Properties

Label 9.2e12_353e3.16t1294.1c1
Dimension 9
Group $S_4\wr C_2$
Conductor $ 2^{12} \cdot 353^{3}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$9$
Group:$S_4\wr C_2$
Conductor:$180170657792= 2^{12} \cdot 353^{3} $
Artin number field: Splitting field of $f= x^{8} - 4 x^{7} + 10 x^{6} + 3 x^{5} - 19 x^{4} + 41 x^{3} - 26 x^{2} + 15 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 16T1294
Parity: Odd
Determinant: 1.2e2_353.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 99.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{3} + 9 x + 92 $
Roots:
$r_{ 1 }$ $=$ $ 5 a^{2} + 3 a + 61 + \left(74 a^{2} + 4 a + 93\right)\cdot 97 + \left(59 a^{2} + 22 a + 23\right)\cdot 97^{2} + \left(83 a^{2} + 57 a + 92\right)\cdot 97^{3} + \left(a^{2} + 42 a + 54\right)\cdot 97^{4} + \left(72 a^{2} + 87 a + 41\right)\cdot 97^{5} + \left(32 a^{2} + 44 a + 30\right)\cdot 97^{6} + \left(70 a^{2} + 13 a + 25\right)\cdot 97^{7} + \left(46 a + 32\right)\cdot 97^{8} + \left(58 a^{2} + 46 a + 40\right)\cdot 97^{9} + \left(59 a^{2} + 35 a + 93\right)\cdot 97^{10} + \left(26 a^{2} + 66 a + 40\right)\cdot 97^{11} + \left(66 a^{2} + 88 a + 42\right)\cdot 97^{12} + \left(64 a^{2} + 14 a + 41\right)\cdot 97^{13} + \left(10 a^{2} + 23 a + 49\right)\cdot 97^{14} + \left(57 a^{2} + 72 a + 88\right)\cdot 97^{15} + \left(47 a^{2} + 38 a + 78\right)\cdot 97^{16} + \left(20 a^{2} + 60 a + 26\right)\cdot 97^{17} + \left(72 a^{2} + 85 a + 27\right)\cdot 97^{18} + \left(8 a^{2} + 11 a + 70\right)\cdot 97^{19} + \left(89 a^{2} + 33 a + 34\right)\cdot 97^{20} + \left(84 a^{2} + 41 a + 29\right)\cdot 97^{21} + \left(34 a^{2} + 86 a + 72\right)\cdot 97^{22} + \left(8 a^{2} + 57 a + 65\right)\cdot 97^{23} + \left(10 a^{2} + 14 a + 95\right)\cdot 97^{24} + \left(86 a^{2} + 67 a + 19\right)\cdot 97^{25} + \left(84 a^{2} + 48 a + 13\right)\cdot 97^{26} + \left(24 a^{2} + 82 a + 19\right)\cdot 97^{27} + \left(81 a^{2} + 25 a + 65\right)\cdot 97^{28} + \left(14 a^{2} + 31 a + 38\right)\cdot 97^{29} + \left(30 a^{2} + 47 a + 6\right)\cdot 97^{30} + \left(56 a^{2} + 14 a + 59\right)\cdot 97^{31} + \left(30 a^{2} + 22 a + 4\right)\cdot 97^{32} + \left(60 a^{2} + 21 a + 69\right)\cdot 97^{33} + \left(17 a^{2} + 26 a + 16\right)\cdot 97^{34} + \left(88 a^{2} + 80 a + 46\right)\cdot 97^{35} + \left(10 a^{2} + 91 a + 45\right)\cdot 97^{36} + \left(51 a^{2} + 35 a + 95\right)\cdot 97^{37} + \left(8 a^{2} + 86 a + 14\right)\cdot 97^{38} + \left(59 a^{2} + 14 a + 86\right)\cdot 97^{39} + \left(20 a^{2} + 68 a + 94\right)\cdot 97^{40} + \left(48 a^{2} + 54 a + 67\right)\cdot 97^{41} + \left(89 a^{2} + 52 a + 55\right)\cdot 97^{42} + \left(15 a^{2} + 24 a + 25\right)\cdot 97^{43} + \left(93 a^{2} + 21 a + 35\right)\cdot 97^{44} + \left(53 a^{2} + 9 a + 15\right)\cdot 97^{45} + \left(37 a^{2} + 22 a + 14\right)\cdot 97^{46} + \left(6 a^{2} + 89 a + 95\right)\cdot 97^{47} + \left(23 a^{2} + 