Properties

Label 150.2
Level 150
Weight 2
Dimension 137
Nonzero newspaces 6
Newform subspaces 12
Sturm bound 2400
Trace bound 1

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 12 \)
Sturm bound: \(2400\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(150))\).

Total New Old
Modular forms 712 137 575
Cusp forms 489 137 352
Eisenstein series 223 0 223

Trace form

\( 137q + 3q^{2} + 7q^{3} + 3q^{4} + 10q^{5} + 7q^{6} + 24q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 137q + 3q^{2} + 7q^{3} + 3q^{4} + 10q^{5} + 7q^{6} + 24q^{7} + 3q^{8} + 3q^{9} + 2q^{10} + 4q^{11} - 9q^{12} + 10q^{13} - 8q^{14} - 4q^{15} + 3q^{16} - 34q^{17} - 23q^{18} - 52q^{19} - 8q^{20} - 32q^{21} - 60q^{22} - 56q^{23} - 13q^{24} - 86q^{25} + 2q^{26} + 7q^{27} - 16q^{28} - 38q^{29} - 20q^{30} - 16q^{31} - 7q^{32} - 20q^{33} - 28q^{34} - 8q^{35} + 3q^{36} - 24q^{37} + 12q^{38} - 54q^{39} - 6q^{40} + 6q^{41} - 8q^{42} - 76q^{43} + 20q^{44} - 70q^{45} - 24q^{46} - 32q^{47} - 9q^{48} - 63q^{49} + 18q^{50} - 30q^{51} + 10q^{52} - 24q^{53} + 7q^{54} - 24q^{55} + 8q^{56} - 20q^{57} + 58q^{58} + 20q^{59} + 32q^{60} + 82q^{61} + 48q^{62} + 84q^{63} + 3q^{64} + 74q^{65} + 48q^{66} + 108q^{67} + 6q^{68} + 180q^{69} + 48q^{70} + 24q^{71} + 19q^{72} + 110q^{73} + 34q^{74} + 164q^{75} - 4q^{76} + 96q^{77} + 118q^{78} + 64q^{79} + 10q^{80} + 19q^{81} + 62q^{82} + 20q^{83} + 68q^{84} + 34q^{85} + 20q^{86} + 62q^{87} + 20q^{88} + 4q^{89} + 54q^{90} + 24q^{92} - 20q^{93} + 80q^{94} + 16q^{95} - 9q^{96} + 30q^{97} + 75q^{98} + 20q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(150))\)

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
150.2.a \(\chi_{150}(1, \cdot)\) 150.2.a.a 1 1
150.2.a.b 1
150.2.a.c 1
150.2.c \(\chi_{150}(49, \cdot)\) 150.2.c.a 2 1
150.2.e \(\chi_{150}(107, \cdot)\) 150.2.e.a 4 2
150.2.e.b 8
150.2.g \(\chi_{150}(31, \cdot)\) 150.2.g.a 4 4
150.2.g.b 4
150.2.g.c 8
150.2.h \(\chi_{150}(19, \cdot)\) 150.2.h.a 8 4
150.2.h.b 16
150.2.l \(\chi_{150}(17, \cdot)\) 150.2.l.a 80 8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(150))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(150)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(75))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( ( 1 + T^{4} )^{2} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T^{2} \))(\( 1 - 4 T + 8 T^{2} - 12 T^{3} + 9 T^{4} \))(\( 1 - 9 T^{4} + 81 T^{8} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))
$5$ (\( 1 - 5 T + 15 T^{2} - 25 T^{3} + 25 T^{4} \))(\( 1 - 5 T + 15 T^{2} - 25 T^{3} + 25 T^{4} \))(\( 1 + 4 T + T^{2} - 16 T^{3} - 39 T^{4} - 80 T^{5} + 25 T^{6} + 500 T^{7} + 625 T^{8} \))(\( 1 + 5 T^{2} + 10 T^{3} + 5 T^{4} + 50 T^{5} + 125 T^{6} + 625 T^{8} \))(\( 1 - 4 T - 4 T^{2} + 6 T^{3} + 161 T^{4} - 260 T^{5} - 580 T^{6} - 520 T^{7} + 7205 T^{8} - 2600 T^{9} - 14500 T^{10} - 32500 T^{11} + 100625 T^{12} + 18750 T^{13} - 62500 T^{14} - 312500 T^{15} + 390625 T^{16} \))
$7$ (\( 1 - 2 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( ( 1 - 2 T + 2 T^{2} - 14 T^{3} + 49 T^{4} )^{2} \))(\( ( 1 - 94 T^{4} + 2401 T^{8} )^{2} \))(\( ( 1 - 3 T + 15 T^{2} - 21 T^{3} + 49 T^{4} )^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{4} \))(\( ( 1 + T + 9 T^{2} + 17 T^{3} + 104 T^{4} + 119 T^{5} + 441 T^{6} + 343 T^{7} + 2401 T^{8} )^{2} \))(\( 1 - 32 T^{2} + 500 T^{4} - 5152 T^{6} + 40294 T^{8} - 252448 T^{10} + 1200500 T^{12} - 3764768 T^{14} + 5764801 T^{16} \))(\( 1 - 30 T^{2} + 527 T^{4} - 6940 T^{6} + 74703 T^{8} - 716280 T^{10} + 6275029 T^{12} - 50516750 T^{14} + 372187880 T^{16} - 2475320750 T^{18} + 15066344629 T^{20} - 84269625720 T^{22} + 430647929103 T^{24} - 1960378228060 T^{26} + 7294358354927 T^{28} - 20346692185470 T^{30} + 33232930569601 T^{32} \))
$11$ (\( 1 - 2 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 - 20 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 5 T^{2} + 121 T^{4} )^{4} \))(\( 1 + 5 T - T^{2} - 55 T^{3} - 184 T^{4} - 605 T^{5} - 121 T^{6} + 6655 T^{7} + 14641 T^{8} \))(\( 1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 374 T^{5} + 1573 T^{6} + 2662 T^{7} + 14641 T^{8} \))(\( 1 + 5 T + 38 T^{2} + 210 T^{3} + 1048 T^{4} + 4455 T^{5} + 19301 T^{6} + 69050 T^{7} + 243280 T^{8} + 759550 T^{9} + 2335421 T^{10} + 5929605 T^{11} + 15343768 T^{12} + 33820710 T^{13} + 67319318 T^{14} + 97435855 T^{15} + 214358881 T^{16} \))(\( 1 - 10 T + 28 T^{2} + 60 T^{3} - 617 T^{4} + 1790 T^{5} - 794 T^{6} - 18400 T^{7} + 94725 T^{8} - 202400 T^{9} - 96074 T^{10} + 2382490 T^{11} - 9033497 T^{12} + 9663060 T^{13} + 49603708 T^{14} - 194871710 T^{15} + 214358881 T^{16} \))(\( 1 - 2 T + 11 T^{2} - 50 T^{3} + 270 T^{4} + 764 T^{5} - 168 T^{6} + 4404 T^{7} - 16480 T^{8} + 222470 T^{9} + 461835 T^{10} - 66610 T^{11} + 2053685 T^{12} + 10233320 T^{13} + 85541270 T^{14} + 166971000 T^{15} - 141281840 T^{16} + 1836681000 T^{17} + 10350493670 T^{18} + 13620548920 T^{19} + 30068002085 T^{20} - 10727607110 T^{21} + 818168874435 T^{22} + 4335310932370 T^{23} - 3532634358880 T^{24} + 10384401631164 T^{25} - 4357487332968 T^{26} + 217978116346804 T^{27} + 847375661714670 T^{28} - 1726135607196550 T^{29} + 4177248169415651 T^{30} - 8354496338831302 T^{31} + 45949729863572161 T^{32} \))
