Properties

Label 35.4
Level 35
Weight 4
Dimension 112
Nonzero newspaces 6
Newform subspaces 11
Sturm bound 384
Trace bound 2

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Defining parameters

Level: \( N \) = \( 35 = 5 \cdot 7 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 11 \)
Sturm bound: \(384\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(35))\).

Total New Old
Modular forms 168 144 24
Cusp forms 120 112 8
Eisenstein series 48 32 16

Trace form

\( 112q + 2q^{2} + 2q^{3} - 22q^{4} - 11q^{5} - 68q^{6} - 60q^{7} + 42q^{8} + 124q^{9} + O(q^{10}) \) \( 112q + 2q^{2} + 2q^{3} - 22q^{4} - 11q^{5} - 68q^{6} - 60q^{7} + 42q^{8} + 124q^{9} - 22q^{10} - 82q^{11} - 128q^{12} + 64q^{13} - 30q^{14} - 70q^{15} - 10q^{16} - 214q^{17} - 154q^{18} + 106q^{19} + 236q^{20} + 426q^{21} + 256q^{22} - 270q^{23} - 1056q^{24} - 941q^{25} - 1240q^{26} - 664q^{27} - 822q^{28} + 160q^{29} + 982q^{30} + 990q^{31} + 2134q^{32} + 1678q^{33} + 2092q^{34} + 1305q^{35} + 3422q^{36} + 1394q^{37} + 1820q^{38} + 896q^{39} - 132q^{40} - 2024q^{41} - 2304q^{42} - 1580q^{43} - 3284q^{44} - 2456q^{45} - 3540q^{46} - 46q^{47} - 1736q^{48} + 344q^{49} + 986q^{50} - 206q^{51} + 332q^{52} + 914q^{53} - 3344q^{54} - 2878q^{55} - 3894q^{56} - 6412q^{57} - 6448q^{58} - 4810q^{59} - 2792q^{60} + 946q^{61} - 624q^{62} + 2088q^{63} + 5158q^{64} + 2590q^{65} + 8588q^{66} + 6126q^{67} + 9244q^{68} + 11556q^{69} + 12666q^{70} + 5608q^{71} + 10866q^{72} + 6394q^{73} + 8968q^{74} + 2237q^{75} + 1508q^{76} + 270q^{77} - 392q^{78} - 4314q^{79} - 6616q^{80} - 7922q^{81} - 8512q^{82} - 7764q^{83} - 15648q^{84} - 8542q^{85} - 12184q^{86} - 12400q^{87} - 17772q^{88} - 9990q^{89} - 12844q^{90} - 2364q^{91} - 8136q^{92} + 174q^{93} - 1952q^{94} + 4435q^{95} + 5924q^{96} + 5612q^{97} + 4258q^{98} + 8624q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(35))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
35.4.a \(\chi_{35}(1, \cdot)\) 35.4.a.a 1 1
35.4.a.b 2
35.4.a.c 3
35.4.b \(\chi_{35}(29, \cdot)\) 35.4.b.a 10 1
35.4.e \(\chi_{35}(11, \cdot)\) 35.4.e.a 2 2
35.4.e.b 4
35.4.e.c 10
35.4.f \(\chi_{35}(13, \cdot)\) 35.4.f.a 4 2
35.4.f.b 16
35.4.j \(\chi_{35}(4, \cdot)\) 35.4.j.a 20 2
35.4.k \(\chi_{35}(3, \cdot)\) 35.4.k.a 40 4

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(35))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(35)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + 8 T^{2} \))(\( 1 - 8 T + 30 T^{2} - 64 T^{3} + 64 T^{4} \))(\( 1 + 3 T + 10 T^{2} + 18 T^{3} + 80 T^{4} + 192 T^{5} + 512 T^{6} \))(\( 1 - 22 T^{2} + 277 T^{4} - 2288 T^{6} + 16804 T^{8} - 119360 T^{10} + 1075456 T^{12} - 9371648 T^{14} + 72613888 T^{16} - 369098752 T^{18} + 1073741824 T^{20} \))(\( 1 + 3 T + T^{2} + 24 T^{3} + 64 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 42 T^{3} + 177 T^{4} - 336 T^{5} + 832 T^{6} - 3072 T^{7} + 4096 T^{8} \))(\( 1 + T - 2 T^{2} - 19 T^{3} - 42 T^{4} - 63 T^{5} + 657 T^{6} + 1560 T^{7} + 688 T^{8} - 12224 T^{9} - 47936 T^{10} - 97792 T^{11} + 44032 T^{12} + 798720 T^{13} + 2691072 T^{14} - 2064384 T^{15} - 11010048 T^{16} - 39845888 T^{17} - 33554432 T^{18} + 134217728 T^{19} + 1073741824 T^{20} \))(\( ( 1 + 4 T + 8 T^{2} + 32 T^{3} + 64 T^{4} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} - 36 T^{3} + 69 T^{4} + 30 T^{5} + 450 T^{6} - 1060 T^{7} - 2140 T^{8} - 8480 T^{9} + 28800 T^{10} + 15360 T^{11} + 282624 T^{12} - 1179648 T^{13} + 524288 T^{14} - 4194304 T^{15} + 16777216 T^{16} )^{2} \))(\( 1 + 25 T^{2} + 258 T^{4} + 33 T^{6} - 25502 T^{8} - 235267 T^{10} + 345717 T^{12} + 20966068 T^{14} + 119154704 T^{16} - 842590720 T^{18} - 14784910336 T^{20} - 53925806080 T^{22} + 488057667584 T^{24} + 5496128929792 T^{26} + 5800168783872 T^{28} - 252616017707008 T^{30} - 1752484095721472 T^{32} + 145135534866432 T^{34} + 72620543991349248 T^{36} + 450359962737049600 T^{38} + 1152921504606846976 T^{40} \))
$3$ (\( 1 + 8 T + 27 T^{2} \))(\( 1 - 2 T + 23 T^{2} - 54 T^{3} + 729 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 40 T^{3} + 54 T^{4} - 1458 T^{5} + 19683 T^{6} \))(\( 1 - 112 T^{2} + 6658 T^{4} - 276902 T^{6} + 9205601 T^{8} - 263150732 T^{10} + 6710883129 T^{12} - 147157075782 T^{14} + 2579445615762 T^{16} - 31632108085872 T^{18} + 205891132094649 T^{20} \))(\( 1 - 2 T - 23 T^{2} - 54 T^{3} + 729 T^{4} \))(\( 1 - 2 T - 33 T^{2} + 34 T^{3} + 532 T^{4} + 918 T^{5} - 24057 T^{6} - 39366 T^{7} + 531441 T^{8} \))(\( 1 - 8 T + 5 T^{2} + 332 T^{3} - 2600 T^{4} + 7952 T^{5} + 33919 T^{6} - 391568 T^{7} + 1475571 T^{8} + 3518628 T^{9} - 49970576 T^{10} + 95002956 T^{11} + 1075691259 T^{12} - 7707232944 T^{13} + 18025947279 T^{14} + 114102508464 T^{15} - 1007293271400 T^{16} + 3472837263396 T^{17} + 1412147682405 T^{18} - 61004779879896 T^{19} + 205891132094649 T^{20} \))(\( ( 1 - 14 T + 98 T^{2} - 378 T^{3} + 729 T^{4} )( 1 + 14 T + 98 T^{2} + 378 T^{3} + 729 T^{4} ) \))(\( 1 - 578 T^{4} + 569733 T^{8} - 419608166 T^{12} + 7995847720 T^{16} - 222996983347206 T^{20} + 160909427107929573 T^{24} - 86754699201665491938 T^{28} + \)\(79\!\cdots\!61\)\( T^{32} \))(\( 1 + 94 T^{2} + 3897 T^{4} + 63714 T^{6} - 1432644 T^{8} - 117895374 T^{10} - 3905926665 T^{12} - 76543542222 T^{14} + 35136880767 T^{16} + 71543747590012 T^{18} + 2847783100372168 T^{20} + 52155391993118748 T^{22} + 18673179051695247 T^{24} - 29654536557439386558 T^{26} - \)\(11\!\cdots\!65\)\( T^{28} - \)\(24\!\cdots\!26\)\( T^{30} - \)\(21\!\cdots\!24\)\( T^{32} + \)\(69\!\cdots\!26\)\( T^{34} + \)\(31\!\cdots\!17\)\( T^{36} + \)\(54\!\cdots\!86\)\( T^{38} + \)\(42\!\cdots\!01\)\( T^{40} \))
$5$ (\( 1 + 5 T \))(\( ( 1 + 5 T )^{2} \))(\( ( 1 - 5 T )^{3} \))(\( 1 - 6 T + 199 T^{2} + 1836 T^{3} + 20 T^{4} + 520100 T^{5} + 2500 T^{6} + 28687500 T^{7} + 388671875 T^{8} - 1464843750 T^{9} + 30517578125 T^{10} \))(\( 1 + 5 T + 25 T^{2} \))(\( ( 1 + 5 T + 25 T^{2} )^{2} \))(\( ( 1 - 5 T + 25 T^{2} )^{5} \))(\( 1 + 70 T^{2} + 15625 T^{4} \))(\( 1 + 80 T^{2} - 27550 T^{4} - 1190000 T^{6} + 402781250 T^{8} - 18593750000 T^{10} - 6726074218750 T^{12} + 305175781250000 T^{14} + 59604644775390625 T^{16} \))(\( 1 - 3 T + 98 T^{2} + 987 T^{3} + 941 T^{4} + 99190 T^{5} + 2224950 T^{6} - 4310000 T^{7} + 319455625 T^{8} + 3205278125 T^{9} - 7141625000 T^{10} + 400659765625 T^{11} + 4991494140625 T^{12} - 8417968750000 T^{13} + 543200683593750 T^{14} + 3027038574218750 T^{15} + 3589630126953125 T^{16} + 470638275146484375 T^{17} + 5841255187988281250 T^{18} - 22351741790771484375 T^{19} + \)\(93\!\cdots\!25\)\( T^{20} \))
