, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(143, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(182, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(286, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1001, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T^{2} )^{6} \))(\( ( 1 + T^{2} )^{9} \))(\( ( 1 + T^{2} )^{9} \))(\( ( 1 + T^{2} )^{10} \))
$3$ (\( ( 1 + T + 10 T^{2} + 10 T^{3} + 57 T^{4} + 50 T^{5} + 207 T^{6} + 150 T^{7} + 513 T^{8} + 270 T^{9} + 810 T^{10} + 243 T^{11} + 729 T^{12} )^{2} \))(\( ( 1 + T + 11 T^{2} + 11 T^{3} + 62 T^{4} + 63 T^{5} + 241 T^{6} + 237 T^{7} + 769 T^{8} + 736 T^{9} + 2307 T^{10} + 2133 T^{11} + 6507 T^{12} + 5103 T^{13} + 15066 T^{14} + 8019 T^{15} + 24057 T^{16} + 6561 T^{17} + 19683 T^{18} )^{2} \))(\( ( 1 - T + 9 T^{2} - 9 T^{3} + 40 T^{4} - 57 T^{5} + 139 T^{6} - 293 T^{7} + 445 T^{8} - 1064 T^{9} + 1335 T^{10} - 2637 T^{11} + 3753 T^{12} - 4617 T^{13} + 9720 T^{14} - 6561 T^{15} + 19683 T^{16} - 6561 T^{17} + 19683 T^{18} )^{2} \))(\( ( 1 - T + 8 T^{2} - 8 T^{3} + 51 T^{4} - 52 T^{5} + 243 T^{6} - 224 T^{7} + 954 T^{8} - 859 T^{9} + 3126 T^{10} - 2577 T^{11} + 8586 T^{12} - 6048 T^{13} + 19683 T^{14} - 12636 T^{15} + 37179 T^{16} - 17496 T^{17} + 52488 T^{18} - 19683 T^{19} + 59049 T^{20} )^{2} \))
$5$ (\( 1 - 27 T^{2} + 338 T^{4} - 2638 T^{6} + 15159 T^{8} - 75238 T^{10} + 369011 T^{12} - 1880950 T^{14} + 9474375 T^{16} - 41218750 T^{18} + 132031250 T^{20} - 263671875 T^{22} + 244140625 T^{24} \))(\( 1 - 37 T^{2} + 711 T^{4} - 9531 T^{6} + 100292 T^{8} - 878461 T^{10} + 6617189 T^{12} - 43762883 T^{14} + 257545231 T^{16} - 1358602560 T^{18} + 6438630775 T^{20} - 27351801875 T^{22} + 103393578125 T^{24} - 343148828125 T^{26} + 979414062500 T^{28} - 2326904296875 T^{30} + 4339599609375 T^{32} - 5645751953125 T^{34} + 3814697265625 T^{36} \))(\( 1 - 33 T^{2} + 567 T^{4} - 6751 T^{6} + 61632 T^{8} - 456281 T^{10} + 2861213 T^{12} - 15843775 T^{14} + 81427647 T^{16} - 407988936 T^{18} + 2035691175 T^{20} - 9902359375 T^{22} + 44706453125 T^{24} - 178234765625 T^{26} + 601875000000 T^{28} - 1648193359375 T^{30} + 3460693359375 T^{32} - 5035400390625 T^{34} + 3814697265625 T^{36} \))(\( 1 - 31 T^{2} + 510 T^{4} - 5786 T^{6} + 49681 T^{8} - 334552 T^{10} + 1770915 T^{12} - 7091382 T^{14} + 18621686 T^{16} - 11604585 T^{18} - 109810546 T^{20} - 290114625 T^{22} + 11638553750 T^{24} - 110802843750 T^{26} + 691763671875 T^{28} - 3267109375000 T^{30} + 12129150390625 T^{32} - 35314941406250 T^{34} + 77819824218750 T^{36} - 118255615234375 T^{38} + 95367431640625 T^{40} \))
$7$ (\( ( 1 + T^{2} )^{6} \))(\( ( 1 + T^{2} )^{9} \))(\( ( 1 + T^{2} )^{9} \))(\( ( 1 + T^{2} )^{10} \))
$11$ (\( ( 1 + T^{2} )^{6} \))(\( ( 1 + T^{2} )^{9} \))(\( ( 1 + T^{2} )^{9} \))(\( ( 1 + T^{2} )^{10} \))
$13$ (\( 1 - 8 T + 26 T^{2} - 56 T^{3} - 89 T^{4} + 1056 T^{5} - 3444 T^{6} + 13728 T^{7} - 15041 T^{8} - 123032 T^{9} + 742586 T^{10} - 2970344 T^{11} + 4826809 T^{12} \))(\( 1 + T^{2} - 40 T^{4} - 512 T^{5} - 2096 T^{6} + 8448 T^{7} + 16562 T^{8} + 47616 T^{9} + 215306 T^{10} + 1427712 T^{11} - 4604912 T^{12} - 14623232 T^{13} - 14851720 T^{14} + 62748517 T^{16} + 10604499373 T^{18} \))(\( 1 + 4 T + 39 T^{2} + 216 T^{3} + 962 T^{4} + 4736 T^{5} + 19558 T^{6} + 76008 T^{7} + 286588 T^{8} + 1096568 T^{9} + 3725644 T^{10} + 12845352 T^{11} + 42968926 T^{12} + 135264896 T^{13} + 357183866 T^{14} + 1042590744 T^{15} + 2447192163 T^{16} + 3262922884 T^{17} + 10604499373 T^{18} \))(\( 1 - 4 T - 52 T^{3} + 501 T^{4} - 720 T^{5} - 768 T^{6} - 11536 T^{7} + 51250 T^{8} - 24152 T^{9} - 215808 T^{10} - 313976 T^{11} + 8661250 T^{12} - 25344592 T^{13} - 21934848 T^{14} - 267330960 T^{15} + 2418231309 T^{16} - 3262922884 T^{17} - 42417997492 T^{19} + 137858491849 T^{20} \))
