# Properties

 Degree 100 Conductor $2^{150} \cdot 751^{50}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 6·3-s + 23·5-s + 12·7-s − 29·9-s − 5·11-s + 36·13-s + 138·15-s + 14·17-s + 9·19-s + 72·21-s + 3·23-s + 175·25-s − 244·27-s + 61·29-s + 27·31-s − 30·33-s + 276·35-s + 56·37-s + 216·39-s + 10·41-s + 19·43-s − 667·45-s + 3·47-s − 62·49-s + 84·51-s + 56·53-s − 115·55-s + ⋯
 L(s)  = 1 + 3.46·3-s + 10.2·5-s + 4.53·7-s − 9.66·9-s − 1.50·11-s + 9.98·13-s + 35.6·15-s + 3.39·17-s + 2.06·19-s + 15.7·21-s + 0.625·23-s + 35·25-s − 46.9·27-s + 11.3·29-s + 4.84·31-s − 5.22·33-s + 46.6·35-s + 9.20·37-s + 34.5·39-s + 1.56·41-s + 2.89·43-s − 99.4·45-s + 0.437·47-s − 8.85·49-s + 11.7·51-s + 7.69·53-s − 15.5·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{150} \cdot 751^{50}\right)^{s/2} \, \Gamma_{\C}(s)^{50} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{150} \cdot 751^{50}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{50} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$100$$ $$N$$ = $$2^{150} \cdot 751^{50}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{6008} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(100,\ 2^{150} \cdot 751^{50} ,\ ( \ : [1/2]^{50} ),\ 1 )$ $L(1)$ $\approx$ $1.747344448e8$ $L(\frac12)$ $\approx$ $1.747344448e8$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;751\}$, $$F_p$$ is a polynomial of degree 100. If $p \in \{2,\;751\}$, then $F_p$ is a polynomial of degree at most 99.
$p$$F_p$
bad2 $$1$$
751 $$( 1 + T )^{50}$$
good3 $$1 - 2 p T + 65 T^{2} - 320 T^{3} + 680 p T^{4} - 8660 T^{5} + 42032 T^{6} - 158912 T^{7} + 647170 T^{8} - 2228408 T^{9} + 8002357 T^{10} - 25502839 T^{11} + 83174210 T^{12} - 248280463 T^{13} + 749668261 T^{14} - 2115191587 T^{15} + 5992231207 T^{16} - 16093755884 T^{17} + 43191758956 T^{18} - 37013433305 p T^{19} + 284362028480 T^{20} - 234309591523 p T^{21} + 63974384702 p^{3} T^{22} - 4120593722147 T^{23} + 9758657914453 T^{24} - 22534234852046 T^{25} + 51614254306589 T^{26} - 115658909416225 T^{27} + 256945565989222 T^{28} - 186639972305794 p T^{29} + 403106303059693 p T^{30} - 2567300944751786 T^{31} + 5401170093492832 T^{32} - 3729159925822724 p T^{33} + 22963703533302401 T^{34} - 46467788703491122 T^{35} + 3451299930697250 p^{3} T^{36} - 184413629459763535 T^{37} + 361709748585427723 T^{38} - 700709271257891141 T^{39} + 448496874259035973 p T^{40} - 2553379319839247297 T^{41} + 533726387305178426 p^{2} T^{42} - 8935428424363136159 T^{43} + 16478709013445581594 T^{44} - 10020248044003448137 p T^{45} + 54370692690704995583 T^{46} - 32433064339530423869 p T^{47} + 57550057125202229590 p T^{48} -$$$$30\!\cdots\!35$$$$T^{49} +$$$$52\!\cdots\!52$$$$T^{50} -$$$$30\!\cdots\!35$$$$p T^{51} + 57550057125202229590 p^{3} T^{52} - 32433064339530423869 p^{4} T^{53} + 54370692690704995583 p^{4} T^{54} - 10020248044003448137 p^{6} T^{55} + 16478709013445581594 p^{6} T^{56} - 8935428424363136159 p^{7} T^{57} + 533726387305178426 p^{10} T^{58} - 2553379319839247297 p^{9} T^{59} + 448496874259035973 p^{11} T^{60} - 700709271257891141 p^{11} T^{61} + 361709748585427723 p^{12} T^{62} - 184413629459763535 p^{13} T^{63} + 3451299930697250 p^{17} T^{64} - 46467788703491122 p^{15} T^{65} + 22963703533302401 p^{16} T^{66} - 3729159925822724 p^{18} T^{67} + 5401170093492832 p^{18} T^{68} - 2567300944751786 p^{19} T^{69} + 403106303059693 p^{21} T^{70} - 186639972305794 p^{22} T^{71} + 256945565989222 p^{22} T^{72} - 115658909416225 p^{23} T^{73} + 51614254306589 p^{24} T^{74} - 22534234852046 p^{25} T^{75} + 9758657914453 p^{26} T^{76} - 4120593722147 p^{27} T^{77} + 63974384702 p^{31} T^{78} - 234309591523 p^{30} T^{79} + 284362028480 p^{30} T^{80} - 37013433305 p^{32} T^{81} + 43191758956 p^{32} T^{82} - 16093755884 p^{33} T^{83} + 5992231207 p^{34} T^{84} - 2115191587 p^{35} T^{85} + 749668261 p^{36} T^{86} - 248280463 p^{37} T^{87} + 83174210 p^{38} T^{88} - 25502839 p^{39} T^{89} + 8002357 p^{40} T^{90} - 2228408 p^{41} T^{91} + 647170 p^{42} T^{92} - 158912 p^{43} T^{93} + 42032 p^{44} T^{94} - 8660 p^{45} T^{95} + 680 p^{47} T^{96} - 320 p^{47} T^{97} + 65 p^{48} T^{98} - 2 p^{50} T^{99} + p^{50} T^{100}$$
5 $$1 - 23 T + 354 T^{2} - 819 p T^{3} + 7921 p T^{4} - 331816 T^{5} + 2486244 T^{6} - 3389349 p T^{7} + 106642483 T^{8} - 625657303 T^{9} + 138045551 p^{2} T^{10} - 18011152928 T^{11} + 89419341229 T^{12} - 424148608876 T^{13} + 1929493922684 T^{14} - 337783132816 p^{2} T^{15} + 7131117079362 p T^{16} - 5823542529842 p^{2} T^{17} + 576098351364803 T^{18} - 2213301466823689 T^{19} + 1653903553504339 p T^{20} - 30092309242738347 T^{21} + 106793296323948378 T^{22} - 370052553420474784 T^{23} + 1253382227702744057 T^{24} - 830727968132135853 p T^{25} + 13479883200248003527 T^{26} - 42875268673219234664 T^{27} +$$$$13\!\cdots\!29$$$$T^{28} -$$$$40\!\cdots\!59$$$$T^{29} +$$$$24\!\cdots\!93$$$$p T^{30} -$$$$36\!\cdots\!63$$$$T^{31} +$$$$10\!\cdots\!02$$$$T^{32} -$$$$30\!\cdots\!12$$$$T^{33} +$$$$84\!\cdots\!73$$$$T^{34} -$$$$23\!\cdots\!68$$$$T^{35} +$$$$64\!\cdots\!12$$$$T^{36} -$$$$17\!\cdots\!57$$$$T^{37} +$$$$45\!\cdots\!17$$$$T^{38} -$$$$11\!\cdots\!63$$$$T^{39} +$$$$30\!\cdots\!22$$$$T^{40} -$$$$77\!\cdots\!24$$$$T^{41} +$$$$19\!\cdots\!22$$$$T^{42} -$$$$47\!\cdots\!66$$$$T^{43} +$$$$11\!\cdots\!24$$$$T^{44} -$$$$27\!\cdots\!11$$$$T^{45} +$$$$13\!\cdots\!44$$$$p T^{46} -$$$$15\!\cdots\!59$$$$T^{47} +$$$$35\!\cdots\!99$$$$T^{48} -$$$$16\!\cdots\!73$$$$p T^{49} +$$$$73\!\cdots\!96$$$$p^{2} T^{50} -$$$$16\!\cdots\!73$$$$p^{2} T^{51} +$$$$35\!\cdots\!99$$$$p^{2} T^{52} -$$$$15\!\cdots\!59$$$$p^{3} T^{53} +$$$$13\!\cdots\!44$$$$p^{5} T^{54} -$$$$27\!\cdots\!11$$$$p^{5} T^{55} +$$$$11\!\cdots\!24$$$$p^{6} T^{56} -$$$$47\!\cdots\!66$$$$p^{7} T^{57} +$$$$19\!\cdots\!22$$$$p^{8} T^{58} -$$$$77\!\cdots\!24$$$$p^{9} T^{59} +$$$$30\!\cdots\!22$$$$p^{10} T^{60} -$$$$11\!\cdots\!63$$$$p^{11} T^{61} +$$$$45\!\cdots\!17$$$$p^{12} T^{62} -$$$$17\!\cdots\!57$$$$p^{13} T^{63} +$$$$64\!\cdots\!12$$$$p^{14} T^{64} -$$$$23\!\cdots\!68$$$$p^{15} T^{65} +$$$$84\!\cdots\!73$$$$p^{16} T^{66} -$$$$30\!\cdots\!12$$$$p^{17} T^{67} +$$$$10\!\cdots\!02$$$$p^{18} T^{68} -$$$$36\!\cdots\!63$$$$p^{19} T^{69} +$$$$24\!\cdots\!93$$$$p^{21} T^{70} -$$$$40\!\cdots\!59$$$$p^{21} T^{71} +$$$$13\!\cdots\!29$$$$p^{22} T^{72} - 42875268673219234664 p^{23} T^{73} + 13479883200248003527 p^{24} T^{74} - 830727968132135853 p^{26} T^{75} + 1253382227702744057 p^{26} T^{76} - 370052553420474784 p^{27} T^{77} + 106793296323948378 p^{28} T^{78} - 30092309242738347 p^{29} T^{79} + 1653903553504339 p^{31} T^{80} - 2213301466823689 p^{31} T^{81} + 576098351364803 p^{32} T^{82} - 5823542529842 p^{35} T^{83} + 7131117079362 p^{35} T^{84} - 337783132816 p^{37} T^{85} + 1929493922684 p^{36} T^{86} - 424148608876 p^{37} T^{87} + 89419341229 p^{38} T^{88} - 18011152928 p^{39} T^{89} + 138045551 p^{42} T^{90} - 625657303 p^{41} T^{91} + 106642483 p^{42} T^{92} - 3389349 p^{44} T^{93} + 2486244 p^{44} T^{94} - 331816 p^{45} T^{95} + 7921 p^{47} T^{96} - 819 p^{48} T^{97} + 354 p^{48} T^{98} - 23 p^{49} T^{99} + p^{50} T^{100}$$
7 $$1 - 12 T + 206 T^{2} - 1894 T^{3} + 19646 T^{4} - 21458 p T^{5} + 1201565 T^{6} - 1142766 p T^{7} + 54005838 T^{8} - 322283713 T^{9} + 1921951001 T^{10} - 10484935505 T^{11} + 56743156259 T^{12} - 40997602183 p T^{13} + 1434518752286 T^{14} - 6796614298484 T^{15} + 31767956852136 T^{16} - 142128875608317 T^{17} + 626815202208107 T^{18} - 2664753744603435 T^{19} + 1594953342211374 p T^{20} - 45327821574052750 T^{21} + 181386794613553177 T^{22} - 706152789382509673 T^{23} + 2710466580254419881 T^{24} - 10152559531169806573 T^{25} + 37508666582484110483 T^{26} -$$$$13\!\cdots\!96$$$$T^{27} + 69061439089053274625 p T^{28} -$$$$16\!\cdots\!64$$$$T^{29} +$$$$58\!\cdots\!74$$$$T^{30} -$$$$19\!\cdots\!94$$$$T^{31} +$$$$66\!\cdots\!09$$$$T^{32} -$$$$31\!\cdots\!89$$$$p T^{33} +$$$$70\!\cdots\!10$$$$T^{34} -$$$$22\!\cdots\!99$$$$T^{35} +$$$$10\!\cdots\!62$$$$p T^{36} -$$$$22\!\cdots\!28$$$$T^{37} +$$$$68\!\cdots\!13$$$$T^{38} -$$$$29\!\cdots\!86$$$$p T^{39} +$$$$62\!\cdots\!45$$$$T^{40} -$$$$26\!\cdots\!23$$$$p T^{41} +$$$$53\!\cdots\!68$$$$T^{42} -$$$$22\!\cdots\!22$$$$p T^{43} +$$$$90\!\cdots\!54$$$$p^{2} T^{44} -$$$$12\!\cdots\!24$$$$T^{45} +$$$$34\!\cdots\!57$$$$T^{46} -$$$$95\!\cdots\!06$$$$T^{47} +$$$$26\!\cdots\!36$$$$T^{48} -$$$$70\!\cdots\!16$$$$T^{49} +$$$$18\!\cdots\!38$$$$T^{50} -$$$$70\!\cdots\!16$$$$p T^{51} +$$$$26\!\cdots\!36$$$$p^{2} T^{52} -$$$$95\!\cdots\!06$$$$p^{3} T^{53} +$$$$34\!\cdots\!57$$$$p^{4} T^{54} -$$$$12\!\cdots\!24$$$$p^{5} T^{55} +$$$$90\!\cdots\!54$$$$p^{8} T^{56} -$$$$22\!\cdots\!22$$$$p^{8} T^{57} +$$$$53\!\cdots\!68$$$$p^{8} T^{58} -$$$$26\!\cdots\!23$$$$p^{10} T^{59} +$$$$62\!\cdots\!45$$$$p^{10} T^{60} -$$$$29\!\cdots\!86$$$$p^{12} T^{61} +$$$$68\!\cdots\!13$$$$p^{12} T^{62} -$$$$22\!\cdots\!28$$$$p^{13} T^{63} +$$$$10\!\cdots\!62$$$$p^{15} T^{64} -$$$$22\!\cdots\!99$$$$p^{15} T^{65} +$$$$70\!\cdots\!10$$$$p^{16} T^{66} -$$$$31\!\cdots\!89$$$$p^{18} T^{67} +$$$$66\!\cdots\!09$$$$p^{18} T^{68} -$$$$19\!\cdots\!94$$$$p^{19} T^{69} +$$$$58\!\cdots\!74$$$$p^{20} T^{70} -$$$$16\!\cdots\!64$$$$p^{21} T^{71} + 69061439089053274625 p^{23} T^{72} -$$$$13\!\cdots\!96$$$$p^{23} T^{73} + 37508666582484110483 p^{24} T^{74} - 10152559531169806573 p^{25} T^{75} + 2710466580254419881 p^{26} T^{76} - 706152789382509673 p^{27} T^{77} + 181386794613553177 p^{28} T^{78} - 45327821574052750 p^{29} T^{79} + 1594953342211374 p^{31} T^{80} - 2664753744603435 p^{31} T^{81} + 626815202208107 p^{32} T^{82} - 142128875608317 p^{33} T^{83} + 31767956852136 p^{34} T^{84} - 6796614298484 p^{35} T^{85} + 1434518752286 p^{36} T^{86} - 40997602183 p^{38} T^{87} + 56743156259 p^{38} T^{88} - 10484935505 p^{39} T^{89} + 1921951001 p^{40} T^{90} - 322283713 p^{41} T^{91} + 54005838 p^{42} T^{92} - 1142766 p^{44} T^{93} + 1201565 p^{44} T^{94} - 21458 p^{46} T^{95} + 19646 p^{46} T^{96} - 1894 p^{47} T^{97} + 206 p^{48} T^{98} - 12 p^{49} T^{99} + p^{50} T^{100}$$
