Properties

Label 19.3
Level 19
Weight 3
Dimension 21
Nonzero newspaces 3
Newform subspaces 4
Sturm bound 90
Trace bound 1

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 19\( 19 \) \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 4 \)
Sturm bound: \(90\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(19))\).

Total New Old
Modular forms 39 39 0
Cusp forms 21 21 0
Eisenstein series 18 18 0

Trace form

\( 21q - 9q^{2} - 9q^{3} - 9q^{4} - 9q^{5} - 9q^{6} - 9q^{7} - 9q^{8} - 9q^{9} + O(q^{10}) \) \( 21q - 9q^{2} - 9q^{3} - 9q^{4} - 9q^{5} - 9q^{6} - 9q^{7} - 9q^{8} - 9q^{9} - 9q^{10} - 9q^{11} + 63q^{12} + 51q^{13} + 63q^{14} + 45q^{15} + 63q^{16} - 30q^{19} - 90q^{20} - 72q^{21} - 117q^{22} - 54q^{23} - 225q^{24} - 135q^{25} - 153q^{26} - 27q^{27} + 114q^{28} + 135q^{29} + 432q^{30} + 99q^{31} + 216q^{32} + 207q^{33} + 126q^{34} + 63q^{35} + 117q^{36} - 72q^{38} - 126q^{39} - 234q^{40} - 81q^{41} - 459q^{42} - 186q^{43} - 144q^{45} - 54q^{46} + 54q^{47} - 198q^{48} - 78q^{49} + 72q^{50} + 90q^{51} + 255q^{52} + 99q^{53} + 180q^{54} + 27q^{55} - 54q^{57} - 90q^{58} - 144q^{59} - 252q^{60} + 231q^{61} + 252q^{62} + 315q^{63} + 255q^{64} + 126q^{65} - 81q^{66} + 336q^{67} - 324q^{68} + 72q^{69} + 27q^{70} - 270q^{71} + 108q^{72} + 102q^{73} + 99q^{74} - 99q^{76} - 45q^{77} + 261q^{78} - 75q^{79} + 243q^{80} - 27q^{81} - 522q^{82} + 99q^{84} - 405q^{85} - 414q^{86} - 369q^{87} - 405q^{88} - 630q^{89} - 504q^{90} - 669q^{91} - 630q^{92} - 477q^{93} + 612q^{95} + 918q^{96} + 486q^{97} + 1188q^{98} + 513q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(19))\)

We only show spaces with odd 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
19.3.b \(\chi_{19}(18, \cdot)\) 19.3.b.a 1 1
19.3.b.b 2
19.3.d \(\chi_{19}(8, \cdot)\) 19.3.d.a 6 2
19.3.f \(\chi_{19}(2, \cdot)\) 19.3.f.a 12 6

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T )( 1 + 2 T ) \))(\( 1 + 5 T^{2} + 16 T^{4} \))(\( 1 + 3 T + 8 T^{2} + 15 T^{3} + 20 T^{4} + 27 T^{5} - T^{6} + 108 T^{7} + 320 T^{8} + 960 T^{9} + 2048 T^{10} + 3072 T^{11} + 4096 T^{12} \))(\( 1 + 6 T + 18 T^{2} + 39 T^{3} + 84 T^{4} + 141 T^{5} + 58 T^{6} - 288 T^{7} - 627 T^{8} - 1113 T^{9} - 2082 T^{10} + 27 T^{11} + 7077 T^{12} + 108 T^{13} - 33312 T^{14} - 71232 T^{15} - 160512 T^{16} - 294912 T^{17} + 237568 T^{18} + 2310144 T^{19} + 5505024 T^{20} + 10223616 T^{21} + 18874368 T^{22} + 25165824 T^{23} + 16777216 T^{24} \))
$3$ (\( ( 1 - 3 T )( 1 + 3 T ) \))(\( 1 - 5 T^{2} + 81 T^{4} \))(\( 1 + 9 T + 47 T^{2} + 180 T^{3} + 463 T^{4} + 891 T^{5} + 2142 T^{6} + 8019 T^{7} + 37503 T^{8} + 131220 T^{9} + 308367 T^{10} + 531441 T^{11} + 531441 T^{12} \))(\( 1 + 12 T^{2} - 18 T^{3} + 24 T^{4} - 306 T^{5} + 134 T^{6} - 3357 T^{7} + 2190 T^{8} - 24768 T^{9} + 38157 T^{10} + 66051 T^{11} + 718057 T^{12} + 594459 T^{13} + 3090717 T^{14} - 18055872 T^{15} + 14368590 T^{16} - 198227493 T^{17} + 71213094 T^{18} - 1463588514 T^{19} + 1033121304 T^{20} - 6973568802 T^{21} + 41841412812 T^{22} + 282429536481 T^{24} \))
