Properties

Label 2016.3.m
Level 2016
Weight 3
Character orbit m
Rep. character \(\chi_{2016}(127,\cdot)\)
Character field \(\Q\)
Dimension 60
Newform subspaces 6
Sturm bound 1152
Trace bound 5

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Defining parameters

Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2016.m (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(1152\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(2016, [\chi])\).

Total New Old
Modular forms 800 60 740
Cusp forms 736 60 676
Eisenstein series 64 0 64

Trace form

\( 60q - 8q^{5} + O(q^{10}) \) \( 60q - 8q^{5} + 24q^{13} + 24q^{17} + 372q^{25} - 40q^{29} - 8q^{37} - 264q^{41} - 420q^{49} + 344q^{53} - 40q^{61} + 240q^{65} + 120q^{73} + 16q^{85} - 232q^{89} - 72q^{97} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(2016, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
2016.3.m.a \(4\) \(54.932\) \(\Q(i, \sqrt{7})\) None \(0\) \(0\) \(8\) \(0\) \(q+(2-\beta _{2})q^{5}+\beta _{3}q^{7}+(-2\beta _{1}+4\beta _{3})q^{11}+\cdots\)
2016.3.m.b \(8\) \(54.932\) 8.0.49787136.1 None \(0\) \(0\) \(-16\) \(0\) \(q+(-2+\beta _{6})q^{5}-\beta _{3}q^{7}+(\beta _{1}-2\beta _{3}+\cdots)q^{11}+\cdots\)
2016.3.m.c \(8\) \(54.932\) 8.0.1997017344.2 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{5}+\beta _{5}q^{7}+(-2\beta _{5}+\beta _{6}+\beta _{7})q^{11}+\cdots\)
2016.3.m.d \(12\) \(54.932\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(-16\) \(0\) \(q+(-1+\beta _{1})q^{5}-\beta _{6}q^{7}+(-\beta _{3}+\beta _{6}+\cdots)q^{11}+\cdots\)
2016.3.m.e \(12\) \(54.932\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(16\) \(0\) \(q+(1-\beta _{1})q^{5}-\beta _{6}q^{7}+(\beta _{3}-\beta _{6})q^{11}+\cdots\)
2016.3.m.f \(16\) \(54.932\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{5}+\beta _{4}q^{7}+(-\beta _{4}+\beta _{11})q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(2016, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(2016, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(672, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(1008, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( ( 1 - 4 T + 26 T^{2} - 100 T^{3} + 625 T^{4} )^{2} \))(\( ( 1 + 8 T + 104 T^{2} + 552 T^{3} + 3902 T^{4} + 13800 T^{5} + 65000 T^{6} + 125000 T^{7} + 390625 T^{8} )^{2} \))(\( ( 1 + 24 T^{2} + 224 T^{3} + 78 T^{4} + 5600 T^{5} + 15000 T^{6} + 390625 T^{8} )^{2} \))(\( ( 1 + 8 T + 74 T^{2} + 360 T^{3} + 1935 T^{4} + 8016 T^{5} + 41452 T^{6} + 200400 T^{7} + 1209375 T^{8} + 5625000 T^{9} + 28906250 T^{10} + 78125000 T^{11} + 244140625 T^{12} )^{2} \))(\( ( 1 - 8 T + 74 T^{2} - 360 T^{3} + 1935 T^{4} - 8016 T^{5} + 41452 T^{6} - 200400 T^{7} + 1209375 T^{8} - 5625000 T^{9} + 28906250 T^{10} - 78125000 T^{11} + 244140625 T^{12} )^{2} \))(\( ( 1 + 84 T^{2} + 80 T^{3} + 4452 T^{4} + 3344 T^{5} + 172204 T^{6} + 137760 T^{7} + 4809334 T^{8} + 3444000 T^{9} + 107627500 T^{10} + 52250000 T^{11} + 1739062500 T^{12} + 781250000 T^{13} + 20507812500 T^{14} + 152587890625 T^{16} )^{2} \))
$7$ (\( ( 1 + 7 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} )^{4} \))(\( ( 1 + 7 T^{2} )^{4} \))(\( ( 1 + 7 T^{2} )^{6} \))(\( ( 1 + 7 T^{2} )^{6} \))(\( ( 1 + 7 T^{2} )^{8} \))
$11$ (\( 1 - 228 T^{2} + 35110 T^{4} - 3338148 T^{6} + 214358881 T^{8} \))(\( 1 - 720 T^{2} + 247100 T^{4} - 52647984 T^{6} + 7630714182 T^{8} - 770819133744 T^{10} + 52968079495100 T^{12} - 2259668431239120 T^{14} + 45949729863572161 T^{16} \))(\( 1 - 296 T^{2} + 72860 T^{4} - 12907416 T^{6} + 1704788486 T^{8} - 188977477656 T^{10} + 15618188069660 T^{12} - 928974799509416 T^{14} + 45949729863572161 T^{16} \))(\( 1 - 644 T^{2} + 191202 T^{4} - 35732660 T^{6} + 5112464783 T^{8} - 660287106440 T^{10} + 81869903390940 T^{12} - 9667263525388040 T^{14} + 1095902230035787823 T^{16} - \)\(11\!\cdots\!60\)\( T^{18} + \)\(87\!\cdots\!22\)\( T^{20} - \)\(43\!\cdots\!44\)\( T^{22} + \)\(98\!\cdots\!41\)\( T^{24} \))(\( 1 - 644 T^{2} + 191202 T^{4} - 35732660 T^{6} + 5112464783 T^{8} - 660287106440 T^{10} + 81869903390940 T^{12} - 9667263525388040 T^{14} + 1095902230035787823 T^{16} - \)\(11\!\cdots\!60\)\( T^{18} + \)\(87\!\cdots\!22\)\( T^{20} - \)\(43\!\cdots\!44\)\( T^{22} + \)\(98\!\cdots\!41\)\( T^{24} \))(\( 1 - 568 T^{2} + 171608 T^{4} - 37528232 T^{6} + 6700418780 T^{8} - 1058153018936 T^{10} + 154812072039528 T^{12} - 21114927937649640 T^{14} + 2662322500884271174 T^{16} - \)\(30\!\cdots\!40\)\( T^{18} + \)\(33\!\cdots\!68\)\( T^{20} - \)\(33\!\cdots\!56\)\( T^{22} + \)\(30\!\cdots\!80\)\( T^{24} - \)\(25\!\cdots\!32\)\( T^{26} + \)\(16\!\cdots\!28\)\( T^{28} - \)\(81\!\cdots\!08\)\( T^{30} + \)\(21\!\cdots\!21\)\( T^{32} \))
