# Properties

 Degree 94 Conductor $2^{47} \cdot 19^{47} \cdot 211^{47}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 47·2-s + 10·3-s + 1.12e3·4-s + 15·5-s + 470·6-s + 1.84e4·8-s + 14·9-s + 705·10-s + 17·11-s + 1.12e4·12-s + 27·13-s + 150·15-s + 2.30e5·16-s + 16·17-s + 658·18-s − 47·19-s + 1.69e4·20-s + 799·22-s + 24·23-s + 1.84e5·24-s + 38·25-s + 1.26e3·26-s − 189·27-s + 58·29-s + 7.05e3·30-s + 21·31-s + 2.34e6·32-s + ⋯
 L(s)  = 1 + 33.2·2-s + 5.77·3-s + 564·4-s + 6.70·5-s + 191.·6-s + 6.51e3·8-s + 14/3·9-s + 222.·10-s + 5.12·11-s + 3.25e3·12-s + 7.48·13-s + 38.7·15-s + 5.75e4·16-s + 3.88·17-s + 155.·18-s − 10.7·19-s + 3.78e3·20-s + 170.·22-s + 5.00·23-s + 3.76e4·24-s + 38/5·25-s + 248.·26-s − 36.3·27-s + 10.7·29-s + 1.28e3·30-s + 3.77·31-s + 4.15e5·32-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{47} \cdot 19^{47} \cdot 211^{47}\right)^{s/2} \, \Gamma_{\C}(s)^{47} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{47} \cdot 19^{47} \cdot 211^{47}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{47} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$94$$ $$N$$ = $$2^{47} \cdot 19^{47} \cdot 211^{47}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{8018} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(94,\ 2^{47} \cdot 19^{47} \cdot 211^{47} ,\ ( \ : [1/2]^{47} ),\ 1 )$ $L(1)$ $\approx$ $1.523963059e21$ $L(\frac12)$ $\approx$ $1.523963059e21$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;19,\;211\}$, $$F_p$$ is a polynomial of degree 94. If $p \in \{2,\;19,\;211\}$, then $F_p$ is a polynomial of degree at most 93.
$p$$F_p$
bad2 $$( 1 - T )^{47}$$
19 $$( 1 + T )^{47}$$
211 $$( 1 + T )^{47}$$
good3 $$1 - 10 T + 86 T^{2} - 59 p^{2} T^{3} + 2929 T^{4} - 1544 p^{2} T^{5} + 60715 T^{6} - 241883 T^{7} + 905230 T^{8} - 1058318 p T^{9} + 10601123 T^{10} - 11232920 p T^{11} + 34313459 p T^{12} - 302411393 T^{13} + 859595501 T^{14} - 2366339942 T^{15} + 6335425433 T^{16} - 16511165306 T^{17} + 14005013948 p T^{18} - 104471856812 T^{19} + 254418042356 T^{20} - 607213302149 T^{21} + 474256145359 p T^{22} - 3274679054833 T^{23} + 7413680664536 T^{24} - 1835212497541 p^{2} T^{25} + 36251088090926 T^{26} - 78410302813075 T^{27} + 167286509403754 T^{28} - 352140570954167 T^{29} + 731886894023504 T^{30} - 1502275942887328 T^{31} + 3047077162341409 T^{32} - 6108397845166894 T^{33} + 1345379583310792 p^{2} T^{34} - 23736989116159931 T^{35} + 46037572606085855 T^{36} - 88347051233728456 T^{37} + 167805135737157160 T^{38} - 315486150488599003 T^{39} + 587263561042850651 T^{40} - 360790980609868655 p T^{41} + 1975633642050482120 T^{42} - 3571253609444070061 T^{43} + 710481873879386800 p^{2} T^{44} - 11340078995440281341 T^{45} + 6640794671528803613 p T^{46} - 34669080274893246754 T^{47} + 6640794671528803613 p^{2} T^{48} - 11340078995440281341 p^{2} T^{49} + 710481873879386800 p^{5} T^{50} - 3571253609444070061 p^{4} T^{51} + 1975633642050482120 p^{5} T^{52} - 360790980609868655 p^{7} T^{53} + 587263561042850651 p^{7} T^{54} - 315486150488599003 p^{8} T^{55} + 167805135737157160 p^{9} T^{56} - 88347051233728456 p^{10} T^{57} + 46037572606085855 p^{11} T^{58} - 23736989116159931 p^{12} T^{59} + 1345379583310792 p^{15} T^{60} - 6108397845166894 p^{14} T^{61} + 3047077162341409 p^{15} T^{62} - 1502275942887328 p^{16} T^{63} + 731886894023504 p^{17} T^{64} - 352140570954167 p^{18} T^{65} + 167286509403754 p^{19} T^{66} - 78410302813075 p^{20} T^{67} + 36251088090926 p^{21} T^{68} - 1835212497541 p^{24} T^{69} + 7413680664536 p^{23} T^{70} - 3274679054833 p^{24} T^{71} + 474256145359 p^{26} T^{72} - 607213302149 p^{26} T^{73} + 254418042356 p^{27} T^{74} - 104471856812 p^{28} T^{75} + 14005013948 p^{30} T^{76} - 16511165306 p^{30} T^{77} + 6335425433 p^{31} T^{78} - 2366339942 p^{32} T^{79} + 859595501 p^{33} T^{80} - 302411393 p^{34} T^{81} + 34313459 p^{36} T^{82} - 11232920 p^{37} T^{83} + 10601123 p^{37} T^{84} - 1058318 p^{39} T^{85} + 905230 p^{39} T^{86} - 241883 p^{40} T^{87} + 60715 p^{41} T^{88} - 1544 p^{44} T^{89} + 2929 p^{43} T^{90} - 59 p^{46} T^{91} + 86 p^{45} T^{92} - 10 p^{46} T^{93} + p^{47} T^{94}$$
5 $$1 - 3 p T + 187 T^{2} - 1689 T^{3} + 13492 T^{4} - 92867 T^{5} + 585932 T^{6} - 3369879 T^{7} + 18149163 T^{8} - 91494503 T^{9} + 87576329 p T^{10} - 1991436969 T^{11} + 8678547037 T^{12} - 36287020583 T^{13} + 146356695232 T^{14} - 570120818634 T^{15} + 2152862266851 T^{16} - 7889304918176 T^{17} + 5626268135331 p T^{18} - 97698000754954 T^{19} + 331125440466844 T^{20} - 1096142569950228 T^{21} + 3549625915526376 T^{22} - 18003832396297 p^{4} T^{23} + 34961577843476164 T^{24} - 106533043479444018 T^{25} + 318685758418225794 T^{26} - 936384590806140524 T^{27} + 2704723680341573314 T^{28} - 7683639517149421084 T^{29} + 859316838983825411 p^{2} T^{30} - 59139078999049340571 T^{31} +$$$$16\!\cdots\!33$$$$T^{32} -$$$$42\!\cdots\!83$$$$T^{33} +$$$$11\!\cdots\!21$$$$T^{34} -$$$$29\!\cdots\!97$$$$T^{35} +$$$$75\!\cdots\!06$$$$T^{36} -$$$$19\!\cdots\!89$$$$T^{37} +$$$$47\!\cdots\!97$$$$T^{38} -$$$$11\!\cdots\!68$$$$T^{39} +$$$$28\!\cdots\!96$$$$T^{40} -$$$$69\!\cdots\!79$$$$T^{41} +$$$$16\!\cdots\!67$$$$T^{42} -$$$$38\!\cdots\!27$$$$T^{43} +$$$$90\!\cdots\!02$$$$T^{44} -$$$$82\!\cdots\!19$$$$p^{2} T^{45} +$$$$47\!\cdots\!24$$$$T^{46} -$$$$10\!\cdots\!78$$$$T^{47} +$$$$47\!\cdots\!24$$$$p T^{48} -$$$$82\!\cdots\!19$$$$p^{4} T^{49} +$$$$90\!\cdots\!02$$$$p^{3} T^{50} -$$$$38\!\cdots\!27$$$$p^{4} T^{51} +$$$$16\!\cdots\!67$$$$p^{5} T^{52} -$$$$69\!\cdots\!79$$$$p^{6} T^{53} +$$$$28\!\cdots\!96$$$$p^{7} T^{54} -$$$$11\!\cdots\!68$$$$p^{8} T^{55} +$$$$47\!\cdots\!97$$$$p^{9} T^{56} -$$$$19\!\cdots\!89$$$$p^{10} T^{57} +$$$$75\!\cdots\!06$$$$p^{11} T^{58} -$$$$29\!\cdots\!97$$$$p^{12} T^{59} +$$$$11\!\cdots\!21$$$$p^{13} T^{60} -$$$$42\!\cdots\!83$$$$p^{14} T^{61} +$$$$16\!\cdots\!33$$$$p^{15} T^{62} - 59139078999049340571 p^{16} T^{63} + 859316838983825411 p^{19} T^{64} - 7683639517149421084 p^{18} T^{65} + 2704723680341573314 p^{19} T^{66} - 936384590806140524 p^{20} T^{67} + 318685758418225794 p^{21} T^{68} - 106533043479444018 p^{22} T^{69} + 34961577843476164 p^{23} T^{70} - 18003832396297 p^{28} T^{71} + 3549625915526376 p^{25} T^{72} - 1096142569950228 p^{26} T^{73} + 331125440466844 p^{27} T^{74} - 97698000754954 p^{28} T^{75} + 5626268135331 p^{30} T^{76} - 7889304918176 p^{30} T^{77} + 2152862266851 p^{31} T^{78} - 570120818634 p^{32} T^{79} + 146356695232 p^{33} T^{80} - 36287020583 p^{34} T^{81} + 8678547037 p^{35} T^{82} - 1991436969 p^{36} T^{83} + 87576329 p^{38} T^{84} - 91494503 p^{38} T^{85} + 18149163 p^{39} T^{86} - 3369879 p^{40} T^{87} + 585932 p^{41} T^{88} - 92867 p^{42} T^{89} + 13492 p^{43} T^{90} - 1689 p^{44} T^{91} + 187 p^{45} T^{92} - 3 p^{47} T^{93} + p^{47} T^{94}$$
7 $$1 + 118 T^{2} + 51 T^{3} + 995 p T^{4} + 6094 T^{5} + 276001 T^{6} + 362841 T^{7} + 8316809 T^{8} + 14403964 T^{9} + 204512517 T^{10} + 430459376 T^{11} + 4293690063 T^{12} + 10366300739 T^{13} + 79333021377 T^{14} + 210215193478 T^{15} + 1316818612983 T^{16} + 3702021034402 T^{17} + 19914537690532 T^{18} + 8273441613760 p T^{19} + 39593020539202 p T^{20} + 818739878547439 T^{21} + 3576200779886543 T^{22} + 1514190596418029 p T^{23} + 43046044626361928 T^{24} + 126972532067866942 T^{25} + 485906383895530664 T^{26} + 202754323851595176 p T^{27} + 5168502211840757717 T^{28} + 14904623105866952735 T^{29} + 52033893777613495067 T^{30} +$$$$14\!\cdots\!14$$$$T^{31} +$$$$49\!\cdots\!79$$$$T^{32} + 28434206877426228126 p^{2} T^{33} +$$$$45\!\cdots\!23$$$$T^{34} +$$$$17\!\cdots\!13$$$$p T^{35} +$$$$39\!\cdots\!79$$$$T^{36} +$$$$10\!\cdots\!66$$$$T^{37} +$$$$33\!\cdots\!93$$$$T^{38} +$$$$88\!\cdots\!53$$$$T^{39} +$$$$26\!\cdots\!32$$$$T^{40} +$$$$70\!\cdots\!50$$$$T^{41} +$$$$20\!\cdots\!86$$$$T^{42} +$$$$53\!\cdots\!20$$$$T^{43} +$$$$15\!\cdots\!88$$$$T^{44} +$$$$39\!\cdots\!67$$$$T^{45} +$$$$11\!\cdots\!01$$$$T^{46} +$$$$40\!\cdots\!66$$$$p T^{47} +$$$$11\!\cdots\!01$$$$p T^{48} +$$$$39\!\cdots\!67$$$$p^{2} T^{49} +$$$$15\!\cdots\!88$$$$p^{3} T^{50} +$$$$53\!\cdots\!20$$$$p^{4} T^{51} +$$$$20\!\cdots\!86$$$$p^{5} T^{52} +$$$$70\!\cdots\!50$$$$p^{6} T^{53} +$$$$26\!\cdots\!32$$$$p^{7} T^{54} +$$$$88\!\cdots\!53$$$$p^{8} T^{55} +$$$$33\!\cdots\!93$$$$p^{9} T^{56} +$$$$10\!\cdots\!66$$$$p^{10} T^{57} +$$$$39\!\cdots\!79$$$$p^{11} T^{58} +$$$$17\!\cdots\!13$$$$p^{13} T^{59} +$$$$45\!\cdots\!23$$$$p^{13} T^{60} + 28434206877426228126 p^{16} T^{61} +$$$$49\!\cdots\!79$$$$p^{15} T^{62} +$$$$14\!\cdots\!14$$$$p^{16} T^{63} + 52033893777613495067 p^{17} T^{64} + 14904623105866952735 p^{18} T^{65} + 5168502211840757717 p^{19} T^{66} + 202754323851595176 p^{21} T^{67} + 485906383895530664 p^{21} T^{68} + 126972532067866942 p^{22} T^{69} + 43046044626361928 p^{23} T^{70} + 1514190596418029 p^{25} T^{71} + 3576200779886543 p^{25} T^{72} + 818739878547439 p^{26} T^{73} + 39593020539202 p^{28} T^{74} + 8273441613760 p^{29} T^{75} + 19914537690532 p^{29} T^{76} + 3702021034402 p^{30} T^{77} + 1316818612983 p^{31} T^{78} + 210215193478 p^{32} T^{79} + 79333021377 p^{33} T^{80} + 10366300739 p^{34} T^{81} + 4293690063 p^{35} T^{82} + 430459376 p^{36} T^{83} + 204512517 p^{37} T^{84} + 14403964 p^{38} T^{85} + 8316809 p^{39} T^{86} + 362841 p^{40} T^{87} + 276001 p^{41} T^{88} + 6094 p^{42} T^{89} + 995 p^{44} T^{90} + 51 p^{44} T^{91} + 118 p^{45} T^{92} + p^{47} T^{94}$$