11 $$1 + 5 T + 247 T^{2} + 1304 T^{3} + 31110 T^{4} + 169199 T^{5} + 2657513 T^{6} + 14607633 T^{7} + 172726887 T^{8} + 945950634 T^{9} + 826100226 p T^{10} + 49081275839 T^{11} + 402179793693 T^{12} + 2127659769186 T^{13} + 15375183967039 T^{14} + 79323866322442 T^{15} + 517653850013286 T^{16} + 2598043882967825 T^{17} + 15579791846274672 T^{18} + 75978305393572350 T^{19} + 424198035047528703 T^{20} + 2009582920692556767 T^{21} + 10551363847290948926 T^{22} + 48573027351235677174 T^{23} +$$$$24\!\cdots\!82$$$$T^{24} +$$$$10\!\cdots\!89$$$$T^{25} +$$$$46\!\cdots\!80$$$$p T^{26} +$$$$22\!\cdots\!27$$$$T^{27} +$$$$10\!\cdots\!06$$$$T^{28} +$$$$43\!\cdots\!85$$$$T^{29} +$$$$18\!\cdots\!86$$$$T^{30} +$$$$78\!\cdots\!01$$$$T^{31} +$$$$33\!\cdots\!49$$$$T^{32} +$$$$13\!\cdots\!70$$$$T^{33} +$$$$54\!\cdots\!34$$$$T^{34} +$$$$19\!\cdots\!73$$$$p T^{35} +$$$$85\!\cdots\!02$$$$T^{36} +$$$$32\!\cdots\!47$$$$T^{37} +$$$$12\!\cdots\!07$$$$T^{38} +$$$$47\!\cdots\!71$$$$T^{39} +$$$$17\!\cdots\!59$$$$T^{40} +$$$$66\!\cdots\!45$$$$T^{41} +$$$$24\!\cdots\!54$$$$T^{42} +$$$$79\!\cdots\!54$$$$p T^{43} +$$$$31\!\cdots\!01$$$$T^{44} +$$$$10\!\cdots\!35$$$$T^{45} +$$$$38\!\cdots\!09$$$$T^{46} +$$$$13\!\cdots\!66$$$$T^{47} +$$$$45\!\cdots\!81$$$$T^{48} +$$$$15\!\cdots\!33$$$$T^{49} +$$$$50\!\cdots\!02$$$$T^{50} +$$$$15\!\cdots\!33$$$$p T^{51} +$$$$45\!\cdots\!81$$$$p^{2} T^{52} +$$$$13\!\cdots\!66$$$$p^{3} T^{53} +$$$$38\!\cdots\!09$$$$p^{4} T^{54} +$$$$10\!\cdots\!35$$$$p^{5} T^{55} +$$$$31\!\cdots\!01$$$$p^{6} T^{56} +$$$$79\!\cdots\!54$$$$p^{8} T^{57} +$$$$24\!\cdots\!54$$$$p^{8} T^{58} +$$$$66\!\cdots\!45$$$$p^{9} T^{59} +$$$$17\!\cdots\!59$$$$p^{10} T^{60} +$$$$47\!\cdots\!71$$$$p^{11} T^{61} +$$$$12\!\cdots\!07$$$$p^{12} T^{62} +$$$$32\!\cdots\!47$$$$p^{13} T^{63} +$$$$85\!\cdots\!02$$$$p^{14} T^{64} +$$$$19\!\cdots\!73$$$$p^{16} T^{65} +$$$$54\!\cdots\!34$$$$p^{16} T^{66} +$$$$13\!\cdots\!70$$$$p^{17} T^{67} +$$$$33\!\cdots\!49$$$$p^{18} T^{68} +$$$$78\!\cdots\!01$$$$p^{19} T^{69} +$$$$18\!\cdots\!86$$$$p^{20} T^{70} +$$$$43\!\cdots\!85$$$$p^{21} T^{71} +$$$$10\!\cdots\!06$$$$p^{22} T^{72} +$$$$22\!\cdots\!27$$$$p^{23} T^{73} +$$$$46\!\cdots\!80$$$$p^{25} T^{74} +$$$$10\!\cdots\!89$$$$p^{25} T^{75} +$$$$24\!\cdots\!82$$$$p^{26} T^{76} + 48573027351235677174 p^{27} T^{77} + 10551363847290948926 p^{28} T^{78} + 2009582920692556767 p^{29} T^{79} + 424198035047528703 p^{30} T^{80} + 75978305393572350 p^{31} T^{81} + 15579791846274672 p^{32} T^{82} + 2598043882967825 p^{33} T^{83} + 517653850013286 p^{34} T^{84} + 79323866322442 p^{35} T^{85} + 15375183967039 p^{36} T^{86} + 2127659769186 p^{37} T^{87} + 402179793693 p^{38} T^{88} + 49081275839 p^{39} T^{89} + 826100226 p^{41} T^{90} + 945950634 p^{41} T^{91} + 172726887 p^{42} T^{92} + 14607633 p^{43} T^{93} + 2657513 p^{44} T^{94} + 169199 p^{45} T^{95} + 31110 p^{46} T^{96} + 1304 p^{47} T^{97} + 247 p^{48} T^{98} + 5 p^{49} T^{99} + p^{50} T^{100}$$
13 $$1 - 36 T + 937 T^{2} - 1392 p T^{3} + 296045 T^{4} - 4176556 T^{5} + 52816588 T^{6} - 605636308 T^{7} + 6405468422 T^{8} - 62967327121 T^{9} + 580721339963 T^{10} - 5051661750648 T^{11} + 41693434372349 T^{12} - 327770197851568 T^{13} + 2464327192257399 T^{14} - 17772557347895306 T^{15} + 123315625792461598 T^{16} - 825132612364057398 T^{17} + 5336625156135167175 T^{18} - 33425255808421870905 T^{19} + 15624504441596789502 p T^{20} -$$$$11\!\cdots\!12$$$$T^{21} +$$$$68\!\cdots\!69$$$$T^{22} -$$$$38\!\cdots\!10$$$$T^{23} +$$$$16\!\cdots\!16$$$$p T^{24} -$$$$11\!\cdots\!73$$$$T^{25} +$$$$45\!\cdots\!11$$$$p T^{26} -$$$$23\!\cdots\!43$$$$p T^{27} +$$$$15\!\cdots\!58$$$$T^{28} -$$$$73\!\cdots\!50$$$$T^{29} +$$$$35\!\cdots\!86$$$$T^{30} -$$$$16\!\cdots\!25$$$$T^{31} +$$$$78\!\cdots\!20$$$$T^{32} -$$$$35\!\cdots\!09$$$$T^{33} +$$$$16\!\cdots\!49$$$$T^{34} -$$$$71\!\cdots\!54$$$$T^{35} +$$$$31\!\cdots\!89$$$$T^{36} -$$$$13\!\cdots\!33$$$$T^{37} +$$$$56\!\cdots\!65$$$$T^{38} -$$$$23\!\cdots\!07$$$$T^{39} +$$$$97\!\cdots\!06$$$$T^{40} -$$$$39\!\cdots\!48$$$$T^{41} +$$$$15\!\cdots\!47$$$$T^{42} -$$$$63\!\cdots\!19$$$$T^{43} +$$$$24\!\cdots\!00$$$$T^{44} -$$$$95\!\cdots\!29$$$$T^{45} +$$$$36\!\cdots\!15$$$$T^{46} -$$$$10\!\cdots\!76$$$$p T^{47} +$$$$50\!\cdots\!66$$$$T^{48} -$$$$18\!\cdots\!58$$$$T^{49} +$$$$67\!\cdots\!68$$$$T^{50} -$$$$18\!\cdots\!58$$$$p T^{51} +$$$$50\!\cdots\!66$$$$p^{2} T^{52} -$$$$10\!\cdots\!76$$$$p^{4} T^{53} +$$$$36\!\cdots\!15$$$$p^{4} T^{54} -$$$$95\!\cdots\!29$$$$p^{5} T^{55} +$$$$24\!\cdots\!00$$$$p^{6} T^{56} -$$$$63\!\cdots\!19$$$$p^{7} T^{57} +$$$$15\!\cdots\!47$$$$p^{8} T^{58} -$$$$39\!\cdots\!48$$$$p^{9} T^{59} +$$$$97\!\cdots\!06$$$$p^{10} T^{60} -$$$$23\!\cdots\!07$$$$p^{11} T^{61} +$$$$56\!\cdots\!65$$$$p^{12} T^{62} -$$$$13\!\cdots\!33$$$$p^{13} T^{63} +$$$$31\!\cdots\!89$$$$p^{14} T^{64} -$$$$71\!\cdots\!54$$$$p^{15} T^{65} +$$$$16\!\cdots\!49$$$$p^{16} T^{66} -$$$$35\!\cdots\!09$$$$p^{17} T^{67} +$$$$78\!\cdots\!20$$$$p^{18} T^{68} -$$$$16\!\cdots\!25$$$$p^{19} T^{69} +$$$$35\!\cdots\!86$$$$p^{20} T^{70} -$$$$73\!\cdots\!50$$$$p^{21} T^{71} +$$$$15\!\cdots\!58$$$$p^{22} T^{72} -$$$$23\!\cdots\!43$$$$p^{24} T^{73} +$$$$45\!\cdots\!11$$$$p^{25} T^{74} -$$$$11\!\cdots\!73$$$$p^{25} T^{75} +$$$$16\!\cdots\!16$$$$p^{27} T^{76} -$$$$38\!\cdots\!10$$$$p^{27} T^{77} +$$$$68\!\cdots\!69$$$$p^{28} T^{78} -$$$$11\!\cdots\!12$$$$p^{29} T^{79} + 15624504441596789502 p^{31} T^{80} - 33425255808421870905 p^{31} T^{81} + 5336625156135167175 p^{32} T^{82} - 825132612364057398 p^{33} T^{83} + 123315625792461598 p^{34} T^{84} - 17772557347895306 p^{35} T^{85} + 2464327192257399 p^{36} T^{86} - 327770197851568 p^{37} T^{87} + 41693434372349 p^{38} T^{88} - 5051661750648 p^{39} T^{89} + 580721339963 p^{40} T^{90} - 62967327121 p^{41} T^{91} + 6405468422 p^{42} T^{92} - 605636308 p^{43} T^{93} + 52816588 p^{44} T^{94} - 4176556 p^{45} T^{95} + 296045 p^{46} T^{96} - 1392 p^{48} T^{97} + 937 p^{48} T^{98} - 36 p^{49} T^{99} + p^{50} T^{100}$$
17 $$1 - 14 T + 480 T^{2} - 5648 T^{3} + 110400 T^{4} - 1133005 T^{5} + 16436492 T^{6} - 150760482 T^{7} + 1795619610 T^{8} - 14974434599 T^{9} + 154268408480 T^{10} - 1184456956752 T^{11} + 10891109000052 T^{12} - 77718676100314 T^{13} + 38306598995580 p T^{14} - 4350441698149463 T^{15} + 33709583018940684 T^{16} - 212006539553171546 T^{17} + 1535805614488232229 T^{18} - 9132252951999652533 T^{19} + 3669098442578983618 p T^{20} -$$$$35\!\cdots\!50$$$$T^{21} +$$$$22\!\cdots\!74$$$$T^{22} -$$$$12\!\cdots\!16$$$$T^{23} +$$$$75\!\cdots\!87$$$$T^{24} -$$$$38\!\cdots\!12$$$$T^{25} +$$$$22\!\cdots\!61$$$$T^{26} -$$$$11\!\cdots\!22$$$$T^{27} +$$$$63\!\cdots\!81$$$$T^{28} -$$$$29\!\cdots\!51$$$$T^{29} +$$$$16\!\cdots\!79$$$$T^{30} -$$$$71\!\cdots\!26$$$$T^{31} +$$$$38\!\cdots\!42$$$$T^{32} -$$$$16\!\cdots\!78$$$$T^{33} +$$$$85\!\cdots\!94$$$$T^{34} -$$$$33\!\cdots\!04$$$$T^{35} +$$$$17\!\cdots\!75$$$$T^{36} -$$$$63\!\cdots\!95$$$$T^{37} +$$$$32\!\cdots\!99$$$$T^{38} -$$$$66\!\cdots\!43$$$$p T^{39} +$$$$58\!\cdots\!83$$$$T^{40} -$$$$18\!\cdots\!91$$$$T^{41} +$$$$97\!\cdots\!19$$$$T^{42} -$$$$15\!\cdots\!36$$$$p T^{43} +$$$$15\!\cdots\!11$$$$T^{44} -$$$$37\!\cdots\!74$$$$T^{45} +$$$$23\!\cdots\!36$$$$T^{46} -$$$$51\!\cdots\!81$$$$T^{47} +$$$$37\!\cdots\!44$$$$T^{48} -$$$$73\!\cdots\!13$$$$T^{49} +$$$$61\!\cdots\!14$$$$T^{50} -$$$$73\!\cdots\!13$$$$p T^{51} +$$$$37\!\cdots\!44$$$$p^{2} T^{52} -$$$$51\!\cdots\!81$$$$p^{3} T^{53} +$$$$23\!\cdots\!36$$$$p^{4} T^{54} -$$$$37\!\cdots\!74$$$$p^{5} T^{55} +$$$$15\!\cdots\!11$$$$p^{6} T^{56} -$$$$15\!\cdots\!36$$$$p^{8} T^{57} +$$$$97\!\cdots\!19$$$$p^{8} T^{58} -$$$$18\!\cdots\!91$$$$p^{9} T^{59} +$$$$58\!\cdots\!83$$$$p^{10} T^{60} -$$$$66\!\cdots\!43$$$$p^{12} T^{61} +$$$$32\!\cdots\!99$$$$p^{12} T^{62} -$$$$63\!\cdots\!95$$$$p^{13} T^{63} +$$$$17\!\cdots\!75$$$$p^{14} T^{64} -$$$$33\!\cdots\!04$$$$p^{15} T^{65} +$$$$85\!\cdots\!94$$$$p^{16} T^{66} -$$$$16\!\cdots\!78$$$$p^{17} T^{67} +$$$$38\!\cdots\!42$$$$p^{18} T^{68} -$$$$71\!\cdots\!26$$$$p^{19} T^{69} +$$$$16\!\cdots\!79$$$$p^{20} T^{70} -$$$$29\!\cdots\!51$$$$p^{21} T^{71} +$$$$63\!\cdots\!81$$$$p^{22} T^{72} -$$$$11\!\cdots\!22$$$$p^{23} T^{73} +$$$$22\!\cdots\!61$$$$p^{24} T^{74} -$$$$38\!\cdots\!12$$$$p^{25} T^{75} +$$$$75\!\cdots\!87$$$$p^{26} T^{76} -$$$$12\!\cdots\!16$$$$p^{27} T^{77} +$$$$22\!\cdots\!74$$$$p^{28} T^{78} -$$$$35\!\cdots\!50$$$$p^{29} T^{79} + 3669098442578983618 p^{31} T^{80} - 9132252951999652533 p^{31} T^{81} + 1535805614488232229 p^{32} T^{82} - 212006539553171546 p^{33} T^{83} + 33709583018940684 p^{34} T^{84} - 4350441698149463 p^{35} T^{85} + 38306598995580 p^{37} T^{86} - 77718676100314 p^{37} T^{87} + 10891109000052 p^{38} T^{88} - 1184456956752 p^{39} T^{89} + 154268408480 p^{40} T^{90} - 14974434599 p^{41} T^{91} + 1795619610 p^{42} T^{92} - 150760482 p^{43} T^{93} + 16436492 p^{44} T^{94} - 1133005 p^{45} T^{95} + 110400 p^{46} T^{96} - 5648 p^{47} T^{97} + 480 p^{48} T^{98} - 14 p^{49} T^{99} + p^{50} T^{100}$$