$5$ (\( 1 + 9 T + 25 T^{2} \))(\( ( 1 - 4 T + 25 T^{2} )^{2} \))(\( 1 + 2 T - 27 T^{2} + 114 T^{3} + 238 T^{4} - 2656 T^{5} + 5401 T^{6} - 66400 T^{7} + 148750 T^{8} + 1781250 T^{9} - 10546875 T^{10} + 19531250 T^{11} + 244140625 T^{12} \))(\( 1 + 6 T + 96 T^{2} + 467 T^{3} + 5685 T^{4} + 26151 T^{5} + 254350 T^{6} + 1092051 T^{7} + 9452817 T^{8} + 38413514 T^{9} + 297623811 T^{10} + 1122741453 T^{11} + 8011385341 T^{12} + 28068536325 T^{13} + 186014881875 T^{14} + 600211156250 T^{15} + 3692506640625 T^{16} + 10664560546875 T^{17} + 62097167968750 T^{18} + 159613037109375 T^{19} + 867462158203125 T^{20} + 1781463623046875 T^{21} + 9155273437500000 T^{22} + 14305114746093750 T^{23} + 59604644775390625 T^{24} \))
$7$ (\( 1 + 5 T + 49 T^{2} \))(\( ( 1 + 5 T + 49 T^{2} )^{2} \))(\( ( 1 + 69 T^{2} - 94 T^{3} + 3381 T^{4} + 117649 T^{6} )^{2} \))(\( 1 - 6 T - 117 T^{2} + 608 T^{3} + 8154 T^{4} - 29562 T^{5} - 205826 T^{6} - 896976 T^{7} - 2284875 T^{8} + 114186218 T^{9} + 743881389 T^{10} - 3902118156 T^{11} - 38601269423 T^{12} - 191203789644 T^{13} + 1786059214989 T^{14} + 13433894361482 T^{15} - 13171849684875 T^{16} - 253373518947024 T^{17} - 2848896779433026 T^{18} - 20049630479562138 T^{19} + 270981315864526554 T^{20} + 990075467529552992 T^{21} - 9335695156820604117 T^{22} - 23458926291497928294 T^{23} + \)\(19\!\cdots\!01\)\( T^{24} \))
$11$ (\( 1 - 3 T + 121 T^{2} \))(\( ( 1 + 10 T + 121 T^{2} )^{2} \))(\( ( 1 - 13 T + 280 T^{2} - 3149 T^{3} + 33880 T^{4} - 190333 T^{5} + 1771561 T^{6} )^{2} \))(\( 1 + 18 T - 228 T^{2} - 5552 T^{3} + 29166 T^{4} + 992034 T^{5} - 2938298 T^{6} - 164354796 T^{7} - 362171790 T^{8} + 16473984520 T^{9} + 121721077314 T^{10} - 623845748532 T^{11} - 15282031103345 T^{12} - 75485335572372 T^{13} + 1782118292954274 T^{14} + 29184668490235720 T^{15} - 77634739634166990 T^{16} - 4262940129062736396 T^{17} - 9221637822462560858 T^{18} + \)\(37\!\cdots\!94\)\( T^{19} + \)\(13\!\cdots\!26\)\( T^{20} - \)\(30\!\cdots\!12\)\( T^{21} - \)\(15\!\cdots\!28\)\( T^{22} + \)\(14\!\cdots\!78\)\( T^{23} + \)\(98\!\cdots\!41\)\( T^{24} \))
$13$ (\( ( 1 - 13 T )( 1 + 13 T ) \))(\( 1 - 325 T^{2} + 28561 T^{4} \))(\( 1 - 30 T + 779 T^{2} - 14370 T^{3} + 239570 T^{4} - 3505902 T^{5} + 48205499 T^{6} - 592497438 T^{7} + 6842358770 T^{8} - 69361245330 T^{9} + 635454231659 T^{10} - 4135754755470 T^{11} + 23298085122481 T^{12} \))(\( 1 - 21 T + 114 T^{2} - 3048 T^{3} + 99153 T^{4} - 1219683 T^{5} + 6059104 T^{6} - 140861934 T^{7} + 4074921219 T^{8} - 34064617704 T^{9} + 57415349265 T^{10} - 4773855887766 T^{11} + 121298781358881 T^{12} - 806781645032454 T^{13} + 1639839790357665 T^{14} - 164423403315226536 T^{15} + 3324038423993068899 T^{16} - 19419013780173375966 T^{17} + \)\(14\!\cdots\!24\)\( T^{18} - \)\(48\!\cdots\!87\)\( T^{19} + \)\(65\!\cdots\!73\)\( T^{20} - \)\(34\!\cdots\!92\)\( T^{21} + \)\(21\!\cdots\!14\)\( T^{22} - \)\(67\!\cdots\!49\)\( T^{23} + \)\(54\!\cdots\!61\)\( T^{24} \))