$17$ (\( ( 1 - T + 74 T^{2} - 102 T^{3} + 2637 T^{4} - 3342 T^{5} + 56937 T^{6} - 56814 T^{7} + 762093 T^{8} - 501126 T^{9} + 6180554 T^{10} - 1419857 T^{11} + 24137569 T^{12} )^{2} \))(\( ( 1 + 5 T + 73 T^{2} + 305 T^{3} + 3090 T^{4} + 11683 T^{5} + 91259 T^{6} + 307887 T^{7} + 2015901 T^{8} + 6020784 T^{9} + 34270317 T^{10} + 88979343 T^{11} + 448355467 T^{12} + 975775843 T^{13} + 4387358130 T^{14} + 7361958545 T^{15} + 29954723129 T^{16} + 34878787205 T^{17} + 118587876497 T^{18} )^{2} \))(\( ( 1 - 5 T + 81 T^{2} - 357 T^{3} + 3542 T^{4} - 13003 T^{5} + 100987 T^{6} - 325115 T^{7} + 2165321 T^{8} - 6150912 T^{9} + 36810457 T^{10} - 93958235 T^{11} + 496149131 T^{12} - 1086023563 T^{13} + 5029133494 T^{14} - 8617112133 T^{15} + 33237432513 T^{16} - 34878787205 T^{17} + 118587876497 T^{18} )^{2} \))(\( ( 1 + T + 82 T^{2} - 86 T^{3} + 2783 T^{4} - 9352 T^{5} + 66281 T^{6} - 302034 T^{7} + 1616600 T^{8} - 5600369 T^{9} + 33144346 T^{10} - 95206273 T^{11} + 467197400 T^{12} - 1483893042 T^{13} + 5535855401 T^{14} - 13278502664 T^{15} + 67174854527 T^{16} - 35289125878 T^{17} + 572012110162 T^{18} + 118587876497 T^{19} + 2015993900449 T^{20} )^{2} \))
$19$ (\( 1 - 127 T^{2} + 8346 T^{4} - 369166 T^{6} + 12193067 T^{8} - 317116722 T^{10} + 6670856231 T^{12} - 114479136642 T^{14} + 1589012684507 T^{16} - 17367739705246 T^{18} + 141744817140186 T^{20} - 778645414740727 T^{22} + 2213314919066161 T^{24} \))(\( 1 - 81 T^{2} + 2631 T^{4} - 54687 T^{6} + 1491672 T^{8} - 45716141 T^{10} + 947108013 T^{12} - 16303574759 T^{14} + 389716744659 T^{16} - 8953448560952 T^{18} + 140687744821899 T^{20} - 2124698166167639 T^{22} + 44557530873744453 T^{24} - 776422962664744781 T^{26} + 9145539866906533272 T^{28} - \)\(12\!\cdots\!07\)\( T^{30} + \)\(21\!\cdots\!51\)\( T^{32} - \)\(23\!\cdots\!61\)\( T^{34} + \)\(10\!\cdots\!41\)\( T^{36} \))(\( 1 - 205 T^{2} + 20647 T^{4} - 1371027 T^{6} + 67781548 T^{8} - 2662731505 T^{10} + 86393213845 T^{12} - 2371314046979 T^{14} + 55881374539227 T^{16} - 1139784885089456 T^{18} + 20173176208660947 T^{20} - 309032017916350259 T^{22} + 4064444857759422445 T^{24} - 45222668376424306705 T^{26} + \)\(41\!\cdots\!48\)\( T^{28} - \)\(30\!\cdots\!47\)\( T^{30} + \)\(16\!\cdots\!87\)\( T^{32} - \)\(59\!\cdots\!05\)\( T^{34} + \)\(10\!\cdots\!41\)\( T^{36} \))(\( 1 - 147 T^{2} + 11478 T^{4} - 633162 T^{6} + 27625501 T^{8} - 1009047156 T^{10} + 31826380991 T^{12} - 883319046602 T^{14} + 21838785818602 T^{16} - 484674830155701 T^{18} + 9694453575428902 T^{20} - 174967613686208061 T^{22} + 2846052406666031242 T^{24} - 41556522751471146362 T^{26} + \)\(54\!\cdots\!31\)\( T^{28} - \)\(61\!\cdots\!56\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} - \)\(50\!\cdots\!02\)\( T^{34} + \)\(33\!\cdots\!18\)\( T^{36} - \)\(15\!\cdots\!27\)\( T^{38} + \)\(37\!\cdots\!01\)\( T^{40} \))
$23$ (\( ( 1 - 12 T + 164 T^{2} - 1274 T^{3} + 10067 T^{4} - 56450 T^{5} + 313956 T^{6} - 1298350 T^{7} + 5325443 T^{8} - 15500758 T^{9} + 45893924 T^{10} - 77236116 T^{11} + 148035889 T^{12} )^{2} \))(\( ( 1 + 4 T + 141 T^{2} + 590 T^{3} + 9326 T^{4} + 40520 T^{5} + 389394 T^{6} + 1693506 T^{7} + 11695272 T^{8} + 47150984 T^{9} + 268991256 T^{10} + 895864674 T^{11} + 4737756798 T^{12} + 11339157320 T^{13} + 60025334818 T^{14} + 87341174510 T^{15} + 480080388027 T^{16} + 313243941124 T^{17} + 1801152661463 T^{18} )^{2} \))(\( ( 1 + 12 T + 185 T^{2} + 1606 T^{3} + 14858 T^{4} + 101464 T^{5} + 709030 T^{6} + 4000394 T^{7} + 22851180 T^{8} + 109058520 T^{9} + 525577140 T^{10} + 2116208426 T^{11} + 8626768010 T^{12} + 28393787224 T^{13} + 95631184294 T^{14} + 237745637734 T^{15} + 629892707695 T^{16} + 939731823372 T^{17} + 1801152661463 T^{18} )^{2} \))(\( ( 1 + 4 T + 108 T^{2} + 350 T^{3} + 5385 T^{4} + 13638 T^{5} + 164724 T^{6} + 308214 T^{7} + 3733590 T^{8} + 5230698 T^{9} + 80585632 T^{10} + 120306054 T^{11} + 1975069110 T^{12} + 3750039738 T^{13} + 46096528884 T^{14} + 87778845834 T^{15} + 797173262265 T^{16} + 1191688906450 T^{17} + 8457586410348 T^{18} + 7204610645852 T^{19} + 41426511213649 T^{20} )^{2} \))
$29$ (\( ( 1 + 2 T + 128 T^{2} + 268 T^{3} + 7707 T^{4} + 14598 T^{5} + 280476 T^{6} + 423342 T^{7} + 6481587 T^{8} + 6536252 T^{9} + 90531968 T^{10} + 41022298 T^{11} + 594823321 T^{12} )^{2} \))(\( ( 1 - 8 T + 115 T^{2} - 906 T^{3} + 8102 T^{4} - 54856 T^{5} + 371574 T^{6} - 2368734 T^{7} + 13354204 T^{8} - 76180400 T^{9} + 387271916 T^{10} - 1992105294 T^{11} + 9062318286 T^{12} - 38798606536 T^{13} + 166181329198 T^{14} - 538909928826 T^{15} + 1983735775535 T^{16} - 4001971303688 T^{17} + 14507145975869 T^{18} )^{2} \))(\( ( 1 - 8 T + 171 T^{2} - 1074 T^{3} + 12206 T^{4} - 62088 T^{5} + 493686 T^{6} - 2163510 T^{7} + 14451548 T^{8} - 62136752 T^{9} + 419094892 T^{10} - 1819511910 T^{11} + 12040507854 T^{12} - 43913662728 T^{13} + 250359084694 T^{14} - 638840246754 T^{15} + 2949728848839 T^{16} - 4001971303688 T^{17} + 14507145975869 T^{18} )^{2} \))(\( ( 1 + 2 T + 172 T^{2} + 348 T^{3} + 13697 T^{4} + 25146 T^{5} + 693604 T^{6} + 1010414 T^{7} + 26237206 T^{8} + 28855570 T^{9} + 820947008 T^{10} + 836811530 T^{11} + 22065490246 T^{12} + 24642987046 T^{13} + 490572930724 T^{14} + 515773352754 T^{15} + 8147295027737 T^{16} + 6002956955532 T^{17} + 86042383029292 T^{18} + 29014291951738 T^{19} + 420707233300201 T^{20} )^{2} \))
