Properties

Label 231.2.j
Level 231
Weight 2
Character orbit j
Rep. character \(\chi_{231}(64,\cdot)\)
Character field \(\Q(\zeta_{5})\)
Dimension 48
Newform subspaces 7
Sturm bound 64
Trace bound 2

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Defining parameters

Level: \( N \) = \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 231.j (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 7 \)
Sturm bound: \(64\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(231, [\chi])\).

Total New Old
Modular forms 144 48 96
Cusp forms 112 48 64
Eisenstein series 32 0 32

Trace form

\( 48q + 4q^{2} - 8q^{4} + 4q^{7} - 8q^{8} - 12q^{9} + O(q^{10}) \) \( 48q + 4q^{2} - 8q^{4} + 4q^{7} - 8q^{8} - 12q^{9} + 16q^{10} - 12q^{11} + 16q^{12} - 4q^{13} - 6q^{14} + 8q^{15} - 44q^{16} + 8q^{17} - 6q^{18} + 8q^{19} - 60q^{20} - 16q^{21} + 36q^{22} - 16q^{23} - 36q^{25} - 52q^{26} + 2q^{28} + 16q^{29} - 8q^{30} + 20q^{31} + 88q^{32} + 8q^{33} + 56q^{34} - 8q^{36} - 28q^{37} + 4q^{38} + 32q^{39} + 4q^{40} + 8q^{41} - 48q^{43} + 18q^{44} - 6q^{46} - 8q^{47} - 16q^{48} - 12q^{49} - 64q^{50} + 24q^{51} - 68q^{52} - 52q^{53} + 36q^{56} - 8q^{57} + 42q^{58} - 4q^{59} - 36q^{60} + 32q^{62} + 4q^{63} - 36q^{64} + 40q^{65} + 80q^{67} + 88q^{68} - 12q^{69} + 20q^{70} + 4q^{71} + 2q^{72} + 8q^{73} - 28q^{74} - 32q^{75} - 16q^{76} + 16q^{77} - 8q^{78} + 36q^{79} + 176q^{80} - 12q^{81} + 4q^{82} - 88q^{83} + 12q^{84} - 12q^{85} + 102q^{86} + 40q^{87} + 84q^{88} - 64q^{89} - 24q^{90} + 10q^{92} - 20q^{93} - 20q^{94} - 128q^{95} - 60q^{96} + 28q^{97} + 4q^{98} - 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(231, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
231.2.j.a \(4\) \(1.845\) \(\Q(\zeta_{10})\) None \(-4\) \(1\) \(-3\) \(-1\) \(q+(-2+\zeta_{10}-\zeta_{10}^{2}+2\zeta_{10}^{3})q^{2}+\cdots\)
231.2.j.b \(4\) \(1.845\) \(\Q(\zeta_{10})\) None \(-2\) \(1\) \(3\) \(1\) \(q+(-\zeta_{10}+\zeta_{10}^{2})q^{2}-\zeta_{10}^{2}q^{3}+(-1+\cdots)q^{4}+\cdots\)
231.2.j.c \(4\) \(1.845\) \(\Q(\zeta_{10})\) None \(1\) \(1\) \(7\) \(-1\) \(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}-\zeta_{10}^{2}q^{3}+\cdots\)
231.2.j.d \(4\) \(1.845\) \(\Q(\zeta_{10})\) None \(2\) \(-1\) \(-1\) \(-1\) \(q+(\zeta_{10}-\zeta_{10}^{2})q^{2}+\zeta_{10}^{2}q^{3}+(-1+\cdots)q^{4}+\cdots\)
231.2.j.e \(4\) \(1.845\) \(\Q(\zeta_{10})\) None \(5\) \(1\) \(-3\) \(-1\) \(q+(1+\zeta_{10}-\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}-\zeta_{10}^{2}q^{3}+\cdots\)
231.2.j.f \(8\) \(1.845\) 8.0.13140625.1 None \(2\) \(2\) \(2\) \(2\) \(q+(-\beta _{3}+\beta _{5}+\beta _{6})q^{2}+(1-\beta _{3}+\beta _{4}+\cdots)q^{3}+\cdots\)
231.2.j.g \(20\) \(1.845\) \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(-5\) \(-5\) \(5\) \(q+\beta _{7}q^{2}-\beta _{4}q^{3}+(\beta _{3}-\beta _{6}-\beta _{8}-2\beta _{9}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(231, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(231, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(33, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 4 T + 4 T^{2} - 7 T^{3} - 21 T^{4} - 14 T^{5} + 16 T^{6} + 32 T^{7} + 16 T^{8} \))(\( 1 + 2 T + 2 T^{2} + 5 T^{3} + 11 T^{4} + 10 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 - T - T^{2} + 3 T^{3} - T^{4} + 6 T^{5} - 4 T^{6} - 8 T^{7} + 16 T^{8} \))(\( 1 - 2 T + 2 T^{2} - 5 T^{3} + 11 T^{4} - 10 T^{5} + 8 T^{6} - 16 T^{7} + 16 T^{8} \))(\( 1 - 5 T + 13 T^{2} - 25 T^{3} + 39 T^{4} - 50 T^{5} + 52 T^{6} - 40 T^{7} + 16 T^{8} \))(\( 1 - 2 T - 3 T^{2} + 6 T^{3} + 3 T^{4} - 6 T^{5} + 9 T^{6} - 2 T^{7} - 27 T^{8} - 4 T^{9} + 36 T^{10} - 48 T^{11} + 48 T^{12} + 192 T^{13} - 192 T^{14} - 256 T^{15} + 256 T^{16} \))(\( 1 + 2 T^{2} + 3 T^{3} + 6 T^{4} - 8 T^{5} + 14 T^{6} + 5 T^{7} + 10 T^{8} + 2 T^{9} + 176 T^{10} + 67 T^{11} + 170 T^{12} + 190 T^{13} + 510 T^{14} - 495 T^{15} + 710 T^{16} + 492 T^{17} + 1850 T^{18} + 927 T^{19} + 8347 T^{20} + 1854 T^{21} + 7400 T^{22} + 3936 T^{23} + 11360 T^{24} - 15840 T^{25} + 32640 T^{26} + 24320 T^{27} + 43520 T^{28} + 34304 T^{29} + 180224 T^{30} + 4096 T^{31} + 40960 T^{32} + 40960 T^{33} + 229376 T^{34} - 262144 T^{35} + 393216 T^{36} + 393216 T^{37} + 524288 T^{38} + 1048576 T^{40} \))
$3$ (\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \))
