Properties

Label 4029.2
Level 4029
Weight 2
Dimension 443711
Nonzero newspaces 40
Sturm bound 2396160

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Defining parameters

Level: \( N \) = \( 4029\( 4029 = 3 \cdot 17 \cdot 79 \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 40 \)
Sturm bound: \(2396160\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(4029))\).

Total New Old
Modular forms 604032 448327 155705
Cusp forms 594049 443711 150338
Eisenstein series 9983 4616 5367

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(4029))\)

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
4029.2.a \(\chi_{4029}(1, \cdot)\) 4029.2.a.a 1 1
4029.2.a.b 1
4029.2.a.c 2
4029.2.a.d 3
4029.2.a.e 18
4029.2.a.f 22
4029.2.a.g 22
4029.2.a.h 25
4029.2.a.i 25
4029.2.a.j 25
4029.2.a.k 31
4029.2.a.l 32
4029.2.b \(\chi_{4029}(3554, \cdot)\) n/a 428 1
4029.2.e \(\chi_{4029}(4028, \cdot)\) n/a 476 1
4029.2.f \(\chi_{4029}(475, \cdot)\) n/a 232 1
4029.2.i \(\chi_{4029}(766, \cdot)\) n/a 428 2
4029.2.k \(\chi_{4029}(1186, \cdot)\) n/a 464 2
4029.2.m \(\chi_{4029}(710, \cdot)\) n/a 952 2
4029.2.p \(\chi_{4029}(1240, \cdot)\) n/a 480 2
4029.2.q \(\chi_{4029}(1478, \cdot)\) n/a 952 2
4029.2.t \(\chi_{4029}(1004, \cdot)\) n/a 852 2
4029.2.w \(\chi_{4029}(712, \cdot)\) n/a 944 4
4029.2.x \(\chi_{4029}(236, \cdot)\) n/a 1904 4
4029.2.y \(\chi_{4029}(293, \cdot)\) n/a 1904 4
4029.2.ba \(\chi_{4029}(55, \cdot)\) n/a 960 4
4029.2.bc \(\chi_{4029}(52, \cdot)\) n/a 2544 12
4029.2.bd \(\chi_{4029}(394, \cdot)\) n/a 1920 8
4029.2.be \(\chi_{4029}(80, \cdot)\) n/a 3744 8
4029.2.bh \(\chi_{4029}(529, \cdot)\) n/a 1920 8
4029.2.bi \(\chi_{4029}(767, \cdot)\) n/a 3808 8
4029.2.bn \(\chi_{4029}(67, \cdot)\) n/a 2880 12
4029.2.bo \(\chi_{4029}(254, \cdot)\) n/a 5712 12
4029.2.br \(\chi_{4029}(137, \cdot)\) n/a 5136 12
4029.2.bs \(\chi_{4029}(256, \cdot)\) n/a 5136 24
4029.2.bv \(\chi_{4029}(23, \cdot)\) n/a 7616 16
4029.2.bw \(\chi_{4029}(214, \cdot)\) n/a 3840 16
4029.2.bx \(\chi_{4029}(140, \cdot)\) n/a 11424 24
4029.2.bz \(\chi_{4029}(64, \cdot)\) n/a 5760 24
4029.2.cb \(\chi_{4029}(35, \cdot)\) n/a 10224 24
4029.2.ce \(\chi_{4029}(305, \cdot)\) n/a 11424 24
4029.2.cf \(\chi_{4029}(16, \cdot)\) n/a 5760 24
4029.2.ci \(\chi_{4029}(185, \cdot)\) n/a 22848 48
4029.2.cj \(\chi_{4029}(100, \cdot)\) n/a 11520 48
4029.2.cn \(\chi_{4029}(4, \cdot)\) n/a 11520 48
4029.2.cp \(\chi_{4029}(47, \cdot)\) n/a 22848 48
4029.2.cs \(\chi_{4029}(62, \cdot)\) n/a 45696 96
4029.2.ct \(\chi_{4029}(58, \cdot)\) n/a 23040 96
4029.2.cw \(\chi_{4029}(53, \cdot)\) n/a 45696 96
4029.2.cx \(\chi_{4029}(19, \cdot)\) n/a 23040 96
4029.2.cy \(\chi_{4029}(7, \cdot)\) n/a 46080 192
4029.2.cz \(\chi_{4029}(5, \cdot)\) n/a 91392 192

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(4029))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(4029)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(17))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(51))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(79))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(237))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(1343))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 + 2 T^{2} + 4 T^{4} \))(\( 1 + 2 T^{2} + 2 T^{3} + 4 T^{4} + 8 T^{6} \))(\( 1 + 6 T + 26 T^{2} + 84 T^{3} + 236 T^{4} + 585 T^{5} + 1341 T^{6} + 2870 T^{7} + 5840 T^{8} + 11329 T^{9} + 21093 T^{10} + 37697 T^{11} + 64969 T^{12} + 108080 T^{13} + 174274 T^{14} + 272538 T^{15} + 414262 T^{16} + 611654 T^{17} + 877762 T^{18} + 1223308 T^{19} + 1657048 T^{20} + 2180304 T^{21} + 2788384 T^{22} + 3458560 T^{23} + 4158016 T^{24} + 4825216 T^{25} + 5399808 T^{26} + 5800448 T^{27} + 5980160 T^{28} + 5877760 T^{29} + 5492736 T^{30} + 4792320 T^{31} + 3866624 T^{32} + 2752512 T^{33} + 1703936 T^{34} + 786432 T^{35} + 262144 T^{36} \))
