Properties

Label 6026.2.a
Level 6026
Weight 2
Character orbit a
Rep. character \(\chi_{6026}(1,\cdot)\)
Character field \(\Q\)
Dimension 241
Newform subspaces 13
Sturm bound 1584
Trace bound 5

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Defining parameters

Level: \( N \) = \( 6026 = 2 \cdot 23 \cdot 131 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6026.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(1584\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6026))\).

Total New Old
Modular forms 796 241 555
Cusp forms 789 241 548
Eisenstein series 7 0 7

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(23\)\(131\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(25\)
\(+\)\(+\)\(-\)\(-\)\(36\)
\(+\)\(-\)\(+\)\(-\)\(34\)
\(+\)\(-\)\(-\)\(+\)\(25\)
\(-\)\(+\)\(+\)\(-\)\(37\)
\(-\)\(+\)\(-\)\(+\)\(23\)
\(-\)\(-\)\(+\)\(+\)\(20\)
\(-\)\(-\)\(-\)\(-\)\(41\)
Plus space\(+\)\(93\)
Minus space\(-\)\(148\)

Trace form

\( 241q + q^{2} + 241q^{4} - 2q^{5} - 4q^{6} + 8q^{7} + q^{8} + 257q^{9} + O(q^{10}) \) \( 241q + q^{2} + 241q^{4} - 2q^{5} - 4q^{6} + 8q^{7} + q^{8} + 257q^{9} + 6q^{10} + 4q^{11} + 18q^{13} + 16q^{15} + 241q^{16} + 18q^{17} - 3q^{18} + 12q^{19} - 2q^{20} + 32q^{21} - 4q^{22} - q^{23} - 4q^{24} + 243q^{25} - 2q^{26} + 8q^{28} + 6q^{29} - 24q^{30} + 16q^{31} + q^{32} - 12q^{33} + 2q^{34} + 4q^{35} + 257q^{36} + 22q^{37} - 12q^{38} + 36q^{39} + 6q^{40} - 14q^{41} + 4q^{43} + 4q^{44} + 58q^{45} + 3q^{46} + 249q^{49} + 39q^{50} + 16q^{51} + 18q^{52} + 18q^{53} - 40q^{54} - 40q^{55} + 40q^{57} + 2q^{58} - 16q^{59} + 16q^{60} + 38q^{61} + 8q^{62} + 28q^{63} + 241q^{64} + 20q^{65} + 16q^{66} - 4q^{67} + 18q^{68} - 4q^{69} - 48q^{70} + 8q^{71} - 3q^{72} + 18q^{73} - 2q^{74} - 12q^{75} + 12q^{76} - 8q^{77} + 8q^{78} + 40q^{79} - 2q^{80} + 297q^{81} + 2q^{82} + 4q^{83} + 32q^{84} + 4q^{85} - 12q^{86} + 24q^{87} - 4q^{88} - 34q^{89} - 2q^{90} + 56q^{91} - q^{92} + 56q^{93} - 16q^{94} + 32q^{95} - 4q^{96} + 10q^{97} + 25q^{98} - 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6026))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 23 131
6026.2.a.a \(1\) \(48.118\) \(\Q\) None \(-1\) \(0\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}-q^{8}-3q^{9}+2q^{11}-2q^{13}+\cdots\)
6026.2.a.b \(1\) \(48.118\) \(\Q\) None \(-1\) \(0\) \(3\) \(-2\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}+3q^{5}-2q^{7}-q^{8}-3q^{9}+\cdots\)
6026.2.a.c \(1\) \(48.118\) \(\Q\) None \(1\) \(-2\) \(3\) \(2\) \(-\) \(+\) \(-\) \(q+q^{2}-2q^{3}+q^{4}+3q^{5}-2q^{6}+2q^{7}+\cdots\)
6026.2.a.d \(1\) \(48.118\) \(\Q\) None \(1\) \(2\) \(-1\) \(2\) \(-\) \(+\) \(-\) \(q+q^{2}+2q^{3}+q^{4}-q^{5}+2q^{6}+2q^{7}+\cdots\)
6026.2.a.e \(2\) \(48.118\) \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(0\) \(4\) \(-\) \(+\) \(+\) \(q+q^{2}+(1+\beta )q^{3}+q^{4}+\beta q^{5}+(1+\beta )q^{6}+\cdots\)
6026.2.a.f \(20\) \(48.118\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(20\) \(-5\) \(-6\) \(-12\) \(-\) \(-\) \(+\) \(q+q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{3}q^{5}-\beta _{1}q^{6}+\cdots\)
6026.2.a.g \(21\) \(48.118\) None \(21\) \(0\) \(-13\) \(-18\) \(-\) \(+\) \(-\)
6026.2.a.h \(24\) \(48.118\) None \(-24\) \(-1\) \(-1\) \(-7\) \(+\) \(+\) \(+\)
6026.2.a.i \(25\) \(48.118\) None \(-25\) \(-4\) \(-3\) \(-11\) \(+\) \(-\) \(-\)
6026.2.a.j \(33\) \(48.118\) None \(-33\) \(3\) \(-4\) \(11\) \(+\) \(-\) \(+\)
6026.2.a.k \(35\) \(48.118\) None \(35\) \(-3\) \(10\) \(14\) \(-\) \(+\) \(+\)
6026.2.a.l \(36\) \(48.118\) None \(-36\) \(4\) \(1\) \(13\) \(+\) \(+\) \(-\)