19 $$1 - 9 T + 447 T^{2} - 187 p T^{3} + 98115 T^{4} - 697987 T^{5} + 14158630 T^{6} - 90871086 T^{7} + 1515180148 T^{8} - 8807748769 T^{9} + 6762978480 p T^{10} - 676795510872 T^{11} + 9008676846399 T^{12} - 42858970794229 T^{13} + 537756850365456 T^{14} - 2294765313391013 T^{15} + 1470571258697873 p T^{16} - 5562698074729302 p T^{17} + 1285854567661472312 T^{18} - 4233901940859659637 T^{19} + 53178791570091977517 T^{20} -$$$$14\!\cdots\!41$$$$T^{21} +$$$$20\!\cdots\!45$$$$T^{22} -$$$$45\!\cdots\!90$$$$T^{23} +$$$$69\!\cdots\!23$$$$T^{24} -$$$$11\!\cdots\!75$$$$T^{25} +$$$$22\!\cdots\!24$$$$T^{26} -$$$$26\!\cdots\!98$$$$T^{27} +$$$$67\!\cdots\!73$$$$T^{28} -$$$$40\!\cdots\!57$$$$T^{29} +$$$$19\!\cdots\!30$$$$T^{30} -$$$$11\!\cdots\!39$$$$T^{31} +$$$$53\!\cdots\!75$$$$T^{32} +$$$$23\!\cdots\!57$$$$T^{33} +$$$$13\!\cdots\!05$$$$T^{34} +$$$$12\!\cdots\!13$$$$T^{35} +$$$$35\!\cdots\!37$$$$T^{36} +$$$$44\!\cdots\!32$$$$T^{37} +$$$$85\!\cdots\!84$$$$T^{38} +$$$$70\!\cdots\!56$$$$p T^{39} +$$$$20\!\cdots\!53$$$$T^{40} +$$$$35\!\cdots\!67$$$$T^{41} +$$$$45\!\cdots\!27$$$$T^{42} +$$$$87\!\cdots\!53$$$$T^{43} +$$$$10\!\cdots\!48$$$$T^{44} +$$$$19\!\cdots\!84$$$$T^{45} +$$$$21\!\cdots\!79$$$$T^{46} +$$$$42\!\cdots\!74$$$$T^{47} +$$$$42\!\cdots\!28$$$$T^{48} +$$$$84\!\cdots\!57$$$$T^{49} +$$$$82\!\cdots\!78$$$$T^{50} +$$$$84\!\cdots\!57$$$$p T^{51} +$$$$42\!\cdots\!28$$$$p^{2} T^{52} +$$$$42\!\cdots\!74$$$$p^{3} T^{53} +$$$$21\!\cdots\!79$$$$p^{4} T^{54} +$$$$19\!\cdots\!84$$$$p^{5} T^{55} +$$$$10\!\cdots\!48$$$$p^{6} T^{56} +$$$$87\!\cdots\!53$$$$p^{7} T^{57} +$$$$45\!\cdots\!27$$$$p^{8} T^{58} +$$$$35\!\cdots\!67$$$$p^{9} T^{59} +$$$$20\!\cdots\!53$$$$p^{10} T^{60} +$$$$70\!\cdots\!56$$$$p^{12} T^{61} +$$$$85\!\cdots\!84$$$$p^{12} T^{62} +$$$$44\!\cdots\!32$$$$p^{13} T^{63} +$$$$35\!\cdots\!37$$$$p^{14} T^{64} +$$$$12\!\cdots\!13$$$$p^{15} T^{65} +$$$$13\!\cdots\!05$$$$p^{16} T^{66} +$$$$23\!\cdots\!57$$$$p^{17} T^{67} +$$$$53\!\cdots\!75$$$$p^{18} T^{68} -$$$$11\!\cdots\!39$$$$p^{19} T^{69} +$$$$19\!\cdots\!30$$$$p^{20} T^{70} -$$$$40\!\cdots\!57$$$$p^{21} T^{71} +$$$$67\!\cdots\!73$$$$p^{22} T^{72} -$$$$26\!\cdots\!98$$$$p^{23} T^{73} +$$$$22\!\cdots\!24$$$$p^{24} T^{74} -$$$$11\!\cdots\!75$$$$p^{25} T^{75} +$$$$69\!\cdots\!23$$$$p^{26} T^{76} -$$$$45\!\cdots\!90$$$$p^{27} T^{77} +$$$$20\!\cdots\!45$$$$p^{28} T^{78} -$$$$14\!\cdots\!41$$$$p^{29} T^{79} + 53178791570091977517 p^{30} T^{80} - 4233901940859659637 p^{31} T^{81} + 1285854567661472312 p^{32} T^{82} - 5562698074729302 p^{34} T^{83} + 1470571258697873 p^{35} T^{84} - 2294765313391013 p^{35} T^{85} + 537756850365456 p^{36} T^{86} - 42858970794229 p^{37} T^{87} + 9008676846399 p^{38} T^{88} - 676795510872 p^{39} T^{89} + 6762978480 p^{41} T^{90} - 8807748769 p^{41} T^{91} + 1515180148 p^{42} T^{92} - 90871086 p^{43} T^{93} + 14158630 p^{44} T^{94} - 697987 p^{45} T^{95} + 98115 p^{46} T^{96} - 187 p^{48} T^{97} + 447 p^{48} T^{98} - 9 p^{49} T^{99} + p^{50} T^{100}$$
23 $$1 - 3 T + 559 T^{2} - 1728 T^{3} + 157110 T^{4} - 496472 T^{5} + 29601768 T^{6} - 94962184 T^{7} + 4205751758 T^{8} - 13616289859 T^{9} + 480514510477 T^{10} - 1562386613719 T^{11} + 45973578062245 T^{12} - 149538689428524 T^{13} + 3787459729298427 T^{14} - 12286289613617705 T^{15} + 274186050158342671 T^{16} - 884972145578172802 T^{17} + 17713722564169528058 T^{18} - 56788634300440057122 T^{19} +$$$$10\!\cdots\!73$$$$T^{20} -$$$$32\!\cdots\!36$$$$T^{21} +$$$$55\!\cdots\!79$$$$T^{22} -$$$$17\!\cdots\!18$$$$T^{23} +$$$$26\!\cdots\!53$$$$T^{24} -$$$$84\!\cdots\!83$$$$T^{25} +$$$$12\!\cdots\!02$$$$T^{26} -$$$$37\!\cdots\!21$$$$T^{27} +$$$$51\!\cdots\!07$$$$T^{28} -$$$$15\!\cdots\!22$$$$T^{29} +$$$$20\!\cdots\!09$$$$T^{30} -$$$$61\!\cdots\!53$$$$T^{31} +$$$$75\!\cdots\!58$$$$T^{32} -$$$$22\!\cdots\!54$$$$T^{33} +$$$$26\!\cdots\!19$$$$T^{34} -$$$$78\!\cdots\!70$$$$T^{35} +$$$$87\!\cdots\!00$$$$T^{36} -$$$$25\!\cdots\!74$$$$T^{37} +$$$$27\!\cdots\!83$$$$T^{38} -$$$$79\!\cdots\!59$$$$T^{39} +$$$$81\!\cdots\!78$$$$T^{40} -$$$$23\!\cdots\!47$$$$T^{41} +$$$$10\!\cdots\!56$$$$p T^{42} -$$$$28\!\cdots\!11$$$$p T^{43} +$$$$62\!\cdots\!83$$$$T^{44} -$$$$17\!\cdots\!39$$$$T^{45} +$$$$16\!\cdots\!30$$$$T^{46} -$$$$44\!\cdots\!72$$$$T^{47} +$$$$39\!\cdots\!15$$$$T^{48} -$$$$10\!\cdots\!93$$$$T^{49} +$$$$93\!\cdots\!26$$$$T^{50} -$$$$10\!\cdots\!93$$$$p T^{51} +$$$$39\!\cdots\!15$$$$p^{2} T^{52} -$$$$44\!\cdots\!72$$$$p^{3} T^{53} +$$$$16\!\cdots\!30$$$$p^{4} T^{54} -$$$$17\!\cdots\!39$$$$p^{5} T^{55} +$$$$62\!\cdots\!83$$$$p^{6} T^{56} -$$$$28\!\cdots\!11$$$$p^{8} T^{57} +$$$$10\!\cdots\!56$$$$p^{9} T^{58} -$$$$23\!\cdots\!47$$$$p^{9} T^{59} +$$$$81\!\cdots\!78$$$$p^{10} T^{60} -$$$$79\!\cdots\!59$$$$p^{11} T^{61} +$$$$27\!\cdots\!83$$$$p^{12} T^{62} -$$$$25\!\cdots\!74$$$$p^{13} T^{63} +$$$$87\!\cdots\!00$$$$p^{14} T^{64} -$$$$78\!\cdots\!70$$$$p^{15} T^{65} +$$$$26\!\cdots\!19$$$$p^{16} T^{66} -$$$$22\!\cdots\!54$$$$p^{17} T^{67} +$$$$75\!\cdots\!58$$$$p^{18} T^{68} -$$$$61\!\cdots\!53$$$$p^{19} T^{69} +$$$$20\!\cdots\!09$$$$p^{20} T^{70} -$$$$15\!\cdots\!22$$$$p^{21} T^{71} +$$$$51\!\cdots\!07$$$$p^{22} T^{72} -$$$$37\!\cdots\!21$$$$p^{23} T^{73} +$$$$12\!\cdots\!02$$$$p^{24} T^{74} -$$$$84\!\cdots\!83$$$$p^{25} T^{75} +$$$$26\!\cdots\!53$$$$p^{26} T^{76} -$$$$17\!\cdots\!18$$$$p^{27} T^{77} +$$$$55\!\cdots\!79$$$$p^{28} T^{78} -$$$$32\!\cdots\!36$$$$p^{29} T^{79} +$$$$10\!\cdots\!73$$$$p^{30} T^{80} - 56788634300440057122 p^{31} T^{81} + 17713722564169528058 p^{32} T^{82} - 884972145578172802 p^{33} T^{83} + 274186050158342671 p^{34} T^{84} - 12286289613617705 p^{35} T^{85} + 3787459729298427 p^{36} T^{86} - 149538689428524 p^{37} T^{87} + 45973578062245 p^{38} T^{88} - 1562386613719 p^{39} T^{89} + 480514510477 p^{40} T^{90} - 13616289859 p^{41} T^{91} + 4205751758 p^{42} T^{92} - 94962184 p^{43} T^{93} + 29601768 p^{44} T^{94} - 496472 p^{45} T^{95} + 157110 p^{46} T^{96} - 1728 p^{47} T^{97} + 559 p^{48} T^{98} - 3 p^{49} T^{99} + p^{50} T^{100}$$
29 $$1 - 61 T + 2615 T^{2} - 83234 T^{3} + 2227777 T^{4} - 51337939 T^{5} + 1056762932 T^{6} - 19691008785 T^{7} + 337657031004 T^{8} - 5373130202378 T^{9} + 80076032563417 T^{10} - 1123954051820811 T^{11} + 14944388050827141 T^{12} - 188988060510580610 T^{13} + 2282139404783750510 T^{14} - 26393426395366326450 T^{15} +$$$$29\!\cdots\!77$$$$T^{16} -$$$$31\!\cdots\!56$$$$T^{17} +$$$$32\!\cdots\!25$$$$T^{18} -$$$$32\!\cdots\!30$$$$T^{19} +$$$$31\!\cdots\!44$$$$T^{20} -$$$$29\!\cdots\!56$$$$T^{21} +$$$$26\!\cdots\!68$$$$T^{22} -$$$$23\!\cdots\!69$$$$T^{23} +$$$$20\!\cdots\!71$$$$T^{24} -$$$$16\!\cdots\!34$$$$T^{25} +$$$$13\!\cdots\!46$$$$T^{26} -$$$$13\!\cdots\!88$$$$p^{2} T^{27} +$$$$85\!\cdots\!10$$$$T^{28} -$$$$65\!\cdots\!82$$$$T^{29} +$$$$48\!\cdots\!58$$$$T^{30} -$$$$35\!\cdots\!57$$$$T^{31} +$$$$25\!\cdots\!36$$$$T^{32} -$$$$17\!\cdots\!36$$$$T^{33} +$$$$12\!\cdots\!61$$$$T^{34} -$$$$82\!\cdots\!77$$$$T^{35} +$$$$54\!\cdots\!50$$$$T^{36} -$$$$35\!\cdots\!40$$$$T^{37} +$$$$22\!\cdots\!18$$$$T^{38} -$$$$14\!\cdots\!05$$$$T^{39} +$$$$88\!\cdots\!06$$$$T^{40} -$$$$53\!\cdots\!24$$$$T^{41} +$$$$32\!\cdots\!73$$$$T^{42} -$$$$19\!\cdots\!97$$$$T^{43} +$$$$11\!\cdots\!60$$$$T^{44} -$$$$64\!\cdots\!65$$$$T^{45} +$$$$36\!\cdots\!50$$$$T^{46} -$$$$70\!\cdots\!34$$$$p T^{47} +$$$$11\!\cdots\!63$$$$T^{48} -$$$$62\!\cdots\!22$$$$T^{49} +$$$$33\!\cdots\!50$$$$T^{50} -$$$$62\!\cdots\!22$$$$p T^{51} +$$$$11\!\cdots\!63$$$$p^{2} T^{52} -$$$$70\!\cdots\!34$$$$p^{4} T^{53} +$$$$36\!\cdots\!50$$$$p^{4} T^{54} -$$$$64\!\cdots\!65$$$$p^{5} T^{55} +$$$$11\!\cdots\!60$$$$p^{6} T^{56} -$$$$19\!\cdots\!97$$$$p^{7} T^{57} +$$$$32\!\cdots\!73$$$$p^{8} T^{58} -$$$$53\!\cdots\!24$$$$p^{9} T^{59} +$$$$88\!\cdots\!06$$$$p^{10} T^{60} -$$$$14\!\cdots\!05$$$$p^{11} T^{61} +$$$$22\!\cdots\!18$$$$p^{12} T^{62} -$$$$35\!\cdots\!40$$$$p^{13} T^{63} +$$$$54\!\cdots\!50$$$$p^{14} T^{64} -$$$$82\!\cdots\!77$$$$p^{15} T^{65} +$$$$12\!\cdots\!61$$$$p^{16} T^{66} -$$$$17\!\cdots\!36$$$$p^{17} T^{67} +$$$$25\!\cdots\!36$$$$p^{18} T^{68} -$$$$35\!\cdots\!57$$$$p^{19} T^{69} +$$$$48\!\cdots\!58$$$$p^{20} T^{70} -$$$$65\!\cdots\!82$$$$p^{21} T^{71} +$$$$85\!\cdots\!10$$$$p^{22} T^{72} -$$$$13\!\cdots\!88$$$$p^{25} T^{73} +$$$$13\!\cdots\!46$$$$p^{24} T^{74} -$$$$16\!\cdots\!34$$$$p^{25} T^{75} +$$$$20\!\cdots\!71$$$$p^{26} T^{76} -$$$$23\!\cdots\!69$$$$p^{27} T^{77} +$$$$26\!\cdots\!68$$$$p^{28} T^{78} -$$$$29\!\cdots\!56$$$$p^{29} T^{79} +$$$$31\!\cdots\!44$$$$p^{30} T^{80} -$$$$32\!\cdots\!30$$$$p^{31} T^{81} +$$$$32\!\cdots\!25$$$$p^{32} T^{82} -$$$$31\!\cdots\!56$$$$p^{33} T^{83} +$$$$29\!\cdots\!77$$$$p^{34} T^{84} - 26393426395366326450 p^{35} T^{85} + 2282139404783750510 p^{36} T^{86} - 188988060510580610 p^{37} T^{87} + 14944388050827141 p^{38} T^{88} - 1123954051820811 p^{39} T^{89} + 80076032563417 p^{40} T^{90} - 5373130202378 p^{41} T^{91} + 337657031004 p^{42} T^{92} - 19691008785 p^{43} T^{93} + 1056762932 p^{44} T^{94} - 51337939 p^{45} T^{95} + 2227777 p^{46} T^{96} - 83234 p^{47} T^{97} + 2615 p^{48} T^{98} - 61 p^{49} T^{99} + p^{50} T^{100}$$