$7$ (\( 1 - 7 T \))(\( ( 1 + 7 T )^{2} \))(\( ( 1 - 7 T )^{3} \))(\( ( 1 + 49 T^{2} )^{5} \))(\( 1 + 28 T + 343 T^{2} \))(\( 1 - 22 T + 357 T^{2} - 7546 T^{3} + 117649 T^{4} \))(\( 1 + 62 T + 1423 T^{2} + 21774 T^{3} + 645505 T^{4} + 16781324 T^{5} + 221408215 T^{6} + 2561689326 T^{7} + 57423182761 T^{8} + 858159806462 T^{9} + 4747561509943 T^{10} \))(\( 1 + 42 T + 882 T^{2} + 14406 T^{3} + 117649 T^{4} \))(\( 1 - 32 T + 512 T^{2} - 11776 T^{3} + 157796 T^{4} - 2653504 T^{5} + 73457664 T^{6} - 1223791072 T^{7} + 21739628806 T^{8} - 419760337696 T^{9} + 8642220711936 T^{10} - 107078457588928 T^{11} + 2184099755168996 T^{12} - 55907284341088768 T^{13} + 833747762130149888 T^{14} - 17873467650665088224 T^{15} + \)\(19\!\cdots\!01\)\( T^{16} \))(\( 1 + 163 T^{2} - 85392 T^{4} + 47561997 T^{6} + 18120709551 T^{8} - 2661841566720 T^{10} + 2131883357965599 T^{12} + 658319260330100397 T^{14} - \)\(13\!\cdots\!08\)\( T^{16} + \)\(31\!\cdots\!63\)\( T^{18} + \)\(22\!\cdots\!49\)\( T^{20} \))
$11$ (\( 1 - 12 T + 1331 T^{2} \))(\( 1 + 14 T + 663 T^{2} + 18634 T^{3} + 1771561 T^{4} \))(\( 1 + 74 T + 5570 T^{2} + 204680 T^{3} + 7413670 T^{4} + 131095514 T^{5} + 2357947691 T^{6} \))(\( ( 1 - 42 T + 4384 T^{2} - 71864 T^{3} + 6091387 T^{4} - 31351772 T^{5} + 8107636097 T^{6} - 127311459704 T^{7} + 10337242677344 T^{8} - 131813991822282 T^{9} + 4177248169415651 T^{10} )^{2} \))(\( 1 - 45 T + 694 T^{2} - 59895 T^{3} + 1771561 T^{4} \))(\( 1 + 28 T - 1874 T^{2} - 112 T^{3} + 4249899 T^{4} - 149072 T^{5} - 3319905314 T^{6} + 66022535348 T^{7} + 3138428376721 T^{8} \))(\( 1 + 47 T - 2290 T^{2} - 69067 T^{3} + 5592863 T^{4} + 81640994 T^{5} - 4895628428 T^{6} + 11964087958 T^{7} + 250747435653 T^{8} - 11800753851543 T^{9} + 6322646842057914 T^{10} - 15706803376403733 T^{11} + 444214377852864333 T^{12} + 28210693575487004978 T^{13} - \)\(15\!\cdots\!88\)\( T^{14} + \)\(34\!\cdots\!94\)\( T^{15} + \)\(31\!\cdots\!03\)\( T^{16} - \)\(51\!\cdots\!37\)\( T^{17} - \)\(22\!\cdots\!90\)\( T^{18} + \)\(61\!\cdots\!37\)\( T^{19} + \)\(17\!\cdots\!01\)\( T^{20} \))(\( ( 1 - 24 T + 1331 T^{2} )^{4} \))(\( ( 1 + 38 T + 5001 T^{2} + 128438 T^{3} + 9570204 T^{4} + 170950978 T^{5} + 8859576561 T^{6} + 89602012258 T^{7} + 3138428376721 T^{8} )^{4} \))(\( ( 1 - 18 T - 2320 T^{2} - 14512 T^{3} + 3066033 T^{4} + 114865384 T^{5} + 843320902 T^{6} - 324387374202 T^{7} - 4324951728847 T^{8} + 184633610607512 T^{9} + 10564962314685174 T^{10} + 245747335718598472 T^{11} - 7661915809707920167 T^{12} - \)\(76\!\cdots\!82\)\( T^{13} + \)\(26\!\cdots\!42\)\( T^{14} + \)\(47\!\cdots\!84\)\( T^{15} + \)\(17\!\cdots\!73\)\( T^{16} - \)\(10\!\cdots\!32\)\( T^{17} - \)\(22\!\cdots\!20\)\( T^{18} - \)\(23\!\cdots\!78\)\( T^{19} + \)\(17\!\cdots\!01\)\( T^{20} )^{2} \))
$13$ (\( 1 + 78 T + 2197 T^{2} \))(\( 1 - 50 T + 4987 T^{2} - 109850 T^{3} + 4826809 T^{4} \))(\( 1 - 44 T + 3100 T^{2} - 238206 T^{3} + 6810700 T^{4} - 212379596 T^{5} + 10604499373 T^{6} \))(\( 1 - 5148 T^{2} + 15695890 T^{4} - 26456649834 T^{6} + 12726832447985 T^{8} + 6726945022008644 T^{10} + 61429989401426029865 T^{12} - \)\(61\!\cdots\!54\)\( T^{14} + \)\(17\!\cdots\!10\)\( T^{16} - \)\(27\!\cdots\!28\)\( T^{18} + \)\(26\!\cdots\!49\)\( T^{20} \))(\( ( 1 - 59 T + 2197 T^{2} )^{2} \))(\( ( 1 + 36 T + 4518 T^{2} + 79092 T^{3} + 4826809 T^{4} )^{2} \))(\( ( 1 - T + 3333 T^{2} - 85816 T^{3} + 12357966 T^{4} - 60104190 T^{5} + 27150451302 T^{6} - 414217441144 T^{7} + 35344796410209 T^{8} - 23298085122481 T^{9} + 51185893014090757 T^{10} )^{2} \))(\( 1 - 9038962 T^{4} + 23298085122481 T^{8} \))(\( 1 - 10320258 T^{4} + 44629878227173 T^{8} - 7691523192689518726 T^{12} - \)\(43\!\cdots\!80\)\( T^{16} - \)\(17\!\cdots\!06\)\( T^{20} + \)\(24\!\cdots\!53\)\( T^{24} - \)\(13\!\cdots\!78\)\( T^{28} + \)\(29\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 18525 T^{2} + 160006609 T^{4} - 849909002004 T^{6} + 3078968672040518 T^{8} - 7957049203639338622 T^{10} + \)\(14\!\cdots\!62\)\( T^{12} - \)\(19\!\cdots\!24\)\( T^{14} + \)\(17\!\cdots\!61\)\( T^{16} - \)\(10\!\cdots\!25\)\( T^{18} + \)\(26\!\cdots\!49\)\( T^{20} )^{2} \))
$17$ (\( 1 + 94 T + 4913 T^{2} \))(\( 1 + 50 T + 6579 T^{2} + 245650 T^{3} + 24137569 T^{4} \))(\( 1 + 52 T + 2992 T^{2} + 454246 T^{3} + 14699696 T^{4} + 1255153588 T^{5} + 118587876497 T^{6} \))(\( 1 - 27808 T^{2} + 398887406 T^{4} - 3863758657518 T^{6} + 27870658319090257 T^{8} - \)\(15\!\cdots\!08\)\( T^{10} + \)\(67\!\cdots\!33\)\( T^{12} - \)\(22\!\cdots\!98\)\( T^{14} + \)\(56\!\cdots\!54\)\( T^{16} - \)\(94\!\cdots\!68\)\( T^{18} + \)\(81\!\cdots\!49\)\( T^{20} \))(\( 1 - 54 T - 1997 T^{2} - 265302 T^{3} + 24137569 T^{4} \))(\( 1 - 76 T + 338 T^{2} + 333488 T^{3} - 22943213 T^{4} + 1638426544 T^{5} + 8158498322 T^{6} - 9012678613772 T^{7} + 582622237229761 T^{8} \))(\( 1 - 2 T - 14261 T^{2} + 138958 T^{3} + 98300987 T^{4} - 1490182020 T^{5} - 460609812434 T^{6} + 8569542609964 T^{7} + 1925726567282549 T^{8} - 18628621470376406 T^{9} - 8447181344461544071 T^{10} - 91522417283959282678 T^{11} + \)\(46\!\cdots\!81\)\( T^{12} + \)\(10\!\cdots\!08\)\( T^{13} - \)\(26\!\cdots\!74\)\( T^{14} - \)\(42\!\cdots\!60\)\( T^{15} + \)\(13\!\cdots\!83\)\( T^{16} + \)\(96\!\cdots\!86\)\( T^{17} - \)\(48\!\cdots\!81\)\( T^{18} - \)\(33\!\cdots\!46\)\( T^{19} + \)\(81\!\cdots\!49\)\( T^{20} \))(\( 1 + 501118 T^{4} + 582622237229761 T^{8} \))(\( 1 + 56587042 T^{4} + 2629953035023393 T^{8} + \)\(89\!\cdots\!54\)\( T^{12} + \)\(23\!\cdots\!20\)\( T^{16} + \)\(51\!\cdots\!94\)\( T^{20} + \)\(89\!\cdots\!53\)\( T^{24} + \)\(11\!\cdots\!02\)\( T^{28} + \)\(11\!\cdots\!41\)\( T^{32} \))(\( 1 + 24871 T^{2} + 310754372 T^{4} + 2565664433651 T^{6} + 15392238159232011 T^{8} + 69187097376257387954 T^{10} + \)\(20\!\cdots\!20\)\( T^{12} - \)\(13\!\cdots\!18\)\( T^{14} - \)\(74\!\cdots\!43\)\( T^{16} - \)\(65\!\cdots\!47\)\( T^{18} - \)\(38\!\cdots\!52\)\( T^{20} - \)\(15\!\cdots\!43\)\( T^{22} - \)\(43\!\cdots\!23\)\( T^{24} - \)\(19\!\cdots\!62\)\( T^{26} + \)\(68\!\cdots\!20\)\( T^{28} + \)\(56\!\cdots\!46\)\( T^{30} + \)\(30\!\cdots\!91\)\( T^{32} + \)\(12\!\cdots\!39\)\( T^{34} + \)\(35\!\cdots\!52\)\( T^{36} + \)\(69\!\cdots\!59\)\( T^{38} + \)\(67\!\cdots\!01\)\( T^{40} \))