6026.2.a.m \(41\) \(48.118\) None \(41\) \(4\) \(9\) \(12\) \(-\) \(-\) \(-\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6026))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6026)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(131))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(262))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3013))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{20} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 - 2 T + 4 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 + 5 T + 43 T^{2} + 170 T^{3} + 870 T^{4} + 2913 T^{5} + 11363 T^{6} + 33431 T^{7} + 108834 T^{8} + 287590 T^{9} + 817133 T^{10} + 1966679 T^{11} + 5003300 T^{12} + 11070057 T^{13} + 25617412 T^{14} + 52423138 T^{15} + 111471658 T^{16} + 211770510 T^{17} + 416452564 T^{18} + 735797283 T^{19} + 1343338098 T^{20} + 2207391849 T^{21} + 3748073076 T^{22} + 5717803770 T^{23} + 9029204298 T^{24} + 12738822534 T^{25} + 18675093348 T^{26} + 24210214659 T^{27} + 32826651300 T^{28} + 38710142757 T^{29} + 48250886517 T^{30} + 50945705730 T^{31} + 57838849794 T^{32} + 53299812213 T^{33} + 54348876747 T^{34} + 41798366091 T^{35} + 37450647270 T^{36} + 21953827710 T^{37} + 16659081027 T^{38} + 5811307335 T^{39} + 3486784401 T^{40} \))
$5$ (\( 1 + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 + 7 T^{2} + 25 T^{4} \))(\( 1 + 6 T + 75 T^{2} + 381 T^{3} + 2690 T^{4} + 11870 T^{5} + 61882 T^{6} + 241840 T^{7} + 1031036 T^{8} + 3622669 T^{9} + 13300861 T^{10} + 42508071 T^{11} + 138513654 T^{12} + 406195814 T^{13} + 1197041252 T^{14} + 3241483406 T^{15} + 8745063885 T^{16} + 21956418274 T^{17} + 54653761616 T^{18} + 127482517206 T^{19} + 294154111133 T^{20} + 637412586030 T^{21} + 1366344040400 T^{22} + 2744552284250 T^{23} + 5465664928125 T^{24} + 10129635643750 T^{25} + 18703769562500 T^{26} + 31734047968750 T^{27} + 54106896093750 T^{28} + 83023576171875 T^{29} + 129891220703125 T^{30} + 176888134765625 T^{31} + 251717773437500 T^{32} + 295214843750000 T^{33} + 377697753906250 T^{34} + 362243652343750 T^{35} + 410461425781250 T^{36} + 290679931640625 T^{37} + 286102294921875 T^{38} + 114440917968750 T^{39} + 95367431640625 T^{40} \))
$7$ (\( 1 + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( 1 + 12 T + 147 T^{2} + 1170 T^{3} + 8891 T^{4} + 55000 T^{5} + 323177 T^{6} + 1668589 T^{7} + 8214171 T^{8} + 36824221 T^{9} + 158146527 T^{10} + 630789928 T^{11} + 2420077368 T^{12} + 8725743805 T^{13} + 30356087150 T^{14} + 99978175894 T^{15} + 318410565688 T^{16} + 964332838117 T^{17} + 2828061993736 T^{18} + 7906019233338 T^{19} + 21415945449380 T^{20} + 55342134633366 T^{21} + 138575037693064 T^{22} + 330766163474131 T^{23} + 764503768216888 T^{24} + 1680333202250458 T^{25} + 3571363297110350 T^{26} + 7186025230401115 T^{27} + 13951264431123768 T^{28} + 25454648854070296 T^{29} + 44672479592810223 T^{30} + 72813516973442203 T^{31} + 113694699929125371 T^{32} + 161667936986005723 T^{33} + 219186098014121273 T^{34} + 261115883046865000 T^{35} + 295473985694322491 T^{36} + 272177701365032190 T^{37} + 239376798892836003 T^{38} + 136786742224477716 T^{39} + 79792266297612001 T^{40} \))
$11$ (\( 1 - 2 T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( 1 + 19 T^{2} + 121 T^{4} \))(\( 1 + 3 T + 131 T^{2} + 321 T^{3} + 8135 T^{4} + 15880 T^{5} + 321429 T^{6} + 481417 T^{7} + 9179096 T^{8} + 10029665 T^{9} + 204842879 T^{10} + 154673252 T^{11} + 3783896214 T^{12} + 1927511248 T^{13} + 60456516650 T^{14} + 22207375194 T^{15} + 859913224225 T^{16} + 265628161228 T^{17} + 11027077922255 T^{18} + 3195229405816 T^{19} + 127695328671322 T^{20} + 35147523463976 T^{21} + 1334276428592855 T^{22} + 353551082594468 T^{23} + 12589989515878225 T^{24} + 3576519982368894 T^{25} + 107102407092990650 T^{26} + 37561741294199408 T^{27} + 811111758253176534 T^{28} + 364711437412861132 T^{29} + 5313096729114266279 T^{30} + 2861580476818675315 T^{31} + 28807935359046224216 T^{32} + 16619820512194830227 T^{33} + \)\(12\!\cdots\!89\)\( T^{34} + 66334700930320537880 T^{35} + \)\(37\!\cdots\!35\)\( T^{36} + \)\(16\!\cdots\!91\)\( T^{37} + \)\(72\!\cdots\!11\)\( T^{38} + \)\(18\!