$5$ (\( 1 + 3 T - T^{2} - 3 T^{3} + 16 T^{4} - 15 T^{5} - 25 T^{6} + 375 T^{7} + 625 T^{8} \))(\( 1 - 3 T - T^{2} + 3 T^{3} + 16 T^{4} + 15 T^{5} - 25 T^{6} - 375 T^{7} + 625 T^{8} \))(\( 1 - 7 T + 29 T^{2} - 93 T^{3} + 236 T^{4} - 465 T^{5} + 725 T^{6} - 875 T^{7} + 625 T^{8} \))(\( 1 + T + T^{2} + 11 T^{3} + 36 T^{4} + 55 T^{5} + 25 T^{6} + 125 T^{7} + 625 T^{8} \))(\( 1 + 3 T - T^{2} - 3 T^{3} + 16 T^{4} - 15 T^{5} - 25 T^{6} + 375 T^{7} + 625 T^{8} \))(\( 1 - 2 T + 10 T^{2} - 42 T^{3} + 99 T^{4} - 338 T^{5} + 900 T^{6} - 2008 T^{7} + 5281 T^{8} - 10040 T^{9} + 22500 T^{10} - 42250 T^{11} + 61875 T^{12} - 131250 T^{13} + 156250 T^{14} - 156250 T^{15} + 390625 T^{16} \))(\( 1 + 5 T + 4 T^{2} - 32 T^{3} - 105 T^{4} - 10 T^{5} + 535 T^{6} + 956 T^{7} - 452 T^{8} - 4251 T^{9} - 8164 T^{10} - 5041 T^{11} + 36064 T^{12} + 140288 T^{13} + 128201 T^{14} - 579422 T^{15} - 1696447 T^{16} - 345724 T^{17} + 4805888 T^{18} + 5640991 T^{19} - 3186474 T^{20} + 28204955 T^{21} + 120147200 T^{22} - 43215500 T^{23} - 1060279375 T^{24} - 1810693750 T^{25} + 2003140625 T^{26} + 10960000000 T^{27} + 14087500000 T^{28} - 9845703125 T^{29} - 79726562500 T^{30} - 207568359375 T^{31} - 110351562500 T^{32} + 1166992187500 T^{33} + 3265380859375 T^{34} - 305175781250 T^{35} - 16021728515625 T^{36} - 24414062500000 T^{37} + 15258789062500 T^{38} + 95367431640625 T^{39} + 95367431640625 T^{40} \))
$7$ (\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5} \))
$11$ (\( 1 - T - 9 T^{2} - 11 T^{3} + 121 T^{4} \))(\( 1 - 9 T + 41 T^{2} - 99 T^{3} + 121 T^{4} \))(\( 1 - T - 9 T^{2} - 11 T^{3} + 121 T^{4} \))(\( 1 + T + 21 T^{2} + 11 T^{3} + 121 T^{4} \))(\( 1 - T + 21 T^{2} - 11 T^{3} + 121 T^{4} \))(\( ( 1 + 11 T + 51 T^{2} + 121 T^{3} + 121 T^{4} )^{2} \))(\( 1 + T - T^{2} + 25 T^{3} + 246 T^{4} + 17 T^{5} - 807 T^{6} + 1153 T^{7} + 25424 T^{8} - 22607 T^{9} - 174843 T^{10} - 248677 T^{11} + 3076304 T^{12} + 1534643 T^{13} - 11815287 T^{14} + 2737867 T^{15} + 435804006 T^{16} + 487179275 T^{17} - 214358881 T^{18} + 2357947691 T^{19} + 25937424601 T^{20} \))
$13$ (\( 1 + 11 T + 48 T^{2} + 145 T^{3} + 491 T^{4} + 1885 T^{5} + 8112 T^{6} + 24167 T^{7} + 28561 T^{8} \))(\( 1 + 7 T + 6 T^{2} - 49 T^{3} - 181 T^{4} - 637 T^{5} + 1014 T^{6} + 15379 T^{7} + 28561 T^{8} \))(\( 1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 650 T^{5} + 507 T^{6} - 8788 T^{7} + 28561 T^{8} \))(\( 1 - T - 12 T^{2} + 25 T^{3} + 131 T^{4} + 325 T^{5} - 2028 T^{6} - 2197 T^{7} + 28561 T^{8} \))(\( 1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 650 T^{5} + 507 T^{6} - 8788 T^{7} + 28561 T^{8} \))(\( 1 + 8 T - 6 T^{2} - 254 T^{3} - 569 T^{4} + 3664 T^{5} + 18260 T^{6} - 21760 T^{7} - 308143 T^{8} - 282880 T^{9} + 3085940 T^{10} + 8049808 T^{11} - 16251209 T^{12} - 94308422 T^{13} - 28960854 T^{14} + 501988136 T^{15} + 815730721 T^{16} \))(\( 1 - 13 T + 66 T^{2} - 139 T^{3} + 51 T^{4} - 2859 T^{5} + 33752 T^{6} - 143455 T^{7} + 202546 T^{8} + 225355 T^{9} + 2636810 T^{10} - 30800939 T^{11} + 94014139 T^{12} + 15525383 T^{13} - 616206028 T^{14} - 489756397 T^{15} + 8014753706 T^{16} + 3731069987 T^{17} - 135009335494 T^{18} + 334766284429 T^{19} - 343304150125 T^{20} + 4351961697577 T^{21} - 22816577698486 T^{22} + 8197160761439 T^{23} + 228909380597066 T^{24} - 181843121911321 T^{25} - 2974308801804652 T^{26} + 974194759107011 T^{27} + 76690221390664219 T^{28} - 326628538313311247 T^{29} + 363506649892361690 T^{30} + 403872305598208135 T^{31} + 4718933949218036626 T^{32} - 43448948416191654115 T^{33} + \)\(13\!\cdots\!28\)\( T^{34} - \)\(14\!\cdots\!63\)\( T^{35} + 33936247068342171891 T^{36} - \)\(12\!\cdots\!87\)\( T^{37} + \)\(74\!\cdots\!14\)\( T^{38} - \)\(19\!\cdots\!01\)\( T^{39} + \)\(19\!\cdots\!01\)\( T^{40} \))
$17$ (\( 1 - 3 T - 8 T^{2} + 75 T^{3} - 89 T^{4} + 1275 T^{5} - 2312 T^{6} - 14739 T^{7} + 83521 T^{8} \))(\( 1 - 9 T + 44 T^{2} - 243 T^{3} + 1279 T^{4} - 4131 T^{5} + 12716 T^{6} - 44217 T^{7} + 83521 T^{8} \))(\( 1 + 12 T + 37 T^{2} - 180 T^{3} - 1619 T^{4} - 3060 T^{5} + 10693 T^{6} + 58956 T^{7} + 83521 T^{8} \))(\( 1 - 7 T + 2 T^{2} + 65 T^{3} - 169 T^{4} + 1105 T^{5} + 578 T^{6} - 34391 T^{7} + 83521 T^{8} \))(\( 1 - 6 T + 19 T^{2} - 132 T^{3} + 829 T^{4} - 2244 T^{5} + 5491 T^{6} - 29478 T^{7} + 83521 T^{8} \))(\( 1 + 4 T - 10 T^{2} - 76 T^{3} - 101 T^{4} + 3812 T^{5} + 15336 T^{6} - 19792 T^{7} - 179499 T^{8} - 336464 T^{9} + 4432104 T^{10} + 18728356 T^{11} - 8435621 T^{12} - 107909132 T^{13} - 241375690 T^{14} + 1641354692 T^{15} + 6975757441 T^{16} \))(\( 1 + T - 34 T^{2} - 41 T^{3} + 811 T^{4} - 3837 T^{5} - 18522 T^{6} + 139403 T^{7} + 389000 T^{8} - 3251903 T^{9} + 2685136 T^{10} + 59936211 T^{11} - 176272043 T^{12} - 1070308079 T^{13} + 4384181682 T^{14} + 6978997953 T^{15} - 68502447326 T^{16} - 33742794795 T^{17} + 1218398812778 T^{18} - 458231992717 T^{19} - 15300717915965 T^{20} - 7789943876189 T^{21} + 352117256892842 T^{22} - 165778350827835 T^{23} - 5721392903114846 T^{24} + 9909179096552721 T^{25} + 105823487857811058 T^{26} - 439188796838039167 T^{27} - 1229631015597521963 T^{28} + 7107707987766132867 T^{29} + 5413217797876026064 T^{30} - \)\(11\!\cdots\!99\)\( T^{31} + \)\(22\!\cdots\!00\)\( T^{32} + \)\(13\!\cdots\!11\)\( T^{33} - \)\(31\!\cdots\!38\)\( T^{34} - \)\(10\!\cdots\!41\)\( T^{35} + \)\(39\!\cdots\!91\)\( T^{36} - \)\(33\!\cdots\!57\)\( T^{37} - \)\(47\!\cdots\!06\)\( T^{38} + \)\(23\!\cdots\!53\)\( T^{39} + \)\(40\!\cdots\!01\)\( T^{40} \))
