Properties

Label 6010.2
Level 6010
Weight 2
Dimension 331101
Nonzero newspaces 48
Sturm bound 4334400

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Defining parameters

Level: \( N \) = \( 6010 = 2 \cdot 5 \cdot 601 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 48 \)
Sturm bound: \(4334400\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(6010))\).

Total New Old
Modular forms 1088400 331101 757299
Cusp forms 1078801 331101 747700
Eisenstein series 9599 0 9599

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(6010))\)

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
6010.2.a \(\chi_{6010}(1, \cdot)\) 6010.2.a.a 1 1
6010.2.a.b 1
6010.2.a.c 16
6010.2.a.d 21
6010.2.a.e 21
6010.2.a.f 22
6010.2.a.g 27
6010.2.a.h 28
6010.2.a.i 29
6010.2.a.j 33
6010.2.b \(\chi_{6010}(4809, \cdot)\) n/a 300 1
6010.2.c \(\chi_{6010}(6009, \cdot)\) n/a 300 1
6010.2.d \(\chi_{6010}(1201, \cdot)\) n/a 198 1
6010.2.e \(\chi_{6010}(3581, \cdot)\) n/a 404 2
6010.2.f \(\chi_{6010}(2279, \cdot)\) n/a 604 2
6010.2.k \(\chi_{6010}(3481, \cdot)\) n/a 396 2
6010.2.l \(\chi_{6010}(4521, \cdot)\) n/a 792 4
6010.2.m \(\chi_{6010}(2981, \cdot)\) n/a 404 2
6010.2.n \(\chi_{6010}(1779, \cdot)\) n/a 600 2
6010.2.o \(\chi_{6010}(2379, \cdot)\) n/a 600 2
6010.2.q \(\chi_{6010}(163, \cdot)\) n/a 1204 4
6010.2.r \(\chi_{6010}(1143, \cdot)\) n/a 1204 4
6010.2.t \(\chi_{6010}(1371, \cdot)\) n/a 792 4
6010.2.u \(\chi_{6010}(169, \cdot)\) n/a 1200 4
6010.2.v \(\chi_{6010}(3319, \cdot)\) n/a 1200 4
6010.2.w \(\chi_{6010}(481, \cdot)\) n/a 808 4
6010.2.bb \(\chi_{6010}(2399, \cdot)\) n/a 1208 4
6010.2.bc \(\chi_{6010}(151, \cdot)\) n/a 1616 8
6010.2.bd \(\chi_{6010}(511, \cdot)\) n/a 1584 8
6010.2.bi \(\chi_{6010}(1189, \cdot)\) n/a 2416 8
6010.2.bk \(\chi_{6010}(387, \cdot)\) n/a 2408 8
6010.2.bl \(\chi_{6010}(273, \cdot)\) n/a 2408 8
6010.2.bn \(\chi_{6010}(301, \cdot)\) n/a 3960 20
6010.2.bo \(\chi_{6010}(619, \cdot)\) n/a 2400 8
6010.2.bp \(\chi_{6010}(199, \cdot)\) n/a 2400 8
6010.2.bq \(\chi_{6010}(1051, \cdot)\) n/a 1616 8
6010.2.bs \(\chi_{6010}(193, \cdot)\) n/a 4816 16
6010.2.bt \(\chi_{6010}(97, \cdot)\) n/a 4816 16
6010.2.bv \(\chi_{6010}(111, \cdot)\) n/a 3960 20
6010.2.bw \(\chi_{6010}(89, \cdot)\) n/a 6000 20
6010.2.bx \(\chi_{6010}(379, \cdot)\) n/a 6000 20
6010.2.by \(\chi_{6010}(289, \cdot)\) n/a 4832 16
6010.2.cd \(\chi_{6010}(441, \cdot)\) n/a 3232 16
6010.2.ce \(\chi_{6010}(81, \cdot)\) n/a 8080 40
6010.2.cf \(\chi_{6010}(119, \cdot)\) n/a 12080 40
6010.2.ck \(\chi_{6010}(101, \cdot)\) n/a 7920 40
6010.2.cm \(\chi_{6010}(87, \cdot)\) n/a 9632 32
6010.2.cn \(\chi_{6010}(17, \cdot)\) n/a 9632 32
6010.2.cp \(\chi_{6010}(651, \cdot)\) n/a 8080 40
6010.2.cq \(\chi_{6010}(9, \cdot)\) n/a 12000 40
6010.2.cr \(\chi_{6010}(179, \cdot)\) n/a 12000 40
6010.2.cs \(\chi_{6010}(123, \cdot)\) n/a 24080 80
6010.2.cv \(\chi_{6010}(63, \cdot)\) n/a 24080 80