\cdots\!73\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 2 T + 24 T^{2} + 26 T^{3} + 169 T^{4} \))(\( 1 + 13 T + 250 T^{2} + 2515 T^{3} + 28423 T^{4} + 235369 T^{5} + 2004624 T^{6} + 14191771 T^{7} + 99780516 T^{8} + 619217464 T^{9} + 3759589966 T^{10} + 20807541823 T^{11} + 111940867944 T^{12} + 559237823792 T^{13} + 2708537158574 T^{14} + 12317082402164 T^{15} + 54249221358905 T^{16} + 225804772898792 T^{17} + 910064764244018 T^{18} + 3478200391442993 T^{19} + 12871278826699238 T^{20} + 45216605088758909 T^{21} + 153800945157239042 T^{22} + 496093086058646024 T^{23} + 1549412011231685705 T^{24} + 4573246476346678052 T^{25} + 13073591533839410366 T^{26} + 35091344093255316464 T^{27} + 91313604917324907624 T^{28} + \)\(22\!\cdots\!79\)\( T^{29} + \)\(51\!\cdots\!34\)\( T^{30} + \)\(11\!\cdots\!68\)\( T^{31} + \)\(23\!\cdots\!96\)\( T^{32} + \)\(42\!\cdots\!63\)\( T^{33} + \)\(78\!\cdots\!36\)\( T^{34} + \)\(12\!\cdots\!33\)\( T^{35} + \)\(18\!\cdots\!43\)\( T^{36} + \)\(21\!\cdots\!95\)\( T^{37} + \)\(28\!\cdots\!50\)\( T^{38} + \)\(19\!\cdots\!01\)\( T^{39} + \)\(19\!\cdots\!01\)\( T^{40} \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 - T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 + 31 T^{2} + 289 T^{4} \))(\( 1 + 14 T + 257 T^{2} + 2549 T^{3} + 27893 T^{4} + 218211 T^{5} + 1801951 T^{6} + 11737184 T^{7} + 79817016 T^{8} + 446660195 T^{9} + 2619243701 T^{10} + 12853016346 T^{11} + 67082057368 T^{12} + 293396794586 T^{13} + 1403055057140 T^{14} + 5571417931583 T^{15} + 25265396829208 T^{16} + 93765328911505 T^{17} + 420595892488941 T^{18} + 1525578603027681 T^{19} + 7009077428022231 T^{20} + 25934836251470577 T^{21} + 121552212929303949 T^{22} + 460669060942224065 T^{23} + 2110191208572281368 T^{24} + 7910616750083643631 T^{25} + 33866338252515692660 T^{26} + \)\(12\!\cdots\!78\)\( T^{27} + \)\(46\!\cdots\!88\)\( T^{28} + \)\(15\!\cdots\!62\)\( T^{29} + \)\(52\!\cdots\!49\)\( T^{30} + \)\(15\!\cdots\!35\)\( T^{31} + \)\(46\!\cdots\!76\)\( T^{32} + \)\(11\!\cdots\!08\)\( T^{33} + \)\(30\!\cdots\!79\)\( T^{34} + \)\(62\!\cdots\!23\)\( T^{35} + \)\(13\!\cdots\!33\)\( T^{36} + \)\(21\!\cdots\!73\)\( T^{37} + \)\(36\!\cdots\!13\)\( T^{38} + \)\(33\!\cdots\!42\)\( T^{39} + \)\(40\!\cdots\!01\)\( T^{40} \))
$19$ (\( 1 - 2 T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 - 10 T + 60 T^{2} - 190 T^{3} + 361 T^{4} \))(\( 1 + 21 T + 451 T^{2} + 6362 T^{3} + 84400 T^{4} + 923179 T^{5} + 9436044 T^{6} + 85628621 T^{7} + 729772014 T^{8} + 5706874732 T^{9} + 42168273342 T^{10} + 290911980160 T^{11} + 1906043983671 T^{12} + 11778286141925 T^{13} + 69391734738037 T^{14} + 387962218052004 T^{15} + 2073638466666164 T^{16} + 10557286158037905 T^{17} + 51473695046135409 T^{18} + 239535359041619767 T^{19} + 1068395447119897552 T^{20} + 4551171821790775573 T^{21} + 18582003911654882649 T^{22} + 72412425757981990395 T^{23} + \)\(27\!\cdots\!44\)\( T^{24} + \)\(96\!\cdots\!96\)\( T^{25} + \)\(32\!\cdots\!97\)\( T^{26} + \)\(10\!\cdots\!75\)\( T^{27} + \)\(32\!\cdots\!11\)\( T^{28} + \)\(93\!\cdots\!40\)\( T^{29} + \)\(25\!\cdots\!42\)\( T^{30} + \)\(66\!\cdots\!08\)\( T^{31} + \)\(16\!\cdots\!54\)\( T^{32} + \)\(36\!\cdots\!39\)\( T^{33} + \)\(75\!\cdots\!24\)\( T^{34} + \)\(14\!\cdots\!21\)\( T^{35} + \)\(24\!\cdots\!00\)\( T^{36} + \)\(34\!\cdots\!18\)\( T^{37} + \)\(46\!\cdots\!91\)\( T^{38} + \)\(41\!\cdots\!59\)\( T^{39} + \)\(37\!\cdots\!01\)\( T^{40} \))
$23$ (\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{20} \))