$19$ (\( 1 + 4 T - 3 T^{2} - 88 T^{3} - 295 T^{4} - 1672 T^{5} - 1083 T^{6} + 27436 T^{7} + 130321 T^{8} \))(\( 1 - 8 T + 45 T^{2} - 268 T^{3} + 1529 T^{4} - 5092 T^{5} + 16245 T^{6} - 54872 T^{7} + 130321 T^{8} \))(\( 1 - T - 13 T^{2} - 53 T^{3} + 400 T^{4} - 1007 T^{5} - 4693 T^{6} - 6859 T^{7} + 130321 T^{8} \))(\( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} \))(\( 1 + 7 T + 15 T^{2} + 107 T^{3} + 824 T^{4} + 2033 T^{5} + 5415 T^{6} + 48013 T^{7} + 130321 T^{8} \))(\( 1 - 10 T^{2} + 70 T^{3} + 319 T^{4} - 280 T^{5} + 300 T^{6} + 20400 T^{7} + 134681 T^{8} + 387600 T^{9} + 108300 T^{10} - 1920520 T^{11} + 41572399 T^{12} + 173326930 T^{13} - 470458810 T^{14} + 16983563041 T^{16} \))(\( 1 - 10 T - 2 T^{2} + 248 T^{3} + 82 T^{4} - 2032 T^{5} - 23994 T^{6} + 41732 T^{7} + 518146 T^{8} - 926242 T^{9} + 540084 T^{10} - 30787074 T^{11} + 50342382 T^{12} + 703180340 T^{13} - 2471883182 T^{14} - 3864155488 T^{15} - 21312045690 T^{16} + 241124991192 T^{17} + 917942443426 T^{18} - 4413260805162 T^{19} - 5762405945050 T^{20} - 83851955298078 T^{21} + 331377222076786 T^{22} + 1653876314585928 T^{23} - 2777407106366490 T^{24} - 9568031539681312 T^{25} - 116291922026273342 T^{26} + 628553033346411260 T^{27} + 854993018331103662 T^{28} - 9934610030411708646 T^{29} + 3311290788778195284 T^{30} - \)\(10\!\cdots\!98\)\( T^{31} + \)\(11\!\cdots\!06\)\( T^{32} + \)\(17\!\cdots\!88\)\( T^{33} - \)\(19\!\cdots\!74\)\( T^{34} - \)\(30\!\cdots\!68\)\( T^{35} + \)\(23\!\cdots\!42\)\( T^{36} + \)\(13\!\cdots\!72\)\( T^{37} - \)\(20\!\cdots\!82\)\( T^{38} - \)\(19\!\cdots\!90\)\( T^{39} + \)\(37\!\cdots\!01\)\( T^{40} \))
$23$ (\( ( 1 + 10 T + 66 T^{2} + 230 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 5 T + 21 T^{2} - 115 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 2 T + 42 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 11 T + 75 T^{2} - 253 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 10 T + 83 T^{2} + 500 T^{3} + 2869 T^{4} + 11500 T^{5} + 43907 T^{6} + 121670 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 + 124 T^{2} + 38 T^{3} + 8330 T^{4} + 3634 T^{5} + 381506 T^{6} + 180342 T^{7} + 12965000 T^{8} + 5936040 T^{9} + 338330610 T^{10} + 136528920 T^{11} + 6858485000 T^{12} + 2194221114 T^{13} + 106761020546 T^{14} + 23389670462 T^{15} + 1233138955370 T^{16} + 129383366986 T^{17} + 9710562174844 T^{18} + 41426511213649 T^{20} )^{2} \))
$29$ (\( 1 + 3 T - 20 T^{2} - 147 T^{3} + 139 T^{4} - 4263 T^{5} - 16820 T^{6} + 73167 T^{7} + 707281 T^{8} \))(\( 1 + 7 T - 10 T^{2} - 113 T^{3} + 59 T^{4} - 3277 T^{5} - 8410 T^{6} + 170723 T^{7} + 707281 T^{8} \))(\( 1 - 2 T - 25 T^{2} + 108 T^{3} + 509 T^{4} + 3132 T^{5} - 21025 T^{6} - 48778 T^{7} + 707281 T^{8} \))(\( 1 - 15 T + 106 T^{2} - 675 T^{3} + 4171 T^{4} - 19575 T^{5} + 89146 T^{6} - 365835 T^{7} + 707281 T^{8} \))(\( 1 - 6 T + 7 T^{2} + 132 T^{3} - 995 T^{4} + 3828 T^{5} + 5887 T^{6} - 146334 T^{7} + 707281 T^{8} \))(\( 1 - 106 T^{2} + 4095 T^{4} - 50924 T^{6} - 177451 T^{8} - 42827084 T^{10} + 2896315695 T^{12} - 63051272026 T^{14} + 500246412961 T^{16} \))(\( 1 - 3 T - 4 T^{2} + 59 T^{3} + 609 T^{4} + 7469 T^{5} - 58340 T^{6} + 18675 T^{7} + 564354 T^{8} + 1565053 T^{9} - 1348916 T^{10} - 228651093 T^{11} + 603011151 T^{12} - 2322397715 T^{13} - 7665582012 T^{14} + 36331461243 T^{15} + 746249961816 T^{16} - 576665139899 T^{17} - 30471253369428 T^{18} + 122834837258411 T^{19} + 929104458856239 T^{20} + 3562210280493919 T^{21} - 25626324083688948 T^{22} - 14064286096996711 T^{23} + 527808419243182296 T^{24} + 745200014942898207 T^{25} - 4559666949775701852 T^{26} - 40061073324054233935 T^{27} + \)\(30\!\cdots\!11\)\( T^{28} - \)\(33\!\cdots\!17\)\( T^{29} - \)\(56\!\cdots\!16\)\( T^{30} + \)\(19\!\cdots\!37\)\( T^{31} + \)\(19\!\cdots\!14\)\( T^{32} + \)\(19\!\cdots\!75\)\( T^{33} - \)\(17\!\cdots\!40\)\( T^{34} + \)\(64\!\cdots\!81\)\( T^{35} + \)\(15\!\cdots\!89\)\( T^{36} + \)\(42\!\cdots\!31\)\( T^{37} - \)\(84\!\cdots\!44\)\( T^{38} - \)\(18\!\cdots\!07\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \))
$31$ (\( 1 + 3 T - 12 T^{2} + 131 T^{3} + 1365 T^{4} + 4061 T^{5} - 11532 T^{6} + 89373 T^{7} + 923521 T^{8} \))(\( 1 + 3 T - 22 T^{2} - 159 T^{3} + 205 T^{4} - 4929 T^{5} - 21142 T^{6} + 89373 T^{7} + 923521 T^{8} \))(\( 1 - 17 T + 153 T^{2} - 1129 T^{3} + 7100 T^{4} - 34999 T^{5} + 147033 T^{6} - 506447 T^{7} + 923521 T^{8} \))(\( 1 - 13 T + 48 T^{2} + 319 T^{3} - 3835 T^{4} + 9889 T^{5} + 46128 T^{6} - 387283 T^{7} + 923521 T^{8} \))(\( 1 + 15 T + 69 T^{2} + 95 T^{3} + 36 T^{4} + 2945 T^{5} + 66309 T^{6} + 446865 T^{7} + 923521 T^{8} \))(\( 1 - 24 T + 238 T^{2} - 1194 T^{3} + 1727 T^{4} + 14880 T^{5} - 38008 T^{6} - 810792 T^{7} + 7640665 T^{8} - 25134552 T^{9} - 36525688 T^{10} + 443290080 T^{11} + 1594920767 T^{12} - 34183206294 T^{13} + 211225876078 T^{14} - 660302738664 T^{15} + 852891037441 T^{16} \))(\( 1 + 13 T + 65 T^{2} + 146 T^{3} + 295 T^{4} + 2802 T^{5} + 48827 T^{6} + 252198 T^{7} + 448847 T^{8} + 5903225 T^{9} + 65860972 T^{10} + 114046885 T^{11} - 1198096047 T^{12} - 10361131234 T^{13} - 7489384555 T^{14} + 470405899010 T^{15} + 3347900754057 T^{16} + 6862384130018 T^{17} + 25638402034775 T^{18} + 239884734794937 T^{19} + 1493178408224822 T^{20} + 7436426778643047 T^{21} + 24638504355418775 T^{22} + 204437285617366238 T^{23} + 3091856652287474697 T^{24} + 13467321514048040510 T^{25} - 6646856360987046955 T^{26} - \)\(28\!