$31$ (\( 1 - 196 T^{2} + 17346 T^{4} - 928808 T^{6} + 34856199 T^{8} - 1062134852 T^{10} + 31607835804 T^{12} - 1020711592772 T^{14} + 32190431756679 T^{16} - 824320518942248 T^{18} + 14794247935451586 T^{20} - 160647144248236996 T^{22} + 787662783788549761 T^{24} \))(\( 1 - 306 T^{2} + 47337 T^{4} - 4889876 T^{6} + 377911636 T^{8} - 23262602288 T^{10} + 1186952103636 T^{12} - 51581812458444 T^{14} + 1944347427433646 T^{16} - 64257574321152380 T^{18} + 1868517877763733806 T^{20} - 47636887023434661324 T^{22} + \)\(10\!\cdots\!16\)\( T^{24} - \)\(19\!\cdots\!08\)\( T^{26} + \)\(30\!\cdots\!36\)\( T^{28} - \)\(38\!\cdots\!36\)\( T^{30} + \)\(35\!\cdots\!77\)\( T^{32} - \)\(22\!\cdots\!86\)\( T^{34} + \)\(69\!\cdots\!41\)\( T^{36} \))(\( 1 - 266 T^{2} + 33625 T^{4} - 2721860 T^{6} + 161301092 T^{8} - 7624555312 T^{10} + 306608071300 T^{12} - 11026004509852 T^{14} + 367840433471870 T^{16} - 11653444028171212 T^{18} + 353494656566467070 T^{20} - 10182746710943028892 T^{22} + \)\(27\!\cdots\!00\)\( T^{24} - \)\(65\!\cdots\!92\)\( T^{26} + \)\(13\!\cdots\!92\)\( T^{28} - \)\(21\!\cdots\!60\)\( T^{30} + \)\(25\!\cdots\!25\)\( T^{32} - \)\(19\!\cdots\!46\)\( T^{34} + \)\(69\!\cdots\!41\)\( T^{36} \))(\( 1 - 252 T^{2} + 31966 T^{4} - 2786816 T^{6} + 189679525 T^{8} - 10737685484 T^{10} + 524824326216 T^{12} - 22719292996452 T^{14} + 885614907408282 T^{16} - 31387405508666548 T^{18} + 1016799190733902388 T^{20} - 30163296693828552628 T^{22} + \)\(81\!\cdots\!22\)\( T^{24} - \)\(20\!\cdots\!12\)\( T^{26} + \)\(44\!\cdots\!56\)\( T^{28} - \)\(88\!\cdots\!84\)\( T^{30} + \)\(14\!\cdots\!25\)\( T^{32} - \)\(21\!\cdots\!36\)\( T^{34} + \)\(23\!\cdots\!46\)\( T^{36} - \)\(17\!\cdots\!32\)\( T^{38} + \)\(67\!\cdots\!01\)\( T^{40} \))
$37$ (\( 1 - 144 T^{2} + 11962 T^{4} - 777764 T^{6} + 39942311 T^{8} - 1756160460 T^{10} + 69389094636 T^{12} - 2404183669740 T^{14} + 74858321526071 T^{16} - 1995529634769476 T^{18} + 42016279227803002 T^{20} - 692436149628170256 T^{22} + 6582952005840035281 T^{24} \))(\( 1 - 422 T^{2} + 85665 T^{4} - 11149716 T^{6} + 1047751908 T^{8} - 76081459248 T^{10} + 4478532397188 T^{12} - 222366017845964 T^{14} + 9642355996177710 T^{16} - 374876936501431700 T^{18} + 13200385358767284990 T^{20} - \)\(41\!\cdots\!04\)\( T^{22} + \)\(11\!\cdots\!92\)\( T^{24} - \)\(26\!\cdots\!08\)\( T^{26} + \)\(50\!\cdots\!92\)\( T^{28} - \)\(73\!\cdots\!96\)\( T^{30} + \)\(77\!\cdots\!85\)\( T^{32} - \)\(52\!\cdots\!02\)\( T^{34} + \)\(16\!\cdots\!29\)\( T^{36} \))(\( 1 - 118 T^{2} + 7921 T^{4} - 412356 T^{6} + 20976308 T^{8} - 1059825296 T^{10} + 49457319860 T^{12} - 2093223864860 T^{14} + 81652699672798 T^{16} - 3062483344297524 T^{18} + 111782545852060462 T^{20} - 3923038531789882460 T^{22} + \)\(12\!\cdots\!40\)\( T^{24} - \)\(37\!\cdots\!16\)\( T^{26} + \)\(10\!\cdots\!92\)\( T^{28} - \)\(27\!\cdots\!36\)\( T^{30} + \)\(71\!\cdots\!69\)\( T^{32} - \)\(14\!\cdots\!38\)\( T^{34} + \)\(16\!\cdots\!29\)\( T^{36} \))(\( 1 - 472 T^{2} + 108998 T^{4} - 16405596 T^{6} + 1811244805 T^{8} - 156665879180 T^{10} + 11079008998568 T^{12} - 659938217912676 T^{14} + 33822110787729322 T^{16} - 1513084910890860380 T^{18} + 59586887482862701860 T^{20} - \)\(20\!\cdots\!20\)\( T^{22} + \)\(63\!\cdots\!42\)\( T^{24} - \)\(16\!\cdots\!84\)\( T^{26} + \)\(38\!\cdots\!28\)\( T^{28} - \)\(75\!\cdots\!20\)\( T^{30} + \)\(11\!\cdots\!05\)\( T^{32} - \)\(14\!\cdots\!44\)\( T^{34} + \)\(13\!\cdots\!18\)\( T^{36} - \)\(79\!\cdots\!88\)\( T^{38} + \)\(23\!\cdots\!01\)\( T^{40} \))