$29$ (\( 1 - 2 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 12 T + 82 T^{2} + 348 T^{3} + 841 T^{4} \))(\( 1 + 27 T + 671 T^{2} + 11017 T^{3} + 165315 T^{4} + 2010743 T^{5} + 22663351 T^{6} + 221757410 T^{7} + 2035049503 T^{8} + 16732130981 T^{9} + 130229187383 T^{10} + 923100618802 T^{11} + 6243404043424 T^{12} + 38808061014968 T^{13} + 232367815425918 T^{14} + 1288269677620003 T^{15} + 7002351817865045 T^{16} + 35777926622125189 T^{17} + 185989094839062606 T^{18} + 939301419801050704 T^{19} + 5068588391553922034 T^{20} + 27239741174230470416 T^{21} + \)\(15\!\cdots\!46\)\( T^{22} + \)\(87\!\cdots\!21\)\( T^{23} + \)\(49\!\cdots\!45\)\( T^{24} + \)\(26\!\cdots\!47\)\( T^{25} + \)\(13\!\cdots\!78\)\( T^{26} + \)\(66\!\cdots\!12\)\( T^{27} + \)\(31\!\cdots\!64\)\( T^{28} + \)\(13\!\cdots\!38\)\( T^{29} + \)\(54\!\cdots\!83\)\( T^{30} + \)\(20\!\cdots\!49\)\( T^{31} + \)\(72\!\cdots\!23\)\( T^{32} + \)\(22\!\cdots\!90\)\( T^{33} + \)\(67\!\cdots\!31\)\( T^{34} + \)\(17\!\cdots\!07\)\( T^{35} + \)\(41\!\cdots\!15\)\( T^{36} + \)\(79\!\cdots\!53\)\( T^{37} + \)\(14\!\cdots\!31\)\( T^{38} + \)\(16\!\cdots\!63\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \))
$31$ (\( 1 + 8 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( 1 + 27 T + 713 T^{2} + 12137 T^{3} + 195684 T^{4} + 2532878 T^{5} + 31207842 T^{6} + 333285818 T^{7} + 3414953124 T^{8} + 31478051657 T^{9} + 280310736443 T^{10} + 2293712746101 T^{11} + 18232824796335 T^{12} + 134951431179116 T^{13} + 974527353169652 T^{14} + 6609003430534677 T^{15} + 43870230254309681 T^{16} + 274973038486893379 T^{17} + 1690597180418362117 T^{18} + 9844637256758697796 T^{19} + 56293976094785761312 T^{20} + \)\(30\!\cdots\!76\)\( T^{21} + \)\(16\!\cdots\!37\)\( T^{22} + \)\(81\!\cdots\!89\)\( T^{23} + \)\(40\!\cdots\!01\)\( T^{24} + \)\(18\!\cdots\!27\)\( T^{25} + \)\(86\!\cdots\!12\)\( T^{26} + \)\(37\!\cdots\!76\)\( T^{27} + \)\(15\!\cdots\!35\)\( T^{28} + \)\(60\!\cdots\!71\)\( T^{29} + \)\(22\!\cdots\!43\)\( T^{30} + \)\(79\!\cdots\!67\)\( T^{31} + \)\(26\!\cdots\!64\)\( T^{32} + \)\(81\!\cdots\!38\)\( T^{33} + \)\(23\!\cdots\!82\)\( T^{34} + \)\(59\!\cdots\!78\)\( T^{35} + \)\(14\!\cdots\!04\)\( T^{36} + \)\(27\!\cdots\!07\)\( T^{37} + \)\(49\!\cdots\!33\)\( T^{38} + \)\(58\!\cdots\!17\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \))
$37$ (\( 1 - 4 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 8 T + 78 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( 1 + 19 T + 647 T^{2} + 9963 T^{3} + 195436 T^{4} + 2548820 T^{5} + 37291257 T^{6} + 423559055 T^{7} + 5089644511 T^{8} + 51308513336 T^{9} + 531205842420 T^{10} + 4816778803176 T^{11} + 44152688761797 T^{12} + 363561122402284 T^{13} + 3000111142403417 T^{14} + 22582633158474953 T^{15} + 169535397164083678 T^{16} + 1171649398494154643 T^{17} + 8054932557801109683 T^{18} + 51227599492139006671 T^{19} + \)\(32\!\cdots\!38\)\( T^{20} + \)\(18\!\cdots\!27\)\( T^{21} + \)\(11\!\cdots\!27\)\( T^{22} + \)\(59\!\cdots\!79\)\( T^{23} + \)\(31\!\cdots\!58\)\( T^{24} + \)\(15\!\cdots\!21\)\( T^{25} + \)\(76\!\cdots\!53\)\( T^{26} + \)\(34\!\cdots\!72\)\( T^{27} + \)\(15\!\cdots\!37\)\( T^{28} + \)\(62\!\cdots\!52\)\( T^{29} + \)\(25\!\cdots\!80\)\( T^{30} + \)\(91\!\cdots\!68\)\( T^{31} + \)\(33\!\cdots\!91\)\( T^{32} + \)\(10\!\cdots\!35\)\( T^{33} + \)\(33\!\cdots\!73\)\( T^{34} + \)\(84\!\cdots\!60\)\( T^{35} + \)\(24\!\cdots\!76\)\( T^{36} + \)\(45\!\cdots\!71\)\( T^{37} + \)\(10\!\cdots\!63\)\( T^{38} + \)\(11\!\cdots\!87\)\( T^{39} + \)\(23\!\cdots\!01\)\( T^{40} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 - 3 T + 41 T^{2} \))(\( 1 + 9 T + 41 T^{2} \))(\( 1 - 7 T + 41 T^{2} \))(\( 1 - 6 T + 79 T^{2} - 246 T^{3} + 1681 T^{4} \))(\( 1 + 17 T + 614 T^{2} + 8450 T^{3} + 172614 T^{4} + 2008685 T^{5} + 30118031 T^{6} + 305182851 T^{7} + 3712005492 T^{8} + 33479163477 T^{9} + 348309302875 T^{10} + 2846798532185 T^{11} + 26203538538417 T^{12} + 197127228039002 T^{13} + 1645086735274430 T^{14} + 11547517225333148 T^{15} + 89050943966169610 T^{16} + 589691058124247137 T^{17} + 4264569325585298427 T^{18} + 26826917650906222371 T^{19} + \)\(18\!\cdots\!49\)\( T^{20} + \)\(10\!\cdots\!11\)\( T^{21} + \)\(71\!\cdots\!87\)\( T^{22} + \)\(40\!\cdots\!77\)\( T^{23} + \)\(25\!\cdots\!10\)\( T^{24} + \)\(13\!\cdots\!48\)\( T^{25} + \)\(78\!\cdots\!30\)\( T^{26} + \)\(38\!\cdots\!62\)\( T^{27} + \)\(20\!\cdots\!57\)\( T^{28} + \)\(93\!\cdots\!85\)\( T^{29} + \)\(46\!\cdots\!75\)\( T^{30} + \)\(18\!\cdots\!57\)\( T^{31} + \)\(83\!\cdots\!52\)\( T^{32} + \)\(28\!\cdots\!71\)\( T^{33} + \)\(11\!\cdots\!91\)\( T^{34} + \)\(31\!\cdots\!85\)\( T^{35} + \)\(11\!\cdots\!74\)\( T^{36} + \)\(22\!\cdots\!50\)\( T^{37} + \)\(65\!\cdots\!94\)\( T^{38} + \)\(74\!\cdots\!37\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \))
