Properties

Label 3024.2.df
Level 3024
Weight 2
Character orbit df
Rep. character \(\chi_{3024}(17,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 92
Newform subspaces 5
Sturm bound 1152
Trace bound 7

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Defining parameters

Level: \( N \) = \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3024.df (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 63 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 5 \)
Sturm bound: \(1152\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3024, [\chi])\).

Total New Old
Modular forms 1224 100 1124
Cusp forms 1080 92 988
Eisenstein series 144 8 136

Trace form

\( 92q + 6q^{5} + q^{7} + O(q^{10}) \) \( 92q + 6q^{5} + q^{7} + 6q^{19} + 74q^{25} - 6q^{29} - 15q^{31} - 15q^{35} - 2q^{37} + 8q^{43} - 3q^{47} - q^{49} - 3q^{59} - 3q^{61} + 27q^{65} - q^{67} - 6q^{73} + 27q^{77} - q^{79} - 30q^{83} + 3q^{85} - 12q^{89} + 12q^{91} - 87q^{95} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(3024, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
3024.2.df.a \(2\) \(24.147\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(6\) \(5\) \(q+3q^{5}+(3-\zeta_{6})q^{7}+(1-2\zeta_{6})q^{11}+\cdots\)
3024.2.df.b \(10\) \(24.147\) 10.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(-3\) \(q+(\beta _{2}+\beta _{8}+\beta _{9})q^{5}+(-1+2\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
3024.2.df.c \(16\) \(24.147\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(-2\) \(q+(-\beta _{11}-\beta _{15})q^{5}+(\beta _{2}-\beta _{5}+\beta _{6}+\cdots)q^{7}+\cdots\)
3024.2.df.d \(16\) \(24.147\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(1\) \(q+(\beta _{9}-\beta _{11})q^{5}-\beta _{10}q^{7}+(-\beta _{2}+\beta _{3}+\cdots)q^{11}+\cdots\)
3024.2.df.e \(48\) \(24.147\) None \(0\) \(0\) \(0\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(3024, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3024, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(378, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(504, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(756, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1008, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1512, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( ( 1 - 3 T + 5 T^{2} )^{2} \))(\( ( 1 + 16 T^{2} + 6 T^{3} + 127 T^{4} + 51 T^{5} + 635 T^{6} + 150 T^{7} + 2000 T^{8} + 3125 T^{10} )^{2} \))(\( ( 1 + 16 T^{2} - 12 T^{3} + 133 T^{4} - 138 T^{5} + 943 T^{6} - 696 T^{7} + 5539 T^{8} - 3480 T^{9} + 23575 T^{10} - 17250 T^{11} + 83125 T^{12} - 37500 T^{13} + 250000 T^{14} + 390625 T^{16} )^{2} \))(\( ( 1 + 16 T^{2} - 12 T^{3} + 151 T^{4} - 165 T^{5} + 1096 T^{6} - 1236 T^{7} + 6142 T^{8} - 6180 T^{9} + 27400 T^{10} - 20625 T^{11} + 94375 T^{12} - 37500 T^{13} + 250000 T^{14} + 390625 T^{16} )^{2} \))
$7$ (\( 1 - 5 T + 7 T^{2} \))(\( 1 + 3 T + 16 T^{2} + 62 T^{3} + 220 T^{4} + 473 T^{5} + 1540 T^{6} + 3038 T^{7} + 5488 T^{8} + 7203 T^{9} + 16807 T^{10} \))(\( 1 + 2 T - 3 T^{2} - 26 T^{3} - 67 T^{4} + 84 T^{5} + 355 T^{6} - 40 T^{7} - 675 T^{8} - 280 T^{9} + 17395 T^{10} + 28812 T^{11} - 160867 T^{12} - 436982 T^{13} - 352947 T^{14} + 1647086 T^{15} + 5764801 T^{16} \))(\( 1 - T - 6 T^{2} + 16 T^{3} - 10 T^{4} - 39 T^{5} + 385 T^{6} + 239 T^{7} - 2880 T^{8} + 1673 T^{9} + 18865 T^{10} - 13377 T^{11} - 24010 T^{12} + 268912 T^{13} - 705894 T^{14} - 823543 T^{15} + 5764801 T^{16} \))
$11$ (\( 1 - 19 T^{2} + 121 T^{4} \))(\( 1 - 66 T^{2} + 2257 T^{4} - 51461 T^{6} + 855052 T^{8} - 10752323 T^{10} + 103461292 T^{12} - 753440501 T^{14} + 3998413177 T^{16} - 14147686146 T^{18} + 25937424601 T^{20} \))(\( 1 - 68 T^{2} + 2622 T^{4} - 71710 T^{6} + 1545179 T^{8} - 27618072 T^{10} + 422760301 T^{12} - 5636332820 T^{14} + 66055843863 T^{16} - 681996271220 T^{18} + 6189633566941 T^{20} - 48927099250392 T^{22} + 331222841384699 T^{24} - 1859972718137710 T^{26} + 8228959203762462 T^{28} - 25822988683660388 T^{30} + 45949729863572161 T^{32} \))(\( 1 - 86 T^{2} + 3801 T^{4} - 115279 T^{6} + 2684372 T^{8} - 50825121 T^{10} + 808796131 T^{12} - 11028714095 T^{14} + 130183029216 T^{16} - 1334474405495 T^{18} + 11841584153971 T^{20} - 90039802183881 T^{22} + 575418978107732 T^{24} - 2990040370578679 T^{26} + 11929166259916521 T^{28} - 32658485688158726 T^{30} + 45949729863572161 T^{32} \))
$13$ (\( ( 1 - 2 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( 1 - 6 T + 68 T^{2} - 336 T^{3} + 2292 T^{4} - 9399 T^{5} + 51837 T^{6} - 187401 T^{7} + 909867 T^{8} - 3004662 T^{9} + 13054461 T^{10} - 39060606 T^{11} + 153767523 T^{12} - 411719997 T^{13} + 1480516557 T^{14} - 3489782907 T^{15} + 11063046228 T^{16} - 21083501712 T^{17} + 55469689028 T^{18} - 63626996238 T^{19} + 137858491849 T^{20} \))(\( 1 + 6 T + 47 T^{2} + 210 T^{3} + 888 T^{4} + 2658 T^{5} + 9181 T^{6} + 12564 T^{7} + 21641 T^{8} - 214770 T^{9} - 1221396 T^{10} - 6263526 T^{11} - 19011605 T^{12} - 72302586 T^{13} - 179114935 T^{14} - 742674348 T^{15} - 2320143696 T^{16} - 9654766524 T^{17} - 30270424015 T^{18} - 158848781442 T^{19} - 542990450405 T^{20} - 2325603359118 T^{21} - 5895445205364 T^{22} - 13476498996090 T^{23} + 17653228533161 T^{24} + 133234930122372 