6010.2.cw \(\chi_{6010}(39, \cdot)\) n/a 24160 80
6010.2.db \(\chi_{6010}(61, \cdot)\) n/a 16160 80
6010.2.dc \(\chi_{6010}(7, \cdot)\) n/a 48160 160
6010.2.df \(\chi_{6010}(33, \cdot)\) n/a 48160 160

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

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(6010))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(6010)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(601))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(1202))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(3005))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( ( 1 - T )^{16} \))
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 8 T + 57 T^{2} + 281 T^{3} + 1254 T^{4} + 4685 T^{5} + 16162 T^{6} + 49691 T^{7} + 143004 T^{8} + 377505 T^{9} + 941243 T^{10} + 2186116 T^{11} + 4823804 T^{12} + 10001048 T^{13} + 19768774 T^{14} + 36880364 T^{15} + 65707223 T^{16} + 110641092 T^{17} + 177918966 T^{18} + 270028296 T^{19} + 390728124 T^{20} + 531226188 T^{21} + 686166147 T^{22} + 825603435 T^{23} + 938249244 T^{24} + 978067953 T^{25} + 954349938 T^{26} + 829933695 T^{27} + 666427014 T^{28} + 448004763 T^{29} + 272629233 T^{30} + 114791256 T^{31} + 43046721 T^{32} \))
$5$ (\( 1 - T \))(\( 1 - T \))(\( ( 1 - T )^{16} \))
$7$ (\( 1 + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 10 T + 118 T^{2} + 806 T^{3} + 5690 T^{4} + 30245 T^{5} + 161206 T^{6} + 711125 T^{7} + 3121801 T^{8} + 11874467 T^{9} + 44938096 T^{10} + 151206868 T^{11} + 507467739 T^{12} + 1537856119 T^{13} + 4662653269 T^{14} + 12875599765 T^{15} + 35644712104 T^{16} + 90129198355 T^{17} + 228470010181 T^{18} + 527484648817 T^{19} + 1218430041339 T^{20} + 2541333830476 T^{21} + 5286922056304 T^{22} + 9779134176581 T^{23} + 17996561526601 T^{24} + 28696458777875 T^{25} + 45536704990294 T^{26} + 59804247342035 T^{27} + 78756924173690 T^{28} + 78092542388042 T^{29} + 80030322596182 T^{30} + 47475615099430 T^{31} + 33232930569601 T^{32} \))
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 14 T + 206 T^{2} + 1890 T^{3} + 16720 T^{4} + 117103 T^{5} + 782197 T^{6} + 4497985 T^{7} + 24724320 T^{8} + 121847359 T^{9} + 576830634 T^{10} + 2504819032 T^{11} + 10496872857 T^{12} + 40889363569 T^{13} + 154224172953 T^{14} + 544510493329 T^{15} + 1864540396174 T^{16} + 5989615426619 T^{17} + 18661124927313 T^{18} + 54423742910339 T^{19} + 153684715499337 T^{20} + 403403609922632 T^{21} + 1021890654799674 T^{22} + 2374460320731389 T^{23} + 5299877568685920 T^{24} + 10606013344902635 T^{25} + 20288175710628397 T^{26} + 33410852563559933 T^{27} + 52474522458775120 T^{28} + 65247925952029590 T^{29} + 78228465718147646 T^{30} + 58481474371819114 T^{31} + 45949729863572161 T^{32} \))