31 $$1 - 27 T + 1125 T^{2} - 24017 T^{3} + 595382 T^{4} - 10670312 T^{5} + 202003199 T^{6} - 3153971703 T^{7} + 49997559945 T^{8} - 22491750393 p T^{9} + 9693140627796 T^{10} - 122892716679132 T^{11} + 1539624092558115 T^{12} - 17980405178181842 T^{13} + 206631968590701587 T^{14} - 2245265454970957037 T^{15} + 23964020246452256499 T^{16} -$$$$24\!\cdots\!65$$$$T^{17} +$$$$24\!\cdots\!85$$$$T^{18} -$$$$23\!\cdots\!47$$$$T^{19} +$$$$22\!\cdots\!15$$$$T^{20} -$$$$20\!\cdots\!32$$$$T^{21} +$$$$18\!\cdots\!32$$$$T^{22} -$$$$15\!\cdots\!74$$$$T^{23} +$$$$13\!\cdots\!52$$$$T^{24} -$$$$11\!\cdots\!94$$$$T^{25} +$$$$29\!\cdots\!52$$$$p T^{26} -$$$$73\!\cdots\!27$$$$T^{27} +$$$$57\!\cdots\!98$$$$T^{28} -$$$$43\!\cdots\!49$$$$T^{29} +$$$$33\!\cdots\!94$$$$T^{30} -$$$$24\!\cdots\!70$$$$T^{31} +$$$$17\!\cdots\!05$$$$T^{32} -$$$$12\!\cdots\!61$$$$T^{33} +$$$$90\!\cdots\!51$$$$T^{34} -$$$$62\!\cdots\!63$$$$T^{35} +$$$$42\!\cdots\!54$$$$T^{36} -$$$$28\!\cdots\!76$$$$T^{37} +$$$$18\!\cdots\!67$$$$T^{38} -$$$$12\!\cdots\!82$$$$T^{39} +$$$$78\!\cdots\!65$$$$T^{40} -$$$$49\!\cdots\!88$$$$T^{41} +$$$$31\!\cdots\!92$$$$T^{42} -$$$$19\!\cdots\!02$$$$T^{43} +$$$$11\!\cdots\!93$$$$T^{44} -$$$$69\!\cdots\!24$$$$T^{45} +$$$$40\!\cdots\!77$$$$T^{46} -$$$$23\!\cdots\!65$$$$T^{47} +$$$$13\!\cdots\!56$$$$T^{48} -$$$$77\!\cdots\!26$$$$T^{49} +$$$$43\!\cdots\!50$$$$T^{50} -$$$$77\!\cdots\!26$$$$p T^{51} +$$$$13\!\cdots\!56$$$$p^{2} T^{52} -$$$$23\!\cdots\!65$$$$p^{3} T^{53} +$$$$40\!\cdots\!77$$$$p^{4} T^{54} -$$$$69\!\cdots\!24$$$$p^{5} T^{55} +$$$$11\!\cdots\!93$$$$p^{6} T^{56} -$$$$19\!\cdots\!02$$$$p^{7} T^{57} +$$$$31\!\cdots\!92$$$$p^{8} T^{58} -$$$$49\!\cdots\!88$$$$p^{9} T^{59} +$$$$78\!\cdots\!65$$$$p^{10} T^{60} -$$$$12\!\cdots\!82$$$$p^{11} T^{61} +$$$$18\!\cdots\!67$$$$p^{12} T^{62} -$$$$28\!\cdots\!76$$$$p^{13} T^{63} +$$$$42\!\cdots\!54$$$$p^{14} T^{64} -$$$$62\!\cdots\!63$$$$p^{15} T^{65} +$$$$90\!\cdots\!51$$$$p^{16} T^{66} -$$$$12\!\cdots\!61$$$$p^{17} T^{67} +$$$$17\!\cdots\!05$$$$p^{18} T^{68} -$$$$24\!\cdots\!70$$$$p^{19} T^{69} +$$$$33\!\cdots\!94$$$$p^{20} T^{70} -$$$$43\!\cdots\!49$$$$p^{21} T^{71} +$$$$57\!\cdots\!98$$$$p^{22} T^{72} -$$$$73\!\cdots\!27$$$$p^{23} T^{73} +$$$$29\!\cdots\!52$$$$p^{25} T^{74} -$$$$11\!\cdots\!94$$$$p^{25} T^{75} +$$$$13\!\cdots\!52$$$$p^{26} T^{76} -$$$$15\!\cdots\!74$$$$p^{27} T^{77} +$$$$18\!\cdots\!32$$$$p^{28} T^{78} -$$$$20\!\cdots\!32$$$$p^{29} T^{79} +$$$$22\!\cdots\!15$$$$p^{30} T^{80} -$$$$23\!\cdots\!47$$$$p^{31} T^{81} +$$$$24\!\cdots\!85$$$$p^{32} T^{82} -$$$$24\!\cdots\!65$$$$p^{33} T^{83} + 23964020246452256499 p^{34} T^{84} - 2245265454970957037 p^{35} T^{85} + 206631968590701587 p^{36} T^{86} - 17980405178181842 p^{37} T^{87} + 1539624092558115 p^{38} T^{88} - 122892716679132 p^{39} T^{89} + 9693140627796 p^{40} T^{90} - 22491750393 p^{42} T^{91} + 49997559945 p^{42} T^{92} - 3153971703 p^{43} T^{93} + 202003199 p^{44} T^{94} - 10670312 p^{45} T^{95} + 595382 p^{46} T^{96} - 24017 p^{47} T^{97} + 1125 p^{48} T^{98} - 27 p^{49} T^{99} + p^{50} T^{100}$$
37 $$1 - 56 T + 69 p T^{2} - 83964 T^{3} + 2412617 T^{4} - 59364625 T^{5} + 1325671368 T^{6} - 26803754236 T^{7} + 503005977523 T^{8} - 8775615804232 T^{9} + 144169678045240 T^{10} - 2235640192813900 T^{11} + 32971074193164748 T^{12} - 463522836386886921 T^{13} + 6242043704838665220 T^{14} - 80682461595021915294 T^{15} +$$$$10\!\cdots\!23$$$$T^{16} -$$$$12\!\cdots\!97$$$$T^{17} +$$$$14\!\cdots\!98$$$$T^{18} -$$$$15\!\cdots\!96$$$$T^{19} +$$$$17\!\cdots\!38$$$$T^{20} -$$$$18\!\cdots\!21$$$$T^{21} +$$$$18\!\cdots\!01$$$$T^{22} -$$$$18\!\cdots\!89$$$$T^{23} +$$$$18\!\cdots\!97$$$$T^{24} -$$$$17\!\cdots\!39$$$$T^{25} +$$$$16\!\cdots\!87$$$$T^{26} -$$$$15\!\cdots\!44$$$$T^{27} +$$$$13\!\cdots\!97$$$$T^{28} -$$$$11\!\cdots\!28$$$$T^{29} +$$$$10\!\cdots\!00$$$$T^{30} -$$$$84\!\cdots\!84$$$$T^{31} +$$$$69\!\cdots\!88$$$$T^{32} -$$$$56\!\cdots\!44$$$$T^{33} +$$$$44\!\cdots\!52$$$$T^{34} -$$$$34\!\cdots\!74$$$$T^{35} +$$$$26\!\cdots\!09$$$$T^{36} -$$$$20\!\cdots\!09$$$$T^{37} +$$$$14\!\cdots\!29$$$$T^{38} -$$$$10\!\cdots\!54$$$$T^{39} +$$$$77\!\cdots\!56$$$$T^{40} -$$$$54\!\cdots\!04$$$$T^{41} +$$$$37\!\cdots\!97$$$$T^{42} -$$$$25\!\cdots\!09$$$$T^{43} +$$$$17\!\cdots\!49$$$$T^{44} -$$$$11\!\cdots\!32$$$$T^{45} +$$$$74\!\cdots\!02$$$$T^{46} -$$$$47\!\cdots\!55$$$$T^{47} +$$$$30\!\cdots\!30$$$$T^{48} -$$$$18\!\cdots\!61$$$$T^{49} +$$$$11\!\cdots\!98$$$$T^{50} -$$$$18\!\cdots\!61$$$$p T^{51} +$$$$30\!\cdots\!30$$$$p^{2} T^{52} -$$$$47\!\cdots\!55$$$$p^{3} T^{53} +$$$$74\!\cdots\!02$$$$p^{4} T^{54} -$$$$11\!\cdots\!32$$$$p^{5} T^{55} +$$$$17\!\cdots\!49$$$$p^{6} T^{56} -$$$$25\!\cdots\!09$$$$p^{7} T^{57} +$$$$37\!\cdots\!97$$$$p^{8} T^{58} -$$$$54\!\cdots\!04$$$$p^{9} T^{59} +$$$$77\!\cdots\!56$$$$p^{10} T^{60} -$$$$10\!\cdots\!54$$$$p^{11} T^{61} +$$$$14\!\cdots\!29$$$$p^{12} T^{62} -$$$$20\!\cdots\!09$$$$p^{13} T^{63} +$$$$26\!\cdots\!09$$$$p^{14} T^{64} -$$$$34\!\cdots\!74$$$$p^{15} T^{65} +$$$$44\!\cdots\!52$$$$p^{16} T^{66} -$$$$56\!\cdots\!44$$$$p^{17} T^{67} +$$$$69\!\cdots\!88$$$$p^{18} T^{68} -$$$$84\!\cdots\!84$$$$p^{19} T^{69} +$$$$10\!\cdots\!00$$$$p^{20} T^{70} -$$$$11\!\cdots\!28$$$$p^{21} T^{71} +$$$$13\!\cdots\!97$$$$p^{22} T^{72} -$$$$15\!\cdots\!44$$$$p^{23} T^{73} +$$$$16\!\cdots\!87$$$$p^{24} T^{74} -$$$$17\!\cdots\!39$$$$p^{25} T^{75} +$$$$18\!\cdots\!97$$$$p^{26} T^{76} -$$$$18\!\cdots\!89$$$$p^{27} T^{77} +$$$$18\!\cdots\!01$$$$p^{28} T^{78} -$$$$18\!\cdots\!21$$$$p^{29} T^{79} +$$$$17\!\cdots\!38$$$$p^{30} T^{80} -$$$$15\!\cdots\!96$$$$p^{31} T^{81} +$$$$14\!\cdots\!98$$$$p^{32} T^{82} -$$$$12\!\cdots\!97$$$$p^{33} T^{83} +$$$$10\!\cdots\!23$$$$p^{34} T^{84} - 80682461595021915294 p^{35} T^{85} + 6242043704838665220 p^{36} T^{86} - 463522836386886921 p^{37} T^{87} + 32971074193164748 p^{38} T^{88} - 2235640192813900 p^{39} T^{89} + 144169678045240 p^{40} T^{90} - 8775615804232 p^{41} T^{91} + 503005977523 p^{42} T^{92} - 26803754236 p^{43} T^{93} + 1325671368 p^{44} T^{94} - 59364625 p^{45} T^{95} + 2412617 p^{46} T^{96} - 83964 p^{47} T^{97} + 69 p^{49} T^{98} - 56 p^{49} T^{99} + p^{50} T^{100}$$
41 $$1 - 10 T + 1098 T^{2} - 9780 T^{3} + 591378 T^{4} - 4746547 T^{5} + 209140406 T^{6} - 1526844536 T^{7} + 54787310918 T^{8} - 366756792251 T^{9} + 11362630179574 T^{10} - 70262439163038 T^{11} + 1946164862877161 T^{12} - 11196519015860919 T^{13} + 283444038154076939 T^{14} - 1528242619483186685 T^{15} + 35862651015111087876 T^{16} -$$$$18\!\cdots\!66$$$$T^{17} +$$$$40\!\cdots\!15$$$$T^{18} -$$$$47\!\cdots\!08$$$$p T^{19} +$$$$40\!\cdots\!12$$$$T^{20} -$$$$18\!\cdots\!90$$$$T^{21} +$$$$36\!\cdots\!26$$$$T^{22} -$$$$16\!\cdots\!54$$$$T^{23} +$$$$29\!\cdots\!44$$$$T^{24} -$$$$13\!\cdots\!15$$$$T^{25} +$$$$22\!\cdots\!51$$$$T^{26} -$$$$98\!\cdots\!64$$$$T^{27} +$$$$15\!\cdots\!32$$$$T^{28} -$$$$68\!\cdots\!96$$$$T^{29} +$$$$10\!\cdots\!49$$$$T^{30} -$$$$44\!\cdots\!22$$$$T^{31} +$$$$64\!\cdots\!07$$$$T^{32} -$$$$27\!\cdots\!95$$$$T^{33} +$$$$37\!\cdots\!46$$$$T^{34} -$$$$16\!\cdots\!61$$$$T^{35} +$$$$20\!\cdots\!88$$$$T^{36} -$$$$89\!\cdots\!59$$$$T^{37} +$$$$11\!\cdots\!97$$$$T^{38} -$$$$47\!\cdots\!66$$$$T^{39} +$$$$55\!\cdots\!08$$$$T^{40} -$$$$23\!\cdots\!65$$$$T^{41} +$$$$26\!\cdots\!35$$$$T^{42} -$$$$11\!\cdots\!71$$$$T^{43} +$$$$12\!\cdots\!05$$$$T^{44} -$$$$53\!\cdots\!65$$$$T^{45} +$$$$56\!\cdots\!19$$$$T^{46} -$$$$23\!\cdots\!17$$$$T^{47} +$$$$24\!\cdots\!54$$$$T^{48} -$$$$10\!\cdots\!96$$$$T^{49} +$$$$10\!\cdots\!50$$$$T^{50} -$$$$10\!\cdots\!96$$$$p T^{51} +$$$$24\!\cdots\!54$$$$p^{2} T^{52} -$$$$23\!\cdots\!17$$$$p^{3} T^{53} +$$$$56\!\cdots\!19$$$$p^{4} T^{54} -$$$$53\!\cdots\!65$$$$p^{5} T^{55} +$$$$12\!\cdots\!05$$$$p^{6} T^{56} -$$$$11\!\cdots\!71$$$$p^{7} T^{57} +$$$$26\!\cdots\!35$$$$p^{8} T^{58} -$$$$23\!\cdots\!65$$$$p^{9} T^{59} +$$$$55\!\cdots\!08$$$$p^{10} T^{60} -$$$$47\!\cdots\!66$$$$p^{11} T^{61} +$$$$11\!\cdots\!97$$$$p^{12} T^{62} -$$$$89\!\cdots\!59$$$$p^{13} T^{63} +$$$$20\!\cdots\!88$$$$p^{14} T^{64} -$$$$16\!\cdots\!61$$$$p^{15} T^{65} +$$$$37\!\cdots\!46$$$$p^{16} T^{66} -$$$$27\!\cdots\!95$$$$p^{17} T^{67} +$$$$64\!\cdots\!07$$$$p^{18} T^{68} -$$$$44\!\cdots\!22$$$$p^{19} T^{69} +$$$$10\!\cdots\!49$$$$p^{20} T^{70} -$$$$68\!\cdots\!96$$$$p^{21} T^{71} +$$$$15\!\cdots\!32$$$$p^{22} T^{72} -$$$$98\!\cdots\!64$$$$p^{23} T^{73} +$$$$22\!\cdots\!51$$$$p^{24} T^{74} -$$$$13\!\cdots\!15$$$$p^{25} T^{75} +$$$$29\!\cdots\!44$$$$p^{26} T^{76} -$$$$16\!\cdots\!54$$$$p^{27} T^{77} +$$$$36\!\cdots\!26$$$$p^{28} T^{78} -$$$$18\!\cdots\!90$$$$p^{29} T^{79} +$$$$40\!\cdots\!12$$$$p^{30} T^{80} -$$$$47\!\cdots\!08$$$$p^{32} T^{81} +$$$$40\!\cdots\!15$$$$p^{32} T^{82} -$$$$18\!\cdots\!66$$$$p^{33} T^{83} + 35862651015111087876 p^{34} T^{84} - 1528242619483186685 p^{35} T^{85} + 283444038154076939 p^{36} T^{86} - 11196519015860919 p^{37} T^{87} + 1946164862877161 p^{38} T^{88} - 70262439163038 p^{39} T^{89} + 11362630179574 p^{40} T^{90} - 366756792251 p^{41} T^{91} + 54787310918 p^{42} T^{92} - 1526844536 p^{43} T^{93} + 209140406 p^{44} T^{94} - 4746547 p^{45} T^{95} + 591378 p^{46} T^{96} - 9780 p^{47} T^{97} + 1098 p^{48} T^{98} - 10 p^{49} T^{99} + p^{50} T^{100}$$