$19$ (\( 1 - 40 T + 6859 T^{2} \))(\( 1 - 36 T + 10170 T^{2} - 246924 T^{3} + 47045881 T^{4} \))(\( 1 - 168 T + 26197 T^{2} - 2333344 T^{3} + 179685223 T^{4} - 7903708008 T^{5} + 322687697779 T^{6} \))(\( ( 1 - 36 T + 16945 T^{2} - 428680 T^{3} + 139308340 T^{4} - 2473195688 T^{5} + 955515904060 T^{6} - 20167628267080 T^{7} + 5467943038865155 T^{8} - 79679337086381796 T^{9} + 15181127029874798299 T^{10} )^{2} \))(\( 1 - 121 T + 7782 T^{2} - 829939 T^{3} + 47045881 T^{4} \))(\( 1 + 160 T + 6834 T^{2} + 807680 T^{3} + 129526475 T^{4} + 5539877120 T^{5} + 321511550754 T^{6} + 51630031644640 T^{7} + 2213314919066161 T^{8} \))(\( 1 - 21 T - 20094 T^{2} + 939517 T^{3} + 167437943 T^{4} - 9799934370 T^{5} - 1287146857848 T^{6} + 7008445096094 T^{7} + 14499989680340221 T^{8} + 155708744157753493 T^{9} - \)\(12\!\cdots\!86\)\( T^{10} + \)\(10\!\cdots\!87\)\( T^{11} + \)\(68\!\cdots\!01\)\( T^{12} + \)\(22\!\cdots\!26\)\( T^{13} - \)\(28\!\cdots\!28\)\( T^{14} - \)\(14\!\cdots\!30\)\( T^{15} + \)\(17\!\cdots\!63\)\( T^{16} + \)\(67\!\cdots\!23\)\( T^{17} - \)\(98\!\cdots\!74\)\( T^{18} - \)\(70\!\cdots\!19\)\( T^{19} + \)\(23\!\cdots\!01\)\( T^{20} \))(\( ( 1 + 7238 T^{2} + 47045881 T^{4} )^{2} \))(\( ( 1 + 34112 T^{2} + 604486178 T^{4} + 6969705935024 T^{6} + 56454362274260770 T^{8} + \)\(32\!\cdots\!44\)\( T^{10} + \)\(13\!\cdots\!58\)\( T^{12} + \)\(35\!\cdots\!92\)\( T^{14} + \)\(48\!\cdots\!21\)\( T^{16} )^{2} \))(\( ( 1 + 96 T - 21064 T^{2} - 1349852 T^{3} + 327425553 T^{4} + 10949009820 T^{5} - 3811487661418 T^{6} - 64847940490544 T^{7} + 34272562484687481 T^{8} + 175161053358359992 T^{9} - \)\(25\!\cdots\!46\)\( T^{10} + \)\(12\!\cdots\!28\)\( T^{11} + \)\(16\!\cdots\!61\)\( T^{12} - \)\(20\!\cdots\!76\)\( T^{13} - \)\(84\!\cdots\!98\)\( T^{14} + \)\(16\!\cdots\!80\)\( T^{15} + \)\(34\!\cdots\!73\)\( T^{16} - \)\(96\!\cdots\!88\)\( T^{17} - \)\(10\!\cdots\!44\)\( T^{18} + \)\(32\!\cdots\!44\)\( T^{19} + \)\(23\!\cdots\!01\)\( T^{20} )^{2} \))
$23$ (\( 1 - 32 T + 12167 T^{2} \))(\( 1 - 244 T + 29970 T^{2} - 2968748 T^{3} + 148035889 T^{4} \))(\( 1 + 124 T + 38233 T^{2} + 2923048 T^{3} + 465180911 T^{4} + 18356450236 T^{5} + 1801152661463 T^{6} \))(\( 1 - 45414 T^{2} + 1274694781 T^{4} - 26147824589064 T^{6} + 429674577520286642 T^{8} - \)\(57\!\cdots\!04\)\( T^{10} + \)\(63\!\cdots\!38\)\( T^{12} - \)\(57\!\cdots\!44\)\( T^{14} + \)\(41\!\cdots\!89\)\( T^{16} - \)\(21\!\cdots\!74\)\( T^{18} + \)\(71\!\cdots\!49\)\( T^{20} \))(\( 1 + 69 T - 7406 T^{2} + 839523 T^{3} + 148035889 T^{4} \))(\( 1 - 22 T + 3407 T^{2} + 599654 T^{3} - 145380788 T^{4} + 7295990218 T^{5} + 504358273823 T^{6} - 39625358552186 T^{7} + 21914624432020321 T^{8} \))(\( 1 + 201 T - 8588 T^{2} - 2602055 T^{3} + 123259362 T^{4} - 4272835029 T^{5} - 7259002378338 T^{6} - 285773779105835 T^{7} + 81100284674964733 T^{8} + 4436119997844018170 T^{9} - \)\(41\!\cdots\!12\)\( T^{10} + \)\(53\!\cdots\!90\)\( T^{11} + \)\(12\!\cdots\!37\)\( T^{12} - \)\(51\!\cdots\!05\)\( T^{13} - \)\(15\!\cdots\!98\)\( T^{14} - \)\(11\!\cdots\!03\)\( T^{15} + \)\(39\!\cdots\!78\)\( T^{16} - \)\(10\!\cdots\!65\)\( T^{17} - \)\(41\!\cdots\!08\)\( T^{18} + \)\(11\!\cdots\!47\)\( T^{19} + \)\(71\!\cdots\!49\)\( T^{20} \))(\( ( 1 + 118 T + 6962 T^{2} + 1435706 T^{3} + 148035889 T^{4} )^{2} \))(\( ( 1 + 36 T + 648 T^{2} - 285388 T^{3} + 164757104 T^{4} + 18803211700 T^{5} + 610876173080 T^{6} + 124249589297060 T^{7} + 7200722074542750 T^{8} + 1511744752977329020 T^{9} + 90431597350815668120 T^{10} + \)\(33\!\cdots\!00\)\( T^{11} + \)\(36\!\cdots\!84\)\( T^{12} - \)\(76\!\cdots\!16\)\( T^{13} + \)\(21\!\cdots\!12\)\( T^{14} + \)\(14\!\cdots\!28\)\( T^{15} + \)\(48\!\cdots\!41\)\( T^{16} )^{2} \))(\( 1 + 86997 T^{2} + 3851234318 T^{4} + 120213119227875 T^{6} + 3024885906662880458 T^{8} + \)\(65\!\cdots\!75\)\( T^{10} + \)\(12\!\cdots\!72\)\( T^{12} + \)\(20\!\cdots\!59\)\( T^{14} + \)\(31\!\cdots\!69\)\( T^{16} + \)\(44\!\cdots\!70\)\( T^{18} + \)\(56\!\cdots\!84\)\( T^{20} + \)\(65\!\cdots\!30\)\( T^{22} + \)\(69\!\cdots\!49\)\( T^{24} + \)\(67\!\cdots\!71\)\( T^{26} + \)\(59\!\cdots\!52\)\( T^{28} + \)\(46\!\cdots\!75\)\( T^{30} + \)\(31\!\cdots\!38\)\( T^{32} + \)\(18\!\cdots\!75\)\( T^{34} + \)\(88\!\cdots\!58\)\( T^{36} + \)\(29\!\cdots\!73\)\( T^{38} + \)\(50\!\cdots\!01\)\( T^{40} \))
$29$ (\( 1 + 50 T + 24389 T^{2} \))(\( 1 + 26 T + 47795 T^{2} + 634114 T^{3} + 594823321 T^{4} \))(\( 1 - 332 T + 80572 T^{2} - 13628846 T^{3} + 1965070508 T^{4} - 197481342572 T^{5} + 14507145975869 T^{6} \))(\( ( 1 - 44 T + 5930 T^{2} + 1541470 T^{3} + 551498365 T^{4} + 1453032508 T^{5} + 13450493623985 T^{6} + 916902304621870 T^{7} + 86027375636903170 T^{8} - 15567850461040637804 T^{9} + \)\(86\!\cdots\!49\)\( T^{10} )^{2} \))(\( ( 1 + 162 T + 24389 T^{2} )^{2} \))(\( ( 1 + 250 T + 57203 T^{2} + 6097250 T^{3} + 594823321 T^{4} )^{2} \))(\( ( 1 - 190 T + 63587 T^{2} - 10646688 T^{3} + 1713461669 T^{4} - 306482100734 T^{5} + 41789616645241 T^{6} - 6332898313810848 T^{7} + 922465891167582103 T^{8} - 67224808809039117790 T^{9} + \)\(86\!\cdots\!49\)\( T^{10} )^{2} \))(\( ( 1 - 46074 T^{2} + 594823321 T^{4} )^{2} \))(\( ( 1 - 112382 T^{2} + 6958299041 T^{4} - 280305393329582 T^{6} + 8070801965751649684 T^{8} - \)\(16\!\cdots\!22\)\( T^{10} + \)\(24\!\cdots\!81\)\( T^{12} - \)\(23\!\cdots\!02\)\( T^{14} + \)\(12\!\cdots\!81\)\( T^{16} )^{2} \))(\( ( 1 - 65 T + 90182 T^{2} - 6713567 T^{3} + 3728949859 T^{4} - 248936343596 T^{5} + 90945358111151 T^{6} - 3993386218696007 T^{7} + 1308283438395818158 T^{8} - 22997960908355487665 T^{9} + \)\(86\!\cdots\!49\)\( T^{10} )^{4} \))