11 $$1 - 17 T + 346 T^{2} - 4172 T^{3} + 51924 T^{4} - 501124 T^{5} + 4813802 T^{6} - 39508776 T^{7} + 319210876 T^{8} - 2309627830 T^{9} + 16401167687 T^{10} - 107119166812 T^{11} + 686574943185 T^{12} - 374133419871 p T^{13} + 24233974400098 T^{14} - 134957431423116 T^{15} + 739375349522289 T^{16} - 351006530333354 p T^{17} + 19866286498515411 T^{18} - 97987342740003818 T^{19} + 476932496754390315 T^{20} - 2234681842087121670 T^{21} + 10348070099410000399 T^{22} - 46274595761757374657 T^{23} +$$$$20\!\cdots\!87$$$$T^{24} -$$$$87\!\cdots\!31$$$$T^{25} +$$$$37\!\cdots\!70$$$$T^{26} -$$$$13\!\cdots\!09$$$$p T^{27} +$$$$62\!\cdots\!62$$$$T^{28} -$$$$24\!\cdots\!44$$$$T^{29} +$$$$98\!\cdots\!53$$$$T^{30} -$$$$37\!\cdots\!48$$$$T^{31} +$$$$13\!\cdots\!15$$$$p T^{32} -$$$$53\!\cdots\!88$$$$T^{33} +$$$$19\!\cdots\!56$$$$T^{34} -$$$$71\!\cdots\!82$$$$T^{35} +$$$$25\!\cdots\!70$$$$T^{36} -$$$$91\!\cdots\!88$$$$T^{37} +$$$$32\!\cdots\!26$$$$T^{38} -$$$$11\!\cdots\!15$$$$T^{39} +$$$$38\!\cdots\!44$$$$T^{40} -$$$$13\!\cdots\!86$$$$T^{41} +$$$$44\!\cdots\!38$$$$T^{42} -$$$$14\!\cdots\!89$$$$T^{43} +$$$$50\!\cdots\!24$$$$T^{44} -$$$$16\!\cdots\!15$$$$T^{45} +$$$$55\!\cdots\!96$$$$T^{46} -$$$$18\!\cdots\!20$$$$T^{47} +$$$$55\!\cdots\!96$$$$p T^{48} -$$$$16\!\cdots\!15$$$$p^{2} T^{49} +$$$$50\!\cdots\!24$$$$p^{3} T^{50} -$$$$14\!\cdots\!89$$$$p^{4} T^{51} +$$$$44\!\cdots\!38$$$$p^{5} T^{52} -$$$$13\!\cdots\!86$$$$p^{6} T^{53} +$$$$38\!\cdots\!44$$$$p^{7} T^{54} -$$$$11\!\cdots\!15$$$$p^{8} T^{55} +$$$$32\!\cdots\!26$$$$p^{9} T^{56} -$$$$91\!\cdots\!88$$$$p^{10} T^{57} +$$$$25\!\cdots\!70$$$$p^{11} T^{58} -$$$$71\!\cdots\!82$$$$p^{12} T^{59} +$$$$19\!\cdots\!56$$$$p^{13} T^{60} -$$$$53\!\cdots\!88$$$$p^{14} T^{61} +$$$$13\!\cdots\!15$$$$p^{16} T^{62} -$$$$37\!\cdots\!48$$$$p^{16} T^{63} +$$$$98\!\cdots\!53$$$$p^{17} T^{64} -$$$$24\!\cdots\!44$$$$p^{18} T^{65} +$$$$62\!\cdots\!62$$$$p^{19} T^{66} -$$$$13\!\cdots\!09$$$$p^{21} T^{67} +$$$$37\!\cdots\!70$$$$p^{21} T^{68} -$$$$87\!\cdots\!31$$$$p^{22} T^{69} +$$$$20\!\cdots\!87$$$$p^{23} T^{70} - 46274595761757374657 p^{24} T^{71} + 10348070099410000399 p^{25} T^{72} - 2234681842087121670 p^{26} T^{73} + 476932496754390315 p^{27} T^{74} - 97987342740003818 p^{28} T^{75} + 19866286498515411 p^{29} T^{76} - 351006530333354 p^{31} T^{77} + 739375349522289 p^{31} T^{78} - 134957431423116 p^{32} T^{79} + 24233974400098 p^{33} T^{80} - 374133419871 p^{35} T^{81} + 686574943185 p^{35} T^{82} - 107119166812 p^{36} T^{83} + 16401167687 p^{37} T^{84} - 2309627830 p^{38} T^{85} + 319210876 p^{39} T^{86} - 39508776 p^{40} T^{87} + 4813802 p^{41} T^{88} - 501124 p^{42} T^{89} + 51924 p^{43} T^{90} - 4172 p^{44} T^{91} + 346 p^{45} T^{92} - 17 p^{46} T^{93} + p^{47} T^{94}$$
13 $$1 - 27 T + 609 T^{2} - 9895 T^{3} + 141722 T^{4} - 1742331 T^{5} + 19539298 T^{6} - 199016524 T^{7} + 1888651003 T^{8} - 16715674645 T^{9} + 139773523987 T^{10} - 1106651241716 T^{11} + 8358546502102 T^{12} - 60363910812746 T^{13} + 32216517508563 p T^{14} - 2797433182601650 T^{15} + 18047338455264894 T^{16} - 112656184452099491 T^{17} + 682055375389266043 T^{18} - 4011109995155422580 T^{19} + 22954901678106347340 T^{20} - 9846153790522824439 p T^{21} +$$$$69\!\cdots\!72$$$$T^{22} -$$$$37\!\cdots\!68$$$$T^{23} +$$$$19\!\cdots\!67$$$$T^{24} -$$$$97\!\cdots\!63$$$$T^{25} +$$$$48\!\cdots\!06$$$$T^{26} -$$$$23\!\cdots\!10$$$$T^{27} +$$$$11\!\cdots\!00$$$$T^{28} -$$$$53\!\cdots\!99$$$$T^{29} +$$$$24\!\cdots\!23$$$$T^{30} -$$$$11\!\cdots\!97$$$$T^{31} +$$$$50\!\cdots\!16$$$$T^{32} -$$$$22\!\cdots\!82$$$$T^{33} +$$$$96\!\cdots\!99$$$$T^{34} -$$$$41\!\cdots\!05$$$$T^{35} +$$$$17\!\cdots\!13$$$$T^{36} -$$$$72\!\cdots\!64$$$$T^{37} +$$$$29\!\cdots\!35$$$$T^{38} -$$$$11\!\cdots\!16$$$$T^{39} +$$$$47\!\cdots\!52$$$$T^{40} -$$$$18\!\cdots\!23$$$$T^{41} +$$$$71\!\cdots\!07$$$$T^{42} -$$$$27\!\cdots\!68$$$$T^{43} +$$$$10\!\cdots\!52$$$$T^{44} -$$$$38\!\cdots\!54$$$$T^{45} +$$$$83\!\cdots\!76$$$$p^{2} T^{46} -$$$$51\!\cdots\!14$$$$T^{47} +$$$$83\!\cdots\!76$$$$p^{3} T^{48} -$$$$38\!\cdots\!54$$$$p^{2} T^{49} +$$$$10\!\cdots\!52$$$$p^{3} T^{50} -$$$$27\!\cdots\!68$$$$p^{4} T^{51} +$$$$71\!\cdots\!07$$$$p^{5} T^{52} -$$$$18\!\cdots\!23$$$$p^{6} T^{53} +$$$$47\!\cdots\!52$$$$p^{7} T^{54} -$$$$11\!\cdots\!16$$$$p^{8} T^{55} +$$$$29\!\cdots\!35$$$$p^{9} T^{56} -$$$$72\!\cdots\!64$$$$p^{10} T^{57} +$$$$17\!\cdots\!13$$$$p^{11} T^{58} -$$$$41\!\cdots\!05$$$$p^{12} T^{59} +$$$$96\!\cdots\!99$$$$p^{13} T^{60} -$$$$22\!\cdots\!82$$$$p^{14} T^{61} +$$$$50\!\cdots\!16$$$$p^{15} T^{62} -$$$$11\!\cdots\!97$$$$p^{16} T^{63} +$$$$24\!\cdots\!23$$$$p^{17} T^{64} -$$$$53\!\cdots\!99$$$$p^{18} T^{65} +$$$$11\!\cdots\!00$$$$p^{19} T^{66} -$$$$23\!\cdots\!10$$$$p^{20} T^{67} +$$$$48\!\cdots\!06$$$$p^{21} T^{68} -$$$$97\!\cdots\!63$$$$p^{22} T^{69} +$$$$19\!\cdots\!67$$$$p^{23} T^{70} -$$$$37\!\cdots\!68$$$$p^{24} T^{71} +$$$$69\!\cdots\!72$$$$p^{25} T^{72} - 9846153790522824439 p^{27} T^{73} + 22954901678106347340 p^{27} T^{74} - 4011109995155422580 p^{28} T^{75} + 682055375389266043 p^{29} T^{76} - 112656184452099491 p^{30} T^{77} + 18047338455264894 p^{31} T^{78} - 2797433182601650 p^{32} T^{79} + 32216517508563 p^{34} T^{80} - 60363910812746 p^{34} T^{81} + 8358546502102 p^{35} T^{82} - 1106651241716 p^{36} T^{83} + 139773523987 p^{37} T^{84} - 16715674645 p^{38} T^{85} + 1888651003 p^{39} T^{86} - 199016524 p^{40} T^{87} + 19539298 p^{41} T^{88} - 1742331 p^{42} T^{89} + 141722 p^{43} T^{90} - 9895 p^{44} T^{91} + 609 p^{45} T^{92} - 27 p^{46} T^{93} + p^{47} T^{94}$$
17 $$1 - 16 T + 488 T^{2} - 6275 T^{3} + 109411 T^{4} - 1189164 T^{5} + 15347809 T^{6} - 145641717 T^{7} + 1535205530 T^{8} - 765377345 p T^{9} + 117895271019 T^{10} - 907715387163 T^{11} + 7295613161674 T^{12} - 51715732120317 T^{13} + 376746185206746 T^{14} - 2485914969278550 T^{15} + 16680494721096703 T^{16} - 103407256517358151 T^{17} + 647268903507099499 T^{18} - 3800059958202029722 T^{19} + 22420978605144955974 T^{20} -$$$$12\!\cdots\!58$$$$T^{21} +$$$$70\!\cdots\!94$$$$T^{22} -$$$$37\!\cdots\!77$$$$T^{23} +$$$$20\!\cdots\!75$$$$T^{24} -$$$$10\!\cdots\!85$$$$T^{25} +$$$$54\!\cdots\!98$$$$T^{26} -$$$$27\!\cdots\!24$$$$T^{27} +$$$$13\!\cdots\!63$$$$T^{28} -$$$$66\!\cdots\!79$$$$T^{29} +$$$$32\!\cdots\!40$$$$T^{30} -$$$$15\!\cdots\!20$$$$T^{31} +$$$$72\!\cdots\!51$$$$T^{32} -$$$$33\!\cdots\!99$$$$T^{33} +$$$$15\!\cdots\!37$$$$T^{34} -$$$$70\!\cdots\!13$$$$T^{35} +$$$$32\!\cdots\!43$$$$T^{36} -$$$$14\!\cdots\!31$$$$T^{37} +$$$$63\!\cdots\!32$$$$T^{38} -$$$$27\!\cdots\!18$$$$T^{39} +$$$$12\!\cdots\!56$$$$T^{40} -$$$$52\!\cdots\!29$$$$T^{41} +$$$$22\!\cdots\!19$$$$T^{42} -$$$$94\!\cdots\!19$$$$T^{43} +$$$$40\!\cdots\!11$$$$T^{44} -$$$$16\!\cdots\!74$$$$T^{45} +$$$$70\!\cdots\!95$$$$T^{46} -$$$$28\!\cdots\!12$$$$T^{47} +$$$$70\!\cdots\!95$$$$p T^{48} -$$$$16\!\cdots\!74$$$$p^{2} T^{49} +$$$$40\!\cdots\!11$$$$p^{3} T^{50} -$$$$94\!\cdots\!19$$$$p^{4} T^{51} +$$$$22\!\cdots\!19$$$$p^{5} T^{52} -$$$$52\!\cdots\!29$$$$p^{6} T^{53} +$$$$12\!\cdots\!56$$$$p^{7} T^{54} -$$$$27\!\cdots\!18$$$$p^{8} T^{55} +$$$$63\!\cdots\!32$$$$p^{9} T^{56} -$$$$14\!\cdots\!31$$$$p^{10} T^{57} +$$$$32\!\cdots\!43$$$$p^{11} T^{58} -$$$$70\!\cdots\!13$$$$p^{12} T^{59} +$$$$15\!\cdots\!37$$$$p^{13} T^{60} -$$$$33\!\cdots\!99$$$$p^{14} T^{61} +$$$$72\!\cdots\!51$$$$p^{15} T^{62} -$$$$15\!\cdots\!20$$$$p^{16} T^{63} +$$$$32\!\cdots\!40$$$$p^{17} T^{64} -$$$$66\!\cdots\!79$$$$p^{18} T^{65} +$$$$13\!\cdots\!63$$$$p^{19} T^{66} -$$$$27\!\cdots\!24$$$$p^{20} T^{67} +$$$$54\!\cdots\!98$$$$p^{21} T^{68} -$$$$10\!\cdots\!85$$$$p^{22} T^{69} +$$$$20\!\cdots\!75$$$$p^{23} T^{70} -$$$$37\!\cdots\!77$$$$p^{24} T^{71} +$$$$70\!\cdots\!94$$$$p^{25} T^{72} -$$$$12\!\cdots\!58$$$$p^{26} T^{73} + 22420978605144955974 p^{27} T^{74} - 3800059958202029722 p^{28} T^{75} + 647268903507099499 p^{29} T^{76} - 103407256517358151 p^{30} T^{77} + 16680494721096703 p^{31} T^{78} - 2485914969278550 p^{32} T^{79} + 376746185206746 p^{33} T^{80} - 51715732120317 p^{34} T^{81} + 7295613161674 p^{35} T^{82} - 907715387163 p^{36} T^{83} + 117895271019 p^{37} T^{84} - 765377345 p^{39} T^{85} + 1535205530 p^{39} T^{86} - 145641717 p^{40} T^{87} + 15347809 p^{41} T^{88} - 1189164 p^{42} T^{89} + 109411 p^{43} T^{90} - 6275 p^{44} T^{91} + 488 p^{45} T^{92} - 16 p^{46} T^{93} + p^{47} T^{94}$$