$3$ (\( 1 - T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{18} \))
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 + 2 T + 9 T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 + 5 T + 20 T^{2} + 49 T^{3} + 100 T^{4} + 125 T^{5} + 125 T^{6} \))(\( 1 + 5 T + 51 T^{2} + 214 T^{3} + 1248 T^{4} + 4520 T^{5} + 19672 T^{6} + 62976 T^{7} + 226625 T^{8} + 653957 T^{9} + 2054349 T^{10} + 5430570 T^{11} + 15414869 T^{12} + 37859104 T^{13} + 99545331 T^{14} + 230161484 T^{15} + 571195125 T^{16} + 1257540163 T^{17} + 2981186163 T^{18} + 6287700815 T^{19} + 14279878125 T^{20} + 28770185500 T^{21} + 62215831875 T^{22} + 118309700000 T^{23} + 240857328125 T^{24} + 424263281250 T^{25} + 802480078125 T^{26} + 1277259765625 T^{27} + 2213134765625 T^{28} + 3075000000000 T^{29} + 4802734375000 T^{30} + 5517578125000 T^{31} + 7617187500000 T^{32} + 6530761718750 T^{33} + 7781982421875 T^{34} + 3814697265625 T^{35} + 3814697265625 T^{36} \))
$7$ (\( 1 - T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( ( 1 - T + 7 T^{2} )^{2} \))(\( ( 1 - T + 7 T^{2} )^{3} \))(\( 1 + 13 T + 140 T^{2} + 1061 T^{3} + 7102 T^{4} + 39918 T^{5} + 205104 T^{6} + 940029 T^{7} + 4027083 T^{8} + 15879338 T^{9} + 59503200 T^{10} + 209378398 T^{11} + 708772045 T^{12} + 2281855639 T^{13} + 7118854825 T^{14} + 21255808435 T^{15} + 61655778091 T^{16} + 171468343445 T^{17} + 463286448717 T^{18} + 1200278404115 T^{19} + 3021133126459 T^{20} + 7290742293205 T^{21} + 17092370434825 T^{22} + 38351147724673 T^{23} + 83386322322205 T^{24} + 172432114024114 T^{25} + 343024106863200 T^{26} + 640788565072166 T^{27} + 1137551273168667 T^{28} + 1858744480895547 T^{29} + 2838903370073904 T^{30} + 3867615517426626 T^{31} + 4816740263373598 T^{32} + 5037162762049523 T^{33} + 4652610279744140 T^{34} + 3024196681833691 T^{35} + 1628413597910449 T^{36} \))
$11$ (\( 1 + 4 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( 1 - 4 T + 17 T^{2} - 56 T^{3} + 187 T^{4} - 484 T^{5} + 1331 T^{6} \))(\( 1 + 27 T + 474 T^{2} + 6124 T^{3} + 65105 T^{4} + 589676 T^{5} + 4709226 T^{6} + 33695597 T^{7} + 219440929 T^{8} + 1312203498 T^{9} + 7267204278 T^{10} + 37470574364 T^{11} + 180803719532 T^{12} + 819033857735 T^{13} + 3494184377912 T^{14} + 14064960404714 T^{15} + 53515932990553 T^{16} + 192638908889659 T^{17} + 656595064222556 T^{18} + 2119027997786249 T^{19} + 6475427891856913 T^{20} + 18720462298674334 T^{21} + 51158353477009592 T^{22} + 131906221822079485 T^{23} + 320304818177829452 T^{24} + 730195490099484244 T^{25} + 1557789777030492918 T^{26} + 3094107208231223118 T^{27} + 5691732550310894329 T^{28} + 9613747072304999767 T^{29} + 14779568510792327946 T^{30} + 20357214806184656356 T^{31} + 24723612915436905305 T^{32} + 25581467789501446724 T^{33} + 21780171955333204314 T^{34} + 13647069769480931817 T^{35} + 5559917313492231481 T^{36} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( 1 - 10 T + 67 T^{2} - 280 T^{3} + 871 T^{4} - 1690 T^{5} + 2197 T^{6} \))(\( 1 + 4 T + 98 