$13$ (\( 1 - 6 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 + 169 T^{4} )^{2} \))(\( ( 1 + 169 T^{4} )^{4} \))(\( 1 - 12 T + 51 T^{2} - 76 T^{3} + 9 T^{4} - 988 T^{5} + 8619 T^{6} - 26364 T^{7} + 28561 T^{8} \))(\( 1 + 6 T + 23 T^{2} + 120 T^{3} + 601 T^{4} + 1560 T^{5} + 3887 T^{6} + 13182 T^{7} + 28561 T^{8} \))(\( 1 - 6 T + 6 T^{2} + 26 T^{3} + 3 T^{4} - 1366 T^{5} + 5284 T^{6} - 2112 T^{7} - 15227 T^{8} - 27456 T^{9} + 892996 T^{10} - 3001102 T^{11} + 85683 T^{12} + 9653618 T^{13} + 28960854 T^{14} - 376491102 T^{15} + 815730721 T^{16} \))(\( 1 + 20 T + 202 T^{2} + 1420 T^{3} + 8095 T^{4} + 40660 T^{5} + 185692 T^{6} + 772160 T^{7} + 2918789 T^{8} + 10038080 T^{9} + 31381948 T^{10} + 89330020 T^{11} + 231201295 T^{12} + 527236060 T^{13} + 975015418 T^{14} + 1254970340 T^{15} + 815730721 T^{16} \))(\( 1 - 20 T + 240 T^{2} - 2100 T^{3} + 14722 T^{4} - 86440 T^{5} + 437000 T^{6} - 1922660 T^{7} + 7352283 T^{8} - 23888120 T^{9} + 61481440 T^{10} - 91205000 T^{11} - 205909876 T^{12} + 2578228580 T^{13} - 14615222400 T^{14} + 65003714260 T^{15} - 249510986035 T^{16} + 845048285380 T^{17} - 2469972585600 T^{18} + 5664368190260 T^{19} - 5880991968436 T^{20} - 33863778065000 T^{21} + 296759167924960 T^{22} - 1498944103918040 T^{23} + 5997483112586043 T^{24} - 20388846764492180 T^{25} + 60244160938013000 T^{26} - 154914344460558280 T^{27} + 342994409173165282 T^{28} - 636037723843731300 T^{29} + 944970332567829360 T^{30} - 1023717860281815140 T^{31} + 665416609183179841 T^{32} \))
$17$ (\( 1 - 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 2 T^{2} + 289 T^{4} \))(\( ( 1 - 16 T^{2} + 289 T^{4} )( 1 + 16 T^{2} + 289 T^{4} ) \))(\( ( 1 + 47 T^{4} + 83521 T^{8} )^{2} \))(\( 1 - 6 T - T^{2} + 18 T^{3} + 169 T^{4} + 306 T^{5} - 289 T^{6} - 29478 T^{7} + 83521 T^{8} \))(\( 1 + 12 T + 37 T^{2} - 180 T^{3} - 1619 T^{4} - 3060 T^{5} + 10693 T^{6} + 58956 T^{7} + 83521 T^{8} \))(\( 1 + 2 T + 44 T^{2} + 12 T^{3} + 1273 T^{4} - 258 T^{5} + 28916 T^{6} - 13984 T^{7} + 544693 T^{8} - 237728 T^{9} + 8356724 T^{10} - 1267554 T^{11} + 106322233 T^{12} + 17038284 T^{13} + 1062053036 T^{14} + 820677346 T^{15} + 6975757441 T^{16} \))(\( 1 - 10 T + 78 T^{2} - 590 T^{3} + 3615 T^{4} - 20330 T^{5} + 105768 T^{6} - 488740 T^{7} + 2089449 T^{8} - 8308580 T^{9} + 30566952 T^{10} - 99881290 T^{11} + 301928415 T^{12} - 837715630 T^{13} + 1882730382 T^{14} - 4103386730 T^{15} + 6975757441 T^{16} \))(\( 1 + 30 T + 500 T^{2} + 5990 T^{3} + 56782 T^{4} + 447460 T^{5} + 3015460 T^{6} + 17669550 T^{7} + 90820363 T^{8} + 409992380 T^{9} + 1611336300 T^{10} + 5357847740 T^{11} + 13820507184 T^{12} + 18014105810 T^{13} - 71676889400 T^{14} - 711604275870 T^{15} - 3544790874535 T^{16} - 12097272689790 T^{17} - 20714621036600 T^{18} + 88503301844530 T^{19} + 1154302580514864 T^{20} + 7607377618573180 T^{21} + 38893741123454700 T^{22} + 168235729149311740 T^{23} + 633540822991571083 T^{24} + 2095394413157566350 T^{25} + 6079148967047941540 T^{26} + 15335302721813462180 T^{27} + 33082455874380289102 T^{28} + 59328422417106562630 T^{29} + 84188913279700464500 T^{30} + 85872691545294473790 T^{31} + 48661191875666868481 T^{32} \))
$19$ (\( 1 + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 22 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} + 361 T^{4} )^{4} \))(\( 1 + 16 T + 117 T^{2} + 578 T^{3} + 2525 T^{4} + 10982 T^{5} + 42237 T^{6} + 109744 T^{7} + 130321 T^{8} \))(\( 1 + 21 T^{2} - 10 T^{3} + 381 T^{4} - 190 T^{5} + 7581 T^{6} + 130321 T^{8} \))(\( ( 1 - 4 T - 3 T^{2} - 62 T^{3} + 605 T^{4} - 1178 T^{5} - 1083 T^{6} - 27436 T^{7} + 130321 T^{8} )^{2} \))(\( 1 + 8 T - 10 T^{2} - 270 T^{3} - 565 T^{4} + 5224 T^{5} + 28422 T^{6} - 55860 T^{7} - 788595 T^{8} - 1061340 T^{9} + 10260342 T^{10} + 35831416 T^{11} - 73631365 T^{12} - 668546730 T^{13} - 470458810 T^{14} + 7150973912 T^{15} + 16983563041 T^{16} \))(\( 1 + 24 T^{2} - 30 T^{3} + 330 T^{4} + 740 T^{5} + 7630 T^{6} + 18810 T^{7} + 160255 T^{8} - 244020 T^{9} + 7105152 T^{10} - 5384250 T^{11} + 141174008 T^{12} - 174364190 T^{13} + 1779959840 T^{14} + 4165821270 T^{15} + 38170716365 T^{16} + 79150604130 T^{17} + 642565502240 T^{18} - 1195963979210 T^{19} + 18397937896568 T^{20} - 13331936040750 T^{21} + 334268135478912 T^{22} - 218122581750780 T^{23} + 2721700895135455 T^{24} + 6069755595222990 T^{25} + 46780035547021630 T^{26} + 86202791584682060 T^{27} + 730393923291833130 