30\right)\cdot 97^{48} + \left(87 a^{2} + 43 a + 20\right)\cdot 97^{49} + \left(33 a^{2} + 28 a + 48\right)\cdot 97^{50} + \left(90 a^{2} + 18 a + 72\right)\cdot 97^{51} + \left(18 a^{2} + 35 a + 91\right)\cdot 97^{52} + \left(78 a^{2} + 23 a + 44\right)\cdot 97^{53} + \left(30 a^{2} + 73 a + 79\right)\cdot 97^{54} + \left(86 a^{2} + a + 42\right)\cdot 97^{55} + \left(24 a^{2} + 52 a + 65\right)\cdot 97^{56} + \left(85 a^{2} + 20 a + 10\right)\cdot 97^{57} + \left(47 a^{2} + 44 a + 87\right)\cdot 97^{58} + \left(25 a^{2} + 64 a + 10\right)\cdot 97^{59} + \left(37 a^{2} + 55 a + 70\right)\cdot 97^{60} + \left(65 a^{2} + 52 a + 56\right)\cdot 97^{61} + \left(84 a^{2} + 11 a + 82\right)\cdot 97^{62} + \left(12 a^{2} + 85 a + 63\right)\cdot 97^{63} + \left(2 a^{2} + 51 a + 92\right)\cdot 97^{64} + \left(74 a^{2} + 74 a + 59\right)\cdot 97^{65} + \left(84 a^{2} + 68 a + 17\right)\cdot 97^{66} + \left(75 a^{2} + 47 a + 83\right)\cdot 97^{67} + \left(3 a^{2} + 56 a + 65\right)\cdot 97^{68} + \left(17 a^{2} + 43 a + 17\right)\cdot 97^{69} + \left(27 a^{2} + 5 a + 96\right)\cdot 97^{70} + \left(77 a^{2} + 8 a + 80\right)\cdot 97^{71} + \left(8 a^{2} + 54 a + 28\right)\cdot 97^{72} + \left(44 a^{2} + 70 a + 48\right)\cdot 97^{73} + \left(a^{2} + 41 a + 91\right)\cdot 97^{74} + \left(87 a^{2} + 55 a + 53\right)\cdot 97^{75} + \left(18 a^{2} + 43 a + 38\right)\cdot 97^{76} + \left(19 a^{2} + 30 a + 66\right)\cdot 97^{77} + \left(42 a^{2} + 9 a + 77\right)\cdot 97^{78} + \left(18 a^{2} + 26 a + 30\right)\cdot 97^{79} + \left(94 a^{2} + 88 a + 23\right)\cdot 97^{80} + \left(21 a^{2} + 76 a + 20\right)\cdot 97^{81} + \left(26 a^{2} + 52 a + 76\right)\cdot 97^{82} + \left(58 a^{2} + 63 a + 39\right)\cdot 97^{83} + \left(96 a^{2} + 65 a + 58\right)\cdot 97^{84} + \left(11 a^{2} + 40 a + 15\right)\cdot 97^{85} + \left(35 a^{2} + 66 a + 90\right)\cdot 97^{86} + \left(8 a^{2} + 45 a + 74\right)\cdot 97^{87} + \left(77 a^{2} + 8 a + 64\right)\cdot 97^{88} + \left(76 a^{2} + 64 a + 11\right)\cdot 97^{89} + \left(76 a^{2} + 83 a + 79\right)\cdot 97^{90} + \left(63 a^{2} + 94 a + 51\right)\cdot 97^{91} + \left(80 a^{2} + 60 a + 64\right)\cdot 97^{92} + \left(93 a^{2} + 77 a + 21\right)\cdot 97^{93} + \left(90 a^{2} + 54 a + 7\right)\cdot 97^{94} + \left(61 a^{2} + 13 a + 91\right)\cdot 97^{95} + \left(68 a^{2} + 50 a + 96\right)\cdot 97^{96} + \left(27 a^{2} + 18 a + 66\right)\cdot 97^{97} + \left(89 a^{2} + 8 a + 28\right)\cdot 97^{98} +O\left(97^{ 99 }\right)$