43 $$1 - 19 T + 1282 T^{2} - 21398 T^{3} + 793662 T^{4} - 11856391 T^{5} + 317989606 T^{6} - 4311141054 T^{7} + 93104035942 T^{8} - 1157830725627 T^{9} + 21308706606423 T^{10} - 245120887841844 T^{11} + 3980279195025856 T^{12} - 42639283812586084 T^{13} + 625387464583426910 T^{14} - 6273311230123655197 T^{15} + 1965878176712812656 p T^{16} -$$$$79\!\cdots\!92$$$$T^{17} +$$$$10\!\cdots\!95$$$$T^{18} -$$$$89\!\cdots\!98$$$$T^{19} +$$$$10\!\cdots\!84$$$$T^{20} -$$$$88\!\cdots\!70$$$$T^{21} +$$$$99\!\cdots\!87$$$$T^{22} -$$$$79\!\cdots\!24$$$$T^{23} +$$$$19\!\cdots\!37$$$$p T^{24} -$$$$64\!\cdots\!21$$$$T^{25} +$$$$66\!\cdots\!22$$$$T^{26} -$$$$48\!\cdots\!44$$$$T^{27} +$$$$48\!\cdots\!33$$$$T^{28} -$$$$33\!\cdots\!10$$$$T^{29} +$$$$32\!\cdots\!48$$$$T^{30} -$$$$21\!\cdots\!99$$$$T^{31} +$$$$20\!\cdots\!23$$$$T^{32} -$$$$13\!\cdots\!95$$$$T^{33} +$$$$12\!\cdots\!20$$$$T^{34} -$$$$75\!\cdots\!42$$$$T^{35} +$$$$69\!\cdots\!38$$$$T^{36} -$$$$40\!\cdots\!42$$$$T^{37} +$$$$37\!\cdots\!13$$$$T^{38} -$$$$21\!\cdots\!05$$$$T^{39} +$$$$19\!\cdots\!95$$$$T^{40} -$$$$10\!\cdots\!50$$$$T^{41} +$$$$96\!\cdots\!13$$$$T^{42} -$$$$50\!\cdots\!60$$$$T^{43} +$$$$46\!\cdots\!99$$$$T^{44} -$$$$23\!\cdots\!79$$$$T^{45} +$$$$21\!\cdots\!48$$$$T^{46} -$$$$10\!\cdots\!57$$$$T^{47} +$$$$96\!\cdots\!20$$$$T^{48} -$$$$46\!\cdots\!38$$$$T^{49} +$$$$42\!\cdots\!38$$$$T^{50} -$$$$46\!\cdots\!38$$$$p T^{51} +$$$$96\!\cdots\!20$$$$p^{2} T^{52} -$$$$10\!\cdots\!57$$$$p^{3} T^{53} +$$$$21\!\cdots\!48$$$$p^{4} T^{54} -$$$$23\!\cdots\!79$$$$p^{5} T^{55} +$$$$46\!\cdots\!99$$$$p^{6} T^{56} -$$$$50\!\cdots\!60$$$$p^{7} T^{57} +$$$$96\!\cdots\!13$$$$p^{8} T^{58} -$$$$10\!\cdots\!50$$$$p^{9} T^{59} +$$$$19\!\cdots\!95$$$$p^{10} T^{60} -$$$$21\!\cdots\!05$$$$p^{11} T^{61} +$$$$37\!\cdots\!13$$$$p^{12} T^{62} -$$$$40\!\cdots\!42$$$$p^{13} T^{63} +$$$$69\!\cdots\!38$$$$p^{14} T^{64} -$$$$75\!\cdots\!42$$$$p^{15} T^{65} +$$$$12\!\cdots\!20$$$$p^{16} T^{66} -$$$$13\!\cdots\!95$$$$p^{17} T^{67} +$$$$20\!\cdots\!23$$$$p^{18} T^{68} -$$$$21\!\cdots\!99$$$$p^{19} T^{69} +$$$$32\!\cdots\!48$$$$p^{20} T^{70} -$$$$33\!\cdots\!10$$$$p^{21} T^{71} +$$$$48\!\cdots\!33$$$$p^{22} T^{72} -$$$$48\!\cdots\!44$$$$p^{23} T^{73} +$$$$66\!\cdots\!22$$$$p^{24} T^{74} -$$$$64\!\cdots\!21$$$$p^{25} T^{75} +$$$$19\!\cdots\!37$$$$p^{27} T^{76} -$$$$79\!\cdots\!24$$$$p^{27} T^{77} +$$$$99\!\cdots\!87$$$$p^{28} T^{78} -$$$$88\!\cdots\!70$$$$p^{29} T^{79} +$$$$10\!\cdots\!84$$$$p^{30} T^{80} -$$$$89\!\cdots\!98$$$$p^{31} T^{81} +$$$$10\!\cdots\!95$$$$p^{32} T^{82} -$$$$79\!\cdots\!92$$$$p^{33} T^{83} + 1965878176712812656 p^{35} T^{84} - 6273311230123655197 p^{35} T^{85} + 625387464583426910 p^{36} T^{86} - 42639283812586084 p^{37} T^{87} + 3980279195025856 p^{38} T^{88} - 245120887841844 p^{39} T^{89} + 21308706606423 p^{40} T^{90} - 1157830725627 p^{41} T^{91} + 93104035942 p^{42} T^{92} - 4311141054 p^{43} T^{93} + 317989606 p^{44} T^{94} - 11856391 p^{45} T^{95} + 793662 p^{46} T^{96} - 21398 p^{47} T^{97} + 1282 p^{48} T^{98} - 19 p^{49} T^{99} + p^{50} T^{100}$$
47 $$1 - 3 T + 1206 T^{2} - 2790 T^{3} + 722058 T^{4} - 1183356 T^{5} + 286607003 T^{6} - 277993817 T^{7} + 84972868321 T^{8} - 25739375056 T^{9} + 20099802176247 T^{10} + 7305919111666 T^{11} + 3956765842028502 T^{12} + 4067337694165730 T^{13} + 667620372996493214 T^{14} + 1126043085903112079 T^{15} + 98685916521050079463 T^{16} +$$$$23\!\cdots\!25$$$$T^{17} +$$$$12\!\cdots\!77$$$$T^{18} +$$$$38\!\cdots\!80$$$$T^{19} +$$$$15\!\cdots\!24$$$$T^{20} +$$$$54\!\cdots\!82$$$$T^{21} +$$$$16\!\cdots\!88$$$$T^{22} +$$$$68\!\cdots\!56$$$$T^{23} +$$$$16\!\cdots\!07$$$$T^{24} +$$$$77\!\cdots\!36$$$$T^{25} +$$$$15\!\cdots\!70$$$$T^{26} +$$$$78\!\cdots\!99$$$$T^{27} +$$$$13\!\cdots\!57$$$$T^{28} +$$$$73\!\cdots\!17$$$$T^{29} +$$$$10\!\cdots\!82$$$$T^{30} +$$$$62\!\cdots\!18$$$$T^{31} +$$$$83\!\cdots\!02$$$$T^{32} +$$$$50\!\cdots\!63$$$$T^{33} +$$$$60\!\cdots\!29$$$$T^{34} +$$$$37\!\cdots\!43$$$$T^{35} +$$$$41\!\cdots\!21$$$$T^{36} +$$$$25\!\cdots\!69$$$$T^{37} +$$$$26\!\cdots\!73$$$$T^{38} +$$$$16\!\cdots\!47$$$$T^{39} +$$$$16\!\cdots\!53$$$$T^{40} +$$$$10\!\cdots\!04$$$$T^{41} +$$$$97\!\cdots\!92$$$$T^{42} +$$$$60\!\cdots\!90$$$$T^{43} +$$$$54\!\cdots\!89$$$$T^{44} +$$$$33\!\cdots\!38$$$$T^{45} +$$$$28\!\cdots\!70$$$$T^{46} +$$$$17\!\cdots\!49$$$$T^{47} +$$$$14\!\cdots\!02$$$$T^{48} +$$$$86\!\cdots\!59$$$$T^{49} +$$$$70\!\cdots\!86$$$$T^{50} +$$$$86\!\cdots\!59$$$$p T^{51} +$$$$14\!\cdots\!02$$$$p^{2} T^{52} +$$$$17\!\cdots\!49$$$$p^{3} T^{53} +$$$$28\!\cdots\!70$$$$p^{4} T^{54} +$$$$33\!\cdots\!38$$$$p^{5} T^{55} +$$$$54\!\cdots\!89$$$$p^{6} T^{56} +$$$$60\!\cdots\!90$$$$p^{7} T^{57} +$$$$97\!\cdots\!92$$$$p^{8} T^{58} +$$$$10\!\cdots\!04$$$$p^{9} T^{59} +$$$$16\!\cdots\!53$$$$p^{10} T^{60} +$$$$16\!\cdots\!47$$$$p^{11} T^{61} +$$$$26\!\cdots\!73$$$$p^{12} T^{62} +$$$$25\!\cdots\!69$$$$p^{13} T^{63} +$$$$41\!\cdots\!21$$$$p^{14} T^{64} +$$$$37\!\cdots\!43$$$$p^{15} T^{65} +$$$$60\!\cdots\!29$$$$p^{16} T^{66} +$$$$50\!\cdots\!63$$$$p^{17} T^{67} +$$$$83\!\cdots\!02$$$$p^{18} T^{68} +$$$$62\!\cdots\!18$$$$p^{19} T^{69} +$$$$10\!\cdots\!82$$$$p^{20} T^{70} +$$$$73\!\cdots\!17$$$$p^{21} T^{71} +$$$$13\!\cdots\!57$$$$p^{22} T^{72} +$$$$78\!\cdots\!99$$$$p^{23} T^{73} +$$$$15\!\cdots\!70$$$$p^{24} T^{74} +$$$$77\!\cdots\!36$$$$p^{25} T^{75} +$$$$16\!\cdots\!07$$$$p^{26} T^{76} +$$$$68\!\cdots\!56$$$$p^{27} T^{77} +$$$$16\!\cdots\!88$$$$p^{28} T^{78} +$$$$54\!\cdots\!82$$$$p^{29} T^{79} +$$$$15\!\cdots\!24$$$$p^{30} T^{80} +$$$$38\!\cdots\!80$$$$p^{31} T^{81} +$$$$12\!\cdots\!77$$$$p^{32} T^{82} +$$$$23\!\cdots\!25$$$$p^{33} T^{83} + 98685916521050079463 p^{34} T^{84} + 1126043085903112079 p^{35} T^{85} + 667620372996493214 p^{36} T^{86} + 4067337694165730 p^{37} T^{87} + 3956765842028502 p^{38} T^{88} + 7305919111666 p^{39} T^{89} + 20099802176247 p^{40} T^{90} - 25739375056 p^{41} T^{91} + 84972868321 p^{42} T^{92} - 277993817 p^{43} T^{93} + 286607003 p^{44} T^{94} - 1183356 p^{45} T^{95} + 722058 p^{46} T^{96} - 2790 p^{47} T^{97} + 1206 p^{48} T^{98} - 3 p^{49} T^{99} + p^{50} T^{100}$$
53 $$1 - 56 T + 54 p T^{2} - 100771 T^{3} + 3225466 T^{4} - 86795762 T^{5} + 2159734888 T^{6} - 48249278013 T^{7} + 1010971225984 T^{8} - 370057673412 p T^{9} + 360666812119787 T^{10} - 6248388823331101 T^{11} + 103423898334627800 T^{12} - 1630639267159308303 T^{13} + 24712017077891882477 T^{14} -$$$$35\!\cdots\!35$$$$T^{15} +$$$$50\!\cdots\!46$$$$T^{16} -$$$$68\!\cdots\!35$$$$T^{17} +$$$$89\!\cdots\!95$$$$T^{18} -$$$$11\!\cdots\!71$$$$T^{19} +$$$$14\!\cdots\!86$$$$T^{20} -$$$$17\!\cdots\!53$$$$T^{21} +$$$$37\!\cdots\!29$$$$p T^{22} -$$$$22\!\cdots\!02$$$$T^{23} +$$$$25\!\cdots\!03$$$$T^{24} -$$$$27\!\cdots\!34$$$$T^{25} +$$$$29\!\cdots\!65$$$$T^{26} -$$$$31\!\cdots\!41$$$$T^{27} +$$$$32\!\cdots\!40$$$$T^{28} -$$$$32\!\cdots\!79$$$$T^{29} +$$$$31\!\cdots\!36$$$$T^{30} -$$$$30\!\cdots\!88$$$$T^{31} +$$$$29\!\cdots\!38$$$$T^{32} -$$$$27\!\cdots\!80$$$$T^{33} +$$$$25\!\cdots\!11$$$$T^{34} -$$$$22\!\cdots\!19$$$$T^{35} +$$$$20\!\cdots\!47$$$$T^{36} -$$$$17\!\cdots\!67$$$$T^{37} +$$$$15\!\cdots\!29$$$$T^{38} -$$$$12\!\cdots\!05$$$$T^{39} +$$$$10\!\cdots\!71$$$$T^{40} -$$$$89\!\cdots\!67$$$$T^{41} +$$$$73\!\cdots\!89$$$$T^{42} -$$$$58\!\cdots\!73$$$$T^{43} +$$$$46\!\cdots\!56$$$$T^{44} -$$$$36\!\cdots\!34$$$$T^{45} +$$$$28\!\cdots\!49$$$$T^{46} -$$$$21\!\cdots\!87$$$$T^{47} +$$$$16\!\cdots\!38$$$$T^{48} -$$$$11\!\cdots\!48$$$$T^{49} +$$$$87\!\cdots\!54$$$$T^{50} -$$$$11\!\cdots\!48$$$$p T^{51} +$$$$16\!\cdots\!38$$$$p^{2} T^{52} -$$$$21\!\cdots\!87$$$$p^{3} T^{53} +$$$$28\!\cdots\!49$$$$p^{4} T^{54} -$$$$36\!\cdots\!34$$$$p^{5} T^{55} +$$$$46\!\cdots\!56$$$$p^{6} T^{56} -$$$$58\!\cdots\!73$$$$p^{7} T^{57} +$$$$73\!\cdots\!89$$$$p^{8} T^{58} -$$$$89\!\cdots\!67$$$$p^{9} T^{59} +$$$$10\!\cdots\!71$$$$p^{10} T^{60} -$$$$12\!\cdots\!05$$$$p^{11} T^{61} +$$$$15\!\cdots\!29$$$$p^{12} T^{62} -$$$$17\!\cdots\!67$$$$p^{13} T^{63} +$$$$20\!\cdots\!47$$$$p^{14} T^{64} -$$$$22\!\cdots\!19$$$$p^{15} T^{65} +$$$$25\!\cdots\!11$$$$p^{16} T^{66} -$$$$27\!\cdots\!80$$$$p^{17} T^{67} +$$$$29\!\cdots\!38$$$$p^{18} T^{68} -$$$$30\!\cdots\!88$$$$p^{19} T^{69} +$$$$31\!\cdots\!36$$$$p^{20} T^{70} -$$$$32\!\cdots\!79$$$$p^{21} T^{71} +$$$$32\!\cdots\!40$$$$p^{22} T^{72} -$$$$31\!\cdots\!41$$$$p^{23} T^{73} +$$$$29\!\cdots\!65$$$$p^{24} T^{74} -$$$$27\!\cdots\!34$$$$p^{25} T^{75} +$$$$25\!\cdots\!03$$$$p^{26} T^{76} -$$$$22\!\cdots\!02$$$$p^{27} T^{77} +$$$$37\!\cdots\!29$$$$p^{29} T^{78} -$$$$17\!\cdots\!53$$$$p^{29} T^{79} +$$$$14\!\cdots\!86$$$$p^{30} T^{80} -$$$$11\!\cdots\!71$$$$p^{31} T^{81} +$$$$89\!\cdots\!95$$$$p^{32} T^{82} -$$$$68\!\cdots\!35$$$$p^{33} T^{83} +$$$$50\!\cdots\!46$$$$p^{34} T^{84} -$$$$35\!\cdots\!35$$$$p^{35} T^{85} + 24712017077891882477 p^{36} T^{86} - 1630639267159308303 p^{37} T^{87} + 103423898334627800 p^{38} T^{88} - 6248388823331101 p^{39} T^{89} + 360666812119787 p^{40} T^{90} - 370057673412 p^{42} T^{91} + 1010971225984 p^{42} T^{92} - 48249278013 p^{43} T^{93} + 2159734888 p^{44} T^{94} - 86795762 p^{45} T^{95} + 3225466 p^{46} T^{96} - 100771 p^{47} T^{97} + 54 p^{49} T^{98} - 56 p^{49} T^{99} + p^{50} T^{100}$$