T^{28} - 1261589503867711770 T^{29} + 19176160458789218904 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( 1 + 4 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( 1 - 158 T^{4} + 279841 T^{8} \))(\( ( 1 - 958 T^{4} + 279841 T^{8} )^{2} \))(\( 1 + 10 T + 37 T^{2} + 200 T^{3} + 1389 T^{4} + 4600 T^{5} + 19573 T^{6} + 121670 T^{7} + 279841 T^{8} \))(\( 1 + 6 T + 13 T^{2} - 60 T^{3} - 659 T^{4} - 1380 T^{5} + 6877 T^{6} + 73002 T^{7} + 279841 T^{8} \))(\( ( 1 + 10 T + 37 T^{2} + 200 T^{3} + 1389 T^{4} + 4600 T^{5} + 19573 T^{6} + 121670 T^{7} + 279841 T^{8} )^{2} \))(\( 1 + 10 T + 116 T^{2} + 950 T^{3} + 6927 T^{4} + 45910 T^{5} + 267168 T^{6} + 1460600 T^{7} + 7269605 T^{8} + 33593800 T^{9} + 141331872 T^{10} + 558586970 T^{11} + 1938458607 T^{12} + 6114525850 T^{13} + 17172163124 T^{14} + 34048254470 T^{15} + 78310985281 T^{16} \))(\( 1 + 10 T + 30 T^{2} - 250 T^{3} - 2718 T^{4} - 8680 T^{5} + 13790 T^{6} + 227450 T^{7} + 942143 T^{8} + 3042820 T^{9} + 4039430 T^{10} - 83513580 T^{11} - 909320196 T^{12} - 3918445730 T^{13} - 578679600 T^{14} + 92103435490 T^{15} + 633292773765 T^{16} + 2118379016270 T^{17} - 306121508400 T^{18} - 47675729196910 T^{19} - 254465072968836 T^{20} - 537522046037940 T^{21} + 597980611103270 T^{22} + 10360270966640540 T^{23} + 73780146605597183 T^{24} + 409672172849759350 T^{25} + 571271589636219710 T^{26} - 8270388698692886360 T^{27} - 59563949206231232478 T^{28} - \)\(12\!\cdots\!50\)\( T^{29} + \)\(34\!\cdots\!70\)\( T^{30} + \)\(26\!\cdots\!70\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 8 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 29 T^{2} )^{8} \))(\( 1 - 6 T + 47 T^{2} - 288 T^{3} + 2365 T^{4} - 8352 T^{5} + 39527 T^{6} - 146334 T^{7} + 707281 T^{8} \))(\( 1 - 19 T^{2} + 120 T^{3} + 721 T^{4} + 3480 T^{5} - 15979 T^{6} + 707281 T^{8} \))(\( 1 + 18 T + 160 T^{2} + 1050 T^{3} + 5925 T^{4} + 26514 T^{5} + 92732 T^{6} + 300180 T^{7} + 1280525 T^{8} + 8705220 T^{9} + 77987612 T^{10} + 646649946 T^{11} + 4190639925 T^{12} + 21536706450 T^{13} + 95171731360 T^{14} + 310497773562 T^{15} + 500246412961 T^{16} \))(\( 1 + 22 T + 290 T^{2} + 2930 T^{3} + 24395 T^{4} + 175526 T^{5} + 1131792 T^{6} + 6698140 T^{7} + 37055745 T^{8} + 194246060 T^{9} + 951837072 T^{10} + 4280903614 T^{11} + 17254119995 T^{12} + 60097666570 T^{13} + 172498763090 T^{14} + 379497278798 T^{15} + 500246412961 T^{16} \))(\( 1 + 10 T - 36 T^{2} - 270 T^{3} + 3910 T^{4} + 4460 T^{5} - 185440 T^{6} + 333090 T^{7} + 4522435 T^{8} - 38473780 T^{9} - 69131068 T^{10} + 1395128220 T^{11} - 3409890032 T^{12} - 36344819210 T^{13} + 245632726680 T^{14} + 543098432830 T^{15} - 7660398480495 T^{16} + 15749854552070 T^{17} + 206577123137880 T^{18} - 886413795712690 T^{19} - 2411750431722992 T^{20} + 28615682794524780 T^{21} - 41120771452036828 T^{22} - 663667946139678020 T^{23} + 2262331886599280035 T^{24} + 4832185253102205210 T^{25} - 78015949343189273440 T^{26} + 54414273555047997340 T^{27} + \)\(13\!\cdots\!10\)\( T^{28} - \)\(27\!\cdots\!30\)\( T^{29} - \)\(10\!\cdots\!16\)\( T^{30} + \)\(86\!\cdots\!90\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( 1 + 8 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 2 T + 31 T^{2} )^{4} \))(\( ( 1 + 2 T + 31 T^{2} )^{8} \))(\( ( 1 - T - 39 T^{2} - 31 T^{3} + 961 T^{4} )( 1 + 4 T + 46 T^{2} + 124 T^{3} + 961 T^{4} ) \))(\( 1 + 12 T + 113 T^{2} + 834 T^{3} + 5605 T^{4} + 25854 T^{5} + 108593 T^{6} + 357492 T^{7} + 923521 T^{8} \))(\( 1 - 9 T - 20 T^{2} + 300 T^{3} + 690 T^{4} - 8217 T^{5} - 2797 T^{6} + 139770 T^{7} - 752500 T^{8} + 4332870 T^{9} - 2687917 T^{10} - 244792647 T^{11} + 637229490 T^{12} + 8588745300 T^{13} - 17750073620 T^{14} - 247613526999 T^{15} + 852891037441 T^{16} \))(\( 1 - 24 T + 230 T^{2} - 990 T^{3} - 685 T^{4} + 32568 T^{5} - 115202 T^{6} - 726180 T^{7} + 8308045 T^{8} - 22511580 T^{9} - 110709122 T^{10} + 970233288 T^{11} - 632611885 T^{12} - 28342859490 T^{13} + 204125846630 T^{14} - 660302738664 T^{15} + 852891037441 T^{16} \))(\( 1 + 18 T + 121 T^{2} + 610 T^{3} + 6060 T^{4} + 54724 T^{5} + 309962 T^{6} + 1456784 T^{7} + 9332270 T^{8} + 64865970 T^{9} + 327433735 T^{10} + 1367357590 T^{11} + 7702756335 T^{12} + 47160385620 T^{13} + 227624335970 T^{14} + 1111476930100 T^{15} + 6249005152660 T^{16} + 34455784833100 T^{17} + 218746986867170 T^{18} + 1404955048005420 T^{19} + 7113657233255535 T^{20} + 39146286915106090 T^{21} + 290598645096078535 T^{22} + 1784632401545702670 T^{23} + 7959409441979521070 T^{24} + 38516818529710942064 T^{25} + \)\(25\!\cdots\!62\)\( T^{26} + \)\(13\!\cdots\!44\)\( T^{27} + \)\(47\!\cdots\!60\)\( T^{28} + \)\(14\!\cdots\!10\)\( T^{29} + \)\(91\!\cdots\!41\)\( T^{30} + \)\(42\!