Properties

Label 81.2
Level 81
Weight 2
Dimension 164
Nonzero newspaces 4
Newform subspaces 5
Sturm bound 972
Trace bound 1

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Defining parameters

Level: \( N \) = \( 81 = 3^{4} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 5 \)
Sturm bound: \(972\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(81))\).

Total New Old
Modular forms 297 220 77
Cusp forms 190 164 26
Eisenstein series 107 56 51

Trace form

\( 164q - 12q^{2} - 18q^{3} - 22q^{4} - 15q^{5} - 18q^{6} - 23q^{7} - 24q^{8} - 18q^{9} + O(q^{10}) \) \( 164q - 12q^{2} - 18q^{3} - 22q^{4} - 15q^{5} - 18q^{6} - 23q^{7} - 24q^{8} - 18q^{9} - 39q^{10} - 21q^{11} - 18q^{12} - 29q^{13} - 33q^{14} - 18q^{15} - 22q^{16} - 27q^{17} - 9q^{18} - 23q^{19} + 21q^{20} + 9q^{21} - 15q^{22} + 21q^{23} + 36q^{24} - 10q^{25} + 75q^{26} + 9q^{27} - 5q^{28} + 15q^{29} + 36q^{30} - 11q^{31} + 36q^{32} + 9q^{33} - 9q^{34} - 3q^{35} + 18q^{36} - 41q^{37} - 51q^{38} - 18q^{39} + 3q^{40} - 15q^{41} + 27q^{42} - 23q^{43} + 51q^{44} + 36q^{45} + 15q^{46} + 51q^{47} + 81q^{48} + 132q^{50} + 45q^{51} + 19q^{52} + 63q^{53} + 108q^{54} + 15q^{55} + 159q^{56} + 36q^{57} + 3q^{58} + 57q^{59} + 99q^{60} - 5q^{61} + 93q^{62} + 36q^{63} - 16q^{64} - 3q^{65} - 18q^{66} - 11q^{67} - 108q^{68} - 72q^{69} + 21q^{70} - 117q^{71} - 234q^{72} - 68q^{73} - 195q^{74} - 108q^{75} + q^{76} - 177q^{77} - 135q^{78} - 23q^{79} - 330q^{80} - 90q^{81} - 120q^{82} - 129q^{83} - 243q^{84} + 9q^{85} - 213q^{86} - 162q^{87} + 33q^{88} - 90q^{89} - 99q^{90} - 34q^{91} - 105q^{92} + 21q^{94} + 51q^{95} - 27q^{96} + 31q^{97} + 126q^{98} + 54q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(81))\)

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
81.2.a \(\chi_{81}(1, \cdot)\) 81.2.a.a 2 1
81.2.c \(\chi_{81}(28, \cdot)\) 81.2.c.a 2 2
81.2.c.b 4
81.2.e \(\chi_{81}(10, \cdot)\) 81.2.e.a 12 6
81.2.g \(\chi_{81}(4, \cdot)\) 81.2.g.a 144 18

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(81))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(81)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T^{2} + 4 T^{4} \))(\( 1 - 2 T^{2} + 4 T^{4} \))(\( 1 - T^{2} - 3 T^{4} - 4 T^{6} + 16 T^{8} \))(\( ( 1 - 3 T + 3 T^{2} - 3 T^{4} + 6 T^{5} - 11 T^{6} + 12 T^{7} - 12 T^{8} + 48 T^{10} - 96 T^{11} + 64 T^{12} )( 1 - 3 T + 9 T^{2} - 18 T^{3} + 36 T^{4} - 57 T^{5} + 91 T^{6} - 114 T^{7} + 144 T^{8} - 144 T^{9} + 144 T^{10} - 96 T^{11} + 64 T^{12} ) \))
$3$ 1
$5$ (\( 1 + 7 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 7 T^{2} + 24 T^{4} - 175 T^{6} + 625 T^{8} \))(\( 1 - 3 T + 3 T^{2} - 18 T^{3} + 87 T^{4} - 147 T^{5} + 323 T^{6} - 1368 T^{7} + 3096 T^{8} - 5562 T^{9} + 16272 T^{10} - 41310 T^{11} + 82629 T^{12} - 206550 T^{13} + 406800 T^{14} - 695250 T^{15} + 1935000 T^{16} - 4275000 T^{17} + 5046875 T^{18} - 11484375 T^{19} + 33984375 T^{20} - 35156250 T^{21} + 29296875 T^{22} - 146484375 T^{23} + 244140625 T^{24} \))