$43$ (\( 1 - 2 T + 43 T^{2} \))(\( 1 - 3 T + 43 T^{2} \))(\( 1 + 5 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 + 8 T + 75 T^{2} + 344 T^{3} + 1849 T^{4} \))(\( 1 + 27 T + 848 T^{2} + 16241 T^{3} + 309305 T^{4} + 4686811 T^{5} + 68377421 T^{6} + 869072333 T^{7} + 10577445269 T^{8} + 116877285139 T^{9} + 1238610560284 T^{10} + 12181830342590 T^{11} + 115255509326132 T^{12} + 1025169834050054 T^{13} + 8797100750992930 T^{14} + 71536734877722021 T^{15} + 562459634866818242 T^{16} + 4211267863165516099 T^{17} + 30531924168395012207 T^{18} + \)\(21\!\cdots\!41\)\( T^{19} + \)\(14\!\cdots\!06\)\( T^{20} + \)\(90\!\cdots\!63\)\( T^{21} + \)\(56\!\cdots\!43\)\( T^{22} + \)\(33\!\cdots\!93\)\( T^{23} + \)\(19\!\cdots\!42\)\( T^{24} + \)\(10\!\cdots\!03\)\( T^{25} + \)\(55\!\cdots\!70\)\( T^{26} + \)\(27\!\cdots\!78\)\( T^{27} + \)\(13\!\cdots\!32\)\( T^{28} + \)\(61\!\cdots\!70\)\( T^{29} + \)\(26\!\cdots\!16\)\( T^{30} + \)\(10\!\cdots\!73\)\( T^{31} + \)\(42\!\cdots\!69\)\( T^{32} + \)\(14\!\cdots\!19\)\( T^{33} + \)\(50\!\cdots\!29\)\( T^{34} + \)\(14\!\cdots\!77\)\( T^{35} + \)\(42\!\cdots\!05\)\( T^{36} + \)\(95\!\cdots\!63\)\( T^{37} + \)\(21\!\cdots\!52\)\( T^{38} + \)\(29\!\cdots\!89\)\( T^{39} + \)\(46\!\cdots\!01\)\( T^{40} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 + 10 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 6 T + 28 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 + 28 T + 993 T^{2} + 20685 T^{3} + 439032 T^{4} + 7360562 T^{5} + 118848621 T^{6} + 1678339164 T^{7} + 22476337393 T^{8} + 275221030231 T^{9} + 3189017382204 T^{10} + 34529640553953 T^{11} + 354602683045808 T^{12} + 3441770986824135 T^{13} + 31792828545292042 T^{14} + 279268232138503491 T^{15} + 2342138326084974085 T^{16} + 18740110031998932636 T^{17} + \)\(14\!\cdots\!31\)\( T^{18} + \)\(10\!\cdots\!93\)\( T^{19} + \)\(73\!\cdots\!26\)\( T^{20} + \)\(49\!\cdots\!71\)\( T^{21} + \)\(31\!\cdots\!79\)\( T^{22} + \)\(19\!\cdots\!28\)\( T^{23} + \)\(11\!\cdots\!85\)\( T^{24} + \)\(64\!\cdots\!37\)\( T^{25} + \)\(34\!\cdots\!18\)\( T^{26} + \)\(17\!\cdots\!05\)\( T^{27} + \)\(84\!\cdots\!88\)\( T^{28} + \)\(38\!\cdots\!51\)\( T^{29} + \)\(16\!\cdots\!96\)\( T^{30} + \)\(68\!\cdots\!93\)\( T^{31} + \)\(26\!\cdots\!13\)\( T^{32} + \)\(91\!\cdots\!28\)\( T^{33} + \)\(30\!\cdots\!49\)\( T^{34} + \)\(88\!\cdots\!66\)\( T^{35} + \)\(24\!\cdots\!72\)\( T^{36} + \)\(55\!\cdots\!95\)\( T^{37} + \)\(12\!\cdots\!77\)\( T^{38} + \)\(16\!\cdots\!24\)\( T^{39} + \)\(27\!\cdots\!01\)\( T^{40} \))
$53$ (\( 1 + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 94 T^{2} + 2809 T^{4} \))(\( 1 + 47 T + 1681 T^{2} + 43968 T^{3} + 981807 T^{4} + 18761686 T^{5} + 320983201 T^{6} + 4945844776 T^{7} + 69979455888 T^{8} + 913949652973 T^{9} + 11132689880723 T^{10} + 126995499372888 T^{11} + 1365445272517663 T^{12} + 13882487655437830 T^{13} + 134049218942923013 T^{14} + 1232441923026567905 T^{15} + 10822039562950719883 T^{16} + 90925914919540226034 T^{17} + \)\(73\!\cdots\!17\)\( T^{18} + \)\(56\!\cdots\!89\)\( T^{19} + \)\(42\!\cdots\!86\)\( T^{20} + \)\(30\!\cdots\!17\)\( T^{21} + \)\(20\!\cdots\!53\)\( T^{22} + \)\(13\!\cdots\!18\)\( T^{23} + \)\(85\!\cdots\!23\)\( T^{24} + \)\(51\!\cdots\!65\)\( T^{25} + \)\(29\!\cdots\!77\)\( T^{26} + \)\(16\!\cdots\!10\)\( T^{27} + \)\(85\!\cdots\!43\)\( T^{28} + \)\(41\!\cdots\!04\)\( T^{29} + \)\(19\!\cdots\!27\)\( T^{30} + \)\(84\!\cdots\!81\)\( T^{31} + \)\(34\!\cdots\!08\)\( T^{32} + \)\(12\!\cdots\!48\)\( T^{33} + \)\(44\!\cdots\!69\)\( T^{34} + \)\(13\!\cdots\!02\)\( T^{35} + \)\(38\!\cdots\!47\)\( T^{36} + \)\(90\!\cdots\!84\)\( T^{37} + \)\(18\!\cdots\!09\)\( T^{38} + \)\(27\!\cdots\!99\)\( T^{39} + \)\(30\!\cdots\!01\)\( T^{40} \))