59 $$1 + T + 1467 T^{2} + 1928 T^{3} + 1077515 T^{4} + 29427 p T^{5} + 528368287 T^{6} + 1001221611 T^{7} + 194591183611 T^{8} + 421974009608 T^{9} + 57412412668907 T^{10} + 139795782576355 T^{11} + 14135513786855475 T^{12} + 38117510750446290 T^{13} + 2987333223424847339 T^{14} + 8828160926753750026 T^{15} +$$$$55\!\cdots\!08$$$$T^{16} +$$$$17\!\cdots\!07$$$$T^{17} +$$$$91\!\cdots\!77$$$$T^{18} +$$$$31\!\cdots\!93$$$$T^{19} +$$$$13\!\cdots\!21$$$$T^{20} +$$$$50\!\cdots\!28$$$$T^{21} +$$$$18\!\cdots\!24$$$$T^{22} +$$$$72\!\cdots\!57$$$$T^{23} +$$$$22\!\cdots\!94$$$$T^{24} +$$$$96\!\cdots\!47$$$$T^{25} +$$$$26\!\cdots\!13$$$$T^{26} +$$$$11\!\cdots\!82$$$$T^{27} +$$$$28\!\cdots\!80$$$$T^{28} +$$$$13\!\cdots\!19$$$$T^{29} +$$$$28\!\cdots\!34$$$$T^{30} +$$$$13\!\cdots\!66$$$$T^{31} +$$$$26\!\cdots\!17$$$$T^{32} +$$$$13\!\cdots\!93$$$$T^{33} +$$$$23\!\cdots\!95$$$$T^{34} +$$$$12\!\cdots\!81$$$$T^{35} +$$$$19\!\cdots\!89$$$$T^{36} +$$$$10\!\cdots\!96$$$$T^{37} +$$$$15\!\cdots\!66$$$$T^{38} +$$$$85\!\cdots\!55$$$$T^{39} +$$$$11\!\cdots\!29$$$$T^{40} +$$$$65\!\cdots\!03$$$$T^{41} +$$$$86\!\cdots\!14$$$$T^{42} +$$$$47\!\cdots\!41$$$$T^{43} +$$$$59\!\cdots\!07$$$$T^{44} +$$$$32\!\cdots\!02$$$$T^{45} +$$$$39\!\cdots\!58$$$$T^{46} +$$$$21\!\cdots\!61$$$$T^{47} +$$$$24\!\cdots\!85$$$$T^{48} +$$$$13\!\cdots\!53$$$$T^{49} +$$$$15\!\cdots\!38$$$$T^{50} +$$$$13\!\cdots\!53$$$$p T^{51} +$$$$24\!\cdots\!85$$$$p^{2} T^{52} +$$$$21\!\cdots\!61$$$$p^{3} T^{53} +$$$$39\!\cdots\!58$$$$p^{4} T^{54} +$$$$32\!\cdots\!02$$$$p^{5} T^{55} +$$$$59\!\cdots\!07$$$$p^{6} T^{56} +$$$$47\!\cdots\!41$$$$p^{7} T^{57} +$$$$86\!\cdots\!14$$$$p^{8} T^{58} +$$$$65\!\cdots\!03$$$$p^{9} T^{59} +$$$$11\!\cdots\!29$$$$p^{10} T^{60} +$$$$85\!\cdots\!55$$$$p^{11} T^{61} +$$$$15\!\cdots\!66$$$$p^{12} T^{62} +$$$$10\!\cdots\!96$$$$p^{13} T^{63} +$$$$19\!\cdots\!89$$$$p^{14} T^{64} +$$$$12\!\cdots\!81$$$$p^{15} T^{65} +$$$$23\!\cdots\!95$$$$p^{16} T^{66} +$$$$13\!\cdots\!93$$$$p^{17} T^{67} +$$$$26\!\cdots\!17$$$$p^{18} T^{68} +$$$$13\!\cdots\!66$$$$p^{19} T^{69} +$$$$28\!\cdots\!34$$$$p^{20} T^{70} +$$$$13\!\cdots\!19$$$$p^{21} T^{71} +$$$$28\!\cdots\!80$$$$p^{22} T^{72} +$$$$11\!\cdots\!82$$$$p^{23} T^{73} +$$$$26\!\cdots\!13$$$$p^{24} T^{74} +$$$$96\!\cdots\!47$$$$p^{25} T^{75} +$$$$22\!\cdots\!94$$$$p^{26} T^{76} +$$$$72\!\cdots\!57$$$$p^{27} T^{77} +$$$$18\!\cdots\!24$$$$p^{28} T^{78} +$$$$50\!\cdots\!28$$$$p^{29} T^{79} +$$$$13\!\cdots\!21$$$$p^{30} T^{80} +$$$$31\!\cdots\!93$$$$p^{31} T^{81} +$$$$91\!\cdots\!77$$$$p^{32} T^{82} +$$$$17\!\cdots\!07$$$$p^{33} T^{83} +$$$$55\!\cdots\!08$$$$p^{34} T^{84} + 8828160926753750026 p^{35} T^{85} + 2987333223424847339 p^{36} T^{86} + 38117510750446290 p^{37} T^{87} + 14135513786855475 p^{38} T^{88} + 139795782576355 p^{39} T^{89} + 57412412668907 p^{40} T^{90} + 421974009608 p^{41} T^{91} + 194591183611 p^{42} T^{92} + 1001221611 p^{43} T^{93} + 528368287 p^{44} T^{94} + 29427 p^{46} T^{95} + 1077515 p^{46} T^{96} + 1928 p^{47} T^{97} + 1467 p^{48} T^{98} + p^{49} T^{99} + p^{50} T^{100}$$
61 $$1 - 67 T + 3881 T^{2} - 160018 T^{3} + 5867214 T^{4} - 183814018 T^{5} + 5265912755 T^{6} - 136661788927 T^{7} + 3305097077852 T^{8} - 74396105931384 T^{9} + 1580828955804242 T^{10} - 31744284178782819 T^{11} + 607264174814441368 T^{12} - 11085265065713482283 T^{13} +$$$$19\!\cdots\!60$$$$T^{14} -$$$$32\!\cdots\!86$$$$T^{15} +$$$$52\!\cdots\!61$$$$T^{16} -$$$$82\!\cdots\!71$$$$T^{17} +$$$$12\!\cdots\!17$$$$T^{18} -$$$$18\!\cdots\!44$$$$T^{19} +$$$$26\!\cdots\!01$$$$T^{20} -$$$$36\!\cdots\!91$$$$T^{21} +$$$$48\!\cdots\!48$$$$T^{22} -$$$$64\!\cdots\!67$$$$T^{23} +$$$$81\!\cdots\!10$$$$T^{24} -$$$$10\!\cdots\!89$$$$T^{25} +$$$$12\!\cdots\!75$$$$T^{26} -$$$$14\!\cdots\!53$$$$T^{27} +$$$$17\!\cdots\!87$$$$T^{28} -$$$$19\!\cdots\!71$$$$T^{29} +$$$$22\!\cdots\!58$$$$T^{30} -$$$$24\!\cdots\!97$$$$T^{31} +$$$$26\!\cdots\!86$$$$T^{32} -$$$$27\!\cdots\!44$$$$T^{33} +$$$$28\!\cdots\!53$$$$T^{34} -$$$$28\!\cdots\!13$$$$T^{35} +$$$$46\!\cdots\!62$$$$p T^{36} -$$$$27\!\cdots\!43$$$$T^{37} +$$$$26\!\cdots\!36$$$$T^{38} -$$$$25\!\cdots\!50$$$$T^{39} +$$$$23\!\cdots\!24$$$$T^{40} -$$$$21\!\cdots\!24$$$$T^{41} +$$$$19\!\cdots\!58$$$$T^{42} -$$$$16\!\cdots\!45$$$$T^{43} +$$$$14\!\cdots\!79$$$$T^{44} -$$$$12\!\cdots\!66$$$$T^{45} +$$$$10\!\cdots\!75$$$$T^{46} -$$$$87\!\cdots\!25$$$$T^{47} +$$$$70\!\cdots\!27$$$$T^{48} -$$$$56\!\cdots\!85$$$$T^{49} +$$$$44\!\cdots\!04$$$$T^{50} -$$$$56\!\cdots\!85$$$$p T^{51} +$$$$70\!\cdots\!27$$$$p^{2} T^{52} -$$$$87\!\cdots\!25$$$$p^{3} T^{53} +$$$$10\!\cdots\!75$$$$p^{4} T^{54} -$$$$12\!\cdots\!66$$$$p^{5} T^{55} +$$$$14\!\cdots\!79$$$$p^{6} T^{56} -$$$$16\!\cdots\!45$$$$p^{7} T^{57} +$$$$19\!\cdots\!58$$$$p^{8} T^{58} -$$$$21\!\cdots\!24$$$$p^{9} T^{59} +$$$$23\!\cdots\!24$$$$p^{10} T^{60} -$$$$25\!\cdots\!50$$$$p^{11} T^{61} +$$$$26\!\cdots\!36$$$$p^{12} T^{62} -$$$$27\!\cdots\!43$$$$p^{13} T^{63} +$$$$46\!\cdots\!62$$$$p^{15} T^{64} -$$$$28\!\cdots\!13$$$$p^{15} T^{65} +$$$$28\!\cdots\!53$$$$p^{16} T^{66} -$$$$27\!\cdots\!44$$$$p^{17} T^{67} +$$$$26\!\cdots\!86$$$$p^{18} T^{68} -$$$$24\!\cdots\!97$$$$p^{19} T^{69} +$$$$22\!\cdots\!58$$$$p^{20} T^{70} -$$$$19\!\cdots\!71$$$$p^{21} T^{71} +$$$$17\!\cdots\!87$$$$p^{22} T^{72} -$$$$14\!\cdots\!53$$$$p^{23} T^{73} +$$$$12\!\cdots\!75$$$$p^{24} T^{74} -$$$$10\!\cdots\!89$$$$p^{25} T^{75} +$$$$81\!\cdots\!10$$$$p^{26} T^{76} -$$$$64\!\cdots\!67$$$$p^{27} T^{77} +$$$$48\!\cdots\!48$$$$p^{28} T^{78} -$$$$36\!\cdots\!91$$$$p^{29} T^{79} +$$$$26\!\cdots\!01$$$$p^{30} T^{80} -$$$$18\!\cdots\!44$$$$p^{31} T^{81} +$$$$12\!\cdots\!17$$$$p^{32} T^{82} -$$$$82\!\cdots\!71$$$$p^{33} T^{83} +$$$$52\!\cdots\!61$$$$p^{34} T^{84} -$$$$32\!\cdots\!86$$$$p^{35} T^{85} +$$$$19\!\cdots\!60$$$$p^{36} T^{86} - 11085265065713482283 p^{37} T^{87} + 607264174814441368 p^{38} T^{88} - 31744284178782819 p^{39} T^{89} + 1580828955804242 p^{40} T^{90} - 74396105931384 p^{41} T^{91} + 3305097077852 p^{42} T^{92} - 136661788927 p^{43} T^{93} + 5265912755 p^{44} T^{94} - 183814018 p^{45} T^{95} + 5867214 p^{46} T^{96} - 160018 p^{47} T^{97} + 3881 p^{48} T^{98} - 67 p^{49} T^{99} + p^{50} T^{100}$$
67 $$1 - 46 T + 2591 T^{2} - 87361 T^{3} + 3026666 T^{4} - 82760355 T^{5} + 2227386559 T^{6} - 52132693711 T^{7} + 1186300691525 T^{8} - 24564197810685 T^{9} + 493144813974115 T^{10} - 9234742580714641 T^{11} + 167750110794092048 T^{12} - 2885425497158851071 T^{13} + 48223121508229000173 T^{14} -$$$$77\!\cdots\!66$$$$T^{15} +$$$$11\!\cdots\!87$$$$T^{16} -$$$$17\!\cdots\!03$$$$T^{17} +$$$$26\!\cdots\!20$$$$T^{18} -$$$$37\!\cdots\!38$$$$T^{19} +$$$$51\!\cdots\!77$$$$T^{20} -$$$$68\!\cdots\!74$$$$T^{21} +$$$$90\!\cdots\!80$$$$T^{22} -$$$$11\!\cdots\!65$$$$T^{23} +$$$$14\!\cdots\!96$$$$T^{24} -$$$$17\!\cdots\!54$$$$T^{25} +$$$$21\!\cdots\!39$$$$T^{26} -$$$$25\!\cdots\!08$$$$T^{27} +$$$$29\!\cdots\!59$$$$T^{28} -$$$$32\!\cdots\!77$$$$T^{29} +$$$$36\!\cdots\!28$$$$T^{30} -$$$$39\!\cdots\!40$$$$T^{31} +$$$$42\!\cdots\!12$$$$T^{32} -$$$$45\!\cdots\!59$$$$T^{33} +$$$$47\!\cdots\!79$$$$T^{34} -$$$$48\!\cdots\!26$$$$T^{35} +$$$$48\!\cdots\!17$$$$T^{36} -$$$$47\!\cdots\!19$$$$T^{37} +$$$$46\!\cdots\!33$$$$T^{38} -$$$$44\!\cdots\!62$$$$T^{39} +$$$$42\!\cdots\!76$$$$T^{40} -$$$$39\!\cdots\!63$$$$T^{41} +$$$$36\!\cdots\!34$$$$T^{42} -$$$$33\!\cdots\!21$$$$T^{43} +$$$$29\!\cdots\!76$$$$T^{44} -$$$$26\!\cdots\!66$$$$T^{45} +$$$$22\!\cdots\!15$$$$T^{46} -$$$$19\!\cdots\!70$$$$T^{47} +$$$$16\!\cdots\!16$$$$T^{48} -$$$$13\!\cdots\!33$$$$T^{49} +$$$$11\!\cdots\!12$$$$T^{50} -$$$$13\!\cdots\!33$$$$p T^{51} +$$$$16\!\cdots\!16$$$$p^{2} T^{52} -$$$$19\!\cdots\!70$$$$p^{3} T^{53} +$$$$22\!\cdots\!15$$$$p^{4} T^{54} -$$$$26\!\cdots\!66$$$$p^{5} T^{55} +$$$$29\!\cdots\!76$$$$p^{6} T^{56} -$$$$33\!\cdots\!21$$$$p^{7} T^{57} +$$$$36\!\cdots\!34$$$$p^{8} T^{58} -$$$$39\!\cdots\!63$$$$p^{9} T^{59} +$$$$42\!\cdots\!76$$$$p^{10} T^{60} -$$$$44\!\cdots\!62$$$$p^{11} T^{61} +$$$$46\!\cdots\!33$$$$p^{12} T^{62} -$$$$47\!\cdots\!19$$$$p^{13} T^{63} +$$$$48\!\cdots\!17$$$$p^{14} T^{64} -$$$$48\!\cdots\!26$$$$p^{15} T^{65} +$$$$47\!\cdots\!79$$$$p^{16} T^{66} -$$$$45\!\cdots\!59$$$$p^{17} T^{67} +$$$$42\!\cdots\!12$$$$p^{18} T^{68} -$$$$39\!\cdots\!40$$$$p^{19} T^{69} +$$$$36\!\cdots\!28$$$$p^{20} T^{70} -$$$$32\!\cdots\!77$$$$p^{21} T^{71} +$$$$29\!\cdots\!59$$$$p^{22} T^{72} -$$$$25\!\cdots\!08$$$$p^{23} T^{73} +$$$$21\!\cdots\!39$$$$p^{24} T^{74} -$$$$17\!\cdots\!54$$$$p^{25} T^{75} +$$$$14\!\cdots\!96$$$$p^{26} T^{76} -$$$$11\!\cdots\!65$$$$p^{27} T^{77} +$$$$90\!\cdots\!80$$$$p^{28} T^{78} -$$$$68\!\cdots\!74$$$$p^{29} T^{79} +$$$$51\!\cdots\!77$$$$p^{30} T^{80} -$$$$37\!\cdots\!38$$$$p^{31} T^{81} +$$$$26\!\cdots\!20$$$$p^{32} T^{82} -$$$$17\!\cdots\!03$$$$p^{33} T^{83} +$$$$11\!\cdots\!87$$$$p^{34} T^{84} -$$$$77\!\cdots\!66$$$$p^{35} T^{85} + 48223121508229000173 p^{36} T^{86} - 2885425497158851071 p^{37} T^{87} + 167750110794092048 p^{38} T^{88} - 9234742580714641 p^{39} T^{89} + 493144813974115 p^{40} T^{90} - 24564197810685 p^{41} T^{91} + 1186300691525 p^{42} T^{92} - 52132693711 p^{43} T^{93} + 2227386559 p^{44} T^{94} - 82760355 p^{45} T^{95} + 3026666 p^{46} T^{96} - 87361 p^{47} T^{97} + 2591 p^{48} T^{98} - 46 p^{49} T^{99} + p^{50} T^{100}$$
