Properties

Label 1190.2.p
Level 1190
Weight 2
Character orbit p
Rep. character \(\chi_{1190}(421,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 72
Newform subspaces 5
Sturm bound 432
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 1190 = 2 \cdot 5 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1190.p (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 17 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 5 \)
Sturm bound: \(432\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1190, [\chi])\).

Total New Old
Modular forms 448 72 376
Cusp forms 416 72 344
Eisenstein series 32 0 32

Trace form

\( 72q - 8q^{3} - 72q^{4} - 8q^{6} + O(q^{10}) \) \( 72q - 8q^{3} - 72q^{4} - 8q^{6} - 16q^{11} + 8q^{12} + 32q^{13} + 72q^{16} + 16q^{17} + 24q^{18} - 24q^{22} + 8q^{24} + 16q^{27} + 8q^{29} - 16q^{30} + 32q^{31} - 16q^{33} - 8q^{34} + 8q^{35} - 16q^{37} + 16q^{38} + 48q^{39} - 56q^{41} + 16q^{44} + 32q^{45} - 8q^{48} + 8q^{50} - 8q^{51} - 32q^{52} - 64q^{54} - 64q^{57} + 16q^{58} + 32q^{61} + 48q^{62} + 32q^{63} - 72q^{64} + 8q^{65} - 16q^{67} - 16q^{68} - 64q^{69} + 32q^{71} - 24q^{72} - 72q^{73} - 8q^{74} + 8q^{75} - 32q^{78} - 16q^{79} - 88q^{81} + 24q^{82} - 48q^{86} + 24q^{88} + 48q^{89} + 40q^{91} + 24q^{95} - 8q^{96} - 24q^{97} - 8q^{98} - 64q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1190, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1190.2.p.a \(4\) \(9.502\) \(\Q(\zeta_{8})\) None \(0\) \(-4\) \(0\) \(0\) \(q-\zeta_{8}^{2}q^{2}+(-1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+\cdots\)
1190.2.p.b \(12\) \(9.502\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(4\) \(0\) \(0\) \(q+\beta _{10}q^{2}-\beta _{5}q^{3}-q^{4}-\beta _{4}q^{5}-\beta _{3}q^{6}+\cdots\)
1190.2.p.c \(16\) \(9.502\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(-4\) \(0\) \(0\) \(q+\beta _{5}q^{2}+\beta _{7}q^{3}-q^{4}+\beta _{3}q^{5}+\beta _{11}q^{6}+\cdots\)
1190.2.p.d \(16\) \(9.502\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{9}q^{2}-\beta _{11}q^{3}-q^{4}+\beta _{3}q^{5}+\beta _{12}q^{6}+\cdots\)
1190.2.p.e \(24\) \(9.502\) None \(0\) \(-4\) \(0\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(1190, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1190, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(34, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(85, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(119, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(170, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(238, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(595, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{6} \))(\( ( 1 + T^{2} )^{8} \))(\( ( 1 + T^{2} )^{8} \))
$3$ (\( 1 + 4 T + 8 T^{2} + 16 T^{3} + 31 T^{4} + 48 T^{5} + 72 T^{6} + 108 T^{7} + 81 T^{8} \))(\( 1 - 4 T + 8 T^{2} - 8 T^{3} + 13 T^{4} - 28 T^{5} + 40 T^{6} + 56 T^{7} - 104 T^{8} - 196 T^{9} + 1040 T^{10} - 1252 T^{11} + 1672 T^{12} - 3756 T^{13} + 9360 T^{14} - 5292 T^{15} - 8424 T^{16} + 13608 T^{17} + 29160 T^{18} - 61236 T^{19} + 85293 T^{20} - 157464 T^{21} + 472392 T^{22} - 708588 T^{23} + 531441 T^{24} \))(\( 1 + 4 T + 8 T^{2} + 12 T^{3} - 8 T^{4} - 56 T^{5} - 88 T^{6} - 48 T^{7} + 356 T^{8} + 808 T^{9} + 672 T^{10} - 884 T^{11} - 6207 T^{12} - 9084 T^{13} - 4680 T^{14} + 13632 T^{15} + 52996 T^{16} + 40896 T^{17} - 42120 T^{18} - 245268 T^{19} - 502767 T^{20} - 214812 T^{21} + 489888 T^{22} + 1767096 T^{23} + 2335716 T^{24} - 944784 T^{25} - 5196312 T^{26} - 9920232 T^{27} - 4251528 T^{28} + 19131876 T^{29} + 38263752 T^{30} + 57395628 T^{31} + 43046721 T^{32} \))(\( 1 - 20 T^{4} - 12 T^{5} - 28 T^{7} + 300 T^{8} + 164 T^{9} + 72 T^{10} + 264 T^{11} - 2243 T^{12} - 1420 T^{13} - 136 T^{14} - 1136 T^{15} + 21736 T^{16} - 3408 T^{17} - 1224 T^{18} - 38340 T^{19} - 181683 T^{20} + 64152 T^{21} + 52488 T^{22} + 358668 T^{23} + 1968300 T^{24} - 551124 T^{25} - 2125764 T^{27} - 10628820 T^{28} + 43046721 T^{32} \))
$5$ (\( 1 + T^{4} \))(\( ( 1 + T^{4} )^{3} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 + T^{4} )^{4} \))
$7$ (\( 1 + T^{4} \))(\( ( 1 + T^{4} )^{3} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 + T^{4} )^{4} \))