$7$ (\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( ( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} )^{2} \))(\( 1 + 6 T + 12 T^{2} - 11 T^{3} - 213 T^{4} - 678 T^{5} - 224 T^{6} + 3942 T^{7} + 15255 T^{8} + 25135 T^{9} - 22044 T^{10} - 210732 T^{11} - 647141 T^{12} - 1475124 T^{13} - 1080156 T^{14} + 8621305 T^{15} + 36627255 T^{16} + 66253194 T^{17} - 26353376 T^{18} - 558362154 T^{19} - 1227902613 T^{20} - 443889677 T^{21} + 3389702988 T^{22} + 11863960458 T^{23} + 13841287201 T^{24} \))
$11$ (\( 1 + 10 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 10 T^{2} - 21 T^{4} - 1210 T^{6} + 14641 T^{8} \))(\( 1 + 3 T - 15 T^{2} - 126 T^{3} - 201 T^{4} + 1488 T^{5} + 7145 T^{6} + 1530 T^{7} - 61974 T^{8} - 202716 T^{9} - 19692 T^{10} + 1304451 T^{11} + 4526883 T^{12} + 14348961 T^{13} - 2382732 T^{14} - 269814996 T^{15} - 907361334 T^{16} + 246408030 T^{17} + 12657803345 T^{18} + 28996910448 T^{19} - 43086135081 T^{20} - 297101409066 T^{21} - 389061369015 T^{22} + 855935011833 T^{23} + 3138428376721 T^{24} \))
$13$ (\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} )^{2} \))(\( 1 + 6 T + 48 T^{2} + 214 T^{3} + 1488 T^{4} + 5928 T^{5} + 32329 T^{6} + 112023 T^{7} + 560277 T^{8} + 1799710 T^{9} + 8467593 T^{10} + 25055985 T^{11} + 112181629 T^{12} + 325727805 T^{13} + 1431023217 T^{14} + 3953962870 T^{15} + 16002071397 T^{16} + 41593355739 T^{17} + 156045908161 T^{18} + 371973208776 T^{19} + 1213807312848 T^{20} + 2269362865822 T^{21} + 6617207608752 T^{22} + 10752962364222 T^{23} + 23298085122481 T^{24} \))
$17$ (\( 1 + 7 T^{2} + 289 T^{4} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} + 289 T^{4} )^{2} \))(\( 1 + 9 T - 30 T^{2} - 423 T^{3} + 1029 T^{4} + 14184 T^{5} - 23521 T^{6} - 296649 T^{7} + 637560 T^{8} + 4620213 T^{9} - 12537675 T^{10} - 28264410 T^{11} + 250681641 T^{12} - 480494970 T^{13} - 3623388075 T^{14} + 22699106469 T^{15} + 53249648760 T^{16} - 421199159193 T^{17} - 567739760449 T^{18} + 5820243737832 T^{19} + 7178054406789 T^{20} - 50162671758231 T^{21} - 60479817013470 T^{22} + 308447066768697 T^{23} + 582622237229761 T^{24} \))
$19$ (\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{4} \))(\( 1 + 3 T - 75 T^{2} - 242 T^{3} + 3012 T^{4} + 9714 T^{5} - 85589 T^{6} - 257166 T^{7} + 1946502 T^{8} + 4391737 T^{9} - 39399504 T^{10} - 33490578 T^{11} + 763159453 T^{12} - 636320982 T^{13} - 14223220944 T^{14} + 30122924083 T^{15} + 253670087142 T^{16} - 636768475434 T^{17} - 4026609908909 T^{18} + 8683070072646 T^{19} + 51154491879492 T^{20} - 78090422862518 T^{21} - 459829969335075 T^{22} + 349470776694657 T^{23} + 2213314919066161 T^{24} \))