$41$ (\( 1 - 336 T^{2} + 56050 T^{4} - 6097444 T^{6} + 479341847 T^{8} - 28599362556 T^{10} + 1325676810140 T^{12} - 48075528456636 T^{14} + 1354505496920567 T^{16} - 28963494603660004 T^{18} + 447555059092232050 T^{20} - 4510013528211206736 T^{22} + 22563490300366186081 T^{24} \))(\( 1 - 450 T^{2} + 94785 T^{4} - 12416068 T^{6} + 1133494924 T^{8} - 76687203088 T^{10} + 4012232636860 T^{12} - 170269102703068 T^{14} + 6399837319139398 T^{16} - 247074188466071068 T^{18} + 10758126533473328038 T^{20} - \)\(48\!\cdots\!48\)\( T^{22} + \)\(19\!\cdots\!60\)\( T^{24} - \)\(61\!\cdots\!48\)\( T^{26} + \)\(15\!\cdots\!24\)\( T^{28} - \)\(28\!\cdots\!08\)\( T^{30} + \)\(35\!\cdots\!85\)\( T^{32} - \)\(28\!\cdots\!50\)\( T^{34} + \)\(10\!\cdots\!21\)\( T^{36} \))(\( 1 - 242 T^{2} + 33713 T^{4} - 3338804 T^{6} + 263065404 T^{8} - 17422369456 T^{10} + 1009285253260 T^{12} - 52271078870380 T^{14} + 2456054239321686 T^{16} - 105277236629926780 T^{18} + 4128627176299754166 T^{20} - \)\(14\!\cdots\!80\)\( T^{22} + \)\(47\!\cdots\!60\)\( T^{24} - \)\(13\!\cdots\!76\)\( T^{26} + \)\(35\!\cdots\!04\)\( T^{28} - \)\(75\!\cdots\!24\)\( T^{30} + \)\(12\!\cdots\!93\)\( T^{32} - \)\(15\!\cdots\!22\)\( T^{34} + \)\(10\!\cdots\!21\)\( T^{36} \))(\( 1 - 368 T^{2} + 70814 T^{4} - 9449972 T^{6} + 978400661 T^{8} - 83243290972 T^{10} + 6016636434728 T^{12} - 377117157988372 T^{14} + 20770552564905290 T^{16} - 1013549047668483244 T^{18} + 44017526478002197492 T^{20} - \)\(17\!\cdots\!64\)\( T^{22} + \)\(58\!\cdots\!90\)\( T^{24} - \)\(17\!\cdots\!52\)\( T^{26} + \)\(48\!\cdots\!88\)\( T^{28} - \)\(11\!\cdots\!72\)\( T^{30} + \)\(22\!\cdots\!41\)\( T^{32} - \)\(35\!\cdots\!92\)\( T^{34} + \)\(45\!\cdots\!74\)\( T^{36} - \)\(39\!\cdots\!28\)\( T^{38} + \)\(18\!\cdots\!01\)\( T^{40} \))
$43$ (\( ( 1 - 9 T + 100 T^{2} - 648 T^{3} + 6397 T^{4} - 40598 T^{5} + 354629 T^{6} - 1745714 T^{7} + 11828053 T^{8} - 51520536 T^{9} + 341880100 T^{10} - 1323075987 T^{11} + 6321363049 T^{12} )^{2} \))(\( ( 1 + 15 T + 411 T^{2} + 4457 T^{3} + 69064 T^{4} + 585077 T^{5} + 6594035 T^{6} + 45565693 T^{7} + 410120541 T^{8} + 2361198340 T^{9} + 17635183263 T^{10} + 84250966357 T^{11} + 524271940745 T^{12} + 2000261832677 T^{13} + 10152991107352 T^{14} + 28174315109393 T^{15} + 111717449164977 T^{16} + 175323004164015 T^{17} + 502592611936843 T^{18} )^{2} \))(\( ( 1 - T + 211 T^{2} + 529 T^{3} + 20036 T^{4} + 105133 T^{5} + 1459783 T^{6} + 7670557 T^{7} + 89928281 T^{8} + 361839708 T^{9} + 3866916083 T^{10} + 14182859893 T^{11} + 116062966981 T^{12} + 359428805533 T^{13} + 2945461163948 T^{14} + 3344001052921 T^{15} + 57353726943577 T^{16} - 11688200277601 T^{17} + 502592611936843 T^{18} )^{2} \))(\( ( 1 + 7 T + 248 T^{2} + 1796 T^{3} + 33903 T^{4} + 227652 T^{5} + 3102053 T^{6} + 18776130 T^{7} + 205209960 T^{8} + 1101546579 T^{9} + 10145514870 T^{10} + 47366502897 T^{11} + 379433216040 T^{12} + 1492833767910 T^{13} + 10605301898453 T^{14} + 33466766065836 T^{15} + 214313171450247 T^{16} + 488186225548172 T^{17} + 2898673668845048 T^{18} + 3518148283557901 T^{19} + 21611482313284249 T^{20} )^{2} \))
$47$ (\( 1 - 260 T^{2} + 28022 T^{4} - 1495784 T^{6} + 23843895 T^{8} + 2170785244 T^{10} - 174577664108 T^{12} + 4795264603996 T^{14} + 116350601397495 T^{16} - 16123377821672936 T^{18} + 667239874835866742 T^{20} - 13675774381315812740 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 - 370 T^{2} + 70029 T^{4} - 8979404 T^{6} + 875730112 T^{8} - 69263175680 T^{10} + 4639243700216 T^{12} - 272329461789460 T^{14} + 14420616607496922 T^{16} - 703404538077349212 T^{18} + 31855142085960700698 T^{20} - \)\(13\!\cdots\!60\)\( T^{22} + \)\(50\!\cdots\!64\)\( T^{24} - \)\(16\!\cdots\!80\)\( T^{26} + \)\(46\!\cdots\!88\)\( T^{28} - \)\(10\!\cdots\!64\)\( T^{30} + \)\(17\!\cdots\!01\)\( T^{32} - \)\(20\!\cdots\!70\)\( T^{34} + \)\(12\!\cdots\!89\)\( T^{36} \))(\( 1 - 634 T^{2} + 193085 T^{4} - 37651388 T^{6} + 5288886256 T^{8} - 571091014656 T^{10} + 49404261154376 T^{12} - 3522120988814116 T^{14} + 210938577336194218 T^{16} - 10734551679023006540 T^{18} + \)\(46\!\cdots\!62\)\( T^{20} - \)\(17\!\cdots\!96\)\( T^{22} + \)\(53\!\cdots\!04\)\( T^{24} - \)\(13\!\cdots\!16\)\( T^{26} + \)\(27\!\cdots\!44\)\( T^{28} - \)\(43\!\cdots\!08\)\( T^{30} + \)\(49\!\cdots\!65\)\( T^{32} - \)\(35\!\cdots\!14\)\( T^{34} + \)\(12\!\cdots\!89\)\( T^{36} \))(\( 1 - 396 T^{2} + 82450 T^{4} - 12001424 T^{6} + 1363818085 T^{8} - 128037323276 T^{10} + 10270941917432 T^{12} - 719180664046980 T^{14} + 44562343413265050 T^{16} - 2464508141674633428 T^{18} + \)\(12\!\cdots\!76\)\( T^{20} - \)\(54\!\cdots\!52\)\( T^{22} + \)\(21\!\cdots\!50\)\( T^{24} - \)\(77\!\cdots\!20\)\( T^{26} + \)\(24\!\cdots\!52\)\( T^{28} - \)\(67\!\cdots\!24\)\( T^{30} + \)\(15\!\cdots\!85\)\( T^{32} - \)\(30\!\cdots\!56\)\( T^{34} + \)\(46\!\cdots\!50\)\( T^{36} - \)\(49\!\cdots\!44\)\( T^{38} + \)\(27\!\cdots\!01\)\( T^{40} \))