$17$ (\( 1 - 15 T + 289 T^{2} \))(\( ( 1 - 15 T + 289 T^{2} )^{2} \))(\( 1 + 42 T + 333 T^{2} + 6726 T^{3} + 513306 T^{4} + 6959454 T^{5} + 21390509 T^{6} + 2011282206 T^{7} + 42871830426 T^{8} + 162349289094 T^{9} + 2322927227853 T^{10} + 84671743818858 T^{11} + 582622237229761 T^{12} \))(\( 1 + 3 T - 84 T^{2} + 3702 T^{3} - 39603 T^{4} - 812499 T^{5} + 4691260 T^{6} - 149973228 T^{7} - 3075033123 T^{8} + 136201755708 T^{9} - 454077765279 T^{10} - 22107577523112 T^{11} + 875532343794093 T^{12} - 6389089904179368 T^{13} - 37925029033867359 T^{14} + 3287579276322993852 T^{15} - 21450685189088718243 T^{16} - \)\(30\!\cdots\!72\)\( T^{17} + \)\(27\!\cdots\!60\)\( T^{18} - \)\(13\!\cdots\!71\)\( T^{19} - \)\(19\!\cdots\!43\)\( T^{20} + \)\(52\!\cdots\!18\)\( T^{21} - \)\(34\!\cdots\!84\)\( T^{22} + \)\(35\!\cdots\!67\)\( T^{23} + \)\(33\!\cdots\!21\)\( T^{24} \))
$19$ (\( 1 + 19 T \))(\( 1 + 12 T + 361 T^{2} \))(\( 1 - 25 T + 1026 T^{2} - 16967 T^{3} + 370386 T^{4} - 3258025 T^{5} + 47045881 T^{6} \))(\( 1 + 24 T - 234 T^{2} - 22458 T^{3} - 279414 T^{4} + 4979634 T^{5} + 206298143 T^{6} + 1797647874 T^{7} - 36413511894 T^{8} - 1056556395498 T^{9} - 3974153751594 T^{10} + 147145590187224 T^{11} + 2213314919066161 T^{12} \))
$23$ (\( 1 + 30 T + 529 T^{2} \))(\( ( 1 - 35 T + 529 T^{2} )^{2} \))(\( 1 - 8 T - 1413 T^{2} + 3876 T^{3} + 1337428 T^{4} - 1325942 T^{5} - 814307171 T^{6} - 701423318 T^{7} + 374267188948 T^{8} + 573787105764 T^{9} - 110653422202053 T^{10} - 331412089709192 T^{11} + 21914624432020321 T^{12} \))(\( 1 + 102 T + 5880 T^{2} + 242044 T^{3} + 8074740 T^{4} + 232105944 T^{5} + 5933501770 T^{6} + 135098714970 T^{7} + 2764757638548 T^{8} + 51583388614096 T^{9} + 904612835742672 T^{10} + 15789113035371534 T^{11} + 320785786069630231 T^{12} + 8352440795711541486 T^{13} + \)\(25\!\cdots\!52\)\( T^{14} + \)\(76\!\cdots\!44\)\( T^{15} + \)\(21\!\cdots\!88\)\( T^{16} + \)\(55\!\cdots\!30\)\( T^{17} + \)\(13\!\cdots\!70\)\( T^{18} + \)\(26\!\cdots\!96\)\( T^{19} + \)\(49\!\cdots\!40\)\( T^{20} + \)\(78\!\cdots\!36\)\( T^{21} + \)\(10\!\cdots\!80\)\( T^{22} + \)\(92\!\cdots\!58\)\( T^{23} + \)\(48\!\cdots\!41\)\( T^{24} \))
$29$ (\( ( 1 - 29 T )( 1 + 29 T ) \))(\( 1 - 1357 T^{2} + 707281 T^{4} \))(\( 1 + 12 T + 1091 T^{2} + 12516 T^{3} + 342470 T^{4} + 25875306 T^{5} + 53097851 T^{6} + 21761132346 T^{7} + 242222524070 T^{8} + 7444808685636 T^{9} + 545768836540451 T^{10} + 5048486799602412 T^{11} + 353814783205469041 T^{12} \))(\( 1 - 147 T + 10638 T^{2} - 548808 T^{3} + 23661696 T^{4} - 919992729 T^{5} + 33943669042 T^{6} - 1192831592220 T^{7} + 39785563065882 T^{8} - 1286899558576728 T^{9} + 40759217279931174 T^{10} - 1260467388512258220 T^{11} + 37512869403908106759 T^{12} - \)\(10\!\cdots\!20\)\( T^{13} + \)\(28\!\cdots\!94\)\( T^{14} - \)\(76\!\cdots\!88\)\( T^{15} + \)\(19\!\cdots\!02\)\( T^{16} - \)\(50\!\cdots\!20\)\( T^{17} + \)\(12\!\cdots\!22\)\( T^{18} - \)\(27\!\cdots\!49\)\( T^{19} + \)\(59\!\cdots\!16\)\( T^{20} - \)\(11\!\cdots\!88\)\( T^{21} + \)\(18\!\cdots\!38\)\( T^{22} - \)\(21\!\cdots\!27\)\( T^{23} + \)\(12\!\cdots\!81\)\( T^{24} \))