$23$ (\( 1 + 34 T^{2} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 34 T^{2} + 627 T^{4} - 17986 T^{6} + 279841 T^{8} \))(\( 1 - 12 T + 48 T^{2} - 153 T^{3} - 336 T^{4} + 12228 T^{5} - 51922 T^{6} + 116820 T^{7} - 165330 T^{8} - 4324509 T^{9} + 14509764 T^{10} + 2453454 T^{11} + 107316369 T^{12} + 56429442 T^{13} + 7675665156 T^{14} - 52616301003 T^{15} - 46266112530 T^{16} + 751893589260 T^{17} - 7686319428658 T^{18} + 41634205565916 T^{19} - 26312491054416 T^{20} - 275576357203839 T^{21} + 1988472538255152 T^{22} - 11433717094967124 T^{23} + 21914624432020321 T^{24} \))
$29$ (\( 1 + 55 T^{2} + 841 T^{4} \))(\( 1 - 29 T^{2} + 841 T^{4} \))(\( 1 - 55 T^{2} + 2184 T^{4} - 46255 T^{6} + 707281 T^{8} \))(\( 1 - 6 T + 21 T^{2} - 252 T^{3} + 249 T^{4} + 984 T^{5} + 18431 T^{6} - 29592 T^{7} + 680634 T^{8} - 5882274 T^{9} + 10161684 T^{10} - 17557326 T^{11} + 254066229 T^{12} - 509162454 T^{13} + 8545976244 T^{14} - 143462780586 T^{15} + 481399496154 T^{16} - 606965921208 T^{17} + 10963188629351 T^{18} + 16973878288056 T^{19} + 124561356827289 T^{20} - 3655800785918988 T^{21} + 8834851899304221 T^{22} - 73203058594234974 T^{23} + 353814783205469041 T^{24} \))
$31$ (\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( ( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} )^{2} \))(\( 1 - 3 T + 84 T^{2} - 434 T^{3} + 5601 T^{4} - 30963 T^{5} + 266473 T^{6} - 1627992 T^{7} + 11453211 T^{8} - 69240287 T^{9} + 408317577 T^{10} - 2527882269 T^{11} + 13547586181 T^{12} - 78364350339 T^{13} + 392393191497 T^{14} - 2062737390017 T^{15} + 10577280875931 T^{16} - 46608028794792 T^{17} + 236495768387113 T^{18} - 851873070718893 T^{19} + 4777042700707041 T^{20} - 11474796017731214 T^{21} + 68848776106387284 T^{22} - 76225430689214493 T^{23} + 787662783788549761 T^{24} \))
$37$ (\( ( 1 + 7 T + 37 T^{2} )^{2} \))(\( ( 1 - 11 T + 37 T^{2} )^{2} \))(\( ( 1 + 7 T + 37 T^{2} )^{4} \))(\( 1 + 3 T - 156 T^{2} - 107 T^{3} + 13731 T^{4} - 9132 T^{5} - 864755 T^{6} + 641043 T^{7} + 43249536 T^{8} - 18536771 T^{9} - 1953626739 T^{10} + 269355786 T^{11} + 78884071369 T^{12} + 9966164082 T^{13} - 2674515005691 T^{14} - 938943061463 T^{15} + 81056593639296 T^{16} + 44452458227151 T^{17} - 2218724740814795 T^{18} - 866917901978556 T^{19} + 48229855381789251 T^{20} - 13905906158073239 T^{21} - 750139162097184444 T^{22} + 533752865338381239 T^{23} + 6582952005840035281 T^{24} \))