$53$ (\( ( 1 + 15 T + 362 T^{2} + 3646 T^{3} + 49087 T^{4} + 363896 T^{5} + 3463567 T^{6} + 19286488 T^{7} + 137885383 T^{8} + 542805542 T^{9} + 2856354122 T^{10} + 6272932395 T^{11} + 22164361129 T^{12} )^{2} \))(\( ( 1 - 19 T + 483 T^{2} - 6817 T^{3} + 100300 T^{4} - 1114149 T^{5} + 12069585 T^{6} - 109090201 T^{7} + 942682199 T^{8} - 7034222540 T^{9} + 49962156547 T^{10} - 306434374609 T^{11} + 1796883606045 T^{12} - 8791171515669 T^{13} + 41945007947900 T^{14} - 151094449816393 T^{15} + 567385480541271 T^{16} - 1182934117815859 T^{17} + 3299763591802133 T^{18} )^{2} \))(\( ( 1 + T + 227 T^{2} + 995 T^{3} + 25240 T^{4} + 188531 T^{5} + 2012061 T^{6} + 18355219 T^{7} + 130678295 T^{8} + 1166988972 T^{9} + 6925949635 T^{10} + 51559810171 T^{11} + 299549605497 T^{12} + 1487600273411 T^{13} + 10555254243320 T^{14} + 22053539323355 T^{15} + 266659428742999 T^{16} + 62259690411361 T^{17} + 3299763591802133 T^{18} )^{2} \))(\( ( 1 + 11 T + 294 T^{2} + 2786 T^{3} + 43289 T^{4} + 359934 T^{5} + 4279611 T^{6} + 31763204 T^{7} + 318024698 T^{8} + 2123597481 T^{9} + 18754773782 T^{10} + 112550666493 T^{11} + 893331376682 T^{12} + 4728810521908 T^{13} + 33768189282891 T^{14} + 150522776577462 T^{15} + 959473028913281 T^{16} + 3272745235585882 T^{17} + 18304348980940134 T^{18} + 36297399509823463 T^{19} + 174887470365513049 T^{20} )^{2} \))
$59$ (\( 1 - 372 T^{2} + 75770 T^{4} - 10569384 T^{6} + 1107525015 T^{8} - 90988359492 T^{10} + 5981879188108 T^{12} - 316730479391652 T^{14} + 13420280423285415 T^{16} - 445822257376647144 T^{18} + 11125342257279402170 T^{20} - \)\(19\!\cdots\!72\)\( T^{22} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 - 490 T^{2} + 124553 T^{4} - 21864436 T^{6} + 2965814636 T^{8} - 329147652224 T^{10} + 30909734652988 T^{12} - 2508945941546860 T^{14} + 178379995647366726 T^{16} - 11190637970043445612 T^{18} + \)\(62\!\cdots\!06\)\( T^{20} - \)\(30\!\cdots\!60\)\( T^{22} + \)\(13\!\cdots\!08\)\( T^{24} - \)\(48\!\cdots\!04\)\( T^{26} + \)\(15\!\cdots\!36\)\( T^{28} - \)\(38\!\cdots\!16\)\( T^{30} + \)\(77\!\cdots\!33\)\( T^{32} - \)\(10\!\cdots\!90\)\( T^{34} + \)\(75\!\cdots\!21\)\( T^{36} \))(\( 1 - 230 T^{2} + 21629 T^{4} - 1564668 T^{6} + 145309144 T^{8} - 11921042000 T^{10} + 768329424992 T^{12} - 51332807997156 T^{14} + 3507791958010562 T^{16} - 216445681789264212 T^{18} + 12210623805834766322 T^{20} - \)\(62\!\cdots\!16\)\( T^{22} + \)\(32\!\cdots\!72\)\( T^{24} - \)\(17\!\cdots\!00\)\( T^{26} + \)\(74\!\cdots\!44\)\( T^{28} - \)\(27\!\cdots\!08\)\( T^{30} + \)\(13\!\cdots\!69\)\( T^{32} - \)\(49\!\cdots\!30\)\( T^{34} + \)\(75\!\cdots\!21\)\( T^{36} \))(\( 1 - 672 T^{2} + 227842 T^{4} - 51609364 T^{6} + 8745885093 T^{8} - 1178820909260 T^{10} + 131304840847064 T^{12} - 12403771606961476 T^{14} + 1011865821318863306 T^{16} - 72163137930336540412 T^{18} + \)\(45\!\cdots\!04\)\( T^{20} - \)\(25\!\cdots\!72\)\( T^{22} + \)\(12\!\cdots\!66\)\( T^{24} - \)\(52\!\cdots\!16\)\( T^{26} + \)\(19\!\cdots\!44\)\( T^{28} - \)\(60\!\cdots\!60\)\( T^{30} + \)\(15\!\cdots\!33\)\( T^{32} - \)\(31\!\cdots\!04\)\( T^{34} + \)\(49\!\cdots\!22\)\( T^{36} - \)\(50\!\cdots\!12\)\( T^{38} + \)\(26\!\cdots\!01\)\( T^{40} \))
$61$ (\( ( 1 - T + 226 T^{2} - 596 T^{3} + 26775 T^{4} - 65942 T^{5} + 2065617 T^{6} - 4022462 T^{7} + 99629775 T^{8} - 135280676 T^{9} + 3129160066 T^{10} - 844596301 T^{11} + 51520374361 T^{12} )^{2} \))(\( ( 1 - 7 T + 305 T^{2} - 1625 T^{3} + 43988 T^{4} - 186611 T^{5} + 4291925 T^{6} - 15251295 T^{7} + 325356137 T^{8} - 1016195404 T^{9} + 19846724357 T^{10} - 56750068695 T^{11} + 974185428425 T^{12} - 2583786234851 T^{13} + 37152102088388 T^{14} - 83720608336625 T^{15} + 958536564986405 T^{16} - 1341951190980967 T^{17} + 11694146092834141 T^{18} )^{2} \))(\( ( 1 + 7 T + 275 T^{2} + 1383 T^{3} + 38058 T^{4} + 154541 T^{5} + 3767967 T^{6} + 13493475 T^{7} + 292044589 T^{8} + 930574604 T^{9} + 17814719929 T^{10} + 50209220475 T^{11} + 855256917627 T^{12} + 2139750113981 T^{13} + 32143646023458 T^{14} + 71252677741263 T^{15} + 864254279905775 T^{16} + 1341951190980967 T^{17} + 11694146092834141 T^{18} )^{2} \))(\( ( 1 - 15 T + 532 T^{2} - 6434 T^{3} + 128701 T^{4} - 1308940 T^{5} + 19040445 T^{6} - 165454776 T^{7} + 1918195280 T^{8} - 14275170323 T^{9} + 137703059122 T^{10} - 870785389703 T^{11} + 7137604636880 T^{12} - 37555090511256 T^{13} + 263630974039245 T^{14} - 1105525882230940 T^{15} + 6630723700635061 T^{16} - 20220407406959114 T^{17} + 101988290514553492 T^{18} - 175412191392512115 T^{19} + 713342911662882601 T^{20} )^{2} \))