\cdots\!18\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 + 12 T + 72 T^{2} + 444 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 1106 T^{4} + 1874161 T^{8} )^{2} \))(\( 1 + 8 T + 27 T^{2} - 80 T^{3} - 1639 T^{4} - 2960 T^{5} + 36963 T^{6} + 405224 T^{7} + 1874161 T^{8} \))(\( 1 - 13 T + 27 T^{2} + 445 T^{3} - 4264 T^{4} + 16465 T^{5} + 36963 T^{6} - 658489 T^{7} + 1874161 T^{8} \))(\( 1 - 21 T + 258 T^{2} - 2462 T^{3} + 19746 T^{4} - 143741 T^{5} + 980785 T^{6} - 6308436 T^{7} + 38864404 T^{8} - 233412132 T^{9} + 1342694665 T^{10} - 7280912873 T^{11} + 37007183106 T^{12} - 170724822134 T^{13} + 661957413522 T^{14} - 1993569419793 T^{15} + 3512479453921 T^{16} \))(\( 1 + 20 T + 275 T^{2} + 3140 T^{3} + 29831 T^{4} + 251620 T^{5} + 1934025 T^{6} + 13441540 T^{7} + 85392936 T^{8} + 497336980 T^{9} + 2647680225 T^{10} + 12745307860 T^{11} + 55908096791 T^{12} + 217740024980 T^{13} + 705574762475 T^{14} + 1898637542660 T^{15} + 3512479453921 T^{16} \))(\( 1 - 20 T + 265 T^{2} - 2300 T^{3} + 17532 T^{4} - 115640 T^{5} + 825500 T^{6} - 5506160 T^{7} + 38914148 T^{8} - 253492260 T^{9} + 1794329215 T^{10} - 11470149820 T^{11} + 74232183599 T^{12} - 424648096560 T^{13} + 2693507120800 T^{14} - 16225080394680 T^{15} + 105486161388440 T^{16} - 600327974603160 T^{17} + 3687411248375200 T^{18} - 21509700035053680 T^{19} + 139123063446085439 T^{20} - 795385575901637740 T^{21} + 4603757853365738935 T^{22} - 24064496080486490580 T^{23} + \)\(13\!\cdots\!08\)\( T^{24} - \)\(71\!\cdots\!20\)\( T^{25} + \)\(39\!\cdots\!00\)\( T^{26} - \)\(20\!\cdots\!20\)\( T^{27} + \)\(11\!\cdots\!92\)\( T^{28} - \)\(56\!\cdots\!00\)\( T^{29} + \)\(23\!\cdots\!85\)\( T^{30} - \)\(66\!\cdots\!60\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( 1 - 2 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 50 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 55 T^{2} + 1681 T^{4} )^{4} \))(\( 1 + 14 T + 95 T^{2} + 786 T^{3} + 6569 T^{4} + 32226 T^{5} + 159695 T^{6} + 964894 T^{7} + 2825761 T^{8} \))(\( 1 + 12 T + 53 T^{2} + 444 T^{3} + 4405 T^{4} + 18204 T^{5} + 89093 T^{6} + 827052 T^{7} + 2825761 T^{8} \))(\( 1 - 2 T + 36 T^{2} - 168 T^{3} + 2201 T^{4} - 17286 T^{5} + 84044 T^{6} - 959864 T^{7} + 2420157 T^{8} - 39354424 T^{9} + 141277964 T^{10} - 1191368406 T^{11} + 6219499961 T^{12} - 19463841768 T^{13} + 171003752676 T^{14} - 389508547762 T^{15} + 7984925229121 T^{16} \))(\( 1 - 22 T + 146 T^{2} + 142 T^{3} - 6829 T^{4} + 52634 T^{5} - 236776 T^{6} - 967444 T^{7} + 17720057 T^{8} - 39665204 T^{9} - 398020456 T^{10} + 3627587914 T^{11} - 19297121869 T^{12} + 16451580542 T^{13} + 693515219186 T^{14} - 4284594025382 T^{15} + 7984925229121 T^{16} \))(\( 1 - 22 T + 316 T^{2} - 3710 T^{3} + 39530 T^{4} - 407436 T^{5} + 3918652 T^{6} - 35382126 T^{7} + 301988495 T^{8} - 2475859380 T^{9} + 19707200260 T^{10} - 150750040860 T^{11} + 1111070917960 T^{12} - 7891095298530 T^{13} + 54497096107120 T^{14} - 366799083339050 T^{15} + 2387443151993985 T^{16} - 15038762416901050 T^{17} + 91609618556068720 T^{18} - 543862179069986130 T^{19} + 3139620868205567560 T^{20} - 17465327034634372860 T^{21} + 93611255533262302660 T^{22} - \)\(48\!\cdots\!80\)\( T^{23} + \)\(24\!\cdots\!95\)\( T^{24} - \)\(11\!\cdots\!86\)\( T^{25} + \)\(52\!\cdots\!52\)\( T^{26} - \)\(22\!\cdots\!76\)\( T^{27} + \)\(89\!\cdots\!30\)\( T^{28} - \)\(34\!\cdots\!10\)\( T^{29} + \)\(11\!\cdots\!76\)\( T^{30} - \)\(34\!\cdots\!22\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( ( 1 + 12 T + 72 T^{2} + 516 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 1778 T^{4} + 3418801 T^{8} )^{2} \))(\( ( 1 + 2 T + 42 T^{2} + 86 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 2 T + 82 T^{2} - 86 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 16 T + 168 T^{2} + 1280 T^{3} + 9326 T^{4} + 55040 T^{5} + 310632 T^{6} + 1272112 T^{7} + 3418801 T^{8} )^{2} \))(\( 1 - 120 T^{2} + 10856 T^{4} - 676920 T^{6} + 33149566 T^{8} - 1251625080 T^{10} + 37114503656 T^{12} - 758563565880 T^{14} + 11688200277601 T^{16} \))(\( 1 - 360 T^{2} + 63972 T^{4} - 7536360 T^{6} + 666842308 T^{8} - 47508203160 T^{10} + 2846509586204 T^{12} - 147613243096600 T^{14} + 6742598283189430 T^{16} - 272936886485613400 T^{18} + 9731649819823821404 T^{20} - \)\(30\!\cdots\!40\)\( T^{22} + \)\(77\!\cdots\!08\)\( T^{24} - \)\(16\!\cdots\!40\)\( T^{26} + \)\(25\!\cdots\!72\)\( T^{28} - \)\(26\!