T^{2} + 250 T^{3} + 4210 T^{4} + 6297 T^{5} + 112437 T^{6} + 75818 T^{7} + 2173969 T^{8} + 18282 T^{9} + 31649589 T^{10} - 21621181 T^{11} + 320274648 T^{12} - 620969333 T^{13} + 1490415485 T^{14} - 11602397658 T^{15} - 14891727576 T^{16} - 171029557263 T^{17} - 373079673670 T^{18} - 2223384244419 T^{19} - 2516701960344 T^{20} - 25490467654626 T^{21} + 42567756667085 T^{22} - 230561566557569 T^{23} + 1545904553438232 T^{24} - 1356697043538577 T^{25} + 25817542054323669 T^{26} + 193871457537186 T^{27} + 299700087666478681 T^{28} + 135878016755097266 T^{29} + 2619566796916396197 T^{30} + 1907204546211417141 T^{31} + 16576354583794006690 T^{32} + 12796473253522689250 T^{33} + 65210827699951624418 T^{34} + 34601663677525351732 T^{35} + \)\(11\!\cdots\!29\)\( T^{36} \))
$17$ (\( 1 - T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 - T )^{18} \))
$19$ (\( 1 - 5 T + 19 T^{2} \))(\( 1 - 3 T + 19 T^{2} \))(\( 1 + 10 T + 55 T^{2} + 190 T^{3} + 361 T^{4} \))(\( 1 + T + 52 T^{2} + 37 T^{3} + 988 T^{4} + 361 T^{5} + 6859 T^{6} \))(\( 1 + 30 T + 613 T^{2} + 9328 T^{3} + 118519 T^{4} + 1295162 T^{5} + 12593080 T^{6} + 110549341 T^{7} + 889612918 T^{8} + 6617301111 T^{9} + 45875324298 T^{10} + 297919680698 T^{11} + 1821262153890 T^{12} + 10513859976151 T^{13} + 57486781497956 T^{14} + 298257016155968 T^{15} + 1470895681109265 T^{16} + 6901127118158326 T^{17} + 30827957285013563 T^{18} + 131121415245008194 T^{19} + 530993340880444665 T^{20} + 2045744873813784512 T^{21} + 7491734851595123876 T^{22} + 26033358173087514949 T^{23} + 85682882561712627090 T^{24} + \)\(26\!\cdots\!22\)\( T^{25} + \)\(77\!\cdots\!18\)\( T^{26} + \)\(21\!\cdots\!69\)\( T^{27} + \)\(54\!\cdots\!18\)\( T^{28} + \)\(12\!\cdots\!79\)\( T^{29} + \)\(27\!\cdots\!80\)\( T^{30} + \)\(54\!\cdots\!58\)\( T^{31} + \)\(94\!\cdots\!99\)\( T^{32} + \)\(14\!\cdots\!72\)\( T^{33} + \)\(17\!\cdots\!53\)\( T^{34} + \)\(16\!\cdots\!70\)\( T^{35} + \)\(10\!\cdots\!41\)\( T^{36} \))
$23$ (\( 1 + 7 T + 23 T^{2} \))(\( 1 + T + 23 T^{2} \))(\( 1 + 6 T + 37 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - T + 48 T^{2} - 75 T^{3} + 1104 T^{4} - 529 T^{5} + 12167 T^{6} \))(\( 1 + 21 T + 399 T^{2} + 5294 T^{3} + 63915 T^{4} + 653182 T^{5} + 6204986 T^{6} + 53085837 T^{7} + 428229778 T^{8} + 3199234231 T^{9} + 22736118492 T^{10} + 151872330331 T^{11} + 970521249804 T^{12} + 5879138020655 T^{13} + 34195289160146 T^{14} + 189473534017856 T^{15} + 1010221396939261 T^{16} + 5144533999925194 T^{17} + 25234931256368241 T^{18} + 118324281998279462 T^{19} + 534407118980869069 T^{20} + 2305324488395253952 T^{21} + 9569243913864416786 T^{22} + 37840148845276664665 T^{23} + \)\(14\!\cdots\!56\)\( T^{24} + \)\(51\!\cdots\!57\)\( T^{25} + \)\(17\!\cdots\!52\)\( T^{26} + \)\(57\!\cdots\!53\)\( T^{27} + \)\(17\!\cdots\!22\)\( T^{28} + \)\(50\!\cdots\!99\)\( T^{29} + \)\(13\!\cdots\!06\)\( T^{30} + \)\(32\!\cdots\!06\)\( T^{31} + \)\(74\!\cdots\!35\)\( T^{32} + \)\(14\!\cdots\!58\)\( T^{33} + \)\(24\!\cdots\!39\)\( T^{34} + \)\(29\!\cdots\!63\)\( T^{35} + \)\(32\!\cdots\!69\)\( T^{36} \))