$31$ (\( ( 1 - 31 T )( 1 + 31 T ) \))(\( 1 - 622 T^{2} + 923521 T^{4} \))(\( 1 - 1222 T^{2} + 2797835 T^{4} - 2253009856 T^{6} + 2583859377035 T^{8} - 1042232847752902 T^{10} + 787662783788549761 T^{12} \))(\( 1 - 99 T + 8727 T^{2} - 540540 T^{3} + 30402945 T^{4} - 1458608319 T^{5} + 64909332043 T^{6} - 2619622606347 T^{7} + 99818681518848 T^{8} - 3553487633552742 T^{9} + 121725452021726463 T^{10} - 3963361277030280903 T^{11} + \)\(12\!\cdots\!15\)\( T^{12} - \)\(38\!\cdots\!83\)\( T^{13} + \)\(11\!\cdots\!23\)\( T^{14} - \)\(31\!\cdots\!02\)\( T^{15} + \)\(85\!\cdots\!68\)\( T^{16} - \)\(21\!\cdots\!47\)\( T^{17} + \)\(51\!\cdots\!23\)\( T^{18} - \)\(11\!\cdots\!99\)\( T^{19} + \)\(22\!\cdots\!45\)\( T^{20} - \)\(37\!\cdots\!40\)\( T^{21} + \)\(58\!\cdots\!27\)\( T^{22} - \)\(63\!\cdots\!39\)\( T^{23} + \)\(62\!\cdots\!21\)\( T^{24} \))
$37$ (\( ( 1 - 37 T )( 1 + 37 T ) \))(\( 1 - 2270 T^{2} + 1874161 T^{4} \))(\( 1 - 5190 T^{2} + 13520751 T^{4} - 22697438888 T^{6} + 25340064214911 T^{8} - 18229768365849990 T^{10} + 6582952005840035281 T^{12} \))(\( 1 - 8874 T^{2} + 39453723 T^{4} - 117970083763 T^{6} + 266195955119445 T^{8} - 481289917598888787 T^{10} + \)\(72\!\cdots\!18\)\( T^{12} - \)\(90\!\cdots\!07\)\( T^{14} + \)\(93\!\cdots\!45\)\( T^{16} - \)\(77\!\cdots\!03\)\( T^{18} + \)\(48\!\cdots\!43\)\( T^{20} - \)\(20\!\cdots\!74\)\( T^{22} + \)\(43\!\cdots\!61\)\( T^{24} \))
$41$ (\( ( 1 - 41 T )( 1 + 41 T ) \))(\( 1 - 2062 T^{2} + 2825761 T^{4} \))(\( 1 - 63 T + 4739 T^{2} - 215208 T^{3} + 9562553 T^{4} - 413399385 T^{5} + 16368671414 T^{6} - 694924366185 T^{7} + 27021489327833 T^{8} - 1022260433497128 T^{9} + 37840560660804419 T^{10} - 845627536539601263 T^{11} + 22563490300366186081 T^{12} \))(\( 1 + 144 T + 11163 T^{2} + 355752 T^{3} - 8464866 T^{4} - 1542494952 T^{5} - 62169863144 T^{6} + 399495011904 T^{7} + 176879623935177 T^{8} + 8978616925749936 T^{9} + 138472442393345793 T^{10} - 8947832283253483704 T^{11} - \)\(65\!\cdots\!47\)\( T^{12} - \)\(15\!\cdots\!24\)\( T^{13} + \)\(39\!\cdots\!73\)\( T^{14} + \)\(42\!\cdots\!76\)\( T^{15} + \)\(14\!\cdots\!17\)\( T^{16} + \)\(53\!\cdots\!04\)\( T^{17} - \)\(14\!\cdots\!64\)\( T^{18} - \)\(58\!\cdots\!72\)\( T^{19} - \)\(53\!\cdots\!06\)\( T^{20} + \)\(38\!\cdots\!92\)\( T^{21} + \)\(20\!\cdots\!63\)\( T^{22} + \)\(43\!\cdots\!64\)\( T^{23} + \)\(50\!\cdots\!61\)\( T^{24} \))