$13$ (\( 1 - 4 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 20 T + 313 T^{2} + 3506 T^{3} + 33864 T^{4} + 276888 T^{5} + 2038693 T^{6} + 13439079 T^{7} + 81710659 T^{8} + 456934474 T^{9} + 2391229326 T^{10} + 11682785938 T^{11} + 53889393551 T^{12} + 234038507716 T^{13} + 964433576632 T^{14} + 3757940186918 T^{15} + 13924887105996 T^{16} + 48853222429934 T^{17} + 162989274450808 T^{18} + 514182601452052 T^{19} + 1539134969210111 T^{20} + 4337736639277834 T^{21} + 11542007231800734 T^{22} + 28671960609675058 T^{23} + 66653894779455139 T^{24} + 142514704829197467 T^{25} + 281051142323113357 T^{26} + 496227707184116856 T^{27} + 788966354587696584 T^{28} + 1061880123712439018 T^{29} + 1232398808723877457 T^{30} + 1023717860281815140 T^{31} + 665416609183179841 T^{32} \))
$17$ (\( 1 - 4 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 27 T + 500 T^{2} + 6749 T^{3} + 75881 T^{4} + 725535 T^{5} + 6154083 T^{6} + 46807141 T^{7} + 325583454 T^{8} + 2084535307 T^{9} + 12422492128 T^{10} + 69190714506 T^{11} + 362723475236 T^{12} + 1794195007725 T^{13} + 8411202177539 T^{14} + 37407922994307 T^{15} + 158212935030678 T^{16} + 635934690903219 T^{17} + 2430837429308771 T^{18} + 8814880072952925 T^{19} + 30295027375185956 T^{20} + 98240920326345642 T^{21} + 299848760891556832 T^{22} + 855365451696027611 T^{23} + 2271191201906981214 T^{24} + 5550759456085665077 T^{25} + 12406593790856883267 T^{26} + 24865460287558508655 T^{27} + 44209957983231494441 T^{28} + 66845997144082168813 T^{29} + 84188913279700464500 T^{30} + 77285422390765026411 T^{31} + 48661191875666868481 T^{32} \))
$19$ (\( 1 + 6 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 17 T + 327 T^{2} + 3664 T^{3} + 41421 T^{4} + 347995 T^{5} + 2882425 T^{6} + 19146171 T^{7} + 125016961 T^{8} + 668348993 T^{9} + 3537389525 T^{10} + 15073727168 T^{11} + 65267623081 T^{12} + 213187296893 T^{13} + 791849657816 T^{14} + 2042124568060 T^{15} + 9835891477592 T^{16} + 38800366793140 T^{17} + 285857726471576 T^{18} + 1462251669389087 T^{19} + 8505741907539001 T^{20} + 37324040766957632 T^{21} + 166419606643796525 T^{22} + 597418276631808827 T^{23} + 2123233438337738401 T^{24} + 6178233841273054209 T^{25} + 17672338658142047425 T^{26} + 40538027645285720905 T^{27} + 91677717262639454781 T^{28} + \)\(15\!\cdots\!76\)\( T^{29} + \)\(26\!\cdots\!67\)\( T^{30} + \)\(25\!\cdots\!83\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( 1 - 6 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 9 T + 237 T^{2} + 1668 T^{3} + 25173 T^{4} + 144102 T^{5} + 1647056 T^{6} + 7839001 T^{7} + 76598936 T^{8} + 306625254 T^{9} + 2764786114 T^{10} + 9384886144 T^{11} + 82645544743 T^{12} + 242143582184 T^{13} + 2160591783873 T^{14} + 5708822051453 T^{15} + 51561782179826 T^{16} + 131302907183419 T^{17} + 1142953053668817 T^{18} + 2946160964432728 T^{19} + 23127611886425863 T^{20} + 60404346238731392 T^{21} + 409287570280845346 T^{22} + 1044005467512038538 T^{23} + 5998538149636261016 T^{24} + 14119237514361118463 T^{25} + 68231783853507867344 T^{26} + \)\(13\!