$31$ (\( 1 + 248 T + 29791 T^{2} \))(\( 1 + 120 T - 1618 T^{2} + 3574920 T^{3} + 887503681 T^{4} \))(\( 1 - 320 T + 113341 T^{2} - 19016064 T^{3} + 3376541731 T^{4} - 284001177920 T^{5} + 26439622160671 T^{6} \))(\( ( 1 - 60 T + 93703 T^{2} - 4540464 T^{3} + 4639455974 T^{4} - 198463065128 T^{5} + 138214032921434 T^{6} - 4029678513447984 T^{7} + 2477471915321354713 T^{8} - 47259767027312985660 T^{9} + \)\(23\!\cdots\!51\)\( T^{10} )^{2} \))(\( 1 - 88 T - 22047 T^{2} - 2621608 T^{3} + 887503681 T^{4} \))(\( 1 - 132 T - 26514 T^{2} + 2065008 T^{3} + 523965779 T^{4} + 61518653328 T^{5} - 23531272598034 T^{6} - 3490030125208572 T^{7} + 787662783788549761 T^{8} \))(\( 1 + 388 T - 22159 T^{2} - 20563236 T^{3} + 2008090307 T^{4} + 1081135761624 T^{5} - 53801367425630 T^{6} - 25898096736714840 T^{7} + 2959646551177751045 T^{8} + \)\(44\!\cdots\!60\)\( T^{9} - \)\(63\!\cdots\!45\)\( T^{10} + \)\(13\!\cdots\!60\)\( T^{11} + \)\(26\!\cdots\!45\)\( T^{12} - \)\(68\!\cdots\!40\)\( T^{13} - \)\(42\!\cdots\!30\)\( T^{14} + \)\(25\!\cdots\!24\)\( T^{15} + \)\(14\!\cdots\!87\)\( T^{16} - \)\(42\!\cdots\!16\)\( T^{17} - \)\(13\!\cdots\!39\)\( T^{18} + \)\(71\!\cdots\!68\)\( T^{19} + \)\(55\!\cdots\!01\)\( T^{20} \))(\( ( 1 - 56202 T^{2} + 887503681 T^{4} )^{2} \))(\( ( 1 - 131748 T^{2} + 6374620088 T^{4} - 129464520417676 T^{6} + 1790519884005949870 T^{8} - \)\(11\!\cdots\!56\)\( T^{10} + \)\(50\!\cdots\!68\)\( T^{12} - \)\(92\!\cdots\!68\)\( T^{14} + \)\(62\!\cdots\!21\)\( T^{16} )^{2} \))(\( ( 1 - 417 T + 56846 T^{2} - 11580411 T^{3} + 3514034027 T^{4} - 410962717846 T^{5} + 101522259384800 T^{6} - 32401580870656750 T^{7} + 3387898207635269725 T^{8} - \)\(49\!\cdots\!35\)\( T^{9} + \)\(14\!\cdots\!70\)\( T^{10} - \)\(14\!\cdots\!85\)\( T^{11} + \)\(30\!\cdots\!25\)\( T^{12} - \)\(85\!\cdots\!50\)\( T^{13} + \)\(79\!\cdots\!00\)\( T^{14} - \)\(96\!\cdots\!46\)\( T^{15} + \)\(24\!\cdots\!07\)\( T^{16} - \)\(24\!\cdots\!41\)\( T^{17} + \)\(35\!\cdots\!66\)\( T^{18} - \)\(77\!\cdots\!87\)\( T^{19} + \)\(55\!\cdots\!01\)\( T^{20} )^{2} \))
$37$ (\( 1 + 434 T + 50653 T^{2} \))(\( 1 - 564 T + 173630 T^{2} - 28568292 T^{3} + 2565726409 T^{4} \))(\( 1 + 54 T + 139843 T^{2} + 5496260 T^{3} + 7083467479 T^{4} + 138549226086 T^{5} + 129961739795077 T^{6} \))(\( 1 - 341874 T^{2} + 58168168629 T^{4} - 6417971253293144 T^{6} + \)\(50\!\cdots\!78\)\( T^{8} - \)\(29\!\cdots\!04\)\( T^{10} + \)\(12\!\cdots\!02\)\( T^{12} - \)\(42\!\cdots\!64\)\( T^{14} + \)\(98\!\cdots\!41\)\( T^{16} - \)\(14\!\cdots\!14\)\( T^{18} + \)\(11\!\cdots\!49\)\( T^{20} \))(\( 1 - 259 T + 16428 T^{2} - 13119127 T^{3} + 2565726409 T^{4} \))(\( 1 - 416 T + 65478 T^{2} - 2609152 T^{3} + 241494107 T^{4} - 132161376256 T^{5} + 167998633808502 T^{6} - 54064083754752032 T^{7} + 6582952005840035281 T^{8} \))(\( 1 - 145 T - 115656 T^{2} + 19216271 T^{3} + 5000432907 T^{4} - 642783651926 T^{5} - 165614030132196 T^{6} - 34861351624072058 T^{7} + 17483682674137096733 T^{8} + \)\(17\!\cdots\!33\)\( T^{9} - \)\(12\!\cdots\!64\)\( T^{10} + \)\(89\!\cdots\!49\)\( T^{11} + \)\(44\!\cdots\!97\)\( T^{12} - \)\(45\!\cdots\!66\)\( T^{13} - \)\(10\!\cdots\!76\)\( T^{14} - \)\(21\!\cdots\!18\)\( T^{15} + \)\(84\!\cdots\!03\)\( T^{16} + \)\(16\!\cdots\!27\)\( T^{17} - \)\(50\!\cdots\!16\)\( T^{18} - \)\(31\!\cdots\!85\)\( T^{19} + \)\(11\!\cdots\!49\)\( T^{20} \))(\( ( 1 - 186 T + 17298 T^{2} - 9421458 T^{3} + 2565726409 T^{4} )^{2} \))(\( ( 1 + 128 T + 8192 T^{2} + 12209344 T^{3} + 2617912324 T^{4} - 578850045760 T^{5} - 21004703160320 T^{6} - 19309031650910720 T^{7} - 10318342014772450650 T^{8} - \)\(97\!\cdots\!60\)\( T^{9} - \)\(53\!\cdots\!80\)\( T^{10} - \)\(75\!\cdots\!20\)\( T^{11} + \)\(17\!\cdots\!44\)\( T^{12} + \)\(40\!\cdots\!92\)\( T^{13} + \)\(13\!\cdots\!68\)\( T^{14} + \)\(10\!\cdots\!36\)\( T^{15} + \)\(43\!\cdots\!61\)\( T^{16} )^{2} \))(\( 1 + 341982 T^{2} + 60045301980 T^{4} + 7262673706709132 T^{6} + \)\(68\!\cdots\!29\)\( T^{8} + \)\(53\!\cdots\!88\)\( T^{10} + \)\(36\!\cdots\!98\)\( T^{12} + \)\(21\!\cdots\!74\)\( T^{14} + \)\(12\!\cdots\!41\)\( T^{16} + \)\(65\!\cdots\!56\)\( T^{18} + \)\(33\!\cdots\!82\)\( T^{20} + \)\(16\!\cdots\!04\)\( T^{22} + \)\(81\!\cdots\!21\)\( T^{24} + \)\(37\!\cdots\!46\)\( T^{26} + \)\(15\!\cdots\!78\)\( T^{28} + \)\(59\!\cdots\!12\)\( T^{30} + \)\(19\!\cdots\!89\)\( T^{32} + \)\(53\!\cdots\!08\)\( T^{34} + \)\(11\!\cdots\!80\)\( T^{36} + \)\(16\!\cdots\!98\)\( T^{38} + \)\(12\!\cdots\!01\)\( T^{40} \))
$41$ (\( 1 - 402 T + 68921 T^{2} \))(\( 1 + 328 T + 133986 T^{2} + 22606088 T^{3} + 4750104241 T^{4} \))(\( 1 - 362 T + 248299 T^{2} - 51434996 T^{3} + 17113015379 T^{4} - 1719537735242 T^{5} + 327381934393961 T^{6} \))(\( ( 1 + 426 T + 295309 T^{2} + 96639912 T^{3} + 38701723370 T^{4} + 9383799279644 T^{5} + 2667361476383770 T^{6} + 459049655841066792 T^{7} + 96678831663946228949 T^{8} + \)\(96\!\cdots\!06\)\( T^{9} + \)\(15\!\cdots\!01\)\( T^{10} )^{2} \))(\( ( 1 - 195 T + 68921 T^{2} )^{2} \))(\( ( 1 + 106 T + 138851 T^{2} + 7305626 T^{3} + 4750104241 T^{4} )^{2} \))(\( ( 1 + 281 T + 269231 T^{2} + 52376686 T^{3} + 31133332221 T^{4} + 4627148668935 T^{5} + 2145740390003541 T^{6} + 248794718298125326 T^{7} + 88141365578820513991 T^{8} + \)\(63\!\cdots\!61\)\( T^{9} + \)\(15\!\cdots\!01\)\( T^{10} )^{2} \))(\( ( 1 - 137122 T^{2} + 4750104241 T^{4} )^{2} \))(\( ( 1 - 150948 T^{2} + 22824779528 T^{4} - 2154287110206316 T^{6} + \)\(17\!\cdots\!70\)\( T^{8} - \)\(10\!\cdots\!56\)\( T^{10} + \)\(51\!\cdots\!68\)\( T^{12} - \)\(16\!\cdots\!08\)\( T^{14} + \)\(50\!\cdots\!61\)\( T^{16} )^{2} \))(\( ( 1 + 306 T + 238411 T^{2} + 72369516 T^{3} + 26948434661 T^{4} + 7098521786510 T^{5} + 1857313065270781 T^{6} + 343762744870717356 T^{7} + 78051454360798635971 T^{8} + \)\(69\!\cdots\!86\)\( T^{9} + \)\(15\!\cdots\!01\)\( T^{10} )^{4} \))
