# Properties

 Degree 80 Conductor $2^{40} \cdot 2003^{40}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 40

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 40·2-s − 3-s + 820·4-s − 23·5-s + 40·6-s + 12·7-s − 1.14e4·8-s − 45·9-s + 920·10-s − 28·11-s − 820·12-s − 12·13-s − 480·14-s + 23·15-s + 1.23e5·16-s − 10·17-s + 1.80e3·18-s + 3·19-s − 1.88e4·20-s − 12·21-s + 1.12e3·22-s − 10·23-s + 1.14e4·24-s + 186·25-s + 480·26-s + 44·27-s + 9.84e3·28-s + ⋯
 L(s)  = 1 − 28.2·2-s − 0.577·3-s + 410·4-s − 10.2·5-s + 16.3·6-s + 4.53·7-s − 4.05e3·8-s − 15·9-s + 290.·10-s − 8.44·11-s − 236.·12-s − 3.32·13-s − 128.·14-s + 5.93·15-s + 3.08e4·16-s − 2.42·17-s + 424.·18-s + 0.688·19-s − 4.21e3·20-s − 2.61·21-s + 238.·22-s − 2.08·23-s + 2.34e3·24-s + 37.1·25-s + 94.1·26-s + 8.46·27-s + 1.85e3·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 2003^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{40} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 2003^{40}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{40} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$80$$ $$N$$ = $$2^{40} \cdot 2003^{40}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{4006} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 40 Selberg data = $(80,\ 2^{40} \cdot 2003^{40} ,\ ( \ : [1/2]^{40} ),\ 1 )$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;2003\}$,$$F_p(T)$$ is a polynomial of degree 80. If $p \in \{2,\;2003\}$, then $F_p(T)$ is a polynomial of degree at most 79.
$p$$F_p(T)$
bad2 $$( 1 + T )^{40}$$
2003 $$( 1 + T )^{40}$$
good3 $$1 + T + 46 T^{2} + 47 T^{3} + 1082 T^{4} + 1130 T^{5} + 5765 p T^{6} + 18431 T^{7} + 23426 p^{2} T^{8} + 76249 p T^{9} + 2087362 T^{10} + 2300384 T^{11} + 5821766 p T^{12} + 2167723 p^{2} T^{13} + 126965396 T^{14} + 143483722 T^{15} + 90945982 p^{2} T^{16} + 934250849 T^{17} + 4754493628 T^{18} + 5472857155 T^{19} + 8401130398 p T^{20} + 29218090465 T^{21} + 123207210766 T^{22} + 143665851344 T^{23} + 560305095490 T^{24} + 656297263099 T^{25} + 795956939768 p T^{26} + 2805568154396 T^{27} + 1066204596707 p^{2} T^{28} + 3763243375660 p T^{29} + 36551125482250 T^{30} + 42970371084356 T^{31} + 44179254433229 p T^{32} + 155283392365532 T^{33} + 459111829461746 T^{34} + 534335002735139 T^{35} + 507799127235806 p T^{36} + 1754488563183947 T^{37} + 4851352150931531 T^{38} + 1834894581039757 p T^{39} + 14844680425034002 T^{40} + 1834894581039757 p^{2} T^{41} + 4851352150931531 p^{2} T^{42} + 1754488563183947 p^{3} T^{43} + 507799127235806 p^{5} T^{44} + 534335002735139 p^{5} T^{45} + 459111829461746 p^{6} T^{46} + 155283392365532 p^{7} T^{47} + 44179254433229 p^{9} T^{48} + 42970371084356 p^{9} T^{49} + 36551125482250 p^{10} T^{50} + 3763243375660 p^{12} T^{51} + 1066204596707 p^{14} T^{52} + 2805568154396 p^{13} T^{53} + 795956939768 p^{15} T^{54} + 656297263099 p^{15} T^{55} + 560305095490 p^{16} T^{56} + 143665851344 p^{17} T^{57} + 123207210766 p^{18} T^{58} + 29218090465 p^{19} T^{59} + 8401130398 p^{21} T^{60} + 5472857155 p^{21} T^{61} + 4754493628 p^{22} T^{62} + 934250849 p^{23} T^{63} + 90945982 p^{26} T^{64} + 143483722 p^{25} T^{65} + 126965396 p^{26} T^{66} + 2167723 p^{29} T^{67} + 5821766 p^{29} T^{68} + 2300384 p^{29} T^{69} + 2087362 p^{30} T^{70} + 76249 p^{32} T^{71} + 23426 p^{34} T^{72} + 18431 p^{33} T^{73} + 5765 p^{35} T^{74} + 1130 p^{35} T^{75} + 1082 p^{36} T^{76} + 47 p^{37} T^{77} + 46 p^{38} T^{78} + p^{39} T^{79} + p^{40} T^{80}$$
5 $$1 + 23 T + 343 T^{2} + 769 p T^{3} + 7158 p T^{4} + 288058 T^{5} + 2066591 T^{6} + 13460647 T^{7} + 80749148 T^{8} + 450732669 T^{9} + 2360461622 T^{10} + 11672517208 T^{11} + 10958962343 p T^{12} + 245255044942 T^{13} + 1050535251506 T^{14} + 4319947329422 T^{15} + 684004812053 p^{2} T^{16} + 65311674268752 T^{17} + 241183626444207 T^{18} + 862693283841958 T^{19} + 598750488534571 p T^{20} + 10093664528570219 T^{21} + 33106600315417239 T^{22} + 105758633662547818 T^{23} + 329387814469741976 T^{24} + 1001155345706476659 T^{25} + 2972158711539757579 T^{26} + 8625000882600375244 T^{27} + 24483439791943169241 T^{28} + 68028805048524351908 T^{29} +$$$$18\!\cdots\!27$$$$T^{30} +$$$$49\!\cdots\!93$$$$T^{31} +$$$$12\!\cdots\!78$$$$T^{32} +$$$$13\!\cdots\!62$$$$p^{2} T^{33} +$$$$83\!\cdots\!21$$$$T^{34} +$$$$20\!\cdots\!92$$$$T^{35} +$$$$49\!\cdots\!73$$$$T^{36} +$$$$11\!\cdots\!14$$$$T^{37} +$$$$55\!\cdots\!29$$$$p T^{38} +$$$$63\!\cdots\!59$$$$T^{39} +$$$$28\!\cdots\!28$$$$p T^{40} +$$$$63\!\cdots\!59$$$$p T^{41} +$$$$55\!\cdots\!29$$$$p^{3} T^{42} +$$$$11\!\cdots\!14$$$$p^{3} T^{43} +$$$$49\!\cdots\!73$$$$p^{4} T^{44} +$$$$20\!\cdots\!92$$$$p^{5} T^{45} +$$$$83\!\cdots\!21$$$$p^{6} T^{46} +$$$$13\!\cdots\!62$$$$p^{9} T^{47} +$$$$12\!\cdots\!78$$$$p^{8} T^{48} +$$$$49\!\cdots\!93$$$$p^{9} T^{49} +$$$$18\!\cdots\!27$$$$p^{10} T^{50} + 68028805048524351908 p^{11} T^{51} + 24483439791943169241 p^{12} T^{52} + 8625000882600375244 p^{13} T^{53} + 2972158711539757579 p^{14} T^{54} + 1001155345706476659 p^{15} T^{55} + 329387814469741976 p^{16} T^{56} + 105758633662547818 p^{17} T^{57} + 33106600315417239 p^{18} T^{58} + 10093664528570219 p^{19} T^{59} + 598750488534571 p^{21} T^{60} + 862693283841958 p^{21} T^{61} + 241183626444207 p^{22} T^{62} + 65311674268752 p^{23} T^{63} + 684004812053 p^{26} T^{64} + 4319947329422 p^{25} T^{65} + 1050535251506 p^{26} T^{66} + 245255044942 p^{27} T^{67} + 10958962343 p^{29} T^{68} + 11672517208 p^{29} T^{69} + 2360461622 p^{30} T^{70} + 450732669 p^{31} T^{71} + 80749148 p^{32} T^{72} + 13460647 p^{33} T^{73} + 2066591 p^{34} T^{74} + 288058 p^{35} T^{75} + 7158 p^{37} T^{76} + 769 p^{38} T^{77} + 343 p^{38} T^{78} + 23 p^{39} T^{79} + p^{40} T^{80}$$
7 $$1 - 12 T + 198 T^{2} - 256 p T^{3} + 17825 T^{4} - 132936 T^{5} + 145227 p T^{6} - 6558107 T^{7} + 42190664 T^{8} - 242642683 T^{9} + 1374607047 T^{10} - 7192095535 T^{11} + 36865423004 T^{12} - 178023918745 T^{13} + 840404738402 T^{14} - 3785759293712 T^{15} + 16663330688881 T^{16} - 70591146291312 T^{17} + 292312344607016 T^{18} - 1171917808283150 T^{19} + 656598177128827 p T^{20} - 17525652476560698 T^{21} + 65435439788070920 T^{22} - 238257884672766363 T^{23} + 850283829478853263 T^{24} - 2965852777419896598 T^{25} + 10148726458665298900 T^{26} - 33999501913835040083 T^{27} +$$$$11\!\cdots\!35$$$$T^{28} -$$$$36\!\cdots\!86$$$$T^{29} +$$$$16\!\cdots\!45$$$$p T^{30} -$$$$35\!\cdots\!97$$$$T^{31} +$$$$10\!\cdots\!56$$$$T^{32} -$$$$32\!\cdots\!17$$$$T^{33} +$$$$96\!\cdots\!79$$$$T^{34} -$$$$27\!\cdots\!83$$$$T^{35} +$$$$11\!\cdots\!97$$$$p T^{36} -$$$$22\!\cdots\!66$$$$T^{37} +$$$$61\!\cdots\!70$$$$T^{38} -$$$$23\!\cdots\!97$$$$p T^{39} +$$$$44\!\cdots\!78$$$$T^{40} -$$$$23\!\cdots\!97$$$$p^{2} T^{41} +$$$$61\!\cdots\!70$$$$p^{2} T^{42} -$$$$22\!\cdots\!66$$$$p^{3} T^{43} +$$$$11\!\cdots\!97$$$$p^{5} T^{44} -$$$$27\!\cdots\!83$$$$p^{5} T^{45} +$$$$96\!\cdots\!79$$$$p^{6} T^{46} -$$$$32\!\cdots\!17$$$$p^{7} T^{47} +$$$$10\!\cdots\!56$$$$p^{8} T^{48} -$$$$35\!\cdots\!97$$$$p^{9} T^{49} +$$$$16\!\cdots\!45$$$$p^{11} T^{50} -$$$$36\!\cdots\!86$$$$p^{11} T^{51} +$$$$11\!\cdots\!35$$$$p^{12} T^{52} - 33999501913835040083 p^{13} T^{53} + 10148726458665298900 p^{14} T^{54} - 2965852777419896598 p^{15} T^{55} + 850283829478853263 p^{16} T^{56} - 238257884672766363 p^{17} T^{57} + 65435439788070920 p^{18} T^{58} - 17525652476560698 p^{19} T^{59} + 656598177128827 p^{21} T^{60} - 1171917808283150 p^{21} T^{61} + 292312344607016 p^{22} T^{62} - 70591146291312 p^{23} T^{63} + 16663330688881 p^{24} T^{64} - 3785759293712 p^{25} T^{65} + 840404738402 p^{26} T^{66} - 178023918745 p^{27} T^{67} + 36865423004 p^{28} T^{68} - 7192095535 p^{29} T^{69} + 1374607047 p^{30} T^{70} - 242642683 p^{31} T^{71} + 42190664 p^{32} T^{72} - 6558107 p^{33} T^{73} + 145227 p^{35} T^{74} - 132936 p^{35} T^{75} + 17825 p^{36} T^{76} - 256 p^{38} T^{77} + 198 p^{38} T^{78} - 12 p^{39} T^{79} + p^{40} T^{80}$$