$59$ (\( 1 + 4 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 - 10 T + 59 T^{2} \))(\( 1 - 18 T + 196 T^{2} - 1062 T^{3} + 3481 T^{4} \))(\( 1 + 16 T + 827 T^{2} + 11877 T^{3} + 335576 T^{4} + 4383449 T^{5} + 88976111 T^{6} + 1067077138 T^{7} + 17310676371 T^{8} + 191886219097 T^{9} + 2629904793335 T^{10} + 27070380562048 T^{11} + 324088376542843 T^{12} + 3106822957430109 T^{13} + 33208019826739010 T^{14} + 296892323633288999 T^{15} + 2876225056598832830 T^{16} + 23979334991434569270 T^{17} + \)\(21\!\cdots\!94\)\( T^{18} + \)\(16\!\cdots\!05\)\( T^{19} + \)\(13\!\cdots\!58\)\( T^{20} + \)\(97\!\cdots\!95\)\( T^{21} + \)\(74\!\cdots\!14\)\( T^{22} + \)\(49\!\cdots\!30\)\( T^{23} + \)\(34\!\cdots\!30\)\( T^{24} + \)\(21\!\cdots\!01\)\( T^{25} + \)\(14\!\cdots\!10\)\( T^{26} + \)\(77\!\cdots\!71\)\( T^{27} + \)\(47\!\cdots\!03\)\( T^{28} + \)\(23\!\cdots\!72\)\( T^{29} + \)\(13\!\cdots\!35\)\( T^{30} + \)\(57\!\cdots\!23\)\( T^{31} + \)\(30\!\cdots\!51\)\( T^{32} + \)\(11\!\cdots\!02\)\( T^{33} + \)\(55\!\cdots\!71\)\( T^{34} + \)\(16\!\cdots\!51\)\( T^{35} + \)\(72\!\cdots\!16\)\( T^{36} + \)\(15\!\cdots\!63\)\( T^{37} + \)\(62\!\cdots\!67\)\( T^{38} + \)\(70\!\cdots\!24\)\( T^{39} + \)\(26\!\cdots\!01\)\( T^{40} \))
$61$ (\( 1 - 12 T + 61 T^{2} \))(\( 1 - 13 T + 61 T^{2} \))(\( 1 + 7 T + 61 T^{2} \))(\( 1 + 3 T + 61 T^{2} \))(\( 1 - 4 T + 99 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( 1 + 9 T + 796 T^{2} + 5942 T^{3} + 302091 T^{4} + 1870209 T^{5} + 73537618 T^{6} + 375202370 T^{7} + 13029424208 T^{8} + 54247970616 T^{9} + 1806973241077 T^{10} + 6080157655422 T^{11} + 205721234015499 T^{12} + 557440679325247 T^{13} + 19860791343813298 T^{14} + 43771868722018452 T^{15} + 1661098684851801070 T^{16} + 3068191863509903970 T^{17} + \)\(12\!\cdots\!13\)\( T^{18} + \)\(19\!\cdots\!64\)\( T^{19} + \)\(79\!\cdots\!13\)\( T^{20} + \)\(12\!\cdots\!04\)\( T^{21} + \)\(45\!\cdots\!73\)\( T^{22} + \)\(69\!\cdots\!70\)\( T^{23} + \)\(22\!\cdots\!70\)\( T^{24} + \)\(36\!\cdots\!52\)\( T^{25} + \)\(10\!\cdots\!78\)\( T^{26} + \)\(17\!\cdots\!87\)\( T^{27} + \)\(39\!\cdots\!19\)\( T^{28} + \)\(71\!\cdots\!02\)\( T^{29} + \)\(12\!\cdots\!77\)\( T^{30} + \)\(23\!\cdots\!76\)\( T^{31} + \)\(34\!\cdots\!68\)\( T^{32} + \)\(60\!\cdots\!70\)\( T^{33} + \)\(72\!\cdots\!38\)\( T^{34} + \)\(11\!\cdots\!09\)\( T^{35} + \)\(11\!\cdots\!51\)\( T^{36} + \)\(13\!\cdots\!82\)\( T^{37} + \)\(10\!\cdots\!76\)\( T^{38} + \)\(75\!\cdots\!69\)\( T^{39} + \)\(50\!\cdots\!01\)\( T^{40} \))
$67$ (\( 1 - 2 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 6 T + 67 T^{2} \))(\( 1 + 10 T + 67 T^{2} \))(\( 1 + 2 T - 12 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 + 8 T + 799 T^{2} + 5806 T^{3} + 301830 T^{4} + 1937254 T^{5} + 71371796 T^{6} + 387292940 T^{7} + 11775592638 T^{8} + 49738934215 T^{9} + 1428010560154 T^{10} + 3805498703355 T^{11} + 130231227462876 T^{12} + 55241662470200 T^{13} + 8941893613771951 T^{14} - 29357008106891146 T^{15} + 454372620941213478 T^{16} - 4793715363180557428 T^{17} + 17319841290126240121 T^{18} - \)\(45\!\cdots\!06\)\( T^{19} + \)\(75\!\cdots\!62\)\( T^{20} - \)\(30\!\cdots\!02\)\( T^{21} + \)\(77\!\cdots\!69\)\( T^{22} - \)\(14\!\cdots\!64\)\( T^{23} + \)\(91\!\cdots\!38\)\( T^{24} - \)\(39\!\cdots\!22\)\( T^{25} + \)\(80\!\cdots\!19\)\( T^{26} + \)\(33\!\cdots\!00\)\( T^{27} + \)\(52\!\cdots\!16\)\( T^{28} + \)\(10\!\cdots\!85\)\( T^{29} + \)\(26\!\cdots\!46\)\( T^{30} + \)\(60\!\cdots\!45\)\( T^{31} + \)\(96\!\cdots\!18\)\( T^{32} + \)\(21\!\cdots\!80\)\( T^{33} + \)\(26\!\cdots\!84\)\( T^{34} + \)\(47\!