\cdots\!74\)\( T^{27} - \)\(10\!\cdots\!27\)\( T^{28} + \)\(30\!\cdots\!35\)\( T^{29} + \)\(53\!\cdots\!72\)\( T^{30} + \)\(14\!\cdots\!75\)\( T^{31} + \)\(35\!\cdots\!67\)\( T^{32} + \)\(61\!\cdots\!18\)\( T^{33} + \)\(36\!\cdots\!67\)\( T^{34} + \)\(65\!\cdots\!02\)\( T^{35} + \)\(21\!\cdots\!95\)\( T^{36} + \)\(32\!\cdots\!06\)\( T^{37} + \)\(45\!\cdots\!65\)\( T^{38} + \)\(28\!\cdots\!23\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \))
$37$ (\( 1 + 6 T + 39 T^{2} + 352 T^{3} + 3309 T^{4} + 13024 T^{5} + 53391 T^{6} + 303918 T^{7} + 1874161 T^{8} \))(\( 1 + 6 T - T^{2} - 228 T^{3} - 1331 T^{4} - 8436 T^{5} - 1369 T^{6} + 303918 T^{7} + 1874161 T^{8} \))(\( 1 + 6 T + 99 T^{2} + 272 T^{3} + 3969 T^{4} + 10064 T^{5} + 135531 T^{6} + 303918 T^{7} + 1874161 T^{8} \))(\( 1 - 22 T + 207 T^{2} - 1220 T^{3} + 6701 T^{4} - 45140 T^{5} + 283383 T^{6} - 1114366 T^{7} + 1874161 T^{8} \))(\( 1 + 6 T - 21 T^{2} - 248 T^{3} - 471 T^{4} - 9176 T^{5} - 28749 T^{6} + 303918 T^{7} + 1874161 T^{8} \))(\( 1 - 6 T + 51 T^{2} - 228 T^{3} + 3191 T^{4} - 8808 T^{5} + 60945 T^{6} - 124230 T^{7} + 3928552 T^{8} - 4596510 T^{9} + 83433705 T^{10} - 446151624 T^{11} + 5980447751 T^{12} - 15810422196 T^{13} + 130852046859 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16} \))(\( 1 + 32 T + 496 T^{2} + 5340 T^{3} + 50712 T^{4} + 444144 T^{5} + 3184776 T^{6} + 17020156 T^{7} + 60430936 T^{8} - 66824016 T^{9} - 4566815524 T^{10} - 57414614544 T^{11} - 482483421256 T^{12} - 3283683956236 T^{13} - 19092317847912 T^{14} - 82953054964976 T^{15} - 130679609958264 T^{16} + 1790702451469140 T^{17} + 24993747942875904 T^{18} + 219578098429763712 T^{19} + 1506082500954951158 T^{20} + 8124389641901257344 T^{21} + 34216440933797112576 T^{22} + 90704451274266348420 T^{23} - \)\(24\!\cdots\!04\)\( T^{24} - \)\(57\!\cdots\!32\)\( T^{25} - \)\(48\!\cdots\!08\)\( T^{26} - \)\(31\!\cdots\!88\)\( T^{27} - \)\(16\!\cdots\!76\)\( T^{28} - \)\(74\!\cdots\!88\)\( T^{29} - \)\(21\!\cdots\!76\)\( T^{30} - \)\(11\!\cdots\!08\)\( T^{31} + \)\(39\!\cdots\!16\)\( T^{32} + \)\(41\!\cdots\!32\)\( T^{33} + \)\(28\!\cdots\!64\)\( T^{34} + \)\(14\!\cdots\!92\)\( T^{35} + \)\(62\!\cdots\!92\)\( T^{36} + \)\(24\!\cdots\!80\)\( T^{37} + \)\(83\!\cdots\!84\)\( T^{38} + \)\(19\!\cdots\!36\)\( T^{39} + \)\(23\!\cdots\!01\)\( T^{40} \))
$41$ (\( 1 + 15 T + 59 T^{2} + 45 T^{3} + 256 T^{4} + 1845 T^{5} + 99179 T^{6} + 1033815 T^{7} + 2825761 T^{8} \))(\( 1 + T + 35 T^{2} + 99 T^{3} + 1904 T^{4} + 4059 T^{5} + 58835 T^{6} + 68921 T^{7} + 2825761 T^{8} \))(\( 1 + 49 T^{2} - 60 T^{3} + 1861 T^{4} - 2460 T^{5} + 82369 T^{6} + 2825761 T^{8} \))(\( 1 - 13 T + 53 T^{2} - 331 T^{3} + 3380 T^{4} - 13571 T^{5} + 89093 T^{6} - 895973 T^{7} + 2825761 T^{8} \))(\( 1 + 6 T + 35 T^{2} + 84 T^{3} - 371 T^{4} + 3444 T^{5} + 58835 T^{6} + 413526 T^{7} + 2825761 T^{8} \))(\( 1 - 20 T + 170 T^{2} - 1070 T^{3} + 8519 T^{4} - 70060 T^{5} + 465660 T^{6} - 2505360 T^{7} + 13832161 T^{8} - 102719760 T^{9} + 782774460 T^{10} - 4828605260 T^{11} + 24072657959 T^{12} - 123966135070 T^{13} + 807517720970 T^{14} - 3895085477620 T^{15} + 7984925229121 T^{16} \))(\( 1 + 3 T - 151 T^{2} + 283 T^{3} + 15260 T^{4} - 53807 T^{5} - 908059 T^{6} + 5105899 T^{7} + 41696524 T^{8} - 332390297 T^{9} - 1424132099 T^{10} + 18190782425 T^{11} + 13662607978 T^{12} - 883993384739 T^{13} + 2395186168445 T^{14} + 33386146780263 T^{15} - 238454984830174 T^{16} - 934346443615571 T^{17} + 13814720376087777 T^{18} + 13438189208809197 T^{19} - 619586511510882716 T^{20} + 550965757561177077 T^{21} + 23222544952203553137 T^{22} - 64396091240428768891 T^{23} - \)\(67\!\cdots\!14\)\( T^{24} + \)\(38\!\cdots\!63\)\( T^{25} + \)\(11\!\cdots\!45\)\( T^{26} - \)\(17\!\cdots\!59\)\( T^{27} + \)\(10\!\cdots\!38\)\( T^{28} + \)\(59\!\cdots\!25\)\( T^{29} - \)\(19\!\cdots\!99\)\( T^{30} - \)\(18\!\cdots\!77\)\( T^{31} + \)\(94\!\cdots\!44\)\( T^{32} + \)\(47\!\cdots\!79\)\( T^{33} - \)\(34\!\cdots\!99\)\( T^{34} - \)\(83\!\cdots\!07\)\( T^{35} + \)\(97\!\cdots\!60\)\( T^{36} + \)\(73\!\cdots\!23\)\( T^{37} - \)\(16\!\cdots\!71\)\( T^{38} + \)\(13\!\cdots\!83\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \))