71 $$1 - 4 T + 1792 T^{2} - 7932 T^{3} + 1604527 T^{4} - 7688761 T^{5} + 957523602 T^{6} - 4876595519 T^{7} + 428521752731 T^{8} - 2282826668527 T^{9} + 153395059383130 T^{10} - 842799025772340 T^{11} + 45735723254483440 T^{12} - 255897463447923993 T^{13} + 11676395032423291684 T^{14} - 65756934918892635042 T^{15} +$$$$26\!\cdots\!38$$$$T^{16} -$$$$14\!\cdots\!57$$$$T^{17} +$$$$51\!\cdots\!04$$$$T^{18} -$$$$28\!\cdots\!59$$$$T^{19} +$$$$91\!\cdots\!99$$$$T^{20} -$$$$49\!\cdots\!20$$$$T^{21} +$$$$14\!\cdots\!43$$$$T^{22} -$$$$75\!\cdots\!62$$$$T^{23} +$$$$21\!\cdots\!59$$$$T^{24} -$$$$10\!\cdots\!51$$$$T^{25} +$$$$28\!\cdots\!63$$$$T^{26} -$$$$13\!\cdots\!72$$$$T^{27} +$$$$35\!\cdots\!54$$$$T^{28} -$$$$14\!\cdots\!07$$$$T^{29} +$$$$40\!\cdots\!69$$$$T^{30} -$$$$15\!\cdots\!59$$$$T^{31} +$$$$42\!\cdots\!94$$$$T^{32} -$$$$13\!\cdots\!28$$$$T^{33} +$$$$42\!\cdots\!61$$$$T^{34} -$$$$10\!\cdots\!52$$$$T^{35} +$$$$39\!\cdots\!58$$$$T^{36} -$$$$72\!\cdots\!37$$$$T^{37} +$$$$33\!\cdots\!58$$$$T^{38} -$$$$34\!\cdots\!40$$$$T^{39} +$$$$27\!\cdots\!54$$$$T^{40} -$$$$24\!\cdots\!91$$$$T^{41} +$$$$21\!\cdots\!31$$$$T^{42} +$$$$19\!\cdots\!48$$$$T^{43} +$$$$15\!\cdots\!08$$$$T^{44} +$$$$29\!\cdots\!06$$$$T^{45} +$$$$11\!\cdots\!12$$$$T^{46} +$$$$30\!\cdots\!90$$$$T^{47} +$$$$82\!\cdots\!57$$$$T^{48} +$$$$25\!\cdots\!27$$$$T^{49} +$$$$58\!\cdots\!46$$$$T^{50} +$$$$25\!\cdots\!27$$$$p T^{51} +$$$$82\!\cdots\!57$$$$p^{2} T^{52} +$$$$30\!\cdots\!90$$$$p^{3} T^{53} +$$$$11\!\cdots\!12$$$$p^{4} T^{54} +$$$$29\!\cdots\!06$$$$p^{5} T^{55} +$$$$15\!\cdots\!08$$$$p^{6} T^{56} +$$$$19\!\cdots\!48$$$$p^{7} T^{57} +$$$$21\!\cdots\!31$$$$p^{8} T^{58} -$$$$24\!\cdots\!91$$$$p^{9} T^{59} +$$$$27\!\cdots\!54$$$$p^{10} T^{60} -$$$$34\!\cdots\!40$$$$p^{11} T^{61} +$$$$33\!\cdots\!58$$$$p^{12} T^{62} -$$$$72\!\cdots\!37$$$$p^{13} T^{63} +$$$$39\!\cdots\!58$$$$p^{14} T^{64} -$$$$10\!\cdots\!52$$$$p^{15} T^{65} +$$$$42\!\cdots\!61$$$$p^{16} T^{66} -$$$$13\!\cdots\!28$$$$p^{17} T^{67} +$$$$42\!\cdots\!94$$$$p^{18} T^{68} -$$$$15\!\cdots\!59$$$$p^{19} T^{69} +$$$$40\!\cdots\!69$$$$p^{20} T^{70} -$$$$14\!\cdots\!07$$$$p^{21} T^{71} +$$$$35\!\cdots\!54$$$$p^{22} T^{72} -$$$$13\!\cdots\!72$$$$p^{23} T^{73} +$$$$28\!\cdots\!63$$$$p^{24} T^{74} -$$$$10\!\cdots\!51$$$$p^{25} T^{75} +$$$$21\!\cdots\!59$$$$p^{26} T^{76} -$$$$75\!\cdots\!62$$$$p^{27} T^{77} +$$$$14\!\cdots\!43$$$$p^{28} T^{78} -$$$$49\!\cdots\!20$$$$p^{29} T^{79} +$$$$91\!\cdots\!99$$$$p^{30} T^{80} -$$$$28\!\cdots\!59$$$$p^{31} T^{81} +$$$$51\!\cdots\!04$$$$p^{32} T^{82} -$$$$14\!\cdots\!57$$$$p^{33} T^{83} +$$$$26\!\cdots\!38$$$$p^{34} T^{84} - 65756934918892635042 p^{35} T^{85} + 11676395032423291684 p^{36} T^{86} - 255897463447923993 p^{37} T^{87} + 45735723254483440 p^{38} T^{88} - 842799025772340 p^{39} T^{89} + 153395059383130 p^{40} T^{90} - 2282826668527 p^{41} T^{91} + 428521752731 p^{42} T^{92} - 4876595519 p^{43} T^{93} + 957523602 p^{44} T^{94} - 7688761 p^{45} T^{95} + 1604527 p^{46} T^{96} - 7932 p^{47} T^{97} + 1792 p^{48} T^{98} - 4 p^{49} T^{99} + p^{50} T^{100}$$
73 $$1 - 62 T + 3825 T^{2} - 157393 T^{3} + 6061851 T^{4} - 194166938 T^{5} + 5821815509 T^{6} - 156165722194 T^{7} + 3951476209196 T^{8} - 92497540689508 T^{9} + 2058213407594241 T^{10} - 43155802449531160 T^{11} + 865766259491653344 T^{12} - 16553249225535260234 T^{13} +$$$$30\!\cdots\!62$$$$T^{14} -$$$$53\!\cdots\!02$$$$T^{15} +$$$$91\!\cdots\!39$$$$T^{16} -$$$$15\!\cdots\!32$$$$T^{17} +$$$$24\!\cdots\!57$$$$T^{18} -$$$$37\!\cdots\!80$$$$T^{19} +$$$$56\!\cdots\!78$$$$T^{20} -$$$$82\!\cdots\!81$$$$T^{21} +$$$$11\!\cdots\!43$$$$T^{22} -$$$$16\!\cdots\!84$$$$T^{23} +$$$$22\!\cdots\!84$$$$T^{24} -$$$$29\!\cdots\!06$$$$T^{25} +$$$$38\!\cdots\!31$$$$T^{26} -$$$$48\!\cdots\!13$$$$T^{27} +$$$$60\!\cdots\!94$$$$T^{28} -$$$$73\!\cdots\!36$$$$T^{29} +$$$$11\!\cdots\!45$$$$p T^{30} -$$$$10\!\cdots\!08$$$$T^{31} +$$$$11\!\cdots\!00$$$$T^{32} -$$$$13\!\cdots\!09$$$$T^{33} +$$$$14\!\cdots\!45$$$$T^{34} -$$$$15\!\cdots\!68$$$$T^{35} +$$$$17\!\cdots\!44$$$$T^{36} -$$$$18\!\cdots\!12$$$$T^{37} +$$$$18\!\cdots\!00$$$$T^{38} -$$$$19\!\cdots\!02$$$$T^{39} +$$$$19\!\cdots\!98$$$$T^{40} -$$$$18\!\cdots\!81$$$$T^{41} +$$$$18\!\cdots\!52$$$$T^{42} -$$$$17\!\cdots\!28$$$$T^{43} +$$$$16\!\cdots\!08$$$$T^{44} -$$$$15\!\cdots\!06$$$$T^{45} +$$$$13\!\cdots\!77$$$$T^{46} -$$$$12\!\cdots\!59$$$$T^{47} +$$$$11\!\cdots\!95$$$$T^{48} -$$$$97\!\cdots\!08$$$$T^{49} +$$$$83\!\cdots\!42$$$$T^{50} -$$$$97\!\cdots\!08$$$$p T^{51} +$$$$11\!\cdots\!95$$$$p^{2} T^{52} -$$$$12\!\cdots\!59$$$$p^{3} T^{53} +$$$$13\!\cdots\!77$$$$p^{4} T^{54} -$$$$15\!\cdots\!06$$$$p^{5} T^{55} +$$$$16\!\cdots\!08$$$$p^{6} T^{56} -$$$$17\!\cdots\!28$$$$p^{7} T^{57} +$$$$18\!\cdots\!52$$$$p^{8} T^{58} -$$$$18\!\cdots\!81$$$$p^{9} T^{59} +$$$$19\!\cdots\!98$$$$p^{10} T^{60} -$$$$19\!\cdots\!02$$$$p^{11} T^{61} +$$$$18\!\cdots\!00$$$$p^{12} T^{62} -$$$$18\!\cdots\!12$$$$p^{13} T^{63} +$$$$17\!\cdots\!44$$$$p^{14} T^{64} -$$$$15\!\cdots\!68$$$$p^{15} T^{65} +$$$$14\!\cdots\!45$$$$p^{16} T^{66} -$$$$13\!\cdots\!09$$$$p^{17} T^{67} +$$$$11\!\cdots\!00$$$$p^{18} T^{68} -$$$$10\!\cdots\!08$$$$p^{19} T^{69} +$$$$11\!\cdots\!45$$$$p^{21} T^{70} -$$$$73\!\cdots\!36$$$$p^{21} T^{71} +$$$$60\!\cdots\!94$$$$p^{22} T^{72} -$$$$48\!\cdots\!13$$$$p^{23} T^{73} +$$$$38\!\cdots\!31$$$$p^{24} T^{74} -$$$$29\!\cdots\!06$$$$p^{25} T^{75} +$$$$22\!\cdots\!84$$$$p^{26} T^{76} -$$$$16\!\cdots\!84$$$$p^{27} T^{77} +$$$$11\!\cdots\!43$$$$p^{28} T^{78} -$$$$82\!\cdots\!81$$$$p^{29} T^{79} +$$$$56\!\cdots\!78$$$$p^{30} T^{80} -$$$$37\!\cdots\!80$$$$p^{31} T^{81} +$$$$24\!\cdots\!57$$$$p^{32} T^{82} -$$$$15\!\cdots\!32$$$$p^{33} T^{83} +$$$$91\!\cdots\!39$$$$p^{34} T^{84} -$$$$53\!\cdots\!02$$$$p^{35} T^{85} +$$$$30\!\cdots\!62$$$$p^{36} T^{86} - 16553249225535260234 p^{37} T^{87} + 865766259491653344 p^{38} T^{88} - 43155802449531160 p^{39} T^{89} + 2058213407594241 p^{40} T^{90} - 92497540689508 p^{41} T^{91} + 3951476209196 p^{42} T^{92} - 156165722194 p^{43} T^{93} + 5821815509 p^{44} T^{94} - 194166938 p^{45} T^{95} + 6061851 p^{46} T^{96} - 157393 p^{47} T^{97} + 3825 p^{48} T^{98} - 62 p^{49} T^{99} + p^{50} T^{100}$$
79 $$1 - 7 T + 2005 T^{2} - 12548 T^{3} + 2013328 T^{4} - 11185597 T^{5} + 1350499977 T^{6} - 6593939648 T^{7} + 680973985722 T^{8} - 2881974733374 T^{9} + 275395442084251 T^{10} - 991463241694572 T^{11} + 93067194610247146 T^{12} - 277744564496782999 T^{13} + 27037676305529556305 T^{14} - 64462373944289013379 T^{15} +$$$$68\!\cdots\!27$$$$T^{16} -$$$$12\!\cdots\!37$$$$T^{17} +$$$$15\!\cdots\!41$$$$T^{18} -$$$$19\!\cdots\!70$$$$T^{19} +$$$$32\!\cdots\!11$$$$T^{20} -$$$$22\!\cdots\!20$$$$T^{21} +$$$$60\!\cdots\!62$$$$T^{22} -$$$$10\!\cdots\!38$$$$T^{23} +$$$$10\!\cdots\!69$$$$T^{24} +$$$$32\!\cdots\!72$$$$T^{25} +$$$$16\!\cdots\!19$$$$T^{26} +$$$$12\!\cdots\!81$$$$T^{27} +$$$$24\!\cdots\!58$$$$T^{28} +$$$$29\!\cdots\!91$$$$T^{29} +$$$$34\!\cdots\!48$$$$T^{30} +$$$$53\!\cdots\!17$$$$T^{31} +$$$$44\!\cdots\!51$$$$T^{32} +$$$$83\!\cdots\!26$$$$T^{33} +$$$$54\!\cdots\!35$$$$T^{34} +$$$$11\!\cdots\!27$$$$T^{35} +$$$$63\!\cdots\!52$$$$T^{36} +$$$$15\!\cdots\!42$$$$T^{37} +$$$$88\!\cdots\!29$$$$p T^{38} +$$$$18\!\cdots\!85$$$$T^{39} +$$$$72\!\cdots\!24$$$$T^{40} +$$$$19\!\cdots\!67$$$$T^{41} +$$$$71\!\cdots\!02$$$$T^{42} +$$$$20\!\cdots\!24$$$$T^{43} +$$$$67\!\cdots\!09$$$$T^{44} +$$$$19\!\cdots\!55$$$$T^{45} +$$$$60\!\cdots\!49$$$$T^{46} +$$$$17\!\cdots\!43$$$$T^{47} +$$$$51\!\cdots\!46$$$$T^{48} +$$$$14\!\cdots\!55$$$$T^{49} +$$$$41\!\cdots\!62$$$$T^{50} +$$$$14\!\cdots\!55$$$$p T^{51} +$$$$51\!\cdots\!46$$$$p^{2} T^{52} +$$$$17\!\cdots\!43$$$$p^{3} T^{53} +$$$$60\!\cdots\!49$$$$p^{4} T^{54} +$$$$19\!\cdots\!55$$$$p^{5} T^{55} +$$$$67\!\cdots\!09$$$$p^{6} T^{56} +$$$$20\!\cdots\!24$$$$p^{7} T^{57} +$$$$71\!\cdots\!02$$$$p^{8} T^{58} +$$$$19\!\cdots\!67$$$$p^{9} T^{59} +$$$$72\!\cdots\!24$$$$p^{10} T^{60} +$$$$18\!\cdots\!85$$$$p^{11} T^{61} +$$$$88\!\cdots\!29$$$$p^{13} T^{62} +$$$$15\!\cdots\!42$$$$p^{13} T^{63} +$$$$63\!\cdots\!52$$$$p^{14} T^{64} +$$$$11\!\cdots\!27$$$$p^{15} T^{65} +$$$$54\!\cdots\!35$$$$p^{16} T^{66} +$$$$83\!\cdots\!26$$$$p^{17} T^{67} +$$$$44\!\cdots\!51$$$$p^{18} T^{68} +$$$$53\!\cdots\!17$$$$p^{19} T^{69} +$$$$34\!\cdots\!48$$$$p^{20} T^{70} +$$$$29\!\cdots\!91$$$$p^{21} T^{71} +$$$$24\!\cdots\!58$$$$p^{22} T^{72} +$$$$12\!\cdots\!81$$$$p^{23} T^{73} +$$$$16\!\cdots\!19$$$$p^{24} T^{74} +$$$$32\!\cdots\!72$$$$p^{25} T^{75} +$$$$10\!\cdots\!69$$$$p^{26} T^{76} -$$$$10\!\cdots\!38$$$$p^{27} T^{77} +$$$$60\!\cdots\!62$$$$p^{28} T^{78} -$$$$22\!\cdots\!20$$$$p^{29} T^{79} +$$$$32\!\cdots\!11$$$$p^{30} T^{80} -$$$$19\!\cdots\!70$$$$p^{31} T^{81} +$$$$15\!\cdots\!41$$$$p^{32} T^{82} -$$$$12\!\cdots\!37$$$$p^{33} T^{83} +$$$$68\!\cdots\!27$$$$p^{34} T^{84} - 64462373944289013379 p^{35} T^{85} + 27037676305529556305 p^{36} T^{86} - 277744564496782999 p^{37} T^{87} + 93067194610247146 p^{38} T^{88} - 991463241694572 p^{39} T^{89} + 275395442084251 p^{40} T^{90} - 2881974733374 p^{41} T^{91} + 680973985722 p^{42} T^{92} - 6593939648 p^{43} T^{93} + 1350499977 p^{44} T^{94} - 11185597 p^{45} T^{95} + 2013328 p^{46} T^{96} - 12548 p^{47} T^{97} + 2005 p^{48} T^{98} - 7 p^{49} T^{99} + p^{50} T^{100}$$