\cdots\!40\)\( T^{30} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 + 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( ( 1 - 47 T^{2} )^{2} \))(\( ( 1 + 2209 T^{4} )^{2} \))(\( ( 1 + 2209 T^{4} )^{4} \))(\( 1 - 20 T + 113 T^{2} + 270 T^{3} - 5851 T^{4} + 12690 T^{5} + 249617 T^{6} - 2076460 T^{7} + 4879681 T^{8} \))(\( 1 + 2 T + 17 T^{2} - 130 T^{3} + 761 T^{4} - 6110 T^{5} + 37553 T^{6} + 207646 T^{7} + 4879681 T^{8} \))(\( 1 + 10 T + 66 T^{2} + 210 T^{3} + 1787 T^{4} - 6510 T^{5} - 94072 T^{6} - 974600 T^{7} - 2764935 T^{8} - 45806200 T^{9} - 207805048 T^{10} - 675887730 T^{11} + 8719989947 T^{12} + 48162451470 T^{13} + 711428211714 T^{14} + 5066231204630 T^{15} + 23811286661761 T^{16} \))(\( 1 - 10 T + 164 T^{2} - 1060 T^{3} + 6767 T^{4} - 10550 T^{5} - 301398 T^{6} + 3376520 T^{7} - 34256075 T^{8} + 158696440 T^{9} - 665788182 T^{10} - 1095332650 T^{11} + 33020801327 T^{12} - 243105707420 T^{13} + 1767791313956 T^{14} - 5066231204630 T^{15} + 23811286661761 T^{16} \))(\( 1 + 50 T + 1290 T^{2} + 23780 T^{3} + 355482 T^{4} + 4545300 T^{5} + 51239280 T^{6} + 520209820 T^{7} + 4818856003 T^{8} + 41087515780 T^{9} + 324882626890 T^{10} + 2399419588970 T^{11} + 16710816340744 T^{12} + 111327617454640 T^{13} + 724257830956800 T^{14} + 4728319081214120 T^{15} + 31780196573425365 T^{16} + 222230996817063640 T^{17} + 1599885548583571200 T^{18} + 11558367226993088720 T^{19} + 81543452992418022664 T^{20} + \)\(55\!\cdots\!90\)\( T^{21} + \)\(35\!\cdots\!10\)\( T^{22} + \)\(20\!\cdots\!40\)\( T^{23} + \)\(11\!\cdots\!83\)\( T^{24} + \)\(58\!\cdots\!40\)\( T^{25} + \)\(26\!\cdots\!20\)\( T^{26} + \)\(11\!\cdots\!00\)\( T^{27} + \)\(41\!\cdots\!62\)\( T^{28} + \)\(12\!\cdots\!60\)\( T^{29} + \)\(33\!\cdots\!10\)\( T^{30} + \)\(60\!\cdots\!50\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( ( 1 - 56 T^{2} + 2809 T^{4} )( 1 + 56 T^{2} + 2809 T^{4} ) \))(\( ( 1 - 718 T^{4} + 7890481 T^{8} )^{2} \))(\( 1 + 16 T + 53 T^{2} - 200 T^{3} - 1759 T^{4} - 10600 T^{5} + 148877 T^{6} + 2382032 T^{7} + 7890481 T^{8} \))(\( 1 + 21 T + 253 T^{2} + 2535 T^{3} + 21196 T^{4} + 134355 T^{5} + 710677 T^{6} + 3126417 T^{7} + 7890481 T^{8} \))(\( 1 - 7 T + 62 T^{2} - 816 T^{3} + 3586 T^{4} + 23943 T^{5} - 77365 T^{6} + 2388872 T^{7} - 30975296 T^{8} + 126610216 T^{9} - 217318285 T^{10} + 3564562011 T^{11} + 28295264866 T^{12} - 341247522288 T^{13} + 1374190389998 T^{14} - 8222977978859 T^{15} + 62259690411361 T^{16} \))(\( 1 + 30 T + 425 T^{2} + 3090 T^{3} + 5031 T^{4} - 120570 T^{5} - 1120625 T^{6} - 3254310 T^{7} + 1334576 T^{8} - 172478430 T^{9} - 3147835625 T^{10} - 17950099890 T^{11} + 39697009911 T^{12} + 1292224073370 T^{13} + 9419853479825 T^{14} + 35241334195110 T^{15} + 62259690411361 T^{16} \))(\( 1 - 30 T + 655 T^{2} - 10630 T^{3} + 152332 T^{4} - 1919860 T^{5} + 22723040 T^{6} - 248684220 T^{7} + 2591225028 T^{8} - 25379094630 T^{9} + 238630725505 T^{10} - 2134501287330 T^{11} + 18442776242119 T^{12} - 152410720886420 T^{13} + 1218238235823200 T^{14} - 9336287074370240 T^{15} + 69349166231444240 T^{16} - 494823214941622720 T^{17} + 3422031204427368800 T^{18} - 22690450893407550340 T^{19} + \)\(14\!\cdots\!39\)\( T^{20} - \)\(89\!\cdots\!90\)\( T^{21} + \)\(52\!\cdots\!45\)\( T^{22} - \)\(29\!\cdots\!10\)\( T^{23} + \)\(16\!\cdots\!08\)\( T^{24} - \)\(82\!\cdots\!60\)\( T^{25} + \)\(39\!\cdots\!60\)\( T^{26} - \)\(17\!\cdots\!20\)\( T^{27} + \)\(74\!\cdots\!12\)\( T^{28} - \)\(27\!\cdots\!90\)\( T^{29} + \)\(90\!\cdots\!95\)\( T^{30} - \)\(21\!\cdots\!10\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 - 10 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 10 T + 59 T^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 20 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 3481 T^{4} )^{4} \))(\( 1 - 5 T - 49 T^{2} + 295 T^{3} + 1736 T^{4} + 17405 T^{5} - 170569 T^{6} - 1026895 T^{7} + 12117361 T^{8} \))(\( 1 - 20 T + 181 T^{2} - 1600 T^{3} + 14601 T^{4} - 94400 T^{5} + 630061 T^{6} - 4107580 T^{7} + 12117361 T^{8} \))(\( 1 + 25 T + 212 T^{2} + 150 T^{3} - 12692 T^{4} - 128475 T^{5} - 252121 T^{6} + 7858000 T^{7} + 98067400 T^{8} + 463622000 T^{9} - 877633201 T^{10} - 26386067025 T^{11} - 153793545812 T^{12} + 107238644850 T^{13} + 8942273131892 T^{14} + 62216287120475 T^{15} + 146830437604321 T^{16} \))(\( 1 + 20 T + 242 T^{2} + 2760 T^{3} + 31483 T^{4} + 335220 T^{5} + 3074684 T^{6} + 25371200 T^{7} + 201486805 T^{8} + 1496900800 T^{9} + 10702975004 T^{10} + 68847148380 T^{11} + 381490876363 T^{12} + 1973191065240 T^{13} + 10207689141122 T^{14} + 49773029696380 T^{15} + 146830437604321 T^{16} \))(\( 1 - 20 T - 111 T^{2} + 3640 T^{3} + 11660 T^{4} - 317820 T^{5} - 1817980 T^{6} + 11679820 T^{7} + 217773860 T^{8} + 627241760 T^{9} - 13630850673 T^{10} - 133224376740 T^{11} + 186113511463 T^{12} + 9371240514920 T^{13} + 48131664631280 T^{14} - 252534601553160 T^{15} - 4526062111550040 T^{16} - 14899541491636440 T^{17} + 167546324581485680 T^{18} + 1924656005713754680 T^{19} + 2255204605374809143 T^{20} - 95245344150556405260 T^{21} - \)\(57\!\cdots\!93\)\( T^{22} + \)\(15\!\cdots\!40\)\( T^{23} + \)\(31\!\cdots\!60\)\( T^{24} + \)\(10\!\cdots\!80\)\( T^{25} - \)\(92\!\cdots\!80\)\( T^{26} - \)\(95\!\cdots\!80\)\( T^{27} + \)\(20\!\cdots\!60\)\( T^{28} + \)\(38\!\cdots\!60\)\( T^{29} - \)\(68\!