$13$ (\( ( 1 + 4 T + 90 T^{2} + 676 T^{3} + 28561 T^{4} )^{2} \))(\( ( 1 + 32 T + 796 T^{2} + 14048 T^{3} + 200422 T^{4} + 2374112 T^{5} + 22734556 T^{6} + 154457888 T^{7} + 815730721 T^{8} )^{2} \))(\( ( 1 - 16 T + 696 T^{2} - 7536 T^{3} + 176398 T^{4} - 1273584 T^{5} + 19878456 T^{6} - 77228944 T^{7} + 815730721 T^{8} )^{2} \))(\( ( 1 - 12 T + 386 T^{2} - 3644 T^{3} + 56479 T^{4} - 216984 T^{5} + 6711132 T^{6} - 36670296 T^{7} + 1613096719 T^{8} - 17588891996 T^{9} + 314872058306 T^{10} - 1654301902188 T^{11} + 23298085122481 T^{12} )^{2} \))(\( ( 1 - 12 T + 386 T^{2} - 3644 T^{3} + 56479 T^{4} - 216984 T^{5} + 6711132 T^{6} - 36670296 T^{7} + 1613096719 T^{8} - 17588891996 T^{9} + 314872058306 T^{10} - 1654301902188 T^{11} + 23298085122481 T^{12} )^{2} \))(\( ( 1 - 8 T + 664 T^{2} - 4824 T^{3} + 225308 T^{4} - 1395464 T^{5} + 53184936 T^{6} - 270740888 T^{7} + 9879976070 T^{8} - 45755210072 T^{9} + 1519014957096 T^{10} - 6735638194376 T^{11} + 183790657287068 T^{12} - 665029364679576 T^{13} + 15469928521327384 T^{14} - 31499011085594312 T^{15} + 665416609183179841 T^{16} )^{2} \))
$17$ (\( ( 1 - 28 T + 662 T^{2} - 8092 T^{3} + 83521 T^{4} )^{2} \))(\( ( 1 + 32 T + 968 T^{2} + 15600 T^{3} + 314654 T^{4} + 4508400 T^{5} + 80848328 T^{6} + 772402208 T^{7} + 6975757441 T^{8} )^{2} \))(\( ( 1 - 8 T + 428 T^{2} + 1416 T^{3} + 75814 T^{4} + 409224 T^{5} + 35746988 T^{6} - 193100552 T^{7} + 6975757441 T^{8} )^{2} \))(\( ( 1 + 40 T + 1610 T^{2} + 37864 T^{3} + 872767 T^{4} + 15204848 T^{5} + 282802860 T^{6} + 4394201072 T^{7} + 72894372607 T^{8} + 913944912616 T^{9} + 11230969480010 T^{10} + 80639756017960 T^{11} + 582622237229761 T^{12} )^{2} \))(\( ( 1 - 40 T + 1610 T^{2} - 37864 T^{3} + 872767 T^{4} - 15204848 T^{5} + 282802860 T^{6} - 4394201072 T^{7} + 72894372607 T^{8} - 913944912616 T^{9} + 11230969480010 T^{10} - 80639756017960 T^{11} + 582622237229761 T^{12} )^{2} \))(\( ( 1 - 8 T + 1460 T^{2} - 16744 T^{3} + 1091556 T^{4} - 13257752 T^{5} + 536903180 T^{6} - 6028156536 T^{7} + 184043682742 T^{8} - 1742137238904 T^{9} + 44842690496780 T^{10} - 320009903684888 T^{11} + 7614429889268196 T^{12} - 33755801869118056 T^{13} + 850628466355451060 T^{14} - 1347022612475207432 T^{15} + 48661191875666868481 T^{16} )^{2} \))
$19$ (\( 1 - 348 T^{2} + 111718 T^{4} - 45351708 T^{6} + 16983563041 T^{8} \))(\( 1 - 2376 T^{2} + 2620892 T^{4} - 1746585720 T^{6} + 767524694790 T^{8} - 227616797616120 T^{10} + 44512084505652572 T^{12} - 5258836247701198536 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} \))(\( 1 - 1936 T^{2} + 1795420 T^{4} - 1065665392 T^{6} + 449725089670 T^{8} - 138878579550832 T^{10} + 30492628755072220 T^{12} - 4284977683312087696 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} \))(\( 1 - 2572 T^{2} + 3295394 T^{4} - 2806678172 T^{6} + 1778495598543 T^{8} - 885164316367896 T^{10} + 354799999408803420 T^{12} - \)\(11\!\cdots\!16\)\( T^{14} + \)\(30\!\cdots\!63\)\( T^{16} - \)\(62\!\cdots\!92\)\( T^{18} + \)\(95\!\cdots\!14\)\( T^{20} - \)\(96\!\cdots\!72\)\( T^{22} + \)\(48\!\cdots\!21\)\( T^{24} \))(\( 1 - 2572 T^{2} + 3295394 T^{4} - 2806678172 T^{6} + 1778495598543 T^{8} - 885164316367896 T^{10} + 354799999408803420 T^{12} - \)\(11\!\cdots\!16\)\( T^{14} + \)\(30\!\cdots\!63\)\( T^{16} - \)\(62\!\cdots\!92\)\( T^{18} + \)\(95\!\cdots\!14\)\( T^{20} - \)\(96\!\cdots\!72\)\( T^{22} + \)\(48\!\cdots\!21\)\( T^{24} \))(\( 1 - 1616 T^{2} + 1503864 T^{4} - 958989808 T^{6} + 454590996380 T^{8} - 158491050727248 T^{10} + 38845189362274888 T^{12} - 5743391356121447024 T^{14} + \)\(76\!\cdots\!10\)\( T^{16} - \)\(74\!\cdots\!04\)\( T^{18} + \)\(65\!\cdots\!08\)\( T^{20} - \)\(35\!\cdots\!28\)\( T^{22} + \)\(13\!\cdots\!80\)\( T^{24} - \)\(36\!\cdots\!08\)\( T^{26} + \)\(73\!\cdots\!44\)\( T^{28} - \)\(10\!\cdots\!56\)\( T^{30} + \)\(83\!\cdots\!61\)\( T^{32} \))