$29$ (\( 1 - 10 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 4 T + 54 T^{2} - 116 T^{3} + 841 T^{4} \))(\( 1 - 4 T + 71 T^{2} - 200 T^{3} + 2059 T^{4} - 3364 T^{5} + 24389 T^{6} \))(\( 1 + 47 T + 1313 T^{2} + 26430 T^{3} + 425112 T^{4} + 5721994 T^{5} + 66669440 T^{6} + 686460856 T^{7} + 6354540166 T^{8} + 53545945404 T^{9} + 415469607526 T^{10} + 2996718045254 T^{11} + 20286957428121 T^{12} + 130012385198156 T^{13} + 795771334712276 T^{14} + 4687735112736730 T^{15} + 26769931718718836 T^{16} + 148964538131643953 T^{17} + 810764484495779482 T^{18} + 4319971605817674637 T^{19} + 22513512575442541076 T^{20} + \)\(11\!\cdots\!70\)\( T^{21} + \)\(56\!\cdots\!56\)\( T^{22} + \)\(26\!\cdots\!44\)\( T^{23} + \)\(12\!\cdots\!41\)\( T^{24} + \)\(51\!\cdots\!86\)\( T^{25} + \)\(20\!\cdots\!86\)\( T^{26} + \)\(77\!\cdots\!76\)\( T^{27} + \)\(26\!\cdots\!66\)\( T^{28} + \)\(83\!\cdots\!24\)\( T^{29} + \)\(23\!\cdots\!40\)\( T^{30} + \)\(58\!\cdots\!66\)\( T^{31} + \)\(12\!\cdots\!72\)\( T^{32} + \)\(22\!\cdots\!70\)\( T^{33} + \)\(32\!\cdots\!73\)\( T^{34} + \)\(34\!\cdots\!23\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \))
$31$ (\( 1 - 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 8 T + 46 T^{2} - 248 T^{3} + 961 T^{4} \))(\( 1 + 16 T + 157 T^{2} + 1024 T^{3} + 4867 T^{4} + 15376 T^{5} + 29791 T^{6} \))(\( 1 + 18 T + 395 T^{2} + 5537 T^{3} + 75131 T^{4} + 857931 T^{5} + 9199665 T^{6} + 89315185 T^{7} + 819667467 T^{8} + 6992698011 T^{9} + 56853618133 T^{10} + 435716674755 T^{11} + 3197795405136 T^{12} + 22314094581645 T^{13} + 149550626955265 T^{14} + 957665897398615 T^{15} + 5902456848639721 T^{16} + 34838474367831399 T^{17} + 198152534832019708 T^{18} + 1079992705402773369 T^{19} + 5672261031542771881 T^{20} + 28529824749402139465 T^{21} + \)\(13\!\cdots\!65\)\( T^{22} + \)\(63\!\cdots\!95\)\( T^{23} + \)\(28\!\cdots\!16\)\( T^{24} + \)\(11\!\cdots\!05\)\( T^{25} + \)\(48\!\cdots\!53\)\( T^{26} + \)\(18\!\cdots\!81\)\( T^{27} + \)\(67\!\cdots\!67\)\( T^{28} + \)\(22\!\cdots\!35\)\( T^{29} + \)\(72\!\cdots\!65\)\( T^{30} + \)\(20\!\cdots\!21\)\( T^{31} + \)\(56\!\cdots\!51\)\( T^{32} + \)\(12\!\cdots\!87\)\( T^{33} + \)\(28\!\cdots\!95\)\( T^{34} + \)\(40\!\cdots\!98\)\( T^{35} + \)\(69\!\cdots\!41\)\( T^{36} \))
$37$ (\( 1 + 5 T + 37 T^{2} \))(\( 1 + 8 T + 37 T^{2} \))(\( 1 + 10 T + 67 T^{2} + 370 T^{3} + 1369 T^{4} \))(\( 1 + 19 T + 218 T^{2} + 1575 T^{3} + 8066 T^{4} + 26011 T^{5} + 50653 T^{6} \))(\( 1 - T + 412 T^{2} - 354 T^{3} + 83444 T^{4} - 67883 T^{5} + 11105147 T^{6} - 9208591 T^{7} + 1094209425 T^{8} - 956006262 T^{9} + 85182936825 T^{10} - 78285856202 T^{11} + 5451516217666 T^{12} - 5153362534950 T^{13} + 294042385236328 T^{14} - 277041579062029 T^{15} + 13569080378057165 T^{16} - 12313692550669995 T^{17} + 539978839130965547 T^{18} - 455606624374789815 T^{19} + 18576071037560258885 T^{20} - 14032987104228954937 T^{21} + \)\(55\!\cdots\!08\)\( T^{22} - \)\(35\!\cdots\!50\)\( T^{23} + \)\(13\!\cdots\!94\)\( T^{24} - \)\(74\!\cdots\!66\)\( T^{25} + \)\(29\!\cdots\!25\)\( T^{26} - \)\(12\!\cdots\!74\)\( T^{27} + \)\(52\!\cdots\!25\)\( T^{28} - \)\(16\!\cdots\!83\)\( T^{29} + \)\(73\!\cdots\!07\)\( T^{30} - \)\(16\!\cdots\!51\)\( T^{31} + \)\(75\!\cdots\!16\)\( T^{32} - \)\(11\!\cdots\!22\)\( T^{33} + \)\(50\!\cdots\!92\)\( T^{34} - \)\(45\!\cdots\!17\)\( T^{35} + \)\(16\!\cdots\!29\)\( T^{36} \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( 1 + 11 T + 41 T^{2} \))(\( 1 - 4 T + 54 T^{2} - 164 T^{3} + 1681 T^{4} \))(\( 1 + 2 T + 39 T^{2} + 60 T^{3} + 1599 T^{4} + 3362 T^{5} + 68921 T^{6} \))(\( 1 + 18 T + 448 T^{2} + 5983 T^{3} + 89359 T^{4} + 969897 T^{5} + 11107708 T^{6} + 102855327 T^{7} + 989203059 T^{8} + 8062923195 T^{9} + 68334607194 T^{10} + 502708222870 T^{11} + 3881844521818 T^{12} + 26379771465264 T^{13} + 190864206279602 T^{14} + 1225857147599468 T^{15} + 8513820028353411 T^{16} + 52756216822433594 T^{17} + 357325936283022160 T^{18} + 2163004889719777354 T^{19} + 14311731467662083891 T^{20} + 84487300469702934028 T^{21} + \)\(53\!