11 $$1 + 28 T + 601 T^{2} + 9436 T^{3} + 127340 T^{4} + 1476368 T^{5} + 15435172 T^{6} + 146213575 T^{7} + 1281807414 T^{8} + 10448328953 T^{9} + 80093482690 T^{10} + 52692539303 p T^{11} + 3987487198606 T^{12} + 26157411621650 T^{13} + 164377945132072 T^{14} + 991964216572646 T^{15} + 5767432900112537 T^{16} + 32369842312264857 T^{17} + 175804493797962963 T^{18} + 925385887397453551 T^{19} + 4729646706853328167 T^{20} + 23501187920303513328 T^{21} +$$$$11\!\cdots\!55$$$$T^{22} + 402746841112384235 p^{3} T^{23} +$$$$24\!\cdots\!64$$$$T^{24} +$$$$11\!\cdots\!36$$$$T^{25} +$$$$48\!\cdots\!17$$$$T^{26} +$$$$20\!\cdots\!72$$$$T^{27} +$$$$87\!\cdots\!53$$$$T^{28} +$$$$36\!\cdots\!77$$$$T^{29} +$$$$14\!\cdots\!93$$$$T^{30} +$$$$57\!\cdots\!89$$$$T^{31} +$$$$22\!\cdots\!90$$$$T^{32} +$$$$84\!\cdots\!06$$$$T^{33} +$$$$31\!\cdots\!99$$$$T^{34} +$$$$10\!\cdots\!94$$$$p T^{35} +$$$$38\!\cdots\!94$$$$p T^{36} +$$$$13\!\cdots\!01$$$$p T^{37} +$$$$51\!\cdots\!70$$$$T^{38} +$$$$17\!\cdots\!17$$$$T^{39} +$$$$58\!\cdots\!84$$$$T^{40} +$$$$17\!\cdots\!17$$$$p T^{41} +$$$$51\!\cdots\!70$$$$p^{2} T^{42} +$$$$13\!\cdots\!01$$$$p^{4} T^{43} +$$$$38\!\cdots\!94$$$$p^{5} T^{44} +$$$$10\!\cdots\!94$$$$p^{6} T^{45} +$$$$31\!\cdots\!99$$$$p^{6} T^{46} +$$$$84\!\cdots\!06$$$$p^{7} T^{47} +$$$$22\!\cdots\!90$$$$p^{8} T^{48} +$$$$57\!\cdots\!89$$$$p^{9} T^{49} +$$$$14\!\cdots\!93$$$$p^{10} T^{50} +$$$$36\!\cdots\!77$$$$p^{11} T^{51} +$$$$87\!\cdots\!53$$$$p^{12} T^{52} +$$$$20\!\cdots\!72$$$$p^{13} T^{53} +$$$$48\!\cdots\!17$$$$p^{14} T^{54} +$$$$11\!\cdots\!36$$$$p^{15} T^{55} +$$$$24\!\cdots\!64$$$$p^{16} T^{56} + 402746841112384235 p^{20} T^{57} +$$$$11\!\cdots\!55$$$$p^{18} T^{58} + 23501187920303513328 p^{19} T^{59} + 4729646706853328167 p^{20} T^{60} + 925385887397453551 p^{21} T^{61} + 175804493797962963 p^{22} T^{62} + 32369842312264857 p^{23} T^{63} + 5767432900112537 p^{24} T^{64} + 991964216572646 p^{25} T^{65} + 164377945132072 p^{26} T^{66} + 26157411621650 p^{27} T^{67} + 3987487198606 p^{28} T^{68} + 52692539303 p^{30} T^{69} + 80093482690 p^{30} T^{70} + 10448328953 p^{31} T^{71} + 1281807414 p^{32} T^{72} + 146213575 p^{33} T^{73} + 15435172 p^{34} T^{74} + 1476368 p^{35} T^{75} + 127340 p^{36} T^{76} + 9436 p^{37} T^{77} + 601 p^{38} T^{78} + 28 p^{39} T^{79} + p^{40} T^{80}$$
13 $$1 + 12 T + 333 T^{2} + 3410 T^{3} + 53382 T^{4} + 479859 T^{5} + 425938 p T^{6} + 44590639 T^{7} + 420336083 T^{8} + 3079103530 T^{9} + 1923403740 p T^{10} + 168612140111 T^{11} + 1217782791812 T^{12} + 7632088300976 T^{13} + 50070411047375 T^{14} + 293944398910248 T^{15} + 1778082671256139 T^{16} + 9842534139804177 T^{17} + 55511716732133921 T^{18} + 291370757452937924 T^{19} + 1545510913134207297 T^{20} + 7729215978422432587 T^{21} + 38825319708376351733 T^{22} +$$$$18\!\cdots\!46$$$$T^{23} +$$$$88\!\cdots\!75$$$$T^{24} +$$$$31\!\cdots\!23$$$$p T^{25} +$$$$18\!\cdots\!15$$$$T^{26} +$$$$63\!\cdots\!07$$$$p T^{27} +$$$$36\!\cdots\!22$$$$T^{28} +$$$$92\!\cdots\!96$$$$p^{2} T^{29} +$$$$65\!\cdots\!18$$$$T^{30} +$$$$27\!\cdots\!25$$$$T^{31} +$$$$85\!\cdots\!76$$$$p T^{32} +$$$$44\!\cdots\!41$$$$T^{33} +$$$$17\!\cdots\!98$$$$T^{34} +$$$$69\!\cdots\!39$$$$T^{35} +$$$$26\!\cdots\!85$$$$T^{36} +$$$$77\!\cdots\!15$$$$p T^{37} +$$$$37\!\cdots\!61$$$$T^{38} +$$$$13\!\cdots\!07$$$$T^{39} +$$$$50\!\cdots\!60$$$$T^{40} +$$$$13\!\cdots\!07$$$$p T^{41} +$$$$37\!\cdots\!61$$$$p^{2} T^{42} +$$$$77\!\cdots\!15$$$$p^{4} T^{43} +$$$$26\!\cdots\!85$$$$p^{4} T^{44} +$$$$69\!\cdots\!39$$$$p^{5} T^{45} +$$$$17\!\cdots\!98$$$$p^{6} T^{46} +$$$$44\!\cdots\!41$$$$p^{7} T^{47} +$$$$85\!\cdots\!76$$$$p^{9} T^{48} +$$$$27\!\cdots\!25$$$$p^{9} T^{49} +$$$$65\!\cdots\!18$$$$p^{10} T^{50} +$$$$92\!\cdots\!96$$$$p^{13} T^{51} +$$$$36\!\cdots\!22$$$$p^{12} T^{52} +$$$$63\!\cdots\!07$$$$p^{14} T^{53} +$$$$18\!\cdots\!15$$$$p^{14} T^{54} +$$$$31\!\cdots\!23$$$$p^{16} T^{55} +$$$$88\!\cdots\!75$$$$p^{16} T^{56} +$$$$18\!\cdots\!46$$$$p^{17} T^{57} + 38825319708376351733 p^{18} T^{58} + 7729215978422432587 p^{19} T^{59} + 1545510913134207297 p^{20} T^{60} + 291370757452937924 p^{21} T^{61} + 55511716732133921 p^{22} T^{62} + 9842534139804177 p^{23} T^{63} + 1778082671256139 p^{24} T^{64} + 293944398910248 p^{25} T^{65} + 50070411047375 p^{26} T^{66} + 7632088300976 p^{27} T^{67} + 1217782791812 p^{28} T^{68} + 168612140111 p^{29} T^{69} + 1923403740 p^{31} T^{70} + 3079103530 p^{31} T^{71} + 420336083 p^{32} T^{72} + 44590639 p^{33} T^{73} + 425938 p^{35} T^{74} + 479859 p^{35} T^{75} + 53382 p^{36} T^{76} + 3410 p^{37} T^{77} + 333 p^{38} T^{78} + 12 p^{39} T^{79} + p^{40} T^{80}$$
17 $$1 + 10 T + 386 T^{2} + 3481 T^{3} + 73282 T^{4} + 603767 T^{5} + 9146366 T^{6} + 69509831 T^{7} + 845429123 T^{8} + 5970385496 T^{9} + 61777764909 T^{10} + 23987206719 p T^{11} + 3719171882102 T^{12} + 23056980595243 T^{13} + 189820826510301 T^{14} + 1109774201720156 T^{15} + 8389469756419921 T^{16} + 46422303926728398 T^{17} + 326467814150797869 T^{18} + 1715445484663845977 T^{19} + 11340070494707988281 T^{20} + 56765411454926886583 T^{21} +$$$$35\!\cdots\!88$$$$T^{22} +$$$$17\!\cdots\!21$$$$T^{23} +$$$$10\!\cdots\!48$$$$T^{24} +$$$$46\!\cdots\!96$$$$T^{25} +$$$$26\!\cdots\!82$$$$T^{26} +$$$$69\!\cdots\!79$$$$p T^{27} +$$$$65\!\cdots\!52$$$$T^{28} +$$$$28\!\cdots\!67$$$$T^{29} +$$$$15\!\cdots\!39$$$$T^{30} +$$$$62\!\cdots\!48$$$$T^{31} +$$$$32\!\cdots\!08$$$$T^{32} +$$$$13\!\cdots\!66$$$$T^{33} +$$$$66\!\cdots\!45$$$$T^{34} +$$$$25\!\cdots\!52$$$$T^{35} +$$$$12\!\cdots\!07$$$$T^{36} +$$$$48\!\cdots\!84$$$$T^{37} +$$$$23\!\cdots\!47$$$$T^{38} +$$$$51\!\cdots\!22$$$$p T^{39} +$$$$41\!\cdots\!46$$$$T^{40} +$$$$51\!\cdots\!22$$$$p^{2} T^{41} +$$$$23\!\cdots\!47$$$$p^{2} T^{42} +$$$$48\!\cdots\!84$$$$p^{3} T^{43} +$$$$12\!\cdots\!07$$$$p^{4} T^{44} +$$$$25\!\cdots\!52$$$$p^{5} T^{45} +$$$$66\!\cdots\!45$$$$p^{6} T^{46} +$$$$13\!\cdots\!66$$$$p^{7} T^{47} +$$$$32\!\cdots\!08$$$$p^{8} T^{48} +$$$$62\!\cdots\!48$$$$p^{9} T^{49} +$$$$15\!\cdots\!39$$$$p^{10} T^{50} +$$$$28\!\cdots\!67$$$$p^{11} T^{51} +$$$$65\!\cdots\!52$$$$p^{12} T^{52} +$$$$69\!\cdots\!79$$$$p^{14} T^{53} +$$$$26\!\cdots\!82$$$$p^{14} T^{54} +$$$$46\!\cdots\!96$$$$p^{15} T^{55} +$$$$10\!\cdots\!48$$$$p^{16} T^{56} +$$$$17\!\cdots\!21$$$$p^{17} T^{57} +$$$$35\!\cdots\!88$$$$p^{18} T^{58} + 56765411454926886583 p^{19} T^{59} + 11340070494707988281 p^{20} T^{60} + 1715445484663845977 p^{21} T^{61} + 326467814150797869 p^{22} T^{62} + 46422303926728398 p^{23} T^{63} + 8389469756419921 p^{24} T^{64} + 1109774201720156 p^{25} T^{65} + 189820826510301 p^{26} T^{66} + 23056980595243 p^{27} T^{67} + 3719171882102 p^{28} T^{68} + 23987206719 p^{30} T^{69} + 61777764909 p^{30} T^{70} + 5970385496 p^{31} T^{71} + 845429123 p^{32} T^{72} + 69509831 p^{33} T^{73} + 9146366 p^{34} T^{74} + 603767 p^{35} T^{75} + 73282 p^{36} T^{76} + 3481 p^{37} T^{77} + 386 p^{38} T^{78} + 10 p^{39} T^{79} + p^{40} T^{80}$$