$23$ (\( 1 - 1604 T^{2} + 1138374 T^{4} - 448864964 T^{6} + 78310985281 T^{8} \))(\( 1 - 2896 T^{2} + 3902332 T^{4} - 3323362480 T^{6} + 2032863959686 T^{8} - 930013079765680 T^{10} + 305595463813575292 T^{12} - 63464752355130849616 T^{14} + \)\(61\!\cdots\!61\)\( T^{16} \))(\( 1 - 1928 T^{2} + 2366876 T^{4} - 1917103032 T^{6} + 1186521008582 T^{8} - 536484029577912 T^{10} + 185352391597952156 T^{12} - 42251395904935178888 T^{14} + \)\(61\!\cdots\!61\)\( T^{16} \))(\( 1 - 3268 T^{2} + 5271042 T^{4} - 5712064692 T^{6} + 4715761171375 T^{8} - 3168464485539592 T^{10} + 1805463479913892636 T^{12} - \)\(88\!\cdots\!72\)\( T^{14} + \)\(36\!\cdots\!75\)\( T^{16} - \)\(12\!\cdots\!32\)\( T^{18} + \)\(32\!\cdots\!62\)\( T^{20} - \)\(56\!\cdots\!68\)\( T^{22} + \)\(48\!\cdots\!41\)\( T^{24} \))(\( 1 - 3268 T^{2} + 5271042 T^{4} - 5712064692 T^{6} + 4715761171375 T^{8} - 3168464485539592 T^{10} + 1805463479913892636 T^{12} - \)\(88\!\cdots\!72\)\( T^{14} + \)\(36\!\cdots\!75\)\( T^{16} - \)\(12\!\cdots\!32\)\( T^{18} + \)\(32\!\cdots\!62\)\( T^{20} - \)\(56\!\cdots\!68\)\( T^{22} + \)\(48\!\cdots\!41\)\( T^{24} \))(\( 1 - 2680 T^{2} + 3819992 T^{4} - 4018450664 T^{6} + 3538843517276 T^{8} - 2722647796005752 T^{10} + 1862831153417521128 T^{12} - \)\(11\!\cdots\!56\)\( T^{14} + \)\(63\!\cdots\!26\)\( T^{16} - \)\(32\!\cdots\!96\)\( T^{18} + \)\(14\!\cdots\!68\)\( T^{20} - \)\(59\!\cdots\!92\)\( T^{22} + \)\(21\!\cdots\!36\)\( T^{24} - \)\(68\!\cdots\!64\)\( T^{26} + \)\(18\!\cdots\!72\)\( T^{28} - \)\(36\!\cdots\!80\)\( T^{30} + \)\(37\!\cdots\!21\)\( T^{32} \))
$29$ (\( ( 1 + 20 T + 774 T^{2} + 16820 T^{3} + 707281 T^{4} )^{2} \))(\( ( 1 + 32 T + 1708 T^{2} + 38048 T^{3} + 1465030 T^{4} + 31998368 T^{5} + 1208035948 T^{6} + 19034346272 T^{7} + 500246412961 T^{8} )^{2} \))(\( ( 1 - 40 T + 3660 T^{2} - 100632 T^{3} + 4741126 T^{4} - 84631512 T^{5} + 2588648460 T^{6} - 23792932840 T^{7} + 500246412961 T^{8} )^{2} \))(\( ( 1 + 3110 T^{2} + 21248 T^{3} + 4689919 T^{4} + 44939520 T^{5} + 4702874772 T^{6} + 37794136320 T^{7} + 3317090600239 T^{8} + 12638805924608 T^{9} + 1555766344308710 T^{10} + 353814783205469041 T^{12} )^{2} \))(\( ( 1 + 3110 T^{2} - 21248 T^{3} + 4689919 T^{4} - 44939520 T^{5} + 4702874772 T^{6} - 37794136320 T^{7} + 3317090600239 T^{8} - 12638805924608 T^{9} + 1555766344308710 T^{10} + 353814783205469041 T^{12} )^{2} \))(\( ( 1 + 8 T + 2152 T^{2} + 6936 T^{3} + 2840252 T^{4} + 35865416 T^{5} + 3038036952 T^{6} + 47174886680 T^{7} + 2401820829254 T^{8} + 39674079697880 T^{9} + 2148745813447512 T^{10} + 21333585854166536 T^{11} + 1420825874905306172 T^{12} + 2918025370170194136 T^{13} + \)\(76\!\cdots\!32\)\( T^{14} + \)\(23\!\cdots\!48\)\( T^{15} + \)\(25\!\cdots\!21\)\( T^{16} )^{2} \))
$31$ (\( ( 1 - 1906 T^{2} + 923521 T^{4} )^{2} \))(\( 1 - 5896 T^{2} + 16111132 T^{4} - 27072775480 T^{6} + 31035743101510 T^{8} - 25002276684065080 T^{10} + 13741040085828893212 T^{12} - \)\(46\!\cdots\!56\)\( T^{14} + \)\(72\!\cdots\!81\)\( T^{16} \))(\( 1 - 3624 T^{2} + 5953500 T^{4} - 5905968152 T^{6} + 5172694104774 T^{8} - 5454285613703192 T^{10} + 5077686791404993500 T^{12} - \)\(28\!\cdots\!64\)\( T^{14} + \)\(72\!\cdots\!81\)\( T^{16} \))(\( 1 - 7340 T^{2} + 24952962 T^{4} - 52480410364 T^{6} + 78028577767919 T^{8} - 91071297834552920 T^{10} + 91962598141207812380 T^{12} - \)\(84\!\cdots\!20\)\( T^{14} + \)\(66\!\cdots\!79\)\( T^{16} - \)\(41\!\cdots\!04\)\( T^{18} + \)\(18\!\cdots\!22\)\( T^{20} - \)\(49\!\cdots\!40\)\( T^{22} + \)\(62\!\cdots\!21\)\( T^{24} \))(\( 1 - 7340 T^{2} + 24952962 T^{4} - 52480410364 T^{6} + 78028577767919 T^{8} - 91071297834552920 T^{10} + 91962598141207812380 T^{12} - \)\(84\!\cdots\!20\)\( T^{14} + \)\(66\!\cdots\!79\)\( T^{16} - \)\(41\!\cdots\!04\)\( T^{18} + \)\(18\!\cdots\!22\)\( T^{20} - \)\(49\!\cdots\!40\)\( T^{22} + \)\(62\!\cdots\!21\)\( T^{24} \))(\( 1 - 2448 T^{2} + 4544376 T^{4} - 6660944560 T^{6} + 9940033472796 T^{8} - 12711964760567440 T^{10} + 15108037137965584456 T^{12} - \)\(15\!\cdots\!88\)\( T^{14} + \)\(16\!\cdots\!18\)\( T^{16} - \)\(14\!\cdots\!48\)\( T^{18} + \)\(12\!\cdots\!96\)\( T^{20} - \)\(10\!\cdots\!40\)\( T^{22} + \)\(72\!\cdots\!76\)\( T^{24} - \)\(44\!\cdots\!60\)\( T^{26} + \)\(28\!\cdots\!96\)\( T^{28} - \)\(14\!\cdots\!68\)\( T^{30} + \)\(52\!\cdots\!61\)\( T^{32} \))