23 $$1 - 24 T + 732 T^{2} - 12902 T^{3} + 240337 T^{4} - 3446566 T^{5} + 49682405 T^{6} - 611880923 T^{7} + 7444671582 T^{8} - 81372813086 T^{9} + 873134975782 T^{10} - 8656626371648 T^{11} + 84083383552670 T^{12} - 767843538759056 T^{13} + 6868187211450109 T^{14} - 58425030922367612 T^{15} + 487092399581479726 T^{16} - 3893045871655911948 T^{17} + 30521690748568237802 T^{18} -$$$$23\!\cdots\!61$$$$T^{19} +$$$$17\!\cdots\!15$$$$T^{20} -$$$$12\!\cdots\!27$$$$T^{21} +$$$$87\!\cdots\!31$$$$T^{22} -$$$$59\!\cdots\!17$$$$T^{23} +$$$$40\!\cdots\!18$$$$T^{24} -$$$$26\!\cdots\!86$$$$T^{25} +$$$$17\!\cdots\!02$$$$T^{26} -$$$$10\!\cdots\!68$$$$T^{27} +$$$$68\!\cdots\!79$$$$T^{28} -$$$$41\!\cdots\!41$$$$T^{29} +$$$$25\!\cdots\!71$$$$T^{30} -$$$$14\!\cdots\!83$$$$T^{31} +$$$$86\!\cdots\!67$$$$T^{32} -$$$$49\!\cdots\!92$$$$T^{33} +$$$$27\!\cdots\!53$$$$T^{34} -$$$$15\!\cdots\!31$$$$T^{35} +$$$$85\!\cdots\!48$$$$T^{36} -$$$$45\!\cdots\!01$$$$T^{37} +$$$$24\!\cdots\!18$$$$T^{38} -$$$$12\!\cdots\!33$$$$T^{39} +$$$$67\!\cdots\!14$$$$T^{40} -$$$$34\!\cdots\!30$$$$T^{41} +$$$$17\!\cdots\!33$$$$T^{42} -$$$$88\!\cdots\!52$$$$T^{43} +$$$$44\!\cdots\!96$$$$T^{44} -$$$$21\!\cdots\!98$$$$T^{45} +$$$$10\!\cdots\!78$$$$T^{46} -$$$$51\!\cdots\!54$$$$T^{47} +$$$$10\!\cdots\!78$$$$p T^{48} -$$$$21\!\cdots\!98$$$$p^{2} T^{49} +$$$$44\!\cdots\!96$$$$p^{3} T^{50} -$$$$88\!\cdots\!52$$$$p^{4} T^{51} +$$$$17\!\cdots\!33$$$$p^{5} T^{52} -$$$$34\!\cdots\!30$$$$p^{6} T^{53} +$$$$67\!\cdots\!14$$$$p^{7} T^{54} -$$$$12\!\cdots\!33$$$$p^{8} T^{55} +$$$$24\!\cdots\!18$$$$p^{9} T^{56} -$$$$45\!\cdots\!01$$$$p^{10} T^{57} +$$$$85\!\cdots\!48$$$$p^{11} T^{58} -$$$$15\!\cdots\!31$$$$p^{12} T^{59} +$$$$27\!\cdots\!53$$$$p^{13} T^{60} -$$$$49\!\cdots\!92$$$$p^{14} T^{61} +$$$$86\!\cdots\!67$$$$p^{15} T^{62} -$$$$14\!\cdots\!83$$$$p^{16} T^{63} +$$$$25\!\cdots\!71$$$$p^{17} T^{64} -$$$$41\!\cdots\!41$$$$p^{18} T^{65} +$$$$68\!\cdots\!79$$$$p^{19} T^{66} -$$$$10\!\cdots\!68$$$$p^{20} T^{67} +$$$$17\!\cdots\!02$$$$p^{21} T^{68} -$$$$26\!\cdots\!86$$$$p^{22} T^{69} +$$$$40\!\cdots\!18$$$$p^{23} T^{70} -$$$$59\!\cdots\!17$$$$p^{24} T^{71} +$$$$87\!\cdots\!31$$$$p^{25} T^{72} -$$$$12\!\cdots\!27$$$$p^{26} T^{73} +$$$$17\!\cdots\!15$$$$p^{27} T^{74} -$$$$23\!\cdots\!61$$$$p^{28} T^{75} + 30521690748568237802 p^{29} T^{76} - 3893045871655911948 p^{30} T^{77} + 487092399581479726 p^{31} T^{78} - 58425030922367612 p^{32} T^{79} + 6868187211450109 p^{33} T^{80} - 767843538759056 p^{34} T^{81} + 84083383552670 p^{35} T^{82} - 8656626371648 p^{36} T^{83} + 873134975782 p^{37} T^{84} - 81372813086 p^{38} T^{85} + 7444671582 p^{39} T^{86} - 611880923 p^{40} T^{87} + 49682405 p^{41} T^{88} - 3446566 p^{42} T^{89} + 240337 p^{43} T^{90} - 12902 p^{44} T^{91} + 732 p^{45} T^{92} - 24 p^{46} T^{93} + p^{47} T^{94}$$
29 $$1 - 2 p T + 2324 T^{2} - 69493 T^{3} + 1743049 T^{4} - 37733691 T^{5} + 729732846 T^{6} - 441303474 p T^{7} + 206733886873 T^{8} - 3104140780861 T^{9} + 43704344891932 T^{10} - 580458459397499 T^{11} + 7313175067979045 T^{12} - 87773854136053280 T^{13} + 1007464055331142890 T^{14} - 11093088919519484017 T^{15} +$$$$11\!\cdots\!14$$$$T^{16} -$$$$12\!\cdots\!62$$$$T^{17} +$$$$11\!\cdots\!22$$$$T^{18} -$$$$11\!\cdots\!78$$$$T^{19} +$$$$10\!\cdots\!37$$$$T^{20} -$$$$94\!\cdots\!07$$$$T^{21} +$$$$82\!\cdots\!71$$$$T^{22} -$$$$70\!\cdots\!79$$$$T^{23} +$$$$58\!\cdots\!96$$$$T^{24} -$$$$47\!\cdots\!90$$$$T^{25} +$$$$37\!\cdots\!21$$$$T^{26} -$$$$29\!\cdots\!12$$$$T^{27} +$$$$22\!\cdots\!59$$$$T^{28} -$$$$16\!\cdots\!32$$$$T^{29} +$$$$12\!\cdots\!52$$$$T^{30} -$$$$85\!\cdots\!11$$$$T^{31} +$$$$59\!\cdots\!29$$$$T^{32} -$$$$41\!\cdots\!98$$$$T^{33} +$$$$27\!\cdots\!88$$$$T^{34} -$$$$18\!\cdots\!77$$$$T^{35} +$$$$11\!\cdots\!31$$$$T^{36} -$$$$76\!\cdots\!57$$$$T^{37} +$$$$47\!\cdots\!24$$$$T^{38} -$$$$29\!\cdots\!83$$$$T^{39} +$$$$17\!\cdots\!74$$$$T^{40} -$$$$10\!\cdots\!22$$$$T^{41} +$$$$62\!\cdots\!94$$$$T^{42} -$$$$36\!\cdots\!57$$$$T^{43} +$$$$20\!\cdots\!50$$$$T^{44} -$$$$11\!\cdots\!02$$$$T^{45} +$$$$63\!\cdots\!18$$$$T^{46} -$$$$34\!\cdots\!48$$$$T^{47} +$$$$63\!\cdots\!18$$$$p T^{48} -$$$$11\!\cdots\!02$$$$p^{2} T^{49} +$$$$20\!\cdots\!50$$$$p^{3} T^{50} -$$$$36\!\cdots\!57$$$$p^{4} T^{51} +$$$$62\!\cdots\!94$$$$p^{5} T^{52} -$$$$10\!\cdots\!22$$$$p^{6} T^{53} +$$$$17\!\cdots\!74$$$$p^{7} T^{54} -$$$$29\!\cdots\!83$$$$p^{8} T^{55} +$$$$47\!\cdots\!24$$$$p^{9} T^{56} -$$$$76\!\cdots\!57$$$$p^{10} T^{57} +$$$$11\!\cdots\!31$$$$p^{11} T^{58} -$$$$18\!\cdots\!77$$$$p^{12} T^{59} +$$$$27\!\cdots\!88$$$$p^{13} T^{60} -$$$$41\!\cdots\!98$$$$p^{14} T^{61} +$$$$59\!\cdots\!29$$$$p^{15} T^{62} -$$$$85\!\cdots\!11$$$$p^{16} T^{63} +$$$$12\!\cdots\!52$$$$p^{17} T^{64} -$$$$16\!\cdots\!32$$$$p^{18} T^{65} +$$$$22\!\cdots\!59$$$$p^{19} T^{66} -$$$$29\!\cdots\!12$$$$p^{20} T^{67} +$$$$37\!\cdots\!21$$$$p^{21} T^{68} -$$$$47\!\cdots\!90$$$$p^{22} T^{69} +$$$$58\!\cdots\!96$$$$p^{23} T^{70} -$$$$70\!\cdots\!79$$$$p^{24} T^{71} +$$$$82\!\cdots\!71$$$$p^{25} T^{72} -$$$$94\!\cdots\!07$$$$p^{26} T^{73} +$$$$10\!\cdots\!37$$$$p^{27} T^{74} -$$$$11\!\cdots\!78$$$$p^{28} T^{75} +$$$$11\!\cdots\!22$$$$p^{29} T^{76} -$$$$12\!\cdots\!62$$$$p^{30} T^{77} +$$$$11\!\cdots\!14$$$$p^{31} T^{78} - 11093088919519484017 p^{32} T^{79} + 1007464055331142890 p^{33} T^{80} - 87773854136053280 p^{34} T^{81} + 7313175067979045 p^{35} T^{82} - 580458459397499 p^{36} T^{83} + 43704344891932 p^{37} T^{84} - 3104140780861 p^{38} T^{85} + 206733886873 p^{39} T^{86} - 441303474 p^{41} T^{87} + 729732846 p^{41} T^{88} - 37733691 p^{42} T^{89} + 1743049 p^{43} T^{90} - 69493 p^{44} T^{91} + 2324 p^{45} T^{92} - 2 p^{47} T^{93} + p^{47} T^{94}$$
31 $$1 - 21 T + 815 T^{2} - 13904 T^{3} + 315082 T^{4} - 4601503 T^{5} + 78692044 T^{6} - 1016342570 T^{7} + 14448229029 T^{8} - 168697283311 T^{9} + 2094505874430 T^{10} - 22459363814965 T^{11} + 250826027744847 T^{12} - 2499357847314207 T^{13} + 25599325961627260 T^{14} - 239217524997219284 T^{15} + 2277997205871094958 T^{16} - 20109287915879189503 T^{17} +$$$$17\!\cdots\!01$$$$T^{18} -$$$$15\!\cdots\!10$$$$T^{19} +$$$$12\!\cdots\!52$$$$T^{20} -$$$$10\!\cdots\!63$$$$T^{21} +$$$$82\!\cdots\!30$$$$T^{22} -$$$$63\!\cdots\!68$$$$T^{23} +$$$$48\!\cdots\!53$$$$T^{24} -$$$$36\!\cdots\!62$$$$T^{25} +$$$$26\!\cdots\!98$$$$T^{26} -$$$$19\!\cdots\!99$$$$T^{27} +$$$$13\!\cdots\!40$$$$T^{28} -$$$$95\!\cdots\!19$$$$T^{29} +$$$$66\!\cdots\!74$$$$T^{30} -$$$$44\!\cdots\!44$$$$T^{31} +$$$$29\!\cdots\!71$$$$T^{32} -$$$$19\!\cdots\!40$$$$T^{33} +$$$$12\!\cdots\!11$$$$T^{34} -$$$$81\!\cdots\!49$$$$T^{35} +$$$$51\!\cdots\!24$$$$T^{36} -$$$$32\!\cdots\!53$$$$T^{37} +$$$$20\!\cdots\!37$$$$T^{38} -$$$$39\!\cdots\!94$$$$p T^{39} +$$$$74\!\cdots\!33$$$$T^{40} -$$$$14\!\cdots\!76$$$$p T^{41} +$$$$26\!\cdots\!86$$$$T^{42} -$$$$15\!\cdots\!69$$$$T^{43} +$$$$88\!\cdots\!48$$$$T^{44} -$$$$50\!\cdots\!26$$$$T^{45} +$$$$28\!\cdots\!16$$$$T^{46} -$$$$15\!\cdots\!68$$$$T^{47} +$$$$28\!\cdots\!16$$$$p T^{48} -$$$$50\!\cdots\!26$$$$p^{2} T^{49} +$$$$88\!\cdots\!48$$$$p^{3} T^{50} -$$$$15\!\cdots\!69$$$$p^{4} T^{51} +$$$$26\!\cdots\!86$$$$p^{5} T^{52} -$$$$14\!\cdots\!76$$$$p^{7} T^{53} +$$$$74\!\cdots\!33$$$$p^{7} T^{54} -$$$$39\!\cdots\!94$$$$p^{9} T^{55} +$$$$20\!\cdots\!37$$$$p^{9} T^{56} -$$$$32\!\cdots\!53$$$$p^{10} T^{57} +$$$$51\!\cdots\!24$$$$p^{11} T^{58} -$$$$81\!\cdots\!49$$$$p^{12} T^{59} +$$$$12\!\cdots\!11$$$$p^{13} T^{60} -$$$$19\!\cdots\!40$$$$p^{14} T^{61} +$$$$29\!\cdots\!71$$$$p^{15} T^{62} -$$$$44\!\cdots\!44$$$$p^{16} T^{63} +$$$$66\!\cdots\!74$$$$p^{17} T^{64} -$$$$95\!\cdots\!19$$$$p^{18} T^{65} +$$$$13\!\cdots\!40$$$$p^{19} T^{66} -$$$$19\!\cdots\!99$$$$p^{20} T^{67} +$$$$26\!\cdots\!98$$$$p^{21} T^{68} -$$$$36\!\cdots\!62$$$$p^{22} T^{69} +$$$$48\!\cdots\!53$$$$p^{23} T^{70} -$$$$63\!\cdots\!68$$$$p^{24} T^{71} +$$$$82\!\cdots\!30$$$$p^{25} T^{72} -$$$$10\!\cdots\!63$$$$p^{26} T^{73} +$$$$12\!\cdots\!52$$$$p^{27} T^{74} -$$$$15\!\cdots\!10$$$$p^{28} T^{75} +$$$$17\!\cdots\!01$$$$p^{29} T^{76} - 20109287915879189503 p^{30} T^{77} + 2277997205871094958 p^{31} T^{78} - 239217524997219284 p^{32} T^{79} + 25599325961627260 p^{33} T^{80} - 2499357847314207 p^{34} T^{81} + 250826027744847 p^{35} T^{82} - 22459363814965 p^{36} T^{83} + 2094505874430 p^{37} T^{84} - 168697283311 p^{38} T^{85} + 14448229029 p^{39} T^{86} - 1016342570 p^{40} T^{87} + 78692044 p^{41} T^{88} - 4601503 p^{42} T^{89} + 315082 p^{43} T^{90} - 13904 p^{44} T^{91} + 815 p^{45} T^{92} - 21 p^{46} T^{93} + p^{47} T^{94}$$