19 $$1 - 3 T + 360 T^{2} - 997 T^{3} + 63829 T^{4} - 165926 T^{5} + 7446343 T^{6} - 18559101 T^{7} + 644082667 T^{8} - 1578663875 T^{9} + 44118416754 T^{10} - 109372566589 T^{11} + 131354783223 p T^{12} - 6439066972381 T^{13} + 120063724379541 T^{14} - 330943789362928 T^{15} + 5021589470238826 T^{16} - 15105230978379177 T^{17} + 185889138670444683 T^{18} - 618904759695745579 T^{19} + 6186677928916903859 T^{20} - 22924372253305775127 T^{21} +$$$$18\!\cdots\!89$$$$T^{22} -$$$$77\!\cdots\!10$$$$T^{23} +$$$$52\!\cdots\!63$$$$T^{24} -$$$$23\!\cdots\!05$$$$T^{25} +$$$$13\!\cdots\!17$$$$T^{26} -$$$$66\!\cdots\!80$$$$T^{27} +$$$$35\!\cdots\!89$$$$T^{28} -$$$$91\!\cdots\!20$$$$p T^{29} +$$$$84\!\cdots\!80$$$$T^{30} -$$$$41\!\cdots\!05$$$$T^{31} +$$$$19\!\cdots\!76$$$$T^{32} -$$$$93\!\cdots\!81$$$$T^{33} +$$$$44\!\cdots\!94$$$$T^{34} -$$$$20\!\cdots\!03$$$$T^{35} +$$$$95\!\cdots\!58$$$$T^{36} -$$$$40\!\cdots\!06$$$$T^{37} +$$$$19\!\cdots\!31$$$$T^{38} -$$$$79\!\cdots\!05$$$$T^{39} +$$$$38\!\cdots\!34$$$$T^{40} -$$$$79\!\cdots\!05$$$$p T^{41} +$$$$19\!\cdots\!31$$$$p^{2} T^{42} -$$$$40\!\cdots\!06$$$$p^{3} T^{43} +$$$$95\!\cdots\!58$$$$p^{4} T^{44} -$$$$20\!\cdots\!03$$$$p^{5} T^{45} +$$$$44\!\cdots\!94$$$$p^{6} T^{46} -$$$$93\!\cdots\!81$$$$p^{7} T^{47} +$$$$19\!\cdots\!76$$$$p^{8} T^{48} -$$$$41\!\cdots\!05$$$$p^{9} T^{49} +$$$$84\!\cdots\!80$$$$p^{10} T^{50} -$$$$91\!\cdots\!20$$$$p^{12} T^{51} +$$$$35\!\cdots\!89$$$$p^{12} T^{52} -$$$$66\!\cdots\!80$$$$p^{13} T^{53} +$$$$13\!\cdots\!17$$$$p^{14} T^{54} -$$$$23\!\cdots\!05$$$$p^{15} T^{55} +$$$$52\!\cdots\!63$$$$p^{16} T^{56} -$$$$77\!\cdots\!10$$$$p^{17} T^{57} +$$$$18\!\cdots\!89$$$$p^{18} T^{58} - 22924372253305775127 p^{19} T^{59} + 6186677928916903859 p^{20} T^{60} - 618904759695745579 p^{21} T^{61} + 185889138670444683 p^{22} T^{62} - 15105230978379177 p^{23} T^{63} + 5021589470238826 p^{24} T^{64} - 330943789362928 p^{25} T^{65} + 120063724379541 p^{26} T^{66} - 6439066972381 p^{27} T^{67} + 131354783223 p^{29} T^{68} - 109372566589 p^{29} T^{69} + 44118416754 p^{30} T^{70} - 1578663875 p^{31} T^{71} + 644082667 p^{32} T^{72} - 18559101 p^{33} T^{73} + 7446343 p^{34} T^{74} - 165926 p^{35} T^{75} + 63829 p^{36} T^{76} - 997 p^{37} T^{77} + 360 p^{38} T^{78} - 3 p^{39} T^{79} + p^{40} T^{80}$$
23 $$1 + 10 T + 492 T^{2} + 4747 T^{3} + 122642 T^{4} + 1128996 T^{5} + 20507794 T^{6} + 179144552 T^{7} + 2575148324 T^{8} + 21302387160 T^{9} + 258097629377 T^{10} + 2021616178723 T^{11} + 21452348707324 T^{12} + 159248120826742 T^{13} + 1517995064172827 T^{14} + 10694743485267225 T^{15} + 93214455652012414 T^{16} + 624293722728788125 T^{17} + 5040392000454394506 T^{18} + 32144143827177478055 T^{19} +$$$$24\!\cdots\!66$$$$T^{20} +$$$$14\!\cdots\!14$$$$T^{21} +$$$$10\!\cdots\!87$$$$T^{22} +$$$$61\!\cdots\!39$$$$T^{23} +$$$$41\!\cdots\!07$$$$T^{24} +$$$$22\!\cdots\!88$$$$T^{25} +$$$$14\!\cdots\!79$$$$T^{26} +$$$$79\!\cdots\!47$$$$T^{27} +$$$$49\!\cdots\!07$$$$T^{28} +$$$$25\!\cdots\!64$$$$T^{29} +$$$$15\!\cdots\!25$$$$T^{30} +$$$$74\!\cdots\!75$$$$T^{31} +$$$$43\!\cdots\!71$$$$T^{32} +$$$$20\!\cdots\!33$$$$T^{33} +$$$$11\!\cdots\!60$$$$T^{34} +$$$$53\!\cdots\!42$$$$T^{35} +$$$$29\!\cdots\!01$$$$T^{36} +$$$$13\!\cdots\!42$$$$T^{37} +$$$$72\!\cdots\!81$$$$T^{38} +$$$$31\!\cdots\!77$$$$T^{39} +$$$$16\!\cdots\!54$$$$T^{40} +$$$$31\!\cdots\!77$$$$p T^{41} +$$$$72\!\cdots\!81$$$$p^{2} T^{42} +$$$$13\!\cdots\!42$$$$p^{3} T^{43} +$$$$29\!\cdots\!01$$$$p^{4} T^{44} +$$$$53\!\cdots\!42$$$$p^{5} T^{45} +$$$$11\!\cdots\!60$$$$p^{6} T^{46} +$$$$20\!\cdots\!33$$$$p^{7} T^{47} +$$$$43\!\cdots\!71$$$$p^{8} T^{48} +$$$$74\!\cdots\!75$$$$p^{9} T^{49} +$$$$15\!\cdots\!25$$$$p^{10} T^{50} +$$$$25\!\cdots\!64$$$$p^{11} T^{51} +$$$$49\!\cdots\!07$$$$p^{12} T^{52} +$$$$79\!\cdots\!47$$$$p^{13} T^{53} +$$$$14\!\cdots\!79$$$$p^{14} T^{54} +$$$$22\!\cdots\!88$$$$p^{15} T^{55} +$$$$41\!\cdots\!07$$$$p^{16} T^{56} +$$$$61\!\cdots\!39$$$$p^{17} T^{57} +$$$$10\!\cdots\!87$$$$p^{18} T^{58} +$$$$14\!\cdots\!14$$$$p^{19} T^{59} +$$$$24\!\cdots\!66$$$$p^{20} T^{60} + 32144143827177478055 p^{21} T^{61} + 5040392000454394506 p^{22} T^{62} + 624293722728788125 p^{23} T^{63} + 93214455652012414 p^{24} T^{64} + 10694743485267225 p^{25} T^{65} + 1517995064172827 p^{26} T^{66} + 159248120826742 p^{27} T^{67} + 21452348707324 p^{28} T^{68} + 2021616178723 p^{29} T^{69} + 258097629377 p^{30} T^{70} + 21302387160 p^{31} T^{71} + 2575148324 p^{32} T^{72} + 179144552 p^{33} T^{73} + 20507794 p^{34} T^{74} + 1128996 p^{35} T^{75} + 122642 p^{36} T^{76} + 4747 p^{37} T^{77} + 492 p^{38} T^{78} + 10 p^{39} T^{79} + p^{40} T^{80}$$
29 $$1 + 37 T + 1223 T^{2} + 27597 T^{3} + 565020 T^{4} + 9625304 T^{5} + 151941161 T^{6} + 2134944511 T^{7} + 28228061059 T^{8} + 343296897080 T^{9} + 3972386302208 T^{10} + 43096422437064 T^{11} + 448520876874339 T^{12} + 4431502570600655 T^{13} + 42264076006470100 T^{14} + 13311035945744930 p T^{15} + 3419692065721119213 T^{16} + 29198584777171782862 T^{17} +$$$$24\!\cdots\!48$$$$T^{18} +$$$$19\!\cdots\!85$$$$T^{19} +$$$$15\!\cdots\!17$$$$T^{20} +$$$$11\!\cdots\!63$$$$T^{21} +$$$$87\!\cdots\!42$$$$T^{22} +$$$$64\!\cdots\!62$$$$T^{23} +$$$$45\!\cdots\!65$$$$T^{24} +$$$$11\!\cdots\!54$$$$p T^{25} +$$$$21\!\cdots\!94$$$$T^{26} +$$$$14\!\cdots\!66$$$$T^{27} +$$$$97\!\cdots\!04$$$$T^{28} +$$$$63\!\cdots\!08$$$$T^{29} +$$$$40\!\cdots\!58$$$$T^{30} +$$$$25\!\cdots\!44$$$$T^{31} +$$$$15\!\cdots\!59$$$$T^{32} +$$$$93\!\cdots\!67$$$$T^{33} +$$$$55\!\cdots\!26$$$$T^{34} +$$$$32\!\cdots\!80$$$$T^{35} +$$$$18\!\cdots\!40$$$$T^{36} +$$$$10\!\cdots\!18$$$$T^{37} +$$$$60\!\cdots\!40$$$$T^{38} +$$$$33\!\cdots\!37$$$$T^{39} +$$$$17\!\cdots\!50$$$$T^{40} +$$$$33\!\cdots\!37$$$$p T^{41} +$$$$60\!\cdots\!40$$$$p^{2} T^{42} +$$$$10\!\cdots\!18$$$$p^{3} T^{43} +$$$$18\!\cdots\!40$$$$p^{4} T^{44} +$$$$32\!\cdots\!80$$$$p^{5} T^{45} +$$$$55\!\cdots\!26$$$$p^{6} T^{46} +$$$$93\!\cdots\!67$$$$p^{7} T^{47} +$$$$15\!\cdots\!59$$$$p^{8} T^{48} +$$$$25\!\cdots\!44$$$$p^{9} T^{49} +$$$$40\!\cdots\!58$$$$p^{10} T^{50} +$$$$63\!\cdots\!08$$$$p^{11} T^{51} +$$$$97\!\cdots\!04$$$$p^{12} T^{52} +$$$$14\!\cdots\!66$$$$p^{13} T^{53} +$$$$21\!\cdots\!94$$$$p^{14} T^{54} +$$$$11\!\cdots\!54$$$$p^{16} T^{55} +$$$$45\!\cdots\!65$$$$p^{16} T^{56} +$$$$64\!\cdots\!62$$$$p^{17} T^{57} +$$$$87\!\cdots\!42$$$$p^{18} T^{58} +$$$$11\!\cdots\!63$$$$p^{19} T^{59} +$$$$15\!\cdots\!17$$$$p^{20} T^{60} +$$$$19\!\cdots\!85$$$$p^{21} T^{61} +$$$$24\!\cdots\!48$$$$p^{22} T^{62} + 29198584777171782862 p^{23} T^{63} + 3419692065721119213 p^{24} T^{64} + 13311035945744930 p^{26} T^{65} + 42264076006470100 p^{26} T^{66} + 4431502570600655 p^{27} T^{67} + 448520876874339 p^{28} T^{68} + 43096422437064 p^{29} T^{69} + 3972386302208 p^{30} T^{70} + 343296897080 p^{31} T^{71} + 28228061059 p^{32} T^{72} + 2134944511 p^{33} T^{73} + 151941161 p^{34} T^{74} + 9625304 p^{35} T^{75} + 565020 p^{36} T^{76} + 27597 p^{37} T^{77} + 1223 p^{38} T^{78} + 37 p^{39} T^{79} + p^{40} T^{80}$$