\cdots\!22\)\( T^{22} + \)\(30\!\cdots\!64\)\( T^{23} + \)\(18\!\cdots\!38\)\( T^{24} + \)\(97\!\cdots\!70\)\( T^{25} + \)\(54\!\cdots\!74\)\( T^{26} + \)\(26\!\cdots\!95\)\( T^{27} + \)\(13\!\cdots\!59\)\( T^{28} + \)\(56\!\cdots\!07\)\( T^{29} + \)\(25\!\cdots\!48\)\( T^{30} + \)\(89\!\cdots\!37\)\( T^{31} + \)\(33\!\cdots\!99\)\( T^{32} + \)\(93\!\cdots\!83\)\( T^{33} + \)\(28\!\cdots\!68\)\( T^{34} + \)\(47\!\cdots\!58\)\( T^{35} + \)\(10\!\cdots\!21\)\( T^{36} \))
$43$ (\( 1 - 6 T + 43 T^{2} \))(\( 1 - 3 T + 43 T^{2} \))(\( 1 - 4 T + 18 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 - 14 T + 133 T^{2} - 860 T^{3} + 5719 T^{4} - 25886 T^{5} + 79507 T^{6} \))(\( 1 + 39 T + 1183 T^{2} + 26421 T^{3} + 505467 T^{4} + 8316424 T^{5} + 122655302 T^{6} + 1633709544 T^{7} + 19991129432 T^{8} + 226023061513 T^{9} + 2382134596419 T^{10} + 23495313142778 T^{11} + 217992338769957 T^{12} + 1907475481819399 T^{13} + 15790821650006012 T^{14} + 123862438254265247 T^{15} + 922306024038269159 T^{16} + 6523797732987249839 T^{17} + 43874778897622692696 T^{18} + \)\(28\!\cdots\!77\)\( T^{19} + \)\(17\!\cdots\!91\)\( T^{20} + \)\(98\!\cdots\!29\)\( T^{21} + \)\(53\!\cdots\!12\)\( T^{22} + \)\(28\!\cdots\!57\)\( T^{23} + \)\(13\!\cdots\!93\)\( T^{24} + \)\(63\!\cdots\!46\)\( T^{25} + \)\(27\!\cdots\!19\)\( T^{26} + \)\(11\!\cdots\!59\)\( T^{27} + \)\(43\!\cdots\!68\)\( T^{28} + \)\(15\!\cdots\!08\)\( T^{29} + \)\(49\!\cdots\!02\)\( T^{30} + \)\(14\!\cdots\!32\)\( T^{31} + \)\(37\!\cdots\!83\)\( T^{32} + \)\(83\!\cdots\!47\)\( T^{33} + \)\(16\!\cdots\!83\)\( T^{34} + \)\(22\!\cdots\!77\)\( T^{35} + \)\(25\!\cdots\!49\)\( T^{36} \))
$47$ (\( 1 + 3 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 6 T + 53 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 9 T + 164 T^{2} - 859 T^{3} + 7708 T^{4} - 19881 T^{5} + 103823 T^{6} \))(\( 1 + 462 T^{2} - 280 T^{3} + 106315 T^{4} - 119557 T^{5} + 16392446 T^{6} - 25076987 T^{7} + 1913187274 T^{8} - 3465273660 T^{9} + 180069700066 T^{10} - 355945625134 T^{11} + 14161759331307 T^{12} - 28952630487794 T^{13} + 950120856387563 T^{14} - 1931670701944198 T^{15} + 55045695249395495 T^{16} - 107724675191143418 T^{17} + 2772053073615354805 T^{18} - 5063059733983740646 T^{19} + \)\(12\!\cdots\!55\)\( T^{20} - \)\(20\!\cdots\!54\)\( T^{21} + \)\(46\!\cdots\!03\)\( T^{22} - \)\(66\!\cdots\!58\)\( T^{23} + \)\(15\!\cdots\!03\)\( T^{24} - \)\(18\!\cdots\!42\)\( T^{25} + \)\(42\!\cdots\!26\)\( T^{26} - \)\(38\!\cdots\!20\)\( T^{27} + \)\(10\!\cdots\!26\)\( T^{28} - \)\(61\!\cdots\!61\)\( T^{29} + \)\(19\!\cdots\!86\)\( T^{30} - \)\(65\!\cdots\!39\)\( T^{31} + \)\(27\!\cdots\!35\)\( T^{32} - \)\(33\!\cdots\!40\)\( T^{33} + \)\(26\!\cdots\!02\)\( T^{34} + \)\(12\!\cdots\!89\)\( T^{36} \))