37 $$1 - 78 T + 3868 T^{2} - 142456 T^{3} + 4314010 T^{4} - 111954488 T^{5} + 2569978891 T^{6} - 53177492854 T^{7} + 1006952234633 T^{8} - 17636094052271 T^{9} + 288234984496769 T^{10} - 4425921145370694 T^{11} + 64223054931355370 T^{12} - 884832141126322722 T^{13} + 11622326062822702004 T^{14} -$$$$14\!\cdots\!42$$$$T^{15} +$$$$17\!\cdots\!23$$$$T^{16} -$$$$20\!\cdots\!87$$$$T^{17} +$$$$22\!\cdots\!24$$$$T^{18} -$$$$24\!\cdots\!85$$$$T^{19} +$$$$25\!\cdots\!45$$$$T^{20} -$$$$26\!\cdots\!56$$$$T^{21} +$$$$25\!\cdots\!81$$$$T^{22} -$$$$24\!\cdots\!26$$$$T^{23} +$$$$23\!\cdots\!87$$$$T^{24} -$$$$21\!\cdots\!05$$$$T^{25} +$$$$18\!\cdots\!03$$$$T^{26} -$$$$16\!\cdots\!07$$$$T^{27} +$$$$13\!\cdots\!84$$$$T^{28} -$$$$11\!\cdots\!81$$$$T^{29} +$$$$93\!\cdots\!76$$$$T^{30} -$$$$74\!\cdots\!99$$$$T^{31} +$$$$58\!\cdots\!00$$$$T^{32} -$$$$44\!\cdots\!13$$$$T^{33} +$$$$33\!\cdots\!13$$$$T^{34} -$$$$25\!\cdots\!24$$$$T^{35} +$$$$18\!\cdots\!92$$$$T^{36} -$$$$13\!\cdots\!11$$$$T^{37} +$$$$91\!\cdots\!16$$$$T^{38} -$$$$63\!\cdots\!35$$$$T^{39} +$$$$43\!\cdots\!58$$$$T^{40} -$$$$28\!\cdots\!71$$$$T^{41} +$$$$19\!\cdots\!05$$$$T^{42} -$$$$12\!\cdots\!58$$$$T^{43} +$$$$21\!\cdots\!81$$$$p T^{44} -$$$$50\!\cdots\!49$$$$T^{45} +$$$$31\!\cdots\!54$$$$T^{46} -$$$$19\!\cdots\!20$$$$T^{47} +$$$$31\!\cdots\!54$$$$p T^{48} -$$$$50\!\cdots\!49$$$$p^{2} T^{49} +$$$$21\!\cdots\!81$$$$p^{4} T^{50} -$$$$12\!\cdots\!58$$$$p^{4} T^{51} +$$$$19\!\cdots\!05$$$$p^{5} T^{52} -$$$$28\!\cdots\!71$$$$p^{6} T^{53} +$$$$43\!\cdots\!58$$$$p^{7} T^{54} -$$$$63\!\cdots\!35$$$$p^{8} T^{55} +$$$$91\!\cdots\!16$$$$p^{9} T^{56} -$$$$13\!\cdots\!11$$$$p^{10} T^{57} +$$$$18\!\cdots\!92$$$$p^{11} T^{58} -$$$$25\!\cdots\!24$$$$p^{12} T^{59} +$$$$33\!\cdots\!13$$$$p^{13} T^{60} -$$$$44\!\cdots\!13$$$$p^{14} T^{61} +$$$$58\!\cdots\!00$$$$p^{15} T^{62} -$$$$74\!\cdots\!99$$$$p^{16} T^{63} +$$$$93\!\cdots\!76$$$$p^{17} T^{64} -$$$$11\!\cdots\!81$$$$p^{18} T^{65} +$$$$13\!\cdots\!84$$$$p^{19} T^{66} -$$$$16\!\cdots\!07$$$$p^{20} T^{67} +$$$$18\!\cdots\!03$$$$p^{21} T^{68} -$$$$21\!\cdots\!05$$$$p^{22} T^{69} +$$$$23\!\cdots\!87$$$$p^{23} T^{70} -$$$$24\!\cdots\!26$$$$p^{24} T^{71} +$$$$25\!\cdots\!81$$$$p^{25} T^{72} -$$$$26\!\cdots\!56$$$$p^{26} T^{73} +$$$$25\!\cdots\!45$$$$p^{27} T^{74} -$$$$24\!\cdots\!85$$$$p^{28} T^{75} +$$$$22\!\cdots\!24$$$$p^{29} T^{76} -$$$$20\!\cdots\!87$$$$p^{30} T^{77} +$$$$17\!\cdots\!23$$$$p^{31} T^{78} -$$$$14\!\cdots\!42$$$$p^{32} T^{79} + 11622326062822702004 p^{33} T^{80} - 884832141126322722 p^{34} T^{81} + 64223054931355370 p^{35} T^{82} - 4425921145370694 p^{36} T^{83} + 288234984496769 p^{37} T^{84} - 17636094052271 p^{38} T^{85} + 1006952234633 p^{39} T^{86} - 53177492854 p^{40} T^{87} + 2569978891 p^{41} T^{88} - 111954488 p^{42} T^{89} + 4314010 p^{43} T^{90} - 142456 p^{44} T^{91} + 3868 p^{45} T^{92} - 78 p^{46} T^{93} + p^{47} T^{94}$$
41 $$1 - 53 T + 2228 T^{2} - 67118 T^{3} + 1758659 T^{4} - 39190825 T^{5} + 792292103 T^{6} - 14432142858 T^{7} + 244263474476 T^{8} - 3832339094757 T^{9} + 56729779266627 T^{10} - 791689374172623 T^{11} + 10535417349426538 T^{12} - 133656199578865193 T^{13} + 1629435063248195324 T^{14} - 19087735131226937697 T^{15} +$$$$21\!\cdots\!35$$$$T^{16} -$$$$23\!\cdots\!74$$$$T^{17} +$$$$25\!\cdots\!64$$$$T^{18} -$$$$25\!\cdots\!36$$$$T^{19} +$$$$26\!\cdots\!73$$$$T^{20} -$$$$25\!\cdots\!17$$$$T^{21} +$$$$24\!\cdots\!08$$$$T^{22} -$$$$22\!\cdots\!26$$$$T^{23} +$$$$20\!\cdots\!77$$$$T^{24} -$$$$18\!\cdots\!84$$$$T^{25} +$$$$16\!\cdots\!00$$$$T^{26} -$$$$14\!\cdots\!39$$$$T^{27} +$$$$11\!\cdots\!53$$$$T^{28} -$$$$99\!\cdots\!36$$$$T^{29} +$$$$81\!\cdots\!75$$$$T^{30} -$$$$65\!\cdots\!25$$$$T^{31} +$$$$51\!\cdots\!12$$$$T^{32} -$$$$40\!\cdots\!05$$$$T^{33} +$$$$31\!\cdots\!02$$$$T^{34} -$$$$23\!\cdots\!98$$$$T^{35} +$$$$17\!\cdots\!56$$$$T^{36} -$$$$12\!\cdots\!61$$$$T^{37} +$$$$93\!\cdots\!74$$$$T^{38} -$$$$66\!\cdots\!05$$$$T^{39} +$$$$47\!\cdots\!49$$$$T^{40} -$$$$32\!\cdots\!69$$$$T^{41} +$$$$22\!\cdots\!60$$$$T^{42} -$$$$15\!\cdots\!64$$$$T^{43} +$$$$10\!\cdots\!37$$$$T^{44} -$$$$16\!\cdots\!56$$$$p T^{45} +$$$$44\!\cdots\!37$$$$T^{46} -$$$$28\!\cdots\!14$$$$T^{47} +$$$$44\!\cdots\!37$$$$p T^{48} -$$$$16\!\cdots\!56$$$$p^{3} T^{49} +$$$$10\!\cdots\!37$$$$p^{3} T^{50} -$$$$15\!\cdots\!64$$$$p^{4} T^{51} +$$$$22\!\cdots\!60$$$$p^{5} T^{52} -$$$$32\!\cdots\!69$$$$p^{6} T^{53} +$$$$47\!\cdots\!49$$$$p^{7} T^{54} -$$$$66\!\cdots\!05$$$$p^{8} T^{55} +$$$$93\!\cdots\!74$$$$p^{9} T^{56} -$$$$12\!\cdots\!61$$$$p^{10} T^{57} +$$$$17\!\cdots\!56$$$$p^{11} T^{58} -$$$$23\!\cdots\!98$$$$p^{12} T^{59} +$$$$31\!\cdots\!02$$$$p^{13} T^{60} -$$$$40\!\cdots\!05$$$$p^{14} T^{61} +$$$$51\!\cdots\!12$$$$p^{15} T^{62} -$$$$65\!\cdots\!25$$$$p^{16} T^{63} +$$$$81\!\cdots\!75$$$$p^{17} T^{64} -$$$$99\!\cdots\!36$$$$p^{18} T^{65} +$$$$11\!\cdots\!53$$$$p^{19} T^{66} -$$$$14\!\cdots\!39$$$$p^{20} T^{67} +$$$$16\!\cdots\!00$$$$p^{21} T^{68} -$$$$18\!\cdots\!84$$$$p^{22} T^{69} +$$$$20\!\cdots\!77$$$$p^{23} T^{70} -$$$$22\!\cdots\!26$$$$p^{24} T^{71} +$$$$24\!\cdots\!08$$$$p^{25} T^{72} -$$$$25\!\cdots\!17$$$$p^{26} T^{73} +$$$$26\!\cdots\!73$$$$p^{27} T^{74} -$$$$25\!\cdots\!36$$$$p^{28} T^{75} +$$$$25\!\cdots\!64$$$$p^{29} T^{76} -$$$$23\!\cdots\!74$$$$p^{30} T^{77} +$$$$21\!\cdots\!35$$$$p^{31} T^{78} - 19087735131226937697 p^{32} T^{79} + 1629435063248195324 p^{33} T^{80} - 133656199578865193 p^{34} T^{81} + 10535417349426538 p^{35} T^{82} - 791689374172623 p^{36} T^{83} + 56729779266627 p^{37} T^{84} - 3832339094757 p^{38} T^{85} + 244263474476 p^{39} T^{86} - 14432142858 p^{40} T^{87} + 792292103 p^{41} T^{88} - 39190825 p^{42} T^{89} + 1758659 p^{43} T^{90} - 67118 p^{44} T^{91} + 2228 p^{45} T^{92} - 53 p^{46} T^{93} + p^{47} T^{94}$$
43 $$1 - 47 T + 2006 T^{2} - 59054 T^{3} + 1582341 T^{4} - 35654947 T^{5} + 744684807 T^{6} - 13970498557 T^{7} + 246446977625 T^{8} - 4028313021006 T^{9} + 62570084713255 T^{10} - 916536153528917 T^{11} + 12857626271625241 T^{12} - 172016521296308676 T^{13} + 2217028071386876538 T^{14} - 27460081396766360152 T^{15} +$$$$32\!\cdots\!62$$$$T^{16} -$$$$38\!\cdots\!44$$$$T^{17} +$$$$42\!\cdots\!63$$$$T^{18} -$$$$46\!\cdots\!09$$$$T^{19} +$$$$49\!\cdots\!33$$$$T^{20} -$$$$51\!\cdots\!61$$$$T^{21} +$$$$52\!\cdots\!90$$$$T^{22} -$$$$51\!\cdots\!00$$$$T^{23} +$$$$49\!\cdots\!67$$$$T^{24} -$$$$46\!\cdots\!59$$$$T^{25} +$$$$43\!\cdots\!33$$$$T^{26} -$$$$39\!\cdots\!22$$$$T^{27} +$$$$34\!\cdots\!29$$$$T^{28} -$$$$30\!\cdots\!95$$$$T^{29} +$$$$26\!\cdots\!66$$$$T^{30} -$$$$22\!\cdots\!77$$$$T^{31} +$$$$18\!\cdots\!40$$$$T^{32} -$$$$14\!\cdots\!82$$$$T^{33} +$$$$11\!\cdots\!15$$$$T^{34} -$$$$93\!\cdots\!13$$$$T^{35} +$$$$72\!\cdots\!04$$$$T^{36} -$$$$55\!\cdots\!31$$$$T^{37} +$$$$41\!\cdots\!66$$$$T^{38} -$$$$30\!\cdots\!63$$$$T^{39} +$$$$22\!\cdots\!67$$$$T^{40} -$$$$16\!\cdots\!57$$$$T^{41} +$$$$11\!\cdots\!04$$$$T^{42} -$$$$80\!\cdots\!97$$$$T^{43} +$$$$12\!\cdots\!32$$$$p T^{44} -$$$$37\!\cdots\!47$$$$T^{45} +$$$$25\!\cdots\!39$$$$T^{46} -$$$$16\!\cdots\!46$$$$T^{47} +$$$$25\!\cdots\!39$$$$p T^{48} -$$$$37\!