\cdots\!71\)\( T^{30} - \)\(73\!\cdots\!80\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 - 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( ( 1 + 10 T + 61 T^{2} )^{2} \))(\( ( 1 + 6 T + 61 T^{2} )^{4} \))(\( ( 1 - 14 T + 61 T^{2} )^{8} \))(\( 1 - 21 T^{2} - 410 T^{3} + 2901 T^{4} - 25010 T^{5} - 78141 T^{6} + 13845841 T^{8} \))(\( 1 + 2 T + 3 T^{2} + 424 T^{3} + 4265 T^{4} + 25864 T^{5} + 11163 T^{6} + 453962 T^{7} + 13845841 T^{8} \))(\( 1 - 10 T - 62 T^{2} + 700 T^{3} + 5543 T^{4} - 26410 T^{5} - 544444 T^{6} + 851800 T^{7} + 32679625 T^{8} + 51959800 T^{9} - 2025876124 T^{10} - 5994568210 T^{11} + 76747496663 T^{12} + 591217410700 T^{13} - 3194263210382 T^{14} - 31427428360210 T^{15} + 191707312997281 T^{16} \))(\( 1 + 118 T^{2} + 10203 T^{4} + 835676 T^{6} + 64821605 T^{8} + 3109550396 T^{10} + 141269115723 T^{12} + 6079404174598 T^{14} + 191707312997281 T^{16} \))(\( 1 - 12 T - 104 T^{2} + 1460 T^{3} + 9310 T^{4} - 110696 T^{5} - 1128308 T^{6} + 9605804 T^{7} + 95707895 T^{8} - 545482080 T^{9} - 8568890440 T^{10} + 31012865040 T^{11} + 647686771260 T^{12} - 1172976299580 T^{13} - 45238788397880 T^{14} + 2951178504100 T^{15} + 3155660575885285 T^{16} + 180021888750100 T^{17} - 168333531628511480 T^{18} - 266243333454967980 T^{19} + 8967768052669329660 T^{20} + 26193351096196217040 T^{21} - \)\(44\!\cdots\!40\)\( T^{22} - \)\(17\!\cdots\!80\)\( T^{23} + \)\(18\!\cdots\!95\)\( T^{24} + \)\(11\!\cdots\!64\)\( T^{25} - \)\(80\!\cdots\!08\)\( T^{26} - \)\(48\!\cdots\!56\)\( T^{27} + \)\(24\!\cdots\!10\)\( T^{28} + \)\(23\!\cdots\!60\)\( T^{29} - \)\(10\!\cdots\!64\)\( T^{30} - \)\(72\!\cdots\!12\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( 1 + 8 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( ( 1 - 8 T + 32 T^{2} - 536 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 2471 T^{4} + 20151121 T^{8} )^{2} \))(\( 1 - 16 T + 29 T^{2} + 868 T^{3} - 9191 T^{4} + 58156 T^{5} + 130181 T^{6} - 4812208 T^{7} + 20151121 T^{8} \))(\( 1 - 18 T + 77 T^{2} - 30 T^{3} + 961 T^{4} - 2010 T^{5} + 345653 T^{6} - 5413734 T^{7} + 20151121 T^{8} \))(\( 1 + 2 T + 34 T^{2} + 232 T^{3} - 637 T^{4} - 10798 T^{5} + 66956 T^{6} + 2488096 T^{7} - 7257587 T^{8} + 166702432 T^{9} + 300565484 T^{10} - 3247638874 T^{11} - 12836264077 T^{12} + 313229024824 T^{13} + 3075584993746 T^{14} + 12121423210646 T^{15} + 406067677556641 T^{16} \))(\( 1 - 10 T + 88 T^{2} - 1660 T^{3} + 11615 T^{4} - 86790 T^{5} + 911158 T^{6} - 5793400 T^{7} + 42525189 T^{8} - 388157800 T^{9} + 4090188262 T^{10} - 26103220770 T^{11} + 234055270415 T^{12} - 2241207677620 T^{13} + 7960337630872 T^{14} - 60607116053230 T^{15} + 406067677556641 T^{16} \))(\( 1 + 50 T + 1390 T^{2} + 30100 T^{3} + 553682 T^{4} + 8918100 T^{5} + 129754300 T^{6} + 1739003620 T^{7} + 21685053163 T^{8} + 253853201940 T^{9} + 2808527300190 T^{10} + 29485013121570 T^{11} + 294663718701784 T^{12} + 2810511152550240 T^{13} + 25630455288876000 T^{14} + 223725991474439120 T^{15} + 1870714073010261365 T^{16} + 14989641428787421040 T^{17} + \)\(11\!\cdots\!00\)\( T^{18} + \)\(84\!\cdots\!20\)\( T^{19} + \)\(59\!\cdots\!64\)\( T^{20} + \)\(39\!\cdots\!90\)\( T^{21} + \)\(25\!\cdots\!10\)\( T^{22} + \)\(15\!\cdots\!20\)\( T^{23} + \)\(88\!\cdots\!83\)\( T^{24} + \)\(47\!\cdots\!40\)\( T^{25} + \)\(23\!\cdots\!00\)\( T^{26} + \)\(10\!\cdots\!00\)\( T^{27} + \)\(45\!\cdots\!02\)\( T^{28} + \)\(16\!\cdots\!00\)\( T^{29} + \)\(51\!\cdots\!10\)\( T^{30} + \)\(12\!\cdots\!50\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( 1 - 12 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 58 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 71 T^{2} )^{8} \))(\( 1 + 20 T + 89 T^{2} - 1420 T^{3} - 22639 T^{4} - 100820 T^{5} + 448649 T^{6} + 7158220 T^{7} + 25411681 T^{8} \))(\( 1 - 28 T + 313 T^{2} - 2126 T^{3} + 14805 T^{4} - 150946 T^{5} + 1577833 T^{6} - 10021508 T^{7} + 25411681 T^{8} \))(\( 1 - 182 T^{2} - 750 T^{3} + 7983 T^{4} + 134400 T^{5} + 952256 T^{6} - 5830800 T^{7} - 134438695 T^{8} - 413986800 T^{9} + 4800322496 T^{10} + 48103238400 T^{11} + 202861449423 T^{12} - 1353172013250 T^{13} - 23314251673622 T^{14} + 645753531245761 T^{16} \))(\( 1 - 20 T + 118 T^{2} + 490 T^{3} - 14397 T^{4} + 149580 T^{5} - 402414 T^{6} - 9696700 T^{7} + 137916525 T^{8} - 688465700 T^{9} - 2028568974 T^{10} + 53536327380 T^{11} - 365851971357 T^{12} + 884072381990 T^{13} + 15115833502678 T^{14} - 181902403167820 T^{15} + 645753531245761 T^{16} \))(\( 1 + 28 T + 256 T^{2} + 1150 T^{3} + 20770 T^{4} + 328604 T^{5} + 2351902 T^{6} + 21115094 T^{7} + 321977895 T^{8} + 2983992020 T^{9} + 18685321160 T^{10} + 180970630790 T^{11} + 2127756781760 T^{12} + 17298444226470 T^{13} + 123520240637920 T^{14} + 1239239342309850 T^{15} + 12274938369025885 T^{16} + 87985993303999350 T^{17} + 622665533055754720 T^{18} + 6191303471540104170 T^{19} + 54069876583671738560 T^{20} + \)\(32\!\cdots\!90\)\( T^{21} + \)\(23\!\cdots\!60\)\( T^{22} + \)\(27\!\cdots\!20\)\( T^{23} + \)\(20\!\cdots\!95\)\( T^{24} + \)\(96\!\cdots\!14\)\( T^{25} + \)\(76\!\cdots\!02\)\( T^{26} + \)\(75\!\cdots\!84\)\( T^{27} + \)\(34\!\cdots\!70\)\( T^{28} + \)\(13\!\cdots\!50\)\( T^{29} + \)\(21\!\cdots\!36\)\( T^{30} + \)\(16\!\cdots\!28\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \))