$37$ (\( ( 1 - 68 T + 3782 T^{2} - 93092 T^{3} + 1874161 T^{4} )^{2} \))(\( ( 1 - 64 T + 4220 T^{2} - 216768 T^{3} + 8557286 T^{4} - 296755392 T^{5} + 7908959420 T^{6} - 164206490176 T^{7} + 3512479453921 T^{8} )^{2} \))(\( ( 1 + 88 T + 3692 T^{2} + 66024 T^{3} + 510982 T^{4} + 90386856 T^{5} + 6919402412 T^{6} + 225783923992 T^{7} + 3512479453921 T^{8} )^{2} \))(\( ( 1 - 4 T + 4562 T^{2} - 79444 T^{3} + 11000959 T^{4} - 208839944 T^{5} + 18676988604 T^{6} - 285901883336 T^{7} + 20617568320399 T^{8} - 203831568836596 T^{9} + 16023931268787602 T^{10} - 19234337489671396 T^{11} + 6582952005840035281 T^{12} )^{2} \))(\( ( 1 - 4 T + 4562 T^{2} - 79444 T^{3} + 11000959 T^{4} - 208839944 T^{5} + 18676988604 T^{6} - 285901883336 T^{7} + 20617568320399 T^{8} - 203831568836596 T^{9} + 16023931268787602 T^{10} - 19234337489671396 T^{11} + 6582952005840035281 T^{12} )^{2} \))(\( ( 1 + 56 T + 8696 T^{2} + 389992 T^{3} + 34664796 T^{4} + 1289117240 T^{5} + 84548927816 T^{6} + 2643048584616 T^{7} + 139161224552326 T^{8} + 3618333512339304 T^{9} + 158458303104562376 T^{10} + 3307522146965191160 T^{11} + \)\(12\!\cdots\!16\)\( T^{12} + \)\(18\!\cdots\!08\)\( T^{13} + \)\(57\!\cdots\!76\)\( T^{14} + \)\(50\!\cdots\!84\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} )^{2} \))
$41$ (\( ( 1 - 60 T + 4150 T^{2} - 100860 T^{3} + 2825761 T^{4} )^{2} \))(\( ( 1 + 5384 T^{2} - 21168 T^{3} + 12368094 T^{4} - 35583408 T^{5} + 15213897224 T^{6} + 7984925229121 T^{8} )^{2} \))(\( ( 1 + 72 T + 4364 T^{2} + 180408 T^{3} + 8880422 T^{4} + 303265848 T^{5} + 12331621004 T^{6} + 342007505352 T^{7} + 7984925229121 T^{8} )^{2} \))(\( ( 1 + 24 T + 5610 T^{2} + 109272 T^{3} + 12333791 T^{4} + 221579664 T^{5} + 19306356972 T^{6} + 372475415184 T^{7} + 34852345589951 T^{8} + 519053390622552 T^{9} + 44795430535368810 T^{10} + 322143823443657624 T^{11} + 22563490300366186081 T^{12} )^{2} \))(\( ( 1 - 24 T + 5610 T^{2} - 109272 T^{3} + 12333791 T^{4} - 221579664 T^{5} + 19306356972 T^{6} - 372475415184 T^{7} + 34852345589951 T^{8} - 519053390622552 T^{9} + 44795430535368810 T^{10} - 322143823443657624 T^{11} + 22563490300366186081 T^{12} )^{2} \))(\( ( 1 + 120 T + 10612 T^{2} + 619224 T^{3} + 32879204 T^{4} + 1430831784 T^{5} + 62637551820 T^{6} + 2429685088776 T^{7} + 102774652510262 T^{8} + 4084300634232456 T^{9} + 176998751068435020 T^{10} + 6796600125335995944 T^{11} + \)\(26\!\cdots\!84\)\( T^{12} + \)\(83\!\cdots\!24\)\( T^{13} + \)\(23\!\cdots\!72\)\( T^{14} + \)\(45\!\cdots\!20\)\( T^{15} + \)\(63\!\cdots\!41\)\( T^{16} )^{2} \))
$43$ (\( 1 - 1508 T^{2} + 5793318 T^{4} - 5155551908 T^{6} + 11688200277601 T^{8} \))(\( 1 - 8200 T^{2} + 33845980 T^{4} - 91985805496 T^{6} + 190535158400902 T^{8} - 314481163815530296 T^{10} + \)\(39\!\cdots\!80\)\( T^{12} - \)\(32\!\cdots\!00\)\( T^{14} + \)\(13\!\cdots\!01\)\( T^{16} \))(\( 1 - 4392 T^{2} + 15655580 T^{4} - 32786290584 T^{6} + 70340912004102 T^{8} - 112089803034869784 T^{10} + \)\(18\!\cdots\!80\)\( T^{12} - \)\(17\!\cdots\!92\)\( T^{14} + \)\(13\!\cdots\!01\)\( T^{16} \))(\( 1 - 14156 T^{2} + 94495970 T^{4} - 406877897820 T^{6} + 1300026263155855 T^{8} - 3294725001750613144 T^{10} + \)\(67\!\cdots\!44\)\( T^{12} - \)\(11\!\cdots\!44\)\( T^{14} + \)\(15\!\cdots\!55\)\( T^{16} - \)\(16\!\cdots\!20\)\( T^{18} + \)\(12\!\cdots\!70\)\( T^{20} - \)\(66\!\cdots\!56\)\( T^{22} + \)\(15\!\cdots\!01\)\( T^{24} \))(\( 1 - 14156 T^{2} + 94495970 T^{4} - 406877897820 T^{6} + 1300026263155855 T^{8} - 3294725001750613144 T^{10} + \)\(67\!\cdots\!44\)\( T^{12} - \)\(11\!\cdots\!44\)\( T^{14} + \)\(15\!\cdots\!55\)\( T^{16} - \)\(16\!\cdots\!20\)\( T^{18} + \)\(12\!\cdots\!70\)\( T^{20} - \)\(66\!\cdots\!56\)\( T^{22} + \)\(15\!\cdots\!01\)\( T^{24} \))(\( 1 - 16272 T^{2} + 136621816 T^{4} - 780160752048 T^{6} + 3370386034053532 T^{8} - 11622652546480860816 T^{10} + \)\(32\!\cdots\!76\)\( T^{12} - \)\(78\!\cdots\!56\)\( T^{14} + \)\(15\!\cdots\!30\)\( T^{16} - \)\(26\!\cdots\!56\)\( T^{18} + \)\(38\!\cdots\!76\)\( T^{20} - \)\(46\!\cdots\!16\)\( T^{22} + \)\(46\!\cdots\!32\)\( T^{24} - \)\(36\!\cdots\!48\)\( T^{26} + \)\(21\!\cdots\!16\)\( T^{28} - \)\(88\!