$43$ (\( 1 + 85 T + 1849 T^{2} \))(\( ( 1 + 20 T + 1849 T^{2} )^{2} \))(\( 1 + 34 T - 3475 T^{2} - 31338 T^{3} + 9673978 T^{4} - 28877782 T^{5} - 22052754731 T^{6} - 53395018918 T^{7} + 33073405660378 T^{8} - 198098875229562 T^{9} - 40616495964663475 T^{10} + 734790398651664466 T^{11} + 39959630797262576401 T^{12} \))(\( 1 + 27 T + 1068 T^{2} - 61858 T^{3} + 227013 T^{4} + 26711865 T^{5} + 3468305848 T^{6} + 113462971236 T^{7} + 2290399946613 T^{8} - 31890759747304 T^{9} - 13892532646710483 T^{10} - 130352073354404160 T^{11} + 1800910079507940253 T^{12} - \)\(24\!\cdots\!40\)\( T^{13} - \)\(47\!\cdots\!83\)\( T^{14} - \)\(20\!\cdots\!96\)\( T^{15} + \)\(26\!\cdots\!13\)\( T^{16} + \)\(24\!\cdots\!64\)\( T^{17} + \)\(13\!\cdots\!48\)\( T^{18} + \)\(19\!\cdots\!85\)\( T^{19} + \)\(31\!\cdots\!13\)\( T^{20} - \)\(15\!\cdots\!42\)\( T^{21} + \)\(49\!\cdots\!68\)\( T^{22} + \)\(23\!\cdots\!23\)\( T^{23} + \)\(15\!\cdots\!01\)\( T^{24} \))
$47$ (\( 1 - 75 T + 2209 T^{2} \))(\( ( 1 - 10 T + 2209 T^{2} )^{2} \))(\( 1 - 58 T - 3429 T^{2} + 99066 T^{3} + 16840372 T^{4} - 217034296 T^{5} - 36013746767 T^{6} - 479428759864 T^{7} + 82175643281332 T^{8} + 1067853745782714 T^{9} - 81648901963178469 T^{10} - 3050749669678142842 T^{11} + \)\(11\!\cdots\!41\)\( T^{12} \))(\( 1 + 99 T + 6018 T^{2} + 294074 T^{3} + 10863897 T^{4} + 565964145 T^{5} + 25757282596 T^{6} + 583337737242 T^{7} + 10177397826039 T^{8} - 1186851009182872 T^{9} - 69183145856627091 T^{10} - 2572193055285963618 T^{11} - \)\(17\!\cdots\!43\)\( T^{12} - \)\(56\!\cdots\!62\)\( T^{13} - \)\(33\!\cdots\!71\)\( T^{14} - \)\(12\!\cdots\!88\)\( T^{15} + \)\(24\!\cdots\!79\)\( T^{16} + \)\(30\!\cdots\!58\)\( T^{17} + \)\(29\!\cdots\!36\)\( T^{18} + \)\(14\!\cdots\!05\)\( T^{19} + \)\(61\!\cdots\!37\)\( T^{20} + \)\(36\!\cdots\!86\)\( T^{21} + \)\(16\!\cdots\!18\)\( T^{22} + \)\(60\!\cdots\!91\)\( T^{23} + \)\(13\!\cdots\!81\)\( T^{24} \))
$53$ (\( ( 1 - 53 T )( 1 + 53 T ) \))(\( 1 + 115 T^{2} + 7890481 T^{4} \))(\( 1 + 12 T + 7659 T^{2} + 91332 T^{3} + 36866070 T^{4} + 510480060 T^{5} + 120117765391 T^{6} + 1433938488540 T^{7} + 290891024879670 T^{8} + 2024315430633828 T^{9} + 476846968860613899 T^{10} + 2098649644386156588 T^{11} + \)\(49\!\cdots\!41\)\( T^{12} \))(\( 1 - 111 T + 13224 T^{2} - 910716 T^{3} + 59831619 T^{4} - 2799852225 T^{5} + 110592030932 T^{6} - 4169273946750 T^{7} + 193735787337603 T^{8} - 18841775739136788 T^{9} + 1586777771876360301 T^{10} - \)\(12\!\cdots\!42\)\( T^{11} + \)\(69\!\cdots\!33\)\( T^{12} - \)\(33\!\cdots\!78\)\( T^{13} + \)\(12\!\cdots\!81\)\( T^{14} - \)\(41\!\cdots\!52\)\( T^{15} + \)\(12\!\cdots\!83\)\( T^{16} - \)\(72\!\cdots\!50\)\( T^{17} + \)\(54\!\cdots\!12\)\( T^{18} - \)\(38\!\cdots\!25\)\( T^{19} + \)\(23\!\cdots\!99\)\( T^{20} - \)\(99\!\cdots\!24\)\( T^{21} + \)\(40\!\cdots\!24\)\( T^{22} - \)\(95\!\cdots\!99\)\( T^{23} + \)\(24\!\cdots\!81\)\( T^{24} \))