$11$ (\( 1 + 8 T + 32 T^{2} + 120 T^{3} + 434 T^{4} + 1320 T^{5} + 3872 T^{6} + 10648 T^{7} + 14641 T^{8} \))(\( 1 + 8 T + 32 T^{2} + 148 T^{3} + 993 T^{4} + 4588 T^{5} + 15880 T^{6} + 67632 T^{7} + 325532 T^{8} + 1152452 T^{9} + 3568160 T^{10} + 13846100 T^{11} + 53056084 T^{12} + 152307100 T^{13} + 431747360 T^{14} + 1533913612 T^{15} + 4766114012 T^{16} + 10892201232 T^{17} + 28132388680 T^{18} + 89407140548 T^{19} + 212858368833 T^{20} + 348976258268 T^{21} + 829997587232 T^{22} + 2282493364888 T^{23} + 3138428376721 T^{24} \))(\( 1 - 8 T + 32 T^{2} - 52 T^{3} + 3 T^{4} - 92 T^{5} + 1992 T^{6} - 6456 T^{7} + 26575 T^{8} - 155620 T^{9} + 728528 T^{10} - 1846024 T^{11} + 3955517 T^{12} - 15503664 T^{13} + 73480040 T^{14} - 193693476 T^{15} + 506831264 T^{16} - 2130628236 T^{17} + 8891084840 T^{18} - 20635376784 T^{19} + 57912724397 T^{20} - 297304011224 T^{21} + 1290631792208 T^{22} - 3032593551020 T^{23} + 5696587262575 T^{24} - 15222910293096 T^{25} + 51667349805192 T^{26} - 26248673696212 T^{27} + 9415285130163 T^{28} - 1795181031484412 T^{29} + 12151994674663712 T^{30} - 33417985355325208 T^{31} + 45949729863572161 T^{32} \))(\( 1 + 4 T + 8 T^{2} + 88 T^{3} - 105 T^{4} - 1820 T^{5} - 2568 T^{6} - 31944 T^{7} - 74557 T^{8} + 315536 T^{9} + 434088 T^{10} + 5299788 T^{11} + 27720093 T^{12} - 14477168 T^{13} - 23237160 T^{14} - 357582644 T^{15} - 4555764384 T^{16} - 3933409084 T^{17} - 2811696360 T^{18} - 19269110608 T^{19} + 405849881613 T^{20} + 853536157188 T^{21} + 769013371368 T^{22} + 6148903988656 T^{23} - 15981955090717 T^{24} - 75322281041304 T^{25} - 66607306375368 T^{26} - 519267240512020 T^{27} - 329534979555705 T^{28} + 3037998668665928 T^{29} + 3037998668665928 T^{30} + 16708992677662604 T^{31} + 45949729863572161 T^{32} \))
$13$ (\( ( 1 + 3 T + 13 T^{2} )^{4} \))(\( ( 1 - 12 T + 106 T^{2} - 644 T^{3} + 3439 T^{4} - 14960 T^{5} + 59076 T^{6} - 194480 T^{7} + 581191 T^{8} - 1414868 T^{9} + 3027466 T^{10} - 4455516 T^{11} + 4826809 T^{12} )^{2} \))(\( ( 1 - 10 T + 103 T^{2} - 638 T^{3} + 3928 T^{4} - 18434 T^{5} + 86933 T^{6} - 338326 T^{7} + 1329606 T^{8} - 4398238 T^{9} + 14691677 T^{10} - 40499498 T^{11} + 112187608 T^{12} - 236884934 T^{13} + 497161327 T^{14} - 627485170 T^{15} + 815730721 T^{16} )^{2} \))(\( ( 1 - 2 T + 51 T^{2} - 50 T^{3} + 1220 T^{4} + 66 T^{5} + 18337 T^{6} + 18098 T^{7} + 235726 T^{8} + 235274 T^{9} + 3098953 T^{10} + 145002 T^{11} + 34844420 T^{12} - 18564650 T^{13} + 246167259 T^{14} - 125497034 T^{15} + 815730721 T^{16} )^{2} \))
$17$ (\( 1 - 8 T + 32 T^{2} - 136 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 8 T^{2} - 12 T^{3} + 223 T^{4} - 1840 T^{5} + 6672 T^{6} - 31280 T^{7} + 64447 T^{8} - 58956 T^{9} + 668168 T^{10} - 5679428 T^{11} + 24137569 T^{12} \))(\( 1 + 4 T + 26 T^{2} + 164 T^{3} + 1328 T^{4} + 4956 T^{5} + 24518 T^{6} + 108796 T^{7} + 597854 T^{8} + 1849532 T^{9} + 7085702 T^{10} + 24348828 T^{11} + 110915888 T^{12} + 232856548 T^{13} + 627576794 T^{14} + 1641354692 T^{15} + 6975757441 T^{16} \))(\( 1 + 4 T - 8 T^{2} - 116 T^{3} - 576 T^{4} - 1740 T^{5} + 1192 T^{6} + 39612 T^{7} + 240382 T^{8} + 673404 T^{9} + 344488 T^{10} - 8548620 T^{11} - 48108096 T^{12} - 164703412 T^{13} - 193100552 T^{14} + 1641354692 T^{15} + 6975757441 T^{16} \))
$19$ (\( 1 - 10 T^{2} + 619 T^{4} - 3610 T^{6} + 130321 T^{8} \))(\( 1 - 118 T^{2} + 6277 T^{4} - 212364 T^{6} + 5691056 T^{8} - 136103810 T^{10} + 2830429484 T^{12} - 49133475410 T^{14} + 741664108976 T^{16} - 9990851472684 T^{18} + 106605825208357 T^{20} - 723465818420518 T^{22} + 2213314919066161 T^{24} \))(\( 1 - 88 T^{2} + 4588 T^{4} - 178610 T^{6} + 5740940 T^{8} - 158466262 T^{10} + 3863466269 T^{12} - 84594774332 T^{14} + 1684414295884 T^{16} - 30538713533852 T^{18} + 503490787642349 T^{20} - 7455184904566822 T^{22} + 97501616404598540 T^{24} - 1095069744305836610 T^{26} + 10154688848675546668 T^{28} - 70312588348893802648 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( 1 - 180 T^{2} + 16744 T^{4} - 1049926 T^{6} + 49240620 T^{8} - 1821189578 T^{10} + 54749792593 T^{12} - 1361531810596 T^{14} + 28269924675768 T^{16} - 491512983625156 T^{18} + 7135047720512353 T^{20} - 85679468165028218 T^{22} + 836281173947925420 T^{24} - 6437165871787972726 T^{26} + 37059745004843799784 T^{28} - \)\(14\!\cdots\!80\)\( T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( 1 + 8 T + 32 T^{2} + 216 T^{3} + 1442 T^{4} + 4968 T^{5} + 16928 T^{6} + 97336 T^{7} + 279841 T^{8} \))(\( 1 - 8 T + 32 T^{2} - 276 T^{3} + 1981 T^{4} - 5928 T^{5} + 22120 T^{6} - 156872 T^{7} + 149184 T^{8} + 2133792 T^{9} - 4796680 T^{10} + 59887308 T^{11} - 598535992 T^{12} + 1377408084 T^{13} - 2537443720 T^{14} + 25961847264 T^{15} + 41747799744 T^{16} - 1009681999096 T^{17} + 3274553864680 T^{18} - 20183805249816 T^{19} + 155134061841661 T^{20} - 497118134563788 T^{21} + 1325648358836768 T^{22} - 7622478063311416 T^{23} + 21914624432020321 T^{24} \))(\( 1 + 84 T^{3} - 33 T^{4} - 2040 T^{5} + 3528 T^{6} + 3264 T^{7} - 25717 T^{8} - 627648 T^{9} + 2471400 T^{10} - 1368576 T^{11} - 138167835 T^{12} - 86171892 T^{13} + 1343054160 T^{14} - 3013724832 T^{15} + 37447934320 T^{16} - 69315671136 