T^{25} + 1265678813665669 T^{26} + 4763562327350346 T^{27} + 20688699588763128 T^{28} + 63603772384373130 T^{29} + 185056690127866583 T^{30} + 307115358084544542 T^{31} + 665416609183179841 T^{32} \))(\( 1 - 3 T + 53 T^{2} - 150 T^{3} + 1317 T^{4} - 4452 T^{5} + 23056 T^{6} - 90522 T^{7} + 327950 T^{8} - 1218621 T^{9} + 4082037 T^{10} - 12551466 T^{11} + 42977824 T^{12} - 136810599 T^{13} + 317458436 T^{14} - 1828010007 T^{15} + 2359677564 T^{16} - 23764130091 T^{17} + 53650475684 T^{18} - 300572886003 T^{19} + 1227489631264 T^{20} - 4660271465538 T^{21} + 19703212929933 T^{22} - 76466660535057 T^{23} + 267518889951950 T^{24} - 959940492242706 T^{25} + 3178465388070544 T^{26} - 7978698074252724 T^{27} + 30683578106307477 T^{28} - 45431265988837950 T^{29} + 208680948442062317 T^{30} - 153557679042272271 T^{31} + 665416609183179841 T^{32} \))
$17$ (\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - 12 T + 26 T^{2} + 36 T^{3} + 1143 T^{4} - 5247 T^{5} - 21540 T^{6} + 73476 T^{7} + 337539 T^{8} + 599625 T^{9} - 13374333 T^{10} + 10193625 T^{11} + 97548771 T^{12} + 360987588 T^{13} - 1799042340 T^{14} - 7449989679 T^{15} + 27589241367 T^{16} + 14772192228 T^{17} + 181369693466 T^{18} - 1423054517964 T^{19} + 2015993900449 T^{20} \))(\( 1 + 18 T + 95 T^{2} - 42 T^{3} - 849 T^{4} + 7584 T^{5} + 29152 T^{6} - 139356 T^{7} - 154873 T^{8} + 2564070 T^{9} - 13186653 T^{10} - 89037906 T^{11} + 222290986 T^{12} + 933503742 T^{13} - 5890959001 T^{14} - 7644891330 T^{15} + 98422426836 T^{16} - 129963152610 T^{17} - 1702487151289 T^{18} + 4586303884446 T^{19} + 18565965441706 T^{20} - 126421094099442 T^{21} - 318293746666557 T^{22} + 1052137081279110 T^{23} - 1080356482159993 T^{24} - 16525932117115932 T^{25} + 58770254185889248 T^{26} + 259918061597088672 T^{27} - 494646279408067089 T^{28} - 415992277382049354 T^{29} + 15995893523143088255 T^{30} + 51523614927176684274 T^{31} + 48661191875666868481 T^{32} \))(\( 1 - 9 T - 31 T^{2} + 498 T^{3} + 6 T^{4} - 11613 T^{5} + 1180 T^{6} + 197118 T^{7} + 356723 T^{8} - 3738756 T^{9} - 13114257 T^{10} + 60977685 T^{11} + 321880693 T^{12} - 548795961 T^{13} - 7936687522 T^{14} + 1772823582 T^{15} + 160937528238 T^{16} + 30138000894 T^{17} - 2293702693858 T^{18} - 2696234556393 T^{19} + 26883797360053 T^{20} + 86579592891045 T^{21} - 316546283221233 T^{22} - 1534156175710788 T^{23} + 2488413121625843 T^{24} + 23375805039335646 T^{25} + 2378872802529820 T^{26} - 397999531820542029 T^{27} + 3495733423378566 T^{28} + 4932479860387156626 T^{29} - 5219712623341428799 T^{30} - 25761807463588342137 T^{31} + 48661191875666868481 T^{32} \))
$19$ (\( ( 1 - 8 T + 19 T^{2} )( 1 - T + 19 T^{2} ) \))(\( 1 + 3 T + 71 T^{2} + 204 T^{3} + 2646 T^{4} + 8547 T^{5} + 74607 T^{6} + 258954 T^{7} + 1743726 T^{8} + 5989488 T^{9} + 35261703 T^{10} + 113800272 T^{11} + 629485086 T^{12} + 1776165486 T^{13} + 9722858847 T^{14} + 21163218153 T^{15} + 124483401126 T^{16} + 182349834756 T^{17} + 1205832975911 T^{18} + 968063093337 T^{19} + 6131066257801 T^{20} \))(\( 1 + 80 T^{2} + 3483 T^{4} + 882 T^{5} + 107770 T^{6} + 26190 T^{7} + 2596658 T^{8} - 477594 T^{9} + 51326154 T^{10} - 58243698 T^{11} + 858124540 T^{12} - 2331197604 T^{13} + 13169269781 T^{14} - 60976736712 T^{15} + 222780924306 T^{16} - 1158557997528 T^{17} + 4754106390941 T^{18} - 15989684365836 T^{19} + 111831648177340 T^{20} - 144217162374102 T^{21} + 2414684133271674 T^{22} - 426907779315966 T^{23} + 44100504838916978 T^{24} + 8451190804832010 T^{25} + 660745010603213770 T^{26} + 102744408348229158 T^{27} + 7708975863107438763 T^{28} + 63920534862630729680 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( 1 + 59 T^{2} + 1482 T^{4} - 279 T^{5} + 21358 T^{6} + 23166 T^{7} + 284075 T^{8} + 1924515 T^{9} + 8259087 T^{10} + 60795468 T^{11} + 210571285 T^{12} + 1125692802 T^{13} + 3051652292 T^{14} + 14070928500 T^{15} + 40349090466 T^{16} + 267347641500 T^{17} + 1101646477412 T^{18} + 7721126928918 T^{19} + 27441860432485 T^{20} + 150535597519332 T^{21} + 388556024170647 T^{22} + 1720269569781585 T^{23} + 4824605670872075 T^{24} + 7475383206748314 T^{25} + 130947313134113758 T^{26} - 32500782232603101 T^{27} + 3280132710056050602 T^{28} + 47141394461190163139 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( 1 - 19 T^{2} + 529 T^{4} \))(\( 1 - 165 T^{2} + 13144 T^{4} - 668429 T^{6} + 24050461 T^{8} - 639494549 T^{10} + 12722693869 T^{12} - 187053839789 T^{14} + 1945783725016 T^{16} - 12921312571365 T^{18} + 41426511213649 T^{20} \))(\( 1 - 224 T^{2} + 24654 T^{4} - 1769830 T^{6} + 93146507 T^{8} - 3841107576 T^{10} + 129779286901 T^{12} - 3708814203716 T^{14} + 91488095720271 T^{16} - 1961962713765764 T^{18} + 36317565425662741 T^{20} - 568621774757795064 T^{22} + 7294394738653563467 T^{24} - 73317882341252409670 T^{26} + \)\(54\!\cdots\!34\)\( T^{28} - \)\(25\!\cdots\!16\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 - 125 T^{2} + 7680 T^{4} - 295747 T^{6} + 8228447 T^{8} - 202400565 T^{10} + 5673034564 T^{12} - 171801092192 T^{14} + 4463691201090 T^{16} - 90882777769568 T^{18} + 1587547665424324 T^{20} - 29962547573877285 T^{22} + 644377791902488607 T^{24} - 12251766411903050803 T^{26} + \)\(16\!