31 $$1 + 16 T + 717 T^{2} + 10149 T^{3} + 255195 T^{4} + 3252564 T^{5} + 60197582 T^{6} + 700503878 T^{7} + 10597238813 T^{8} + 113831749611 T^{9} + 1485839718367 T^{10} + 14862243565911 T^{11} + 172880638067922 T^{12} + 1621722488923659 T^{13} + 17169626123686037 T^{14} + 151925206301243282 T^{15} + 1485666392239909051 T^{16} + 12460105638343032576 T^{17} +$$$$11\!\cdots\!33$$$$T^{18} +$$$$90\!\cdots\!57$$$$T^{19} +$$$$25\!\cdots\!38$$$$p T^{20} +$$$$59\!\cdots\!61$$$$T^{21} +$$$$48\!\cdots\!43$$$$T^{22} +$$$$35\!\cdots\!33$$$$T^{23} +$$$$27\!\cdots\!96$$$$T^{24} +$$$$19\!\cdots\!98$$$$T^{25} +$$$$14\!\cdots\!94$$$$T^{26} +$$$$95\!\cdots\!15$$$$T^{27} +$$$$68\!\cdots\!91$$$$T^{28} +$$$$44\!\cdots\!07$$$$T^{29} +$$$$30\!\cdots\!39$$$$T^{30} +$$$$19\!\cdots\!62$$$$T^{31} +$$$$12\!\cdots\!08$$$$T^{32} +$$$$76\!\cdots\!44$$$$T^{33} +$$$$48\!\cdots\!64$$$$T^{34} +$$$$28\!\cdots\!70$$$$T^{35} +$$$$17\!\cdots\!66$$$$T^{36} +$$$$10\!\cdots\!07$$$$T^{37} +$$$$59\!\cdots\!76$$$$T^{38} +$$$$33\!\cdots\!22$$$$T^{39} +$$$$19\!\cdots\!30$$$$T^{40} +$$$$33\!\cdots\!22$$$$p T^{41} +$$$$59\!\cdots\!76$$$$p^{2} T^{42} +$$$$10\!\cdots\!07$$$$p^{3} T^{43} +$$$$17\!\cdots\!66$$$$p^{4} T^{44} +$$$$28\!\cdots\!70$$$$p^{5} T^{45} +$$$$48\!\cdots\!64$$$$p^{6} T^{46} +$$$$76\!\cdots\!44$$$$p^{7} T^{47} +$$$$12\!\cdots\!08$$$$p^{8} T^{48} +$$$$19\!\cdots\!62$$$$p^{9} T^{49} +$$$$30\!\cdots\!39$$$$p^{10} T^{50} +$$$$44\!\cdots\!07$$$$p^{11} T^{51} +$$$$68\!\cdots\!91$$$$p^{12} T^{52} +$$$$95\!\cdots\!15$$$$p^{13} T^{53} +$$$$14\!\cdots\!94$$$$p^{14} T^{54} +$$$$19\!\cdots\!98$$$$p^{15} T^{55} +$$$$27\!\cdots\!96$$$$p^{16} T^{56} +$$$$35\!\cdots\!33$$$$p^{17} T^{57} +$$$$48\!\cdots\!43$$$$p^{18} T^{58} +$$$$59\!\cdots\!61$$$$p^{19} T^{59} +$$$$25\!\cdots\!38$$$$p^{21} T^{60} +$$$$90\!\cdots\!57$$$$p^{21} T^{61} +$$$$11\!\cdots\!33$$$$p^{22} T^{62} + 12460105638343032576 p^{23} T^{63} + 1485666392239909051 p^{24} T^{64} + 151925206301243282 p^{25} T^{65} + 17169626123686037 p^{26} T^{66} + 1621722488923659 p^{27} T^{67} + 172880638067922 p^{28} T^{68} + 14862243565911 p^{29} T^{69} + 1485839718367 p^{30} T^{70} + 113831749611 p^{31} T^{71} + 10597238813 p^{32} T^{72} + 700503878 p^{33} T^{73} + 60197582 p^{34} T^{74} + 3252564 p^{35} T^{75} + 255195 p^{36} T^{76} + 10149 p^{37} T^{77} + 717 p^{38} T^{78} + 16 p^{39} T^{79} + p^{40} T^{80}$$
37 $$1 + 17 T + 859 T^{2} + 13099 T^{3} + 360225 T^{4} + 4973009 T^{5} + 98476341 T^{6} + 1241406954 T^{7} + 534582871 p T^{8} + 229527677087 T^{9} + 3120471054065 T^{10} + 33586094405317 T^{11} + 403835340308266 T^{12} + 4060211916171800 T^{13} + 44227061670431192 T^{14} + 418138286906229304 T^{15} + 4197699963992164936 T^{16} + 37548763933917370950 T^{17} +$$$$35\!\cdots\!63$$$$T^{18} +$$$$29\!\cdots\!41$$$$T^{19} +$$$$26\!\cdots\!45$$$$T^{20} +$$$$21\!\cdots\!42$$$$T^{21} +$$$$18\!\cdots\!77$$$$T^{22} +$$$$14\!\cdots\!73$$$$T^{23} +$$$$11\!\cdots\!76$$$$T^{24} +$$$$85\!\cdots\!60$$$$T^{25} +$$$$65\!\cdots\!37$$$$T^{26} +$$$$47\!\cdots\!69$$$$T^{27} +$$$$35\!\cdots\!63$$$$T^{28} +$$$$25\!\cdots\!24$$$$T^{29} +$$$$17\!\cdots\!63$$$$T^{30} +$$$$12\!\cdots\!15$$$$T^{31} +$$$$85\!\cdots\!10$$$$T^{32} +$$$$56\!\cdots\!17$$$$T^{33} +$$$$38\!\cdots\!98$$$$T^{34} +$$$$24\!\cdots\!72$$$$T^{35} +$$$$16\!\cdots\!67$$$$T^{36} +$$$$10\!\cdots\!73$$$$T^{37} +$$$$65\!\cdots\!77$$$$T^{38} +$$$$39\!\cdots\!53$$$$T^{39} +$$$$24\!\cdots\!28$$$$T^{40} +$$$$39\!\cdots\!53$$$$p T^{41} +$$$$65\!\cdots\!77$$$$p^{2} T^{42} +$$$$10\!\cdots\!73$$$$p^{3} T^{43} +$$$$16\!\cdots\!67$$$$p^{4} T^{44} +$$$$24\!\cdots\!72$$$$p^{5} T^{45} +$$$$38\!\cdots\!98$$$$p^{6} T^{46} +$$$$56\!\cdots\!17$$$$p^{7} T^{47} +$$$$85\!\cdots\!10$$$$p^{8} T^{48} +$$$$12\!\cdots\!15$$$$p^{9} T^{49} +$$$$17\!\cdots\!63$$$$p^{10} T^{50} +$$$$25\!\cdots\!24$$$$p^{11} T^{51} +$$$$35\!\cdots\!63$$$$p^{12} T^{52} +$$$$47\!\cdots\!69$$$$p^{13} T^{53} +$$$$65\!\cdots\!37$$$$p^{14} T^{54} +$$$$85\!\cdots\!60$$$$p^{15} T^{55} +$$$$11\!\cdots\!76$$$$p^{16} T^{56} +$$$$14\!\cdots\!73$$$$p^{17} T^{57} +$$$$18\!\cdots\!77$$$$p^{18} T^{58} +$$$$21\!\cdots\!42$$$$p^{19} T^{59} +$$$$26\!\cdots\!45$$$$p^{20} T^{60} +$$$$29\!\cdots\!41$$$$p^{21} T^{61} +$$$$35\!\cdots\!63$$$$p^{22} T^{62} + 37548763933917370950 p^{23} T^{63} + 4197699963992164936 p^{24} T^{64} + 418138286906229304 p^{25} T^{65} + 44227061670431192 p^{26} T^{66} + 4060211916171800 p^{27} T^{67} + 403835340308266 p^{28} T^{68} + 33586094405317 p^{29} T^{69} + 3120471054065 p^{30} T^{70} + 229527677087 p^{31} T^{71} + 534582871 p^{33} T^{72} + 1241406954 p^{33} T^{73} + 98476341 p^{34} T^{74} + 4973009 p^{35} T^{75} + 360225 p^{36} T^{76} + 13099 p^{37} T^{77} + 859 p^{38} T^{78} + 17 p^{39} T^{79} + p^{40} T^{80}$$
41 $$1 + 58 T + 2537 T^{2} + 80629 T^{3} + 2192425 T^{4} + 50908215 T^{5} + 1062951177 T^{6} + 488073917 p T^{7} + 347835588218 T^{8} + 5599082024527 T^{9} + 84582648463764 T^{10} + 1202181281735400 T^{11} + 16212615194039343 T^{12} + 207885807891221484 T^{13} + 2549296094133312766 T^{14} + 29947239312977017901 T^{15} +$$$$33\!\cdots\!51$$$$T^{16} +$$$$36\!\cdots\!66$$$$T^{17} +$$$$38\!\cdots\!54$$$$T^{18} +$$$$39\!\cdots\!93$$$$T^{19} +$$$$38\!\cdots\!10$$$$T^{20} +$$$$37\!\cdots\!35$$$$T^{21} +$$$$85\!\cdots\!85$$$$p T^{22} +$$$$31\!\cdots\!99$$$$T^{23} +$$$$28\!\cdots\!03$$$$T^{24} +$$$$24\!\cdots\!83$$$$T^{25} +$$$$49\!\cdots\!61$$$$p T^{26} +$$$$16\!\cdots\!05$$$$T^{27} +$$$$13\!\cdots\!35$$$$T^{28} +$$$$10\!\cdots\!96$$$$T^{29} +$$$$83\!\cdots\!62$$$$T^{30} +$$$$63\!\cdots\!89$$$$T^{31} +$$$$47\!\cdots\!58$$$$T^{32} +$$$$84\!\cdots\!08$$$$p T^{33} +$$$$25\!\cdots\!97$$$$T^{34} +$$$$17\!\cdots\!24$$$$T^{35} +$$$$12\!\cdots\!91$$$$T^{36} +$$$$83\!\cdots\!67$$$$T^{37} +$$$$55\!\cdots\!53$$$$T^{38} +$$$$36\!\cdots\!68$$$$T^{39} +$$$$23\!\cdots\!90$$$$T^{40} +$$$$36\!\cdots\!68$$$$p T^{41} +$$$$55\!\cdots\!53$$$$p^{2} T^{42} +$$$$83\!\cdots\!67$$$$p^{3} T^{43} +$$$$12\!\cdots\!91$$$$p^{4} T^{44} +$$$$17\!\cdots\!24$$$$p^{5} T^{45} +$$$$25\!\cdots\!97$$$$p^{6} T^{46} +$$$$84\!\cdots\!08$$$$p^{8} T^{47} +$$$$47\!\cdots\!58$$$$p^{8} T^{48} +$$$$63\!\cdots\!89$$$$p^{9} T^{49} +$$$$83\!\cdots\!62$$$$p^{10} T^{50} +$$$$10\!\cdots\!96$$$$p^{11} T^{51} +$$$$13\!\cdots\!35$$$$p^{12} T^{52} +$$$$16\!\cdots\!05$$$$p^{13} T^{53} +$$$$49\!\cdots\!61$$$$p^{15} T^{54} +$$$$24\!\cdots\!83$$$$p^{15} T^{55} +$$$$28\!\cdots\!03$$$$p^{16} T^{56} +$$$$31\!\cdots\!99$$$$p^{17} T^{57} +$$$$85\!\cdots\!85$$$$p^{19} T^{58} +$$$$37\!\cdots\!35$$$$p^{19} T^{59} +$$$$38\!\cdots\!10$$$$p^{20} T^{60} +$$$$39\!\cdots\!93$$$$p^{21} T^{61} +$$$$38\!\cdots\!54$$$$p^{22} T^{62} +$$$$36\!\cdots\!66$$$$p^{23} T^{63} +$$$$33\!\cdots\!51$$$$p^{24} T^{64} + 29947239312977017901 p^{25} T^{65} + 2549296094133312766 p^{26} T^{66} + 207885807891221484 p^{27} T^{67} + 16212615194039343 p^{28} T^{68} + 1202181281735400 p^{29} T^{69} + 84582648463764 p^{30} T^{70} + 5599082024527 p^{31} T^{71} + 347835588218 p^{32} T^{72} + 488073917 p^{34} T^{73} + 1062951177 p^{34} T^{74} + 50908215 p^{35} T^{75} + 2192425 p^{36} T^{76} + 80629 p^{37} T^{77} + 2537 p^{38} T^{78} + 58 p^{39} T^{79} + p^{40} T^{80}$$