T^{17} + 710475650640 T^{18} - 1048453409964 T^{19} - 38665025114235 T^{20} - 8808624557568 T^{21} + 365855896074600 T^{22} - 2137031882158656 T^{23} - 2013923608471477 T^{24} + 5878962287015232 T^{25} + 146152731561753672 T^{26} - 1943731906144411080 T^{27} - 723182606256670593 T^{28} + 42339054402663260172 T^{29} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 - 132 T^{3} - 1233 T^{4} + 1256 T^{5} + 8712 T^{6} + 175904 T^{7} + 453099 T^{8} - 3107504 T^{9} - 11688664 T^{10} - 81893824 T^{11} + 289859605 T^{12} + 2135655380 T^{13} + 4018547152 T^{14} - 9809509232 T^{15} - 321784302672 T^{16} - 225618712336 T^{17} + 2125811443408 T^{18} + 25984519008460 T^{19} + 81114601722805 T^{20} - 527096740845632 T^{21} - 1730341766462296 T^{22} - 10580508695854288 T^{23} + 35482629119835819 T^{24} + 316829957761987552 T^{25} + 360907765693310088 T^{26} + 1196729055939892312 T^{27} - 27020731924681055793 T^{28} - 66532799775613694556 T^{29} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 + 8 T + 32 T^{2} + 288 T^{3} + 2543 T^{4} + 8352 T^{5} + 26912 T^{6} + 195112 T^{7} + 707281 T^{8} \))(\( 1 + 12 T + 72 T^{2} + 572 T^{3} + 2381 T^{4} - 3504 T^{5} - 49888 T^{6} - 538232 T^{7} - 4020924 T^{8} - 9237232 T^{9} - 4286624 T^{10} + 198892256 T^{11} + 2408480908 T^{12} + 5767875424 T^{13} - 3605050784 T^{14} - 225286851248 T^{15} - 2843923147644 T^{16} - 11039756748568 T^{17} - 29674545838048 T^{18} - 60443566586736 T^{19} + 1191086709260141 T^{20} + 8298087498197068 T^{21} + 30290920797614472 T^{22} + 146406117188469948 T^{23} + 353814783205469041 T^{24} \))(\( 1 - 24 T + 288 T^{2} - 2328 T^{3} + 15360 T^{4} - 104640 T^{5} + 797472 T^{6} - 5976072 T^{7} + 40738480 T^{8} - 258728280 T^{9} + 1614686976 T^{10} - 9967679208 T^{11} + 59280699905 T^{12} - 340614573600 T^{13} + 1947321809856 T^{14} - 11141827618536 T^{15} + 61699214311088 T^{16} - 323113000937544 T^{17} + 1637697642088896 T^{18} - 8307248835530400 T^{19} + 41928112709508305 T^{20} - 204448553419489992 T^{21} + 960453469439767296 T^{22} - 4463030827640318520 T^{23} + 20379278489483439280 T^{24} - 86695748866303406568 T^{25} + \)\(33\!\cdots\!72\)\( T^{26} - \)\(12\!\cdots\!60\)\( T^{27} + \)\(54\!\cdots\!60\)\( T^{28} - \)\(23\!\cdots\!92\)\( T^{29} + \)\(85\!\cdots\!28\)\( T^{30} - \)\(20\!\cdots\!76\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( 1 - 16 T + 128 T^{2} - 928 T^{3} + 8272 T^{4} - 65800 T^{5} + 424576 T^{6} - 2740384 T^{7} + 19068016 T^{8} - 122515280 T^{9} + 715342112 T^{10} - 4333954496 T^{11} + 27165742017 T^{12} - 156730497552 T^{13} + 855608500992 T^{14} - 4881954852824 T^{15} + 27492243586864 T^{16} - 141576690731896 T^{17} + 719566749334272 T^{18} - 3822500104795728 T^{19} + 19213813179525777 T^{20} - 88894386426675904 T^{21} + 425502170710993952 T^{22} - 2113373425962501520 T^{23} + 9538706606282955376 T^{24} - 39755150717935793696 T^{25} + \)\(17\!\cdots\!76\)\( T^{26} - \)\(80\!\cdots\!00\)\( T^{27} + \)\(29\!\cdots\!52\)\( T^{28} - \)\(95\!\cdots\!92\)\( T^{29} + \)\(38\!\cdots\!68\)\( T^{30} - \)\(13\!\cdots\!84\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( 1 - 8 T + 32 T^{2} - 240 T^{3} + 1799 T^{4} - 7440 T^{5} + 30752 T^{6} - 238328 T^{7} + 923521 T^{8} \))(\( 1 + 380 T^{3} + 2301 T^{4} - 4584 T^{5} + 72200 T^{6} + 909512 T^{7} + 685328 T^{8} + 765584 T^{9} + 189988888 T^{10} + 681915348 T^{11} - 1137538520 T^{12} + 21139375788 T^{13} + 182579321368 T^{14} + 22807512944 T^{15} + 632914799888 T^{16} + 26038556384312 T^{17} + 64077765768200 T^{18} - 126117823084824 T^{19} + 1962502277151741 T^{20} + 10047056421054980 T^{21} + 787662783788549761 T^{24} \))(\( 1 - 8 T + 32 T^{2} - 340 T^{3} + 772 T^{4} + 7320 T^{5} - 25464 T^{6} + 338988 T^{7} - 1936564 T^{8} - 5917200 T^{9} + 40501136 T^{10} - 504090192 T^{11} + 3693875301 T^{12} - 698908052 T^{13} - 7659094552 T^{14} + 290158768532 T^{15} - 4479255233224 T^{16} + 8994921824492 T^{17} - 7360389864472 T^{18} - 20821169777132 T^{19} + 3411371411854821 T^{20} - 14431674224386992 T^{21} + 35944907284681616 T^{22} - 162797640217609200 T^{23} - 1651678079030892724 T^{24} + 8962714637001540948 T^{25} - 20871014699679116664 T^{26} + \)\(18\!\cdots\!20\)\( T^{27} + \)\(60\!\cdots\!92\)\( T^{28} - \)\(83\!\cdots\!40\)\( T^{29} + \)\(24\!\cdots\!72\)\( T^{30} - \)\(18\!\cdots\!08\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( 1 + 16 T + 128 T^{2} + 748 T^{3} + 3556 T^{4} + 16984 T^{5} + 96328 T^{6} + 485748 T^{7} + 3031324 T^{8} + 27726608 T^{9} + 220641520 T^{10} + 1485206056 T^{11} + 8702652469 T^{12} + 43226587484 T^{13} + 200901660232 T^{14} + 825731214028 T^{15} + 3535706388184 T^{16} + 25597667634868 T^{17} + 193066495482952 T^{18} + 1287763267735844 T^{19} + 8037082310823349 T^{20} + 42520188443338456 T^{21} + 195820161181435120 T^{22} + 762831466510965488 T^{23} + 2585389071179801884 T^{24} + 12842993585301616908 T^{25} + 78953153628286598728 T^{26} + \)\(43\!