$53$ (\( 1 - 3 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 22 T + 225 T^{2} + 1166 T^{3} + 2809 T^{4} \))(\( 1 - 17 T + 68 T^{2} + 47 T^{3} + 3604 T^{4} - 47753 T^{5} + 148877 T^{6} \))(\( 1 + 9 T + 539 T^{2} + 3791 T^{3} + 137142 T^{4} + 736084 T^{5} + 22291906 T^{6} + 86968448 T^{7} + 2659984730 T^{8} + 6982771218 T^{9} + 254874594104 T^{10} + 407890373453 T^{11} + 20842966776803 T^{12} + 18727247042940 T^{13} + 1502309104097726 T^{14} + 771732602740687 T^{15} + 96105659181356924 T^{16} + 34132637516256824 T^{17} + 5429441445485800205 T^{18} + 1809029788361611672 T^{19} + \)\(26\!\cdots\!16\)\( T^{20} + \)\(11\!\cdots\!99\)\( T^{21} + \)\(11\!\cdots\!06\)\( T^{22} + \)\(78\!\cdots\!20\)\( T^{23} + \)\(46\!\cdots\!87\)\( T^{24} + \)\(47\!\cdots\!61\)\( T^{25} + \)\(15\!\cdots\!44\)\( T^{26} + \)\(23\!\cdots\!94\)\( T^{27} + \)\(46\!\cdots\!70\)\( T^{28} + \)\(80\!\cdots\!56\)\( T^{29} + \)\(10\!\cdots\!46\)\( T^{30} + \)\(19\!\cdots\!32\)\( T^{31} + \)\(18\!\cdots\!98\)\( T^{32} + \)\(27\!\cdots\!87\)\( T^{33} + \)\(20\!\cdots\!19\)\( T^{34} + \)\(18\!\cdots\!17\)\( T^{35} + \)\(10\!\cdots\!89\)\( T^{36} \))
$59$ (\( 1 + 9 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 10 T + 141 T^{2} + 590 T^{3} + 3481 T^{4} \))(\( 1 + T + 164 T^{2} + 95 T^{3} + 9676 T^{4} + 3481 T^{5} + 205379 T^{6} \))(\( 1 + 42 T + 1360 T^{2} + 31025 T^{3} + 612208 T^{4} + 10145493 T^{5} + 152736852 T^{6} + 2055827762 T^{7} + 25836509031 T^{8} + 299604939052 T^{9} + 3298991321185 T^{10} + 34132677066868 T^{11} + 338901264367870 T^{12} + 3196245623845163 T^{13} + 29112314717343492 T^{14} + 253410850248199794 T^{15} + 2137331268684060507 T^{16} + 17277688776576138780 T^{17} + \)\(13\!\cdots\!09\)\( T^{18} + \)\(10\!\cdots\!20\)\( T^{19} + \)\(74\!\cdots\!67\)\( T^{20} + \)\(52\!\cdots\!26\)\( T^{21} + \)\(35\!\cdots\!12\)\( T^{22} + \)\(22\!\cdots\!37\)\( T^{23} + \)\(14\!\cdots\!70\)\( T^{24} + \)\(84\!\cdots\!92\)\( T^{25} + \)\(48\!\cdots\!85\)\( T^{26} + \)\(25\!\cdots\!28\)\( T^{27} + \)\(13\!\cdots\!31\)\( T^{28} + \)\(61\!\cdots\!58\)\( T^{29} + \)\(27\!\cdots\!12\)\( T^{30} + \)\(10\!\cdots\!47\)\( T^{31} + \)\(37\!\cdots\!88\)\( T^{32} + \)\(11\!\cdots\!75\)\( T^{33} + \)\(29\!\cdots\!60\)\( T^{34} + \)\(53\!\cdots\!98\)\( T^{35} + \)\(75\!\cdots\!21\)\( T^{36} \))
$61$ (\( 1 - 9 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 - 10 T + 115 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 + 17 T + 242 T^{2} + 1965 T^{3} + 14762 T^{4} + 63257 T^{5} + 226981 T^{6} \))(\( 1 + 43 T + 1236 T^{2} + 25823 T^{3} + 454265 T^{4} + 6917604 T^{5} + 96199752 T^{6} + 1232649011 T^{7} + 14874379074 T^{8} + 169027953844 T^{9} + 1829604952738 T^{10} + 18856237695016 T^{11} + 186566959862853 T^{12} + 1770499415004349 T^{13} + 16203255957729501 T^{14} + 142747671198317857 T^{15} + 1215480166445993409 T^{16} + 9983323912115608909 T^{17} + 79357672547095432243 T^{18} + \)\(60\!\cdots\!49\)\( T^{19} + \)\(45\!\cdots\!89\)\( T^{20} + \)\(32\!\cdots\!17\)\( T^{21} + \)\(22\!\cdots\!41\)\( T^{22} + \)\(14\!\cdots\!49\)\( T^{23} + \)\(96\!\cdots\!33\)\( T^{24} + \)\(59\!\cdots\!36\)\( T^{25} + \)\(35\!\cdots\!78\)\( T^{26} + \)\(19\!\cdots\!04\)\( T^{27} + \)\(10\!\cdots\!74\)\( T^{28} + \)\(53\!\cdots\!71\)\( T^{29} + \)\(25\!\cdots\!92\)\( T^{30} + \)\(11\!\cdots\!24\)\( T^{31} + \)\(44\!\cdots\!65\)\( T^{32} + \)\(15\!\cdots\!23\)\( T^{33} + \)\(45\!\cdots\!96\)\( T^{34} + \)\(96\!\cdots\!03\)\( T^{35} + \)\(13\!\cdots\!81\)\( T^{36} \))