\cdots\!54\)\( T^{27} + \)\(55\!\cdots\!33\)\( T^{28} + \)\(84\!\cdots\!44\)\( T^{29} + \)\(27\!\cdots\!33\)\( T^{30} + \)\(23\!\cdots\!63\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 + 5 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 + 23 T + 489 T^{2} + 7105 T^{3} + 93596 T^{4} + 1030547 T^{5} + 10453324 T^{6} + 94787388 T^{7} + 803782159 T^{8} + 6294926398 T^{9} + 46619896638 T^{10} + 324745781394 T^{11} + 2154669917430 T^{12} + 13582048934175 T^{13} + 81857640164297 T^{14} + 470924233426233 T^{15} + 2593957059143673 T^{16} + 13656802769360757 T^{17} + 68842275378173777 T^{18} + 331252591455594075 T^{19} + 1523957093869807830 T^{20} + 6660909109293761706 T^{21} + 27730601742891894798 T^{22} + \)\(10\!\cdots\!82\)\( T^{23} + \)\(40\!\cdots\!99\)\( T^{24} + \)\(13\!\cdots\!72\)\( T^{25} + \)\(43\!\cdots\!24\)\( T^{26} + \)\(12\!\cdots\!63\)\( T^{27} + \)\(33\!\cdots\!36\)\( T^{28} + \)\(72\!\cdots\!45\)\( T^{29} + \)\(14\!\cdots\!09\)\( T^{30} + \)\(19\!\cdots\!27\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( 1 + 7 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 21 T + 485 T^{2} + 7222 T^{3} + 102711 T^{4} + 1206611 T^{5} + 13272274 T^{6} + 130135654 T^{7} + 1198353921 T^{8} + 10143279239 T^{9} + 81148138328 T^{10} + 605640943981 T^{11} + 4293843883418 T^{12} + 28630425417982 T^{13} + 181887472875013 T^{14} + 1091456003324614 T^{15} + 6248429943176185 T^{16} + 33835136103063034 T^{17} + 174793861432887493 T^{18} + 852929003627101762 T^{19} + 3965454997058074778 T^{20} + 17338986037014590131 T^{21} + 72019271472397185368 T^{22} + \)\(27\!\cdots\!29\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} + \)\(34\!\cdots\!34\)\( T^{25} + \)\(10\!\cdots\!74\)\( T^{26} + \)\(30\!\cdots\!41\)\( T^{27} + \)\(80\!\cdots\!71\)\( T^{28} + \)\(17\!\cdots\!02\)\( T^{29} + \)\(36\!\cdots\!85\)\( T^{30} + \)\(49\!\cdots\!71\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( 1 + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 + 16 T + 381 T^{2} + 4500 T^{3} + 66637 T^{4} + 664615 T^{5} + 7711576 T^{6} + 67965331 T^{7} + 667463365 T^{8} + 5309263545 T^{9} + 45898688075 T^{10} + 333632041417 T^{11} + 2595110754065 T^{12} + 17346318997023 T^{13} + 122987155212956 T^{14} + 758118044827282 T^{15} + 4936852544954080 T^{16} + 28050367658609434 T^{17} + 168369415486536764 T^{18} + 878643096156206019 T^{19} + 4863655365949214465 T^{20} + 23135365933842667069 T^{21} + \)\(11\!\cdots\!75\)\( T^{22} + \)\(50\!\cdots\!85\)\( T^{23} + \)\(23\!\cdots\!65\)\( T^{24} + \)\(88\!\cdots\!87\)\( T^{25} + \)\(37\!\cdots\!24\)\( T^{26} + \)\(11\!\cdots\!95\)\( T^{27} + \)\(43\!\cdots\!97\)\( T^{28} + \)\(10\!\cdots\!00\)\( T^{29} + \)\(34\!\cdots\!09\)\( T^{30} + \)\(53\!