\cdots\!04\)\( T^{27} + \)\(28\!\cdots\!16\)\( T^{28} + \)\(18\!\cdots\!68\)\( T^{29} + \)\(96\!\cdots\!88\)\( T^{30} + \)\(37\!\cdots\!16\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( 1 - 16 T + 128 T^{2} - 528 T^{3} + 2162 T^{4} - 19536 T^{5} + 175232 T^{6} - 810448 T^{7} + 1874161 T^{8} \))(\( 1 + 20 T + 200 T^{2} + 1300 T^{3} + 8746 T^{4} + 84404 T^{5} + 783880 T^{6} + 5700724 T^{7} + 35998239 T^{8} + 235767528 T^{9} + 1632847088 T^{10} + 10819359816 T^{11} + 67428978188 T^{12} + 400316313192 T^{13} + 2235367663472 T^{14} + 11942332595784 T^{15} + 67466495602479 T^{16} + 395310759924868 T^{17} + 2011221617486920 T^{18} + 8012630157533732 T^{19} + 30720145303993066 T^{20} + 168950261733600100 T^{21} + 961716874483569800 T^{22} + 3558352435589208260 T^{23} + 6582952005840035281 T^{24} \))(\( 1 + 4 T + 8 T^{2} + 388 T^{3} - 644 T^{4} - 12404 T^{5} + 30808 T^{6} - 645620 T^{7} - 2122364 T^{8} + 32787572 T^{9} - 5601400 T^{10} + 908836372 T^{11} + 10627532420 T^{12} - 22251051652 T^{13} + 42098083992 T^{14} + 28466647644 T^{15} - 15976583904202 T^{16} + 1053265962828 T^{17} + 57632276985048 T^{18} - 1127082519328756 T^{19} + 19917706787799620 T^{20} + 63022310300004004 T^{21} - 14371659907372600 T^{22} + 3112585756593391076 T^{23} - 7454759943741589244 T^{24} - 83905898446497612740 T^{25} + \)\(14\!\cdots\!92\)\( T^{26} - \)\(22\!\cdots\!52\)\( T^{27} - \)\(42\!\cdots\!64\)\( T^{28} + \)\(94\!\cdots\!36\)\( T^{29} + \)\(72\!\cdots\!12\)\( T^{30} + \)\(13\!\cdots\!72\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( 1 + 280 T^{3} + 696 T^{4} - 9384 T^{5} + 39200 T^{6} + 168112 T^{7} - 1636388 T^{8} + 955696 T^{9} + 63817888 T^{10} + 354359048 T^{11} + 529060040 T^{12} - 4975173368 T^{13} + 124112264000 T^{14} + 235301571552 T^{15} - 4268808757754 T^{16} + 8706158147424 T^{17} + 169909689416000 T^{18} - 252007456609304 T^{19} + 991543693626440 T^{20} + 24572658587072936 T^{21} + 163739240608204192 T^{22} + 90726015248499568 T^{23} - 5747779228642877348 T^{24} + 21848128000429984624 T^{25} + \)\(18\!\cdots\!00\)\( T^{26} - \)\(16\!\cdots\!92\)\( T^{27} + \)\(45\!\cdots\!76\)\( T^{28} + \)\(68\!\cdots\!60\)\( T^{29} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( 1 + 4 T + 8 T^{2} + 28 T^{3} - 1058 T^{4} + 1148 T^{5} + 13448 T^{6} + 275684 T^{7} + 2825761 T^{8} \))(\( 1 - 4 T + 8 T^{2} + 44 T^{3} - 3367 T^{4} + 6948 T^{5} + 112 T^{6} - 284416 T^{7} + 5540024 T^{8} - 2973292 T^{9} - 20426872 T^{10} + 373653440 T^{11} - 7513850496 T^{12} + 15319791040 T^{13} - 34337571832 T^{14} - 204922257932 T^{15} + 15654783758264 T^{16} - 32951357263616 T^{17} + 532011674992 T^{18} + 1353152694925188 T^{19} - 26885243246450407 T^{20} + 14404805113334284 T^{21} + 107381274481219208 T^{22} - 2201316126864993764 T^{23} + 22563490300366186081 T^{24} \))(\( 1 + 12 T + 72 T^{2} + 284 T^{3} + 1171 T^{4} + 14748 T^{5} + 132992 T^{6} + 1095704 T^{7} + 9997523 T^{8} + 61310920 T^{9} + 280107440 T^{10} + 1349142684 T^{11} + 9218528969 T^{12} + 77398261964 T^{13} + 495102264968 T^{14} + 3489834967892 T^{15} + 24568807713568 T^{16} + 143083233683572 T^{17} + 832266907411208 T^{18} + 5334365612820844 T^{19} + 26049359637970409 T^{20} + 156306545975183484 T^{21} + 1330539538679653040 T^{22} + 11940563705576080520 T^{23} + 79829473631417467283 T^{24} + \)\(35\!\cdots\!44\)\( T^{25} + \)\(17\!\cdots\!92\)\( T^{26} + \)\(81\!\cdots\!68\)\( T^{27} + \)\(26\!\cdots\!51\)\( T^{28} + \)\(26\!\cdots\!64\)\( T^{29} + \)\(27\!\cdots\!92\)\( T^{30} + \)\(18\!\cdots\!12\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( 1 + 16 T + 128 T^{2} + 552 T^{3} + 759 T^{4} - 668 T^{5} + 44512 T^{6} - 69880 T^{7} - 6629801 T^{8} - 40500100 T^{9} + 128477672 T^{10} + 3485075488 T^{11} + 18694594985 T^{12} - 10711256300 T^{13} - 413439766440 T^{14} + 22019003724 T^{15} + 12803020836368 T^{16} + 902779152684 T^{17} - 694992247385640 T^{18} - 738230495452300 T^{19} + 52826457419408585 T^{20} + 403767606237901088 T^{21} + 610282334641006952 T^{22} - 7887567567607888100 T^{23} - 52938465268951634921 T^{24} - 22877449575449994680 T^{25} + \)\(59\!\cdots\!12\)\( T^{26} - \)\(36\!\cdots\!88\)\( T^{27} + \)\(17\!\cdots\!79\)\( T^{28} + \)\(51\!\cdots\!92\)\( T^{29} + \)\(48\!\cdots\!08\)\( T^{30} + \)\(24\!