\cdots\!80\)\( T^{28} - \)\(14\!\cdots\!25\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 - 9 T + 56 T^{2} - 261 T^{3} + 841 T^{4} \))(\( 1 - 15 T + 150 T^{2} - 1125 T^{3} + 6691 T^{4} - 30108 T^{5} + 81631 T^{6} + 232971 T^{7} - 5137202 T^{8} + 44535417 T^{9} - 275752187 T^{10} + 1291527093 T^{11} - 4320386882 T^{12} + 5681929719 T^{13} + 57736055311 T^{14} - 617549674092 T^{15} + 3979962840811 T^{16} - 19406110847625 T^{17} + 75036961944150 T^{18} - 217607189638035 T^{19} + 420707233300201 T^{20} \))(\( 1 + 6 T + 196 T^{2} + 1104 T^{3} + 19989 T^{4} + 108858 T^{5} + 1435478 T^{6} + 7594740 T^{7} + 81322766 T^{8} + 416245488 T^{9} + 3844091772 T^{10} + 18872529852 T^{11} + 155981060614 T^{12} + 727660406082 T^{13} + 5509975202215 T^{14} + 24244157784798 T^{15} + 170557455776958 T^{16} + 703080575759142 T^{17} + 4633889145062815 T^{18} + 17746909643933898 T^{19} + 110322440532130534 T^{20} + 387097271801319948 T^{21} + 2286555434049814812 T^{22} + 7180183182179343792 T^{23} + 40681421983566770126 T^{24} + \)\(11\!\cdots\!60\)\( T^{25} + \)\(60\!\cdots\!78\)\( T^{26} + \)\(13\!\cdots\!82\)\( T^{27} + \)\(70\!\cdots\!49\)\( T^{28} + \)\(11\!\cdots\!56\)\( T^{29} + \)\(58\!\cdots\!76\)\( T^{30} + \)\(51\!\cdots\!94\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( 1 + 6 T + 106 T^{2} + 564 T^{3} + 5445 T^{4} + 23700 T^{5} + 153455 T^{6} + 499140 T^{7} + 2268674 T^{8} + 3839766 T^{9} + 25367247 T^{10} + 108756381 T^{11} + 2283543946 T^{12} + 15452150286 T^{13} + 152754700246 T^{14} + 806878620189 T^{15} + 5726182499856 T^{16} + 23399479985481 T^{17} + 128466702906886 T^{18} + 376862493325254 T^{19} + 1615107245670826 T^{20} + 2230718335391769 T^{21} + 15089030105167287 T^{22} + 66235488555503694 T^{23} + 1134896030677883714 T^{24} + 7241096842395252660 T^{25} + 64559628486082344455 T^{26} + \)\(28\!\cdots\!00\)\( T^{27} + \)\(19\!\cdots\!45\)\( T^{28} + \)\(57\!\cdots\!96\)\( T^{29} + \)\(31\!\cdots\!86\)\( T^{30} + \)\(51\!\cdots\!94\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( 1 - 6 T + 43 T^{2} - 186 T^{3} + 961 T^{4} \))(\( 1 - 9 T + 149 T^{2} - 1098 T^{3} + 10878 T^{4} - 60723 T^{5} + 461409 T^{6} - 2027286 T^{7} + 13421802 T^{8} - 50078664 T^{9} + 374531595 T^{10} - 1552438584 T^{11} + 12898351722 T^{12} - 60394877226 T^{13} + 426120901089 T^{14} - 1738447936173 T^{15} + 9654265041918 T^{16} - 30208850293878 T^{17} + 127080764578709 T^{18} - 237956599446039 T^{19} + 819628286980801 T^{20} \))(\( 1 + 6 T + 164 T^{2} + 912 T^{3} + 14145 T^{4} + 82056 T^{5} + 882730 T^{6} + 5457276 T^{7} + 44537675 T^{8} + 289281588 T^{9} + 1950112812 T^{10} + 12838947000 T^{11} + 76837130221 T^{12} + 493000882878 T^{13} + 2756895706292 T^{14} + 16890347347524 T^{15} + 89911869890409 T^{16} + 523600767773244 T^{17} + 2649376773746612 T^{18} + 14686989301818498 T^{19} + 70960703338828141 T^{20} + 367568152343997000 T^{21} + 1730732299015260972 T^{22} + 7958892700061288268 T^{23} + 37985783835960089675 T^{24} + \)\(14\!\cdots\!96\)\( T^{25} + \)\(72\!\cdots\!30\)\( T^{26} + \)\(20\!\cdots\!36\)\( T^{27} + \)\(11\!\cdots\!45\)\( T^{28} + \)\(22\!\cdots\!92\)\( T^{29} + \)\(12\!\cdots\!44\)\( T^{30} + \)\(14\!\cdots\!06\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( 1 + 6 T + 128 T^{2} + 696 T^{3} + 7467 T^{4} + 41745 T^{5} + 298711 T^{6} + 1778985 T^{7} + 10180130 T^{8} + 58737741 T^{9} + 320070333 T^{10} + 1687070526 T^{11} + 8941505953 T^{12} + 50372743224 T^{13} + 219778671713 T^{14} + 1637243989362 T^{15} + 5894911299924 T^{16} + 50754563670222 T^{17} + 211207303516193 T^{18} + 1500654393386184 T^{19} + 8257668519220513 T^{20} + 48299396836503426 T^{21} + 284063598716395773 T^{22} + 1616028801884863251 T^{23} + 8682541636984247330 T^{24} + 47035691229501298935 T^{25} + \)\(24\!\cdots\!11\)\( T^{26} + \)\(10\!\cdots\!95\)\( T^{27} + \)\(58\!\cdots\!87\)\( T^{28} + \)\(16\!\cdots\!36\)\( T^{29} + \)\(96\!\cdots\!88\)\( T^{30} + \)\(14\!\cdots\!06\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( 1 - 6 T - 97 T^{2} + 194 T^{3} + 7179 T^{4} + 3556 T^{5} - 323794 T^{6} - 533292 T^{7} + 10739317 T^{8} + 10946526 T^{9} - 345629139 T^{10} + 405021462 T^{11} + 14702124973 T^{12} - 27012839676 T^{13} - 606842086834 T^{14} + 246587111092 T^{15} + 18419349890211 T^{16} + 18416784163802 T^{17} - 340710507030337 T^{18} - 779770438770462 T^{19} + 4808584372417849 T^{20} \))(\( 1 + 2 T - 116 T^{2} - 656 T^{3} + 3854 T^{4} + 48538 T^{5} + 98460 T^{6} - 747702 T^{7} - 5251023 T^{8} - 50514234 T^{9} - 417738792 T^{10} + 801423570 T^{11} + 29966075382 T^{12} + 116541321036 T^{13} - 322717550496 T^{14} - 3343790405826 T^{15} - 14327838321804 T^{16} - 123720245015562 T^{17} - 441800326629024 T^{18} + 5903167534436508 T^{19} + 56161249804004502 T^{20} + 55573881576866490 T^{21} - 1071803450698157928 T^{22} - 4795411055555611122 T^{23} - 18444110399566611183 T^{24} - 97172652768258663054 T^{25} + \)\(47\!\cdots\!40\)\( T^{26} + \)\(86\!\cdots\!94\)\( T^{27} + \)\(25\!\cdots\!74\)\( T^{28} - \)\(15\!\cdots\!32\)\( T^{29} - \)\(10\!\cdots\!24\)\( T^{30} + \)\(66\!