$r_{ 2 }$ $=$ $ 45 a^{2} + 62 a + 10 + \left(a^{2} + 46 a + 46\right)\cdot 97 + \left(75 a^{2} + a + 18\right)\cdot 97^{2} + \left(66 a^{2} + 55 a + 88\right)\cdot 97^{3} + \left(78 a^{2} + 93 a + 30\right)\cdot 97^{4} + \left(70 a^{2} + 60 a + 34\right)\cdot 97^{5} + \left(72 a^{2} + 49 a + 76\right)\cdot 97^{6} + \left(7 a^{2} + 77 a + 37\right)\cdot 97^{7} + \left(48 a^{2} + 22 a + 25\right)\cdot 97^{8} + \left(90 a^{2} + 26 a + 41\right)\cdot 97^{9} + \left(7 a^{2} + 34 a + 74\right)\cdot 97^{10} + \left(94 a^{2} + 84 a + 57\right)\cdot 97^{11} + \left(50 a^{2} + 34 a + 47\right)\cdot 97^{12} + \left(23 a^{2} + 47 a + 85\right)\cdot 97^{13} + \left(75 a^{2} + 64 a + 48\right)\cdot 97^{14} + \left(84 a^{2} + 58 a + 60\right)\cdot 97^{15} + \left(18 a^{2} + 9 a + 3\right)\cdot 97^{16} + \left(48 a^{2} + 39 a + 96\right)\cdot 97^{17} + \left(67 a^{2} + 30 a + 95\right)\cdot 97^{18} + \left(8 a^{2} + 11 a + 69\right)\cdot 97^{19} + \left(71 a^{2} + 58 a + 23\right)\cdot 97^{20} + \left(16 a^{2} + 3 a + 8\right)\cdot 97^{21} + \left(a^{2} + 87 a + 64\right)\cdot 97^{22} + \left(65 a^{2} + a + 17\right)\cdot 97^{23} + \left(18 a^{2} + 32 a + 50\right)\cdot 97^{24} + \left(58 a^{2} + 16 a + 46\right)\cdot 97^{25} + \left(74 a^{2} + 6 a + 48\right)\cdot 97^{26} + \left(77 a^{2} + 61 a + 45\right)\cdot 97^{27} + \left(51 a^{2} + 12 a + 82\right)\cdot 97^{28} + \left(44 a^{2} + 70 a + 22\right)\cdot 97^{29} + \left(3 a^{2} + 62 a + 40\right)\cdot 97^{30} + \left(32 a^{2} + 6 a + 10\right)\cdot 97^{31} + \left(93 a^{2} + 69 a + 90\right)\cdot 97^{32} + \left(69 a^{2} + 2 a + 29\right)\cdot 97^{33} + \left(56 a^{2} + 80 a + 57\right)\cdot 97^{34} + \left(9 a^{2} + 62 a + 59\right)\cdot 97^{35} + \left(89 a^{2} + 61 a + 29\right)\cdot 97^{36} + \left(79 a^{2} + 67 a + 74\right)\cdot 97^{37} + \left(82 a^{2} + 10 a + 72\right)\cdot 97^{38} + \left(84 a^{2} + 81 a + 46\right)\cdot 97^{39} + \left(18 a^{2} + 80 a + 84\right)\cdot 97^{40} + \left(3 a^{2} + 82 a + 88\right)\cdot 97^{41} + \left(29 a^{2} + 54 a + 80\right)\cdot 97^{42} + \left(47 a^{2} + 22 a + 19\right)\cdot 97^{43} + \left(8 a^{2} + 9 a + 12\right)\cdot 97^{44} + \left(38 a^{2} + 45 a + 17\right)\cdot 97^{45} + \left(45 a^{2} + 22 a + 61\right)\cdot 97^{46} + \left(52 a^{2} + 63 a + 80\right)\cdot 97^{47} + \left(92 a^{2} + 30 a + 59\right)\cdot 97^{48} + \left(82 a^{2} + 26 a + 91\right)\cdot 97^{49} + \left(45 a^{2} + 42 a + 22\right)\cdot 97^{50} + \left(65 a^{2} + 65 a + 20\right)\cdot 97^{51} + \left(42 a^{2} + 75 a + 40\right)\cdot 97^{52} + \left(55 a^{2} + 23 a + 5\right)\cdot 97^{53} + \left(13 a^{2} + 42 a + 73\right)\cdot 97^{54} + \left(96 a^{2} + 94 a + 4\right)\cdot 97^{55} + \left(57 a^{2} + 82 a + 70\right)\cdot 97^{56} + \left(19 a^{2} + 34 a + 4\right)\cdot 97^{57} + \left(93 a^{2} + 29 a + 68\right)\cdot 97^{58} + \left(36 a^{2} + 81 a + 79\right)\cdot 97^{59} + \left(60 a^{2} + 43 a + 14\right)\cdot 97^{60} + \left(23 a^{2} + 39 a\right)\cdot 97^{61} + \left(56 a^{2} + 56 a + 9\right)\cdot 97^{62} + \left(17 a^{2} + 15 a + 92\right)\cdot 97^{63} + \left(67 a^{2} + 93 a + 94\right)\cdot 97^{64} + \left(57 a^{2} + 55 a + 58\right)\cdot 97^{65} + \left(80 a^{2} + 40 a + 89\right)\cdot 97^{66} + \left(8 a^{2} + 85 a + 68\right)\cdot 97^{67} + \left(10 a^{2} + 87 a + 6\right)\cdot 97^{68} + \left(35 