\cdots\!88\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( 1 + 8 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 35 T + 918 T^{2} + 16953 T^{3} + 265894 T^{4} + 3462826 T^{5} + 40228682 T^{6} + 411359916 T^{7} + 3868417020 T^{8} + 33137711708 T^{9} + 268064983137 T^{10} + 2030644003787 T^{11} + 14882307580052 T^{12} + 104241616709409 T^{13} + 716854118767773 T^{14} + 4748163995608115 T^{15} + 30972925423677248 T^{16} + 194674723819932715 T^{17} + 1205031773648626413 T^{18} + 7184436465229177689 T^{19} + 42053844349715319572 T^{20} + \)\(23\!\cdots\!87\)\( T^{21} + \)\(12\!\cdots\!17\)\( T^{22} + \)\(64\!\cdots\!48\)\( T^{23} + \)\(30\!\cdots\!20\)\( T^{24} + \)\(13\!\cdots\!76\)\( T^{25} + \)\(53\!\cdots\!82\)\( T^{26} + \)\(19\!\cdots\!66\)\( T^{27} + \)\(59\!\cdots\!14\)\( T^{28} + \)\(15\!\cdots\!13\)\( T^{29} + \)\(34\!\cdots\!98\)\( T^{30} + \)\(54\!\cdots\!35\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( 1 - 3 T + 354 T^{2} - 1113 T^{3} + 64244 T^{4} - 211484 T^{5} + 7976330 T^{6} - 26847813 T^{7} + 758139324 T^{8} - 2539386149 T^{9} + 58296039761 T^{10} - 190085305456 T^{11} + 3735486959641 T^{12} - 11658127381433 T^{13} + 202833349633862 T^{14} - 596501842164914 T^{15} + 9417908409262430 T^{16} - 25649579213091302 T^{17} + 375038863473010838 T^{18} - 926902733715593531 T^{19} + 12770886553107610441 T^{20} - 27944144792265965008 T^{21} + \)\(36\!\cdots\!89\)\( T^{22} - \)\(69\!\cdots\!43\)\( T^{23} + \)\(88\!\cdots\!24\)\( T^{24} - \)\(13\!\cdots\!59\)\( T^{25} + \)\(17\!\cdots\!70\)\( T^{26} - \)\(19\!\cdots\!88\)\( T^{27} + \)\(25\!\cdots\!44\)\( T^{28} - \)\(19\!\cdots\!59\)\( T^{29} + \)\(26\!\cdots\!46\)\( T^{30} - \)\(95\!\cdots\!21\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 25 T + 826 T^{2} + 14891 T^{3} + 291686 T^{4} + 4187199 T^{5} + 61686372 T^{6} + 742437886 T^{7} + 8969985253 T^{8} + 93282920111 T^{9} + 964317209983 T^{10} + 8825889445727 T^{11} + 79925527883701 T^{12} + 650963278176669 T^{13} + 5234242779291429 T^{14} + 38159276068392131 T^{15} + 274399066247815802 T^{16} + 1793485975214430157 T^{17} + 11562442299454766661 T^{18} + 67584960430136305587 T^{19} + \)\(39\!\cdots\!81\)\( T^{20} + \)\(20\!\cdots\!89\)\( T^{21} + \)\(10\!\cdots\!07\)\( T^{22} + \)\(47\!\cdots\!93\)\( T^{23} + \)\(21\!\cdots\!33\)\( T^{24} + \)\(83\!\cdots\!62\)\( T^{25} + \)\(32\!\cdots\!28\)\( T^{26} + \)\(10\!\cdots\!97\)\( T^{27} + \)\(33\!\cdots\!26\)\( T^{28} + \)\(81\!\cdots\!57\)\( T^{29} + \)\(21\!\cdots\!94\)\( T^{30} + \)\(30\!\cdots\!75\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 + 5 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 + 39 T + 1231 T^{2} + 27480 T^{3} + 534272 T^{4} + 8776504 T^{5} + 130274242 T^{6} + 1733054869 T^{7} + 21271112460 T^{8} + 240527691552 T^{9} + 2541484963057 T^{10} + 25116215063922 T^{11} + 233797384267797 T^{12} + 2053197049645603 T^{13} + 17065559253270357 T^{14} + 134434383159445269 T^{15} + 1004577971299609801 T^{16} + 7125022307450599257 T^{17} + 47937155942436432813 T^{18} + \)\(30\!