\cdots\!47$$$$p^{2} T^{49} +$$$$12\!\cdots\!32$$$$p^{4} T^{50} -$$$$80\!\cdots\!97$$$$p^{4} T^{51} +$$$$11\!\cdots\!04$$$$p^{5} T^{52} -$$$$16\!\cdots\!57$$$$p^{6} T^{53} +$$$$22\!\cdots\!67$$$$p^{7} T^{54} -$$$$30\!\cdots\!63$$$$p^{8} T^{55} +$$$$41\!\cdots\!66$$$$p^{9} T^{56} -$$$$55\!\cdots\!31$$$$p^{10} T^{57} +$$$$72\!\cdots\!04$$$$p^{11} T^{58} -$$$$93\!\cdots\!13$$$$p^{12} T^{59} +$$$$11\!\cdots\!15$$$$p^{13} T^{60} -$$$$14\!\cdots\!82$$$$p^{14} T^{61} +$$$$18\!\cdots\!40$$$$p^{15} T^{62} -$$$$22\!\cdots\!77$$$$p^{16} T^{63} +$$$$26\!\cdots\!66$$$$p^{17} T^{64} -$$$$30\!\cdots\!95$$$$p^{18} T^{65} +$$$$34\!\cdots\!29$$$$p^{19} T^{66} -$$$$39\!\cdots\!22$$$$p^{20} T^{67} +$$$$43\!\cdots\!33$$$$p^{21} T^{68} -$$$$46\!\cdots\!59$$$$p^{22} T^{69} +$$$$49\!\cdots\!67$$$$p^{23} T^{70} -$$$$51\!\cdots\!00$$$$p^{24} T^{71} +$$$$52\!\cdots\!90$$$$p^{25} T^{72} -$$$$51\!\cdots\!61$$$$p^{26} T^{73} +$$$$49\!\cdots\!33$$$$p^{27} T^{74} -$$$$46\!\cdots\!09$$$$p^{28} T^{75} +$$$$42\!\cdots\!63$$$$p^{29} T^{76} -$$$$38\!\cdots\!44$$$$p^{30} T^{77} +$$$$32\!\cdots\!62$$$$p^{31} T^{78} - 27460081396766360152 p^{32} T^{79} + 2217028071386876538 p^{33} T^{80} - 172016521296308676 p^{34} T^{81} + 12857626271625241 p^{35} T^{82} - 916536153528917 p^{36} T^{83} + 62570084713255 p^{37} T^{84} - 4028313021006 p^{38} T^{85} + 246446977625 p^{39} T^{86} - 13970498557 p^{40} T^{87} + 744684807 p^{41} T^{88} - 35654947 p^{42} T^{89} + 1582341 p^{43} T^{90} - 59054 p^{44} T^{91} + 2006 p^{45} T^{92} - 47 p^{46} T^{93} + p^{47} T^{94}$$
47 $$1 - 16 T + 1172 T^{2} - 17768 T^{3} + 689571 T^{4} - 9902045 T^{5} + 271126722 T^{6} - 3690726550 T^{7} + 80047491890 T^{8} - 1034600628101 T^{9} + 18913823381959 T^{10} - 232584441367247 T^{11} + 3723559710436636 T^{12} - 43664675392696914 T^{13} + 628010217811322738 T^{14} - 7039485650220903908 T^{15} + 92612033891442351698 T^{16} -$$$$99\!\cdots\!31$$$$T^{17} +$$$$12\!\cdots\!29$$$$T^{18} -$$$$12\!\cdots\!29$$$$T^{19} +$$$$14\!\cdots\!80$$$$T^{20} -$$$$14\!\cdots\!11$$$$T^{21} +$$$$15\!\cdots\!11$$$$T^{22} -$$$$14\!\cdots\!46$$$$T^{23} +$$$$14\!\cdots\!30$$$$T^{24} -$$$$13\!\cdots\!49$$$$T^{25} +$$$$13\!\cdots\!34$$$$T^{26} -$$$$12\!\cdots\!11$$$$T^{27} +$$$$11\!\cdots\!45$$$$T^{28} -$$$$98\!\cdots\!34$$$$T^{29} +$$$$88\!\cdots\!90$$$$T^{30} -$$$$74\!\cdots\!84$$$$T^{31} +$$$$64\!\cdots\!87$$$$T^{32} -$$$$52\!\cdots\!48$$$$T^{33} +$$$$44\!\cdots\!65$$$$T^{34} -$$$$35\!\cdots\!39$$$$T^{35} +$$$$28\!\cdots\!91$$$$T^{36} -$$$$22\!\cdots\!62$$$$T^{37} +$$$$17\!\cdots\!26$$$$T^{38} -$$$$13\!\cdots\!36$$$$T^{39} +$$$$10\!\cdots\!47$$$$T^{40} -$$$$15\!\cdots\!64$$$$p T^{41} +$$$$55\!\cdots\!37$$$$T^{42} -$$$$40\!\cdots\!50$$$$T^{43} +$$$$29\!\cdots\!09$$$$T^{44} -$$$$20\!\cdots\!01$$$$T^{45} +$$$$14\!\cdots\!02$$$$T^{46} -$$$$98\!\cdots\!12$$$$T^{47} +$$$$14\!\cdots\!02$$$$p T^{48} -$$$$20\!\cdots\!01$$$$p^{2} T^{49} +$$$$29\!\cdots\!09$$$$p^{3} T^{50} -$$$$40\!\cdots\!50$$$$p^{4} T^{51} +$$$$55\!\cdots\!37$$$$p^{5} T^{52} -$$$$15\!\cdots\!64$$$$p^{7} T^{53} +$$$$10\!\cdots\!47$$$$p^{7} T^{54} -$$$$13\!\cdots\!36$$$$p^{8} T^{55} +$$$$17\!\cdots\!26$$$$p^{9} T^{56} -$$$$22\!\cdots\!62$$$$p^{10} T^{57} +$$$$28\!\cdots\!91$$$$p^{11} T^{58} -$$$$35\!\cdots\!39$$$$p^{12} T^{59} +$$$$44\!\cdots\!65$$$$p^{13} T^{60} -$$$$52\!\cdots\!48$$$$p^{14} T^{61} +$$$$64\!\cdots\!87$$$$p^{15} T^{62} -$$$$74\!\cdots\!84$$$$p^{16} T^{63} +$$$$88\!\cdots\!90$$$$p^{17} T^{64} -$$$$98\!\cdots\!34$$$$p^{18} T^{65} +$$$$11\!\cdots\!45$$$$p^{19} T^{66} -$$$$12\!\cdots\!11$$$$p^{20} T^{67} +$$$$13\!\cdots\!34$$$$p^{21} T^{68} -$$$$13\!\cdots\!49$$$$p^{22} T^{69} +$$$$14\!\cdots\!30$$$$p^{23} T^{70} -$$$$14\!\cdots\!46$$$$p^{24} T^{71} +$$$$15\!\cdots\!11$$$$p^{25} T^{72} -$$$$14\!\cdots\!11$$$$p^{26} T^{73} +$$$$14\!\cdots\!80$$$$p^{27} T^{74} -$$$$12\!\cdots\!29$$$$p^{28} T^{75} +$$$$12\!\cdots\!29$$$$p^{29} T^{76} -$$$$99\!\cdots\!31$$$$p^{30} T^{77} + 92612033891442351698 p^{31} T^{78} - 7039485650220903908 p^{32} T^{79} + 628010217811322738 p^{33} T^{80} - 43664675392696914 p^{34} T^{81} + 3723559710436636 p^{35} T^{82} - 232584441367247 p^{36} T^{83} + 18913823381959 p^{37} T^{84} - 1034600628101 p^{38} T^{85} + 80047491890 p^{39} T^{86} - 3690726550 p^{40} T^{87} + 271126722 p^{41} T^{88} - 9902045 p^{42} T^{89} + 689571 p^{43} T^{90} - 17768 p^{44} T^{91} + 1172 p^{45} T^{92} - 16 p^{46} T^{93} + p^{47} T^{94}$$
53 $$1 - 43 T + 1994 T^{2} - 57508 T^{3} + 1629373 T^{4} - 36603990 T^{5} + 15018379 p T^{6} - 14951819118 T^{7} + 271740988219 T^{8} - 4447622992652 T^{9} + 70638289970403 T^{10} - 1034864063688580 T^{11} + 14764651376348770 T^{12} - 197318417301828380 T^{13} + 2576942794120638136 T^{14} - 31858966819964573351 T^{15} +$$$$38\!\cdots\!92$$$$T^{16} -$$$$44\!\cdots\!14$$$$T^{17} +$$$$50\!\cdots\!30$$$$T^{18} -$$$$55\!\cdots\!54$$$$T^{19} +$$$$59\!\cdots\!29$$$$T^{20} -$$$$61\!\cdots\!77$$$$T^{21} +$$$$62\!\cdots\!30$$$$T^{22} -$$$$61\!\cdots\!54$$$$T^{23} +$$$$60\!\cdots\!38$$$$T^{24} -$$$$57\!\cdots\!00$$$$T^{25} +$$$$53\!\cdots\!12$$$$T^{26} -$$$$48\!\cdots\!24$$$$T^{27} +$$$$43\!\cdots\!67$$$$T^{28} -$$$$38\!\cdots\!31$$$$T^{29} +$$$$33\!\cdots\!14$$$$T^{30} -$$$$28\!\cdots\!95$$$$T^{31} +$$$$24\!\cdots\!66$$$$T^{32} -$$$$19\!\cdots\!96$$$$T^{33} +$$$$16\!\cdots\!39$$$$T^{34} -$$$$13\!\cdots\!80$$$$T^{35} +$$$$10\!\cdots\!48$$$$T^{36} -$$$$82\!\cdots\!89$$$$T^{37} +$$$$64\!\cdots\!79$$$$T^{38} -$$$$49\!\cdots\!21$$$$T^{39} +$$$$37\!\cdots\!33$$$$T^{40} -$$$$28\!\cdots\!30$$$$T^{41} +$$$$21\!\cdots\!97$$$$T^{42} -$$$$15\!\cdots\!54$$$$T^{43} +$$$$11\!\cdots\!46$$$$T^{44} -$$$$87\!\cdots\!10$$$$T^{45} +$$$$64\!\cdots\!17$$$$T^{46} -$$$$46\!\cdots\!46$$$$T^{47} +$$$$64\!\cdots\!17$$$$p T^{48} -$$$$87\!\cdots\!10$$$$p^{2} T^{49} +$$$$11\!\cdots\!46$$$$p^{3} T^{50} -$$$$15\!\cdots\!54$$$$p^{4} T^{51} +$$$$21\!\cdots\!97$$$$p^{5} T^{52} -$$$$28\!\cdots\!30$$$$p^{6} T^{53} +$$$$37\!\cdots\!33$$$$p^{7} T^{54} -$$$$49\!\cdots\!21$$$$p^{8} T^{55} +$$$$64\!\cdots\!79$$$$p^{9} T^{56} -$$$$82\!\cdots\!89$$$$p^{10} T^{57} +$$$$10\!\cdots\!48$$$$p^{11} T^{58} -$$$$13\!\cdots\!80$$$$p^{12} T^{59} +$$$$16\!\cdots\!39$$$$p^{13} T^{60} -$$$$19\!\cdots\!96$$$$p^{14} T^{61} +$$$$24\!\cdots\!66$$$$p^{15} T^{62} -$$$$28\!\cdots\!95$$$$p^{16} T^{63} +$$$$33\!\cdots\!14$$$$p^{17} T^{64} -$$$$38\!\cdots\!31$$$$p^{18} T^{65} +$$$$43\!\cdots\!67$$$$p^{19} T^{66} -$$$$48\!\cdots\!24$$$$p^{20} T^{67} +$$$$53\!\cdots\!12$$$$p^{21} T^{68} -$$$$57\!\cdots\!00$$$$p^{22} T^{69} +$$$$60\!\cdots\!38$$$$p^{23} T^{70} -$$$$61\!\cdots\!54$$$$p^{24} T^{71} +$$$$62\!\cdots\!30$$$$p^{25} T^{72} -$$$$61\!\cdots\!77$$$$p^{26} T^{73} +$$$$59\!\cdots\!29$$$$p^{27} T^{74} -$$$$55\!\cdots\!54$$$$p^{28} T^{75} +$$$$50\!\cdots\!30$$$$p^{29} T^{76} -$$$$44\!\cdots\!14$$$$p^{30} T^{77} +$$$$38\!\cdots\!92$$$$p^{31} T^{78} - 31858966819964573351 p^{32} T^{79} + 2576942794120638136 p^{33} T^{80} - 197318417301828380 p^{34} T^{81} + 14764651376348770 p^{35} T^{82} - 1034864063688580 p^{36} T^{83} + 70638289970403 p^{37} T^{84} - 4447622992652 p^{38} T^{85} + 271740988219 p^{39} T^{86} - 14951819118 p^{40} T^{87} + 15018379 p^{42} T^{88} - 36603990 p^{42} T^{89} + 1629373 p^{43} T^{90} - 57508 p^{44} T^{91} + 1994 p^{45} T^{92} - 43 p^{46} T^{93} + p^{47} T^{94}$$