a^{2} + 65 a + 29\right)\cdot 97^{69} + \left(67 a^{2} + 56 a + 46\right)\cdot 97^{70} + \left(30 a^{2} + 48 a + 92\right)\cdot 97^{71} + \left(67 a^{2} + 93 a + 88\right)\cdot 97^{72} + \left(13 a^{2} + 91 a + 59\right)\cdot 97^{73} + \left(61 a^{2} + 71 a + 61\right)\cdot 97^{74} + \left(68 a^{2} + 63 a + 40\right)\cdot 97^{75} + \left(40 a^{2} + 57 a + 72\right)\cdot 97^{76} + \left(58 a^{2} + 34 a + 10\right)\cdot 97^{77} + \left(30 a^{2} + 77 a + 8\right)\cdot 97^{78} + \left(76 a^{2} + 45 a + 87\right)\cdot 97^{79} + \left(83 a^{2} + 95 a + 57\right)\cdot 97^{80} + \left(72 a^{2} + 82 a + 34\right)\cdot 97^{81} + \left(16 a^{2} + 33 a + 19\right)\cdot 97^{82} + \left(24 a^{2} + 21 a + 29\right)\cdot 97^{83} + \left(29 a^{2} + 4 a + 42\right)\cdot 97^{84} + \left(77 a^{2} + 26 a + 19\right)\cdot 97^{85} + \left(95 a^{2} + 65 a + 66\right)\cdot 97^{86} + \left(39 a^{2} + 43 a + 70\right)\cdot 97^{87} + \left(6 a^{2} + 54 a + 28\right)\cdot 97^{88} + \left(13 a^{2} + 20 a + 17\right)\cdot 97^{89} + \left(48 a^{2} + 76 a + 4\right)\cdot 97^{90} + \left(13 a^{2} + a + 41\right)\cdot 97^{91} + \left(83 a^{2} + 34 a + 79\right)\cdot 97^{92} + \left(86 a^{2} + 30 a + 76\right)\cdot 97^{93} + \left(63 a^{2} + 24 a + 38\right)\cdot 97^{94} + \left(95 a^{2} + 28 a + 2\right)\cdot 97^{95} + \left(95 a^{2} + 75 a + 67\right)\cdot 97^{96} + \left(86 a^{2} + 55 a + 34\right)\cdot 97^{97} + \left(79 a^{2} + 4 a + 69\right)\cdot 97^{98} +O\left(97^{ 99 }\right)$
$r_{ 3 }$ $=$ $ 6 + 82\cdot 97 + 33\cdot 97^{2} + 64\cdot 97^{3} + 62\cdot 97^{4} + 7\cdot 97^{5} + 13\cdot 97^{6} + 26\cdot 97^{7} + 13\cdot 97^{8} + 50\cdot 97^{9} + 16\cdot 97^{10} + 65\cdot 97^{11} + 95\cdot 97^{12} + 69\cdot 97^{13} + 43\cdot 97^{14} + 83\cdot 97^{15} + 37\cdot 97^{16} + 94\cdot 97^{17} + 53\cdot 97^{18} + 43\cdot 97^{19} + 44\cdot 97^{20} + 82\cdot 97^{21} + 22\cdot 97^{22} + 50\cdot 97^{23} + 88\cdot 97^{24} + 34\cdot 97^{25} + 33\cdot 97^{26} + 2\cdot 97^{27} + 6\cdot 97^{28} + 54\cdot 97^{29} + 38\cdot 97^{30} + 60\cdot 97^{31} + 51\cdot 97^{32} + 5\cdot 97^{33} + 73\cdot 97^{34} + 90\cdot 97^{35} + 59\cdot 97^{36} + 51\cdot 97^{37} + 11\cdot 97^{38} + 29\cdot 97^{39} + 86\cdot 97^{40} + 81\cdot 97^{41} + 85\cdot 97^{42} + 15\cdot 97^{43} + 19\cdot 97^{44} + 52\cdot 97^{45} + 51\cdot 97^{46} + 23\cdot 97^{47} + 31\cdot 97^{48} + 54\cdot 97^{49} + 77\cdot 97^{50} + 50\cdot 97^{51} + 65\cdot 97^{52} + 11\cdot 97^{53} + 25\cdot 97^{54} + 67\cdot 97^{55} + 57\cdot 97^{56} + 47\cdot 97^{57} + 18\cdot 97^{58} + 38\cdot 97^{59} + 72\cdot 97^{60} + 36\cdot 97^{61} + 15\cdot 97^{62} + 40\cdot 97^{63} + 51\cdot 97^{64} + 85\cdot 97^{65} + 17\cdot 97^{66} + 49\cdot 97^{67} + 64\cdot 97^{68} + 59\cdot 97^{69} + 6\cdot 97^{70} + 81\cdot 97^{71} + 71\cdot 97^{72} + 66\cdot 97^{73} + 42\cdot 97^{74} + 46\cdot 97^{75} + 30\cdot 97^{76} + 49\cdot 97^{77} + 41\cdot 97^{78} + 45\cdot 97^{79} + 73\cdot 97^{80} + 43\cdot 97^{81} + 49\cdot 97^{82} + 56\cdot 97^{83} + 11\cdot 97^{84} + 72\cdot 97^{85} + 70\cdot 97^{86} + 22\cdot 97^{87} + 29\cdot 97^{88} + 86\cdot 97^{89} + 77\cdot 97^{90} + 22\cdot 97^{91} + 94\cdot 97^{92} + 71\cdot 97^{93} + 63\cdot 97^{94} + 65\cdot 97^{95} + 71\cdot 97^{96} + 6\cdot 97^{97} + 66\cdot 97^{98} +O\left(97^{ 99 }\right)$