$59$ (\( ( 1 - 59 T )( 1 + 59 T ) \))(\( 1 - 6637 T^{2} + 12117361 T^{4} \))(\( 1 + 147 T + 16901 T^{2} + 1425606 T^{3} + 106851335 T^{4} + 7433603271 T^{5} + 456697800146 T^{6} + 25876372986351 T^{7} + 1294756199526935 T^{8} + 60132821841811446 T^{9} + 2481581225950629221 T^{10} + 75134162735194285947 T^{11} + \)\(17\!\cdots\!81\)\( T^{12} \))(\( 1 - 3 T - 5478 T^{2} - 236796 T^{3} + 20289939 T^{4} + 538841139 T^{5} - 70573726772 T^{6} + 365767344396 T^{7} + 479729033250141 T^{8} + 1699477547609316 T^{9} - 1883805256432996599 T^{10} - 14979381777275838012 T^{11} + \)\(59\!\cdots\!37\)\( T^{12} - \)\(52\!\cdots\!72\)\( T^{13} - \)\(22\!\cdots\!39\)\( T^{14} + \)\(71\!\cdots\!56\)\( T^{15} + \)\(70\!\cdots\!61\)\( T^{16} + \)\(18\!\cdots\!96\)\( T^{17} - \)\(12\!\cdots\!32\)\( T^{18} + \)\(33\!\cdots\!79\)\( T^{19} + \)\(43\!\cdots\!99\)\( T^{20} - \)\(17\!\cdots\!16\)\( T^{21} - \)\(14\!\cdots\!78\)\( T^{22} - \)\(27\!\cdots\!43\)\( T^{23} + \)\(31\!\cdots\!61\)\( T^{24} \))
$61$ (\( 1 - 103 T + 3721 T^{2} \))(\( ( 1 + 40 T + 3721 T^{2} )^{2} \))(\( 1 - 58 T - 5011 T^{2} + 428958 T^{3} + 12450718 T^{4} - 1018447028 T^{5} + 4642755289 T^{6} - 3789641391188 T^{7} + 172390661763838 T^{8} + 22100076745145838 T^{9} - 960645345429375091 T^{10} - 41373888876447190858 T^{11} + \)\(26\!\cdots\!21\)\( T^{12} \))(\( 1 - 150 T + 9552 T^{2} - 165800 T^{3} - 47534514 T^{4} + 4657049670 T^{5} - 161253582314 T^{6} - 6697565984088 T^{7} + 1572100685257950 T^{8} - 92881003919900360 T^{9} + 1478892268641949590 T^{10} + \)\(23\!\cdots\!00\)\( T^{11} - \)\(26\!\cdots\!65\)\( T^{12} + \)\(87\!\cdots\!00\)\( T^{13} + \)\(20\!\cdots\!90\)\( T^{14} - \)\(47\!\cdots\!60\)\( T^{15} + \)\(30\!\cdots\!50\)\( T^{16} - \)\(47\!\cdots\!88\)\( T^{17} - \)\(42\!\cdots\!94\)\( T^{18} + \)\(45\!\cdots\!70\)\( T^{19} - \)\(17\!\cdots\!54\)\( T^{20} - \)\(22\!\cdots\!00\)\( T^{21} + \)\(48\!\cdots\!52\)\( T^{22} - \)\(28\!\cdots\!50\)\( T^{23} + \)\(70\!\cdots\!41\)\( T^{24} \))
$67$ (\( ( 1 - 67 T )( 1 + 67 T ) \))(\( 1 - 7405 T^{2} + 20151121 T^{4} \))(\( 1 - 201 T + 28649 T^{2} - 3051582 T^{3} + 282765251 T^{4} - 23237371929 T^{5} + 1659244019666 T^{6} - 104312562589281 T^{7} + 5698036787496371 T^{8} - 276041170776041358 T^{9} + 11633432894320208009 T^{10} - \)\(36\!\cdots\!49\)\( T^{11} + \)\(81\!\cdots\!61\)\( T^{12} \))(\( 1 - 135 T - 4890 T^{2} + 1851210 T^{3} - 82497135 T^{4} - 6137913249 T^{5} + 840261936376 T^{6} - 28744747210134 T^{7} - 2010008421697905 T^{8} + 265516527753593460 T^{9} - 7719241585001434479 T^{10} - \)\(56\!\cdots\!30\)\( T^{11} + \)\(67\!\cdots\!21\)\( T^{12} - \)\(25\!\cdots\!70\)\( T^{13} - \)\(15\!\cdots\!59\)\( T^{14} + \)\(24\!\cdots\!40\)\( T^{15} - \)\(81\!\cdots\!05\)\( T^{16} - \)\(52\!\cdots\!66\)\( T^{17} + \)\(68\!\cdots\!36\)\( T^{18} - \)\(22\!\cdots\!21\)\( T^{19} - \)\(13\!\cdots\!35\)\( T^{20} + \)\(13\!\cdots\!90\)\( T^{21} - \)\(16\!\cdots\!90\)\( T^{22} - \)\(20\!\cdots\!15\)\( T^{23} + \)\(66\!\cdots\!21\)\( T^{24} \))