59 $$1 - 3 T + 1167 T^{2} - 4199 T^{3} + 688611 T^{4} - 2873281 T^{5} + 273710148 T^{6} - 1291578291 T^{7} + 82402127259 T^{8} - 431039701274 T^{9} + 20036495707946 T^{10} - 114255379707139 T^{11} + 4098654887436056 T^{12} - 25109128735795329 T^{13} + 725564243608817009 T^{14} - 4713304203304476647 T^{15} +$$$$11\!\cdots\!37$$$$T^{16} -$$$$77\!\cdots\!10$$$$T^{17} +$$$$15\!\cdots\!72$$$$T^{18} -$$$$11\!\cdots\!64$$$$T^{19} +$$$$20\!\cdots\!12$$$$T^{20} -$$$$14\!\cdots\!46$$$$T^{21} +$$$$23\!\cdots\!31$$$$T^{22} -$$$$17\!\cdots\!02$$$$T^{23} +$$$$25\!\cdots\!13$$$$T^{24} -$$$$19\!\cdots\!83$$$$T^{25} +$$$$26\!\cdots\!18$$$$T^{26} -$$$$19\!\cdots\!33$$$$T^{27} +$$$$25\!\cdots\!09$$$$T^{28} -$$$$18\!\cdots\!51$$$$T^{29} +$$$$22\!\cdots\!37$$$$T^{30} -$$$$16\!\cdots\!62$$$$T^{31} +$$$$19\!\cdots\!15$$$$T^{32} -$$$$14\!\cdots\!28$$$$T^{33} +$$$$15\!\cdots\!90$$$$T^{34} -$$$$11\!\cdots\!14$$$$T^{35} +$$$$11\!\cdots\!23$$$$T^{36} -$$$$84\!\cdots\!16$$$$T^{37} +$$$$86\!\cdots\!40$$$$T^{38} -$$$$61\!\cdots\!79$$$$T^{39} +$$$$60\!\cdots\!39$$$$T^{40} -$$$$42\!\cdots\!64$$$$T^{41} +$$$$40\!\cdots\!54$$$$T^{42} -$$$$27\!\cdots\!27$$$$T^{43} +$$$$25\!\cdots\!36$$$$T^{44} -$$$$17\!\cdots\!31$$$$T^{45} +$$$$15\!\cdots\!11$$$$T^{46} -$$$$10\!\cdots\!78$$$$T^{47} +$$$$15\!\cdots\!11$$$$p T^{48} -$$$$17\!\cdots\!31$$$$p^{2} T^{49} +$$$$25\!\cdots\!36$$$$p^{3} T^{50} -$$$$27\!\cdots\!27$$$$p^{4} T^{51} +$$$$40\!\cdots\!54$$$$p^{5} T^{52} -$$$$42\!\cdots\!64$$$$p^{6} T^{53} +$$$$60\!\cdots\!39$$$$p^{7} T^{54} -$$$$61\!\cdots\!79$$$$p^{8} T^{55} +$$$$86\!\cdots\!40$$$$p^{9} T^{56} -$$$$84\!\cdots\!16$$$$p^{10} T^{57} +$$$$11\!\cdots\!23$$$$p^{11} T^{58} -$$$$11\!\cdots\!14$$$$p^{12} T^{59} +$$$$15\!\cdots\!90$$$$p^{13} T^{60} -$$$$14\!\cdots\!28$$$$p^{14} T^{61} +$$$$19\!\cdots\!15$$$$p^{15} T^{62} -$$$$16\!\cdots\!62$$$$p^{16} T^{63} +$$$$22\!\cdots\!37$$$$p^{17} T^{64} -$$$$18\!\cdots\!51$$$$p^{18} T^{65} +$$$$25\!\cdots\!09$$$$p^{19} T^{66} -$$$$19\!\cdots\!33$$$$p^{20} T^{67} +$$$$26\!\cdots\!18$$$$p^{21} T^{68} -$$$$19\!\cdots\!83$$$$p^{22} T^{69} +$$$$25\!\cdots\!13$$$$p^{23} T^{70} -$$$$17\!\cdots\!02$$$$p^{24} T^{71} +$$$$23\!\cdots\!31$$$$p^{25} T^{72} -$$$$14\!\cdots\!46$$$$p^{26} T^{73} +$$$$20\!\cdots\!12$$$$p^{27} T^{74} -$$$$11\!\cdots\!64$$$$p^{28} T^{75} +$$$$15\!\cdots\!72$$$$p^{29} T^{76} -$$$$77\!\cdots\!10$$$$p^{30} T^{77} +$$$$11\!\cdots\!37$$$$p^{31} T^{78} - 4713304203304476647 p^{32} T^{79} + 725564243608817009 p^{33} T^{80} - 25109128735795329 p^{34} T^{81} + 4098654887436056 p^{35} T^{82} - 114255379707139 p^{36} T^{83} + 20036495707946 p^{37} T^{84} - 431039701274 p^{38} T^{85} + 82402127259 p^{39} T^{86} - 1291578291 p^{40} T^{87} + 273710148 p^{41} T^{88} - 2873281 p^{42} T^{89} + 688611 p^{43} T^{90} - 4199 p^{44} T^{91} + 1167 p^{45} T^{92} - 3 p^{46} T^{93} + p^{47} T^{94}$$
61 $$1 - 12 T + 1271 T^{2} - 13609 T^{3} + 795113 T^{4} - 7713536 T^{5} + 328433199 T^{6} - 47971832 p T^{7} + 101289918048 T^{8} - 839466407900 T^{9} + 24983707859653 T^{10} - 194967944504176 T^{11} + 5151776091626559 T^{12} - 38300977093230272 T^{13} + 916037509721161239 T^{14} - 6559328129030652888 T^{15} +$$$$14\!\cdots\!87$$$$T^{16} -$$$$10\!\cdots\!16$$$$T^{17} +$$$$20\!\cdots\!57$$$$T^{18} -$$$$13\!\cdots\!25$$$$T^{19} +$$$$25\!\cdots\!84$$$$T^{20} -$$$$17\!\cdots\!28$$$$T^{21} +$$$$30\!\cdots\!43$$$$T^{22} -$$$$20\!\cdots\!78$$$$T^{23} +$$$$33\!\cdots\!42$$$$T^{24} -$$$$22\!\cdots\!49$$$$T^{25} +$$$$34\!\cdots\!34$$$$T^{26} -$$$$22\!\cdots\!48$$$$T^{27} +$$$$33\!\cdots\!97$$$$T^{28} -$$$$21\!\cdots\!35$$$$T^{29} +$$$$30\!\cdots\!14$$$$T^{30} -$$$$19\!\cdots\!58$$$$T^{31} +$$$$25\!\cdots\!37$$$$T^{32} -$$$$17\!\cdots\!33$$$$T^{33} +$$$$21\!\cdots\!04$$$$T^{34} -$$$$13\!\cdots\!55$$$$T^{35} +$$$$16\!\cdots\!59$$$$T^{36} -$$$$10\!\cdots\!30$$$$T^{37} +$$$$12\!\cdots\!96$$$$T^{38} -$$$$80\!\cdots\!96$$$$T^{39} +$$$$90\!\cdots\!27$$$$T^{40} -$$$$57\!\cdots\!86$$$$T^{41} +$$$$62\!\cdots\!68$$$$T^{42} -$$$$38\!\cdots\!95$$$$T^{43} +$$$$41\!\cdots\!57$$$$T^{44} -$$$$25\!\cdots\!51$$$$T^{45} +$$$$26\!\cdots\!59$$$$T^{46} -$$$$15\!\cdots\!64$$$$T^{47} +$$$$26\!\cdots\!59$$$$p T^{48} -$$$$25\!\cdots\!51$$$$p^{2} T^{49} +$$$$41\!\cdots\!57$$$$p^{3} T^{50} -$$$$38\!\cdots\!95$$$$p^{4} T^{51} +$$$$62\!\cdots\!68$$$$p^{5} T^{52} -$$$$57\!\cdots\!86$$$$p^{6} T^{53} +$$$$90\!\cdots\!27$$$$p^{7} T^{54} -$$$$80\!\cdots\!96$$$$p^{8} T^{55} +$$$$12\!\cdots\!96$$$$p^{9} T^{56} -$$$$10\!\cdots\!30$$$$p^{10} T^{57} +$$$$16\!\cdots\!59$$$$p^{11} T^{58} -$$$$13\!\cdots\!55$$$$p^{12} T^{59} +$$$$21\!\cdots\!04$$$$p^{13} T^{60} -$$$$17\!\cdots\!33$$$$p^{14} T^{61} +$$$$25\!\cdots\!37$$$$p^{15} T^{62} -$$$$19\!\cdots\!58$$$$p^{16} T^{63} +$$$$30\!\cdots\!14$$$$p^{17} T^{64} -$$$$21\!\cdots\!35$$$$p^{18} T^{65} +$$$$33\!\cdots\!97$$$$p^{19} T^{66} -$$$$22\!\cdots\!48$$$$p^{20} T^{67} +$$$$34\!\cdots\!34$$$$p^{21} T^{68} -$$$$22\!\cdots\!49$$$$p^{22} T^{69} +$$$$33\!\cdots\!42$$$$p^{23} T^{70} -$$$$20\!\cdots\!78$$$$p^{24} T^{71} +$$$$30\!\cdots\!43$$$$p^{25} T^{72} -$$$$17\!\cdots\!28$$$$p^{26} T^{73} +$$$$25\!\cdots\!84$$$$p^{27} T^{74} -$$$$13\!\cdots\!25$$$$p^{28} T^{75} +$$$$20\!\cdots\!57$$$$p^{29} T^{76} -$$$$10\!\cdots\!16$$$$p^{30} T^{77} +$$$$14\!\cdots\!87$$$$p^{31} T^{78} - 6559328129030652888 p^{32} T^{79} + 916037509721161239 p^{33} T^{80} - 38300977093230272 p^{34} T^{81} + 5151776091626559 p^{35} T^{82} - 194967944504176 p^{36} T^{83} + 24983707859653 p^{37} T^{84} - 839466407900 p^{38} T^{85} + 101289918048 p^{39} T^{86} - 47971832 p^{41} T^{87} + 328433199 p^{41} T^{88} - 7713536 p^{42} T^{89} + 795113 p^{43} T^{90} - 13609 p^{44} T^{91} + 1271 p^{45} T^{92} - 12 p^{46} T^{93} + p^{47} T^{94}$$
67 $$1 - 68 T + 3822 T^{2} - 151075 T^{3} + 5245464 T^{4} - 153313140 T^{5} + 4072457953 T^{6} - 96712877785 T^{7} + 2130999802125 T^{8} - 43222454646800 T^{9} + 824664913724401 T^{10} - 14733767601302637 T^{11} + 250059942337815723 T^{12} - 4019381240337990167 T^{13} + 61825291791908334749 T^{14} - 13551974143937199265 p T^{15} +$$$$12\!\cdots\!99$$$$T^{16} -$$$$17\!\cdots\!84$$$$T^{17} +$$$$22\!\cdots\!76$$$$T^{18} -$$$$29\!\cdots\!14$$$$T^{19} +$$$$35\!\cdots\!46$$$$T^{20} -$$$$42\!\cdots\!85$$$$T^{21} +$$$$49\!\cdots\!14$$$$T^{22} -$$$$55\!\cdots\!09$$$$T^{23} +$$$$61\!\cdots\!74$$$$T^{24} -$$$$66\!\cdots\!55$$$$T^{25} +$$$$70\!\cdots\!04$$$$T^{26} -$$$$72\!\cdots\!36$$$$T^{27} +$$$$10\!\cdots\!10$$$$p T^{28} -$$$$72\!\cdots\!67$$$$T^{29} +$$$$70\!\cdots\!80$$$$T^{30} -$$$$67\!\cdots\!03$$$$T^{31} +$$$$64\!\cdots\!19$$$$T^{32} -$$$$59\!\cdots\!70$$$$T^{33} +$$$$54\!\cdots\!63$$$$T^{34} -$$$$49\!\cdots\!77$$$$T^{35} +$$$$44\!\cdots\!61$$$$T^{36} -$$$$39\!\cdots\!76$$$$T^{37} +$$$$34\!\cdots\!93$$$$T^{38} -$$$$29\!\cdots\!61$$$$T^{39} +$$$$25\!\cdots\!23$$$$T^{40} -$$$$21\!\cdots\!81$$$$T^{41} +$$$$18\!\cdots\!24$$$$T^{42} -$$$$15\!\cdots\!33$$$$T^{43} +$$$$12\!\cdots\!02$$$$T^{44} -$$$$10\!\cdots\!83$$$$T^{45} +$$$$88\!\cdots\!44$$$$T^{46} -$$$$72\!\cdots\!06$$$$T^{47} +$$$$88\!\cdots\!44$$$$p T^{48} -$$$$10\!\cdots\!83$$$$p^{2} T^{49} +$$$$12\!\cdots\!02$$$$p^{3} T^{50} -$$$$15\!\cdots\!33$$$$p^{4} T^{51} +$$$$18\!\cdots\!24$$$$p^{5} T^{52} -$$$$21\!\cdots\!81$$$$p^{6} T^{53} +$$$$25\!\cdots\!23$$$$p^{7} T^{54} -$$$$29\!\cdots\!61$$$$p^{8} T^{55} +$$$$34\!\cdots\!93$$$$p^{9} T^{56} -$$$$39\!\cdots\!76$$$$p^{10} T^{57} +$$$$44\!\cdots\!61$$$$p^{11} T^{58} -$$$$49\!\cdots\!77$$$$p^{12} T^{59} +$$$$54\!\cdots\!63$$$$p^{13} T^{60} -$$$$59\!\cdots\!70$$$$p^{14} T^{61} +$$$$64\!\cdots\!19$$$$p^{15} T^{62} -$$$$67\!\cdots\!03$$$$p^{16} T^{63} +$$$$70\!\cdots\!80$$$$p^{17} T^{64} -$$$$72\!\cdots\!67$$$$p^{18} T^{65} +$$$$10\!\cdots\!10$$$$p^{20} T^{66} -$$$$72\!\cdots\!36$$$$p^{20} T^{67} +$$$$70\!\cdots\!04$$$$p^{21} T^{68} -$$$$66\!\cdots\!55$$$$p^{22} T^{69} +$$$$61\!\cdots\!74$$$$p^{23} T^{70} -$$$$55\!\cdots\!09$$$$p^{24} T^{71} +$$$$49\!\cdots\!14$$$$p^{25} T^{72} -$$$$42\!\cdots\!85$$$$p^{26} T^{73} +$$$$35\!\cdots\!46$$$$p^{27} T^{74} -$$$$29\!\cdots\!14$$$$p^{28} T^{75} +$$$$22\!\cdots\!76$$$$p^{29} T^{76} -$$$$17\!\cdots\!84$$$$p^{30} T^{77} +$$$$12\!\cdots\!99$$$$p^{31} T^{78} - 13551974143937199265 p^{33} T^{79} + 61825291791908334749 p^{33} T^{80} - 4019381240337990167 p^{34} T^{81} + 250059942337815723 p^{35} T^{82} - 14733767601302637 p^{36} T^{83} + 824664913724401 p^{37} T^{84} - 43222454646800 p^{38} T^{85} + 2130999802125 p^{39} T^{86} - 96712877785 p^{40} T^{87} + 4072457953 p^{41} T^{88} - 153313140 p^{42} T^{89} + 5245464 p^{43} T^{90} - 151075 p^{44} T^{91} + 3822 p^{45} T^{92} - 68 p^{46} T^{93} + p^{47} T^{94}$$