$67$ (\( 1 - 395 T^{2} + 81522 T^{4} - 11833586 T^{6} + 1326745347 T^{8} - 119361018670 T^{10} + 8801230256487 T^{12} - 535811612809630 T^{14} + 26735406023583987 T^{16} - 1070447044817728034 T^{18} + 33103449209772487602 T^{20} - \)\(72\!\cdots\!55\)\( T^{22} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 - 397 T^{2} + 88663 T^{4} - 14227895 T^{6} + 1797875464 T^{8} - 189657373633 T^{10} + 17343424194701 T^{12} - 1416700123736095 T^{14} + 105823428219730163 T^{16} - 7337339073691700472 T^{18} + \)\(47\!\cdots\!07\)\( T^{20} - \)\(28\!\cdots\!95\)\( T^{22} + \)\(15\!\cdots\!69\)\( T^{24} - \)\(77\!\cdots\!53\)\( T^{26} + \)\(32\!\cdots\!36\)\( T^{28} - \)\(11\!\cdots\!95\)\( T^{30} + \)\(32\!\cdots\!27\)\( T^{32} - \)\(65\!\cdots\!57\)\( T^{34} + \)\(74\!\cdots\!09\)\( T^{36} \))(\( 1 - 437 T^{2} + 109823 T^{4} - 20164087 T^{6} + 2949559096 T^{8} - 360479441065 T^{10} + 37781056110941 T^{12} - 3451499370879311 T^{14} + 277566724269382795 T^{16} - 19751782604406871288 T^{18} + \)\(12\!\cdots\!55\)\( T^{20} - \)\(69\!\cdots\!31\)\( T^{22} + \)\(34\!\cdots\!29\)\( T^{24} - \)\(14\!\cdots\!65\)\( T^{26} + \)\(53\!\cdots\!04\)\( T^{28} - \)\(16\!\cdots\!07\)\( T^{30} + \)\(40\!\cdots\!67\)\( T^{32} - \)\(72\!\cdots\!97\)\( T^{34} + \)\(74\!\cdots\!09\)\( T^{36} \))(\( 1 - 875 T^{2} + 376798 T^{4} - 106178246 T^{6} + 21980355473 T^{8} - 3560736121824 T^{10} + 469854872362795 T^{12} - 51926635989765174 T^{14} + 4903953820619603602 T^{16} - \)\(40\!\cdots\!57\)\( T^{18} + \)\(28\!\cdots\!86\)\( T^{20} - \)\(18\!\cdots\!73\)\( T^{22} + \)\(98\!\cdots\!42\)\( T^{24} - \)\(46\!\cdots\!06\)\( T^{26} + \)\(19\!\cdots\!95\)\( T^{28} - \)\(64\!\cdots\!76\)\( T^{30} + \)\(17\!\cdots\!53\)\( T^{32} - \)\(39\!\cdots\!34\)\( T^{34} + \)\(62\!\cdots\!38\)\( T^{36} - \)\(64\!\cdots\!75\)\( T^{38} + \)\(33\!\cdots\!01\)\( T^{40} \))
$71$ (\( 1 - 199 T^{2} + 20970 T^{4} - 1400338 T^{6} + 16034827 T^{8} + 6277888686 T^{10} - 653666534945 T^{12} + 31646836866126 T^{14} + 407471908614187 T^{16} - 179383695385365298 T^{18} + 13541451550223608170 T^{20} - \)\(64\!\cdots\!99\)\( T^{22} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( 1 - 929 T^{2} + 423959 T^{4} - 126146523 T^{6} + 27404485512 T^{8} - 4613990064985 T^{10} + 623804405062133 T^{12} - 69238433408040899 T^{14} + 6395648676742896283 T^{16} - \)\(49\!\cdots\!08\)\( T^{18} + \)\(32\!\cdots\!03\)\( T^{20} - \)\(17\!\cdots\!19\)\( T^{22} + \)\(79\!\cdots\!93\)\( T^{24} - \)\(29\!\cdots\!85\)\( T^{26} + \)\(89\!\cdots\!12\)\( T^{28} - \)\(20\!\cdots\!43\)\( T^{30} + \)\(35\!\cdots\!79\)\( T^{32} - \)\(38\!\cdots\!09\)\( T^{34} + \)\(21\!\cdots\!61\)\( T^{36} \))(\( 1 - 1009 T^{2} + 494103 T^{4} - 156049291 T^{6} + 35623704072 T^{8} - 6243165779017 T^{10} + 870600731832533 T^{12} - 98782248258083731 T^{14} + 9245714221221074427 T^{16} - \)\(71\!\cdots\!00\)\( T^{18} + \)\(46\!\cdots\!07\)\( T^{20} - \)\(25\!\cdots\!11\)\( T^{22} + \)\(11\!\cdots\!93\)\( T^{24} - \)\(40\!\cdots\!37\)\( T^{26} + \)\(11\!\cdots\!72\)\( T^{28} - \)\(25\!\cdots\!31\)\( T^{30} + \)\(40\!\cdots\!43\)\( T^{32} - \)\(42\!\cdots\!89\)\( T^{34} + \)\(21\!\cdots\!61\)\( T^{36} \))(\( 1 - 847 T^{2} + 354590 T^{4} - 97547486 T^{6} + 19797947329 T^{8} - 3159244452532 T^{10} + 413121389535731 T^{12} - 45628697215411378 T^{14} + 4358652026179297714 T^{16} - \)\(36\!\cdots\!05\)\( T^{18} + \)\(27\!\cdots\!42\)\( T^{20} - \)\(18\!\cdots\!05\)\( T^{22} + \)\(11\!\cdots\!34\)\( T^{24} - \)\(58\!\cdots\!38\)\( T^{26} + \)\(26\!\cdots\!91\)\( T^{28} - \)\(10\!\cdots\!32\)\( T^{30} + \)\(32\!\cdots\!89\)\( T^{32} - \)\(80\!\cdots\!66\)\( T^{34} + \)\(14\!\cdots\!90\)\( T^{36} - \)\(17\!\cdots\!67\)\( T^{38} + \)\(10\!\cdots\!01\)\( T^{40} \))