\cdots\!16\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( 1 - 76 T^{2} + 3094 T^{4} - 140524 T^{6} + 3418801 T^{8} \))(\( 1 - 342 T^{2} + 57821 T^{4} - 6367292 T^{6} + 508974752 T^{8} - 31182125146 T^{10} + 1504538144796 T^{12} - 57655749394954 T^{14} + 1740083391112352 T^{16} - 40249964370993308 T^{18} + 675823428251167421 T^{20} - 7391126951143213158 T^{22} + 39959630797262576401 T^{24} \))(\( 1 + 22 T^{2} + 1979 T^{4} - 159336 T^{6} - 4152665 T^{8} - 409831852 T^{10} + 7528956497 T^{12} + 218571242030 T^{14} + 44392565101272 T^{16} + 404138226513470 T^{18} + 25740004000900097 T^{20} - 2590695925536036748 T^{22} - 48537180205783956665 T^{24} - \)\(34\!\cdots\!64\)\( T^{26} + \)\(79\!\cdots\!79\)\( T^{28} + \)\(16\!\cdots\!78\)\( T^{30} + \)\(13\!\cdots\!01\)\( T^{32} \))(\( 1 - 298 T^{2} + 47579 T^{4} - 5316920 T^{6} + 461795799 T^{8} - 32923473740 T^{10} + 1991021833873 T^{12} - 104256884732194 T^{14} + 4781422425056696 T^{16} - 192770979869826706 T^{18} + 6806907436666846273 T^{20} - \)\(20\!\cdots\!60\)\( T^{22} + \)\(53\!\cdots\!99\)\( T^{24} - \)\(11\!\cdots\!80\)\( T^{26} + \)\(19\!\cdots\!79\)\( T^{28} - \)\(22\!\cdots\!02\)\( T^{30} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( ( 1 - 18 T + 157 T^{2} - 846 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 18 T + 377 T^{2} + 4388 T^{3} + 50924 T^{4} + 419554 T^{5} + 3328722 T^{6} + 19719038 T^{7} + 112491116 T^{8} + 455575324 T^{9} + 1839639737 T^{10} + 4128210126 T^{11} + 10779215329 T^{12} )^{2} \))(\( ( 1 - 32 T + 704 T^{2} - 10742 T^{3} + 134810 T^{4} - 1386066 T^{5} + 12557009 T^{6} - 99505948 T^{7} + 721079232 T^{8} - 4676779556 T^{9} + 27738432881 T^{10} - 143905530318 T^{11} + 657829795610 T^{12} - 2463624065194 T^{13} + 7588567591616 T^{14} - 16211939854816 T^{15} + 23811286661761 T^{16} )^{2} \))(\( ( 1 + 4 T + 154 T^{2} + 1082 T^{3} + 15646 T^{4} + 103602 T^{5} + 1193259 T^{6} + 6785880 T^{7} + 64408682 T^{8} + 318936360 T^{9} + 2635909131 T^{10} + 10756270446 T^{11} + 76347488926 T^{12} + 248151297574 T^{13} + 1659999160666 T^{14} + 2026492481852 T^{15} + 23811286661761 T^{16} )^{2} \))
$53$ (\( 1 + 34 T^{2} + 4939 T^{4} + 95506 T^{6} + 7890481 T^{8} \))(\( 1 - 446 T^{2} + 98449 T^{4} - 14095328 T^{6} + 1450741216 T^{8} - 112741255822 T^{10} + 6772475661300 T^{12} - 316690187603998 T^{14} + 11447046000764896 T^{16} - 312413940023705312 T^{18} + 6129404261308079089 T^{20} - 77999811783018819854 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 - 428 T^{2} + 96944 T^{4} - 15115398 T^{6} + 1799571120 T^{8} - 172500975162 T^{10} + 13723739030857 T^{12} - 922345924823968 T^{14} + 52859706183124580 T^{16} - 2590869702830526112 T^{18} + \)\(10\!\cdots\!17\)\( T^{20} - \)\(38\!\cdots\!98\)\( T^{22} + \)\(11\!\cdots\!20\)\( T^{24} - \)\(26\!\cdots\!02\)\( T^{26} + \)\(47\!\cdots\!04\)\( T^{28} - \)\(59\!\cdots\!32\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 - 412 T^{2} + 85856 T^{4} - 12095750 T^{6} + 1293951216 T^{8} - 112068551370 T^{10} + 8196382550985 T^{12} - 520895583115568 T^{14} + 29271052268650020 T^{16} - 1463195692971630512 T^{18} + 64673400787278673785 T^{20} - \)\(24\!\cdots\!30\)\( T^{22} + \)\(80\!\cdots\!76\)\( T^{24} - \)\(21\!\cdots\!50\)\( T^{26} + \)\(42\!\cdots\!96\)\( T^{28} - \)\(56\!\cdots\!28\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( ( 1 - 109 T^{2} + 3481 T^{4} )^{2} \))(\( 1 - 178 T^{2} + 18485 T^{4} - 1181692 T^{6} + 47757944 T^{8} - 456136294 T^{10} - 32739860484 T^{12} - 1587810439414 T^{14} + 578700248065784 T^{16} - 49844399159300572 T^{18} + 2714160639115873685 T^{20} - 90978782087514169378 T^{22} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 - 428 T^{2} + 95156 T^{4} - 14678798 T^{6} + 1758413580 T^{8} - 173204961086 T^{10} + 14489003079981 T^{12} - 1047825854261196 T^{14} + 66099951142483324 T^{16} - 3647481798683223276 T^{18} + \)\(17\!\cdots\!41\)\( T^{20} - \)\(73\!\cdots\!26\)\( T^{22} + \)\(25\!\cdots\!80\)\( T^{24} - \)\(75\!\cdots\!98\)\( T^{26} + \)\(16\!\cdots\!36\)\( T^{28} - \)\(26\!\cdots\!08\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 - 480 T^{2} + 110400 T^{4} - 16498266 T^{6} + 1847523388 T^{8} - 169804274458 T^{10} + 13539093853665 T^{12} - 957187503869924 T^{14} + 60035211726703560 T^{16} - 3331969700971205444 T^{18} + \)\(16\!\cdots\!65\)\( T^{20} - \)\(71\!\cdots\!78\)\( T^{22} + \)\(27\!\cdots\!48\)\( T^{24} - \)\(84\!\cdots\!66\)\( T^{26} + \)\(19\!\cdots\!00\)\( T^{28} - \)\(29\!\cdots\!80\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 - 12 T + 72 T^{2} - 936 T^{3} + 11879 T^{4} - 57096 T^{5} + 267912 T^{6} - 2723772 T^{7} + 13845841 T^{8} \))(\( 1 + 12 T + 72 T^{2} - 484 T^{3} - 8902 T^{4} + 4988 T^{5} + 817928 T^{6} + 8359596 T^{7} - 8150977 T^{8} - 586224968 T^{9} - 3200238608 T^{10} + 16462544824 T^{11} + 363701662572 T^{12} + 1004215234264 T^{13} - 11908087860368 T^{14} - 133061929461608 T^{15} - 112857131536657 T^{16} + 7060483859454396 T^{17} + 42139956760344008 T^{18} + 15676001266072748 T^{19} - 1706578500301795462 T^{20} - 5659966708931724244 T^{21} + 51360689639727547272 T^{22} + \)\(52\!