$71$ (\( ( 1 - 71 T )( 1 + 71 T ) \))(\( ( 1 - 92 T + 5041 T^{2} )( 1 + 92 T + 5041 T^{2} ) \))(\( 1 + 102 T + 19395 T^{2} + 1624554 T^{3} + 208411794 T^{4} + 13817724054 T^{5} + 1288263668227 T^{6} + 69655146956214 T^{7} + 5296094025765714 T^{8} + 208105828644996234 T^{9} + 12524389738511534595 T^{10} + \)\(33\!\cdots\!02\)\( T^{11} + \)\(16\!\cdots\!41\)\( T^{12} \))(\( 1 + 168 T + 7788 T^{2} - 460248 T^{3} - 78112236 T^{4} - 4942596282 T^{5} - 129339672166 T^{6} + 21694723507134 T^{7} + 3947673961555920 T^{8} + 264008734899691344 T^{9} + 2103086375048613636 T^{10} - \)\(10\!\cdots\!86\)\( T^{11} - \)\(10\!\cdots\!97\)\( T^{12} - \)\(51\!\cdots\!26\)\( T^{13} + \)\(53\!\cdots\!16\)\( T^{14} + \)\(33\!\cdots\!24\)\( T^{15} + \)\(25\!\cdots\!20\)\( T^{16} + \)\(70\!\cdots\!34\)\( T^{17} - \)\(21\!\cdots\!06\)\( T^{18} - \)\(40\!\cdots\!42\)\( T^{19} - \)\(32\!\cdots\!56\)\( T^{20} - \)\(96\!\cdots\!28\)\( T^{21} + \)\(82\!\cdots\!88\)\( T^{22} + \)\(89\!\cdots\!88\)\( T^{23} + \)\(26\!\cdots\!81\)\( T^{24} \))
$73$ (\( 1 + 25 T + 5329 T^{2} \))(\( ( 1 - 105 T + 5329 T^{2} )^{2} \))(\( 1 - 7 T - 12713 T^{2} + 22960 T^{3} + 94466917 T^{4} + 36786071 T^{5} - 563497579498 T^{6} + 196032972359 T^{7} + 2682694275492997 T^{8} + 3474633835595440 T^{9} - 10252527148249451753 T^{10} - 30083380807924903543 T^{11} + \)\(22\!\cdots\!21\)\( T^{12} \))(\( 1 + 90 T - 1881 T^{2} - 503037 T^{3} - 10955619 T^{4} - 1624145481 T^{5} - 104644983377 T^{6} + 1641371272965 T^{7} - 250831885102848 T^{8} - 82541675177265258 T^{9} + 1186525377744958692 T^{10} + \)\(50\!\cdots\!97\)\( T^{11} + \)\(46\!\cdots\!20\)\( T^{12} + \)\(26\!\cdots\!13\)\( T^{13} + \)\(33\!\cdots\!72\)\( T^{14} - \)\(12\!\cdots\!62\)\( T^{15} - \)\(20\!\cdots\!88\)\( T^{16} + \)\(70\!\cdots\!85\)\( T^{17} - \)\(23\!\cdots\!17\)\( T^{18} - \)\(19\!\cdots\!29\)\( T^{19} - \)\(71\!\cdots\!59\)\( T^{20} - \)\(17\!\cdots\!53\)\( T^{21} - \)\(34\!\cdots\!81\)\( T^{22} + \)\(88\!\cdots\!10\)\( T^{23} + \)\(52\!\cdots\!41\)\( T^{24} \))
$79$ (\( ( 1 - 79 T )( 1 + 79 T ) \))(\( 1 - 11182 T^{2} + 38950081 T^{4} \))(\( 1 + 12035 T^{2} + 69730790 T^{4} + 4384013520 T^{5} + 462010537511 T^{6} + 27360628378320 T^{7} + 2716019918693990 T^{8} + 18258404527225461635 T^{10} + \)\(59\!\cdots\!41\)\( T^{12} \))(\( 1 + 75 T - 2808 T^{2} + 1317084 T^{3} + 113789640 T^{4} - 2453884791 T^{5} + 938263725814 T^{6} + 65842204081428 T^{7} - 934380072746862 T^{8} + 572878760809167420 T^{9} + 18887866671028404426 T^{10} + \)\(18\!\cdots\!16\)\( T^{11} + \)\(30\!\cdots\!83\)\( T^{12} + \)\(11\!\cdots\!56\)\( T^{13} + \)\(73\!\cdots\!06\)\( T^{14} + \)\(13\!\cdots\!20\)\( T^{15} - \)\(14\!\cdots\!82\)\( T^{16} + \)\(62\!\cdots\!28\)\( T^{17} + \)\(55\!\cdots\!74\)\( T^{18} - \)\(90\!\cdots\!71\)\( T^{19} + \)\(26\!\cdots\!40\)\( T^{20} + \)\(18\!\cdots\!24\)\( T^{21} - \)\(25\!\cdots\!08\)\( T^{22} + \)\(41\!\cdots\!75\)\( T^{23} + \)\(34\!\cdots\!81\)\( T^{24} \))