43 $$1 - 37 T + 1530 T^{2} - 39816 T^{3} + 1018057 T^{4} - 21100709 T^{5} + 419486680 T^{6} - 7369650655 T^{7} + 123820239829 T^{8} - 1912831697689 T^{9} + 28328962342346 T^{10} - 394126960831711 T^{11} + 5275776326212114 T^{12} - 67203202985286934 T^{13} + 826601677050563092 T^{14} - 9756607641126861125 T^{15} +$$$$11\!\cdots\!02$$$$T^{16} -$$$$12\!\cdots\!99$$$$T^{17} +$$$$13\!\cdots\!06$$$$T^{18} -$$$$13\!\cdots\!16$$$$T^{19} +$$$$13\!\cdots\!56$$$$T^{20} -$$$$13\!\cdots\!02$$$$T^{21} +$$$$13\!\cdots\!36$$$$T^{22} -$$$$12\!\cdots\!93$$$$T^{23} +$$$$11\!\cdots\!59$$$$T^{24} -$$$$99\!\cdots\!40$$$$T^{25} +$$$$86\!\cdots\!92$$$$T^{26} -$$$$74\!\cdots\!02$$$$T^{27} +$$$$61\!\cdots\!04$$$$T^{28} -$$$$50\!\cdots\!97$$$$T^{29} +$$$$40\!\cdots\!78$$$$T^{30} -$$$$31\!\cdots\!80$$$$T^{31} +$$$$24\!\cdots\!69$$$$T^{32} -$$$$18\!\cdots\!53$$$$T^{33} +$$$$13\!\cdots\!70$$$$T^{34} -$$$$99\!\cdots\!89$$$$T^{35} +$$$$70\!\cdots\!97$$$$T^{36} -$$$$49\!\cdots\!92$$$$T^{37} +$$$$33\!\cdots\!46$$$$T^{38} -$$$$22\!\cdots\!93$$$$T^{39} +$$$$15\!\cdots\!40$$$$T^{40} -$$$$22\!\cdots\!93$$$$p T^{41} +$$$$33\!\cdots\!46$$$$p^{2} T^{42} -$$$$49\!\cdots\!92$$$$p^{3} T^{43} +$$$$70\!\cdots\!97$$$$p^{4} T^{44} -$$$$99\!\cdots\!89$$$$p^{5} T^{45} +$$$$13\!\cdots\!70$$$$p^{6} T^{46} -$$$$18\!\cdots\!53$$$$p^{7} T^{47} +$$$$24\!\cdots\!69$$$$p^{8} T^{48} -$$$$31\!\cdots\!80$$$$p^{9} T^{49} +$$$$40\!\cdots\!78$$$$p^{10} T^{50} -$$$$50\!\cdots\!97$$$$p^{11} T^{51} +$$$$61\!\cdots\!04$$$$p^{12} T^{52} -$$$$74\!\cdots\!02$$$$p^{13} T^{53} +$$$$86\!\cdots\!92$$$$p^{14} T^{54} -$$$$99\!\cdots\!40$$$$p^{15} T^{55} +$$$$11\!\cdots\!59$$$$p^{16} T^{56} -$$$$12\!\cdots\!93$$$$p^{17} T^{57} +$$$$13\!\cdots\!36$$$$p^{18} T^{58} -$$$$13\!\cdots\!02$$$$p^{19} T^{59} +$$$$13\!\cdots\!56$$$$p^{20} T^{60} -$$$$13\!\cdots\!16$$$$p^{21} T^{61} +$$$$13\!\cdots\!06$$$$p^{22} T^{62} -$$$$12\!\cdots\!99$$$$p^{23} T^{63} +$$$$11\!\cdots\!02$$$$p^{24} T^{64} - 9756607641126861125 p^{25} T^{65} + 826601677050563092 p^{26} T^{66} - 67203202985286934 p^{27} T^{67} + 5275776326212114 p^{28} T^{68} - 394126960831711 p^{29} T^{69} + 28328962342346 p^{30} T^{70} - 1912831697689 p^{31} T^{71} + 123820239829 p^{32} T^{72} - 7369650655 p^{33} T^{73} + 419486680 p^{34} T^{74} - 21100709 p^{35} T^{75} + 1018057 p^{36} T^{76} - 39816 p^{37} T^{77} + 1530 p^{38} T^{78} - 37 p^{39} T^{79} + p^{40} T^{80}$$
47 $$1 + 34 T + 1494 T^{2} + 37854 T^{3} + 1004636 T^{4} + 20682870 T^{5} + 421450841 T^{6} + 7404670635 T^{7} + 126480877975 T^{8} + 1956952068883 T^{9} + 29292358394980 T^{10} + 407900278084648 T^{11} + 5494578450045535 T^{12} + 69969548144930180 T^{13} + 863418137012803200 T^{14} + 10178357579922818549 T^{15} +$$$$11\!\cdots\!98$$$$T^{16} +$$$$12\!\cdots\!44$$$$T^{17} +$$$$13\!\cdots\!38$$$$T^{18} +$$$$14\!\cdots\!28$$$$T^{19} +$$$$14\!\cdots\!85$$$$T^{20} +$$$$14\!\cdots\!44$$$$T^{21} +$$$$13\!\cdots\!72$$$$T^{22} +$$$$12\!\cdots\!07$$$$T^{23} +$$$$11\!\cdots\!29$$$$T^{24} +$$$$10\!\cdots\!85$$$$T^{25} +$$$$93\!\cdots\!38$$$$T^{26} +$$$$80\!\cdots\!73$$$$T^{27} +$$$$67\!\cdots\!56$$$$T^{28} +$$$$56\!\cdots\!97$$$$T^{29} +$$$$45\!\cdots\!02$$$$T^{30} +$$$$78\!\cdots\!95$$$$p T^{31} +$$$$28\!\cdots\!93$$$$T^{32} +$$$$10\!\cdots\!11$$$$p^{2} T^{33} +$$$$17\!\cdots\!01$$$$T^{34} +$$$$12\!\cdots\!87$$$$T^{35} +$$$$93\!\cdots\!53$$$$T^{36} +$$$$67\!\cdots\!09$$$$T^{37} +$$$$48\!\cdots\!74$$$$T^{38} +$$$$34\!\cdots\!31$$$$T^{39} +$$$$23\!\cdots\!02$$$$T^{40} +$$$$34\!\cdots\!31$$$$p T^{41} +$$$$48\!\cdots\!74$$$$p^{2} T^{42} +$$$$67\!\cdots\!09$$$$p^{3} T^{43} +$$$$93\!\cdots\!53$$$$p^{4} T^{44} +$$$$12\!\cdots\!87$$$$p^{5} T^{45} +$$$$17\!\cdots\!01$$$$p^{6} T^{46} +$$$$10\!\cdots\!11$$$$p^{9} T^{47} +$$$$28\!\cdots\!93$$$$p^{8} T^{48} +$$$$78\!\cdots\!95$$$$p^{10} T^{49} +$$$$45\!\cdots\!02$$$$p^{10} T^{50} +$$$$56\!\cdots\!97$$$$p^{11} T^{51} +$$$$67\!\cdots\!56$$$$p^{12} T^{52} +$$$$80\!\cdots\!73$$$$p^{13} T^{53} +$$$$93\!\cdots\!38$$$$p^{14} T^{54} +$$$$10\!\cdots\!85$$$$p^{15} T^{55} +$$$$11\!\cdots\!29$$$$p^{16} T^{56} +$$$$12\!\cdots\!07$$$$p^{17} T^{57} +$$$$13\!\cdots\!72$$$$p^{18} T^{58} +$$$$14\!\cdots\!44$$$$p^{19} T^{59} +$$$$14\!\cdots\!85$$$$p^{20} T^{60} +$$$$14\!\cdots\!28$$$$p^{21} T^{61} +$$$$13\!\cdots\!38$$$$p^{22} T^{62} +$$$$12\!\cdots\!44$$$$p^{23} T^{63} +$$$$11\!\cdots\!98$$$$p^{24} T^{64} + 10178357579922818549 p^{25} T^{65} + 863418137012803200 p^{26} T^{66} + 69969548144930180 p^{27} T^{67} + 5494578450045535 p^{28} T^{68} + 407900278084648 p^{29} T^{69} + 29292358394980 p^{30} T^{70} + 1956952068883 p^{31} T^{71} + 126480877975 p^{32} T^{72} + 7404670635 p^{33} T^{73} + 421450841 p^{34} T^{74} + 20682870 p^{35} T^{75} + 1004636 p^{36} T^{76} + 37854 p^{37} T^{77} + 1494 p^{38} T^{78} + 34 p^{39} T^{79} + p^{40} T^{80}$$
53 $$1 + 43 T + 2015 T^{2} + 59260 T^{3} + 1706576 T^{4} + 39456814 T^{5} + 876633912 T^{6} + 17033640107 T^{7} + 317419102760 T^{8} + 5387302055476 T^{9} + 87881974034417 T^{10} + 25202037282614 p T^{11} + 19574337033977533 T^{12} + 271088891577783854 T^{13} + 3631501465475442188 T^{14} + 46416829014730917711 T^{15} +$$$$57\!\cdots\!92$$$$T^{16} +$$$$68\!\cdots\!90$$$$T^{17} +$$$$79\!\cdots\!65$$$$T^{18} +$$$$88\!\cdots\!89$$$$T^{19} +$$$$97\!\cdots\!60$$$$T^{20} +$$$$10\!\cdots\!98$$$$T^{21} +$$$$10\!\cdots\!81$$$$T^{22} +$$$$10\!\cdots\!98$$$$T^{23} +$$$$10\!\cdots\!84$$$$T^{24} +$$$$10\!\cdots\!64$$$$T^{25} +$$$$94\!\cdots\!15$$$$T^{26} +$$$$87\!\cdots\!18$$$$T^{27} +$$$$78\!\cdots\!34$$$$T^{28} +$$$$69\!\cdots\!13$$$$T^{29} +$$$$60\!\cdots\!94$$$$T^{30} +$$$$51\!\cdots\!39$$$$T^{31} +$$$$43\!\cdots\!58$$$$T^{32} +$$$$35\!\cdots\!75$$$$T^{33} +$$$$28\!\cdots\!97$$$$T^{34} +$$$$22\!\cdots\!32$$$$T^{35} +$$$$17\!\cdots\!79$$$$T^{36} +$$$$13\!\cdots\!24$$$$T^{37} +$$$$10\!\cdots\!20$$$$T^{38} +$$$$77\!\cdots\!65$$$$T^{39} +$$$$56\!\cdots\!18$$$$T^{40} +$$$$77\!\cdots\!65$$$$p T^{41} +$$$$10\!\cdots\!20$$$$p^{2} T^{42} +$$$$13\!\cdots\!24$$$$p^{3} T^{43} +$$$$17\!\cdots\!79$$$$p^{4} T^{44} +$$$$22\!\cdots\!32$$$$p^{5} T^{45} +$$$$28\!\cdots\!97$$$$p^{6} T^{46} +$$$$35\!\cdots\!75$$$$p^{7} T^{47} +$$$$43\!\cdots\!58$$$$p^{8} T^{48} +$$$$51\!\cdots\!39$$$$p^{9} T^{49} +$$$$60\!\cdots\!94$$$$p^{10} T^{50} +$$$$69\!\cdots\!13$$$$p^{11} T^{51} +$$$$78\!\cdots\!34$$$$p^{12} T^{52} +$$$$87\!\cdots\!18$$$$p^{13} T^{53} +$$$$94\!\cdots\!15$$$$p^{14} T^{54} +$$$$10\!\cdots\!64$$$$p^{15} T^{55} +$$$$10\!\cdots\!84$$$$p^{16} T^{56} +$$$$10\!\cdots\!98$$$$p^{17} T^{57} +$$$$10\!\cdots\!81$$$$p^{18} T^{58} +$$$$10\!\cdots\!98$$$$p^{19} T^{59} +$$$$97\!\cdots\!60$$$$p^{20} T^{60} +$$$$88\!\cdots\!89$$$$p^{21} T^{61} +$$$$79\!\cdots\!65$$$$p^{22} T^{62} +$$$$68\!\cdots\!90$$$$p^{23} T^{63} +$$$$57\!\cdots\!92$$$$p^{24} T^{64} + 46416829014730917711 p^{25} T^{65} + 3631501465475442188 p^{26} T^{66} + 271088891577783854 p^{27} T^{67} + 19574337033977533 p^{28} T^{68} + 25202037282614 p^{30} T^{69} + 87881974034417 p^{30} T^{70} + 5387302055476 p^{31} T^{71} + 317419102760 p^{32} T^{72} + 17033640107 p^{33} T^{73} + 876633912 p^{34} T^{74} + 39456814 p^{35} T^{75} + 1706576 p^{36} T^{76} + 59260 p^{37} T^{77} + 2015 p^{38} T^{78} + 43 p^{39} T^{79} + p^{40} T^{80}$$