\cdots\!32\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( 1 + 16 T + 128 T^{2} + 716 T^{3} + 4241 T^{4} + 6352 T^{5} - 184888 T^{6} - 4812628 T^{7} - 31688058 T^{8} + 4776588 T^{9} + 1490939144 T^{10} + 9880864968 T^{11} + 101746257711 T^{12} + 723713309996 T^{13} + 5059811732976 T^{14} - 25104871691768 T^{15} - 466027565022446 T^{16} - 1531397173197848 T^{17} + 18827559458403696 T^{18} + 164269170816202076 T^{19} + 1408762506611529951 T^{20} + 8345342002653283368 T^{21} + 76813742848348886984 T^{22} + 15011587717623876348 T^{23} - \)\(60\!\cdots\!98\)\( T^{24} - \)\(56\!\cdots\!48\)\( T^{25} - \)\(13\!\cdots\!88\)\( T^{26} + \)\(27\!\cdots\!72\)\( T^{27} + \)\(11\!\cdots\!61\)\( T^{28} + \)\(11\!\cdots\!96\)\( T^{29} + \)\(12\!\cdots\!48\)\( T^{30} + \)\(96\!\cdots\!16\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( 1 - 16 T + 128 T^{2} - 1164 T^{3} + 10913 T^{4} - 93136 T^{5} + 770760 T^{6} - 8408940 T^{7} + 55580998 T^{8} - 44414828 T^{9} - 689870968 T^{10} + 9859011416 T^{11} - 154347573473 T^{12} + 1697172171892 T^{13} - 14693447396240 T^{14} + 174899610543256 T^{15} - 1890140071311982 T^{16} + 10668876243138616 T^{17} - 54674317761409040 T^{18} + 385225836748218052 T^{19} - 2137071961042975793 T^{20} + 8326884573470372216 T^{21} - 35542410532145451448 T^{22} - \)\(13\!\cdots\!88\)\( T^{23} + \)\(10\!\cdots\!38\)\( T^{24} - \)\(98\!\cdots\!40\)\( T^{25} + \)\(54\!\cdots\!60\)\( T^{26} - \)\(40\!\cdots\!96\)\( T^{27} + \)\(28\!\cdots\!73\)\( T^{28} - \)\(18\!\cdots\!84\)\( T^{29} + \)\(12\!\cdots\!48\)\( T^{30} - \)\(96\!\cdots\!16\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( ( 1 - 4 T + 136 T^{2} - 268 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 2 T + 239 T^{2} + 920 T^{3} + 26482 T^{4} + 133790 T^{5} + 2007644 T^{6} + 8963930 T^{7} + 118877698 T^{8} + 276701960 T^{9} + 4816117919 T^{10} + 2700250214 T^{11} + 90458382169 T^{12} )^{2} \))(\( ( 1 + 22 T + 435 T^{2} + 5872 T^{3} + 72639 T^{4} + 737088 T^{5} + 7279643 T^{6} + 63350534 T^{7} + 544942620 T^{8} + 4244485778 T^{9} + 32678317427 T^{10} + 221688798144 T^{11} + 1463757278319 T^{12} + 7927934628304 T^{13} + 39349396243515 T^{14} + 133335655317106 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 + 10 T + 411 T^{2} + 3720 T^{3} + 80975 T^{4} + 645632 T^{5} + 9795435 T^{6} + 67083522 T^{7} + 793396156 T^{8} + 4494595974 T^{9} + 43971707715 T^{10} + 194182217216 T^{11} + 1631737022975 T^{12} + 5022465398040 T^{13} + 37178395071459 T^{14} + 60607116053230 T^{15} + 406067677556641 T^{16} )^{2} \))
$71$ (\( 1 - 20 T + 200 T^{2} - 800 T^{3} + 2239 T^{4} - 56800 T^{5} + 1008200 T^{6} - 7158220 T^{7} + 25411681 T^{8} \))(\( 1 + 28 T + 392 T^{2} + 4220 T^{3} + 30714 T^{4} + 101492 T^{5} - 293912 T^{6} - 8341996 T^{7} - 43578833 T^{8} + 586480024 T^{9} + 12478646288 T^{10} + 157323948120 T^{11} + 1561230125420 T^{12} + 11170000316520 T^{13} + 62904855937808 T^{14} + 209907651869864 T^{15} - 1107411402548273 T^{16} - 15050874029124596 T^{17} - 37650210647788952 T^{18} + 923081935115419372 T^{19} + 19833673958682303354 T^{20} + \)\(19\!\cdots\!20\)\( T^{21} + \)\(12\!\cdots\!92\)\( T^{22} + \)\(64\!\cdots\!88\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( 1 - 28 T + 392 T^{2} - 4576 T^{3} + 47101 T^{4} - 412456 T^{5} + 3555064 T^{6} - 33468548 T^{7} + 284297810 T^{8} - 1697416132 T^{9} + 6847892328 T^{10} + 9811724552 T^{11} - 838473877341 T^{12} + 11809995132944 T^{13} - 119954720751336 T^{14} + 1226255122473636 T^{15} - 11528951107881638 T^{16} + 87064113695628156 T^{17} - 604691747307484776 T^{18} + 4226927168027119984 T^{19} - 21307030697822620221 T^{20} + 17702601420645725752 T^{21} + \)\(87\!\cdots\!88\)\( T^{22} - \)\(15\!\cdots\!12\)\( T^{23} + \)\(18\!\cdots\!10\)\( T^{24} - \)\(15\!\cdots\!88\)\( T^{25} + \)\(11\!\cdots\!64\)\( T^{26} - \)\(95\!\cdots\!76\)\( T^{27} + \)\(77\!\cdots\!41\)\( T^{28} - \)\(53\!\cdots\!36\)\( T^{29} + \)\(32\!\cdots\!52\)\( T^{30} - \)\(16\!\cdots\!28\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \))(\( 1 - 980 T^{3} - 5143 T^{4} + 37120 T^{5} + 480200 T^{6} + 4163108 T^{7} + 16283382 T^{8} - 419714740 T^{9} - 921230040 T^{10} - 33110845128 T^{11} + 174785033735 T^{12} + 1489799348740 T^{13} + 12977385054352 T^{14} - 63336510408248 T^{15} - 1303869445947534 T^{16} - 4496892238985608 T^{17} + 65418998058988432 T^{18} + 533215574706882140 T^{19} + 4441581520848058535 T^{20} - 59739558616352951928 T^{21} - \)\(11\!\cdots\!40\)\( T^{22} - \)\(38\!\cdots\!40\)\( T^{23} + \)\(10\!\cdots\!02\)\( T^{24} + \)\(19\!\cdots\!48\)\( T^{25} + \)\(15\!\cdots\!00\)\( T^{26} + \)\(85\!\cdots\!20\)\( T^{27} - \)\(84\!\cdots\!63\)\( T^{28} - \)\(11\!\cdots\!80\)\( T^{29} + \)\(41\!\cdots\!21\)\( T^{32} \))