\cdots\!72\)\( T^{30} + \)\(18\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 - 5348 T^{2} + 14587206 T^{4} - 26096533988 T^{6} + 23811286661761 T^{8} \))(\( 1 - 11176 T^{2} + 64536028 T^{4} - 240115798168 T^{6} + 627402380719558 T^{8} - 1171688498120224408 T^{10} + \)\(15\!\cdots\!08\)\( T^{12} - \)\(12\!\cdots\!16\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} \))(\( 1 - 6696 T^{2} + 34041564 T^{4} - 108976927256 T^{6} + 286725887177670 T^{8} - 531772641369485336 T^{10} + \)\(81\!\cdots\!04\)\( T^{12} - \)\(77\!\cdots\!36\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} \))(\( 1 - 14828 T^{2} + 113393538 T^{4} - 585955403836 T^{6} + 2269661951584367 T^{8} - 6918101399845148888 T^{10} + \)\(16\!\cdots\!88\)\( T^{12} - \)\(33\!\cdots\!28\)\( T^{14} + \)\(54\!\cdots\!87\)\( T^{16} - \)\(68\!\cdots\!76\)\( T^{18} + \)\(64\!\cdots\!98\)\( T^{20} - \)\(41\!\cdots\!28\)\( T^{22} + \)\(13\!\cdots\!81\)\( T^{24} \))(\( 1 - 14828 T^{2} + 113393538 T^{4} - 585955403836 T^{6} + 2269661951584367 T^{8} - 6918101399845148888 T^{10} + \)\(16\!\cdots\!88\)\( T^{12} - \)\(33\!\cdots\!28\)\( T^{14} + \)\(54\!\cdots\!87\)\( T^{16} - \)\(68\!\cdots\!76\)\( T^{18} + \)\(64\!\cdots\!98\)\( T^{20} - \)\(41\!\cdots\!28\)\( T^{22} + \)\(13\!\cdots\!81\)\( T^{24} \))(\( 1 - 9264 T^{2} + 59936568 T^{4} - 267369406864 T^{6} + 980248186718364 T^{8} - 2939328460179000880 T^{10} + \)\(78\!\cdots\!52\)\( T^{12} - \)\(18\!\cdots\!48\)\( T^{14} + \)\(42\!\cdots\!42\)\( T^{16} - \)\(91\!\cdots\!88\)\( T^{18} + \)\(18\!\cdots\!72\)\( T^{20} - \)\(34\!\cdots\!80\)\( T^{22} + \)\(55\!\cdots\!44\)\( T^{24} - \)\(73\!\cdots\!64\)\( T^{26} + \)\(80\!\cdots\!08\)\( T^{28} - \)\(61\!\cdots\!04\)\( T^{30} + \)\(32\!\cdots\!41\)\( T^{32} \))
$53$ (\( ( 1 - 140 T + 10070 T^{2} - 393260 T^{3} + 7890481 T^{4} )^{2} \))(\( ( 1 - 80 T + 7772 T^{2} - 498480 T^{3} + 33336230 T^{4} - 1400230320 T^{5} + 61324818332 T^{6} - 1773148890320 T^{7} + 62259690411361 T^{8} )^{2} \))(\( ( 1 + 24 T + 6108 T^{2} + 242920 T^{3} + 18930726 T^{4} + 682362280 T^{5} + 48195057948 T^{6} + 531944667096 T^{7} + 62259690411361 T^{8} )^{2} \))(\( ( 1 - 128 T + 17654 T^{2} - 1495296 T^{3} + 124154463 T^{4} - 7734950016 T^{5} + 461963749876 T^{6} - 21727474594944 T^{7} + 979638431366703 T^{8} - 33142280538749184 T^{9} + 1099132574522167094 T^{10} - 22385596206785670272 T^{11} + \)\(49\!\cdots\!41\)\( T^{12} )^{2} \))(\( ( 1 + 128 T + 17654 T^{2} + 1495296 T^{3} + 124154463 T^{4} + 7734950016 T^{5} + 461963749876 T^{6} + 21727474594944 T^{7} + 979638431366703 T^{8} + 33142280538749184 T^{9} + 1099132574522167094 T^{10} + 22385596206785670272 T^{11} + \)\(49\!\cdots\!41\)\( T^{12} )^{2} \))(\( ( 1 + 24 T + 10392 T^{2} + 420232 T^{3} + 58443420 T^{4} + 3234882712 T^{5} + 221678878888 T^{6} + 14475939716424 T^{7} + 680170196938246 T^{8} + 40662914663435016 T^{9} + 1749152981967065128 T^{10} + 71699108638726901848 T^{11} + \)\(36\!\cdots\!20\)\( T^{12} + \)\(73\!\cdots\!68\)\( T^{13} + \)\(51\!\cdots\!72\)\( T^{14} + \)\(33\!\cdots\!56\)\( T^{15} + \)\(38\!\cdots\!21\)\( T^{16} )^{2} \))
$59$ (\( 1 - 9692 T^{2} + 45396006 T^{4} - 117441462812 T^{6} + 146830437604321 T^{8} \))(\( 1 - 10792 T^{2} + 87239068 T^{4} - 443336106136 T^{6} + 1836453342677254 T^{8} - 5372063642384227096 T^{10} + \)\(12\!\cdots\!28\)\( T^{12} - \)\(19\!\cdots\!52\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \))(\( 1 - 21392 T^{2} + 213726428 T^{4} - 1316320178544 T^{6} + 5495654048256518 T^{8} - 15950326795002102384 T^{10} + \)\(31\!\cdots\!88\)\( T^{12} - \)\(38\!\cdots\!52\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \))(\( 1 - 15628 T^{2} + 144668898 T^{4} - 990246600732 T^{6} + 5397356836371727 T^{8} - 24345088021389180952 T^{10} + \)\(92\!\cdots\!36\)\( T^{12} - \)\(29\!\cdots\!72\)\( T^{14} + \)\(79\!\cdots\!67\)\( T^{16} - \)\(17\!\cdots\!92\)\( T^{18} + \)\(31\!\cdots\!18\)\( T^{20} - \)\(40\!\cdots\!28\)\( T^{22} + \)\(31\!\cdots\!61\)\( T^{24} \))(\( 1 - 15628 T^{2} + 144668898 T^{4} - 990246600732 T^{6} + 5397356836371727 T^{8} - 24345088021389180952 T^{10} + \)\(92\!\cdots\!36\)\( T^{12} - \)\(29\!\cdots\!72\)\( T^{14} + \)\(79\!\cdots\!67\)\( T^{16} - \)\(17\!\cdots\!92\)\( T^{18} + \)\(31\!\cdots\!18\)\( T^{20} - \)\(40\!\cdots\!28\)\( T^{22} + \)\(31\!\cdots\!61\)\( T^{24} \))(\( 1 - 32944 T^{2} + 529003448 T^{4} - 5524852963856 T^{6} + 42286305386402588 T^{8} - \)\(25\!\cdots\!04\)\( T^{10} + \)\(12\!\cdots\!40\)\( T^{12} - \)\(53\!\cdots\!68\)\( T^{14} + \)\(19\!\cdots\!66\)\( T^{16} - \)\(64\!\cdots\!48\)\( T^{18} + \)\(18\!\cdots\!40\)\( T^{20} - \)\(45\!\cdots\!24\)\( T^{22} + \)\(91\!\cdots\!08\)\( T^{24} - \)\(14\!\cdots\!56\)\( T^{26} + \)\(16\!\cdots\!28\)\( T^{28} - \)\(12\!\cdots\!24\)\( T^{30} + \)\(46\!\cdots\!81\)\( T^{32} \))