$67$ (\( 1 - 12 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( ( 1 + 10 T + 67 T^{2} )^{2} \))(\( 1 + 12 T + 149 T^{2} + 1004 T^{3} + 9983 T^{4} + 53868 T^{5} + 300763 T^{6} \))(\( 1 + 620 T^{2} - 694 T^{3} + 195835 T^{4} - 414839 T^{5} + 41966104 T^{6} - 124017117 T^{7} + 6854173238 T^{8} - 24649096438 T^{9} + 906665321488 T^{10} - 3652672065348 T^{11} + 100553749300529 T^{12} - 428309205455551 T^{13} + 9542743300208266 T^{14} - 41079869245922669 T^{15} + 784417051405701689 T^{16} - 3280455697875365096 T^{17} + 56222487252751681224 T^{18} - \)\(21\!\cdots\!32\)\( T^{19} + \)\(35\!\cdots\!21\)\( T^{20} - \)\(12\!\cdots\!47\)\( T^{21} + \)\(19\!\cdots\!86\)\( T^{22} - \)\(57\!\cdots\!57\)\( T^{23} + \)\(90\!\cdots\!01\)\( T^{24} - \)\(22\!\cdots\!04\)\( T^{25} + \)\(36\!\cdots\!08\)\( T^{26} - \)\(67\!\cdots\!86\)\( T^{27} + \)\(12\!\cdots\!62\)\( T^{28} - \)\(15\!\cdots\!11\)\( T^{29} + \)\(34\!\cdots\!44\)\( T^{30} - \)\(22\!\cdots\!93\)\( T^{31} + \)\(71\!\cdots\!15\)\( T^{32} - \)\(17\!\cdots\!42\)\( T^{33} + \)\(10\!\cdots\!20\)\( T^{34} + \)\(74\!\cdots\!09\)\( T^{36} \))
$71$ (\( 1 + 6 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 12 T + 128 T^{2} + 852 T^{3} + 5041 T^{4} \))(\( 1 + 8 T + 221 T^{2} + 1138 T^{3} + 15691 T^{4} + 40328 T^{5} + 357911 T^{6} \))(\( 1 - 9 T + 291 T^{2} - 1590 T^{3} + 50941 T^{4} - 266473 T^{5} + 7865537 T^{6} - 35758854 T^{7} + 969153546 T^{8} - 4041636128 T^{9} + 108417058876 T^{10} - 429588450768 T^{11} + 10657558124541 T^{12} - 38510988047844 T^{13} + 946786285111541 T^{14} - 3333395129584239 T^{15} + 77466044278610343 T^{16} - 254903646300667229 T^{17} + 5700836094854813860 T^{18} - 18098158887347373259 T^{19} + \)\(39\!\cdots\!63\)\( T^{20} - \)\(11\!\cdots\!29\)\( T^{21} + \)\(24\!\cdots\!21\)\( T^{22} - \)\(69\!\cdots\!44\)\( T^{23} + \)\(13\!\cdots\!61\)\( T^{24} - \)\(39\!\cdots\!88\)\( T^{25} + \)\(70\!\cdots\!36\)\( T^{26} - \)\(18\!\cdots\!68\)\( T^{27} + \)\(31\!\cdots\!46\)\( T^{28} - \)\(82\!\cdots\!34\)\( T^{29} + \)\(12\!\cdots\!17\)\( T^{30} - \)\(31\!\cdots\!03\)\( T^{31} + \)\(42\!\cdots\!21\)\( T^{32} - \)\(93\!\cdots\!90\)\( T^{33} + \)\(12\!\cdots\!11\)\( T^{34} - \)\(26\!\cdots\!19\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \))
$73$ (\( 1 + 4 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 4 T - 12 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( 1 + 14 T + 261 T^{2} + 2054 T^{3} + 19053 T^{4} + 74606 T^{5} + 389017 T^{6} \))(\( 1 - 19 T + 749 T^{2} - 10601 T^{3} + 244073 T^{4} - 2830323 T^{5} + 49093824 T^{6} - 494314534 T^{7} + 7050835601 T^{8} - 63967945732 T^{9} + 774857518306 T^{10} - 6520491473601 T^{11} + 67787832545490 T^{12} - 548291694762135 T^{13} + 4932260733670244 T^{14} - 40370827287664084 T^{15} + 324419015683971685 T^{16} - 2838905010118790909 T^{17} + 22206332169056364108 T^{18} - \)\(20\!\cdots\!57\)\( T^{19} + \)\(17\!\cdots\!65\)\( T^{20} - \)\(15\!\cdots\!28\)\( T^{21} + \)\(14\!\cdots\!04\)\( T^{22} - \)\(11\!\cdots\!55\)\( T^{23} + \)\(10\!\cdots\!10\)\( T^{24} - \)\(72\!\cdots\!97\)\( T^{25} + \)\(62\!\cdots\!86\)\( T^{26} - \)\(37\!\cdots\!16\)\( T^{27} + \)\(30\!\cdots\!49\)\( T^{28} - \)\(15\!\cdots\!18\)\( T^{29} + \)\(11\!\cdots\!04\)\( T^{30} - \)\(47\!\cdots\!59\)\( T^{31} + \)\(29\!\cdots\!57\)\( T^{32} - \)\(94\!\cdots\!57\)\( T^{33} + \)\(48\!\cdots\!89\)\( T^{34} - \)\(90\!\cdots\!07\)\( T^{35} + \)\(34\!\cdots\!69\)\( T^{36} \))
$79$ (\( 1 + T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{18} \))