59 $$1 + 92 T + 5301 T^{2} + 226046 T^{3} + 7901970 T^{4} + 236394621 T^{5} + 6250362841 T^{6} + 148910089103 T^{7} + 3246178400551 T^{8} + 65453057175101 T^{9} + 1231555171280102 T^{10} + 21771381716340424 T^{11} + 363672723214933365 T^{12} + 5766647833259124296 T^{13} + 87144804376684627521 T^{14} +$$$$12\!\cdots\!62$$$$T^{15} +$$$$17\!\cdots\!60$$$$T^{16} +$$$$23\!\cdots\!03$$$$T^{17} +$$$$29\!\cdots\!83$$$$T^{18} +$$$$36\!\cdots\!04$$$$T^{19} +$$$$44\!\cdots\!82$$$$T^{20} +$$$$86\!\cdots\!92$$$$p T^{21} +$$$$57\!\cdots\!33$$$$T^{22} +$$$$10\!\cdots\!18$$$$p T^{23} +$$$$66\!\cdots\!80$$$$T^{24} +$$$$68\!\cdots\!60$$$$T^{25} +$$$$69\!\cdots\!95$$$$T^{26} +$$$$68\!\cdots\!40$$$$T^{27} +$$$$65\!\cdots\!74$$$$T^{28} +$$$$61\!\cdots\!39$$$$T^{29} +$$$$55\!\cdots\!33$$$$T^{30} +$$$$50\!\cdots\!51$$$$T^{31} +$$$$44\!\cdots\!25$$$$T^{32} +$$$$37\!\cdots\!75$$$$T^{33} +$$$$32\!\cdots\!35$$$$T^{34} +$$$$26\!\cdots\!00$$$$T^{35} +$$$$21\!\cdots\!82$$$$T^{36} +$$$$17\!\cdots\!70$$$$T^{37} +$$$$14\!\cdots\!76$$$$T^{38} +$$$$11\!\cdots\!25$$$$T^{39} +$$$$85\!\cdots\!44$$$$T^{40} +$$$$11\!\cdots\!25$$$$p T^{41} +$$$$14\!\cdots\!76$$$$p^{2} T^{42} +$$$$17\!\cdots\!70$$$$p^{3} T^{43} +$$$$21\!\cdots\!82$$$$p^{4} T^{44} +$$$$26\!\cdots\!00$$$$p^{5} T^{45} +$$$$32\!\cdots\!35$$$$p^{6} T^{46} +$$$$37\!\cdots\!75$$$$p^{7} T^{47} +$$$$44\!\cdots\!25$$$$p^{8} T^{48} +$$$$50\!\cdots\!51$$$$p^{9} T^{49} +$$$$55\!\cdots\!33$$$$p^{10} T^{50} +$$$$61\!\cdots\!39$$$$p^{11} T^{51} +$$$$65\!\cdots\!74$$$$p^{12} T^{52} +$$$$68\!\cdots\!40$$$$p^{13} T^{53} +$$$$69\!\cdots\!95$$$$p^{14} T^{54} +$$$$68\!\cdots\!60$$$$p^{15} T^{55} +$$$$66\!\cdots\!80$$$$p^{16} T^{56} +$$$$10\!\cdots\!18$$$$p^{18} T^{57} +$$$$57\!\cdots\!33$$$$p^{18} T^{58} +$$$$86\!\cdots\!92$$$$p^{20} T^{59} +$$$$44\!\cdots\!82$$$$p^{20} T^{60} +$$$$36\!\cdots\!04$$$$p^{21} T^{61} +$$$$29\!\cdots\!83$$$$p^{22} T^{62} +$$$$23\!\cdots\!03$$$$p^{23} T^{63} +$$$$17\!\cdots\!60$$$$p^{24} T^{64} +$$$$12\!\cdots\!62$$$$p^{25} T^{65} + 87144804376684627521 p^{26} T^{66} + 5766647833259124296 p^{27} T^{67} + 363672723214933365 p^{28} T^{68} + 21771381716340424 p^{29} T^{69} + 1231555171280102 p^{30} T^{70} + 65453057175101 p^{31} T^{71} + 3246178400551 p^{32} T^{72} + 148910089103 p^{33} T^{73} + 6250362841 p^{34} T^{74} + 236394621 p^{35} T^{75} + 7901970 p^{36} T^{76} + 226046 p^{37} T^{77} + 5301 p^{38} T^{78} + 92 p^{39} T^{79} + p^{40} T^{80}$$
61 $$1 + 37 T + 30 p T^{2} + 50126 T^{3} + 1492227 T^{4} + 33231488 T^{5} + 757224306 T^{6} + 14432078073 T^{7} + 275125797360 T^{8} + 4634015152987 T^{9} + 77340466198046 T^{10} + 1176592913751682 T^{11} + 17668433989184443 T^{12} + 246624253705519815 T^{13} + 3393515104355467554 T^{14} + 43978330892770759617 T^{15} +$$$$56\!\cdots\!29$$$$T^{16} +$$$$68\!\cdots\!48$$$$T^{17} +$$$$81\!\cdots\!75$$$$T^{18} +$$$$93\!\cdots\!29$$$$T^{19} +$$$$10\!\cdots\!96$$$$T^{20} +$$$$11\!\cdots\!08$$$$T^{21} +$$$$12\!\cdots\!80$$$$T^{22} +$$$$12\!\cdots\!82$$$$T^{23} +$$$$13\!\cdots\!14$$$$T^{24} +$$$$13\!\cdots\!23$$$$T^{25} +$$$$12\!\cdots\!17$$$$T^{26} +$$$$12\!\cdots\!09$$$$T^{27} +$$$$11\!\cdots\!70$$$$T^{28} +$$$$10\!\cdots\!04$$$$T^{29} +$$$$99\!\cdots\!63$$$$T^{30} +$$$$88\!\cdots\!41$$$$T^{31} +$$$$78\!\cdots\!28$$$$T^{32} +$$$$67\!\cdots\!44$$$$T^{33} +$$$$58\!\cdots\!56$$$$T^{34} +$$$$48\!\cdots\!53$$$$T^{35} +$$$$40\!\cdots\!56$$$$T^{36} +$$$$33\!\cdots\!86$$$$T^{37} +$$$$26\!\cdots\!73$$$$T^{38} +$$$$21\!\cdots\!76$$$$T^{39} +$$$$16\!\cdots\!92$$$$T^{40} +$$$$21\!\cdots\!76$$$$p T^{41} +$$$$26\!\cdots\!73$$$$p^{2} T^{42} +$$$$33\!\cdots\!86$$$$p^{3} T^{43} +$$$$40\!\cdots\!56$$$$p^{4} T^{44} +$$$$48\!\cdots\!53$$$$p^{5} T^{45} +$$$$58\!\cdots\!56$$$$p^{6} T^{46} +$$$$67\!\cdots\!44$$$$p^{7} T^{47} +$$$$78\!\cdots\!28$$$$p^{8} T^{48} +$$$$88\!\cdots\!41$$$$p^{9} T^{49} +$$$$99\!\cdots\!63$$$$p^{10} T^{50} +$$$$10\!\cdots\!04$$$$p^{11} T^{51} +$$$$11\!\cdots\!70$$$$p^{12} T^{52} +$$$$12\!\cdots\!09$$$$p^{13} T^{53} +$$$$12\!\cdots\!17$$$$p^{14} T^{54} +$$$$13\!\cdots\!23$$$$p^{15} T^{55} +$$$$13\!\cdots\!14$$$$p^{16} T^{56} +$$$$12\!\cdots\!82$$$$p^{17} T^{57} +$$$$12\!\cdots\!80$$$$p^{18} T^{58} +$$$$11\!\cdots\!08$$$$p^{19} T^{59} +$$$$10\!\cdots\!96$$$$p^{20} T^{60} +$$$$93\!\cdots\!29$$$$p^{21} T^{61} +$$$$81\!\cdots\!75$$$$p^{22} T^{62} +$$$$68\!\cdots\!48$$$$p^{23} T^{63} +$$$$56\!\cdots\!29$$$$p^{24} T^{64} + 43978330892770759617 p^{25} T^{65} + 3393515104355467554 p^{26} T^{66} + 246624253705519815 p^{27} T^{67} + 17668433989184443 p^{28} T^{68} + 1176592913751682 p^{29} T^{69} + 77340466198046 p^{30} T^{70} + 4634015152987 p^{31} T^{71} + 275125797360 p^{32} T^{72} + 14432078073 p^{33} T^{73} + 757224306 p^{34} T^{74} + 33231488 p^{35} T^{75} + 1492227 p^{36} T^{76} + 50126 p^{37} T^{77} + 30 p^{39} T^{78} + 37 p^{39} T^{79} + p^{40} T^{80}$$
67 $$1 + 10 T + 1644 T^{2} + 15877 T^{3} + 1341105 T^{4} + 12589472 T^{5} + 724448550 T^{6} + 6644633810 T^{7} + 291722801871 T^{8} + 2624529318786 T^{9} + 93448908093386 T^{10} + 826921463834688 T^{11} + 24812148858223926 T^{12} + 216317623649675985 T^{13} + 5617186644759536305 T^{14} + 48284145273635469239 T^{15} +$$$$11\!\cdots\!55$$$$T^{16} +$$$$93\!\cdots\!61$$$$T^{17} +$$$$19\!\cdots\!98$$$$T^{18} +$$$$16\!\cdots\!30$$$$T^{19} +$$$$30\!\cdots\!42$$$$T^{20} +$$$$24\!\cdots\!39$$$$T^{21} +$$$$42\!\cdots\!49$$$$T^{22} +$$$$34\!\cdots\!74$$$$T^{23} +$$$$54\!\cdots\!46$$$$T^{24} +$$$$42\!\cdots\!28$$$$T^{25} +$$$$63\!\cdots\!33$$$$T^{26} +$$$$49\!\cdots\!03$$$$T^{27} +$$$$68\!\cdots\!30$$$$T^{28} +$$$$51\!\cdots\!84$$$$T^{29} +$$$$68\!\cdots\!94$$$$T^{30} +$$$$50\!\cdots\!20$$$$T^{31} +$$$$63\!\cdots\!11$$$$T^{32} +$$$$45\!\cdots\!80$$$$T^{33} +$$$$54\!\cdots\!77$$$$T^{34} +$$$$37\!\cdots\!31$$$$T^{35} +$$$$43\!\cdots\!07$$$$T^{36} +$$$$29\!\cdots\!89$$$$T^{37} +$$$$32\!\cdots\!96$$$$T^{38} +$$$$21\!\cdots\!90$$$$T^{39} +$$$$22\!\cdots\!16$$$$T^{40} +$$$$21\!\cdots\!90$$$$p T^{41} +$$$$32\!\cdots\!96$$$$p^{2} T^{42} +$$$$29\!\cdots\!89$$$$p^{3} T^{43} +$$$$43\!\cdots\!07$$$$p^{4} T^{44} +$$$$37\!\cdots\!31$$$$p^{5} T^{45} +$$$$54\!\cdots\!77$$$$p^{6} T^{46} +$$$$45\!\cdots\!80$$$$p^{7} T^{47} +$$$$63\!\cdots\!11$$$$p^{8} T^{48} +$$$$50\!\cdots\!20$$$$p^{9} T^{49} +$$$$68\!\cdots\!94$$$$p^{10} T^{50} +$$$$51\!\cdots\!84$$$$p^{11} T^{51} +$$$$68\!\cdots\!30$$$$p^{12} T^{52} +$$$$49\!\cdots\!03$$$$p^{13} T^{53} +$$$$63\!\cdots\!33$$$$p^{14} T^{54} +$$$$42\!\cdots\!28$$$$p^{15} T^{55} +$$$$54\!\cdots\!46$$$$p^{16} T^{56} +$$$$34\!\cdots\!74$$$$p^{17} T^{57} +$$$$42\!\cdots\!49$$$$p^{18} T^{58} +$$$$24\!\cdots\!39$$$$p^{19} T^{59} +$$$$30\!\cdots\!42$$$$p^{20} T^{60} +$$$$16\!\cdots\!30$$$$p^{21} T^{61} +$$$$19\!\cdots\!98$$$$p^{22} T^{62} +$$$$93\!\cdots\!61$$$$p^{23} T^{63} +$$$$11\!\cdots\!55$$$$p^{24} T^{64} + 48284145273635469239 p^{25} T^{65} + 5617186644759536305 p^{26} T^{66} + 216317623649675985 p^{27} T^{67} + 24812148858223926 p^{28} T^{68} + 826921463834688 p^{29} T^{69} + 93448908093386 p^{30} T^{70} + 2624529318786 p^{31} T^{71} + 291722801871 p^{32} T^{72} + 6644633810 p^{33} T^{73} + 724448550 p^{34} T^{74} + 12589472 p^{35} T^{75} + 1341105 p^{36} T^{76} + 15877 p^{37} T^{77} + 1644 p^{38} T^{78} + 10 p^{39} T^{79} + p^{40} T^{80}$$