$43$ (\( 1 + 68 T + 79507 T^{2} \))(\( 1 + 260 T + 166666 T^{2} + 20671820 T^{3} + 6321363049 T^{4} \))(\( 1 + 16 T + 149005 T^{2} + 1019664 T^{3} + 11846940535 T^{4} + 101141808784 T^{5} + 502592611936843 T^{6} \))(\( 1 - 571726 T^{2} + 155217037109 T^{4} - 26607197126991720 T^{6} + \)\(32\!\cdots\!58\)\( T^{8} - \)\(29\!\cdots\!88\)\( T^{10} + \)\(20\!\cdots\!42\)\( T^{12} - \)\(10\!\cdots\!20\)\( T^{14} + \)\(39\!\cdots\!41\)\( T^{16} - \)\(91\!\cdots\!26\)\( T^{18} + \)\(10\!\cdots\!49\)\( T^{20} \))(\( ( 1 + 286 T + 79507 T^{2} )^{2} \))(\( ( 1 + 666 T + 269853 T^{2} + 52951662 T^{3} + 6321363049 T^{4} )^{2} \))(\( ( 1 - 568 T + 328987 T^{2} - 103968566 T^{3} + 38129237297 T^{4} - 9424317397102 T^{5} + 3031541269772579 T^{6} - 657223051369917734 T^{7} + \)\(16\!\cdots\!41\)\( T^{8} - \)\(22\!\cdots\!68\)\( T^{9} + \)\(31\!\cdots\!07\)\( T^{10} )^{2} \))(\( ( 1 - 442 T + 97682 T^{2} - 35142094 T^{3} + 6321363049 T^{4} )^{2} \))(\( ( 1 + 156 T + 12168 T^{2} + 7825772 T^{3} + 5356042944 T^{4} + 572935296700 T^{5} + 54826929440600 T^{6} + 21251541125198220 T^{7} - 3798023677929840290 T^{8} + \)\(16\!\cdots\!40\)\( T^{9} + \)\(34\!\cdots\!00\)\( T^{10} + \)\(28\!\cdots\!00\)\( T^{11} + \)\(21\!\cdots\!44\)\( T^{12} + \)\(24\!\cdots\!04\)\( T^{13} + \)\(30\!\cdots\!32\)\( T^{14} + \)\(31\!\cdots\!08\)\( T^{15} + \)\(15\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 498625 T^{2} + 123288342332 T^{4} - 20092366583326983 T^{6} + \)\(23\!\cdots\!19\)\( T^{8} - \)\(21\!\cdots\!84\)\( T^{10} + \)\(15\!\cdots\!31\)\( T^{12} - \)\(80\!\cdots\!83\)\( T^{14} + \)\(31\!\cdots\!68\)\( T^{16} - \)\(79\!\cdots\!25\)\( T^{18} + \)\(10\!\cdots\!49\)\( T^{20} )^{2} \))
$47$ (\( 1 - 536 T + 103823 T^{2} \))(\( 1 + 350 T + 203423 T^{2} + 36338050 T^{3} + 10779215329 T^{4} \))(\( 1 + 730 T + 427526 T^{2} + 156550492 T^{3} + 44387031898 T^{4} + 7868827190170 T^{5} + 1119130473102767 T^{6} \))(\( 1 - 790788 T^{2} + 293544900286 T^{4} - 68085993582616986 T^{6} + \)\(11\!\cdots\!37\)\( T^{8} - \)\(13\!\cdots\!72\)\( T^{10} + \)\(11\!\cdots\!73\)\( T^{12} - \)\(79\!\cdots\!26\)\( T^{14} + \)\(36\!\cdots\!54\)\( T^{16} - \)\(10\!\cdots\!28\)\( T^{18} + \)\(14\!\cdots\!49\)\( T^{20} \))(\( 1 + 45 T - 101798 T^{2} + 4672035 T^{3} + 10779215329 T^{4} \))(\( 1 - 196 T - 170642 T^{2} - 276752 T^{3} + 28937567667 T^{4} - 28733222896 T^{5} - 1839386862171218 T^{6} - 219349572728142332 T^{7} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( 1 - 473 T - 199186 T^{2} + 58080717 T^{3} + 49165993755 T^{4} - 5867004935190 T^{5} - 5795165507082528 T^{6} + 101648572991175666 T^{7} + \)\(44\!\cdots\!01\)\( T^{8} + \)\(23\!\cdots\!89\)\( T^{9} - \)\(43\!\cdots\!94\)\( T^{10} + \)\(24\!\cdots\!47\)\( T^{11} + \)\(48\!\cdots\!29\)\( T^{12} + \)\(11\!\cdots\!22\)\( T^{13} - \)\(67\!\cdots\!48\)\( T^{14} - \)\(70\!\cdots\!70\)\( T^{15} + \)\(61\!\cdots\!95\)\( T^{16} + \)\(75\!\cdots\!99\)\( T^{17} - \)\(26\!\cdots\!66\)\( T^{18} - \)\(66\!\cdots\!99\)\( T^{19} + \)\(14\!\cdots\!49\)\( T^{20} \))(\( 1 + 19389520078 T^{4} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( 1 + 7709822402 T^{4} + \)\(18\!\cdots\!53\)\( T^{8} + \)\(15\!\cdots\!34\)\( T^{12} + \)\(37\!\cdots\!20\)\( T^{16} + \)\(17\!\cdots\!94\)\( T^{20} + \)\(24\!\cdots\!93\)\( T^{24} + \)\(12\!\cdots\!42\)\( T^{28} + \)\(18\!\cdots\!61\)\( T^{32} \))(\( 1 + 586566 T^{2} + 176578551908 T^{4} + 36486163260401340 T^{6} + \)\(57\!\cdots\!69\)\( T^{8} + \)\(67\!\cdots\!20\)\( T^{10} + \)\(56\!\cdots\!06\)\( T^{12} + \)\(16\!\cdots\!82\)\( T^{14} - \)\(40\!\cdots\!95\)\( T^{16} - \)\(95\!\cdots\!20\)\( T^{18} - \)\(12\!\cdots\!18\)\( T^{20} - \)\(10\!\cdots\!80\)\( T^{22} - \)\(47\!\cdots\!95\)\( T^{24} + \)\(20\!\cdots\!98\)\( T^{26} + \)\(75\!\cdots\!86\)\( T^{28} + \)\(98\!\cdots\!80\)\( T^{30} + \)\(89\!\cdots\!49\)\( T^{32} + \)\(61\!\cdots\!60\)\( T^{34} + \)\(32\!\cdots\!88\)\( T^{36} + \)\(11\!\cdots\!54\)\( T^{38} + \)\(21\!\cdots\!01\)\( T^{40} \))
$53$ (\( 1 - 22 T + 148877 T^{2} \))(\( 1 + 56 T + 265770 T^{2} + 8337112 T^{3} + 22164361129 T^{4} \))(\( 1 - 110 T + 105959 T^{2} + 57565396 T^{3} + 15774858043 T^{4} - 2438079724190 T^{5} + 3299763591802133 T^{6} \))(\( 1 - 972802 T^{2} + 426448129813 T^{4} - 113796393195924760 T^{6} + \)\(21\!\cdots\!46\)\( T^{8} - \)\(34\!\cdots\!56\)\( T^{10} + \)\(48\!\cdots\!34\)\( T^{12} - \)\(55\!\cdots\!60\)\( T^{14} + \)\(46\!\cdots\!57\)\( T^{16} - \)\(23\!\cdots\!62\)\( T^{18} + \)\(53\!\cdots\!49\)\( T^{20} \))(\( 1 + 597 T + 207532 T^{2} + 88879569 T^{3} + 22164361129 T^{4} \))(\( 1 - 952 T + 422302 T^{2} - 177308096 T^{3} + 77165754267 T^{4} - 26397097408192 T^{5} + 9360054033498958 T^{6} - 3141374939395630616 T^{7} + \)\(49\!\cdots\!41\)\( T^{8} \))(\( 1 + 351 T - 584980 T^{2} - 134936885 T^{3} + 229763822323 T^{4} + 33792782771134 T^{5} - 61716154410935792 T^{6} - 4669968185983210370 T^{7} + \)\(12\!\cdots\!13\)\( T^{8} + \)\(30\!\cdots\!53\)\( T^{9} - \)\(21\!\cdots\!12\)\( T^{10} + \)\(44\!\cdots\!81\)\( T^{11} + \)\(28\!\cdots\!77\)\( T^{12} - \)\(15\!\cdots\!10\)\( T^{13} - \)\(30\!\cdots\!72\)\( T^{14} + \)\(24\!\cdots\!38\)\( T^{15} + \)\(25\!\cdots\!47\)\( T^{16} - \)\(21\!\cdots\!05\)\( T^{17} - \)\(14\!\cdots\!80\)\( T^{18} + \)\(12\!\cdots\!87\)\( T^{19} + \)\(53\!\cdots\!49\)\( T^{20} \))(\( ( 1 + 174 T + 15138 T^{2} + 25904598 T^{3} + 22164361129 T^{4} )^{2} \))(\( ( 1 - 884 T + 390728 T^{2} - 186770028 T^{3} + 56931497936 T^{4} + 1204571535772 T^{5} - 5868049883599464 T^{6} + 4928363332779316004 T^{7} - \)\(28\!\cdots\!94\)\( T^{8} + \)\(73\!\cdots\!08\)\( T^{9} - \)\(13\!\cdots\!56\)\( T^{10} + \)\(39\!\cdots\!76\)\( T^{11} + \)\(27\!\cdots\!76\)\( T^{12} - \)\(13\!\cdots\!96\)\( T^{13} + \)\(42\!\cdots\!92\)\( T^{14} - \)\(14\!\cdots\!52\)\( T^{15} + \)\(24\!\cdots\!81\)\( T^{16} )^{2} \))(\( 1 + 296626 T^{2} + 51825139476 T^{4} - 234602868312748 T^{6} - \)\(10\!\cdots\!55\)\( T^{8} - \)\(17\!\cdots\!32\)\( T^{10} + \)\(16\!\cdots\!94\)\( T^{12} + \)\(69\!\cdots\!02\)\( T^{14} + \)\(80\!\cdots\!09\)\( T^{16} - \)\(61\!\cdots\!84\)\( T^{18} - \)\(20\!\cdots\!90\)\( T^{20} - \)\(13\!\cdots\!36\)\( T^{22} + \)\(39\!\cdots\!69\)\( T^{24} + \)\(76\!\cdots\!78\)\( T^{26} + \)\(40\!\cdots\!14\)\( T^{28} - \)\(96\!\cdots\!68\)\( T^{30} - \)\(12\!\cdots\!55\)\( T^{32} - \)\(61\!\cdots\!32\)\( T^{34} + \)\(30\!\cdots\!36\)\( T^{36} + \)\(38\!\cdots\!94\)\( T^{38} + \)\(28\!\cdots\!01\)\( T^{40} \))