\cdots\!86\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( 1 - T - 188 T^{2} + 55 T^{3} + 18431 T^{4} + 6958 T^{5} - 1220598 T^{6} - 1466820 T^{7} + 61023285 T^{8} + 128763363 T^{9} - 2426865150 T^{10} - 7087005807 T^{11} + 80710069782 T^{12} + 247085609469 T^{13} - 2444398029834 T^{14} - 3801878390997 T^{15} + 81128049964254 T^{16} - 140669500466889 T^{17} - 3346380902842746 T^{18} + 12515627376433257 T^{19} + 151263665092702902 T^{20} - 491441025939358299 T^{21} - 6226672006436746350 T^{22} + 12223747755547878279 T^{23} + \)\(21\!\cdots\!85\)\( T^{24} - \)\(19\!\cdots\!40\)\( T^{25} - \)\(58\!\cdots\!02\)\( T^{26} + \)\(12\!\cdots\!54\)\( T^{27} + \)\(12\!\cdots\!11\)\( T^{28} + \)\(13\!\cdots\!35\)\( T^{29} - \)\(16\!\cdots\!32\)\( T^{30} - \)\(33\!\cdots\!93\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \))(\( 1 - 9 T - 34 T^{2} + 747 T^{3} - 2085 T^{4} - 20394 T^{5} + 110775 T^{6} + 623979 T^{7} - 5992218 T^{8} - 18494757 T^{9} + 381591615 T^{10} - 758285037 T^{11} - 10072918458 T^{12} + 43005256659 T^{13} + 313023674775 T^{14} - 2362771363194 T^{15} - 9903967342485 T^{16} + 145481442589107 T^{17} - 271487457790114 T^{18} - 2946437409545649 T^{19} + 13422659310152401 T^{20} \))(\( 1 + 6 T - 223 T^{2} - 1686 T^{3} + 25980 T^{4} + 231654 T^{5} - 1971341 T^{6} - 20408106 T^{7} + 109216031 T^{8} + 1268388768 T^{9} - 4836103872 T^{10} - 57438167556 T^{11} + 192545389345 T^{12} + 1824668193534 T^{13} - 7677147470143 T^{14} - 27964729912410 T^{15} + 313424888729076 T^{16} - 1146553926408810 T^{17} - 12905284897310383 T^{18} + 125757956566556814 T^{19} + 544087251940916545 T^{20} - 6654567885439614756 T^{21} - 22971997512303721152 T^{22} + \)\(24\!\cdots\!08\)\( T^{23} + \)\(87\!\cdots\!51\)\( T^{24} - \)\(66\!\cdots\!66\)\( T^{25} - \)\(26\!\cdots\!41\)\( T^{26} + \)\(12\!\cdots\!14\)\( T^{27} + \)\(58\!\cdots\!80\)\( T^{28} - \)\(15\!\cdots\!06\)\( T^{29} - \)\(84\!\cdots\!03\)\( T^{30} + \)\(93\!\cdots\!06\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( 1 + 6 T - 142 T^{2} - 1146 T^{3} + 7575 T^{4} + 89148 T^{5} - 186515 T^{6} - 3858942 T^{7} + 5165546 T^{8} + 137269398 T^{9} - 383379603 T^{10} - 6275799867 T^{11} + 14048576428 T^{12} + 262470296220 T^{13} + 63020850350 T^{14} - 4818424458177 T^{15} - 18114488698896 T^{16} - 197555402785257 T^{17} + 105938049438350 T^{18} + 18089715285778620 T^{19} + 39697919375761708 T^{20} - 727090330826925267 T^{21} - 1821093078123196323 T^{22} + 26733801933571993638 T^{23} + 41246498577585065066 T^{24} - \)\(12\!\cdots\!62\)\( T^{25} - \)\(25\!\cdots\!15\)\( T^{26} + \)\(49\!\cdots\!68\)\( T^{27} + \)\(17\!\cdots\!75\)\( T^{28} - \)\(10\!\cdots\!66\)\( T^{29} - \)\(53\!\cdots\!62\)\( T^{30} + \)\(93\!\cdots\!06\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 + 3 T - 79 T^{2} - 1100 T^{3} + 1674 T^{4} + 79931 T^{5} + 324899 T^{6} - 3229902 T^{7} - 28512986 T^{8} + 52724394 T^{9} + 1438527201 T^{10} + 2267148942 T^{11} - 52720511114 T^{12} - 256799818314 T^{13} + 1110765026099 T^{14} + 11750531857433 T^{15} + 10581961744026 T^{16} - 299000472217700 T^{17} - 923367821930479 T^{18} + 1507777835810529 T^{19} + 21611482313284249 T^{20} \))(\( 1 - 2 T - 209 T^{2} + 602 T^{3} + 20774 T^{4} - 72052 T^{5} - 1457073 T^{6} + 4933350 T^{7} + 91147851 T^{8} - 235722522 T^{9} - 5466637218 T^{10} + 9144255228 T^{11} + 301177025103 T^{12} - 290829011802 T^{13} - 14773623661707 T^{14} + 4808085837138 T^{15} + 658882369500660 T^{16} + 206747690996934 T^{17} - 27316430150496243 T^{18} - 23122942241341614 T^{19} + 1029664314599161503 T^{20} + 1344282723462890004 T^{21} - 34556598512153357682 T^{22} - 64073768536679251854 T^{23} + \)\(10\!\cdots\!51\)\( T^{24} + \)\(24\!\cdots\!50\)\( T^{25} - \)\(31\!\cdots\!77\)\( T^{26} - \)\(66\!\cdots\!64\)\( T^{27} + \)\(83\!\cdots\!74\)\( T^{28} + \)\(10\!\cdots\!86\)\( T^{29} - \)\(15\!\cdots\!41\)\( T^{30} - \)\(63\!\cdots\!14\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \))(\( 1 - 2 T - 137 T^{2} + 620 T^{3} + 8660 T^{4} - 62377 T^{5} - 131688 T^{6} + 3324402 T^{7} - 16902063 T^{8} - 47132751 T^{9} + 1398120027 T^{10} - 5063156292 T^{11} - 36164979717 T^{12} + 388120390392 T^{13} - 815732360646 T^{14} - 8046953157708 T^{15} + 93229603031718 T^{16} - 346018985781444 T^{17} - 1508289134834454 T^{18} + 30858287878896744 T^{19} - 123640868821459317 T^{20} - 744326723152573356 T^{21} + 8838024276744682323 T^{22} - 12811558914472065357 T^{23} - \)\(19\!\cdots\!63\)\( T^{24} + \)\(16\!\cdots\!86\)\( T^{25} - \)\(28\!\cdots\!12\)\( T^{26} - \)\(57\!\cdots\!39\)\( T^{27} + \)\(34\!\cdots\!60\)\( T^{28} + \)\(10\!\cdots\!60\)\( T^{29} - \)\(10\!\cdots\!13\)\( T^{30} - \)\(63\!\cdots\!14\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 15 T - 49 T^{2} + 1068 T^{3} + 9486 T^{4} - 83265 T^{5} - 760914 T^{6} + 2887821 T^{7} + 57371082 T^{8} - 59131839 T^{9} - 3026317959 T^{10} - 2779196433 T^{11} + 126732720138 T^{12} + 299822239683 T^{13} - 3713017588434 T^{14} - 19096412007855 T^{15} + 102251636610894 T^{16} + 541073492654484 T^{17} - 1166753046426289 T^{18} - 16786957096541505 T^{19} + 52599132235830049 T^{20} \))(\( 1 + 18 T - 55 T^{2} - 1722 T^{3} + 12957 T^{4} + 149790 T^{5} - 1485620 T^{6} - 10153800 T^{7} + 96840002 T^{8} + 357022848 T^{9} - 6832958010 T^{10} - 20400467106 T^{11} + 370230517390 T^{12} + 853239339180 T^{13} - 20043242784151 T^{14} - 7467517424682 T^{15} + 1132351959046185 T^{16} - 350973318960054 T^{17} - 44275523310189559 T^{18} + 88585867911685140 T^{19} + 1806606821328152590 T^{20} - 4678745271228839742 T^{21} - 73653925723805335290 T^{22} + \)\(18\!