71 $$1 + 87 T + 4869 T^{2} + 199449 T^{3} + 6677844 T^{4} + 189429193 T^{5} + 4713651455 T^{6} + 1474013947 p T^{7} + 2108579680508 T^{8} + 38947349476052 T^{9} + 666432450752406 T^{10} + 10640943810061526 T^{11} + 159772386136535588 T^{12} + 2269421893210303759 T^{13} + 30693238249781882654 T^{14} +$$$$39\!\cdots\!76$$$$T^{15} +$$$$49\!\cdots\!53$$$$T^{16} +$$$$59\!\cdots\!25$$$$T^{17} +$$$$69\!\cdots\!03$$$$T^{18} +$$$$79\!\cdots\!32$$$$T^{19} +$$$$88\!\cdots\!91$$$$T^{20} +$$$$97\!\cdots\!16$$$$T^{21} +$$$$10\!\cdots\!81$$$$T^{22} +$$$$10\!\cdots\!73$$$$T^{23} +$$$$15\!\cdots\!88$$$$p T^{24} +$$$$11\!\cdots\!19$$$$T^{25} +$$$$11\!\cdots\!54$$$$T^{26} +$$$$11\!\cdots\!81$$$$T^{27} +$$$$11\!\cdots\!64$$$$T^{28} +$$$$10\!\cdots\!12$$$$T^{29} +$$$$10\!\cdots\!48$$$$T^{30} +$$$$95\!\cdots\!13$$$$T^{31} +$$$$88\!\cdots\!12$$$$T^{32} +$$$$81\!\cdots\!79$$$$T^{33} +$$$$73\!\cdots\!86$$$$T^{34} +$$$$65\!\cdots\!46$$$$T^{35} +$$$$58\!\cdots\!05$$$$T^{36} +$$$$50\!\cdots\!22$$$$T^{37} +$$$$44\!\cdots\!92$$$$T^{38} +$$$$37\!\cdots\!15$$$$T^{39} +$$$$32\!\cdots\!48$$$$T^{40} +$$$$37\!\cdots\!15$$$$p T^{41} +$$$$44\!\cdots\!92$$$$p^{2} T^{42} +$$$$50\!\cdots\!22$$$$p^{3} T^{43} +$$$$58\!\cdots\!05$$$$p^{4} T^{44} +$$$$65\!\cdots\!46$$$$p^{5} T^{45} +$$$$73\!\cdots\!86$$$$p^{6} T^{46} +$$$$81\!\cdots\!79$$$$p^{7} T^{47} +$$$$88\!\cdots\!12$$$$p^{8} T^{48} +$$$$95\!\cdots\!13$$$$p^{9} T^{49} +$$$$10\!\cdots\!48$$$$p^{10} T^{50} +$$$$10\!\cdots\!12$$$$p^{11} T^{51} +$$$$11\!\cdots\!64$$$$p^{12} T^{52} +$$$$11\!\cdots\!81$$$$p^{13} T^{53} +$$$$11\!\cdots\!54$$$$p^{14} T^{54} +$$$$11\!\cdots\!19$$$$p^{15} T^{55} +$$$$15\!\cdots\!88$$$$p^{17} T^{56} +$$$$10\!\cdots\!73$$$$p^{17} T^{57} +$$$$10\!\cdots\!81$$$$p^{18} T^{58} +$$$$97\!\cdots\!16$$$$p^{19} T^{59} +$$$$88\!\cdots\!91$$$$p^{20} T^{60} +$$$$79\!\cdots\!32$$$$p^{21} T^{61} +$$$$69\!\cdots\!03$$$$p^{22} T^{62} +$$$$59\!\cdots\!25$$$$p^{23} T^{63} +$$$$49\!\cdots\!53$$$$p^{24} T^{64} +$$$$39\!\cdots\!76$$$$p^{25} T^{65} + 30693238249781882654 p^{26} T^{66} + 2269421893210303759 p^{27} T^{67} + 159772386136535588 p^{28} T^{68} + 10640943810061526 p^{29} T^{69} + 666432450752406 p^{30} T^{70} + 38947349476052 p^{31} T^{71} + 2108579680508 p^{32} T^{72} + 1474013947 p^{34} T^{73} + 4713651455 p^{34} T^{74} + 189429193 p^{35} T^{75} + 6677844 p^{36} T^{76} + 199449 p^{37} T^{77} + 4869 p^{38} T^{78} + 87 p^{39} T^{79} + p^{40} T^{80}$$
73 $$1 - 12 T + 1791 T^{2} - 19866 T^{3} + 1588762 T^{4} - 16391967 T^{5} + 932136055 T^{6} - 8993542637 T^{7} + 407371961466 T^{8} - 3692544723069 T^{9} + 141565341364987 T^{10} - 1210380012760930 T^{11} + 40767151873610149 T^{12} - 329944641293509484 T^{13} + 10008726431143219881 T^{14} - 76919111078127637104 T^{15} +$$$$21\!\cdots\!73$$$$T^{16} -$$$$15\!\cdots\!23$$$$T^{17} +$$$$40\!\cdots\!21$$$$T^{18} -$$$$28\!\cdots\!72$$$$T^{19} +$$$$68\!\cdots\!62$$$$T^{20} -$$$$45\!\cdots\!63$$$$T^{21} +$$$$10\!\cdots\!31$$$$T^{22} -$$$$66\!\cdots\!07$$$$T^{23} +$$$$14\!\cdots\!86$$$$T^{24} -$$$$88\!\cdots\!42$$$$T^{25} +$$$$18\!\cdots\!29$$$$T^{26} -$$$$10\!\cdots\!55$$$$T^{27} +$$$$21\!\cdots\!77$$$$T^{28} -$$$$12\!\cdots\!53$$$$T^{29} +$$$$23\!\cdots\!65$$$$T^{30} -$$$$12\!\cdots\!01$$$$T^{31} +$$$$23\!\cdots\!54$$$$T^{32} -$$$$12\!\cdots\!92$$$$T^{33} +$$$$29\!\cdots\!32$$$$p T^{34} -$$$$11\!\cdots\!08$$$$T^{35} +$$$$18\!\cdots\!51$$$$T^{36} -$$$$93\!\cdots\!45$$$$T^{37} +$$$$15\!\cdots\!46$$$$T^{38} -$$$$73\!\cdots\!33$$$$T^{39} +$$$$11\!\cdots\!05$$$$T^{40} -$$$$73\!\cdots\!33$$$$p T^{41} +$$$$15\!\cdots\!46$$$$p^{2} T^{42} -$$$$93\!\cdots\!45$$$$p^{3} T^{43} +$$$$18\!\cdots\!51$$$$p^{4} T^{44} -$$$$11\!\cdots\!08$$$$p^{5} T^{45} +$$$$29\!\cdots\!32$$$$p^{7} T^{46} -$$$$12\!\cdots\!92$$$$p^{7} T^{47} +$$$$23\!\cdots\!54$$$$p^{8} T^{48} -$$$$12\!\cdots\!01$$$$p^{9} T^{49} +$$$$23\!\cdots\!65$$$$p^{10} T^{50} -$$$$12\!\cdots\!53$$$$p^{11} T^{51} +$$$$21\!\cdots\!77$$$$p^{12} T^{52} -$$$$10\!\cdots\!55$$$$p^{13} T^{53} +$$$$18\!\cdots\!29$$$$p^{14} T^{54} -$$$$88\!\cdots\!42$$$$p^{15} T^{55} +$$$$14\!\cdots\!86$$$$p^{16} T^{56} -$$$$66\!\cdots\!07$$$$p^{17} T^{57} +$$$$10\!\cdots\!31$$$$p^{18} T^{58} -$$$$45\!\cdots\!63$$$$p^{19} T^{59} +$$$$68\!\cdots\!62$$$$p^{20} T^{60} -$$$$28\!\cdots\!72$$$$p^{21} T^{61} +$$$$40\!\cdots\!21$$$$p^{22} T^{62} -$$$$15\!\cdots\!23$$$$p^{23} T^{63} +$$$$21\!\cdots\!73$$$$p^{24} T^{64} - 76919111078127637104 p^{25} T^{65} + 10008726431143219881 p^{26} T^{66} - 329944641293509484 p^{27} T^{67} + 40767151873610149 p^{28} T^{68} - 1210380012760930 p^{29} T^{69} + 141565341364987 p^{30} T^{70} - 3692544723069 p^{31} T^{71} + 407371961466 p^{32} T^{72} - 8993542637 p^{33} T^{73} + 932136055 p^{34} T^{74} - 16391967 p^{35} T^{75} + 1588762 p^{36} T^{76} - 19866 p^{37} T^{77} + 1791 p^{38} T^{78} - 12 p^{39} T^{79} + p^{40} T^{80}$$
79 $$1 - 14 T + 1405 T^{2} - 17821 T^{3} + 983077 T^{4} - 11484758 T^{5} + 459535778 T^{6} - 5000906844 T^{7} + 162005264270 T^{8} - 1654875442942 T^{9} + 46029536426434 T^{10} - 443562686723881 T^{11} + 10988582368584097 T^{12} - 100210456595300242 T^{13} + 2267830851351188629 T^{14} - 19607849882216284821 T^{15} +$$$$41\!\cdots\!20$$$$T^{16} -$$$$33\!\cdots\!69$$$$T^{17} +$$$$67\!\cdots\!85$$$$T^{18} -$$$$52\!\cdots\!80$$$$T^{19} +$$$$99\!\cdots\!50$$$$T^{20} -$$$$73\!\cdots\!73$$$$T^{21} +$$$$13\!\cdots\!02$$$$T^{22} -$$$$95\!\cdots\!93$$$$T^{23} +$$$$21\!\cdots\!40$$$$p T^{24} -$$$$11\!\cdots\!42$$$$T^{25} +$$$$19\!\cdots\!15$$$$T^{26} -$$$$12\!\cdots\!40$$$$T^{27} +$$$$21\!\cdots\!82$$$$T^{28} -$$$$12\!\cdots\!07$$$$T^{29} +$$$$22\!\cdots\!27$$$$T^{30} -$$$$12\!\cdots\!79$$$$T^{31} +$$$$21\!\cdots\!98$$$$T^{32} -$$$$11\!\cdots\!52$$$$T^{33} +$$$$20\!\cdots\!61$$$$T^{34} -$$$$10\!\cdots\!80$$$$T^{35} +$$$$17\!\cdots\!29$$$$T^{36} -$$$$88\!\cdots\!74$$$$T^{37} +$$$$15\!\cdots\!71$$$$T^{38} -$$$$72\!\cdots\!19$$$$T^{39} +$$$$12\!\cdots\!07$$$$T^{40} -$$$$72\!\cdots\!19$$$$p T^{41} +$$$$15\!\cdots\!71$$$$p^{2} T^{42} -$$$$88\!\cdots\!74$$$$p^{3} T^{43} +$$$$17\!\cdots\!29$$$$p^{4} T^{44} -$$$$10\!\cdots\!80$$$$p^{5} T^{45} +$$$$20\!\cdots\!61$$$$p^{6} T^{46} -$$$$11\!\cdots\!52$$$$p^{7} T^{47} +$$$$21\!\cdots\!98$$$$p^{8} T^{48} -$$$$12\!\cdots\!79$$$$p^{9} T^{49} +$$$$22\!\cdots\!27$$$$p^{10} T^{50} -$$$$12\!\cdots\!07$$$$p^{11} T^{51} +$$$$21\!\cdots\!82$$$$p^{12} T^{52} -$$$$12\!\cdots\!40$$$$p^{13} T^{53} +$$$$19\!\cdots\!15$$$$p^{14} T^{54} -$$$$11\!\cdots\!42$$$$p^{15} T^{55} +$$$$21\!\cdots\!40$$$$p^{17} T^{56} -$$$$95\!\cdots\!93$$$$p^{17} T^{57} +$$$$13\!\cdots\!02$$$$p^{18} T^{58} -$$$$73\!\cdots\!73$$$$p^{19} T^{59} +$$$$99\!\cdots\!50$$$$p^{20} T^{60} -$$$$52\!\cdots\!80$$$$p^{21} T^{61} +$$$$67\!\cdots\!85$$$$p^{22} T^{62} -$$$$33\!\cdots\!69$$$$p^{23} T^{63} +$$$$41\!\cdots\!20$$$$p^{24} T^{64} - 19607849882216284821 p^{25} T^{65} + 2267830851351188629 p^{26} T^{66} - 100210456595300242 p^{27} T^{67} + 10988582368584097 p^{28} T^{68} - 443562686723881 p^{29} T^{69} + 46029536426434 p^{30} T^{70} - 1654875442942 p^{31} T^{71} + 162005264270 p^{32} T^{72} - 5000906844 p^{33} T^{73} + 459535778 p^{34} T^{74} - 11484758 p^{35} T^{75} + 983077 p^{36} T^{76} - 17821 p^{37} T^{77} + 1405 p^{38} T^{78} - 14 p^{39} T^{79} + p^{40} T^{80}$$