$59$ (\( 1 + 560 T + 205379 T^{2} \))(\( ( 1 + 616 T + 205379 T^{2} )^{2} \))(\( 1 + 180 T + 3577 T^{2} - 128522760 T^{3} + 734640683 T^{4} + 7592496055380 T^{5} + 8662995818654939 T^{6} \))(\( ( 1 - 432 T + 700701 T^{2} - 268108032 T^{3} + 242314293152 T^{4} - 78692992197632 T^{5} + 49766267213264608 T^{6} - 11308939863198304512 T^{7} + \)\(60\!\cdots\!39\)\( T^{8} - \)\(76\!\cdots\!92\)\( T^{9} + \)\(36\!\cdots\!99\)\( T^{10} )^{2} \))(\( 1 - 360 T - 75779 T^{2} - 73936440 T^{3} + 42180533641 T^{4} \))(\( 1 + 840 T + 259954 T^{2} + 29305920 T^{3} + 11504401275 T^{4} + 6018820543680 T^{5} + 10964998442112514 T^{6} + 7276916487670148760 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} \))(\( 1 + 708 T - 594963 T^{2} - 347355028 T^{3} + 339076027067 T^{4} + 128485265745528 T^{5} - 124220569667051830 T^{6} - 22490966070804129944 T^{7} + \)\(39\!\cdots\!17\)\( T^{8} + \)\(24\!\cdots\!56\)\( T^{9} - \)\(88\!\cdots\!49\)\( T^{10} + \)\(49\!\cdots\!24\)\( T^{11} + \)\(16\!\cdots\!97\)\( T^{12} - \)\(19\!\cdots\!16\)\( T^{13} - \)\(22\!\cdots\!30\)\( T^{14} + \)\(46\!\cdots\!72\)\( T^{15} + \)\(25\!\cdots\!07\)\( T^{16} - \)\(53\!\cdots\!52\)\( T^{17} - \)\(18\!\cdots\!43\)\( T^{18} + \)\(46\!\cdots\!52\)\( T^{19} + \)\(13\!\cdots\!01\)\( T^{20} \))(\( ( 1 + 234038 T^{2} + 42180533641 T^{4} )^{2} \))(\( ( 1 + 564992 T^{2} + 242975513538 T^{4} + 70586293205624784 T^{6} + \)\(16\!\cdots\!70\)\( T^{8} + \)\(29\!\cdots\!44\)\( T^{10} + \)\(43\!\cdots\!78\)\( T^{12} + \)\(42\!\cdots\!32\)\( T^{14} + \)\(31\!\cdots\!61\)\( T^{16} )^{2} \))(\( ( 1 + 1257 T + 28812 T^{2} - 249631107 T^{3} + 284096537737 T^{4} + 190598573007302 T^{5} - 67351197370118230 T^{6} - 17879205302394815916 T^{7} + \)\(35\!\cdots\!17\)\( T^{8} + \)\(50\!\cdots\!09\)\( T^{9} - \)\(62\!\cdots\!14\)\( T^{10} + \)\(10\!\cdots\!11\)\( T^{11} + \)\(15\!\cdots\!97\)\( T^{12} - \)\(15\!\cdots\!24\)\( T^{13} - \)\(11\!\cdots\!30\)\( T^{14} + \)\(69\!\cdots\!98\)\( T^{15} + \)\(21\!\cdots\!77\)\( T^{16} - \)\(38\!\cdots\!13\)\( T^{17} + \)\(91\!\cdots\!32\)\( T^{18} + \)\(81\!\cdots\!83\)\( T^{19} + \)\(13\!\cdots\!01\)\( T^{20} )^{2} \))
$61$ (\( 1 + 278 T + 226981 T^{2} \))(\( 1 - 336 T + 458858 T^{2} - 76265616 T^{3} + 51520374361 T^{4} \))(\( 1 - 1222 T + 1103759 T^{2} - 593135356 T^{3} + 250532321579 T^{4} - 62957897469142 T^{5} + 11694146092834141 T^{6} \))(\( ( 1 + 442 T + 798959 T^{2} + 288407548 T^{3} + 306199000900 T^{4} + 84731818304260 T^{5} + 69501355423282900 T^{6} + 14858864841498076828 T^{7} + \)\(93\!\cdots\!19\)\( T^{8} + \)\(11\!\cdots\!82\)\( T^{9} + \)\(60\!\cdots\!01\)\( T^{10} )^{2} \))(\( 1 + 392 T - 73317 T^{2} + 88976552 T^{3} + 51520374361 T^{4} \))(\( 1 + 98 T + 203041 T^{2} - 63445102 T^{3} - 16282426916 T^{4} - 14400832697062 T^{5} + 10460748330631801 T^{6} + 1146026317097745818 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \))(\( 1 + 1944 T + 1201957 T^{2} + 465031984 T^{3} + 642573181668 T^{4} + 544248825687640 T^{5} + 179953145332374091 T^{6} + 96783385662450749352 T^{7} + \)\(99\!\cdots\!11\)\( T^{8} + \)\(41\!\cdots\!76\)\( T^{9} + \)\(10\!\cdots\!72\)\( T^{10} + \)\(94\!\cdots\!56\)\( T^{11} + \)\(51\!\cdots\!71\)\( T^{12} + \)\(11\!\cdots\!32\)\( T^{13} + \)\(47\!\cdots\!11\)\( T^{14} + \)\(32\!\cdots\!40\)\( T^{15} + \)\(87\!\cdots\!08\)\( T^{16} + \)\(14\!\cdots\!24\)\( T^{17} + \)\(84\!\cdots\!37\)\( T^{18} + \)\(31\!\cdots\!24\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} \))(\( ( 1 - 338442 T^{2} + 51520374361 T^{4} )^{2} \))(\( ( 1 - 1255428 T^{2} + 726056294438 T^{4} - 264100181504479396 T^{6} + \)\(69\!\cdots\!70\)\( T^{8} - \)\(13\!\cdots\!56\)\( T^{10} + \)\(19\!\cdots\!98\)\( T^{12} - \)\(17\!\cdots\!68\)\( T^{14} + \)\(70\!\cdots\!41\)\( T^{16} )^{2} \))(\( ( 1 - 256 T - 751153 T^{2} + 204835764 T^{3} + 291945445208 T^{4} - 69908506373860 T^{5} - 90828760031755899 T^{6} + 11422899003515573152 T^{7} + \)\(26\!\cdots\!11\)\( T^{8} - \)\(72\!\cdots\!04\)\( T^{9} - \)\(65\!\cdots\!08\)\( T^{10} - \)\(16\!\cdots\!24\)\( T^{11} + \)\(13\!\cdots\!71\)\( T^{12} + \)\(13\!\cdots\!32\)\( T^{13} - \)\(24\!\cdots\!79\)\( T^{14} - \)\(42\!\cdots\!60\)\( T^{15} + \)\(39\!\cdots\!48\)\( T^{16} + \)\(63\!\cdots\!04\)\( T^{17} - \)\(52\!\cdots\!73\)\( T^{18} - \)\(40\!\cdots\!76\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \))
$67$ (\( 1 + 164 T + 300763 T^{2} \))(\( 1 + 152 T + 599110 T^{2} + 45715976 T^{3} + 90458382169 T^{4} \))(\( 1 - 204 T + 92049 T^{2} + 202232824 T^{3} + 27684933387 T^{4} - 18453509962476 T^{5} + 27206534396294947 T^{6} \))(\( 1 - 1699398 T^{2} + 1483840354293 T^{4} - 859383478287164840 T^{6} + \)\(36\!\cdots\!46\)\( T^{8} - \)\(12\!\cdots\!24\)\( T^{10} + \)\(33\!\cdots\!74\)\( T^{12} - \)\(70\!\cdots\!40\)\( T^{14} + \)\(10\!\cdots\!37\)\( T^{16} - \)\(11\!\cdots\!58\)\( T^{18} + \)\(60\!\cdots\!49\)\( T^{20} \))(\( 1 - 280 T - 222363 T^{2} - 84213640 T^{3} + 90458382169 T^{4} \))(\( 1 - 1286 T + 648343 T^{2} - 519450122 T^{3} + 423076706692 T^{4} - 156231377043086 T^{5} + 58648058870595967 T^{6} - 34987603233635301842 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} \))(\( 1 + 1118 T - 89755 T^{2} - 544724314 T^{3} - 95454149156 T^{4} + 123793400461058 T^{5} + 10648899628386803 T^{6} - 41531516553755134674 T^{7} - \)\(95\!\cdots\!45\)\( T^{8} + \)\(76\!\cdots\!64\)\( T^{9} + \)\(63\!\cdots\!60\)\( T^{10} + \)\(23\!\cdots\!32\)\( T^{11} - \)\(86\!\cdots\!05\)\( T^{12} - \)\(11\!\cdots\!78\)\( T^{13} + \)\(87\!\cdots\!83\)\( T^{14} + \)\(30\!\cdots\!94\)\( T^{15} - \)\(70\!\cdots\!04\)\( T^{16} - \)\(12\!\cdots\!38\)\( T^{17} - \)\(60\!\cdots\!55\)\( T^{18} + \)\(22\!\cdots\!14\)\( T^{19} + \)\(60\!\cdots\!49\)\( T^{20} \))(\( ( 1 + 554 T + 153458 T^{2} + 166622702 T^{3} + 90458382169 T^{4} )^{2} \))(\( ( 1 - 2252 T + 2535752 T^{2} - 2225077036 T^{3} + 1815966182624 T^{4} - 1326362080610140 T^{5} + 857611434080134680 T^{6} - \)\(51\!