$83$ (\( 1 - 90 T + 6889 T^{2} \))(\( ( 1 + 40 T + 6889 T^{2} )^{2} \))(\( ( 1 - 73 T + 15082 T^{2} - 608183 T^{3} + 103899898 T^{4} - 3464457433 T^{5} + 326940373369 T^{6} )^{2} \))(\( 1 + 156 T - 21894 T^{2} - 3167984 T^{3} + 487243866 T^{4} + 48063809730 T^{5} - 7192617372338 T^{6} - 432106800861834 T^{7} + 89789242349485116 T^{8} + 2881436654419447720 T^{9} - \)\(84\!\cdots\!16\)\( T^{10} - \)\(71\!\cdots\!22\)\( T^{11} + \)\(66\!\cdots\!03\)\( T^{12} - \)\(49\!\cdots\!58\)\( T^{13} - \)\(40\!\cdots\!36\)\( T^{14} + \)\(94\!\cdots\!80\)\( T^{15} + \)\(20\!\cdots\!56\)\( T^{16} - \)\(67\!\cdots\!66\)\( T^{17} - \)\(76\!\cdots\!18\)\( T^{18} + \)\(35\!\cdots\!70\)\( T^{19} + \)\(24\!\cdots\!46\)\( T^{20} - \)\(11\!\cdots\!56\)\( T^{21} - \)\(52\!\cdots\!94\)\( T^{22} + \)\(25\!\cdots\!84\)\( T^{23} + \)\(11\!\cdots\!21\)\( T^{24} \))
$89$ (\( ( 1 - 89 T )( 1 + 89 T ) \))(\( ( 1 - 89 T )^{2}( 1 + 89 T )^{2} \))(\( 1 + 72 T + 9723 T^{2} + 575640 T^{3} + 40073838 T^{4} + 8635069188 T^{5} + 306038573143 T^{6} + 68398383038148 T^{7} + 2514322401590958 T^{8} + 286082310328790040 T^{9} + 38275452957841333563 T^{10} + \)\(22\!\cdots\!72\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} \))(\( 1 + 558 T + 156564 T^{2} + 30167973 T^{3} + 4436202555 T^{4} + 520050206223 T^{5} + 50823330783392 T^{6} + 4459983482310345 T^{7} + 400421887982378415 T^{8} + 41564349081369265482 T^{9} + \)\(47\!\cdots\!03\)\( T^{10} + \)\(51\!\cdots\!21\)\( T^{11} + \)\(49\!\cdots\!65\)\( T^{12} + \)\(41\!\cdots\!41\)\( T^{13} + \)\(29\!\cdots\!23\)\( T^{14} + \)\(20\!\cdots\!02\)\( T^{15} + \)\(15\!\cdots\!15\)\( T^{16} + \)\(13\!\cdots\!45\)\( T^{17} + \)\(12\!\cdots\!32\)\( T^{18} + \)\(10\!\cdots\!43\)\( T^{19} + \)\(68\!\cdots\!55\)\( T^{20} + \)\(37\!\cdots\!13\)\( T^{21} + \)\(15\!\cdots\!64\)\( T^{22} + \)\(42\!\cdots\!18\)\( T^{23} + \)\(61\!\cdots\!41\)\( T^{24} \))
$97$ (\( ( 1 - 97 T )( 1 + 97 T ) \))(\( 1 - 3790 T^{2} + 88529281 T^{4} \))(\( 1 - 21 T + 12695 T^{2} - 263508 T^{3} + 43921757 T^{4} - 6267660039 T^{5} - 57670086154 T^{6} - 58972413306951 T^{7} + 3888361567466717 T^{8} - 219494787074830932 T^{9} + 99496219480615519895 T^{10} - \)\(15\!\cdots\!29\)\( T^{11} + \)\(69\!\cdots\!41\)\( T^{12} \))(\( 1 - 465 T + 120888 T^{2} - 22632888 T^{3} + 3339144633 T^{4} - 402070083927 T^{5} + 39631627117156 T^{6} - 2930901400124028 T^{7} + 96086821819014021 T^{8} + 15804430080957435372 T^{9} - \)\(40\!\cdots\!67\)\( T^{10} + \)\(57\!\cdots\!16\)\( T^{11} - \)\(62\!\cdots\!95\)\( T^{12} + \)\(54\!\cdots\!44\)\( T^{13} - \)\(35\!\cdots\!27\)\( T^{14} + \)\(13\!\cdots\!88\)\( T^{15} + \)\(75\!\cdots\!81\)\( T^{16} - \)\(21\!\cdots\!72\)\( T^{17} + \)\(27\!\cdots\!96\)\( T^{18} - \)\(26\!\cdots\!63\)\( T^{19} + \)\(20\!\cdots\!93\)\( T^{20} - \)\(13\!\cdots\!32\)\( T^{21} + \)\(65\!\cdots\!88\)\( T^{22} - \)\(23\!\cdots\!85\)\( T^{23} + \)\(48\!\cdots\!81\)\( T^{24} \))
show more
show less