$73$ (\( 1 + 4 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 4 T + 73 T^{2} \))(\( 1 - 142 T^{2} + 5329 T^{4} \))(\( ( 1 - 16 T + 73 T^{2} )^{2}( 1 + 6 T + 73 T^{2} )^{2} \))(\( ( 1 - 5617 T^{4} + 28398241 T^{8} )^{2} \))(\( 1 - 2 T + 51 T^{2} - 376 T^{3} + 6389 T^{4} - 27448 T^{5} + 271779 T^{6} - 778034 T^{7} + 28398241 T^{8} \))(\( 1 - 14 T + 3 T^{2} + 980 T^{3} - 9259 T^{4} + 71540 T^{5} + 15987 T^{6} - 5446238 T^{7} + 28398241 T^{8} \))(\( 1 + 24 T + 226 T^{2} + 1116 T^{3} + 5643 T^{4} + 16584 T^{5} - 622036 T^{6} - 12920832 T^{7} - 137738827 T^{8} - 943220736 T^{9} - 3314829844 T^{10} + 6451457928 T^{11} + 160251273963 T^{12} + 2313547897788 T^{13} + 34201535141314 T^{14} + 265137564458328 T^{15} + 806460091894081 T^{16} \))(\( 1 + 20 T + 282 T^{2} + 3780 T^{3} + 46315 T^{4} + 505780 T^{5} + 5254892 T^{6} + 48564720 T^{7} + 416447549 T^{8} + 3545224560 T^{9} + 28003319468 T^{10} + 196757018260 T^{11} + 1315264531915 T^{12} + 7836210621540 T^{13} + 42676251813498 T^{14} + 220947970381940 T^{15} + 806460091894081 T^{16} \))(\( 1 - 20 T + 380 T^{2} - 7420 T^{3} + 107282 T^{4} - 1509240 T^{5} + 20338260 T^{6} - 240254580 T^{7} + 2794851843 T^{8} - 31155740720 T^{9} + 320911861980 T^{10} - 3274927660920 T^{11} + 32181004976604 T^{12} - 299417039187980 T^{13} + 2775262759264800 T^{14} - 24764197420905260 T^{15} + 211758644546106765 T^{16} - 1807786411726083980 T^{17} + 14789375244122119200 T^{18} - \)\(11\!\cdots\!60\)\( T^{19} + \)\(91\!\cdots\!64\)\( T^{20} - \)\(67\!\cdots\!60\)\( T^{21} + \)\(48\!\cdots\!20\)\( T^{22} - \)\(34\!\cdots\!40\)\( T^{23} + \)\(22\!\cdots\!83\)\( T^{24} - \)\(14\!\cdots\!40\)\( T^{25} + \)\(87\!\cdots\!40\)\( T^{26} - \)\(47\!\cdots\!80\)\( T^{27} + \)\(24\!\cdots\!22\)\( T^{28} - \)\(12\!\cdots\!60\)\( T^{29} + \)\(46\!\cdots\!20\)\( T^{30} - \)\(17\!\cdots\!40\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 122 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 + 38 T^{2} + 6241 T^{4} )^{4} \))(\( 1 - 2 T - 55 T^{2} + 578 T^{3} + 3679 T^{4} + 45662 T^{5} - 343255 T^{6} - 986078 T^{7} + 38950081 T^{8} \))(\( 1 - 79 T^{2} + 6241 T^{4} - 493039 T^{6} + 38950081 T^{8} \))(\( 1 + 6 T + 34 T^{2} + 336 T^{3} + 4761 T^{4} - 34122 T^{5} + 60506 T^{6} - 838512 T^{7} + 1917227 T^{8} - 66242448 T^{9} + 377617946 T^{10} - 16823476758 T^{11} + 185441335641 T^{12} + 1033890950064 T^{13} + 8264973487714 T^{14} + 115223453916954 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 16 T + 274 T^{2} - 3476 T^{3} + 48131 T^{4} - 506768 T^{5} + 5479676 T^{6} - 51684448 T^{7} + 496487797 T^{8} - 4083071392 T^{9} + 34198657916 T^{10} - 249856387952 T^{11} + 1874706348611 T^{12} - 10695848042924 T^{13} + 66605962812754 T^{14} - 307262543778544 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 20 T + 224 T^{2} + 1780 T^{3} + 6070 T^{4} + 5760 T^{5} + 875320 T^{6} + 20333540 T^{7} + 305719495 T^{8} + 3198483920 T^{9} + 22059896192 T^{10} + 116979470440 T^{11} + 660067500148 T^{12} + 7893347694940 T^{13} + 139701887310360 T^{14} + 1828379420794980 T^{15} + 18018691485968485 T^{16} + 144441974242803420 T^{17} + 871879478703956760 T^{18} + 3891728254165522660 T^{19} + 25709682596232111988 T^{20} + \)\(35\!\cdots\!60\)\( T^{21} + \)\(53\!\cdots\!32\)\( T^{22} + \)\(61\!\cdots\!80\)\( T^{23} + \)\(46\!\cdots\!95\)\( T^{24} + \)\(24\!\cdots\!60\)\( T^{25} + \)\(82\!\cdots\!20\)\( T^{26} + \)\(43\!\cdots\!40\)\( T^{27} + \)\(35\!\cdots\!70\)\( T^{28} + \)\(83\!\cdots\!20\)\( T^{29} + \)\(82\!\cdots\!44\)\( T^{30} + \)\(58\!\cdots\!80\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 13294 T^{4} + 47458321 T^{8} \))(\( ( 1 + 10871 T^{4} + 47458321 T^{8} )^{2} \))(\( 1 + 13 T - 19 T^{2} - 951 T^{3} - 6196 T^{4} - 78933 T^{5} - 130891 T^{6} + 7433231 T^{7} + 47458321 T^{8} \))(\( 1 + 6 T - 47 T^{2} - 780 T^{3} - 779 T^{4} - 64740 T^{5} - 323783 T^{6} + 3430722 T^{7} + 47458321 T^{8} \))(\( 1 - 11 T - 4 T^{2} - 264 T^{3} + 5188 T^{4} + 21759 T^{5} + 513329 T^{6} - 7685372 T^{7} + 27478708 T^{8} - 637885876 T^{9} + 3536323481 T^{10} + 12441513333 T^{11} + 246213769348 T^{12} - 1039906729752 T^{13} - 1307761493476 T^{14} - 298496560885897 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 70 T + 2432 T^{2} - 56730 T^{3} + 1010935 T^{4} - 14768130 T^{5} + 184247072 T^{6} - 2010015520 T^{7} + 19419043189 T^{8} - 166831288160 T^{9} + 1269278079008 T^{10} - 8444224748310 T^{11} + 47977277740135 T^{12} - 223461775677390 T^{13} + 795118988033408 T^{14} - 1899523569273890 T^{15} + 2252292232139041 T^{16} \))(\( 1 + 30 T + 535 T^{2} + 5780 T^{3} + 43882 T^{4} + 205010 T^{5} + 590510 T^{6} + 2468290 T^{7} + 123659138 T^{8} + 2415976440 T^{9} + 27231081235 T^{10} + 158815212750 T^{11} + 405339323209 T^{12} + 3244201552980 T^{13} + 126271405899100 T^{14} + 2149314211865960 T^{15} + 23236430083058540 T^{16} + 178393079584874680 T^{17} + 869883715238899900 T^{18} + 1854992273373775260 T^{19} + 19236723714775472089 T^{20} + \)\(62\!