\cdots\!22\)\( T^{35} + \)\(49\!\cdots\!30\)\( T^{36} + \)\(64\!\cdots\!62\)\( T^{37} + \)\(59\!\cdots\!91\)\( T^{38} + \)\(39\!\cdots\!24\)\( T^{39} + \)\(33\!\cdots\!01\)\( T^{40} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 - 2 T + 71 T^{2} \))(\( 1 + 2 T + 71 T^{2} \))(\( 1 - 18 T + 220 T^{2} - 1278 T^{3} + 5041 T^{4} \))(\( 1 + 30 T + 992 T^{2} + 20245 T^{3} + 423314 T^{4} + 6961967 T^{5} + 114875051 T^{6} + 1624601233 T^{7} + 22857241197 T^{8} + 287457100846 T^{9} + 3585730364979 T^{10} + 40953613841521 T^{11} + 463695638168566 T^{12} + 4879948332713678 T^{13} + 50953110655371853 T^{14} + 499437946117170912 T^{15} + 4863790579276901412 T^{16} + 44755863471035471874 T^{17} + \)\(40\!\cdots\!20\)\( T^{18} + \)\(35\!\cdots\!28\)\( T^{19} + \)\(30\!\cdots\!68\)\( T^{20} + \)\(25\!\cdots\!88\)\( T^{21} + \)\(20\!\cdots\!20\)\( T^{22} + \)\(16\!\cdots\!14\)\( T^{23} + \)\(12\!\cdots\!72\)\( T^{24} + \)\(90\!\cdots\!12\)\( T^{25} + \)\(65\!\cdots\!13\)\( T^{26} + \)\(44\!\cdots\!98\)\( T^{27} + \)\(29\!\cdots\!26\)\( T^{28} + \)\(18\!\cdots\!51\)\( T^{29} + \)\(11\!\cdots\!79\)\( T^{30} + \)\(66\!\cdots\!66\)\( T^{31} + \)\(37\!\cdots\!77\)\( T^{32} + \)\(18\!\cdots\!63\)\( T^{33} + \)\(95\!\cdots\!31\)\( T^{34} + \)\(40\!\cdots\!17\)\( T^{35} + \)\(17\!\cdots\!94\)\( T^{36} + \)\(59\!\cdots\!95\)\( T^{37} + \)\(20\!\cdots\!12\)\( T^{38} + \)\(44\!\cdots\!30\)\( T^{39} + \)\(10\!\cdots\!01\)\( T^{40} \))
$73$ (\( 1 - 6 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 4 T + 73 T^{2} \))(\( 1 + 16 T + 73 T^{2} \))(\( 1 + 2 T + 120 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( 1 + 26 T + 1197 T^{2} + 24592 T^{3} + 657890 T^{4} + 11341711 T^{5} + 227311894 T^{6} + 3409169339 T^{7} + 56312125437 T^{8} + 752189598694 T^{9} + 10746088856380 T^{10} + 129888755624308 T^{11} + 1650660275351722 T^{12} + 18250686856715718 T^{13} + 209964989934409142 T^{14} + 2139004944961456839 T^{15} + 22530176407805990018 T^{16} + \)\(21\!\cdots\!22\)\( T^{17} + \)\(20\!\cdots\!55\)\( T^{18} + \)\(18\!\cdots\!31\)\( T^{19} + \)\(16\!\cdots\!00\)\( T^{20} + \)\(13\!\cdots\!63\)\( T^{21} + \)\(10\!\cdots\!95\)\( T^{22} + \)\(82\!\cdots\!74\)\( T^{23} + \)\(63\!\cdots\!38\)\( T^{24} + \)\(44\!\cdots\!27\)\( T^{25} + \)\(31\!\cdots\!38\)\( T^{26} + \)\(20\!\cdots\!46\)\( T^{27} + \)\(13\!\cdots\!82\)\( T^{28} + \)\(76\!\cdots\!04\)\( T^{29} + \)\(46\!\cdots\!20\)\( T^{30} + \)\(23\!\cdots\!38\)\( T^{31} + \)\(12\!\cdots\!77\)\( T^{32} + \)\(56\!\cdots\!87\)\( T^{33} + \)\(27\!\cdots\!46\)\( T^{34} + \)\(10\!\cdots\!27\)\( T^{35} + \)\(42\!\cdots\!90\)\( T^{36} + \)\(11\!\cdots\!76\)\( T^{37} + \)\(41\!\cdots\!93\)\( T^{38} + \)\(65\!\cdots\!62\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \))
$79$ (\( 1 + 4 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 + 20 T + 246 T^{2} + 1580 T^{3} + 6241 T^{4} \))(\( 1 + 35 T + 1358 T^{2} + 32138 T^{3} + 763329 T^{4} + 14201535 T^{5} + 259877581 T^{6} + 4078954210 T^{7} + 62660095384 T^{8} + 863796466425 T^{9} + 11643015327162 T^{10} + 144609897419381 T^{11} + 1756305094950960 T^{12} + 19992624445757313 T^{13} + 222610909075945230 T^{14} + 2349868391539924272 T^{15} + 24271098688322182345 T^{16} + \)\(23\!\cdots\!68\)\( T^{17} + \)\(23\!\cdots\!49\)\( T^{18} + \)\(21\!\cdots\!15\)\( T^{19} + \)\(19\!\cdots\!54\)\( T^{20} + \)\(16\!\cdots\!85\)\( T^{21} + \)\(14\!\cdots\!09\)\( T^{22} + \)\(11\!\cdots\!52\)\( T^{23} + \)\(94\!\cdots\!45\)\( T^{24} + \)\(72\!\cdots\!28\)\( T^{25} + \)\(54\!\cdots\!30\)\( T^{26} + \)\(38\!\cdots\!67\)\( T^{27} + \)\(26\!\cdots\!60\)\( T^{28} + \)\(17\!\cdots\!39\)\( T^{29} + \)\(11\!\cdots\!62\)\( T^{30} + \)\(64\!\cdots\!75\)\( T^{31} + \)\(37\!\cdots\!44\)\( T^{32} + \)\(19\!\cdots\!90\)\( T^{33} + \)\(95\!\cdots\!61\)\( T^{34} + \)\(41\!\cdots\!65\)\( T^{35} + \)\(17\!\cdots\!09\)\( T^{36} + \)\(58\!\cdots\!42\)\( T^{37} + \)\(19\!\cdots\!38\)\( T^{38} + \)\(39\!\cdots\!65\)\( T^{39} + \)\(89\!\cdots\!01\)\( T^{40} \))