$r_{ 4 }$ $=$ $ 60 a^{2} + 19 a + 94 + \left(46 a^{2} + 49 a + 38\right)\cdot 97 + \left(20 a^{2} + 86 a + 44\right)\cdot 97^{2} + \left(31 a^{2} + 35 a\right)\cdot 97^{3} + \left(39 a^{2} + 9 a + 42\right)\cdot 97^{4} + \left(2 a^{2} + 40 a + 60\right)\cdot 97^{5} + \left(79 a^{2} + 6 a + 94\right)\cdot 97^{6} + \left(62 a^{2} + 31 a + 7\right)\cdot 97^{7} + \left(31 a^{2} + 45 a + 90\right)\cdot 97^{8} + \left(47 a^{2} + 48 a + 5\right)\cdot 97^{9} + \left(a^{2} + 86 a + 9\right)\cdot 97^{10} + \left(31 a^{2} + 69 a + 6\right)\cdot 97^{11} + \left(15 a^{2} + 81 a + 81\right)\cdot 97^{12} + \left(87 a^{2} + 60 a + 61\right)\cdot 97^{13} + \left(75 a^{2} + 10 a + 72\right)\cdot 97^{14} + \left(86 a^{2} + 16 a + 11\right)\cdot 97^{15} + \left(28 a^{2} + 11 a + 26\right)\cdot 97^{16} + \left(13 a^{2} + 2 a + 34\right)\cdot 97^{17} + \left(9 a^{2} + 42 a + 46\right)\cdot 97^{18} + \left(60 a^{2} + 83 a + 91\right)\cdot 97^{19} + \left(12 a^{2} + 72 a + 95\right)\cdot 97^{20} + \left(43 a^{2} + 79 a + 73\right)\cdot 97^{21} + \left(73 a^{2} + 24 a + 25\right)\cdot 97^{22} + \left(69 a^{2} + 6 a + 10\right)\cdot 97^{23} + \left(2 a^{2} + 76 a + 80\right)\cdot 97^{24} + \left(18 a^{2} + 51 a + 51\right)\cdot 97^{25} + \left(34 a^{2} + 28 a + 56\right)\cdot 97^{26} + \left(10 a^{2} + 41 a + 79\right)\cdot 97^{27} + \left(53 a^{2} + 20 a + 60\right)\cdot 97^{28} + \left(17 a^{2} + 17\right)\cdot 97^{29} + \left(8 a^{2} + 72 a + 62\right)\cdot 97^{30} + \left(49 a^{2} + 91 a + 89\right)\cdot 97^{31} + \left(22 a^{2} + 68 a + 65\right)\cdot 97^{32} + \left(65 a^{2} + 46 a + 42\right)\cdot 97^{33} + \left(16 a^{2} + 43 a + 89\right)\cdot 97^{34} + \left(10 a^{2} + 41 a + 43\right)\cdot 97^{35} + \left(33 a^{2} + 26 a + 61\right)\cdot 97^{36} + \left(37 a^{2} + 81 a\right)\cdot 97^{37} + \left(55 a^{2} + 4 a + 96\right)\cdot 97^{38} + \left(10 a^{2} + 35 a + 53\right)\cdot 97^{39} + \left(89 a^{2} + 76 a + 21\right)\cdot 97^{40} + \left(75 a^{2} + 9 a + 73\right)\cdot 97^{41} + \left(95 a^{2} + 41 a + 81\right)\cdot 97^{42} + \left(88 a^{2} + 77 a + 87\right)\cdot 97^{43} + \left(22 a^{2} + 39 a + 66\right)\cdot 97^{44} + \left(84 a^{2} + 2 a + 69\right)\cdot 97^{45} + \left(28 a^{2} + 11 a + 93\right)\cdot 97^{46} + \left(90 a^{2} + 95 a + 24\right)\cdot 97^{47} + \left(35 a^{2} + 50 a + 44\right)\cdot 97^{48} + \left(58 a^{2} + 42 a + 25\right)\cdot 97^{49} + \left(5 a + 83\right)\cdot 97^{50} + \left(39 a^{2} + 89 a + 65\right)\cdot 97^{51} + \left(37 a^{2} + 13 a + 40\right)\cdot 97^{52} + \left(59 a^{2} + 