$43$ (\( ( 1 + 9 T + 43 T^{2} )^{4} \))(\( ( 1 + 14 T + 115 T^{2} + 602 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 2 T + 82 T^{2} - 86 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + T + 43 T^{2} )^{4} \))(\( ( 1 - 6 T + 50 T^{2} - 258 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 4 T + 69 T^{2} + 392 T^{3} + 4097 T^{4} + 16856 T^{5} + 127581 T^{6} + 318028 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 - 6 T + 312 T^{2} - 1828 T^{3} + 47319 T^{4} - 259736 T^{5} + 4576717 T^{6} - 22838020 T^{7} + 311402464 T^{8} - 1377461130 T^{9} + 15540959670 T^{10} - 59230828590 T^{11} + 575783155936 T^{12} - 1815782456140 T^{13} + 15646884656317 T^{14} - 38183384951048 T^{15} + 299120578115631 T^{16} - 496884421103596 T^{17} + 3646718486611512 T^{18} - 3015555671621058 T^{19} + 21611482313284249 T^{20} )^{2} \))
$47$ (\( 1 - 8 T + 67 T^{2} - 610 T^{3} + 6231 T^{4} - 28670 T^{5} + 148003 T^{6} - 830584 T^{7} + 4879681 T^{8} \))(\( 1 - 16 T + 59 T^{2} + 738 T^{3} - 9721 T^{4} + 34686 T^{5} + 130331 T^{6} - 1661168 T^{7} + 4879681 T^{8} \))(\( 1 + 2 T - 43 T^{2} - 180 T^{3} + 1661 T^{4} - 8460 T^{5} - 94987 T^{6} + 207646 T^{7} + 4879681 T^{8} \))(\( 1 + 18 T + 137 T^{2} + 990 T^{3} + 7951 T^{4} + 46530 T^{5} + 302633 T^{6} + 1868814 T^{7} + 4879681 T^{8} \))(\( 1 + 10 T + 13 T^{2} + 200 T^{3} + 3549 T^{4} + 9400 T^{5} + 28717 T^{6} + 1038230 T^{7} + 4879681 T^{8} \))(\( 1 + 22 T + 222 T^{2} + 1554 T^{3} + 9963 T^{4} + 18726 T^{5} - 507516 T^{6} - 6126008 T^{7} - 43788087 T^{8} - 287922376 T^{9} - 1121102844 T^{10} + 1944189498 T^{11} + 48616261803 T^{12} + 356402140878 T^{13} + 2392985803038 T^{14} + 11145708650186 T^{15} + 23811286661761 T^{16} \))(\( 1 - 20 T + 141 T^{2} - 502 T^{3} + 4037 T^{4} + 6218 T^{5} - 801001 T^{6} + 7197116 T^{7} - 33989974 T^{8} + 317744608 T^{9} - 1914149671 T^{10} - 9965865086 T^{11} + 157004029439 T^{12} - 867584843046 T^{13} + 9623990203403 T^{14} - 81333165828100 T^{15} + 137676757256356 T^{16} + 1184018410587624 T^{17} - 6791369921638187 T^{18} + 141773649130018030 T^{19} - 1649383233650553737 T^{20} + 6663361509110847410 T^{21} - 15002136156898755083 T^{22} + \)\(12\!\cdots\!52\)\( T^{23} + \)\(67\!\cdots\!36\)\( T^{24} - \)\(18\!\cdots\!00\)\( T^{25} + \)\(10\!\cdots\!87\)\( T^{26} - \)\(43\!\cdots\!98\)\( T^{27} + \)\(37\!\cdots\!79\)\( T^{28} - \)\(11\!\cdots\!62\)\( T^{29} - \)\(10\!\cdots\!79\)\( T^{30} + \)\(78\!\cdots\!24\)\( T^{31} - \)\(39\!\cdots\!34\)\( T^{32} + \)\(39\!\cdots\!32\)\( T^{33} - \)\(20\!\cdots\!69\)\( T^{34} + \)\(75\!\cdots\!74\)\( T^{35} + \)\(22\!\cdots\!77\)\( T^{36} - \)\(13\!\cdots\!74\)\( T^{37} + \)\(17\!\cdots\!49\)\( T^{38} - \)\(11\!\cdots\!60\)\( T^{39} + \)\(27\!\cdots\!01\)\( T^{40} \))
$53$ (\( 1 + 11 T + 43 T^{2} + 565 T^{3} + 6936 T^{4} + 29945 T^{5} + 120787 T^{6} + 1637647 T^{7} + 7890481 T^{8} \))(\( 1 - 25 T + 257 T^{2} - 1635 T^{3} + 10244 T^{4} - 86655 T^{5} + 721913 T^{6} - 3721925 T^{7} + 7890481 T^{8} \))(\( 1 + 26 T + 323 T^{2} + 2870 T^{3} + 22001 T^{4} + 152110 T^{5} + 907307 T^{6} + 3870802 T^{7} + 7890481 T^{8} \))(\( 1 + 9 T - 7 T^{2} - 525 T^{3} - 3884 T^{4} - 27825 T^{5} - 19663 T^{6} + 1339893 T^{7} + 7890481 T^{8} \))(\( 1 + 14 T + 43 T^{2} - 650 T^{3} - 8799 T^{4} - 34450 T^{5} + 120787 T^{6} + 2084278 T^{7} + 7890481 T^{8} \))(\( 1 + 20 T + 117 T^{2} + 500 T^{3} + 10595 T^{4} + 96940 T^{5} + 362607 T^{6} + 3666700 T^{7} + 43463984 T^{8} + 194335100 T^{9} + 1018563063 T^{10} + 14432136380 T^{11} + 83599646195 T^{12} + 209097746500 T^{13} + 2593230252093 T^{14} + 23494222796740 T^{15} + 62259690411361 T^{16} \))(\( 1 - 3 T - 126 T^{2} + 1842 T^{3} - 51 T^{4} - 176024 T^{5} + 1687499 T^{6} + 693786 T^{7} - 129427770 T^{8} + 1030515651 T^{9} + 22821416 T^{10} - 58177807579 T^{11} + 439171427598 T^{12} - 484180552566 T^{13} - 15615841075935 T^{14} + 125664083280776 T^{15} - 350599946515409 T^{16} - 1684175373629766 T^{17} + 16682975693935818 T^{18} - 76739832358344117 T^{19} + 165299081934488990 T^{20} - 4067211114992238201 T^{21} + 46862478724265712762 T^{22} - \)\(25\!\cdots\!82\)\( T^{23} - \)\(27\!\cdots\!29\)\( T^{24} + \)\(52\!\cdots\!68\)\( T^{25} - \)\(34\!\cdots\!15\)\( T^{26} - \)\(56\!\cdots\!42\)\( T^{27} + \)\(27\!\cdots\!78\)\( T^{28} - \)\(19\!\cdots\!07\)\( T^{29} + \)\(39\!\cdots\!84\)\( T^{30} + \)\(95\!\cdots\!47\)\( T^{31} - \)\(63\!\cdots\!70\)\( T^{32} + \)\(18\!\cdots\!78\)\( T^{33} + \)\(23\!\cdots\!31\)\( T^{34} - \)\(12\!\cdots\!68\)\( T^{35} - \)\(19\!\cdots\!71\)\( T^{36} + \)\(37\!\cdots\!46\)\( T^{37} - \)\(13\!\cdots\!14\)\( T^{38} - \)\(17\!\cdots\!51\)\( T^{39} + \)\(30\!\cdots\!01\)\( T^{40} \))