$73$ (\( 1 - 6433 T^{4} + 28398241 T^{8} \))(\( 1 + 48 T + 1152 T^{2} + 19536 T^{3} + 281534 T^{4} + 3718224 T^{5} + 44975232 T^{6} + 497542704 T^{7} + 5156421727 T^{8} + 51118101024 T^{9} + 484003176192 T^{10} + 4359246766560 T^{11} + 37783073022212 T^{12} + 318225013958880 T^{13} + 2579252925927168 T^{14} + 19885810306053408 T^{15} + 146433306900982207 T^{16} + 1031441645966807472 T^{17} + 6806291936888274048 T^{18} + 41076702311270923728 T^{19} + \)\(22\!\cdots\!54\)\( T^{20} + \)\(11\!\cdots\!68\)\( T^{21} + \)\(49\!\cdots\!48\)\( T^{22} + \)\(15\!\cdots\!96\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 + 36 T + 648 T^{2} + 7924 T^{3} + 70233 T^{4} + 481880 T^{5} + 3231584 T^{6} + 28241152 T^{7} + 319277158 T^{8} + 3708522760 T^{9} + 39538437552 T^{10} + 397255494304 T^{11} + 3755465860887 T^{12} + 32271569859380 T^{13} + 253257414732264 T^{14} + 1889168467651316 T^{15} + 14945359488791826 T^{16} + 137909298138546068 T^{17} + 1349608763108234856 T^{18} + 12554189291986429460 T^{19} + \)\(10\!\cdots\!67\)\( T^{20} + \)\(82\!\cdots\!72\)\( T^{21} + \)\(59\!\cdots\!28\)\( T^{22} + \)\(40\!\cdots\!20\)\( T^{23} + \)\(25\!\cdots\!98\)\( T^{24} + \)\(16\!\cdots\!76\)\( T^{25} + \)\(13\!\cdots\!16\)\( T^{26} + \)\(15\!\cdots\!60\)\( T^{27} + \)\(16\!\cdots\!93\)\( T^{28} + \)\(13\!\cdots\!92\)\( T^{29} + \)\(79\!\cdots\!32\)\( T^{30} + \)\(32\!\cdots\!52\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 + 16 T + 128 T^{2} + 1152 T^{3} + 7805 T^{4} + 54816 T^{5} + 541568 T^{6} + 7146512 T^{7} + 86717378 T^{8} + 500740592 T^{9} + 2420265600 T^{10} + 10517433760 T^{11} - 87904119197 T^{12} - 816524698944 T^{13} - 2565201036928 T^{14} - 7109009093136 T^{15} + 83596401317242 T^{16} - 518957663798928 T^{17} - 13669956325789312 T^{18} - 317641988809098048 T^{19} - 2496322361849132477 T^{20} + 21803393159115179680 T^{21} + \)\(36\!\cdots\!00\)\( T^{22} + \)\(55\!\cdots\!24\)\( T^{23} + \)\(69\!\cdots\!18\)\( T^{24} + \)\(42\!\cdots\!56\)\( T^{25} + \)\(23\!\cdots\!32\)\( T^{26} + \)\(17\!\cdots\!32\)\( T^{27} + \)\(17\!\cdots\!05\)\( T^{28} + \)\(19\!\cdots\!16\)\( T^{29} + \)\(15\!\cdots\!52\)\( T^{30} + \)\(14\!\cdots\!12\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 - 8 T + 32 T^{2} - 184 T^{3} - 2078 T^{4} - 14536 T^{5} + 199712 T^{6} - 3944312 T^{7} + 38950081 T^{8} \))(\( 1 + 48 T + 1152 T^{2} + 20872 T^{3} + 343830 T^{4} + 5129496 T^{5} + 67943840 T^{6} + 833039952 T^{7} + 9687007279 T^{8} + 105044318112 T^{9} + 1064638861920 T^{10} + 10297866839696 T^{11} + 94546106295284 T^{12} + 813531480335984 T^{13} + 6644411137242720 T^{14} + 51790945557622368 T^{15} + 377309718164639599 T^{16} + 2563310914924252848 T^{17} + 16516295183925940640 T^{18} + 98506374328866645864 T^{19} + \)\(52\!\cdots\!30\)\( T^{20} + \)\(25\!\cdots\!68\)\( T^{21} + \)\(10\!\cdots\!52\)\( T^{22} + \)\(35\!\cdots\!92\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( 1 + 20 T + 200 T^{2} + 2012 T^{3} + 5452 T^{4} - 105764 T^{5} - 1181608 T^{6} - 9931596 T^{7} + 30868644 T^{8} + 1091747236 T^{9} + 7713983432 T^{10} + 24475309548 T^{11} - 710333850124 T^{12} - 7105147071380 T^{13} - 22523104678632 T^{14} + 233493779010692 T^{15} + 5644809254802998 T^{16} + 18446008541844668 T^{17} - 140566696299342312 T^{18} - 3503114606926123820 T^{19} - 27667560999371660044 T^{20} + 75311907862179197652 T^{21} + \)\(18\!\cdots\!72\)\( T^{22} + \)\(20\!\cdots\!24\)\( T^{23} + \)\(46\!\cdots\!84\)\( T^{24} - \)\(11\!\cdots\!24\)\( T^{25} - \)\(11\!\cdots\!08\)\( T^{26} - \)\(79\!\cdots\!56\)\( T^{27} + \)\(32\!\cdots\!32\)\( T^{28} + \)\(93\!\cdots\!68\)\( T^{29} + \)\(73\!\cdots\!00\)\( T^{30} + \)\(58\!\cdots\!80\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( 1 - 48 T + 1152 T^{2} - 19944 T^{3} + 296696 T^{4} - 3941880 T^{5} + 46298016 T^{6} - 485214464 T^{7} + 4549428316 T^{8} - 36856167648 T^{9} + 238044709152 T^{10} - 869359096056 T^{11} - 6341679897080 T^{12} + 189718840156952 T^{13} - 2719681100237632 T^{14} + 30434001497226480 T^{15} - 290524579858067386 T^{16} + 2404286118280891920 T^{17} - 16973529746583061312 T^{18} + 93538787232143457128 T^{19} - \)\(24\!\cdots\!80\)\( T^{20} - \)\(26\!\cdots\!44\)\( T^{21} + \)\(57\!\cdots\!92\)\( T^{22} - \)\(70\!\cdots\!32\)\( T^{23} + \)\(69\!\cdots\!76\)\( T^{24} - \)\(58\!\cdots\!16\)\( T^{25} + \)\(43\!\cdots\!16\)\( T^{26} - \)\(29\!\cdots\!20\)\( T^{27} + \)\(17\!