$41$ (\( 1 + 34 T^{2} + 1681 T^{4} \))(\( 1 - 41 T^{2} + 1681 T^{4} \))(\( 1 - 34 T^{2} - 525 T^{4} - 57154 T^{6} + 2825761 T^{8} \))(\( 1 + 15 T + 93 T^{2} + 90 T^{3} - 2460 T^{4} - 12513 T^{5} + 27971 T^{6} + 441396 T^{7} + 3206862 T^{8} + 12736494 T^{9} - 41813613 T^{10} - 1731506832 T^{11} - 16389887967 T^{12} - 70991780112 T^{13} - 70288683453 T^{14} + 877811902974 T^{15} + 9061825571982 T^{16} + 51138463696596 T^{17} + 132865165725011 T^{18} - 2436960229072953 T^{19} - 19642916063637660 T^{20} + 29464374095456490 T^{21} + 1248307315844173293 T^{22} + 8254935475743726615 T^{23} + 22563490300366186081 T^{24} \))
$43$ (\( ( 1 - 2 T + 43 T^{2} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( ( 1 + 2 T - 39 T^{2} + 86 T^{3} + 1849 T^{4} )^{2} \))(\( 1 - 3 T - 60 T^{2} + 16 T^{3} + 606 T^{4} + 5874 T^{5} + 128269 T^{6} - 73818 T^{7} - 5307417 T^{8} - 22910987 T^{9} - 79924800 T^{10} + 580797228 T^{11} + 14492410483 T^{12} + 24974280804 T^{13} - 147780955200 T^{14} - 1821583843409 T^{15} - 18145002547017 T^{16} - 10851869245374 T^{17} + 810834916932181 T^{18} + 1596662521642518 T^{19} + 7083049368226206 T^{20} + 8041481790989488 T^{21} - 1296688938797054940 T^{22} - 2787881218413668121 T^{23} + 39959630797262576401 T^{24} \))
$47$ (\( 1 + 46 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 46 T^{2} - 93 T^{4} - 101614 T^{6} + 4879681 T^{8} \))(\( 1 - 15 T + 111 T^{2} - 873 T^{3} + 6828 T^{4} - 69612 T^{5} + 654227 T^{6} - 4732173 T^{7} + 31522707 T^{8} - 170376048 T^{9} + 1258782219 T^{10} - 11485670769 T^{11} + 82734051465 T^{12} - 539826526143 T^{13} + 2780649921771 T^{14} - 17688952431504 T^{15} + 153820754416467 T^{16} - 1085300249810211 T^{17} + 7052053707045683 T^{18} - 35267048661670356 T^{19} + 162583465326504108 T^{20} - 977000903018715591 T^{21} + 5838503678177135439 T^{22} - 37082388226260184545 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))
$53$ (\( ( 1 + 53 T^{2} )^{2} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 - 9 T + 210 T^{2} - 1872 T^{3} + 23856 T^{4} - 168327 T^{5} + 1634317 T^{6} - 8921331 T^{7} + 67011504 T^{8} - 278697744 T^{9} + 1657001010 T^{10} - 3763759437 T^{11} + 22164361129 T^{12} )^{2} \))
$59$ (\( 1 - 74 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 74 T^{2} + 1995 T^{4} + 257594 T^{6} + 12117361 T^{8} \))(\( 1 - 12 T + 192 T^{2} - 2349 T^{3} + 25089 T^{4} - 223824 T^{5} + 1972808 T^{6} - 12709350 T^{7} + 71501877 T^{8} - 339681357 T^{9} - 107943444 T^{10} + 13247965206 T^{11} - 122980417173 T^{12} + 781629947154 T^{13} - 375751128564 T^{14} - 69763417419303 T^{15} + 866414055786597 T^{16} - 9086223139495650 T^{17} + 83214094211233928 T^{18} - 557019929938127856 T^{19} + 3683828849054809569 T^{20} - 20349377178020451711 T^{21} + 98134416633723148992 T^{22} - \)\(36\!\cdots\!08\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))