\cdots\!50\)\( T^{21} + \)\(89\!\cdots\!15\)\( T^{22} + \)\(65\!\cdots\!80\)\( T^{23} + \)\(27\!\cdots\!58\)\( T^{24} + \)\(46\!\cdots\!70\)\( T^{25} + \)\(91\!\cdots\!90\)\( T^{26} + \)\(26\!\cdots\!70\)\( T^{27} + \)\(46\!\cdots\!02\)\( T^{28} + \)\(51\!\cdots\!40\)\( T^{29} + \)\(39\!\cdots\!15\)\( T^{30} + \)\(18\!\cdots\!10\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( 1 + 10 T + 89 T^{2} \))(\( 1 - 18 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( ( 1 + 18 T + 89 T^{2} )^{2} \))(\( ( 1 + 170 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 65 T^{2} + 7921 T^{4} )^{4} \))(\( 1 - 2 T + 35 T^{2} - 632 T^{3} + 8789 T^{4} - 56248 T^{5} + 277235 T^{6} - 1409938 T^{7} + 62742241 T^{8} \))(\( 1 + 5 T - 79 T^{2} - 445 T^{3} + 5276 T^{4} - 39605 T^{5} - 625759 T^{6} + 3524845 T^{7} + 62742241 T^{8} \))(\( 1 - 9 T - 166 T^{2} + 1356 T^{3} + 12786 T^{4} - 119727 T^{5} + 426031 T^{6} + 2824668 T^{7} - 98110168 T^{8} + 251395452 T^{9} + 3374591551 T^{10} - 84403823463 T^{11} + 802222293426 T^{12} + 7571984612844 T^{13} - 82498894299526 T^{14} - 398082014059761 T^{15} + 3936588805702081 T^{16} \))(\( 1 + 34 T + 409 T^{2} + 1014 T^{3} - 27449 T^{4} - 344318 T^{5} - 919769 T^{6} + 22713862 T^{7} + 350578752 T^{8} + 2021533718 T^{9} - 7285490249 T^{10} - 242733516142 T^{11} - 1722211773209 T^{12} + 5662236281286 T^{13} + 203265348003049 T^{14} + 1503865386447986 T^{15} + 3936588805702081 T^{16} \))(\( 1 - 70 T + 2119 T^{2} - 35330 T^{3} + 343460 T^{4} - 2032360 T^{5} + 10870620 T^{6} - 64684440 T^{7} - 584089940 T^{8} + 17002587830 T^{9} - 99745085123 T^{10} - 366225143690 T^{11} + 5693869232523 T^{12} - 24776539056740 T^{13} + 308063609313680 T^{14} - 3148141734399680 T^{15} + 20769024208268760 T^{16} - 280184614361571520 T^{17} + 2440171849373659280 T^{18} - 17466691962290941060 T^{19} + \)\(35\!\cdots\!43\)\( T^{20} - \)\(20\!\cdots\!10\)\( T^{21} - \)\(49\!\cdots\!03\)\( T^{22} + \)\(75\!\cdots\!70\)\( T^{23} - \)\(22\!\cdots\!40\)\( T^{24} - \)\(22\!\cdots\!60\)\( T^{25} + \)\(33\!\cdots\!20\)\( T^{26} - \)\(56\!\cdots\!40\)\( T^{27} + \)\(84\!\cdots\!60\)\( T^{28} - \)\(77\!\cdots\!70\)\( T^{29} + \)\(41\!\cdots\!79\)\( T^{30} - \)\(12\!\cdots\!30\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( 1 + 8 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 - 190 T^{2} + 9409 T^{4} \))(\( ( 1 + 6 T + 18 T^{2} + 582 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 2498 T^{4} + 88529281 T^{8} )^{2} \))(\( 1 - 7 T - 73 T^{2} + 835 T^{3} + 1816 T^{4} + 80995 T^{5} - 686857 T^{6} - 6388711 T^{7} + 88529281 T^{8} \))(\( 1 - 18 T + 47 T^{2} + 1740 T^{3} - 27059 T^{4} + 168780 T^{5} + 442223 T^{6} - 16428114 T^{7} + 88529281 T^{8} \))(\( 1 - T + 28 T^{2} - 632 T^{3} + 5996 T^{4} - 78691 T^{5} + 707075 T^{6} - 3586376 T^{7} + 72919024 T^{8} - 347878472 T^{9} + 6652868675 T^{10} - 71819151043 T^{11} + 530821568876 T^{12} - 5427199042424 T^{13} + 23323216138012 T^{14} - 80798284478113 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 60 T + 1990 T^{2} - 47940 T^{3} + 920391 T^{4} - 14754060 T^{5} + 202723580 T^{6} - 2424996600 T^{7} + 25463108381 T^{8} - 235224670200 T^{9} + 1907426164220 T^{10} - 13465632202380 T^{11} + 81481553468871 T^{12} - 411677091920580 T^{13} + 1657614289808710 T^{14} - 4847897068686780 T^{15} + 7837433594376961 T^{16} \))(\( 1 + 10 T + 205 T^{2} + 4320 T^{3} + 44952 T^{4} + 667870 T^{5} + 8897460 T^{6} + 68756910 T^{7} + 834554408 T^{8} + 7477618800 T^{9} + 22708693755 T^{10} + 118605463990 T^{11} - 2491804851081 T^{12} - 84263627884560 T^{13} - 895266623177600 T^{14} - 11056342779615180 T^{15} - 138673183182585960 T^{16} - 1072465249622672460 T^{17} - 8423563657478038400 T^{18} - 76905138052285028880 T^{19} - \)\(22\!\cdots\!61\)\( T^{20} + \)\(10\!\cdots\!30\)\( T^{21} + \)\(18\!\cdots\!95\)\( T^{22} + \)\(60\!\cdots\!00\)\( T^{23} + \)\(65\!\cdots\!88\)\( T^{24} + \)\(52\!\cdots\!70\)\( T^{25} + \)\(65\!\cdots\!40\)\( T^{26} + \)\(47\!\cdots\!10\)\( T^{27} + \)\(31\!\cdots\!32\)\( T^{28} + \)\(29\!\cdots\!40\)\( T^{29} + \)\(13\!\cdots\!45\)\( T^{30} + \)\(63\!\cdots\!30\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))
show more
show less