\cdots\!31\)\( T^{19} + \)\(18\!\cdots\!57\)\( T^{20} + \)\(10\!\cdots\!46\)\( T^{21} + \)\(56\!\cdots\!53\)\( T^{22} + \)\(28\!\cdots\!24\)\( T^{23} + \)\(13\!\cdots\!60\)\( T^{24} + \)\(57\!\cdots\!77\)\( T^{25} + \)\(22\!\cdots\!58\)\( T^{26} + \)\(81\!\cdots\!88\)\( T^{27} + \)\(26\!\cdots\!52\)\( T^{28} + \)\(71\!\cdots\!40\)\( T^{29} + \)\(16\!\cdots\!39\)\( T^{30} + \)\(28\!\cdots\!23\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 + 6 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 32 T + 983 T^{2} + 19540 T^{3} + 366535 T^{4} + 5498224 T^{5} + 78577823 T^{6} + 965605537 T^{7} + 11429639059 T^{8} + 120485947854 T^{9} + 1237084034994 T^{10} + 11552879250744 T^{11} + 106325909342398 T^{12} + 903389276357180 T^{13} + 7651720191074063 T^{14} + 60453390771496632 T^{15} + 479843599712766792 T^{16} + 3566750055518301288 T^{17} + 26635637985128813303 T^{18} + \)\(18\!\cdots\!20\)\( T^{19} + \)\(12\!\cdots\!78\)\( T^{20} + \)\(82\!\cdots\!56\)\( T^{21} + \)\(52\!\cdots\!54\)\( T^{22} + \)\(29\!\cdots\!26\)\( T^{23} + \)\(16\!\cdots\!39\)\( T^{24} + \)\(83\!\cdots\!43\)\( T^{25} + \)\(40\!\cdots\!23\)\( T^{26} + \)\(16\!\cdots\!16\)\( T^{27} + \)\(65\!\cdots\!35\)\( T^{28} + \)\(20\!\cdots\!60\)\( T^{29} + \)\(60\!\cdots\!63\)\( T^{30} + \)\(11\!\cdots\!68\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 + 8 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 38 T + 1295 T^{2} + 30011 T^{3} + 626470 T^{4} + 10856077 T^{5} + 172651410 T^{6} + 2437497950 T^{7} + 32038642080 T^{8} + 385823989688 T^{9} + 4368737882993 T^{10} + 46102639836199 T^{11} + 460384541326451 T^{12} + 4325555972001790 T^{13} + 38601174150957933 T^{14} + 325667276808755064 T^{15} + 2614044405016921478 T^{16} + 19865703885334058904 T^{17} + \)\(14\!\cdots\!93\)\( T^{18} + \)\(98\!\cdots\!90\)\( T^{19} + \)\(63\!\cdots\!91\)\( T^{20} + \)\(38\!\cdots\!99\)\( T^{21} + \)\(22\!\cdots\!73\)\( T^{22} + \)\(12\!\cdots\!48\)\( T^{23} + \)\(61\!\cdots\!80\)\( T^{24} + \)\(28\!\cdots\!50\)\( T^{25} + \)\(12\!\cdots\!10\)\( T^{26} + \)\(47\!\cdots\!97\)\( T^{27} + \)\(16\!\cdots\!70\)\( T^{28} + \)\(48\!\cdots\!91\)\( T^{29} + \)\(12\!\cdots\!95\)\( T^{30} + \)\(22\!\cdots\!38\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( 1 + 11 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 5 T + 402 T^{2} - 1602 T^{3} + 74350 T^{4} - 169243 T^{5} + 7767398 T^{6} + 2498004 T^{7} + 453831496 T^{8} + 2197391981 T^{9} + 12154540398 T^{10} + 171390935532 T^{11} + 332235246970 T^{12} - 3893732693746 T^{13} + 84000552760227 T^{14} - 1543979087410183 T^{15} + 8882609109952675 T^{16} - 103446598856482261 T^{17} + 377078481340659003 T^{18} - 1171090726169128198 T^{19} + 6694912662157353370 T^{20} + \)\(23\!\cdots\!24\)\( T^{21} + \)\(10\!\cdots\!62\)\( T^{22} + \)\(13\!\cdots\!63\)\( T^{23} + \)\(18\!