\cdots\!36\)\( T^{28} - \)\(93\!\cdots\!16\)\( T^{29} + \)\(42\!\cdots\!12\)\( T^{30} - \)\(13\!\cdots\!52\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( 1 - 180 T^{2} + 20726 T^{4} - 1240020 T^{6} + 47458321 T^{8} \))(\( 1 - 28 T^{2} + 18146 T^{4} - 704956 T^{6} + 116248831 T^{8} - 8556378760 T^{10} + 533387322908 T^{12} - 58944893277640 T^{14} + 5516974337472751 T^{16} - 230478577848716764 T^{18} + 40870094844395037986 T^{20} - \)\(43\!\cdots\!72\)\( T^{22} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 - 828 T^{2} + 342524 T^{4} - 94075988 T^{6} + 19195035668 T^{8} - 3081982578236 T^{10} + 402380154128388 T^{12} - 43537915650417268 T^{14} + 3943448834455630742 T^{16} - \)\(29\!\cdots\!52\)\( T^{18} + \)\(19\!\cdots\!48\)\( T^{20} - \)\(10\!\cdots\!84\)\( T^{22} + \)\(43\!\cdots\!88\)\( T^{24} - \)\(14\!\cdots\!12\)\( T^{26} + \)\(36\!\cdots\!64\)\( T^{28} - \)\(60\!\cdots\!12\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 - 844 T^{2} + 344604 T^{4} - 91150276 T^{6} + 17668583636 T^{8} - 2690740216332 T^{10} + 336111874946020 T^{12} - 35339696511651876 T^{14} + 3169518577685933590 T^{16} - \)\(24\!\cdots\!64\)\( T^{18} + \)\(15\!\cdots\!20\)\( T^{20} - \)\(87\!\cdots\!08\)\( T^{22} + \)\(39\!\cdots\!76\)\( T^{24} - \)\(14\!\cdots\!24\)\( T^{26} + \)\(36\!\cdots\!44\)\( T^{28} - \)\(62\!\cdots\!76\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( ( 1 + 2 T + 17 T^{2} + 178 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 12 T + 340 T^{2} - 2548 T^{3} + 50723 T^{4} - 304656 T^{5} + 5315024 T^{6} - 27114384 T^{7} + 401776883 T^{8} - 1796261012 T^{9} + 21332361940 T^{10} - 67008713388 T^{11} + 496981290961 T^{12} )^{2} \))(\( ( 1 - 22 T + 505 T^{2} - 7670 T^{3} + 111036 T^{4} - 1313986 T^{5} + 14684011 T^{6} - 149247410 T^{7} + 1453938158 T^{8} - 13283019490 T^{9} + 116312051131 T^{10} - 926319396434 T^{11} + 6966647471676 T^{12} - 42829735973830 T^{13} + 250975551935305 T^{14} - 973089367701638 T^{15} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 - 2 T + 397 T^{2} - 726 T^{3} + 86392 T^{4} - 145586 T^{5} + 12506639 T^{6} - 18741318 T^{7} + 1300302358 T^{8} - 1667977302 T^{9} + 99065087519 T^{10} - 102633616834 T^{11} + 5420427684472 T^{12} - 4054027159974 T^{13} + 197301572511517 T^{14} - 88462669791058 T^{15} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( 1 + 8 T + 32 T^{2} + 832 T^{3} + 21583 T^{4} + 80704 T^{5} + 301088 T^{6} + 7301384 T^{7} + 88529281 T^{8} \))(\( 1 + 20 T + 200 T^{2} + 3124 T^{3} + 46602 T^{4} + 333956 T^{5} + 2238408 T^{6} + 24420932 T^{7} + 177705615 T^{8} + 648739368 T^{9} + 5173671280 T^{10} + 83157304520 T^{11} + 1241948910092 T^{12} + 8066258538440 T^{13} + 48679073073520 T^{14} + 592086905210664 T^{15} + 15732150325612815 T^{16} + 209710852477059524 T^{17} + 1864531199609113032 T^{18} + 26983071891172705028 T^{19} + \)\(36\!\cdots\!22\)\( T^{20} + \)\(23\!\cdots\!08\)\( T^{21} + \)\(14\!\cdots\!00\)\( T^{22} + \)\(14\!\cdots\!60\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 - 16 T + 128 T^{2} - 1304 T^{3} - 8747 T^{4} + 282264 T^{5} - 2546400 T^{6} + 25347912 T^{7} - 46534438 T^{8} - 1230942648 T^{9} + 10160935232 T^{10} - 79841469624 T^{11} - 415328029989 T^{12} + 4708876664120 T^{13} + 26123170938272 T^{14} - 1012407482206592 T^{15} + 18377547368018666 T^{16} - 98203525774039424 T^{17} + 245792915358201248 T^{18} + 4297664591672392760 T^{19} - 36768691874072607909 T^{20} - \)\(68\!\cdots\!68\)\( T^{21} + \)\(84\!\cdots\!28\)\( T^{22} - \)\(99\!\cdots\!24\)\( T^{23} - \)\(36\!\cdots\!18\)\( T^{24} + \)\(19\!\cdots\!04\)\( T^{25} - \)\(18\!\cdots\!00\)\( T^{26} + \)\(20\!\cdots\!92\)\( T^{27} - \)\(60\!\cdots\!27\)\( T^{28} - \)\(87\!\cdots\!08\)\( T^{29} + \)\(83\!\cdots\!32\)\( T^{30} - \)\(10\!\cdots\!88\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( 1 - 12 T + 72 T^{2} - 52 T^{3} - 35559 T^{4} + 311744 T^{5} - 1179328 T^{6} - 10495752 T^{7} + 796323494 T^{8} - 5466511168 T^{9} + 15465057968 T^{10} + 261428633848 T^{11} - 11355948729225 T^{12} + 57422324188924 T^{13} - 45056933441080 T^{14} - 4833646649041196 T^{15} + 126256184960263378 T^{16} - 468863724956996012 T^{17} - 423940686747121720 T^{18} + 52407804884477833852 T^{19} - \)\(10\!\cdots\!25\)\( T^{20} + \)\(22\!\cdots\!36\)\( T^{21} + \)\(12\!\cdots\!72\)\( T^{22} - \)\(44\!\cdots\!84\)\( T^{23} + \)\(62\!\cdots\!34\)\( T^{24} - \)\(79\!\cdots\!84\)\( T^{25} - \)\(86\!\cdots\!72\)\( T^{26} + \)\(22\!\cdots\!32\)\( T^{27} - \)\(24\!\cdots\!19\)\( T^{28} - \)\(34\!\cdots\!04\)\( T^{29} + \)\(47\!\cdots\!68\)\( T^{30} - \)\(75\!\cdots\!16\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))
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