$61$ (\( ( 1 + 7 T + 61 T^{2} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( ( 1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4} )^{2} \))(\( 1 - 12 T - 51 T^{2} + 583 T^{3} + 2127 T^{4} + 45474 T^{5} - 455363 T^{6} - 2399139 T^{7} + 11507670 T^{8} + 53383966 T^{9} + 844033821 T^{10} - 2578276122 T^{11} - 56950876769 T^{12} - 157274843442 T^{13} + 3140649847941 T^{14} + 12117145986646 T^{15} + 159333369100470 T^{16} - 2026303924984839 T^{17} - 23460472230148043 T^{18} + 142913087725218954 T^{19} + 407761454745216687 T^{20} + 6817687172122304203 T^{21} - 36380488494807012651 T^{22} - \)\(52\!\cdots\!32\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))
$67$ (\( ( 1 + 10 T + 67 T^{2} )^{2} \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4} )^{2} \))(\( 1 + 15 T + 255 T^{2} + 2968 T^{3} + 36174 T^{4} + 397221 T^{5} + 4107115 T^{6} + 41367024 T^{7} + 386429292 T^{8} + 3556146616 T^{9} + 31289775603 T^{10} + 264760435272 T^{11} + 2216964278029 T^{12} + 17738949163224 T^{13} + 140459802681867 T^{14} + 1069557324668008 T^{15} + 7786983421036332 T^{16} + 55850657704271568 T^{17} + 371522978282032435 T^{18} + 2407441924578007383 T^{19} + 14689092167933931534 T^{20} + 80748994088203402696 T^{21} + \)\(46\!\cdots\!95\)\( T^{22} + \)\(18\!\cdots\!45\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))
$71$ (\( 1 + 34 T^{2} + 5041 T^{4} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 34 T^{2} + 5041 T^{4} )^{2} \))(\( 1 + 27 T + 78 T^{2} - 2565 T^{3} + 13071 T^{4} + 524664 T^{5} - 751711 T^{6} - 30297321 T^{7} + 410765508 T^{8} + 3391054713 T^{9} - 30034133541 T^{10} - 13624108308 T^{11} + 3600759258249 T^{12} - 967311689868 T^{13} - 151402067180181 T^{14} + 1213695783384543 T^{15} + 10438242055098948 T^{16} - 54663315804868671 T^{17} - 96294392526538831 T^{18} + 4771882122782055624 T^{19} + 8440644406913342031 T^{20} - \)\(11\!\cdots\!15\)\( T^{21} + \)\(25\!\cdots\!78\)\( T^{22} + \)\(62\!\cdots\!17\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))
$73$ (\( ( 1 + 7 T + 73 T^{2} )^{2} \))(\( ( 1 + 7 T + 73 T^{2} )^{2} \))(\( ( 1 + 7 T + 73 T^{2} )^{4} \))(\( 1 - 6 T - 228 T^{2} + 2296 T^{3} + 24945 T^{4} - 381255 T^{5} - 980072 T^{6} + 40200363 T^{7} - 102286134 T^{8} - 2648934335 T^{9} + 21743689350 T^{10} + 78452536893 T^{11} - 2017821540323 T^{12} + 5727035193189 T^{13} + 115872120546150 T^{14} - 1030480488198695 T^{15} - 2904746284290294 T^{16} + 83338230563588259 T^{17} - 148318437827512808 T^{18} - 4211875922398326735 T^{19} + 20117146992297850545 T^{20} + \)\(13\!\cdots\!48\)\( T^{21} - \)\(97\!\cdots\!72\)\( T^{22} - \)\(18\!\cdots\!62\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))