83 $$1 + T + 2194 T^{2} + 4108 T^{3} + 2412101 T^{4} + 6499374 T^{5} + 1773710196 T^{6} + 6142459035 T^{7} + 982141842112 T^{8} + 4100128613812 T^{9} + 437009569361682 T^{10} + 2109125411089860 T^{11} + 162793866453333577 T^{12} + 881590226713430209 T^{13} + 52218841089641695993 T^{14} +$$$$31\!\cdots\!81$$$$T^{15} +$$$$14\!\cdots\!20$$$$T^{16} +$$$$94\!\cdots\!42$$$$T^{17} +$$$$37\!\cdots\!20$$$$T^{18} +$$$$25\!\cdots\!41$$$$T^{19} +$$$$84\!\cdots\!62$$$$T^{20} +$$$$59\!\cdots\!61$$$$T^{21} +$$$$17\!\cdots\!63$$$$T^{22} +$$$$12\!\cdots\!05$$$$T^{23} +$$$$33\!\cdots\!68$$$$T^{24} +$$$$25\!\cdots\!56$$$$T^{25} +$$$$58\!\cdots\!93$$$$T^{26} +$$$$45\!\cdots\!09$$$$T^{27} +$$$$95\!\cdots\!70$$$$T^{28} +$$$$75\!\cdots\!17$$$$T^{29} +$$$$14\!\cdots\!65$$$$T^{30} +$$$$11\!\cdots\!70$$$$T^{31} +$$$$20\!\cdots\!61$$$$T^{32} +$$$$16\!\cdots\!02$$$$T^{33} +$$$$27\!\cdots\!25$$$$T^{34} +$$$$22\!\cdots\!74$$$$T^{35} +$$$$34\!\cdots\!71$$$$T^{36} +$$$$27\!\cdots\!30$$$$T^{37} +$$$$41\!\cdots\!63$$$$T^{38} +$$$$32\!\cdots\!32$$$$T^{39} +$$$$46\!\cdots\!52$$$$T^{40} +$$$$35\!\cdots\!56$$$$T^{41} +$$$$48\!\cdots\!61$$$$T^{42} +$$$$44\!\cdots\!73$$$$p T^{43} +$$$$48\!\cdots\!21$$$$T^{44} +$$$$36\!\cdots\!58$$$$T^{45} +$$$$46\!\cdots\!68$$$$T^{46} +$$$$34\!\cdots\!86$$$$T^{47} +$$$$41\!\cdots\!76$$$$T^{48} +$$$$29\!\cdots\!62$$$$T^{49} +$$$$35\!\cdots\!66$$$$T^{50} +$$$$29\!\cdots\!62$$$$p T^{51} +$$$$41\!\cdots\!76$$$$p^{2} T^{52} +$$$$34\!\cdots\!86$$$$p^{3} T^{53} +$$$$46\!\cdots\!68$$$$p^{4} T^{54} +$$$$36\!\cdots\!58$$$$p^{5} T^{55} +$$$$48\!\cdots\!21$$$$p^{6} T^{56} +$$$$44\!\cdots\!73$$$$p^{8} T^{57} +$$$$48\!\cdots\!61$$$$p^{8} T^{58} +$$$$35\!\cdots\!56$$$$p^{9} T^{59} +$$$$46\!\cdots\!52$$$$p^{10} T^{60} +$$$$32\!\cdots\!32$$$$p^{11} T^{61} +$$$$41\!\cdots\!63$$$$p^{12} T^{62} +$$$$27\!\cdots\!30$$$$p^{13} T^{63} +$$$$34\!\cdots\!71$$$$p^{14} T^{64} +$$$$22\!\cdots\!74$$$$p^{15} T^{65} +$$$$27\!\cdots\!25$$$$p^{16} T^{66} +$$$$16\!\cdots\!02$$$$p^{17} T^{67} +$$$$20\!\cdots\!61$$$$p^{18} T^{68} +$$$$11\!\cdots\!70$$$$p^{19} T^{69} +$$$$14\!\cdots\!65$$$$p^{20} T^{70} +$$$$75\!\cdots\!17$$$$p^{21} T^{71} +$$$$95\!\cdots\!70$$$$p^{22} T^{72} +$$$$45\!\cdots\!09$$$$p^{23} T^{73} +$$$$58\!\cdots\!93$$$$p^{24} T^{74} +$$$$25\!\cdots\!56$$$$p^{25} T^{75} +$$$$33\!\cdots\!68$$$$p^{26} T^{76} +$$$$12\!\cdots\!05$$$$p^{27} T^{77} +$$$$17\!\cdots\!63$$$$p^{28} T^{78} +$$$$59\!\cdots\!61$$$$p^{29} T^{79} +$$$$84\!\cdots\!62$$$$p^{30} T^{80} +$$$$25\!\cdots\!41$$$$p^{31} T^{81} +$$$$37\!\cdots\!20$$$$p^{32} T^{82} +$$$$94\!\cdots\!42$$$$p^{33} T^{83} +$$$$14\!\cdots\!20$$$$p^{34} T^{84} +$$$$31\!\cdots\!81$$$$p^{35} T^{85} + 52218841089641695993 p^{36} T^{86} + 881590226713430209 p^{37} T^{87} + 162793866453333577 p^{38} T^{88} + 2109125411089860 p^{39} T^{89} + 437009569361682 p^{40} T^{90} + 4100128613812 p^{41} T^{91} + 982141842112 p^{42} T^{92} + 6142459035 p^{43} T^{93} + 1773710196 p^{44} T^{94} + 6499374 p^{45} T^{95} + 2412101 p^{46} T^{96} + 4108 p^{47} T^{97} + 2194 p^{48} T^{98} + p^{49} T^{99} + p^{50} T^{100}$$
89 $$1 - 19 T + 2664 T^{2} - 48709 T^{3} + 3536567 T^{4} - 62343574 T^{5} + 3120350221 T^{6} - 53115998714 T^{7} + 2058844287945 T^{8} - 33886294168795 T^{9} + 1083682695829938 T^{10} - 17264610206239470 T^{11} + 473983990086692321 T^{12} - 7315994777159624768 T^{13} +$$$$17\!\cdots\!42$$$$T^{14} -$$$$26\!\cdots\!32$$$$T^{15} +$$$$57\!\cdots\!45$$$$T^{16} -$$$$83\!\cdots\!93$$$$T^{17} +$$$$16\!\cdots\!68$$$$T^{18} -$$$$23\!\cdots\!38$$$$T^{19} +$$$$43\!\cdots\!45$$$$T^{20} -$$$$59\!\cdots\!50$$$$T^{21} +$$$$10\!\cdots\!70$$$$T^{22} -$$$$13\!\cdots\!89$$$$T^{23} +$$$$21\!\cdots\!11$$$$T^{24} -$$$$28\!\cdots\!19$$$$T^{25} +$$$$42\!\cdots\!90$$$$T^{26} -$$$$53\!\cdots\!11$$$$T^{27} +$$$$77\!\cdots\!57$$$$T^{28} -$$$$95\!\cdots\!16$$$$T^{29} +$$$$13\!\cdots\!78$$$$T^{30} -$$$$15\!\cdots\!81$$$$T^{31} +$$$$20\!\cdots\!65$$$$T^{32} -$$$$23\!\cdots\!83$$$$T^{33} +$$$$30\!\cdots\!97$$$$T^{34} -$$$$34\!\cdots\!72$$$$T^{35} +$$$$41\!\cdots\!27$$$$T^{36} -$$$$45\!\cdots\!36$$$$T^{37} +$$$$53\!\cdots\!99$$$$T^{38} -$$$$57\!\cdots\!34$$$$T^{39} +$$$$65\!\cdots\!39$$$$T^{40} -$$$$68\!\cdots\!87$$$$T^{41} +$$$$75\!\cdots\!43$$$$T^{42} -$$$$76\!\cdots\!34$$$$T^{43} +$$$$81\!\cdots\!33$$$$T^{44} -$$$$80\!\cdots\!21$$$$T^{45} +$$$$83\!\cdots\!97$$$$T^{46} -$$$$80\!\cdots\!03$$$$T^{47} +$$$$80\!\cdots\!56$$$$T^{48} -$$$$75\!\cdots\!72$$$$T^{49} +$$$$74\!\cdots\!02$$$$T^{50} -$$$$75\!\cdots\!72$$$$p T^{51} +$$$$80\!\cdots\!56$$$$p^{2} T^{52} -$$$$80\!\cdots\!03$$$$p^{3} T^{53} +$$$$83\!\cdots\!97$$$$p^{4} T^{54} -$$$$80\!\cdots\!21$$$$p^{5} T^{55} +$$$$81\!\cdots\!33$$$$p^{6} T^{56} -$$$$76\!\cdots\!34$$$$p^{7} T^{57} +$$$$75\!\cdots\!43$$$$p^{8} T^{58} -$$$$68\!\cdots\!87$$$$p^{9} T^{59} +$$$$65\!\cdots\!39$$$$p^{10} T^{60} -$$$$57\!\cdots\!34$$$$p^{11} T^{61} +$$$$53\!\cdots\!99$$$$p^{12} T^{62} -$$$$45\!\cdots\!36$$$$p^{13} T^{63} +$$$$41\!\cdots\!27$$$$p^{14} T^{64} -$$$$34\!\cdots\!72$$$$p^{15} T^{65} +$$$$30\!\cdots\!97$$$$p^{16} T^{66} -$$$$23\!\cdots\!83$$$$p^{17} T^{67} +$$$$20\!\cdots\!65$$$$p^{18} T^{68} -$$$$15\!\cdots\!81$$$$p^{19} T^{69} +$$$$13\!\cdots\!78$$$$p^{20} T^{70} -$$$$95\!\cdots\!16$$$$p^{21} T^{71} +$$$$77\!\cdots\!57$$$$p^{22} T^{72} -$$$$53\!\cdots\!11$$$$p^{23} T^{73} +$$$$42\!\cdots\!90$$$$p^{24} T^{74} -$$$$28\!\cdots\!19$$$$p^{25} T^{75} +$$$$21\!\cdots\!11$$$$p^{26} T^{76} -$$$$13\!\cdots\!89$$$$p^{27} T^{77} +$$$$10\!\cdots\!70$$$$p^{28} T^{78} -$$$$59\!\cdots\!50$$$$p^{29} T^{79} +$$$$43\!\cdots\!45$$$$p^{30} T^{80} -$$$$23\!\cdots\!38$$$$p^{31} T^{81} +$$$$16\!\cdots\!68$$$$p^{32} T^{82} -$$$$83\!\cdots\!93$$$$p^{33} T^{83} +$$$$57\!\cdots\!45$$$$p^{34} T^{84} -$$$$26\!\cdots\!32$$$$p^{35} T^{85} +$$$$17\!\cdots\!42$$$$p^{36} T^{86} - 7315994777159624768 p^{37} T^{87} + 473983990086692321 p^{38} T^{88} - 17264610206239470 p^{39} T^{89} + 1083682695829938 p^{40} T^{90} - 33886294168795 p^{41} T^{91} + 2058844287945 p^{42} T^{92} - 53115998714 p^{43} T^{93} + 3120350221 p^{44} T^{94} - 62343574 p^{45} T^{95} + 3536567 p^{46} T^{96} - 48709 p^{47} T^{97} + 2664 p^{48} T^{98} - 19 p^{49} T^{99} + p^{50} T^{100}$$
97 $$1 - 81 T + 6164 T^{2} - 317807 T^{3} + 15041518 T^{4} - 593038013 T^{5} + 21700648470 T^{6} - 708465700033 T^{7} + 21716930553702 T^{8} - 613313726422507 T^{9} + 16414412306825882 T^{10} - 412113562958163165 T^{11} + 9875944589010335095 T^{12} -$$$$22\!\cdots\!18$$$$T^{13} +$$$$49\!\cdots\!86$$$$T^{14} -$$$$10\!\cdots\!10$$$$T^{15} +$$$$20\!\cdots\!07$$$$T^{16} -$$$$39\!\cdots\!03$$$$T^{17} +$$$$74\!\cdots\!15$$$$T^{18} -$$$$13\!\cdots\!35$$$$T^{19} +$$$$23\!\cdots\!95$$$$T^{20} -$$$$40\!\cdots\!43$$$$T^{21} +$$$$66\!\cdots\!66$$$$T^{22} -$$$$10\!\cdots\!36$$$$T^{23} +$$$$16\!\cdots\!38$$$$T^{24} -$$$$25\!\cdots\!49$$$$T^{25} +$$$$38\!\cdots\!58$$$$T^{26} -$$$$55\!\cdots\!28$$$$T^{27} +$$$$78\!\cdots\!00$$$$T^{28} -$$$$10\!\cdots\!47$$$$T^{29} +$$$$14\!\cdots\!08$$$$T^{30} -$$$$19\!\cdots\!09$$$$T^{31} +$$$$25\!\cdots\!50$$$$T^{32} -$$$$32\!\cdots\!44$$$$T^{33} +$$$$41\!\cdots\!28$$$$T^{34} -$$$$50\!\cdots\!45$$$$T^{35} +$$$$61\!\cdots\!67$$$$T^{36} -$$$$73\!\cdots\!28$$$$T^{37} +$$$$85\!\cdots\!69$$$$T^{38} -$$$$98\!\cdots\!84$$$$T^{39} +$$$$11\!\cdots\!71$$$$T^{40} -$$$$12\!\cdots\!55$$$$T^{41} +$$$$13\!\cdots\!97$$$$T^{42} -$$$$14\!\cdots\!62$$$$T^{43} +$$$$15\!\cdots\!69$$$$T^{44} -$$$$16\!\cdots\!91$$$$T^{45} +$$$$17\!\cdots\!98$$$$T^{46} -$$$$17\!\cdots\!52$$$$T^{47} +$$$$18\!\cdots\!71$$$$T^{48} -$$$$18\!\cdots\!67$$$$T^{49} +$$$$18\!\cdots\!94$$$$T^{50} -$$$$18\!\cdots\!67$$$$p T^{51} +$$$$18\!\cdots\!71$$$$p^{2} T^{52} -$$$$17\!\cdots\!52$$$$p^{3} T^{53} +$$$$17\!\cdots\!98$$$$p^{4} T^{54} -$$$$16\!\cdots\!91$$$$p^{5} T^{55} +$$$$15\!\cdots\!69$$$$p^{6} T^{56} -$$$$14\!\cdots\!62$$$$p^{7} T^{57} +$$$$13\!\cdots\!97$$$$p^{8} T^{58} -$$$$12\!\cdots\!55$$$$p^{9} T^{59} +$$$$11\!\cdots\!71$$$$p^{10} T^{60} -$$$$98\!\cdots\!84$$$$p^{11} T^{61} +$$$$85\!\cdots\!69$$$$p^{12} T^{62} -$$$$73\!\cdots\!28$$$$p^{13} T^{63} +$$$$61\!\cdots\!67$$$$p^{14} T^{64} -$$$$50\!\cdots\!45$$$$p^{15} T^{65} +$$$$41\!\cdots\!28$$$$p^{16} T^{66} -$$$$32\!\cdots\!44$$$$p^{17} T^{67} +$$$$25\!\cdots\!50$$$$p^{18} T^{68} -$$$$19\!\cdots\!09$$$$p^{19} T^{69} +$$$$14\!\cdots\!08$$$$p^{20} T^{70} -$$$$10\!\cdots\!47$$$$p^{21} T^{71} +$$$$78\!\cdots\!00$$$$p^{22} T^{72} -$$$$55\!\cdots\!28$$$$p^{23} T^{73} +$$$$38\!\cdots\!58$$$$p^{24} T^{74} -$$$$25\!\cdots\!49$$$$p^{25} T^{75} +$$$$16\!\cdots\!38$$$$p^{26} T^{76} -$$$$10\!\cdots\!36$$$$p^{27} T^{77} +$$$$66\!\cdots\!66$$$$p^{28} T^{78} -$$$$40\!\cdots\!43$$$$p^{29} T^{79} +$$$$23\!\cdots\!95$$$$p^{30} T^{80} -$$$$13\!\cdots\!35$$$$p^{31} T^{81} +$$$$74\!\cdots\!15$$$$p^{32} T^{82} -$$$$39\!\cdots\!03$$$$p^{33} T^{83} +$$$$20\!\cdots\!07$$$$p^{34} T^{84} -$$$$10\!\cdots\!10$$$$p^{35} T^{85} +$$$$49\!\cdots\!86$$$$p^{36} T^{86} -$$$$22\!\cdots\!18$$$$p^{37} T^{87} + 9875944589010335095 p^{38} T^{88} - 412113562958163165 p^{39} T^{89} + 16414412306825882 p^{40} T^{90} - 613313726422507 p^{41} T^{91} + 21716930553702 p^{42} T^{92} - 708465700033 p^{43} T^{93} + 21700648470 p^{44} T^{94} - 593038013 p^{45} T^{95} + 15041518 p^{46} T^{96} - 317807 p^{47} T^{97} + 6164 p^{48} T^{98} - 81 p^{49} T^{99} + p^{50} T^{100}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{100} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}