$83$ (\( 1 + 6 T + 83 T^{2} \))(\( 1 + 14 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 6 T + 148 T^{2} - 498 T^{3} + 6889 T^{4} \))(\( 1 - 2 T + 1054 T^{2} - 2189 T^{3} + 549076 T^{4} - 1221114 T^{5} + 188559139 T^{6} - 454561506 T^{7} + 47980488365 T^{8} - 125031361088 T^{9} + 9632903576114 T^{10} - 26783048587634 T^{11} + 1585644441482620 T^{12} - 4612318693332748 T^{13} + 219438930094848336 T^{14} - 651863119546389719 T^{15} + 25963241327252864126 T^{16} - 76632059974815401333 T^{17} + \)\(26\!\cdots\!16\)\( T^{18} - \)\(75\!\cdots\!51\)\( T^{19} + \)\(23\!\cdots\!76\)\( T^{20} - \)\(62\!\cdots\!33\)\( T^{21} + \)\(18\!\cdots\!24\)\( T^{22} - \)\(43\!\cdots\!71\)\( T^{23} + \)\(12\!\cdots\!46\)\( T^{24} - \)\(25\!\cdots\!17\)\( T^{25} + \)\(71\!\cdots\!84\)\( T^{26} - \)\(12\!\cdots\!96\)\( T^{27} + \)\(35\!\cdots\!20\)\( T^{28} - \)\(50\!\cdots\!02\)\( T^{29} + \)\(14\!\cdots\!86\)\( T^{30} - \)\(16\!\cdots\!96\)\( T^{31} + \)\(51\!\cdots\!65\)\( T^{32} - \)\(40\!\cdots\!78\)\( T^{33} + \)\(13\!\cdots\!31\)\( T^{34} - \)\(74\!\cdots\!98\)\( T^{35} + \)\(27\!\cdots\!56\)\( T^{36} - \)\(92\!\cdots\!47\)\( T^{37} + \)\(36\!\cdots\!86\)\( T^{38} - \)\(58\!\cdots\!94\)\( T^{39} + \)\(24\!\cdots\!01\)\( T^{40} \))
$89$ (\( 1 - 18 T + 89 T^{2} \))(\( 1 - 4 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 16 T + 89 T^{2} \))(\( 1 - 12 T + 202 T^{2} - 1068 T^{3} + 7921 T^{4} \))(\( 1 + 25 T + 1122 T^{2} + 25128 T^{3} + 647461 T^{4} + 12646106 T^{5} + 249354041 T^{6} + 4264554695 T^{7} + 71079322173 T^{8} + 1081231776971 T^{9} + 15902633104810 T^{10} + 218513406337704 T^{11} + 2899837771736837 T^{12} + 36411084279396553 T^{13} + 441808943107334142 T^{14} + 5109424918721731119 T^{15} + 57157923736166926274 T^{16} + \)\(61\!\cdots\!00\)\( T^{17} + \)\(63\!\cdots\!73\)\( T^{18} + \)\(63\!\cdots\!03\)\( T^{19} + \)\(60\!\cdots\!40\)\( T^{20} + \)\(56\!\cdots\!67\)\( T^{21} + \)\(50\!\cdots\!33\)\( T^{22} + \)\(43\!\cdots\!00\)\( T^{23} + \)\(35\!\cdots\!34\)\( T^{24} + \)\(28\!\cdots\!31\)\( T^{25} + \)\(21\!\cdots\!62\)\( T^{26} + \)\(16\!\cdots\!37\)\( T^{27} + \)\(11\!\cdots\!97\)\( T^{28} + \)\(76\!\cdots\!36\)\( T^{29} + \)\(49\!\cdots\!10\)\( T^{30} + \)\(30\!\cdots\!19\)\( T^{31} + \)\(17\!\cdots\!33\)\( T^{32} + \)\(93\!\cdots\!55\)\( T^{33} + \)\(48\!\cdots\!81\)\( T^{34} + \)\(22\!\cdots\!94\)\( T^{35} + \)\(10\!\cdots\!21\)\( T^{36} + \)\(34\!\cdots\!12\)\( T^{37} + \)\(13\!\cdots\!82\)\( T^{38} + \)\(27\!\cdots\!25\)\( T^{39} + \)\(97\!\cdots\!01\)\( T^{40} \))
$97$ (\( 1 - 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 8 T + 198 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 - 2 T + 1027 T^{2} - 1383 T^{3} + 524261 T^{4} - 464294 T^{5} + 177455904 T^{6} - 103803620 T^{7} + 44747258506 T^{8} - 18322151133 T^{9} + 8955705191589 T^{10} - 2941979482999 T^{11} + 1482863476002801 T^{12} - 467189237697526 T^{13} + 209629148472931131 T^{14} - 71617841081070944 T^{15} + 25982714719775057886 T^{16} - 9836112063168080376 T^{17} + \)\(28\!\cdots\!04\)\( T^{18} - \)\(11\!\cdots\!71\)\( T^{19} + \)\(29\!\cdots\!04\)\( T^{20} - \)\(11\!\cdots\!87\)\( T^{21} + \)\(27\!\cdots\!36\)\( T^{22} - \)\(89\!\cdots\!48\)\( T^{23} + \)\(23\!\cdots\!66\)\( T^{24} - \)\(61\!\cdots\!08\)\( T^{25} + \)\(17\!\cdots\!99\)\( T^{26} - \)\(37\!\cdots\!38\)\( T^{27} + \)\(11\!\cdots\!61\)\( T^{28} - \)\(22\!\cdots\!83\)\( T^{29} + \)\(66\!\cdots\!61\)\( T^{30} - \)\(13\!\cdots\!49\)\( T^{31} + \)\(31\!\cdots\!46\)\( T^{32} - \)\(69\!\cdots\!40\)\( T^{33} + \)\(11\!\cdots\!76\)\( T^{34} - \)\(29\!\cdots\!42\)\( T^{35} + \)\(32\!\cdots\!81\)\( T^{36} - \)\(82\!\cdots\!71\)\( T^{37} + \)\(59\!\cdots\!03\)\( T^{38} - \)\(11\!\cdots\!66\)\( T^{39} + \)\(54\!\cdots\!01\)\( T^{40} \))
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