32 a + 20\right)\cdot 97^{53} + \left(25 a^{2} + 43 a + 44\right)\cdot 97^{54} + \left(34 a^{2} + 14 a + 89\right)\cdot 97^{55} + \left(51 a^{2} + 50 a + 79\right)\cdot 97^{56} + \left(95 a^{2} + 26 a + 75\right)\cdot 97^{57} + \left(61 a^{2} + 32 a + 15\right)\cdot 97^{58} + \left(36 a^{2} + 8 a + 5\right)\cdot 97^{59} + \left(87 a^{2} + 32 a + 14\right)\cdot 97^{60} + \left(37 a^{2} + 82 a + 67\right)\cdot 97^{61} + \left(19 a^{2} + 18 a + 88\right)\cdot 97^{62} + \left(64 a^{2} + 27 a + 14\right)\cdot 97^{63} + \left(5 a^{2} + 80 a + 88\right)\cdot 97^{64} + \left(74 a^{2} + 65 a + 93\right)\cdot 97^{65} + \left(13 a^{2} + 14 a + 17\right)\cdot 97^{66} + \left(55 a^{2} + 45 a + 44\right)\cdot 97^{67} + \left(62 a^{2} + 83 a + 7\right)\cdot 97^{68} + \left(67 a^{2} + 48 a + 67\right)\cdot 97^{69} + \left(63 a^{2} + 65 a + 89\right)\cdot 97^{70} + \left(18 a^{2} + 50 a + 57\right)\cdot 97^{71} + \left(44 a^{2} + 47 a + 69\right)\cdot 97^{72} + \left(63 a^{2} + 23 a + 80\right)\cdot 97^{73} + \left(65 a^{2} + 62 a + 20\right)\cdot 97^{74} + \left(80 a^{2} + 67 a + 34\right)\cdot 97^{75} + \left(93 a^{2} + 47 a + 68\right)\cdot 97^{76} + \left(34 a^{2} + 71 a + 16\right)\cdot 97^{77} + \left(91 a^{2} + 73 a + 74\right)\cdot 97^{78} + \left(94 a^{2} + 27 a + 24\right)\cdot 97^{79} + \left(95 a^{2} + 74 a + 94\right)\cdot 97^{80} + \left(69 a^{2} + 33 a + 94\right)\cdot 97^{81} + \left(66 a^{2} + 53 a + 30\right)\cdot 97^{82} + \left(32 a^{2} + 26 a + 50\right)\cdot 97^{83} + \left(21 a^{2} + 65 a + 35\right)\cdot 97^{84} + \left(81 a^{2} + 51 a + 82\right)\cdot 97^{85} + \left(43 a^{2} + 95 a + 22\right)\cdot 97^{86} + \left(31 a^{2} + 44 a + 91\right)\cdot 97^{87} + \left(64 a^{2} + 91 a + 15\right)\cdot 97^{88} + \left(54 a^{2} + 66 a + 77\right)\cdot 97^{89} + \left(64 a^{2} + 50 a + 5\right)\cdot 97^{90} + \left(83 a^{2} + 91 a + 19\right)\cdot 97^{91} + \left(37 a^{2} + 34 a + 80\right)\cdot 97^{92} + \left(79 a^{2} + 34 a + 88\right)\cdot 97^{93} + \left(64 a^{2} + 43 a + 30\right)\cdot 97^{94} + \left(59 a^{2} + 43 a + 48\right)\cdot 97^{95} + \left(63 a^{2} + 93 a + 29\right)\cdot 97^{96} + \left(47 a^{2} + 7 a + 87\right)\cdot 97^{97} + \left(35 a^{2} + 87 a + 81\right)\cdot 97^{98} +O\left(97^{ 99 }\right)$
$r_{ 5 }$ $=$ $ 52 a^{2} + 79 a + 46 + \left(96 a^{2} + 89 a + 47\right)\cdot 97 + \left(13 a^{2} + 2 a + 5\right)\cdot 97^{2} + \left(25 a^{2} + 91 a + 61\right)\cdot 97^{3} + \left(33 a^{2} + 93 a + 5\right)\cdot 97^{4} + \left(80 a^{2} + 14 a + 43\right)\cdot 97^{5} + \left(20 a^{2} + 19 a + 36\right)\cdot 97^{6} + \left(a^{2} + 45 a + 26\right)\cdot 97^{7} + \left(42 a^{2} + 18 a + 55\right)\cdot 97^{8} + \left(57 a^{2} + 64 a + 66\right)\cdot 97^{9} + \left(55 a^{2} + 5 a + 42\right)\cdot 97^{10} + \left(23 a^{2} + 54 a + 58\right)\cdot 97^{11} + \left(84 a^{2} + 76 a + 9\right)\cdot 97^{12} + \left(15 a^{2} + 36 a + 22\right)\cdot 