$59$ (\( 1 + 9 T + 77 T^{2} + 477 T^{3} + 700 T^{4} + 28143 T^{5} + 268037 T^{6} + 1848411 T^{7} + 12117361 T^{8} \))(\( 1 - T - 43 T^{2} + 347 T^{3} + 2540 T^{4} + 20473 T^{5} - 149683 T^{6} - 205379 T^{7} + 12117361 T^{8} \))(\( 1 + 14 T + 37 T^{2} - 758 T^{3} - 10095 T^{4} - 44722 T^{5} + 128797 T^{6} + 2875306 T^{7} + 12117361 T^{8} \))(\( 1 - 15 T + 31 T^{2} + 885 T^{3} - 10424 T^{4} + 52215 T^{5} + 107911 T^{6} - 3080685 T^{7} + 12117361 T^{8} \))(\( 1 + 6 T - 43 T^{2} - 102 T^{3} + 3025 T^{4} - 6018 T^{5} - 149683 T^{6} + 1232274 T^{7} + 12117361 T^{8} \))(\( 1 - 18 T + 22 T^{2} + 1598 T^{3} - 11653 T^{4} + 17790 T^{5} + 84228 T^{6} - 3028056 T^{7} + 40795865 T^{8} - 178655304 T^{9} + 293197668 T^{10} + 3653692410 T^{11} - 141203607733 T^{12} + 1142449029802 T^{13} + 927971740102 T^{14} - 44795726726742 T^{15} + 146830437604321 T^{16} \))(\( 1 + 9 T - 11 T^{2} - 2009 T^{3} - 16566 T^{4} - 39875 T^{5} + 1481949 T^{6} + 16040319 T^{7} + 91259286 T^{8} - 482457503 T^{9} - 9939821477 T^{10} - 75582121635 T^{11} - 12007856396 T^{12} + 4148742606411 T^{13} + 36852203890145 T^{14} + 106193218742209 T^{15} - 1378906768809232 T^{16} - 14879013797272333 T^{17} - 71002723235317991 T^{18} + 551840607836296325 T^{19} + 6375429074335094320 T^{20} + 32558595862341483175 T^{21} - \)\(24\!\cdots\!71\)\( T^{22} - \)\(30\!\cdots\!07\)\( T^{23} - \)\(16\!\cdots\!52\)\( T^{24} + \)\(75\!\cdots\!91\)\( T^{25} + \)\(15\!\cdots\!45\)\( T^{26} + \)\(10\!\cdots\!09\)\( T^{27} - \)\(17\!\cdots\!16\)\( T^{28} - \)\(65\!\cdots\!65\)\( T^{29} - \)\(50\!\cdots\!77\)\( T^{30} - \)\(14\!\cdots\!77\)\( T^{31} + \)\(16\!\cdots\!66\)\( T^{32} + \)\(16\!\cdots\!01\)\( T^{33} + \)\(91\!\cdots\!89\)\( T^{34} - \)\(14\!\cdots\!25\)\( T^{35} - \)\(35\!\cdots\!06\)\( T^{36} - \)\(25\!\cdots\!71\)\( T^{37} - \)\(82\!\cdots\!31\)\( T^{38} + \)\(39\!\cdots\!51\)\( T^{39} + \)\(26\!\cdots\!01\)\( T^{40} \))
$61$ (\( 1 - 15 T + 74 T^{2} - 675 T^{3} + 8491 T^{4} - 41175 T^{5} + 275354 T^{6} - 3404715 T^{7} + 13845841 T^{8} \))(\( 1 - 19 T + 180 T^{2} - 1781 T^{3} + 17099 T^{4} - 108641 T^{5} + 669780 T^{6} - 4312639 T^{7} + 13845841 T^{8} \))(\( 1 + 20 T + 179 T^{2} + 1600 T^{3} + 15001 T^{4} + 97600 T^{5} + 666059 T^{6} + 4539620 T^{7} + 13845841 T^{8} \))(\( 1 - 3 T + 18 T^{2} + 209 T^{3} + 675 T^{4} + 12749 T^{5} + 66978 T^{6} - 680943 T^{7} + 13845841 T^{8} \))(\( 1 - 61 T^{2} + 3721 T^{4} - 226981 T^{6} + 13845841 T^{8} \))(\( 1 + 2 T - 74 T^{2} - 552 T^{3} + 1931 T^{4} + 48386 T^{5} + 199304 T^{6} - 1537696 T^{7} - 12160643 T^{8} - 93799456 T^{9} + 741610184 T^{10} + 10982702666 T^{11} + 26736318971 T^{12} - 466217158152 T^{13} - 3812507702714 T^{14} + 6285485672042 T^{15} + 191707312997281 T^{16} \))(\( 1 + 15 T - 90 T^{2} - 3165 T^{3} - 19585 T^{4} + 92683 T^{5} + 2977530 T^{6} + 30435495 T^{7} + 58060720 T^{8} - 2575904645 T^{9} - 32871179972 T^{10} - 119504505635 T^{11} + 1285823633985 T^{12} + 22572191661785 T^{13} + 150060548339790 T^{14} - 142916662288619 T^{15} - 12943544265704350 T^{16} - 115634960295586275 T^{17} - 240942316112795150 T^{18} + 4864605484258100025 T^{19} + 61052179162441039695 T^{20} + \)\(29\!\cdots\!25\)\( T^{21} - \)\(89\!\cdots\!50\)\( T^{22} - \)\(26\!\cdots\!75\)\( T^{23} - \)\(17\!\cdots\!50\)\( T^{24} - \)\(12\!\cdots\!19\)\( T^{25} + \)\(77\!\cdots\!90\)\( T^{26} + \)\(70\!\cdots\!85\)\( T^{27} + \)\(24\!\cdots\!85\)\( T^{28} - \)\(13\!\cdots\!35\)\( T^{29} - \)\(23\!\cdots\!72\)\( T^{30} - \)\(11\!\cdots\!45\)\( T^{31} + \)\(15\!\cdots\!20\)\( T^{32} + \)\(49\!\cdots\!95\)\( T^{33} + \)\(29\!\cdots\!30\)\( T^{34} + \)\(55\!\cdots\!83\)\( T^{35} - \)\(71\!\cdots\!85\)\( T^{36} - \)\(70\!\cdots\!65\)\( T^{37} - \)\(12\!\cdots\!90\)\( T^{38} + \)\(12\!\cdots\!15\)\( T^{39} + \)\(50\!\cdots\!01\)\( T^{40} \))
$67$ (\( ( 1 - 6 T + 98 T^{2} - 402 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 22 T + 250 T^{2} - 1474 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 8 T + 67 T^{2} )^{4} \))(\( ( 1 + 14 T + 178 T^{2} + 938 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 67 T^{2} )^{4} \))(\( ( 1 + 28 T + 541 T^{2} + 6696 T^{3} + 64817 T^{4} + 448632 T^{5} + 2428549 T^{6} + 8421364 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 - 38 T + 939 T^{2} - 16920 T^{3} + 253022 T^{4} - 3231648 T^{5} + 37034650 T^{6} - 384823180 T^{7} + 3720320353 T^{8} - 33446868814 T^{9} + 283134556406 T^{10} - 2240940210538 T^{11} + 16700518064617 T^{12} - 115740574086340 T^{13} + 746289713342650 T^{14} - 4363129101786336 T^{15} + 22887960773164718 T^{16} - 102547240362065160 T^{17} + 381297549225685899 T^{18} - 1033848307059207986 T^{19} + 1822837804551761449 T^{20} )^{2} \))
$71$ (\( 1 + 15 T + 44 T^{2} - 945 T^{3} - 13979 T^{4} - 67095 T^{5} + 221804 T^{6} + 5368665 T^{7} + 25411681 T^{8} \))(\( 1 + T + 70 T^{2} + 119 T^{3} + 5709 T^{4} + 8449 T^{5} + 352870 T^{6} + 357911 T^{7} + 25411681 T^{8} \))(\( 1 + 20 T + 169 T^{2} + 1600 T^{3} + 17121 T^{4} + 113600 T^{5} + 851929 T^{6} + 7158220 T^{7} + 25411681 T^{8} \))(\( 1 - 13 T - 2 T^{2} + 349 T^{3} + 405 T^{4} + 24779 T^{5} - 10082 T^{6} - 4652843 T^{7} + 25411681 T^{8} \))(\( 1 + 24 T + 185 T^{2} + 456 T^{3} + 49 T^{4} + 32376 T^{5} + 932585 T^{6} + 8589864 T^{7} + 25411681 T^{8} \))(\( 1 - 14 T - 77 T^{2} + 2716 T^{3} - 11793 T^{4} - 191640 T^{5} + 2135337 T^{6} + 4898798 T^{7} - 179455320 T^{8} + 347814658 T^{9} + 10764233817 T^{10} - 68590064040 T^{11} - 299679954033 T^{12} + 4900286917316 T^{13} - 9863721861917 T^{14} - 127331682217474 T^{15} + 645753531245761 T^{16} \))(\( 1 - 37 T + 515 T^{2} - 3274 T^{3} + 13215 T^{4} - 211688 T^{5} + 4684917 T^{6} - 60912592 T^{7} + 467488647 T^{8} - 2980461565 T^{9} + 33268020472 T^{10} - 428318804705 T^{11} + 4339324633273 T^{12} - 35873672109364 T^{13} + 288523918504695 T^{14} - 2785853372108440 T^{15} + 26650571131092837 T^{16} - 220452230480408382 T^{17} + 1885088405364310445 T^{18} - 17288624106824488873 T^{19} + \)\(15\!