$61$ (\( ( 1 + 4 T + 7194 T^{2} + 14884 T^{3} + 13845841 T^{4} )^{2} \))(\( ( 1 - 16 T + 5644 T^{2} - 245296 T^{3} + 27126022 T^{4} - 912746416 T^{5} + 78145926604 T^{6} - 824325989776 T^{7} + 191707312997281 T^{8} )^{2} \))(\( ( 1 + 96 T + 12728 T^{2} + 895872 T^{3} + 70065870 T^{4} + 3333539712 T^{5} + 176229864248 T^{6} + 4945955938656 T^{7} + 191707312997281 T^{8} )^{2} \))(\( ( 1 - 20 T + 15746 T^{2} - 308196 T^{3} + 121552351 T^{4} - 2024507560 T^{5} + 567759898204 T^{6} - 7533192630760 T^{7} + 1682994525122191 T^{8} - 15878373296562756 T^{9} + 3018623350455186626 T^{10} - 14266858233257652020 T^{11} + \)\(26\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 - 20 T + 15746 T^{2} - 308196 T^{3} + 121552351 T^{4} - 2024507560 T^{5} + 567759898204 T^{6} - 7533192630760 T^{7} + 1682994525122191 T^{8} - 15878373296562756 T^{9} + 3018623350455186626 T^{10} - 14266858233257652020 T^{11} + \)\(26\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 - 24 T + 4552 T^{2} - 539976 T^{3} + 38633596 T^{4} - 1157398488 T^{5} + 249536906872 T^{6} - 11308110799176 T^{7} + 545612994218182 T^{8} - 42077480283733896 T^{9} + 3455048336181519352 T^{10} - 59629603386615366168 T^{11} + \)\(74\!\cdots\!76\)\( T^{12} - \)\(38\!\cdots\!76\)\( T^{13} + \)\(12\!\cdots\!92\)\( T^{14} - \)\(23\!\cdots\!84\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} )^{2} \))
$67$ (\( 1 - 11204 T^{2} + 65233446 T^{4} - 225773159684 T^{6} + 406067677556641 T^{8} \))(\( 1 - 7912 T^{2} + 95063452 T^{4} - 484990938328 T^{6} + 3019816096004230 T^{8} - 9773111082151065688 T^{10} + \)\(38\!\cdots\!32\)\( T^{12} - \)\(64\!\cdots\!32\)\( T^{14} + \)\(16\!\cdots\!81\)\( T^{16} \))(\( 1 - 16200 T^{2} + 120433884 T^{4} - 596093100152 T^{6} + 2634880977729030 T^{8} - 12011944188428070392 T^{10} + \)\(48\!\cdots\!44\)\( T^{12} - \)\(13\!\cdots\!00\)\( T^{14} + \)\(16\!\cdots\!81\)\( T^{16} \))(\( 1 - 18572 T^{2} + 219805986 T^{4} - 1924138298908 T^{6} + 13416169674171215 T^{8} - 77308694226120992024 T^{10} + \)\(37\!\cdots\!12\)\( T^{12} - \)\(15\!\cdots\!04\)\( T^{14} + \)\(54\!\cdots\!15\)\( T^{16} - \)\(15\!\cdots\!88\)\( T^{18} + \)\(36\!\cdots\!66\)\( T^{20} - \)\(61\!\cdots\!72\)\( T^{22} + \)\(66\!\cdots\!21\)\( T^{24} \))(\( 1 - 18572 T^{2} + 219805986 T^{4} - 1924138298908 T^{6} + 13416169674171215 T^{8} - 77308694226120992024 T^{10} + \)\(37\!\cdots\!12\)\( T^{12} - \)\(15\!\cdots\!04\)\( T^{14} + \)\(54\!\cdots\!15\)\( T^{16} - \)\(15\!\cdots\!88\)\( T^{18} + \)\(36\!\cdots\!66\)\( T^{20} - \)\(61\!\cdots\!72\)\( T^{22} + \)\(66\!\cdots\!21\)\( T^{24} \))(\( 1 - 51696 T^{2} + 1310952504 T^{4} - 21631496299984 T^{6} + 259796851945905948 T^{8} - \)\(24\!\cdots\!00\)\( T^{10} + \)\(17\!\cdots\!80\)\( T^{12} - \)\(10\!\cdots\!88\)\( T^{14} + \)\(52\!\cdots\!62\)\( T^{16} - \)\(21\!\cdots\!48\)\( T^{18} + \)\(72\!\cdots\!80\)\( T^{20} - \)\(19\!\cdots\!00\)\( T^{22} + \)\(42\!\cdots\!88\)\( T^{24} - \)\(71\!\cdots\!84\)\( T^{26} + \)\(87\!\cdots\!84\)\( T^{28} - \)\(69\!\cdots\!36\)\( T^{30} + \)\(27\!\cdots\!61\)\( T^{32} \))
$71$ (\( 1 - 8388 T^{2} + 39052870 T^{4} - 213153180228 T^{6} + 645753531245761 T^{8} \))(\( 1 - 35856 T^{2} + 578979068 T^{4} - 5523144764400 T^{6} + 34153538131631238 T^{8} - \)\(14\!\cdots\!00\)\( T^{10} + \)\(37\!\cdots\!48\)\( T^{12} - \)\(58\!\cdots\!96\)\( T^{14} + \)\(41\!\cdots\!21\)\( T^{16} \))(\( 1 - 21000 T^{2} + 247457180 T^{4} - 1991232124728 T^{6} + 11615880585488070 T^{8} - 50600555550540147768 T^{10} + \)\(15\!\cdots\!80\)\( T^{12} - \)\(34\!\cdots\!00\)\( T^{14} + \)\(41\!\cdots\!21\)\( T^{16} \))(\( 1 - 35076 T^{2} + 589219074 T^{4} - 6491139790068 T^{6} + 54120580943264559 T^{8} - \)\(36\!\cdots\!52\)\( T^{10} + \)\(20\!\cdots\!88\)\( T^{12} - \)\(92\!\cdots\!12\)\( T^{14} + \)\(34\!\cdots\!99\)\( T^{16} - \)\(10\!\cdots\!88\)\( T^{18} + \)\(24\!