83 $$1 + 42 T + 2594 T^{2} + 81760 T^{3} + 3000940 T^{4} + 77333822 T^{5} + 2152067270 T^{6} + 47530430535 T^{7} + 1097695510103 T^{8} + 21397419599000 T^{9} + 429428277634827 T^{10} + 7536415183116980 T^{11} + 135115434575699416 T^{12} + 2165554068095163004 T^{13} + 35327051879781894267 T^{14} +$$$$52\!\cdots\!72$$$$T^{15} +$$$$78\!\cdots\!33$$$$T^{16} +$$$$10\!\cdots\!25$$$$T^{17} +$$$$15\!\cdots\!52$$$$T^{18} +$$$$19\!\cdots\!98$$$$T^{19} +$$$$25\!\cdots\!40$$$$T^{20} +$$$$31\!\cdots\!73$$$$T^{21} +$$$$39\!\cdots\!15$$$$T^{22} +$$$$45\!\cdots\!84$$$$T^{23} +$$$$53\!\cdots\!93$$$$T^{24} +$$$$59\!\cdots\!37$$$$T^{25} +$$$$67\!\cdots\!91$$$$T^{26} +$$$$71\!\cdots\!46$$$$T^{27} +$$$$78\!\cdots\!21$$$$T^{28} +$$$$79\!\cdots\!06$$$$T^{29} +$$$$85\!\cdots\!91$$$$T^{30} +$$$$84\!\cdots\!96$$$$T^{31} +$$$$87\!\cdots\!98$$$$T^{32} +$$$$84\!\cdots\!18$$$$T^{33} +$$$$85\!\cdots\!12$$$$T^{34} +$$$$80\!\cdots\!55$$$$T^{35} +$$$$79\!\cdots\!95$$$$T^{36} +$$$$72\!\cdots\!02$$$$T^{37} +$$$$70\!\cdots\!29$$$$T^{38} +$$$$63\!\cdots\!07$$$$T^{39} +$$$$59\!\cdots\!24$$$$T^{40} +$$$$63\!\cdots\!07$$$$p T^{41} +$$$$70\!\cdots\!29$$$$p^{2} T^{42} +$$$$72\!\cdots\!02$$$$p^{3} T^{43} +$$$$79\!\cdots\!95$$$$p^{4} T^{44} +$$$$80\!\cdots\!55$$$$p^{5} T^{45} +$$$$85\!\cdots\!12$$$$p^{6} T^{46} +$$$$84\!\cdots\!18$$$$p^{7} T^{47} +$$$$87\!\cdots\!98$$$$p^{8} T^{48} +$$$$84\!\cdots\!96$$$$p^{9} T^{49} +$$$$85\!\cdots\!91$$$$p^{10} T^{50} +$$$$79\!\cdots\!06$$$$p^{11} T^{51} +$$$$78\!\cdots\!21$$$$p^{12} T^{52} +$$$$71\!\cdots\!46$$$$p^{13} T^{53} +$$$$67\!\cdots\!91$$$$p^{14} T^{54} +$$$$59\!\cdots\!37$$$$p^{15} T^{55} +$$$$53\!\cdots\!93$$$$p^{16} T^{56} +$$$$45\!\cdots\!84$$$$p^{17} T^{57} +$$$$39\!\cdots\!15$$$$p^{18} T^{58} +$$$$31\!\cdots\!73$$$$p^{19} T^{59} +$$$$25\!\cdots\!40$$$$p^{20} T^{60} +$$$$19\!\cdots\!98$$$$p^{21} T^{61} +$$$$15\!\cdots\!52$$$$p^{22} T^{62} +$$$$10\!\cdots\!25$$$$p^{23} T^{63} +$$$$78\!\cdots\!33$$$$p^{24} T^{64} +$$$$52\!\cdots\!72$$$$p^{25} T^{65} + 35327051879781894267 p^{26} T^{66} + 2165554068095163004 p^{27} T^{67} + 135115434575699416 p^{28} T^{68} + 7536415183116980 p^{29} T^{69} + 429428277634827 p^{30} T^{70} + 21397419599000 p^{31} T^{71} + 1097695510103 p^{32} T^{72} + 47530430535 p^{33} T^{73} + 2152067270 p^{34} T^{74} + 77333822 p^{35} T^{75} + 3000940 p^{36} T^{76} + 81760 p^{37} T^{77} + 2594 p^{38} T^{78} + 42 p^{39} T^{79} + p^{40} T^{80}$$
89 $$1 + 118 T + 8357 T^{2} + 435413 T^{3} + 18431415 T^{4} + 665187064 T^{5} + 21135623804 T^{6} + 603779688483 T^{7} + 15751775792214 T^{8} + 379664111187907 T^{9} + 8532653929965743 T^{10} + 180116129648043464 T^{11} + 3592738929481574804 T^{12} + 68057102147116105170 T^{13} +$$$$12\!\cdots\!26$$$$T^{14} +$$$$21\!\cdots\!64$$$$T^{15} +$$$$35\!\cdots\!21$$$$T^{16} +$$$$56\!\cdots\!40$$$$T^{17} +$$$$87\!\cdots\!17$$$$T^{18} +$$$$12\!\cdots\!21$$$$T^{19} +$$$$18\!\cdots\!22$$$$T^{20} +$$$$26\!\cdots\!49$$$$T^{21} +$$$$36\!\cdots\!89$$$$T^{22} +$$$$47\!\cdots\!00$$$$T^{23} +$$$$62\!\cdots\!74$$$$T^{24} +$$$$78\!\cdots\!79$$$$T^{25} +$$$$96\!\cdots\!37$$$$T^{26} +$$$$11\!\cdots\!44$$$$T^{27} +$$$$13\!\cdots\!05$$$$T^{28} +$$$$16\!\cdots\!75$$$$T^{29} +$$$$18\!\cdots\!10$$$$T^{30} +$$$$20\!\cdots\!93$$$$T^{31} +$$$$22\!\cdots\!32$$$$T^{32} +$$$$23\!\cdots\!11$$$$T^{33} +$$$$24\!\cdots\!25$$$$T^{34} +$$$$25\!\cdots\!33$$$$T^{35} +$$$$26\!\cdots\!31$$$$T^{36} +$$$$26\!\cdots\!32$$$$T^{37} +$$$$25\!\cdots\!87$$$$T^{38} +$$$$24\!\cdots\!88$$$$T^{39} +$$$$23\!\cdots\!72$$$$T^{40} +$$$$24\!\cdots\!88$$$$p T^{41} +$$$$25\!\cdots\!87$$$$p^{2} T^{42} +$$$$26\!\cdots\!32$$$$p^{3} T^{43} +$$$$26\!\cdots\!31$$$$p^{4} T^{44} +$$$$25\!\cdots\!33$$$$p^{5} T^{45} +$$$$24\!\cdots\!25$$$$p^{6} T^{46} +$$$$23\!\cdots\!11$$$$p^{7} T^{47} +$$$$22\!\cdots\!32$$$$p^{8} T^{48} +$$$$20\!\cdots\!93$$$$p^{9} T^{49} +$$$$18\!\cdots\!10$$$$p^{10} T^{50} +$$$$16\!\cdots\!75$$$$p^{11} T^{51} +$$$$13\!\cdots\!05$$$$p^{12} T^{52} +$$$$11\!\cdots\!44$$$$p^{13} T^{53} +$$$$96\!\cdots\!37$$$$p^{14} T^{54} +$$$$78\!\cdots\!79$$$$p^{15} T^{55} +$$$$62\!\cdots\!74$$$$p^{16} T^{56} +$$$$47\!\cdots\!00$$$$p^{17} T^{57} +$$$$36\!\cdots\!89$$$$p^{18} T^{58} +$$$$26\!\cdots\!49$$$$p^{19} T^{59} +$$$$18\!\cdots\!22$$$$p^{20} T^{60} +$$$$12\!\cdots\!21$$$$p^{21} T^{61} +$$$$87\!\cdots\!17$$$$p^{22} T^{62} +$$$$56\!\cdots\!40$$$$p^{23} T^{63} +$$$$35\!\cdots\!21$$$$p^{24} T^{64} +$$$$21\!\cdots\!64$$$$p^{25} T^{65} +$$$$12\!\cdots\!26$$$$p^{26} T^{66} + 68057102147116105170 p^{27} T^{67} + 3592738929481574804 p^{28} T^{68} + 180116129648043464 p^{29} T^{69} + 8532653929965743 p^{30} T^{70} + 379664111187907 p^{31} T^{71} + 15751775792214 p^{32} T^{72} + 603779688483 p^{33} T^{73} + 21135623804 p^{34} T^{74} + 665187064 p^{35} T^{75} + 18431415 p^{36} T^{76} + 435413 p^{37} T^{77} + 8357 p^{38} T^{78} + 118 p^{39} T^{79} + p^{40} T^{80}$$
97 $$1 - 2 T + 2080 T^{2} - 5408 T^{3} + 2148147 T^{4} - 6877089 T^{5} + 1469228646 T^{6} - 5602322755 T^{7} + 748981494229 T^{8} - 3326766355455 T^{9} + 303724425889172 T^{10} - 1546133431653172 T^{11} + 102135375697367268 T^{12} - 588142307953997336 T^{13} + 29324916943826178101 T^{14} -$$$$18\!\cdots\!52$$$$T^{15} +$$$$73\!\cdots\!61$$$$T^{16} -$$$$52\!\cdots\!07$$$$T^{17} +$$$$16\!\cdots\!52$$$$T^{18} -$$$$12\!\cdots\!06$$$$T^{19} +$$$$32\!\cdots\!77$$$$T^{20} -$$$$27\!\cdots\!62$$$$T^{21} +$$$$61\!\cdots\!35$$$$p T^{22} -$$$$53\!\cdots\!88$$$$T^{23} +$$$$10\!\cdots\!63$$$$T^{24} -$$$$93\!\cdots\!11$$$$T^{25} +$$$$15\!\cdots\!21$$$$T^{26} -$$$$14\!\cdots\!60$$$$T^{27} +$$$$22\!\cdots\!61$$$$T^{28} -$$$$21\!\cdots\!90$$$$T^{29} +$$$$30\!\cdots\!54$$$$T^{30} -$$$$29\!\cdots\!45$$$$T^{31} +$$$$39\!\cdots\!06$$$$T^{32} -$$$$37\!\cdots\!94$$$$T^{33} +$$$$47\!\cdots\!21$$$$T^{34} -$$$$44\!\cdots\!14$$$$T^{35} +$$$$53\!\cdots\!47$$$$T^{36} -$$$$48\!\cdots\!11$$$$T^{37} +$$$$56\!\cdots\!38$$$$T^{38} -$$$$50\!\cdots\!43$$$$T^{39} +$$$$56\!\cdots\!00$$$$T^{40} -$$$$50\!\cdots\!43$$$$p T^{41} +$$$$56\!\cdots\!38$$$$p^{2} T^{42} -$$$$48\!\cdots\!11$$$$p^{3} T^{43} +$$$$53\!\cdots\!47$$$$p^{4} T^{44} -$$$$44\!\cdots\!14$$$$p^{5} T^{45} +$$$$47\!\cdots\!21$$$$p^{6} T^{46} -$$$$37\!\cdots\!94$$$$p^{7} T^{47} +$$$$39\!\cdots\!06$$$$p^{8} T^{48} -$$$$29\!\cdots\!45$$$$p^{9} T^{49} +$$$$30\!\cdots\!54$$$$p^{10} T^{50} -$$$$21\!\cdots\!90$$$$p^{11} T^{51} +$$$$22\!\cdots\!61$$$$p^{12} T^{52} -$$$$14\!\cdots\!60$$$$p^{13} T^{53} +$$$$15\!\cdots\!21$$$$p^{14} T^{54} -$$$$93\!\cdots\!11$$$$p^{15} T^{55} +$$$$10\!\cdots\!63$$$$p^{16} T^{56} -$$$$53\!\cdots\!88$$$$p^{17} T^{57} +$$$$61\!\cdots\!35$$$$p^{19} T^{58} -$$$$27\!\cdots\!62$$$$p^{19} T^{59} +$$$$32\!\cdots\!77$$$$p^{20} T^{60} -$$$$12\!\cdots\!06$$$$p^{21} T^{61} +$$$$16\!\cdots\!52$$$$p^{22} T^{62} -$$$$52\!\cdots\!07$$$$p^{23} T^{63} +$$$$73\!\cdots\!61$$$$p^{24} T^{64} -$$$$18\!\cdots\!52$$$$p^{25} T^{65} + 29324916943826178101 p^{26} T^{66} - 588142307953997336 p^{27} T^{67} + 102135375697367268 p^{28} T^{68} - 1546133431653172 p^{29} T^{69} + 303724425889172 p^{30} T^{70} - 3326766355455 p^{31} T^{71} + 748981494229 p^{32} T^{72} - 5602322755 p^{33} T^{73} + 1469228646 p^{34} T^{74} - 6877089 p^{35} T^{75} + 2148147 p^{36} T^{76} - 5408 p^{37} T^{77} + 2080 p^{38} T^{78} - 2 p^{39} T^{79} + p^{40} T^{80}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{80} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}