71 $$1 + 13 T + 1670 T^{2} + 21683 T^{3} + 1413431 T^{4} + 18209351 T^{5} + 807052393 T^{6} + 10262831307 T^{7} + 349184597052 T^{8} + 4365416828592 T^{9} + 121931635468312 T^{10} + 1494206453502710 T^{11} + 35746213462665411 T^{12} + 428491296259048232 T^{13} + 9038782595093870413 T^{14} +$$$$10\!\cdots\!22$$$$T^{15} +$$$$20\!\cdots\!11$$$$T^{16} +$$$$22\!\cdots\!22$$$$T^{17} +$$$$39\!\cdots\!91$$$$T^{18} +$$$$44\!\cdots\!86$$$$T^{19} +$$$$71\!\cdots\!76$$$$T^{20} +$$$$77\!\cdots\!82$$$$T^{21} +$$$$11\!\cdots\!89$$$$T^{22} +$$$$12\!\cdots\!90$$$$T^{23} +$$$$17\!\cdots\!27$$$$T^{24} +$$$$25\!\cdots\!06$$$$p T^{25} +$$$$24\!\cdots\!55$$$$T^{26} +$$$$24\!\cdots\!05$$$$T^{27} +$$$$31\!\cdots\!18$$$$T^{28} +$$$$30\!\cdots\!38$$$$T^{29} +$$$$37\!\cdots\!35$$$$T^{30} +$$$$36\!\cdots\!29$$$$T^{31} +$$$$42\!\cdots\!68$$$$T^{32} +$$$$39\!\cdots\!60$$$$T^{33} +$$$$44\!\cdots\!49$$$$T^{34} +$$$$40\!\cdots\!93$$$$T^{35} +$$$$44\!\cdots\!23$$$$T^{36} +$$$$39\!\cdots\!00$$$$T^{37} +$$$$41\!\cdots\!33$$$$T^{38} +$$$$35\!\cdots\!21$$$$T^{39} +$$$$36\!\cdots\!23$$$$T^{40} +$$$$30\!\cdots\!60$$$$T^{41} +$$$$30\!\cdots\!13$$$$T^{42} +$$$$25\!\cdots\!86$$$$T^{43} +$$$$24\!\cdots\!00$$$$T^{44} +$$$$19\!\cdots\!78$$$$T^{45} +$$$$18\!\cdots\!67$$$$T^{46} +$$$$14\!\cdots\!28$$$$T^{47} +$$$$18\!\cdots\!67$$$$p T^{48} +$$$$19\!\cdots\!78$$$$p^{2} T^{49} +$$$$24\!\cdots\!00$$$$p^{3} T^{50} +$$$$25\!\cdots\!86$$$$p^{4} T^{51} +$$$$30\!\cdots\!13$$$$p^{5} T^{52} +$$$$30\!\cdots\!60$$$$p^{6} T^{53} +$$$$36\!\cdots\!23$$$$p^{7} T^{54} +$$$$35\!\cdots\!21$$$$p^{8} T^{55} +$$$$41\!\cdots\!33$$$$p^{9} T^{56} +$$$$39\!\cdots\!00$$$$p^{10} T^{57} +$$$$44\!\cdots\!23$$$$p^{11} T^{58} +$$$$40\!\cdots\!93$$$$p^{12} T^{59} +$$$$44\!\cdots\!49$$$$p^{13} T^{60} +$$$$39\!\cdots\!60$$$$p^{14} T^{61} +$$$$42\!\cdots\!68$$$$p^{15} T^{62} +$$$$36\!\cdots\!29$$$$p^{16} T^{63} +$$$$37\!\cdots\!35$$$$p^{17} T^{64} +$$$$30\!\cdots\!38$$$$p^{18} T^{65} +$$$$31\!\cdots\!18$$$$p^{19} T^{66} +$$$$24\!\cdots\!05$$$$p^{20} T^{67} +$$$$24\!\cdots\!55$$$$p^{21} T^{68} +$$$$25\!\cdots\!06$$$$p^{23} T^{69} +$$$$17\!\cdots\!27$$$$p^{23} T^{70} +$$$$12\!\cdots\!90$$$$p^{24} T^{71} +$$$$11\!\cdots\!89$$$$p^{25} T^{72} +$$$$77\!\cdots\!82$$$$p^{26} T^{73} +$$$$71\!\cdots\!76$$$$p^{27} T^{74} +$$$$44\!\cdots\!86$$$$p^{28} T^{75} +$$$$39\!\cdots\!91$$$$p^{29} T^{76} +$$$$22\!\cdots\!22$$$$p^{30} T^{77} +$$$$20\!\cdots\!11$$$$p^{31} T^{78} +$$$$10\!\cdots\!22$$$$p^{32} T^{79} + 9038782595093870413 p^{33} T^{80} + 428491296259048232 p^{34} T^{81} + 35746213462665411 p^{35} T^{82} + 1494206453502710 p^{36} T^{83} + 121931635468312 p^{37} T^{84} + 4365416828592 p^{38} T^{85} + 349184597052 p^{39} T^{86} + 10262831307 p^{40} T^{87} + 807052393 p^{41} T^{88} + 18209351 p^{42} T^{89} + 1413431 p^{43} T^{90} + 21683 p^{44} T^{91} + 1670 p^{45} T^{92} + 13 p^{46} T^{93} + p^{47} T^{94}$$
73 $$1 - 35 T + 2396 T^{2} - 70211 T^{3} + 2754486 T^{4} - 69902507 T^{5} + 2045946347 T^{6} - 46039825740 T^{7} + 1111220319440 T^{8} - 22561318911530 T^{9} + 472583509962269 T^{10} - 8772521352458548 T^{11} + 164372017575263637 T^{12} - 2818935318429424265 T^{13} + 48188127191191420278 T^{14} -$$$$76\!\cdots\!98$$$$T^{15} +$$$$12\!\cdots\!15$$$$T^{16} -$$$$18\!\cdots\!31$$$$T^{17} +$$$$26\!\cdots\!44$$$$T^{18} -$$$$38\!\cdots\!30$$$$T^{19} +$$$$53\!\cdots\!02$$$$T^{20} -$$$$71\!\cdots\!66$$$$T^{21} +$$$$93\!\cdots\!30$$$$T^{22} -$$$$11\!\cdots\!63$$$$T^{23} +$$$$15\!\cdots\!32$$$$T^{24} -$$$$18\!\cdots\!89$$$$T^{25} +$$$$22\!\cdots\!19$$$$T^{26} -$$$$25\!\cdots\!91$$$$T^{27} +$$$$29\!\cdots\!93$$$$T^{28} -$$$$33\!\cdots\!30$$$$T^{29} +$$$$37\!\cdots\!74$$$$T^{30} -$$$$40\!\cdots\!82$$$$T^{31} +$$$$43\!\cdots\!09$$$$T^{32} -$$$$45\!\cdots\!67$$$$T^{33} +$$$$46\!\cdots\!03$$$$T^{34} -$$$$47\!\cdots\!17$$$$T^{35} +$$$$47\!\cdots\!65$$$$T^{36} -$$$$47\!\cdots\!43$$$$T^{37} +$$$$45\!\cdots\!69$$$$T^{38} -$$$$43\!\cdots\!52$$$$T^{39} +$$$$41\!\cdots\!29$$$$T^{40} -$$$$38\!\cdots\!80$$$$T^{41} +$$$$48\!\cdots\!31$$$$p T^{42} -$$$$32\!\cdots\!70$$$$T^{43} +$$$$28\!\cdots\!36$$$$T^{44} -$$$$25\!\cdots\!17$$$$T^{45} +$$$$22\!\cdots\!26$$$$T^{46} -$$$$19\!\cdots\!62$$$$T^{47} +$$$$22\!\cdots\!26$$$$p T^{48} -$$$$25\!\cdots\!17$$$$p^{2} T^{49} +$$$$28\!\cdots\!36$$$$p^{3} T^{50} -$$$$32\!\cdots\!70$$$$p^{4} T^{51} +$$$$48\!\cdots\!31$$$$p^{6} T^{52} -$$$$38\!\cdots\!80$$$$p^{6} T^{53} +$$$$41\!\cdots\!29$$$$p^{7} T^{54} -$$$$43\!\cdots\!52$$$$p^{8} T^{55} +$$$$45\!\cdots\!69$$$$p^{9} T^{56} -$$$$47\!\cdots\!43$$$$p^{10} T^{57} +$$$$47\!\cdots\!65$$$$p^{11} T^{58} -$$$$47\!\cdots\!17$$$$p^{12} T^{59} +$$$$46\!\cdots\!03$$$$p^{13} T^{60} -$$$$45\!\cdots\!67$$$$p^{14} T^{61} +$$$$43\!\cdots\!09$$$$p^{15} T^{62} -$$$$40\!\cdots\!82$$$$p^{16} T^{63} +$$$$37\!\cdots\!74$$$$p^{17} T^{64} -$$$$33\!\cdots\!30$$$$p^{18} T^{65} +$$$$29\!\cdots\!93$$$$p^{19} T^{66} -$$$$25\!\cdots\!91$$$$p^{20} T^{67} +$$$$22\!\cdots\!19$$$$p^{21} T^{68} -$$$$18\!\cdots\!89$$$$p^{22} T^{69} +$$$$15\!\cdots\!32$$$$p^{23} T^{70} -$$$$11\!\cdots\!63$$$$p^{24} T^{71} +$$$$93\!\cdots\!30$$$$p^{25} T^{72} -$$$$71\!\cdots\!66$$$$p^{26} T^{73} +$$$$53\!\cdots\!02$$$$p^{27} T^{74} -$$$$38\!\cdots\!30$$$$p^{28} T^{75} +$$$$26\!\cdots\!44$$$$p^{29} T^{76} -$$$$18\!\cdots\!31$$$$p^{30} T^{77} +$$$$12\!\cdots\!15$$$$p^{31} T^{78} -$$$$76\!\cdots\!98$$$$p^{32} T^{79} + 48188127191191420278 p^{33} T^{80} - 2818935318429424265 p^{34} T^{81} + 164372017575263637 p^{35} T^{82} - 8772521352458548 p^{36} T^{83} + 472583509962269 p^{37} T^{84} - 22561318911530 p^{38} T^{85} + 1111220319440 p^{39} T^{86} - 46039825740 p^{40} T^{87} + 2045946347 p^{41} T^{88} - 69902507 p^{42} T^{89} + 2754486 p^{43} T^{90} - 70211 p^{44} T^{91} + 2396 p^{45} T^{92} - 35 p^{46} T^{93} + p^{47} T^{94}$$
79 $$1 - 21 T + 1936 T^{2} - 38464 T^{3} + 1870097 T^{4} - 35183861 T^{5} + 1201511507 T^{6} - 21424974878 T^{7} + 577330048393 T^{8} - 9766077972767 T^{9} + 221121240271801 T^{10} - 3551753864478293 T^{11} + 70244194448571476 T^{12} - 1072444565471744899 T^{13} + 19011956127247992063 T^{14} -$$$$27\!\cdots\!55$$$$T^{15} +$$$$44\!\cdots\!03$$$$T^{16} -$$$$61\!\cdots\!52$$$$T^{17} +$$$$92\!\cdots\!90$$$$T^{18} -$$$$12\!\cdots\!58$$$$T^{19} +$$$$17\!\cdots\!82$$$$T^{20} -$$$$21\!\cdots\!58$$$$T^{21} +$$$$28\!\cdots\!41$$$$T^{22} -$$$$33\!\cdots\!42$$$$T^{23} +$$$$41\!\cdots\!40$$$$T^{24} -$$$$47\!\cdots\!17$$$$T^{25} +$$$$56\!\cdots\!74$$$$T^{26} -$$$$61\!\cdots\!68$$$$T^{27} +$$$$68\!\cdots\!36$$$$T^{28} -$$$$71\!\cdots\!43$$$$T^{29} +$$$$74\!\cdots\!25$$$$T^{30} -$$$$74\!\cdots\!68$$$$T^{31} +$$$$73\!\cdots\!68$$$$T^{32} -$$$$68\!\cdots\!56$$$$T^{33} +$$$$64\!\cdots\!95$$$$T^{34} -$$$$56\!\cdots\!05$$$$T^{35} +$$$$49\!\cdots\!52$$$$T^{36} -$$$$38\!\cdots\!68$$$$T^{37} +$$$$30\!\cdots\!66$$$$T^{38} -$$$$20\!\cdots\!46$$$$T^{39} +$$$$13\!\cdots\!73$$$$T^{40} -$$$$51\!\cdots\!90$$$$T^{41} +$$$$32\!\cdots\!18$$$$p T^{42} +$$$$50\!\cdots\!39$$$$T^{43} -$$$$71\!\cdots\!72$$$$T^{44} +$$$$94\!\cdots\!90$$$$T^{45} -$$$$90\!\cdots\!63$$$$T^{46} +$$$$90\!\cdots\!36$$$$T^{47} -$$$$90\!\cdots\!63$$$$p T^{48} +$$$$94\!\cdots\!90$$$$p^{2} T^{49} -$$$$71\!\cdots\!72$$$$p^{3} T^{50} +$$$$50\!\cdots\!39$$$$p^{4} T^{51} +$$$$32\!\cdots\!18$$$$p^{6} T^{52} -$$$$51\!\cdots\!90$$$$p^{6} T^{53} +$$$$13\!\cdots\!73$$$$p^{7} T^{54} -$$$$20\!\cdots\!46$$$$p^{8} T^{55} +$$$$30\!\cdots\!66$$$$p^{9} T^{56} -$$$$38\!\cdots\!68$$$$p^{10} T^{57} +$$$$49\!\cdots\!52$$$$p^{11} T^{58} -$$$$56\!\cdots\!05$$$$p^{12} T^{59} +$$$$64\!\cdots\!95$$$$p^{13} T^{60} -$$$$68\!\cdots\!56$$$$p^{14} T^{61} +$$$$73\!\cdots\!68$$$$p^{15} T^{62} -$$$$74\!\cdots\!68$$$$p^{16} T^{63} +$$$$74\!\cdots\!25$$$$p^{17} T^{64} -$$$$71\!\cdots\!43$$$$p^{18} T^{65} +$$$$68\!\cdots\!36$$$$p^{19} T^{66} -$$$$61\!\cdots\!68$$$$p^{20} T^{67} +$$$$56\!\cdots\!74$$$$p^{21} T^{68} -$$$$47\!\cdots\!17$$$$p^{22} T^{69} +$$$$41\!\cdots\!40$$$$p^{23} T^{70} -$$$$33\!\cdots\!42$$$$p^{24} T^{71} +$$$$28\!\cdots\!41$$$$p^{25} T^{72} -$$$$21\!\cdots\!58$$$$p^{26} T^{73} +$$$$17\!\cdots\!82$$$$p^{27} T^{74} -$$$$12\!\cdots\!58$$$$p^{28} T^{75} +$$$$92\!\cdots\!90$$$$p^{29} T^{76} -$$$$61\!\cdots\!52$$$$p^{30} T^{77} +$$$$44\!\cdots\!03$$$$p^{31} T^{78} -$$$$27\!\cdots\!55$$$$p^{32} T^{79} + 19011956127247992063 p^{33} T^{80} - 1072444565471744899 p^{34} T^{81} + 70244194448571476 p^{35} T^{82} - 3551753864478293 p^{36} T^{83} + 221121240271801 p^{37} T^{84} - 9766077972767 p^{38} T^{85} + 577330048393 p^{39} T^{86} - 21424974878 p^{40} T^{87} + 1201511507 p^{41} T^{88} - 35183861 p^{42} T^{89} + 1870097 p^{43} T^{90} - 38464 p^{44} T^{91} + 1936 p^{45} T^{92} - 21 p^{46} T^{93} + p^{47} T^{94}$$