\cdots\!24\)\( T^{23} + \)\(23\!\cdots\!22\)\( T^{24} - \)\(11\!\cdots\!00\)\( T^{25} - \)\(78\!\cdots\!80\)\( T^{26} + \)\(37\!\cdots\!70\)\( T^{27} + \)\(15\!\cdots\!37\)\( T^{28} - \)\(94\!\cdots\!94\)\( T^{29} - \)\(14\!\cdots\!95\)\( T^{30} + \)\(21\!\cdots\!74\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 + 18 T + 62 T^{2} - 192 T^{3} + 3579 T^{4} + 45093 T^{5} + 305398 T^{6} + 2890521 T^{7} + 680339 T^{8} - 61965303 T^{9} + 840301383 T^{10} + 4943829948 T^{11} + 3397740844 T^{12} + 111292595154 T^{13} - 711051013816 T^{14} - 459408198168 T^{15} + 101830826927016 T^{16} - 21592185313896 T^{17} - 1570711689519544 T^{18} + 11554731106673742 T^{19} + 16579891439390764 T^{20} + 1133842714030869636 T^{21} + 9057789548613500007 T^{22} - 31393055166295295289 T^{23} + 16199746956175816979 T^{24} + \)\(32\!\cdots\!07\)\( T^{25} + \)\(16\!\cdots\!02\)\( T^{26} + \)\(11\!\cdots\!79\)\( T^{27} + \)\(41\!\cdots\!39\)\( T^{28} - \)\(10\!\cdots\!84\)\( T^{29} + \)\(15\!\cdots\!78\)\( T^{30} + \)\(21\!\cdots\!74\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 - 15 T + 128 T^{2} - 795 T^{3} + 2809 T^{4} \))(\( 1 - 9 T + 237 T^{2} - 1890 T^{3} + 28720 T^{4} - 208689 T^{5} + 2523571 T^{6} - 16852668 T^{7} + 178203742 T^{8} - 1088604978 T^{9} + 10361882797 T^{10} - 57696063834 T^{11} + 500574311278 T^{12} - 2508974653836 T^{13} + 19912189027651 T^{14} - 87272799238677 T^{15} + 636560451624880 T^{16} - 2220204054291930 T^{17} + 14755546627492557 T^{18} - 29697872326219197 T^{19} + 174887470365513049 T^{20} \))(\( 1 + 36 T + 766 T^{2} + 12024 T^{3} + 153999 T^{4} + 1662408 T^{5} + 15312632 T^{6} + 121481640 T^{7} + 835010600 T^{8} + 5038052724 T^{9} + 28358007246 T^{10} + 182105031078 T^{11} + 1622318702806 T^{12} + 17289317216124 T^{13} + 177226593270661 T^{14} + 1605696832118286 T^{15} + 12594715934262750 T^{16} + 85101932102269158 T^{17} + 497829500497286749 T^{18} + 2573981679184892748 T^{19} + 12800874900435389686 T^{20} + 76155503249444531454 T^{21} + \)\(62\!\cdots\!34\)\( T^{22} + \)\(59\!\cdots\!88\)\( T^{23} + \)\(51\!\cdots\!00\)\( T^{24} + \)\(40\!\cdots\!20\)\( T^{25} + \)\(26\!\cdots\!68\)\( T^{26} + \)\(15\!\cdots\!76\)\( T^{27} + \)\(75\!\cdots\!59\)\( T^{28} + \)\(31\!\cdots\!52\)\( T^{29} + \)\(10\!\cdots\!54\)\( T^{30} + \)\(26\!\cdots\!52\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 + 271 T^{2} + 39798 T^{4} + 8019 T^{5} + 3882470 T^{6} + 3112830 T^{7} + 272618249 T^{8} + 573793713 T^{9} + 13751201223 T^{10} + 69938411256 T^{11} + 461723008303 T^{12} + 6218775316710 T^{13} + 7202514917530 T^{14} + 423085276358208 T^{15} - 24546838659498 T^{16} + 22423519646985024 T^{17} + 20231864403341770 T^{18} + 925832612825834670 T^{19} + 3643216624277663743 T^{20} + 29247928374839669208 T^{21} + \)\(30\!\cdots\!67\)\( T^{22} + \)\(67\!\cdots\!81\)\( T^{23} + \)\(16\!\cdots\!89\)\( T^{24} + \)\(10\!\cdots\!90\)\( T^{25} + \)\(67\!\cdots\!30\)\( T^{26} + \)\(74\!\cdots\!43\)\( T^{27} + \)\(19\!\cdots\!18\)\( T^{28} + \)\(37\!\cdots\!99\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 18 T - 55 T^{2} - 1536 T^{3} + 19971 T^{4} + 205494 T^{5} - 1764945 T^{6} - 8798931 T^{7} + 181121100 T^{8} + 308804295 T^{9} - 11121159681 T^{10} + 18219453405 T^{11} + 630482549100 T^{12} - 1807115649849 T^{13} - 21386475710145 T^{14} + 146912653898706 T^{15} + 842387437344411 T^{16} - 3822568680681984 T^{17} - 8075674068237655 T^{18} + 155933924735788902 T^{19} + 511116753300641401 T^{20} \))(\( 1 - 30 T + 200 T^{2} + 1272 T^{3} + 1905 T^{4} - 286356 T^{5} - 330074 T^{6} + 21958908 T^{7} + 129465491 T^{8} - 1641971784 T^{9} - 13037705556 T^{10} + 82617584088 T^{11} + 950143418149 T^{12} - 1952014889118 T^{13} - 70581518213368 T^{14} + 49596128462484 T^{15} + 4103530612541721 T^{16} + 2926171579286556 T^{17} - 245694264900734008 T^{18} - 400902865912165722 T^{19} + 11513230799485384789 T^{20} + 59065318389186954312 T^{21} - \)\(54\!\cdots\!96\)\( T^{22} - \)\(40\!\cdots\!96\)\( T^{23} + \)\(19\!\cdots\!11\)\( T^{24} + \)\(19\!\cdots\!12\)\( T^{25} - \)\(16\!\cdots\!74\)\( T^{26} - \)\(86\!\cdots\!04\)\( T^{27} + \)\(33\!\cdots\!05\)\( T^{28} + \)\(13\!\cdots\!88\)\( T^{29} + \)\(12\!\cdots\!00\)\( T^{30} - \)\(10\!\cdots\!70\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 + 15 T - 79 T^{2} - 1428 T^{3} + 12489 T^{4} + 104847 T^{5} - 1041029 T^{6} + 349611 T^{7} + 90472109 T^{8} - 405874929 T^{9} - 2700289995 T^{10} + 51789934650 T^{11} - 105592258130 T^{12} - 2611128991413 T^{13} + 30211727298980 T^{14} + 82730812140738 T^{15} - 2120569425031284 T^{16} + 4881117916303542 T^{17} + 105167022727749380 T^{18} - 536271061127410527 T^{19} - 1279499510566394930 T^{20} + 37025882724907060350 T^{21} - \)\(11\!\cdots\!95\)\( T^{22} - \)\(10\!\cdots\!51\)\( T^{23} + \)\(13\!\cdots\!89\)\( T^{24} + \)\(30\!\cdots\!29\)\( T^{25} - \)\(53\!\cdots\!29\)\( T^{26} + \)\(31\!\cdots\!73\)\( T^{27} + \)\(22\!\cdots\!09\)\( T^{28} - \)\(14\!\cdots\!12\)\( T^{29} - \)\(48\!