\cdots\!54\)\( T^{20} - \)\(37\!\cdots\!76\)\( T^{22} + \)\(26\!\cdots\!81\)\( T^{24} \))(\( 1 - 35076 T^{2} + 589219074 T^{4} - 6491139790068 T^{6} + 54120580943264559 T^{8} - \)\(36\!\cdots\!52\)\( T^{10} + \)\(20\!\cdots\!88\)\( T^{12} - \)\(92\!\cdots\!12\)\( T^{14} + \)\(34\!\cdots\!99\)\( T^{16} - \)\(10\!\cdots\!88\)\( T^{18} + \)\(24\!\cdots\!54\)\( T^{20} - \)\(37\!\cdots\!76\)\( T^{22} + \)\(26\!\cdots\!81\)\( T^{24} \))(\( 1 - 26808 T^{2} + 407905624 T^{4} - 4599225914280 T^{6} + 41967348192175708 T^{8} - \)\(32\!\cdots\!32\)\( T^{10} + \)\(22\!\cdots\!28\)\( T^{12} - \)\(13\!\cdots\!32\)\( T^{14} + \)\(70\!\cdots\!38\)\( T^{16} - \)\(33\!\cdots\!92\)\( T^{18} + \)\(14\!\cdots\!08\)\( T^{20} - \)\(53\!\cdots\!12\)\( T^{22} + \)\(17\!\cdots\!68\)\( T^{24} - \)\(48\!\cdots\!80\)\( T^{26} + \)\(10\!\cdots\!44\)\( T^{28} - \)\(18\!\cdots\!88\)\( T^{30} + \)\(17\!\cdots\!41\)\( T^{32} \))
$73$ (\( ( 1 + 76 T + 4934 T^{2} + 405004 T^{3} + 28398241 T^{4} )^{2} \))(\( ( 1 + 56 T + 19724 T^{2} + 823368 T^{3} + 154541606 T^{4} + 4387728072 T^{5} + 560126905484 T^{6} + 8474716672184 T^{7} + 806460091894081 T^{8} )^{2} \))(\( ( 1 - 136 T + 21660 T^{2} - 1811512 T^{3} + 170528390 T^{4} - 9653547448 T^{5} + 615105900060 T^{6} - 20581454775304 T^{7} + 806460091894081 T^{8} )^{2} \))(\( ( 1 - 36 T + 12418 T^{2} - 695252 T^{3} + 107776207 T^{4} - 4912411976 T^{5} + 749125005212 T^{6} - 26178243420104 T^{7} + 3060654700451887 T^{8} - 105215423495879828 T^{9} + 10014621421140697858 T^{10} - \)\(15\!\cdots\!64\)\( T^{11} + \)\(22\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 - 36 T + 12418 T^{2} - 695252 T^{3} + 107776207 T^{4} - 4912411976 T^{5} + 749125005212 T^{6} - 26178243420104 T^{7} + 3060654700451887 T^{8} - 105215423495879828 T^{9} + 10014621421140697858 T^{10} - \)\(15\!\cdots\!64\)\( T^{11} + \)\(22\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 + 16 T + 17704 T^{2} - 586256 T^{3} + 135622140 T^{4} - 9840320304 T^{5} + 1112431939224 T^{6} - 54182229509328 T^{7} + 7900193312584966 T^{8} - 288737101055208912 T^{9} + 31591110306180504984 T^{10} - \)\(14\!\cdots\!56\)\( T^{11} + \)\(10\!\cdots\!40\)\( T^{12} - \)\(25\!\cdots\!44\)\( T^{13} + \)\(40\!\cdots\!84\)\( T^{14} + \)\(19\!\cdots\!44\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} )^{2} \))
$79$ (\( ( 1 - 1282 T^{2} + 38950081 T^{4} )^{2} \))(\( 1 - 14632 T^{2} + 199664860 T^{4} - 1702783803160 T^{6} + 12474458954560966 T^{8} - 66323567058570055960 T^{10} + \)\(30\!\cdots\!60\)\( T^{12} - \)\(86\!\cdots\!12\)\( T^{14} + \)\(23\!\cdots\!21\)\( T^{16} \))(\( 1 - 26248 T^{2} + 289540636 T^{4} - 1851543245752 T^{6} + 10296312398061382 T^{8} - 72117759397043305912 T^{10} + \)\(43\!\cdots\!96\)\( T^{12} - \)\(15\!\cdots\!68\)\( T^{14} + \)\(23\!\cdots\!21\)\( T^{16} \))(\( 1 - 43020 T^{2} + 906603714 T^{4} - 12711826056668 T^{6} + 134359925922862191 T^{8} - \)\(11\!\cdots\!40\)\( T^{10} + \)\(78\!\cdots\!80\)\( T^{12} - \)\(44\!\cdots\!40\)\( T^{14} + \)\(20\!\cdots\!51\)\( T^{16} - \)\(75\!\cdots\!88\)\( T^{18} + \)\(20\!\cdots\!94\)\( T^{20} - \)\(38\!\cdots\!20\)\( T^{22} + \)\(34\!\cdots\!81\)\( T^{24} \))(\( 1 - 43020 T^{2} + 906603714 T^{4} - 12711826056668 T^{6} + 134359925922862191 T^{8} - \)\(11\!\cdots\!40\)\( T^{10} + \)\(78\!\cdots\!80\)\( T^{12} - \)\(44\!\cdots\!40\)\( T^{14} + \)\(20\!\cdots\!51\)\( T^{16} - \)\(75\!\cdots\!88\)\( T^{18} + \)\(20\!\cdots\!94\)\( T^{20} - \)\(38\!\cdots\!20\)\( T^{22} + \)\(34\!\cdots\!81\)\( T^{24} \))(\( 1 - 54704 T^{2} + 1486108728 T^{4} - 26654340076048 T^{6} + 355897016960757404 T^{8} - \)\(37\!\cdots\!80\)\( T^{10} + \)\(33\!\cdots\!36\)\( T^{12} - \)\(25\!\cdots\!84\)\( T^{14} + \)\(17\!\cdots\!50\)\( T^{16} - \)\(10\!\cdots\!04\)\( T^{18} + \)\(51\!\cdots\!96\)\( T^{20} - \)\(22\!\cdots\!80\)\( T^{22} + \)\(81\!\cdots\!84\)\( T^{24} - \)\(23\!\cdots\!48\)\( T^{26} + \)\(51\!\cdots\!68\)\( T^{28} - \)\(74\!\cdots\!44\)\( T^{30} + \)\(52\!\cdots\!41\)\( T^{32} \))