$73$ (\( 1 - 328 T^{2} + 56678 T^{4} - 7131916 T^{6} + 749460343 T^{8} - 68654541004 T^{10} + 5429359653844 T^{12} - 365860049010316 T^{14} + 21283355440456663 T^{16} - 1079302989818139724 T^{18} + 45708545088372722918 T^{20} - \)\(14\!\cdots\!72\)\( T^{22} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 - 642 T^{2} + 206469 T^{4} - 45119996 T^{6} + 7599171496 T^{8} - 1049281661072 T^{10} + 122712339470928 T^{12} - 12396041164609252 T^{14} + 1095063425825948178 T^{16} - 85136110203436834332 T^{18} + \)\(58\!\cdots\!62\)\( T^{20} - \)\(35\!\cdots\!32\)\( T^{22} + \)\(18\!\cdots\!92\)\( T^{24} - \)\(84\!\cdots\!32\)\( T^{26} + \)\(32\!\cdots\!04\)\( T^{28} - \)\(10\!\cdots\!16\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} - \)\(41\!\cdots\!62\)\( T^{34} + \)\(34\!\cdots\!69\)\( T^{36} \))(\( 1 - 810 T^{2} + 328773 T^{4} - 88738700 T^{6} + 17837433320 T^{8} - 2834364335120 T^{10} + 368805921182864 T^{12} - 40162806685570772 T^{14} + 3709341724976969426 T^{16} - \)\(29\!\cdots\!28\)\( T^{18} + \)\(19\!\cdots\!54\)\( T^{20} - \)\(11\!\cdots\!52\)\( T^{22} + \)\(55\!\cdots\!96\)\( T^{24} - \)\(22\!\cdots\!20\)\( T^{26} + \)\(76\!\cdots\!80\)\( T^{28} - \)\(20\!\cdots\!00\)\( T^{30} + \)\(40\!\cdots\!57\)\( T^{32} - \)\(52\!\cdots\!10\)\( T^{34} + \)\(34\!\cdots\!69\)\( T^{36} \))(\( 1 - 720 T^{2} + 265394 T^{4} - 66543012 T^{6} + 12734799301 T^{8} - 1978083032252 T^{10} + 258731823163096 T^{12} - 29170150260618004 T^{14} + 2877774606850055706 T^{16} - \)\(25\!\cdots\!28\)\( T^{18} + \)\(19\!\cdots\!76\)\( T^{20} - \)\(13\!\cdots\!12\)\( T^{22} + \)\(81\!\cdots\!46\)\( T^{24} - \)\(44\!\cdots\!56\)\( T^{26} + \)\(20\!\cdots\!76\)\( T^{28} - \)\(85\!\cdots\!48\)\( T^{30} + \)\(29\!\cdots\!21\)\( T^{32} - \)\(81\!\cdots\!08\)\( T^{34} + \)\(17\!\cdots\!34\)\( T^{36} - \)\(24\!\cdots\!80\)\( T^{38} + \)\(18\!\cdots\!01\)\( T^{40} \))
$79$ (\( ( 1 + 7 T + 402 T^{2} + 2192 T^{3} + 70353 T^{4} + 301164 T^{5} + 7084207 T^{6} + 23791956 T^{7} + 439073073 T^{8} + 1080741488 T^{9} + 15657932562 T^{10} + 21539394793 T^{11} + 243087455521 T^{12} )^{2} \))(\( ( 1 - 13 T + 601 T^{2} - 6471 T^{3} + 165242 T^{4} - 1498385 T^{5} + 27530999 T^{6} - 212025963 T^{7} + 3092875493 T^{8} - 20165316088 T^{9} + 244337163947 T^{10} - 1323254035083 T^{11} + 13573856215961 T^{12} - 58362217119185 T^{13} + 508458953483558 T^{14} - 1573018924676391 T^{15} + 11541549300681559 T^{16} - 19722414528785293 T^{17} + 119851595982618319 T^{18} )^{2} \))(\( ( 1 + 7 T + 245 T^{2} + 2997 T^{3} + 38118 T^{4} + 450875 T^{5} + 4920995 T^{6} + 49629269 T^{7} + 431802321 T^{8} + 4649325360 T^{9} + 34112383359 T^{10} + 309736267829 T^{11} + 2426242453805 T^{12} + 17561617770875 T^{13} + 117291235817082 T^{14} + 728533104196437 T^{15} + 4704957701608955 T^{16} + 10619761669345927 T^{17} + 119851595982618319 T^{18} )^{2} \))(\( ( 1 + 3 T + 330 T^{2} + 812 T^{3} + 61055 T^{4} + 93870 T^{5} + 7551067 T^{6} + 4460732 T^{7} + 734930060 T^{8} - 41356905 T^{9} + 60858106126 T^{10} - 3267195495 T^{11} + 4586698504460 T^{12} + 2199314844548 T^{13} + 294114671286427 T^{14} + 288843284174130 T^{15} + 14841704596834655 T^{16} + 15593574096761108 T^{17} + 500645907269165130 T^{18} + 359554787947854957 T^{19} + 9468276082626847201 T^{20} )^{2} \))
$83$ (\( 1 - 731 T^{2} + 253478 T^{4} - 55598498 T^{6} + 8685067895 T^{8} - 1029729013566 T^{10} + 95898872194575 T^{12} - 7093803174456174 T^{14} + 412178740067704295 T^{16} - 18177393694875599762 T^{18} + \)\(57\!\cdots\!98\)\( T^{20} - \)\(11\!\cdots\!19\)\( T^{22} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 - 653 T^{2} + 222219 T^{4} - 52799487 T^{6} + 9824103152 T^{8} - 1512585585217 T^{10} + 198522740494685 T^{12} - 22615478694026247 T^{14} + 2261188642121831591 T^{16} - \)\(19\!\cdots\!72\)\( T^{18} + \)\(15\!\cdots\!99\)\( T^{20} - \)\(10\!\cdots\!87\)\( T^{22} + \)\(64\!\cdots\!65\)\( T^{24} - \)\(34\!\cdots\!97\)\( T^{26} + \)\(15\!\cdots\!48\)\( T^{28} - \)\(56\!\cdots\!07\)\( T^{30} + \)\(16\!\cdots\!51\)\( T^{32} - \)\(33\!\cdots\!93\)\( T^{34} + \)\(34\!\cdots\!09\)\( T^{36} \))(\( 1 - 697 T^{2} + 254731 T^{4} - 64828227 T^{6} + 12789108532 T^{8} - 2062211905173 T^{10} + 280299611912949 T^{12} - 32728367663237987 T^{14} + 3320147218351189423 T^{16} - \)\(29\!\cdots\!60\)\( T^{18} + \)\(22\!\cdots\!47\)\( T^{20} - \)\(15\!\cdots\!27\)\( T^{22} + \)\(91\!\cdots\!81\)\( T^{24} - \)\(46\!\cdots\!93\)\( T^{26} + \)\(19\!\cdots\!68\)\( T^{28} - \)\(69\!\cdots\!47\)\( T^{30} + \)\(18\!\cdots\!99\)\( T^{32} - \)\(35\!\cdots\!57\)\( T^{34} + \)\(34\!\cdots\!09\)\( T^{36} \))(\( 1 - 447 T^{2} + 95682 T^{4} - 13546254 T^{6} + 1476082217 T^{8} - 141626412216 T^{10} + 13671055159679 T^{12} - 1373723820019902 T^{14} + 135456642153657502 T^{16} - 12457448576589235917 T^{18} + \)\(10\!\cdots\!06\)\( T^{20} - \)\(85\!\cdots\!13\)\( T^{22} + \)\(64\!\cdots\!42\)\( T^{24} - \)\(44\!\cdots\!38\)\( T^{26} + \)\(30\!\cdots\!39\)\( T^{28} - \)\(21\!\cdots\!84\)\( T^{30} + \)\(15\!\cdots\!37\)\( T^{32} - \)\(99\!\cdots\!66\)\( T^{34} + \)\(48\!\cdots\!42\)\( T^{36} - \)\(15\!\cdots\!23\)\( T^{38} + \)\(24\!\cdots\!01\)\( T^{40} \))