97^{13} + \left(80 a^{2} + 7 a + 1\right)\cdot 97^{14} + \left(3 a^{2} + 96 a + 96\right)\cdot 97^{15} + \left(10 a^{2} + 38 a + 9\right)\cdot 97^{16} + \left(33 a^{2} + 95 a + 56\right)\cdot 97^{17} + \left(20 a^{2} + 51 a + 16\right)\cdot 97^{18} + \left(51 a^{2} + 47 a + 38\right)\cdot 97^{19} + \left(54 a^{2} + 47 a + 56\right)\cdot 97^{20} + \left(48 a^{2} + 52 a + 9\right)\cdot 97^{21} + \left(35 a^{2} + 96 a + 89\right)\cdot 97^{22} + \left(87 a^{2} + 82 a + 18\right)\cdot 97^{23} + \left(24 a^{2} + 51 a + 19\right)\cdot 97^{24} + \left(90 a^{2} + 10 a\right)\cdot 97^{25} + \left(37 a^{2} + 31 a + 79\right)\cdot 97^{26} + \left(13 a^{2} + 76 a\right)\cdot 97^{27} + \left(67 a^{2} + 75 a + 48\right)\cdot 97^{28} + \left(83 a^{2} + 88 a + 26\right)\cdot 97^{29} + \left(51 a^{2} + 12 a + 33\right)\cdot 97^{30} + \left(9 a^{2} + 59 a + 46\right)\cdot 97^{31} + \left(9 a^{2} + 74 a + 82\right)\cdot 97^{32} + \left(60 a^{2} + 88 a + 11\right)\cdot 97^{33} + \left(70 a^{2} + 51 a + 25\right)\cdot 97^{34} + \left(62 a^{2} + 46 a + 68\right)\cdot 97^{35} + \left(18 a^{2} + 61 a + 71\right)\cdot 97^{36} + \left(57 a^{2} + 66 a + 22\right)\cdot 97^{37} + \left(95 a^{2} + 12 a + 46\right)\cdot 97^{38} + \left(8 a^{2} + 9 a + 44\right)\cdot 97^{39} + \left(26 a^{2} + 16 a + 31\right)\cdot 97^{40} + \left(58 a^{2} + 36 a + 64\right)\cdot 97^{41} + \left(78 a^{2} + 61 a + 75\right)\cdot 97^{42} + \left(45 a^{2} + 40 a + 22\right)\cdot 97^{43} + \left(17 a^{2} + 9 a + 34\right)\cdot 97^{44} + \left(84 a^{2} + 69 a + 69\right)\cdot 97^{45} + \left(8 a^{2} + 65 a + 70\right)\cdot 97^{46} + \left(5 a^{2} + 31 a + 95\right)\cdot 97^{47} + \left(30 a^{2} + 34 a + 8\right)\cdot 97^{48} + \left(46 a^{2} + 83 a + 50\right)\cdot 97^{49} + \left(4 a^{2} + 37 a + 9\right)\cdot 97^{50} + \left(35 a^{2} + 49 a + 42\right)\cdot 97^{51} + \left(62 a^{2} + 78 a + 93\right)\cdot 97^{52} + \left(37 a^{2} + 7 a + 83\right)\cdot 97^{53} + \left(29 a^{2} + 72 a + 66\right)\cdot 97^{54} + \left(20 a^{2} + 37 a + 5\right)\cdot 97^{55} + \left(28 a^{2} + 73 a + 38\right)\cdot 97^{56} + \left(a^{2} + 55 a + 92\right)\cdot 97^{57} + \left(69 a^{2} + 46 a + 57\right)\cdot 97^{58} + \left(83 a^{2} + 21 a + 93\right)\cdot 97^{59} + \left(87 a^{2} + 18 a + 16\right)\cdot 97^{60} + \left(60 a^{2} + 64 a + 11\right)\cdot 97^{61} + \left(61 a^{2} + 68 a + 51\right)\cdot 97^{62} + \left(40 a^{2} + 8 a + 67\right)\cdot 97^{63} + \left(67 a^{2} + 68 a + 70\right)\cdot 97^{64} + \left(47 a^{2} + 31 a + 32\right)\cdot 97^{65} + \left(26 a^{2} + 60 a + 94\right)\cdot 97^{66} + \left(18 a^{2} + 93 a + 16\right)\cdot 97^{67} + \left(30 a^{2} + 83 a + 7\right)\cdot 97^{68} + \left(28 a^{2} + 79 a + 25\right)\cdot 97^{69} + \left(86 a^{2} + 56 a + 31\right)\cdot 97^{70} + \left(56 a^{2} + 80 a + 93\right)\cdot 97^{71} + \left(40 a^{2} + 29 a + 47\right)\cdot 97^{72} + \left(7 a^{2} + 64 a + 35\right)\cdot 