\cdots\!36\)\( T^{24} + \)\(67\!\cdots\!88\)\( T^{25} + \)\(14\!\cdots\!02\)\( T^{26} - \)\(20\!\cdots\!69\)\( T^{27} + \)\(60\!\cdots\!50\)\( T^{28} - \)\(87\!\cdots\!74\)\( T^{29} + \)\(14\!\cdots\!58\)\( T^{30} - \)\(12\!\cdots\!15\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( 1 - T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 16 T + 526 T^{2} + 6928 T^{3} + 137047 T^{4} + 1567044 T^{5} + 23635059 T^{6} + 240945062 T^{7} + 3015281495 T^{8} + 27888700314 T^{9} + 303183651837 T^{10} + 2592926994325 T^{11} + 25416086053769 T^{12} + 206209951235329 T^{13} + 1896794466723343 T^{14} + 15052634697965535 T^{15} + 135344181343551961 T^{16} + 1068737063555552985 T^{17} + 9561740906752372063 T^{18} + 73804809856587837719 T^{19} + \)\(64\!\cdots\!89\)\( T^{20} + \)\(46\!\cdots\!75\)\( T^{21} + \)\(38\!\cdots\!77\)\( T^{22} + \)\(25\!\cdots\!74\)\( T^{23} + \)\(19\!\cdots\!95\)\( T^{24} + \)\(11\!\cdots\!22\)\( T^{25} + \)\(76\!\cdots\!59\)\( T^{26} + \)\(36\!\cdots\!24\)\( T^{27} + \)\(22\!\cdots\!27\)\( T^{28} + \)\(80\!\cdots\!08\)\( T^{29} + \)\(43\!\cdots\!06\)\( T^{30} + \)\(93\!\cdots\!16\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \))
$73$ (\( 1 - 10 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 + 17 T + 556 T^{2} + 7081 T^{3} + 144872 T^{4} + 1566328 T^{5} + 25483539 T^{6} + 247130981 T^{7} + 3488766241 T^{8} + 31287429462 T^{9} + 398799856772 T^{10} + 3352212666941 T^{11} + 39357196665648 T^{12} + 311733617992366 T^{13} + 3414522244375559 T^{14} + 25552881000882903 T^{15} + 263564644611155338 T^{16} + 1865360313064451919 T^{17} + 18195989040277353911 T^{18} + \)\(12\!\cdots\!22\)\( T^{19} + \)\(11\!\cdots\!68\)\( T^{20} + \)\(69\!\cdots\!13\)\( T^{21} + \)\(60\!\cdots\!08\)\( T^{22} + \)\(34\!\cdots\!14\)\( T^{23} + \)\(28\!\cdots\!21\)\( T^{24} + \)\(14\!\cdots\!53\)\( T^{25} + \)\(10\!\cdots\!11\)\( T^{26} + \)\(49\!\cdots\!56\)\( T^{27} + \)\(33\!\cdots\!12\)\( T^{28} + \)\(11\!\cdots\!73\)\( T^{29} + \)\(67\!\cdots\!04\)\( T^{30} + \)\(15\!\cdots\!69\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 + 7 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 + 40 T + 1414 T^{2} + 34396 T^{3} + 762120 T^{4} + 14088318 T^{5} + 242425320 T^{6} + 3710981623 T^{7} + 53592229522 T^{8} + 709191792584 T^{9} + 8928044506566 T^{10} + 104622541303753 T^{11} + 1172423954448043 T^{12} + 12337003849558400 T^{13} + 124524935747908654 T^{14} + 1185545136155475404 T^{15} + 10842313620985010147 T^{16} + 93658065756282556916 T^{17} + \)\(77\!\cdots\!14\)\( T^{18} + \)\(60\!\cdots\!00\)\( T^{19} + \)\(45\!\cdots\!83\)\( T^{20} + \)\(32\!\cdots\!47\)\( T^{21} + \)\(21\!\cdots\!86\)\( T^{22} + \)\(13\!\cdots\!56\)\( T^{23} + \)\(81\!\cdots\!42\)\( T^{24} + \)\(44\!\cdots\!37\)\( T^{25} + \)\(22\!\cdots\!20\)\( T^{26} + \)\(10\!\cdots\!22\)\( T^{27} + \)\(45\!\cdots\!20\)\( T^{28} + \)\(16\!\cdots\!44\)\( T^{29} + \)\(52\!