\cdots\!19\)\( T^{30} + \)\(54\!\cdots\!85\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 - 24 T + 253 T^{2} - 1464 T^{3} + 3721 T^{4} \))(\( 1 - 12 T + 251 T^{2} - 2436 T^{3} + 34779 T^{4} - 347730 T^{5} + 3716049 T^{6} - 34819755 T^{7} + 301038894 T^{8} - 2709237273 T^{9} + 20488848807 T^{10} - 165263473653 T^{11} + 1120165724574 T^{12} - 7903422809655 T^{13} + 51451823602209 T^{14} - 293691471746730 T^{15} + 1791827099901219 T^{16} - 7655721548547156 T^{17} + 48118535562317531 T^{18} - 140329753114009692 T^{19} + 713342911662882601 T^{20} \))(\( 1 + 60 T + 2036 T^{2} + 50160 T^{3} + 994527 T^{4} + 16748022 T^{5} + 247695826 T^{6} + 3290908266 T^{7} + 39930949007 T^{8} + 448092194298 T^{9} + 4697101745298 T^{10} + 46372941826116 T^{11} + 434191490396749 T^{12} + 3878280141400200 T^{13} + 33210009451501436 T^{14} + 273658668372668376 T^{15} + 2175265354887641205 T^{16} + 16693178770732770936 T^{17} + \)\(12\!\cdots\!56\)\( T^{18} + \)\(88\!\cdots\!00\)\( T^{19} + \)\(60\!\cdots\!09\)\( T^{20} + \)\(39\!\cdots\!16\)\( T^{21} + \)\(24\!\cdots\!78\)\( T^{22} + \)\(14\!\cdots\!58\)\( T^{23} + \)\(76\!\cdots\!67\)\( T^{24} + \)\(38\!\cdots\!06\)\( T^{25} + \)\(17\!\cdots\!26\)\( T^{26} + \)\(72\!\cdots\!42\)\( T^{27} + \)\(26\!\cdots\!67\)\( T^{28} + \)\(81\!\cdots\!60\)\( T^{29} + \)\(20\!\cdots\!76\)\( T^{30} + \)\(36\!\cdots\!60\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( 1 - 3 T + 377 T^{2} - 1122 T^{3} + 74481 T^{4} - 239253 T^{5} + 10537021 T^{6} - 36111681 T^{7} + 1196449751 T^{8} - 4163017557 T^{9} + 114788648541 T^{10} - 392960339076 T^{11} + 9549397216930 T^{12} - 31727661059931 T^{13} + 697503406538990 T^{14} - 2222103948983802 T^{15} + 45117478421428590 T^{16} - 135548340888011922 T^{17} + 2595410175731581790 T^{18} - 7201576235044198311 T^{19} + \)\(13\!\cdots\!30\)\( T^{20} - \)\(33\!\cdots\!76\)\( T^{21} + \)\(59\!\cdots\!01\)\( T^{22} - \)\(13\!\cdots\!97\)\( T^{23} + \)\(22\!\cdots\!31\)\( T^{24} - \)\(42\!\cdots\!21\)\( T^{25} + \)\(75\!\cdots\!21\)\( T^{26} - \)\(10\!\cdots\!33\)\( T^{27} + \)\(19\!\cdots\!01\)\( T^{28} - \)\(18\!\cdots\!82\)\( T^{29} + \)\(37\!\cdots\!57\)\( T^{30} - \)\(18\!\cdots\!03\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 - 10 T - 182 T^{2} + 2448 T^{3} + 18867 T^{4} - 319605 T^{5} - 772530 T^{6} + 23213154 T^{7} - 12219093 T^{8} - 697872289 T^{9} + 3674653819 T^{10} - 46757443363 T^{11} - 54851508477 T^{12} + 6981657836502 T^{13} - 15567345506130 T^{14} - 431506734822735 T^{15} + 1706678296382523 T^{16} + 14836622009830704 T^{17} - 73904317315308662 T^{18} - 272065343962949470 T^{19} + 1822837804551761449 T^{20} \))(\( 1 + 14 T - 239 T^{2} - 5102 T^{3} + 20453 T^{4} + 910354 T^{5} + 617796 T^{6} - 105135588 T^{7} - 361795050 T^{8} + 8998911156 T^{9} + 52479815982 T^{10} - 604126066254 T^{11} - 5379501807402 T^{12} + 30548371355232 T^{13} + 457126004374545 T^{14} - 769012940073186 T^{15} - 33126987823382439 T^{16} - 51523866984903462 T^{17} + 2052038633637332505 T^{18} + 9187819813913642016 T^{19} - \)\(10\!\cdots\!42\)\( T^{20} - \)\(81\!\cdots\!78\)\( T^{21} + \)\(47\!\cdots\!58\)\( T^{22} + \)\(54\!\cdots\!88\)\( T^{23} - \)\(14\!\cdots\!50\)\( T^{24} - \)\(28\!\cdots\!36\)\( T^{25} + \)\(11\!\cdots\!04\)\( T^{26} + \)\(11\!\cdots\!82\)\( T^{27} + \)\(16\!\cdots\!33\)\( T^{28} - \)\(27\!\cdots\!74\)\( T^{29} - \)\(87\!\cdots\!31\)\( T^{30} + \)\(34\!\cdots\!02\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))(\( 1 - 7 T - 266 T^{2} + 955 T^{3} + 40565 T^{4} - 13739 T^{5} - 4302801 T^{6} - 7895916 T^{7} + 333320832 T^{8} + 1176952818 T^{9} - 19419549429 T^{10} - 109762768995 T^{11} + 793012008891 T^{12} + 6957819527658 T^{13} - 12193565486865 T^{14} - 198908265810519 T^{15} - 461225363558112 T^{16} - 13326853809304773 T^{17} - 54736915470536985 T^{18} + 2092654674597003054 T^{19} + 15980080945615616811 T^{20} - \)\(14\!\cdots\!65\)\( T^{21} - \)\(17\!\cdots\!01\)\( T^{22} + \)\(71\!\cdots\!14\)\( T^{23} + \)\(13\!\cdots\!12\)\( T^{24} - \)\(21\!\cdots\!52\)\( T^{25} - \)\(78\!\cdots\!49\)\( T^{26} - \)\(16\!\cdots\!37\)\( T^{27} + \)\(33\!\cdots\!65\)\( T^{28} + \)\(52\!\cdots\!85\)\( T^{29} - \)\(97\!\cdots\!14\)\( T^{30} - \)\(17\!\cdots\!01\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 351 T^{2} + 63037 T^{4} - 7800935 T^{6} + 747809113 T^{8} - 58386380555 T^{10} + 3769705738633 T^{12} - 198234871721735 T^{14} + 8075057597528077 T^{16} - 226659489467262111 T^{18} + 3255243551009881201 T^{20} \))(\( 1 - 650 T^{2} + 199389 T^{4} - 38930632 T^{6} + 5566228364 T^{8} - 640511863116 T^{10} + 63163988645884 T^{12} - 5475157404521894 T^{14} + 416179213677825948 T^{16} - 27600268476194867654 T^{18} + \)\(16\!\cdots\!04\)\( T^{20} - \)\(82\!\cdots\!36\)\( T^{22} + \)\(35\!\cdots\!04\)\( T^{24} - \)\(12\!\cdots\!32\)\( T^{26} + \)\(32\!\cdots\!49\)\( T^{28} - \)\(53\!\cdots\!50\)\( T^{30} + \)\(41\!\cdots\!21\)\( T^{32} \))(\( 1 - 389 T^{2} + 95718 T^{4} - 16982206 T^{6} + 2417461733 T^{8} - 285171142929 T^{10} + 28739565041956 T^{12} - 2501948752776044 T^{14} + 190114711967546784 T^{16} - 12612323662744037804 T^{18} + \)\(73\!\cdots\!36\)\( T^{20} - \)\(36\!\cdots\!09\)\( T^{22} + \)\(15\!\cdots\!13\)\( T^{24} - \)\(55\!\cdots\!06\)\( T^{26} + \)\(15\!\cdots\!38\)\( T^{28} - \)\(32\!\cdots\!09\)\( T^{30} + \)\(41\!\cdots\!21\)\( T^{32} \))