$83$ (\( 1 - 11804 T^{2} + 67754214 T^{4} - 560198021084 T^{6} + 2252292232139041 T^{8} \))(\( 1 - 1672 T^{2} + 123445084 T^{4} - 156070492984 T^{6} + 7827929741188102 T^{8} - 7406843554662919864 T^{10} + \)\(27\!\cdots\!44\)\( T^{12} - \)\(17\!\cdots\!92\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} \))(\( 1 - 19280 T^{2} + 168996572 T^{4} - 1078781742000 T^{6} + 6945567152070790 T^{8} - 51197170200775182000 T^{10} + \)\(38\!\cdots\!52\)\( T^{12} - \)\(20\!\cdots\!80\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} \))(\( 1 - 48364 T^{2} + 1138161762 T^{4} - 17767706400252 T^{6} + 208480181069188815 T^{8} - \)\(19\!\cdots\!92\)\( T^{10} + \)\(14\!\cdots\!12\)\( T^{12} - \)\(92\!\cdots\!32\)\( T^{14} + \)\(46\!\cdots\!15\)\( T^{16} - \)\(18\!\cdots\!72\)\( T^{18} + \)\(57\!\cdots\!22\)\( T^{20} - \)\(11\!\cdots\!64\)\( T^{22} + \)\(11\!\cdots\!21\)\( T^{24} \))(\( 1 - 48364 T^{2} + 1138161762 T^{4} - 17767706400252 T^{6} + 208480181069188815 T^{8} - \)\(19\!\cdots\!92\)\( T^{10} + \)\(14\!\cdots\!12\)\( T^{12} - \)\(92\!\cdots\!32\)\( T^{14} + \)\(46\!\cdots\!15\)\( T^{16} - \)\(18\!\cdots\!72\)\( T^{18} + \)\(57\!\cdots\!22\)\( T^{20} - \)\(11\!\cdots\!64\)\( T^{22} + \)\(11\!\cdots\!21\)\( T^{24} \))(\( 1 - 85264 T^{2} + 3543197432 T^{4} - 95028175099952 T^{6} + 1837921879888014492 T^{8} - \)\(27\!\cdots\!04\)\( T^{10} + \)\(31\!\cdots\!40\)\( T^{12} - \)\(29\!\cdots\!96\)\( T^{14} + \)\(22\!\cdots\!62\)\( T^{16} - \)\(14\!\cdots\!16\)\( T^{18} + \)\(71\!\cdots\!40\)\( T^{20} - \)\(29\!\cdots\!44\)\( T^{22} + \)\(93\!\cdots\!52\)\( T^{24} - \)\(22\!\cdots\!52\)\( T^{26} + \)\(40\!\cdots\!72\)\( T^{28} - \)\(46\!\cdots\!24\)\( T^{30} + \)\(25\!\cdots\!61\)\( T^{32} \))
$89$ (\( ( 1 + 68 T + 15206 T^{2} + 538628 T^{3} + 62742241 T^{4} )^{2} \))(\( ( 1 + 176 T + 28520 T^{2} + 3175008 T^{3} + 333744158 T^{4} + 25149238368 T^{5} + 1789408713320 T^{6} + 87468707209136 T^{7} + 3936588805702081 T^{8} )^{2} \))(\( ( 1 - 40 T + 25916 T^{2} - 837912 T^{3} + 285914566 T^{4} - 6637100952 T^{5} + 1626027917756 T^{6} - 19879251638440 T^{7} + 3936588805702081 T^{8} )^{2} \))(\( ( 1 + 24 T + 25962 T^{2} + 191768 T^{3} + 253674143 T^{4} - 2941553072 T^{5} + 1755043758060 T^{6} - 23300041883312 T^{7} + 15916084215574463 T^{8} + 95305108205009048 T^{9} + \)\(10\!\cdots\!22\)\( T^{10} + \)\(74\!\cdots\!24\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 - 24 T + 25962 T^{2} - 191768 T^{3} + 253674143 T^{4} + 2941553072 T^{5} + 1755043758060 T^{6} + 23300041883312 T^{7} + 15916084215574463 T^{8} - 95305108205009048 T^{9} + \)\(10\!\cdots\!22\)\( T^{10} - \)\(74\!\cdots\!24\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 - 88 T + 33428 T^{2} - 2848088 T^{3} + 652685604 T^{4} - 50244275176 T^{5} + 8431061696684 T^{6} - 571359294604392 T^{7} + 78644602485944758 T^{8} - 4525736972561389032 T^{9} + \)\(52\!\cdots\!44\)\( T^{10} - \)\(24\!\cdots\!36\)\( T^{11} + \)\(25\!\cdots\!24\)\( T^{12} - \)\(88\!\cdots\!88\)\( T^{13} + \)\(82\!\cdots\!88\)\( T^{14} - \)\(17\!\cdots\!08\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} )^{2} \))
$97$ (\( ( 1 + 300 T + 41206 T^{2} + 2822700 T^{3} + 88529281 T^{4} )^{2} \))(\( ( 1 + 120 T + 34316 T^{2} + 2972040 T^{3} + 480186342 T^{4} + 27963924360 T^{5} + 3037970806796 T^{6} + 99956640591480 T^{7} + 7837433594376961 T^{8} )^{2} \))(\( ( 1 - 264 T + 52364 T^{2} - 7260024 T^{3} + 808802534 T^{4} - 68309565816 T^{5} + 4635747270284 T^{6} - 219904609301256 T^{7} + 7837433594376961 T^{8} )^{2} \))(\( ( 1 + 156 T + 46434 T^{2} + 6606636 T^{3} + 998831087 T^{4} + 116839901496 T^{5} + 12239702422620 T^{6} + 1099346633175864 T^{7} + 88425797972558447 T^{8} + 5503142834756108844 T^{9} + \)\(36\!\cdots\!74\)\( T^{10} + \)\(11\!\cdots\!44\)\( T^{11} + \)\(69\!\cdots\!41\)\( T^{12} )^{2} \))(\( ( 1 + 156 T + 46434 T^{2} + 6606636 T^{3} + 998831087 T^{4} + 116839901496 T^{5} + 12239702422620 T^{6} + 1099346633175864 T^{7} + 88425797972558447 T^{8} + 5503142834756108844 T^{9} + \)\(36\!\cdots\!74\)\( T^{10} + \)\(11\!\cdots\!44\)\( T^{11} + \)\(69\!\cdots\!41\)\( T^{12} )^{2} \))(\( ( 1 - 432 T + 114280 T^{2} - 21434448 T^{3} + 3301939196 T^{4} - 437740327152 T^{5} + 53096494516824 T^{6} - 5875910744827152 T^{7} + 598998309831134342 T^{8} - 55286444198078673168 T^{9} + \)\(47\!\cdots\!44\)\( T^{10} - \)\(36\!\cdots\!08\)\( T^{11} + \)\(25\!\cdots\!56\)\( T^{12} - \)\(15\!\cdots\!52\)\( T^{13} + \)\(79\!\cdots\!80\)\( T^{14} - \)\(28\!\cdots\!08\)\( T^{15} + \)\(61\!\cdots\!21\)\( T^{16} )^{2} \))
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