\cdots\!20\)\( T^{7} + \)\(29\!\cdots\!50\)\( T^{8} - \)\(15\!\cdots\!60\)\( T^{9} + \)\(77\!\cdots\!20\)\( T^{10} - \)\(36\!\cdots\!80\)\( T^{11} + \)\(14\!\cdots\!64\)\( T^{12} - \)\(54\!\cdots\!48\)\( T^{13} + \)\(18\!\cdots\!68\)\( T^{14} - \)\(50\!\cdots\!84\)\( T^{15} + \)\(66\!\cdots\!21\)\( T^{16} )^{2} \))(\( 1 + 1063794 T^{2} + 583098793137 T^{4} + 259526981150249198 T^{6} + \)\(10\!\cdots\!64\)\( T^{8} + \)\(33\!\cdots\!86\)\( T^{10} + \)\(10\!\cdots\!43\)\( T^{12} + \)\(33\!\cdots\!50\)\( T^{14} + \)\(10\!\cdots\!79\)\( T^{16} + \)\(32\!\cdots\!28\)\( T^{18} + \)\(99\!\cdots\!52\)\( T^{20} + \)\(29\!\cdots\!32\)\( T^{22} + \)\(84\!\cdots\!19\)\( T^{24} + \)\(24\!\cdots\!50\)\( T^{26} + \)\(71\!\cdots\!03\)\( T^{28} + \)\(20\!\cdots\!14\)\( T^{30} + \)\(54\!\cdots\!84\)\( T^{32} + \)\(12\!\cdots\!22\)\( T^{34} + \)\(26\!\cdots\!17\)\( T^{36} + \)\(43\!\cdots\!26\)\( T^{38} + \)\(36\!\cdots\!01\)\( T^{40} \))
$71$ (\( 1 - 672 T + 357911 T^{2} \))(\( ( 1 + 952 T + 357911 T^{2} )^{2} \))(\( 1 + 136 T + 900677 T^{2} + 112926832 T^{3} + 322362205747 T^{4} + 17421638613256 T^{5} + 45848500718449031 T^{6} \))(\( ( 1 - 440 T + 1467407 T^{2} - 492789840 T^{3} + 952423975862 T^{4} - 247789000795120 T^{5} + 340883017624744282 T^{6} - 63126518417384162640 T^{7} + \)\(67\!\cdots\!17\)\( T^{8} - \)\(72\!\cdots\!40\)\( T^{9} + \)\(58\!\cdots\!51\)\( T^{10} )^{2} \))(\( ( 1 - 48 T + 357911 T^{2} )^{2} \))(\( ( 1 - 1064 T + 753846 T^{2} - 380817304 T^{3} + 128100283921 T^{4} )^{2} \))(\( ( 1 - 864 T + 1690919 T^{2} - 959528208 T^{3} + 1093514996270 T^{4} - 459054202031136 T^{5} + 391381045829991970 T^{6} - \)\(12\!\cdots\!68\)\( T^{7} + \)\(77\!\cdots\!89\)\( T^{8} - \)\(14\!\cdots\!24\)\( T^{9} + \)\(58\!\cdots\!51\)\( T^{10} )^{2} \))(\( ( 1 + 916 T + 357911 T^{2} )^{4} \))(\( ( 1 - 1592 T + 1740546 T^{2} - 1313608192 T^{3} + 887743954594 T^{4} - 470154821606912 T^{5} + 222964436777560866 T^{6} - 72990813143770857352 T^{7} + \)\(16\!\cdots\!41\)\( T^{8} )^{4} \))(\( ( 1 + 736 T + 1120739 T^{2} + 619752192 T^{3} + 535581261770 T^{4} + 254623498821824 T^{5} + 191690424981362470 T^{6} + 79390431755862104832 T^{7} + \)\(51\!\cdots\!09\)\( T^{8} + \)\(12\!\cdots\!76\)\( T^{9} + \)\(58\!\cdots\!51\)\( T^{10} )^{4} \))
$73$ (\( 1 - 82 T + 389017 T^{2} \))(\( 1 - 676 T + 655606 T^{2} - 262975492 T^{3} + 151334226289 T^{4} \))(\( 1 - 310 T + 450279 T^{2} - 192471924 T^{3} + 175166185743 T^{4} - 46913610149590 T^{5} + 58871586708267913 T^{6} \))(\( 1 - 2526314 T^{2} + 3202004346045 T^{4} - 2657464253457184440 T^{6} + \)\(15\!\cdots\!10\)\( T^{8} - \)\(70\!\cdots\!52\)\( T^{10} + \)\(24\!\cdots\!90\)\( T^{12} - \)\(60\!\cdots\!40\)\( T^{14} + \)\(11\!\cdots\!05\)\( T^{16} - \)\(13\!\cdots\!74\)\( T^{18} + \)\(79\!\cdots\!49\)\( T^{20} \))(\( 1 + 668 T + 57207 T^{2} + 259863356 T^{3} + 151334226289 T^{4} \))(\( 1 - 172 T - 727998 T^{2} + 3517744 T^{3} + 411087581507 T^{4} + 1368462217648 T^{5} - 110171014069939422 T^{6} - 10125912913822081036 T^{7} + \)\(22\!\cdots\!21\)\( T^{8} \))(\( 1 - 1652 T + 187071 T^{2} + 576882732 T^{3} + 434002580187 T^{4} - 531471598604920 T^{5} - 187348919748357658 T^{6} + \)\(14\!\cdots\!36\)\( T^{7} + \)\(11\!\cdots\!81\)\( T^{8} - \)\(15\!\cdots\!24\)\( T^{9} - \)\(60\!\cdots\!27\)\( T^{10} - \)\(59\!\cdots\!08\)\( T^{11} + \)\(17\!\cdots\!09\)\( T^{12} + \)\(87\!\cdots\!68\)\( T^{13} - \)\(42\!\cdots\!18\)\( T^{14} - \)\(47\!\cdots\!40\)\( T^{15} + \)\(15\!\cdots\!03\)\( T^{16} + \)\(77\!\cdots\!36\)\( T^{17} + \)\(98\!\cdots\!11\)\( T^{18} - \)\(33\!\cdots\!44\)\( T^{19} + \)\(79\!\cdots\!49\)\( T^{20} \))(\( 1 + 283756682078 T^{4} + \)\(22\!\cdots\!21\)\( T^{8} \))(\( 1 - 454504886008 T^{4} + \)\(58\!\cdots\!08\)\( T^{8} + \)\(30\!\cdots\!64\)\( T^{12} - \)\(14\!\cdots\!30\)\( T^{16} + \)\(68\!\cdots\!44\)\( T^{20} + \)\(30\!\cdots\!28\)\( T^{24} - \)\(54\!\cdots\!88\)\( T^{28} + \)\(27\!\cdots\!81\)\( T^{32} \))(\( 1 + 2627747 T^{2} + 3676489000168 T^{4} + 3499491410340553131 T^{6} + \)\(24\!\cdots\!99\)\( T^{8} + \)\(13\!\cdots\!14\)\( T^{10} + \)\(58\!\cdots\!36\)\( T^{12} + \)\(18\!\cdots\!34\)\( T^{14} + \)\(32\!\cdots\!05\)\( T^{16} - \)\(20\!\cdots\!67\)\( T^{18} - \)\(32\!\cdots\!88\)\( T^{20} - \)\(31\!\cdots\!63\)\( T^{22} + \)\(74\!\cdots\!05\)\( T^{24} + \)\(62\!\cdots\!46\)\( T^{26} + \)\(30\!\cdots\!76\)\( T^{28} + \)\(10\!\cdots\!86\)\( T^{30} + \)\(29\!\cdots\!39\)\( T^{32} + \)\(63\!\cdots\!99\)\( T^{34} + \)\(10\!\cdots\!08\)\( T^{36} + \)\(10\!\cdots\!23\)\( T^{38} + \)\(63\!\cdots\!01\)\( T^{40} \))
$79$ (\( 1 + 1000 T + 493039 T^{2} \))(\( 1 - 1014 T + 1120119 T^{2} - 499941546 T^{3} + 243087455521 T^{4} \))(\( 1 + 1034 T + 1094262 T^{2} + 675989052 T^{3} + 539513842218 T^{4} + 251352429008714 T^{5} + 119851595982618319 T^{6} \))(\( ( 1 + 1790 T + 3366800 T^{2} + 3670804696 T^{3} + 3768250665575 T^{4} + 2749245121712708 T^{5} + 1857894539904432425 T^{6} + \)\(89\!\cdots\!16\)\( T^{7} + \)\(40\!\cdots\!00\)\( T^{8} + \)\(10\!\cdots\!90\)\( T^{9} + \)\(29\!\cdots\!99\)\( T^{10} )^{2} \))(\( 1 + 782 T + 118485 T^{2} + 385556498 T^{3} + 243087455521 T^{4} \))(\( 1 - 1240 T + 258714 T^{2} - 363081920 T^{3} + 634365179075 T^{4} - 179013546754880 T^{5} + 62890127967659994 T^{6} - \)\(14\!\cdots\!60\)\( T^{7} + \)\(59\!\cdots\!41\)\( T^{8} \))(\( 1 + 218 T - 1453667 T^{2} - 1963370 T^{3} + 1085327492763 T^{4} - 157694661811260 T^{5} - 575577812372860958 T^{6} + 81206950854078977836 T^{7} + \)\(26\!\cdots\!29\)\( T^{8} - \)\(96\!\cdots\!90\)\( T^{9} - \)\(12\!\cdots\!41\)\( T^{10} - \)\(47\!\cdots\!10\)\( T^{11} + \)\(64\!\cdots\!09\)\( T^{12} + \)\(97\!\cdots\!84\)\( T^{13} - \)\(34\!\cdots\!78\)\( T^{14} - \)\(45\!\cdots\!40\)\( T^{15} + \)\(15\!\cdots\!43\)\( T^{16} - \)\(13\!\cdots\!30\)\( T^{17} - \)\(50\!\cdots\!27\)\( T^{18} + \)\(37\!\cdots\!62\)\( T^{19} + \)\(84\!\cdots\!01\)\( T^{20} \))(\( ( 1 - 757594 T^{2} + 243087455521 T^{4} )^{2} \))(\( ( 1 - 3309042 T^{2} + 5035453141141 T^{4} - 4602570291031234982 T^{6} + \)\(27\!\cdots\!04\)\( T^{8} - \)\(11\!\cdots\!22\)\( T^{10} + \)\(29\!\cdots\!81\)\( T^{12} - \)\(47\!\cdots\!62\)\( T^{14} + \)\(34\!\cdots\!81\)\( T^{16} )^{2} \))(\( ( 1 - 23 T - 490582 T^{2} + 501202135 T^{3} - 21157688697 T^{4} - 158473640720330 T^{5} + 191577977326579332 T^{6} - 87280947293160331886 T^{7} - \)\(63\!\cdots\!11\)\( T^{8} + \)\(24\!\cdots\!75\)\( T^{9} - \)\(17\!\cdots\!66\)\( T^{10} + \)\(11\!\cdots\!25\)\( T^{11} - \)\(15\!\cdots\!31\)\( T^{12} - \)\(10\!\cdots\!34\)\( T^{13} + \)\(11\!\cdots\!12\)\( T^{14} - \)\(46\!\cdots\!70\)\( T^{15} - \)\(30\!\cdots\!17\)\( T^{16} + \)\(35\!\cdots\!65\)\( T^{17} - \)\(17\!\cdots\!42\)\( T^{18} - \)\(39\!\cdots\!57\)\( T^{19} + \)\(84\!\cdots\!01\)\( T^{20} )^{2} \))