\cdots\!82\)\( T^{20} - \)\(12\!\cdots\!83\)\( T^{21} + \)\(95\!\cdots\!45\)\( T^{22} - \)\(78\!\cdots\!02\)\( T^{23} + \)\(67\!\cdots\!97\)\( T^{24} - \)\(50\!\cdots\!40\)\( T^{25} + \)\(36\!\cdots\!95\)\( T^{26} - \)\(32\!\cdots\!24\)\( T^{27} + \)\(28\!\cdots\!53\)\( T^{28} - \)\(19\!\cdots\!55\)\( T^{29} + \)\(10\!\cdots\!72\)\( T^{30} - \)\(68\!\cdots\!15\)\( T^{31} + \)\(76\!\cdots\!27\)\( T^{32} - \)\(70\!\cdots\!12\)\( T^{33} + \)\(38\!\cdots\!77\)\( T^{34} - \)\(12\!\cdots\!88\)\( T^{35} + \)\(55\!\cdots\!15\)\( T^{36} - \)\(96\!\cdots\!34\)\( T^{37} + \)\(10\!\cdots\!15\)\( T^{38} - \)\(55\!\cdots\!47\)\( T^{39} + \)\(10\!\cdots\!01\)\( T^{40} \))
$73$ (\( 1 + 7 T - 39 T^{2} + 161 T^{3} + 6764 T^{4} + 11753 T^{5} - 207831 T^{6} + 2723119 T^{7} + 28398241 T^{8} \))(\( 1 + 11 T + 3 T^{2} - 815 T^{3} - 8464 T^{4} - 59495 T^{5} + 15987 T^{6} + 4279187 T^{7} + 28398241 T^{8} \))(\( 1 + 2 T - 49 T^{2} + 406 T^{3} + 5929 T^{4} + 29638 T^{5} - 261121 T^{6} + 778034 T^{7} + 28398241 T^{8} \))(\( 1 - 21 T + 233 T^{2} - 2595 T^{3} + 26956 T^{4} - 189435 T^{5} + 1241657 T^{6} - 8169357 T^{7} + 28398241 T^{8} \))(\( 1 + 22 T + 111 T^{2} - 1054 T^{3} - 17431 T^{4} - 76942 T^{5} + 591519 T^{6} + 8558374 T^{7} + 28398241 T^{8} \))(\( 1 - 2 T - 146 T^{2} + 446 T^{3} + 13211 T^{4} - 19066 T^{5} - 955660 T^{6} + 409840 T^{7} + 76882177 T^{8} + 29918320 T^{9} - 5092712140 T^{10} - 7416998122 T^{11} + 375169161851 T^{12} + 924589930478 T^{13} - 22094797038194 T^{14} - 22094797038194 T^{15} + 806460091894081 T^{16} \))(\( 1 - 27 T + 117 T^{2} + 3825 T^{3} - 60386 T^{4} + 206645 T^{5} + 5076633 T^{6} - 88912135 T^{7} + 412271510 T^{8} + 5307403509 T^{9} - 94198639093 T^{10} + 498629925791 T^{11} + 2922251864844 T^{12} - 76647599821181 T^{13} + 554955524573989 T^{14} + 735710433225271 T^{15} - 49117193322293732 T^{16} + 443373919368750715 T^{17} - 1123922652740353551 T^{18} - 21964116604269227841 T^{19} + \)\(31\!\cdots\!04\)\( T^{20} - \)\(16\!\cdots\!93\)\( T^{21} - \)\(59\!\cdots\!79\)\( T^{22} + \)\(17\!\cdots\!55\)\( T^{23} - \)\(13\!\cdots\!12\)\( T^{24} + \)\(15\!\cdots\!03\)\( T^{25} + \)\(83\!\cdots\!21\)\( T^{26} - \)\(84\!\cdots\!57\)\( T^{27} + \)\(23\!\cdots\!64\)\( T^{28} + \)\(29\!\cdots\!83\)\( T^{29} - \)\(40\!\cdots\!57\)\( T^{30} + \)\(16\!\cdots\!93\)\( T^{31} + \)\(94\!\cdots\!10\)\( T^{32} - \)\(14\!\cdots\!55\)\( T^{33} + \)\(61\!\cdots\!97\)\( T^{34} + \)\(18\!\cdots\!65\)\( T^{35} - \)\(39\!\cdots\!46\)\( T^{36} + \)\(18\!\cdots\!25\)\( T^{37} + \)\(40\!\cdots\!73\)\( T^{38} - \)\(68\!\cdots\!99\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \))
$79$ (\( 1 + 11 T - 33 T^{2} - 557 T^{3} + 80 T^{4} - 44003 T^{5} - 205953 T^{6} + 5423429 T^{7} + 38950081 T^{8} \))(\( 1 - 13 T + 15 T^{2} - 293 T^{3} + 8624 T^{4} - 23147 T^{5} + 93615 T^{6} - 6409507 T^{7} + 38950081 T^{8} \))(\( 1 - 14 T + 117 T^{2} - 532 T^{3} - 1795 T^{4} - 42028 T^{5} + 730197 T^{6} - 6902546 T^{7} + 38950081 T^{8} \))(\( 1 + 15 T + 11 T^{2} - 1185 T^{3} - 13064 T^{4} - 93615 T^{5} + 68651 T^{6} + 7395585 T^{7} + 38950081 T^{8} \))(\( 1 - 10 T + 21 T^{2} + 580 T^{3} - 7459 T^{4} + 45820 T^{5} + 131061 T^{6} - 4930390 T^{7} + 38950081 T^{8} \))(\( 1 - 20 T + 25 T^{2} + 2320 T^{3} - 19901 T^{4} + 85160 T^{5} - 408605 T^{6} - 11176060 T^{7} + 215428736 T^{8} - 882908740 T^{9} - 2550103805 T^{10} + 41987201240 T^{11} - 775145561981 T^{12} + 7138770845680 T^{13} + 6077186388025 T^{14} - 384078179723180 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 5 T - 226 T^{2} + 1188 T^{3} + 28293 T^{4} - 182558 T^{5} - 2094253 T^{6} + 20887148 T^{7} + 66066202 T^{8} - 1685198093 T^{9} + 5203173736 T^{10} + 81171391019 T^{11} - 899452563978 T^{12} + 259180666124 T^{13} + 50685690960293 T^{14} - 453111325450158 T^{15} + 2532348314805923 T^{16} + 41547309775010276 T^{17} - 818375975431692878 T^{18} - 1321261704586397645 T^{19} + 85108949992387444270 T^{20} - \)\(10\!\cdots\!55\)\( T^{21} - \)\(51\!\cdots\!98\)\( T^{22} + \)\(20\!\cdots\!64\)\( T^{23} + \)\(98\!\cdots\!63\)\( T^{24} - \)\(13\!\cdots\!42\)\( T^{25} + \)\(12\!\cdots\!53\)\( T^{26} + \)\(49\!\cdots\!16\)\( T^{27} - \)\(13\!\cdots\!58\)\( T^{28} + \)\(97\!\cdots\!61\)\( T^{29} + \)\(49\!\cdots\!36\)\( T^{30} - \)\(12\!\cdots\!47\)\( T^{31} + \)\(39\!\cdots\!82\)\( T^{32} + \)\(97\!\cdots\!72\)\( T^{33} - \)\(77\!\cdots\!93\)\( T^{34} - \)\(53\!\cdots\!42\)\( T^{35} + \)\(65\!\cdots\!53\)\( T^{36} + \)\(21\!\cdots\!92\)\( T^{37} - \)\(32\!\cdots\!86\)\( T^{38} - \)\(56\!\cdots\!95\)\( T^{39} + \)\(89\!\cdots\!01\)\( T^{40} \))