83 $$1 - 23 T + 2167 T^{2} - 46999 T^{3} + 2343376 T^{4} - 47834227 T^{5} + 1681982138 T^{6} - 32304045222 T^{7} + 899687925487 T^{8} - 16271425080364 T^{9} + 381941080465337 T^{10} - 6514889503855263 T^{11} + 133880773983532870 T^{12} - 2158144305332721474 T^{13} + 39816592010642696252 T^{14} -$$$$60\!\cdots\!14$$$$T^{15} +$$$$10\!\cdots\!65$$$$T^{16} -$$$$14\!\cdots\!45$$$$T^{17} +$$$$23\!\cdots\!31$$$$T^{18} -$$$$31\!\cdots\!59$$$$T^{19} +$$$$46\!\cdots\!28$$$$T^{20} -$$$$61\!\cdots\!59$$$$T^{21} +$$$$84\!\cdots\!64$$$$T^{22} -$$$$10\!\cdots\!86$$$$T^{23} +$$$$13\!\cdots\!51$$$$T^{24} -$$$$16\!\cdots\!40$$$$T^{25} +$$$$20\!\cdots\!08$$$$T^{26} -$$$$23\!\cdots\!07$$$$T^{27} +$$$$28\!\cdots\!20$$$$T^{28} -$$$$31\!\cdots\!75$$$$T^{29} +$$$$36\!\cdots\!54$$$$T^{30} -$$$$39\!\cdots\!93$$$$T^{31} +$$$$42\!\cdots\!63$$$$T^{32} -$$$$44\!\cdots\!21$$$$T^{33} +$$$$47\!\cdots\!65$$$$T^{34} -$$$$48\!\cdots\!87$$$$T^{35} +$$$$50\!\cdots\!86$$$$T^{36} -$$$$49\!\cdots\!61$$$$T^{37} +$$$$50\!\cdots\!58$$$$T^{38} -$$$$48\!\cdots\!28$$$$T^{39} +$$$$47\!\cdots\!57$$$$T^{40} -$$$$45\!\cdots\!82$$$$T^{41} +$$$$43\!\cdots\!40$$$$T^{42} -$$$$40\!\cdots\!37$$$$T^{43} +$$$$38\!\cdots\!20$$$$T^{44} -$$$$35\!\cdots\!25$$$$T^{45} +$$$$33\!\cdots\!74$$$$T^{46} -$$$$29\!\cdots\!18$$$$T^{47} +$$$$33\!\cdots\!74$$$$p T^{48} -$$$$35\!\cdots\!25$$$$p^{2} T^{49} +$$$$38\!\cdots\!20$$$$p^{3} T^{50} -$$$$40\!\cdots\!37$$$$p^{4} T^{51} +$$$$43\!\cdots\!40$$$$p^{5} T^{52} -$$$$45\!\cdots\!82$$$$p^{6} T^{53} +$$$$47\!\cdots\!57$$$$p^{7} T^{54} -$$$$48\!\cdots\!28$$$$p^{8} T^{55} +$$$$50\!\cdots\!58$$$$p^{9} T^{56} -$$$$49\!\cdots\!61$$$$p^{10} T^{57} +$$$$50\!\cdots\!86$$$$p^{11} T^{58} -$$$$48\!\cdots\!87$$$$p^{12} T^{59} +$$$$47\!\cdots\!65$$$$p^{13} T^{60} -$$$$44\!\cdots\!21$$$$p^{14} T^{61} +$$$$42\!\cdots\!63$$$$p^{15} T^{62} -$$$$39\!\cdots\!93$$$$p^{16} T^{63} +$$$$36\!\cdots\!54$$$$p^{17} T^{64} -$$$$31\!\cdots\!75$$$$p^{18} T^{65} +$$$$28\!\cdots\!20$$$$p^{19} T^{66} -$$$$23\!\cdots\!07$$$$p^{20} T^{67} +$$$$20\!\cdots\!08$$$$p^{21} T^{68} -$$$$16\!\cdots\!40$$$$p^{22} T^{69} +$$$$13\!\cdots\!51$$$$p^{23} T^{70} -$$$$10\!\cdots\!86$$$$p^{24} T^{71} +$$$$84\!\cdots\!64$$$$p^{25} T^{72} -$$$$61\!\cdots\!59$$$$p^{26} T^{73} +$$$$46\!\cdots\!28$$$$p^{27} T^{74} -$$$$31\!\cdots\!59$$$$p^{28} T^{75} +$$$$23\!\cdots\!31$$$$p^{29} T^{76} -$$$$14\!\cdots\!45$$$$p^{30} T^{77} +$$$$10\!\cdots\!65$$$$p^{31} T^{78} -$$$$60\!\cdots\!14$$$$p^{32} T^{79} + 39816592010642696252 p^{33} T^{80} - 2158144305332721474 p^{34} T^{81} + 133880773983532870 p^{35} T^{82} - 6514889503855263 p^{36} T^{83} + 381941080465337 p^{37} T^{84} - 16271425080364 p^{38} T^{85} + 899687925487 p^{39} T^{86} - 32304045222 p^{40} T^{87} + 1681982138 p^{41} T^{88} - 47834227 p^{42} T^{89} + 2343376 p^{43} T^{90} - 46999 p^{44} T^{91} + 2167 p^{45} T^{92} - 23 p^{46} T^{93} + p^{47} T^{94}$$
89 $$1 - 24 T + 2059 T^{2} - 45951 T^{3} + 2127152 T^{4} - 44205799 T^{5} + 1468339704 T^{6} - 28509987036 T^{7} + 761571885743 T^{8} - 13875249215820 T^{9} + 316610467906192 T^{10} - 5437909709441752 T^{11} + 109942763007427615 T^{12} - 1788325813789449987 T^{13} + 32816008934327469316 T^{14} -$$$$50\!\cdots\!75$$$$T^{15} +$$$$85\!\cdots\!67$$$$T^{16} -$$$$12\!\cdots\!16$$$$T^{17} +$$$$20\!\cdots\!11$$$$T^{18} -$$$$28\!\cdots\!65$$$$T^{19} +$$$$42\!\cdots\!52$$$$T^{20} -$$$$57\!\cdots\!61$$$$T^{21} +$$$$81\!\cdots\!56$$$$T^{22} -$$$$10\!\cdots\!95$$$$T^{23} +$$$$14\!\cdots\!09$$$$T^{24} -$$$$18\!\cdots\!93$$$$T^{25} +$$$$23\!\cdots\!36$$$$T^{26} -$$$$29\!\cdots\!31$$$$T^{27} +$$$$36\!\cdots\!84$$$$T^{28} -$$$$43\!\cdots\!42$$$$T^{29} +$$$$52\!\cdots\!44$$$$T^{30} -$$$$60\!\cdots\!97$$$$T^{31} +$$$$71\!\cdots\!98$$$$T^{32} -$$$$80\!\cdots\!65$$$$T^{33} +$$$$90\!\cdots\!94$$$$T^{34} -$$$$99\!\cdots\!85$$$$T^{35} +$$$$10\!\cdots\!78$$$$T^{36} -$$$$11\!\cdots\!54$$$$T^{37} +$$$$12\!\cdots\!25$$$$T^{38} -$$$$12\!\cdots\!24$$$$T^{39} +$$$$13\!\cdots\!05$$$$T^{40} -$$$$13\!\cdots\!18$$$$T^{41} +$$$$13\!\cdots\!92$$$$T^{42} -$$$$13\!\cdots\!15$$$$T^{43} +$$$$13\!\cdots\!34$$$$T^{44} -$$$$13\!\cdots\!80$$$$T^{45} +$$$$12\!\cdots\!02$$$$T^{46} -$$$$12\!\cdots\!52$$$$T^{47} +$$$$12\!\cdots\!02$$$$p T^{48} -$$$$13\!\cdots\!80$$$$p^{2} T^{49} +$$$$13\!\cdots\!34$$$$p^{3} T^{50} -$$$$13\!\cdots\!15$$$$p^{4} T^{51} +$$$$13\!\cdots\!92$$$$p^{5} T^{52} -$$$$13\!\cdots\!18$$$$p^{6} T^{53} +$$$$13\!\cdots\!05$$$$p^{7} T^{54} -$$$$12\!\cdots\!24$$$$p^{8} T^{55} +$$$$12\!\cdots\!25$$$$p^{9} T^{56} -$$$$11\!\cdots\!54$$$$p^{10} T^{57} +$$$$10\!\cdots\!78$$$$p^{11} T^{58} -$$$$99\!\cdots\!85$$$$p^{12} T^{59} +$$$$90\!\cdots\!94$$$$p^{13} T^{60} -$$$$80\!\cdots\!65$$$$p^{14} T^{61} +$$$$71\!\cdots\!98$$$$p^{15} T^{62} -$$$$60\!\cdots\!97$$$$p^{16} T^{63} +$$$$52\!\cdots\!44$$$$p^{17} T^{64} -$$$$43\!\cdots\!42$$$$p^{18} T^{65} +$$$$36\!\cdots\!84$$$$p^{19} T^{66} -$$$$29\!\cdots\!31$$$$p^{20} T^{67} +$$$$23\!\cdots\!36$$$$p^{21} T^{68} -$$$$18\!\cdots\!93$$$$p^{22} T^{69} +$$$$14\!\cdots\!09$$$$p^{23} T^{70} -$$$$10\!\cdots\!95$$$$p^{24} T^{71} +$$$$81\!\cdots\!56$$$$p^{25} T^{72} -$$$$57\!\cdots\!61$$$$p^{26} T^{73} +$$$$42\!\cdots\!52$$$$p^{27} T^{74} -$$$$28\!\cdots\!65$$$$p^{28} T^{75} +$$$$20\!\cdots\!11$$$$p^{29} T^{76} -$$$$12\!\cdots\!16$$$$p^{30} T^{77} +$$$$85\!\cdots\!67$$$$p^{31} T^{78} -$$$$50\!\cdots\!75$$$$p^{32} T^{79} + 32816008934327469316 p^{33} T^{80} - 1788325813789449987 p^{34} T^{81} + 109942763007427615 p^{35} T^{82} - 5437909709441752 p^{36} T^{83} + 316610467906192 p^{37} T^{84} - 13875249215820 p^{38} T^{85} + 761571885743 p^{39} T^{86} - 28509987036 p^{40} T^{87} + 1468339704 p^{41} T^{88} - 44205799 p^{42} T^{89} + 2127152 p^{43} T^{90} - 45951 p^{44} T^{91} + 2059 p^{45} T^{92} - 24 p^{46} T^{93} + p^{47} T^{94}$$
97 $$1 - 88 T + 6168 T^{2} - 312745 T^{3} + 13774994 T^{4} - 519210814 T^{5} + 17718001326 T^{6} - 546727990066 T^{7} + 15636299228757 T^{8} - 415184673418736 T^{9} + 10370287856603780 T^{10} - 244191045014560016 T^{11} + 5463771260177454741 T^{12} -$$$$11\!\cdots\!31$$$$T^{13} +$$$$23\!\cdots\!05$$$$T^{14} -$$$$46\!\cdots\!67$$$$T^{15} +$$$$87\!\cdots\!54$$$$T^{16} -$$$$15\!\cdots\!74$$$$T^{17} +$$$$27\!\cdots\!44$$$$T^{18} -$$$$47\!\cdots\!82$$$$T^{19} +$$$$77\!\cdots\!09$$$$T^{20} -$$$$12\!\cdots\!18$$$$T^{21} +$$$$19\!\cdots\!75$$$$T^{22} -$$$$29\!\cdots\!05$$$$T^{23} +$$$$44\!\cdots\!62$$$$p T^{24} -$$$$61\!\cdots\!60$$$$T^{25} +$$$$86\!\cdots\!97$$$$T^{26} -$$$$11\!\cdots\!56$$$$T^{27} +$$$$15\!\cdots\!56$$$$T^{28} -$$$$20\!\cdots\!71$$$$T^{29} +$$$$27\!\cdots\!20$$$$T^{30} -$$$$34\!\cdots\!13$$$$T^{31} +$$$$42\!\cdots\!40$$$$T^{32} -$$$$51\!\cdots\!99$$$$T^{33} +$$$$61\!\cdots\!74$$$$T^{34} -$$$$72\!\cdots\!57$$$$T^{35} +$$$$84\!\cdots\!84$$$$T^{36} -$$$$96\!\cdots\!75$$$$T^{37} +$$$$10\!\cdots\!19$$$$T^{38} -$$$$11\!\cdots\!43$$$$T^{39} +$$$$13\!\cdots\!51$$$$T^{40} -$$$$13\!\cdots\!65$$$$T^{41} +$$$$14\!\cdots\!50$$$$T^{42} -$$$$15\!\cdots\!00$$$$T^{43} +$$$$15\!\cdots\!68$$$$T^{44} -$$$$16\!\cdots\!91$$$$T^{45} +$$$$16\!\cdots\!86$$$$T^{46} -$$$$16\!\cdots\!30$$$$T^{47} +$$$$16\!\cdots\!86$$$$p T^{48} -$$$$16\!\cdots\!91$$$$p^{2} T^{49} +$$$$15\!\cdots\!68$$$$p^{3} T^{50} -$$$$15\!\cdots\!00$$$$p^{4} T^{51} +$$$$14\!\cdots\!50$$$$p^{5} T^{52} -$$$$13\!\cdots\!65$$$$p^{6} T^{53} +$$$$13\!\cdots\!51$$$$p^{7} T^{54} -$$$$11\!\cdots\!43$$$$p^{8} T^{55} +$$$$10\!\cdots\!19$$$$p^{9} T^{56} -$$$$96\!\cdots\!75$$$$p^{10} T^{57} +$$$$84\!\cdots\!84$$$$p^{11} T^{58} -$$$$72\!\cdots\!57$$$$p^{12} T^{59} +$$$$61\!\cdots\!74$$$$p^{13} T^{60} -$$$$51\!\cdots\!99$$$$p^{14} T^{61} +$$$$42\!\cdots\!40$$$$p^{15} T^{62} -$$$$34\!\cdots\!13$$$$p^{16} T^{63} +$$$$27\!\cdots\!20$$$$p^{17} T^{64} -$$$$20\!\cdots\!71$$$$p^{18} T^{65} +$$$$15\!\cdots\!56$$$$p^{19} T^{66} -$$$$11\!\cdots\!56$$$$p^{20} T^{67} +$$$$86\!\cdots\!97$$$$p^{21} T^{68} -$$$$61\!\cdots\!60$$$$p^{22} T^{69} +$$$$44\!\cdots\!62$$$$p^{24} T^{70} -$$$$29\!\cdots\!05$$$$p^{24} T^{71} +$$$$19\!\cdots\!75$$$$p^{25} T^{72} -$$$$12\!\cdots\!18$$$$p^{26} T^{73} +$$$$77\!\cdots\!09$$$$p^{27} T^{74} -$$$$47\!\cdots\!82$$$$p^{28} T^{75} +$$$$27\!\cdots\!44$$$$p^{29} T^{76} -$$$$15\!\cdots\!74$$$$p^{30} T^{77} +$$$$87\!\cdots\!54$$$$p^{31} T^{78} -$$$$46\!\cdots\!67$$$$p^{32} T^{79} +$$$$23\!\cdots\!05$$$$p^{33} T^{80} -$$$$11\!\cdots\!31$$$$p^{34} T^{81} + 5463771260177454741 p^{35} T^{82} - 244191045014560016 p^{36} T^{83} + 10370287856603780 p^{37} T^{84} - 415184673418736 p^{38} T^{85} + 15636299228757 p^{39} T^{86} - 546727990066 p^{40} T^{87} + 17718001326 p^{41} T^{88} - 519210814 p^{42} T^{89} + 13774994 p^{43} T^{90} - 312745 p^{44} T^{91} + 6168 p^{45} T^{92} - 88 p^{46} T^{93} + p^{47} T^{94}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{94} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}