$73$ (\( 1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4} \))(\( 1 - 3 T + 260 T^{2} - 771 T^{3} + 36639 T^{4} - 155724 T^{5} + 3663555 T^{6} - 21043473 T^{7} + 293486934 T^{8} - 2074196103 T^{9} + 21799556757 T^{10} - 151416315519 T^{11} + 1563991871286 T^{12} - 8186268736041 T^{13} + 104038517806755 T^{14} - 322827000748332 T^{15} + 5544734717002671 T^{16} - 8517544258223787 T^{17} + 209679623892461060 T^{18} - 176614760124803739 T^{19} + 4297625829703557649 T^{20} \))(\( 1 + 434 T^{2} + 101253 T^{4} + 70380 T^{5} + 16361620 T^{6} + 29833866 T^{7} + 2027081504 T^{8} + 6817588704 T^{9} + 204710173188 T^{10} + 1064159524062 T^{11} + 17706469884892 T^{12} + 124668495857988 T^{13} + 1383495353685575 T^{14} + 11427374402737884 T^{15} + 102528286806790386 T^{16} + 834198331399865532 T^{17} + 7372646739790429175 T^{18} + 48498164253186917796 T^{19} + \)\(50\!\cdots\!72\)\( T^{20} + \)\(22\!\cdots\!66\)\( T^{21} + \)\(30\!\cdots\!32\)\( T^{22} + \)\(75\!\cdots\!88\)\( T^{23} + \)\(16\!\cdots\!24\)\( T^{24} + \)\(17\!\cdots\!58\)\( T^{25} + \)\(70\!\cdots\!80\)\( T^{26} + \)\(22\!\cdots\!60\)\( T^{27} + \)\(23\!\cdots\!13\)\( T^{28} + \)\(52\!\cdots\!06\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 + 278 T^{2} + 39699 T^{4} - 160362 T^{5} + 4049527 T^{6} - 43185312 T^{7} + 316493258 T^{8} - 5978310966 T^{9} + 30516141699 T^{10} - 573490190745 T^{11} + 3892519182646 T^{12} - 40210747978752 T^{13} + 436575836052170 T^{14} - 2488085380264887 T^{15} + 37348173312201540 T^{16} - 181630232759336751 T^{17} + 2326512630322013930 T^{18} - 15642664546450166784 T^{19} + \)\(11\!\cdots\!86\)\( T^{20} - \)\(11\!\cdots\!85\)\( T^{21} + \)\(46\!\cdots\!11\)\( T^{22} - \)\(66\!\cdots\!02\)\( T^{23} + \)\(25\!\cdots\!98\)\( T^{24} - \)\(25\!\cdots\!56\)\( T^{25} + \)\(17\!\cdots\!23\)\( T^{26} - \)\(50\!\cdots\!74\)\( T^{27} + \)\(90\!\cdots\!79\)\( T^{28} + \)\(33\!\cdots\!02\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 + 20 T + 46 T^{2} + 144 T^{3} + 21153 T^{4} + 101181 T^{5} - 106944 T^{6} + 9264000 T^{7} + 7962453 T^{8} - 230795113 T^{9} + 4723714795 T^{10} - 18232813927 T^{11} + 49693669173 T^{12} + 4567513296000 T^{13} - 4165477462464 T^{14} + 311339643507219 T^{15} + 5142028946635713 T^{16} + 2765362894006896 T^{17} + 69787005255701806 T^{18} + 2397031919652366380 T^{19} + 9468276082626847201 T^{20} \))(\( 1 - 16 T - 227 T^{2} + 5356 T^{3} + 15491 T^{4} - 818138 T^{5} + 465174 T^{6} + 79955730 T^{7} - 161428446 T^{8} - 5129891178 T^{9} + 6184544046 T^{10} + 147661653486 T^{11} + 2275304241738 T^{12} + 3934261822074 T^{13} - 453211437102513 T^{14} - 289666157971230 T^{15} + 45106559543464353 T^{16} - 22883626479727170 T^{17} - 2828492578956783633 T^{18} + 1939744514493542886 T^{19} + 88623284515338680778 T^{20} + \)\(45\!\cdots\!14\)\( T^{21} + \)\(15\!\cdots\!66\)\( T^{22} - \)\(98\!\cdots\!02\)\( T^{23} - \)\(24\!\cdots\!06\)\( T^{24} + \)\(95\!\cdots\!70\)\( T^{25} + \)\(44\!\cdots\!74\)\( T^{26} - \)\(61\!\cdots\!02\)\( T^{27} + \)\(91\!\cdots\!31\)\( T^{28} + \)\(25\!\cdots\!84\)\( T^{29} - \)\(83\!\cdots\!87\)\( T^{30} - \)\(46\!\cdots\!84\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( 1 - T - 488 T^{2} - 305 T^{3} + 130067 T^{4} + 217447 T^{5} - 23515767 T^{6} - 58439790 T^{7} + 3200266602 T^{8} + 9493225932 T^{9} - 346875510927 T^{10} - 1047490944351 T^{11} + 31673904979161 T^{12} + 75618180243888 T^{13} - 2595852946891617 T^{14} - 2451851231878899 T^{15} + 205361771983476300 T^{16} - 193696247318433021 T^{17} - 16200718241550581697 T^{18} + 37282711969266295632 T^{19} + \)\(12\!\cdots\!41\)\( T^{20} - \)\(32\!\cdots\!49\)\( T^{21} - \)\(84\!\cdots\!67\)\( T^{22} + \)\(18\!\cdots\!88\)\( T^{23} + \)\(48\!\cdots\!22\)\( T^{24} - \)\(70\!\cdots\!10\)\( T^{25} - \)\(22\!\cdots\!67\)\( T^{26} + \)\(16\!\cdots\!13\)\( T^{27} + \)\(76\!\cdots\!47\)\( T^{28} - \)\(14\!\cdots\!95\)\( T^{29} - \)\(17\!\cdots\!28\)\( T^{30} - \)\(29\!\cdots\!99\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( 1 + 15 T + 142 T^{2} + 1245 T^{3} + 6889 T^{4} \))(\( 1 + 15 T - 136 T^{2} - 1773 T^{3} + 22674 T^{4} + 93717 T^{5} - 3687774 T^{6} - 10067337 T^{7} + 346135869 T^{8} + 496605294 T^{9} - 27460905396 T^{10} + 41218239402 T^{11} + 2384530001541 T^{12} - 5756372421219 T^{13} - 175015562267454 T^{14} + 369155071940031 T^{15} + 7413046025768706 T^{16} - 48112218404608671 T^{17} - 306311743570909576 T^{18} + 2804103829013106045 T^{19} + 15516041187205853449 T^{20} \))(\( 1 - 487 T^{2} - 312 T^{3} + 123774 T^{4} + 132990 T^{5} - 22183883 T^{6} - 29138634 T^{7} + 3170469341 T^{8} + 4110229572 T^{9} - 386467088226 T^{10} - 393115428402 T^{11} + 41656592194789 T^{12} + 25282823866380 T^{13} - 4030130568645907 T^{14} - 780079655467782 T^{15} + 351851707607703156 T^{16} - 64746611403825906 T^{17} - 27763569487401653323 T^{18} + 14456390010085821060 T^{19} + \)\(19\!\cdots\!69\)\( T^{20} - \)\(15\!\cdots\!86\)\( T^{21} - \)\(12\!\cdots\!94\)\( T^{22} + \)\(11\!\cdots\!44\)\( T^{23} + \)\(71\!\cdots\!81\)\( T^{24} - \)\(54\!\cdots\!02\)\( T^{25} - \)\(34\!