$79$ (\( ( 1 - 2 T + 79 T^{2} )^{2} \))(\( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4} )^{2} \))(\( 1 + 42 T + 813 T^{2} + 9520 T^{3} + 72840 T^{4} + 356811 T^{5} + 973207 T^{6} - 893781 T^{7} - 62793603 T^{8} - 1355379536 T^{9} - 23955645108 T^{10} - 329298299862 T^{11} - 3388931313773 T^{12} - 26014565689098 T^{13} - 149507181119028 T^{14} - 668254971049904 T^{15} - 2445815923131843 T^{16} - 2750214545354619 T^{17} + 236574413325225847 T^{18} + 6852165969260378949 T^{19} + \)\(11\!\cdots\!40\)\( T^{20} + \)\(11\!\cdots\!80\)\( T^{21} + \)\(76\!\cdots\!13\)\( T^{22} + \)\(31\!\cdots\!18\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))
$83$ (\( 1 - 26 T^{2} + 6889 T^{4} \))(\( 1 - 83 T^{2} + 6889 T^{4} \))(\( 1 + 26 T^{2} - 6213 T^{4} + 179114 T^{6} + 47458321 T^{8} \))(\( 1 + 39 T + 912 T^{2} + 16200 T^{3} + 251079 T^{4} + 3515997 T^{5} + 45358019 T^{6} + 541131408 T^{7} + 6100532325 T^{8} + 65514800025 T^{9} + 671478204717 T^{10} + 6541571603403 T^{11} + 60933732837525 T^{12} + 542950443082449 T^{13} + 4625813352295413 T^{14} + 37460510961894675 T^{15} + 289521021350726325 T^{16} + 2131538609315815344 T^{17} + 14829367667138196011 T^{18} + 95410273871375563119 T^{19} + \)\(56\!\cdots\!39\)\( T^{20} + \)\(30\!\cdots\!00\)\( T^{21} + \)\(14\!\cdots\!88\)\( T^{22} + \)\(50\!\cdots\!13\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))
$89$ (\( 1 + 151 T^{2} + 7921 T^{4} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 + 151 T^{2} + 7921 T^{4} )^{2} \))(\( 1 + 9 T - 273 T^{2} - 2772 T^{3} + 38802 T^{4} + 449316 T^{5} - 3561871 T^{6} - 54551502 T^{7} + 157767516 T^{8} + 4371660207 T^{9} + 3816883044 T^{10} - 152630961444 T^{11} - 900621732009 T^{12} - 13584155568516 T^{13} + 30233530591524 T^{14} + 3081884924468583 T^{15} + 9898687510843356 T^{16} - 304618830200242398 T^{17} - 1770183247816548031 T^{18} + 19873846469919508164 T^{19} + \)\(15\!\cdots\!62\)\( T^{20} - \)\(97\!\cdots\!48\)\( T^{21} - \)\(85\!\cdots\!73\)\( T^{22} + \)\(24\!\cdots\!01\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))
$97$ (\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )( 1 - 5 T + 97 T^{2} ) \))(\( ( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} )^{2} \))(\( 1 - 3 T + 102 T^{2} - 1010 T^{3} + 15132 T^{4} - 13512 T^{5} + 1127323 T^{6} - 7762230 T^{7} + 23311719 T^{8} + 575063737 T^{9} + 1596748254 T^{10} + 54554445012 T^{11} - 1453572795209 T^{12} + 5291781166164 T^{13} + 15023804321886 T^{14} + 524845146039001 T^{15} + 2063769721944039 T^{16} - 66656910163093110 T^{17} + 939028499512575067 T^{18} - 1091746419868262856 T^{19} + \)\(11\!\cdots\!52\)\( T^{20} - \)\(76\!\cdots\!70\)\( T^{21} + \)\(75\!\cdots\!98\)\( T^{22} - \)\(21\!\cdots\!59\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))
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