$83$ (\( 1 + 16 T + 83 T^{2} \))(\( 1 - 14 T + 83 T^{2} \))(\( 1 - 8 T + 180 T^{2} - 664 T^{3} + 6889 T^{4} \))(\( 1 - 16 T + 285 T^{2} - 2646 T^{3} + 23655 T^{4} - 110224 T^{5} + 571787 T^{6} \))(\( 1 + 61 T + 2659 T^{2} + 85042 T^{3} + 2290011 T^{4} + 52717719 T^{5} + 1082078720 T^{6} + 19989335932 T^{7} + 338697934016 T^{8} + 5294435238489 T^{9} + 77142424786147 T^{10} + 1051473457010882 T^{11} + 13490168591220219 T^{12} + 163260992813337951 T^{13} + 1871016380051029068 T^{14} + 20327143595012317476 T^{15} + \)\(20\!\cdots\!15\)\( T^{16} + \)\(20\!\cdots\!46\)\( T^{17} + \)\(19\!\cdots\!94\)\( T^{18} + \)\(17\!\cdots\!18\)\( T^{19} + \)\(14\!\cdots\!35\)\( T^{20} + \)\(11\!\cdots\!12\)\( T^{21} + \)\(88\!\cdots\!28\)\( T^{22} + \)\(64\!\cdots\!93\)\( T^{23} + \)\(44\!\cdots\!11\)\( T^{24} + \)\(28\!\cdots\!14\)\( T^{25} + \)\(17\!\cdots\!27\)\( T^{26} + \)\(98\!\cdots\!67\)\( T^{27} + \)\(52\!\cdots\!84\)\( T^{28} + \)\(25\!\cdots\!44\)\( T^{29} + \)\(11\!\cdots\!20\)\( T^{30} + \)\(46\!\cdots\!97\)\( T^{31} + \)\(16\!\cdots\!19\)\( T^{32} + \)\(51\!\cdots\!94\)\( T^{33} + \)\(13\!\cdots\!79\)\( T^{34} + \)\(25\!\cdots\!03\)\( T^{35} + \)\(34\!\cdots\!09\)\( T^{36} \))
$89$ (\( 1 - 12 T + 89 T^{2} \))(\( 1 + 4 T + 89 T^{2} \))(\( 1 + 24 T + 290 T^{2} + 2136 T^{3} + 7921 T^{4} \))(\( 1 + 4 T + 139 T^{2} + 1128 T^{3} + 12371 T^{4} + 31684 T^{5} + 704969 T^{6} \))(\( 1 - 10 T + 836 T^{2} - 5722 T^{3} + 323153 T^{4} - 1317081 T^{5} + 79815303 T^{6} - 119637155 T^{7} + 14743271160 T^{8} + 12653680757 T^{9} + 2244215899085 T^{10} + 6303048874800 T^{11} + 297230779694685 T^{12} + 1238330185305012 T^{13} + 35028745646944059 T^{14} + 171155973199386036 T^{15} + 3692138215615098409 T^{16} + 18646992610590805947 T^{17} + \)\(34\!\cdots\!22\)\( T^{18} + \)\(16\!\cdots\!83\)\( T^{19} + \)\(29\!\cdots\!89\)\( T^{20} + \)\(12\!\cdots\!84\)\( T^{21} + \)\(21\!\cdots\!19\)\( T^{22} + \)\(69\!\cdots\!88\)\( T^{23} + \)\(14\!\cdots\!85\)\( T^{24} + \)\(27\!\cdots\!00\)\( T^{25} + \)\(88\!\cdots\!85\)\( T^{26} + \)\(44\!\cdots\!13\)\( T^{27} + \)\(45\!\cdots\!60\)\( T^{28} - \)\(33\!\cdots\!95\)\( T^{29} + \)\(19\!\cdots\!63\)\( T^{30} - \)\(28\!\cdots\!89\)\( T^{31} + \)\(63\!\cdots\!73\)\( T^{32} - \)\(99\!\cdots\!78\)\( T^{33} + \)\(12\!\cdots\!96\)\( T^{34} - \)\(13\!\cdots\!90\)\( T^{35} + \)\(12\!\cdots\!81\)\( T^{36} \))
$97$ (\( 1 - 2 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 - 32 T + 448 T^{2} - 3104 T^{3} + 9409 T^{4} \))(\( 1 + 4 T + 281 T^{2} + 738 T^{3} + 27257 T^{4} + 37636 T^{5} + 912673 T^{6} \))(\( 1 + 9 T + 1028 T^{2} + 8118 T^{3} + 518160 T^{4} + 3592116 T^{5} + 172002077 T^{6} + 1049394365 T^{7} + 42525205864 T^{8} + 229573122687 T^{9} + 8373979285693 T^{10} + 40339892166513 T^{11} + 1367347895433451 T^{12} + 5941223599913869 T^{13} + 189747033224810021 T^{14} + 752519037609646140 T^{15} + 22704059575290630160 T^{16} + 83151135478609032095 T^{17} + \)\(23\!\cdots\!88\)\( T^{18} + \)\(80\!\cdots\!15\)\( T^{19} + \)\(21\!\cdots\!40\)\( T^{20} + \)\(68\!\cdots\!20\)\( T^{21} + \)\(16\!\cdots\!01\)\( T^{22} + \)\(51\!\cdots\!33\)\( T^{23} + \)\(11\!\cdots\!79\)\( T^{24} + \)\(32\!\cdots\!69\)\( T^{25} + \)\(65\!\cdots\!73\)\( T^{26} + \)\(17\!\cdots\!79\)\( T^{27} + \)\(31\!\cdots\!36\)\( T^{28} + \)\(75\!\cdots\!45\)\( T^{29} + \)\(11\!\cdots\!57\)\( T^{30} + \)\(24\!\cdots\!32\)\( T^{31} + \)\(33\!\cdots\!40\)\( T^{32} + \)\(51\!\cdots\!74\)\( T^{33} + \)\(63\!\cdots\!88\)\( T^{34} + \)\(53\!\cdots\!33\)\( T^{35} + \)\(57\!\cdots\!89\)\( T^{36} \))
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