$89$ (\( 1 - 523 T^{2} + 118478 T^{4} - 15550282 T^{6} + 1416928431 T^{8} - 112744613530 T^{10} + 9526409115619 T^{12} - 893050083771130 T^{14} + 88901265097553871 T^{16} - 7728199223167601002 T^{18} + \)\(46\!\cdots\!18\)\( T^{20} - \)\(16\!\cdots\!23\)\( T^{22} + \)\(24\!\cdots\!21\)\( T^{24} \))(\( 1 - 741 T^{2} + 260019 T^{4} - 58860343 T^{6} + 9962526056 T^{8} - 1389384456765 T^{10} + 170561732827577 T^{12} - 19100828506757231 T^{14} + 1964734600358753883 T^{16} - \)\(18\!\cdots\!04\)\( T^{18} + \)\(15\!\cdots\!43\)\( T^{20} - \)\(11\!\cdots\!71\)\( T^{22} + \)\(84\!\cdots\!97\)\( T^{24} - \)\(54\!\cdots\!65\)\( T^{26} + \)\(31\!\cdots\!56\)\( T^{28} - \)\(14\!\cdots\!03\)\( T^{30} + \)\(50\!\cdots\!79\)\( T^{32} - \)\(11\!\cdots\!01\)\( T^{34} + \)\(12\!\cdots\!81\)\( T^{36} \))(\( 1 - 909 T^{2} + 427443 T^{4} - 136046319 T^{6} + 32599280984 T^{8} - 6217497059557 T^{10} + 975610467233417 T^{12} - 128578383596929143 T^{14} + 14415772731678709595 T^{16} - \)\(13\!\cdots\!68\)\( T^{18} + \)\(11\!\cdots\!95\)\( T^{20} - \)\(80\!\cdots\!63\)\( T^{22} + \)\(48\!\cdots\!37\)\( T^{24} - \)\(24\!\cdots\!17\)\( T^{26} + \)\(10\!\cdots\!84\)\( T^{28} - \)\(33\!\cdots\!99\)\( T^{30} + \)\(83\!\cdots\!63\)\( T^{32} - \)\(14\!\cdots\!49\)\( T^{34} + \)\(12\!\cdots\!81\)\( T^{36} \))(\( 1 - 731 T^{2} + 273154 T^{4} - 70649446 T^{6} + 14305790437 T^{8} - 2411785982772 T^{10} + 350612782868959 T^{12} - 44944971722843370 T^{14} + 5151500895013985086 T^{16} - \)\(53\!\cdots\!73\)\( T^{18} + \)\(49\!\cdots\!86\)\( T^{20} - \)\(42\!\cdots\!33\)\( T^{22} + \)\(32\!\cdots\!26\)\( T^{24} - \)\(22\!\cdots\!70\)\( T^{26} + \)\(13\!\cdots\!79\)\( T^{28} - \)\(75\!\cdots\!72\)\( T^{30} + \)\(35\!\cdots\!77\)\( T^{32} - \)\(13\!\cdots\!86\)\( T^{34} + \)\(42\!\cdots\!94\)\( T^{36} - \)\(89\!\cdots\!11\)\( T^{38} + \)\(97\!\cdots\!01\)\( T^{40} \))
$97$ (\( 1 - 716 T^{2} + 266162 T^{4} - 65809452 T^{6} + 11932054991 T^{8} - 1660108421640 T^{10} + 181221300702460 T^{12} - 15619960139210760 T^{14} + 1056336249205691471 T^{16} - 54817431175718788908 T^{18} + \)\(20\!\cdots\!82\)\( T^{20} - \)\(52\!\cdots\!84\)\( T^{22} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 - 1302 T^{2} + 820697 T^{4} - 334170144 T^{6} + 98836048692 T^{8} - 22606667256840 T^{10} + 4151407935348676 T^{12} - 626502559402143584 T^{14} + 78817868656863876846 T^{16} - \)\(83\!\cdots\!00\)\( T^{18} + \)\(74\!\cdots\!14\)\( T^{20} - \)\(55\!\cdots\!04\)\( T^{22} + \)\(34\!\cdots\!04\)\( T^{24} - \)\(17\!\cdots\!40\)\( T^{26} + \)\(72\!\cdots\!08\)\( T^{28} - \)\(23\!\cdots\!04\)\( T^{30} + \)\(53\!\cdots\!93\)\( T^{32} - \)\(79\!\cdots\!42\)\( T^{34} + \)\(57\!\cdots\!89\)\( T^{36} \))(\( 1 - 758 T^{2} + 292793 T^{4} - 77831776 T^{6} + 16116896980 T^{8} - 2770391202056 T^{10} + 409341085755300 T^{12} - 53024470434108704 T^{14} + 6091416748388557710 T^{16} - \)\(62\!\cdots\!88\)\( T^{18} + \)\(57\!\cdots\!90\)\( T^{20} - \)\(46\!\cdots\!24\)\( T^{22} + \)\(34\!\cdots\!00\)\( T^{24} - \)\(21\!\cdots\!16\)\( T^{26} + \)\(11\!\cdots\!20\)\( T^{28} - \)\(54\!\cdots\!16\)\( T^{30} + \)\(19\!\cdots\!17\)\( T^{32} - \)\(46\!\cdots\!18\)\( T^{34} + \)\(57\!\cdots\!89\)\( T^{36} \))(\( 1 - 84 T^{2} + 22446 T^{4} - 2628420 T^{6} + 400790413 T^{8} - 52701349056 T^{10} + 5898907321576 T^{12} - 733380644252896 T^{14} + 74209256942587634 T^{16} - 8056143392879863240 T^{18} + \)\(77\!\cdots\!52\)\( T^{20} - \)\(75\!\cdots\!60\)\( T^{22} + \)\(65\!\cdots\!54\)\( T^{24} - \)\(61\!\cdots\!84\)\( T^{26} + \)\(46\!\cdots\!36\)\( T^{28} - \)\(38\!\cdots\!44\)\( T^{30} + \)\(27\!\cdots\!33\)\( T^{32} - \)\(17\!\cdots\!80\)\( T^{34} + \)\(13\!\cdots\!66\)\( T^{36} - \)\(48\!\cdots\!76\)\( T^{38} + \)\(54\!\cdots\!01\)\( T^{40} \))
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