\cdots\!34\)\( T^{30} + \)\(11\!\cdots\!60\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( 1 + 14 T + 83 T^{2} \))(\( 1 - 10 T + 83 T^{2} \))(\( 1 + 22 T + 753 T^{2} + 13577 T^{3} + 284515 T^{4} + 4373625 T^{5} + 71313698 T^{6} + 966845185 T^{7} + 13359298838 T^{8} + 163128798226 T^{9} + 1992200753793 T^{10} + 22222733647137 T^{11} + 245584559524393 T^{12} + 2525956690794771 T^{13} + 25620092390282092 T^{14} + 244236171327730249 T^{15} + 2291614794037582042 T^{16} + 20271602220201610667 T^{17} + \)\(17\!\cdots\!88\)\( T^{18} + \)\(14\!\cdots\!77\)\( T^{19} + \)\(11\!\cdots\!53\)\( T^{20} + \)\(87\!\cdots\!91\)\( T^{21} + \)\(65\!\cdots\!17\)\( T^{22} + \)\(44\!\cdots\!02\)\( T^{23} + \)\(30\!\cdots\!58\)\( T^{24} + \)\(18\!\cdots\!55\)\( T^{25} + \)\(11\!\cdots\!02\)\( T^{26} + \)\(56\!\cdots\!75\)\( T^{27} + \)\(30\!\cdots\!15\)\( T^{28} + \)\(12\!\cdots\!51\)\( T^{29} + \)\(55\!\cdots\!37\)\( T^{30} + \)\(13\!\cdots\!54\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( 1 - 3 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 46 T + 1859 T^{2} + 50844 T^{3} + 1258616 T^{4} + 25614133 T^{5} + 482866054 T^{6} + 7998133233 T^{7} + 124655146668 T^{8} + 1762933364465 T^{9} + 23716682648774 T^{10} + 294851095385419 T^{11} + 3513368509717137 T^{12} + 39107608206469575 T^{13} + 419240006454285163 T^{14} + 4222833281253600358 T^{15} + 41060968152247598033 T^{16} + \)\(37\!\cdots\!62\)\( T^{17} + \)\(33\!\cdots\!23\)\( T^{18} + \)\(27\!\cdots\!75\)\( T^{19} + \)\(22\!\cdots\!17\)\( T^{20} + \)\(16\!\cdots\!31\)\( T^{21} + \)\(11\!\cdots\!14\)\( T^{22} + \)\(77\!\cdots\!85\)\( T^{23} + \)\(49\!\cdots\!08\)\( T^{24} + \)\(28\!\cdots\!97\)\( T^{25} + \)\(15\!\cdots\!54\)\( T^{26} + \)\(71\!\cdots\!37\)\( T^{27} + \)\(31\!\cdots\!36\)\( T^{28} + \)\(11\!\cdots\!36\)\( T^{29} + \)\(36\!\cdots\!19\)\( T^{30} + \)\(80\!\cdots\!54\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( 1 + 12 T + 97 T^{2} \))(\( 1 - 6 T + 97 T^{2} \))(\( 1 + 21 T + 818 T^{2} + 13020 T^{3} + 314631 T^{4} + 4271050 T^{5} + 81488077 T^{6} + 987240827 T^{7} + 16148746124 T^{8} + 178735883258 T^{9} + 2612735716840 T^{10} + 26785355520465 T^{11} + 357743623579067 T^{12} + 3421075873165877 T^{13} + 42324738100511689 T^{14} + 379141757202932689 T^{15} + 4380977082083325178 T^{16} + 36776750448684470833 T^{17} + \)\(39\!\cdots\!01\)\( T^{18} + \)\(31\!\cdots\!21\)\( T^{19} + \)\(31\!\cdots\!27\)\( T^{20} + \)\(23\!\cdots\!05\)\( T^{21} + \)\(21\!\cdots\!60\)\( T^{22} + \)\(14\!\cdots\!54\)\( T^{23} + \)\(12\!\cdots\!64\)\( T^{24} + \)\(75\!\cdots\!59\)\( T^{25} + \)\(60\!\cdots\!73\)\( T^{26} + \)\(30\!\cdots\!50\)\( T^{27} + \)\(21\!\cdots\!71\)\( T^{28} + \)\(87\!\cdots\!40\)\( T^{29} + \)\(53\!\cdots\!42\)\( T^{30} + \)\(13\!\cdots\!53\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))
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