$83$ (\( 1 + 448 T + 571787 T^{2} \))(\( 1 + 376 T + 458918 T^{2} + 214991912 T^{3} + 326940373369 T^{4} \))(\( 1 + 1660 T + 2204369 T^{2} + 1855605736 T^{3} + 1260429537403 T^{4} + 542721019792540 T^{5} + 186940255267540403 T^{6} \))(\( 1 - 4190178 T^{2} + 8255172061705 T^{4} - 10209824219182076768 T^{6} + \)\(89\!\cdots\!30\)\( T^{8} - \)\(58\!\cdots\!68\)\( T^{10} + \)\(29\!\cdots\!70\)\( T^{12} - \)\(10\!\cdots\!48\)\( T^{14} + \)\(28\!\cdots\!45\)\( T^{16} - \)\(47\!\cdots\!38\)\( T^{18} + \)\(37\!\cdots\!49\)\( T^{20} \))(\( ( 1 - 768 T + 571787 T^{2} )^{2} \))(\( ( 1 + 1906 T + 2051733 T^{2} + 1089826022 T^{3} + 326940373369 T^{4} )^{2} \))(\( ( 1 + 1502 T + 2544639 T^{2} + 2916185966 T^{3} + 2925083074177 T^{4} + 2334912952153996 T^{5} + 1672524475734444299 T^{6} + \)\(95\!\cdots\!54\)\( T^{7} + \)\(47\!\cdots\!17\)\( T^{8} + \)\(16\!\cdots\!22\)\( T^{9} + \)\(61\!\cdots\!07\)\( T^{10} )^{2} \))(\( 1 - 142535619682 T^{4} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 - 501476514268 T^{4} + \)\(49\!\cdots\!28\)\( T^{8} - \)\(16\!\cdots\!96\)\( T^{12} + \)\(82\!\cdots\!70\)\( T^{16} - \)\(17\!\cdots\!56\)\( T^{20} + \)\(56\!\cdots\!88\)\( T^{24} - \)\(61\!\cdots\!08\)\( T^{28} + \)\(13\!\cdots\!41\)\( T^{32} \))(\( ( 1 - 3839289 T^{2} + 6907901943196 T^{4} - 7907786449656750719 T^{6} + \)\(65\!\cdots\!67\)\( T^{8} - \)\(42\!\cdots\!64\)\( T^{10} + \)\(21\!\cdots\!23\)\( T^{12} - \)\(84\!\cdots\!59\)\( T^{14} + \)\(24\!\cdots\!64\)\( T^{16} - \)\(43\!\cdots\!69\)\( T^{18} + \)\(37\!\cdots\!49\)\( T^{20} )^{2} \))
$89$ (\( 1 + 870 T + 704969 T^{2} \))(\( 1 + 216 T + 1417730 T^{2} + 152273304 T^{3} + 496981290961 T^{4} \))(\( 1 - 242 T + 1427227 T^{2} - 347564516 T^{3} + 1006150790963 T^{4} - 120269472412562 T^{5} + 350356403707485209 T^{6} \))(\( ( 1 + 746 T + 1733225 T^{2} - 126840160 T^{3} + 408516498430 T^{4} - 989188619900372 T^{5} + 287991467381698670 T^{6} - 63037186462499793760 T^{7} + \)\(60\!\cdots\!25\)\( T^{8} + \)\(18\!\cdots\!66\)\( T^{9} + \)\(17\!\cdots\!49\)\( T^{10} )^{2} \))(\( 1 - 1194 T + 720667 T^{2} - 841732986 T^{3} + 496981290961 T^{4} \))(\( 1 + 650 T - 883111 T^{2} - 67812550 T^{3} + 909789389860 T^{4} - 47805745560950 T^{5} - 438889644841859671 T^{6} + \)\(22\!\cdots\!50\)\( T^{7} + \)\(24\!\cdots\!21\)\( T^{8} \))(\( 1 + 2322 T + 885553 T^{2} - 27254778 T^{3} + 2336719735260 T^{4} + 1546137721242198 T^{5} - 1544579435132215297 T^{6} - \)\(76\!\cdots\!66\)\( T^{7} + \)\(50\!\cdots\!63\)\( T^{8} - \)\(79\!\cdots\!44\)\( T^{9} - \)\(15\!\cdots\!20\)\( T^{10} - \)\(55\!\cdots\!36\)\( T^{11} + \)\(24\!\cdots\!43\)\( T^{12} - \)\(26\!\cdots\!94\)\( T^{13} - \)\(38\!\cdots\!37\)\( T^{14} + \)\(26\!\cdots\!02\)\( T^{15} + \)\(28\!\cdots\!60\)\( T^{16} - \)\(23\!\cdots\!42\)\( T^{17} + \)\(54\!\cdots\!73\)\( T^{18} + \)\(99\!\cdots\!38\)\( T^{19} + \)\(30\!\cdots\!01\)\( T^{20} \))(\( ( 1 + 1409918 T^{2} + 496981290961 T^{4} )^{2} \))(\( ( 1 + 2745112 T^{2} + 4127085754748 T^{4} + 4427611292803540904 T^{6} + \)\(35\!\cdots\!70\)\( T^{8} + \)\(22\!\cdots\!44\)\( T^{10} + \)\(10\!\cdots\!08\)\( T^{12} + \)\(33\!\cdots\!72\)\( T^{14} + \)\(61\!\cdots\!41\)\( T^{16} )^{2} \))(\( ( 1 + 2938 T + 2564153 T^{2} + 1165926318 T^{3} + 2792164847900 T^{4} + 3018079716233822 T^{5} - 308766378555648137 T^{6} - \)\(71\!\cdots\!34\)\( T^{7} + \)\(10\!\cdots\!23\)\( T^{8} - \)\(25\!\cdots\!56\)\( T^{9} - \)\(15\!\cdots\!40\)\( T^{10} - \)\(18\!\cdots\!64\)\( T^{11} + \)\(50\!\cdots\!03\)\( T^{12} - \)\(25\!\cdots\!06\)\( T^{13} - \)\(76\!\cdots\!77\)\( T^{14} + \)\(52\!\cdots\!78\)\( T^{15} + \)\(34\!\cdots\!00\)\( T^{16} + \)\(10\!\cdots\!02\)\( T^{17} + \)\(15\!\cdots\!73\)\( T^{18} + \)\(12\!\cdots\!02\)\( T^{19} + \)\(30\!\cdots\!01\)\( T^{20} )^{2} \))
$97$ (\( 1 - 1026 T + 912673 T^{2} \))(\( 1 - 2742 T + 3608187 T^{2} - 2502549366 T^{3} + 832972004929 T^{4} \))(\( 1 - 100 T + 2266056 T^{2} - 184512618 T^{3} + 2068168127688 T^{4} - 83297200492900 T^{5} + 760231058654565217 T^{6} \))(\( 1 - 7632832 T^{2} + 27232168586510 T^{4} - 59864233929174443022 T^{6} + \)\(89\!\cdots\!65\)\( T^{8} - \)\(96\!\cdots\!52\)\( T^{10} + \)\(74\!\cdots\!85\)\( T^{12} - \)\(41\!\cdots\!02\)\( T^{14} + \)\(15\!\cdots\!90\)\( T^{16} - \)\(36\!\cdots\!92\)\( T^{18} + \)\(40\!\cdots\!49\)\( T^{20} \))(\( ( 1 - 902 T + 912673 T^{2} )^{2} \))(\( ( 1 + 628 T + 1423942 T^{2} + 573158644 T^{3} + 832972004929 T^{4} )^{2} \))(\( ( 1 - 598 T + 4425037 T^{2} - 2054168072 T^{3} + 7929732195026 T^{4} - 2752469564393540 T^{5} + 7237252471630964498 T^{6} - \)\(17\!\cdots\!88\)\( T^{7} + \)\(33\!\cdots\!29\)\( T^{8} - \)\(41\!\cdots\!18\)\( T^{9} + \)\(63\!\cdots\!93\)\( T^{10} )^{2} \))(\( 1 - 1665717085762 T^{4} + \)\(69\!\cdots\!41\)\( T^{8} \))(\( 1 - 1933745575038 T^{4} + \)\(56\!\cdots\!93\)\( T^{8} + \)\(12\!\cdots\!34\)\( T^{12} - \)\(15\!\cdots\!80\)\( T^{16} + \)\(86\!\cdots\!94\)\( T^{20} + \)\(27\!\cdots\!33\)\( T^{24} - \)\(64\!\cdots\!98\)\( T^{28} + \)\(23\!\cdots\!61\)\( T^{32} \))(\( ( 1 - 3893530 T^{2} + 9172577050349 T^{4} - 15424484400745116216 T^{6} + \)\(19\!\cdots\!18\)\( T^{8} - \)\(20\!\cdots\!68\)\( T^{10} + \)\(16\!\cdots\!22\)\( T^{12} - \)\(10\!\cdots\!56\)\( T^{14} + \)\(53\!\cdots\!61\)\( T^{16} - \)\(18\!\cdots\!30\)\( T^{18} + \)\(40\!\cdots\!49\)\( T^{20} )^{2} \))
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