\cdots\!67\)\( T^{26} + \)\(17\!\cdots\!30\)\( T^{27} + \)\(13\!\cdots\!14\)\( T^{28} - \)\(27\!\cdots\!56\)\( T^{29} - \)\(35\!\cdots\!23\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 - 334 T^{2} + 1272 T^{3} + 54816 T^{4} - 395796 T^{5} - 5397056 T^{6} + 60133572 T^{7} + 310945691 T^{8} - 6455120304 T^{9} - 3389068248 T^{10} + 563598757704 T^{11} - 2368219763012 T^{12} - 39202056035964 T^{13} + 437295746267318 T^{14} + 1352773893773628 T^{15} - 45259898774669136 T^{16} + 112280233183211124 T^{17} + 3012530396035553702 T^{18} - 22415226014635747668 T^{19} - \)\(11\!\cdots\!52\)\( T^{20} + \)\(22\!\cdots\!72\)\( T^{21} - \)\(11\!\cdots\!12\)\( T^{22} - \)\(17\!\cdots\!08\)\( T^{23} + \)\(70\!\cdots\!31\)\( T^{24} + \)\(11\!\cdots\!16\)\( T^{25} - \)\(83\!\cdots\!44\)\( T^{26} - \)\(50\!\cdots\!32\)\( T^{27} + \)\(58\!\cdots\!76\)\( T^{28} + \)\(11\!\cdots\!36\)\( T^{29} - \)\(24\!\cdots\!86\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} \))(\( 1 + 24 T + 44 T^{2} - 2592 T^{3} - 1287 T^{4} + 278721 T^{5} - 235110 T^{6} - 13705920 T^{7} + 186157425 T^{8} + 904992183 T^{9} - 14602879521 T^{10} + 80544304287 T^{11} + 1474552963425 T^{12} - 9662248716480 T^{13} - 14751328281510 T^{14} + 1556394633684729 T^{15} - 639614921466807 T^{16} - 114647620049211168 T^{17} + 173209907450891564 T^{18} + 8408553688979645016 T^{19} + 31181719929966183601 T^{20} \))(\( 1 + 24 T - 10 T^{2} - 3636 T^{3} + 1197 T^{4} + 543420 T^{5} + 1792912 T^{6} - 58468350 T^{7} - 544062388 T^{8} + 3510821196 T^{9} + 61278252588 T^{10} - 296095488138 T^{11} - 6613804791944 T^{12} + 16297764313308 T^{13} + 597103648531739 T^{14} + 218011613471172 T^{15} - 39896652829458150 T^{16} + 19403033598934308 T^{17} + 4729658000019904619 T^{18} + 11489418610188427452 T^{19} - \)\(41\!\cdots\!04\)\( T^{20} - \)\(16\!\cdots\!62\)\( T^{21} + \)\(30\!\cdots\!68\)\( T^{22} + \)\(15\!\cdots\!84\)\( T^{23} - \)\(21\!\cdots\!28\)\( T^{24} - \)\(20\!\cdots\!50\)\( T^{25} + \)\(55\!\cdots\!12\)\( T^{26} + \)\(15\!\cdots\!80\)\( T^{27} + \)\(29\!\cdots\!37\)\( T^{28} - \)\(79\!\cdots\!84\)\( T^{29} - \)\(19\!\cdots\!10\)\( T^{30} + \)\(41\!\cdots\!76\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( 1 - 21 T - 253 T^{2} + 6084 T^{3} + 64134 T^{4} - 1086543 T^{5} - 14444834 T^{6} + 154075794 T^{7} + 2457441293 T^{8} - 16916386392 T^{9} - 343956627405 T^{10} + 1437006555873 T^{11} + 41653185355711 T^{12} - 92739461592729 T^{13} - 4399434577615834 T^{14} + 3097874885577318 T^{15} + 411982470004053510 T^{16} + 275710864816381302 T^{17} - 34847921289295021114 T^{18} - 65378445499564570401 T^{19} + \)\(26\!\cdots\!51\)\( T^{20} + \)\(80\!\cdots\!77\)\( T^{21} - \)\(17\!\cdots\!05\)\( T^{22} - \)\(74\!\cdots\!68\)\( T^{23} + \)\(96\!\cdots\!33\)\( T^{24} + \)\(53\!\cdots\!46\)\( T^{25} - \)\(45\!\cdots\!34\)\( T^{26} - \)\(30\!\cdots\!27\)\( T^{27} + \)\(15\!\cdots\!14\)\( T^{28} + \)\(13\!\cdots\!96\)\( T^{29} - \)\(49\!\cdots\!73\)\( T^{30} - \)\(36\!\cdots\!29\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( 1 + 3 T + 100 T^{2} + 291 T^{3} + 9409 T^{4} \))(\( 1 - 6 T + 311 T^{2} - 1794 T^{3} + 51903 T^{4} - 385032 T^{5} + 6010353 T^{6} - 63309837 T^{7} + 574248354 T^{8} - 8264282925 T^{9} + 54719955099 T^{10} - 801635443725 T^{11} + 5403102762786 T^{12} - 57781178864301 T^{13} + 532092229646193 T^{14} - 3306400793833224 T^{15} + 43233745971829887 T^{16} - 144952122353734722 T^{17} + 2437441847851234871 T^{18} - 4561386351927391302 T^{19} + 73742412689492826049 T^{20} \))(\( 1 + 6 T + 395 T^{2} + 2298 T^{3} + 68379 T^{4} + 571872 T^{5} + 8922340 T^{6} + 109039644 T^{7} + 1233764039 T^{8} + 14418445794 T^{9} + 171203907675 T^{10} + 1605167799666 T^{11} + 21002714173786 T^{12} + 194363495836338 T^{13} + 2133943680044999 T^{14} + 22669145285986746 T^{15} + 200280216945639852 T^{16} + 2198907092740714362 T^{17} + 20078276085543395591 T^{18} + \)\(17\!\cdots\!74\)\( T^{19} + \)\(18\!\cdots\!66\)\( T^{20} + \)\(13\!\cdots\!62\)\( T^{21} + \)\(14\!\cdots\!75\)\( T^{22} + \)\(11\!\cdots\!22\)\( T^{23} + \)\(96\!\cdots\!79\)\( T^{24} + \)\(82\!\cdots\!48\)\( T^{25} + \)\(65\!\cdots\!60\)\( T^{26} + \)\(40\!\cdots\!16\)\( T^{27} + \)\(47\!\cdots\!39\)\( T^{28} + \)\(15\!\cdots\!46\)\( T^{29} + \)\(25\!\cdots\!55\)\( T^{30} + \)\(37\!\cdots\!58\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( 1 - 3 T + 392 T^{2} - 1167 T^{3} + 75243 T^{4} - 248802 T^{5} + 9281977 T^{6} - 31870980 T^{7} + 787182476 T^{8} - 2086019346 T^{9} + 40966656945 T^{10} + 99548858823 T^{11} - 1131878124686 T^{12} + 48785193558327 T^{13} - 618766130152816 T^{14} + 7413615555033420 T^{15} - 81803373889459032 T^{16} + 719120708838241740 T^{17} - 5821970518607845744 T^{18} + 44524928960458978071 T^{19} - \)\(10\!\cdots\!66\)\( T^{20} + \)\(85\!\cdots\!11\)\( T^{21} + \)\(34\!\cdots\!05\)\( T^{22} - \)\(16\!\cdots\!98\)\( T^{23} + \)\(61\!\cdots\!36\)\( T^{24} - \)\(24\!\cdots\!60\)\( T^{25} + \)\(68\!\cdots\!73\)\( T^{26} - \)\(17\!\cdots\!06\)\( T^{27} + \)\(52\!\cdots\!63\)\( T^{28} - \)\(78\!\cdots\!59\)\( T^{29} + \)\(25\!\cdots\!48\)\( T^{30} - \)\(18\!\cdots\!79\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))
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