$83$ (\( 1 + 14 T + 193 T^{2} + 2140 T^{3} + 26421 T^{4} + 177620 T^{5} + 1329577 T^{6} + 8005018 T^{7} + 47458321 T^{8} \))(\( 1 + 14 T - 7 T^{2} - 640 T^{3} - 2059 T^{4} - 53120 T^{5} - 48223 T^{6} + 8005018 T^{7} + 47458321 T^{8} \))(\( 1 + 14 T + 193 T^{2} + 2140 T^{3} + 26421 T^{4} + 177620 T^{5} + 1329577 T^{6} + 8005018 T^{7} + 47458321 T^{8} \))(\( 1 - 6 T - 47 T^{2} + 780 T^{3} - 779 T^{4} + 64740 T^{5} - 323783 T^{6} - 3430722 T^{7} + 47458321 T^{8} \))(\( 1 + 2 T + 121 T^{2} + 736 T^{3} + 7989 T^{4} + 61088 T^{5} + 833569 T^{6} + 1143574 T^{7} + 47458321 T^{8} \))(\( 1 + 8 T - 10 T^{2} - 98 T^{3} + 5199 T^{4} + 59504 T^{5} + 80364 T^{6} - 1623536 T^{7} + 2309081 T^{8} - 134753488 T^{9} + 553627596 T^{10} + 34023613648 T^{11} + 246735810879 T^{12} - 386025983014 T^{13} - 3269403733690 T^{14} + 217088407917016 T^{15} + 2252292232139041 T^{16} \))(\( 1 + 42 T + 661 T^{2} + 3786 T^{3} - 19549 T^{4} - 561934 T^{5} - 5968953 T^{6} - 20177190 T^{7} + 432561126 T^{8} + 6143013970 T^{9} + 28916165545 T^{10} + 66576463426 T^{11} + 144701828969 T^{12} - 5596879899702 T^{13} - 23882904399413 T^{14} - 248862974645262 T^{15} - 21903508692713044 T^{16} - 235757216137931078 T^{17} + 483849794965596541 T^{18} + 23802972835548797594 T^{19} + \)\(24\!\cdots\!45\)\( T^{20} + \)\(19\!\cdots\!02\)\( T^{21} + \)\(33\!\cdots\!49\)\( T^{22} - \)\(13\!\cdots\!86\)\( T^{23} - \)\(10\!\cdots\!24\)\( T^{24} - \)\(98\!\cdots\!66\)\( T^{25} - \)\(78\!\cdots\!97\)\( T^{26} - \)\(15\!\cdots\!54\)\( T^{27} + \)\(32\!\cdots\!29\)\( T^{28} + \)\(12\!\cdots\!78\)\( T^{29} + \)\(44\!\cdots\!05\)\( T^{30} + \)\(79\!\cdots\!90\)\( T^{31} + \)\(46\!\cdots\!86\)\( T^{32} - \)\(17\!\cdots\!70\)\( T^{33} - \)\(43\!\cdots\!37\)\( T^{34} - \)\(34\!\cdots\!38\)\( T^{35} - \)\(99\!\cdots\!69\)\( T^{36} + \)\(15\!\cdots\!78\)\( T^{37} + \)\(23\!\cdots\!49\)\( T^{38} + \)\(12\!\cdots\!74\)\( T^{39} + \)\(24\!\cdots\!01\)\( T^{40} \))
$89$ (\( ( 1 - 21 T + 227 T^{2} - 1869 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 13 T + 159 T^{2} + 1157 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 9 T + 167 T^{2} + 801 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 15 T + 233 T^{2} + 1335 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 9 T + 197 T^{2} - 801 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 16 T + 308 T^{2} + 3096 T^{3} + 38678 T^{4} + 275544 T^{5} + 2439668 T^{6} + 11279504 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 9 T + 640 T^{2} + 4050 T^{3} + 178355 T^{4} + 698876 T^{5} + 29313265 T^{6} + 51732294 T^{7} + 3369419368 T^{8} + 1036172651 T^{9} + 317703552166 T^{10} + 92219365939 T^{11} + 26689170813928 T^{12} + 36469663568886 T^{13} + 1839179937126865 T^{14} + 3902565131479324 T^{15} + 88639098149349155 T^{16} + 179136906326892450 T^{17} + 2519416835649331840 T^{18} + 3153207633367366881 T^{19} + 31181719929966183601 T^{20} )^{2} \))
$97$ (\( 1 - 33 T + 452 T^{2} - 4035 T^{3} + 36031 T^{4} - 391395 T^{5} + 4252868 T^{6} - 30118209 T^{7} + 88529281 T^{8} \))(\( 1 + 3 T + 182 T^{2} - 345 T^{3} + 16591 T^{4} - 33465 T^{5} + 1712438 T^{6} + 2738019 T^{7} + 88529281 T^{8} \))(\( 1 + 2 T + 27 T^{2} + 760 T^{3} + 10181 T^{4} + 73720 T^{5} + 254043 T^{6} + 1825346 T^{7} + 88529281 T^{8} \))(\( 1 - 17 T + 192 T^{2} - 1615 T^{3} + 8831 T^{4} - 156655 T^{5} + 1806528 T^{6} - 15515441 T^{7} + 88529281 T^{8} \))(\( 1 - 6 T - 61 T^{2} + 948 T^{3} + 229 T^{4} + 91956 T^{5} - 573949 T^{6} - 5476038 T^{7} + 88529281 T^{8} \))(\( 1 - 4 T + 26 T^{2} - 1532 T^{3} + 8091 T^{4} + 62828 T^{5} + 1010040 T^{6} - 5954240 T^{7} - 108387563 T^{8} - 577561280 T^{9} + 9503466360 T^{10} + 57341419244 T^{11} + 716290412571 T^{12} - 13155805273724 T^{13} + 21657272128154 T^{14} - 323193137912452 T^{15} + 7837433594376961 T^{16} \))(\( 1 + 27 T - 146 T^{2} - 10565 T^{3} - 42323 T^{4} + 1720741 T^{5} + 15722390 T^{6} - 118678851 T^{7} - 2079209414 T^{8} - 3980410181 T^{9} + 86169204810 T^{10} + 1330688850015 T^{11} + 13895090921091 T^{12} - 18497220903723 T^{13} - 2497628718197930 T^{14} - 21619068182342995 T^{15} + 116376235825961792 T^{16} + 3142408121720289719 T^{17} + 13315354121489765710 T^{18} - \)\(14\!\cdots\!49\)\( T^{19} - \)\(24\!\cdots\!21\)\( T^{20} - \)\(14\!\cdots\!53\)\( T^{21} + \)\(12\!\cdots\!90\)\( T^{22} + \)\(28\!\cdots\!87\)\( T^{23} + \)\(10\!\cdots\!52\)\( T^{24} - \)\(18\!\cdots\!15\)\( T^{25} - \)\(20\!\cdots\!70\)\( T^{26} - \)\(14\!\cdots\!99\)\( T^{27} + \)\(10\!\cdots\!51\)\( T^{28} + \)\(10\!\cdots\!55\)\( T^{29} + \)\(63\!\cdots\!90\)\( T^{30} - \)\(28\!\cdots\!93\)\( T^{31} - \)\(14\!\cdots\!74\)\( T^{32} - \)\(79\!\cdots\!27\)\( T^{33} + \)\(10\!\cdots\!10\)\( T^{34} + \)\(10\!\cdots\!13\)\( T^{35} - \)\(25\!\cdots\!83\)\( T^{36} - \)\(62\!\cdots\!05\)\( T^{37} - \)\(84\!\cdots\!94\)\( T^{38} + \)\(15\!\cdots\!91\)\( T^{39} + \)\(54\!\cdots\!01\)\( T^{40} \))
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