97^{73} + \left(44 a^{2} + 42 a + 85\right)\cdot 97^{74} + \left(59 a^{2} + 78 a + 3\right)\cdot 97^{75} + \left(68 a^{2} + 23 a + 14\right)\cdot 97^{76} + \left(95 a^{2} + 6 a + 90\right)\cdot 97^{77} + \left(80 a^{2} + 46 a + 11\right)\cdot 97^{78} + \left(15 a^{2} + 69 a + 35\right)\cdot 97^{79} + \left(43 a^{2} + 88 a + 68\right)\cdot 97^{80} + \left(54 a^{2} + 27 a + 1\right)\cdot 97^{81} + \left(71 a^{2} + 54 a + 60\right)\cdot 97^{82} + \left(49 a^{2} + 55\right)\cdot 97^{83} + \left(43 a^{2} + 40 a + 71\right)\cdot 97^{84} + \left(74 a^{2} + 28 a + 41\right)\cdot 97^{85} + \left(83 a^{2} + 70 a + 68\right)\cdot 97^{86} + \left(59 a^{2} + 11 a + 67\right)\cdot 97^{87} + \left(64 a^{2} + 60 a + 17\right)\cdot 97^{88} + \left(45 a^{2} + 74 a + 23\right)\cdot 97^{89} + \left(57 a^{2} + 50 a + 60\right)\cdot 97^{90} + \left(21 a^{2} + 92 a + 34\right)\cdot 97^{91} + \left(7 a^{2} + 13 a + 90\right)\cdot 97^{92} + \left(17 a^{2} + 61 a + 5\right)\cdot 97^{93} + \left(23 a^{2} + 79 a + 72\right)\cdot 97^{94} + \left(a^{2} + 90 a + 85\right)\cdot 97^{95} + \left(46 a^{2} + 16 a + 20\right)\cdot 97^{96} + \left(83 a^{2} + 82 a + 11\right)\cdot 97^{97} + \left(17 a^{2} + 91 a + 73\right)\cdot 97^{98} +O\left(97^{ 99 }\right)$
$r_{ 6 }$ $=$ $ 82 a^{2} + 96 a + 32 + \left(50 a^{2} + 54 a + 64\right)\cdot 97 + \left(62 a^{2} + 7 a + 5\right)\cdot 97^{2} + \left(40 a^{2} + 67 a + 57\right)\cdot 97^{3} + \left(24 a^{2} + 90 a + 49\right)\cdot 97^{4} + \left(14 a^{2} + 41 a + 34\right)\cdot 97^{5} + \left(94 a^{2} + 71 a + 88\right)\cdot 97^{6} + \left(32 a^{2} + 20 a + 22\right)\cdot 97^{7} + \left(23 a^{2} + 33 a + 40\right)\cdot 97^{8} + \left(89 a^{2} + 81 a + 63\right)\cdot 97^{9} + \left(39 a^{2} + 4 a + 45\right)\cdot 97^{10} + \left(42 a^{2} + 70 a + 74\right)\cdot 97^{11} + \left(94 a^{2} + 35 a + 70\right)\cdot 97^{12} + \left(90 a^{2} + 96 a + 84\right)\cdot 97^{13} + \left(37 a^{2} + 78 a + 38\right)\cdot 97^{14} + \left(6 a^{2} + 81 a + 14\right)\cdot 97^{15} + \left(58 a^{2} + 46 a + 7\right)\cdot 97^{16} + \left(50 a^{2} + 96 a + 64\right)\cdot 97^{17} + \left(67 a^{2} + 2 a + 8\right)\cdot 97^{18} + \left(82 a^{2} + 63 a + 33\right)\cdot 97^{19} + \left(29 a^{2} + 73 a + 5\right)\cdot 97^{20} + \left(5 a^{2} + 61 a + 41\right)\cdot 97^{21} + \left(85 a^{2} + 72 a + 95\right)\cdot 97^{22} + \left(36 a^{2} + 7 a + 6\right)\cdot 97^{23} + \left(69 a^{2} + 66 a + 92\right)\cdot 97^{24} + \left(85 a^{2} + 34 a + 69\right)\cdot 97^{25} + \left(24 a^{2} + 37 a\right)\cdot 97^{26} + \left(73 a^{2} + 76 a + 69\right)\cdot 97^{27} + \left(73 a^{2} + 87\right)\cdot 97^{28} + \left(92 a^{2} + 8 a + 80\right)\cdot 97^{29} + \left(36 a^{2} + 12 a + 40\right)\cdot 97^{30} + \left(38 a^{2} + 43 a + 25\right)\cdot 97^{31} + \left(65 a^{2} + 50 a + 32\right)\cdot 97^{32} + \left(68 a^{2} + 58 a + 63\right)\cdot 97^{33} + \left(9 a^{2} + a + 47\right)\cdot 97^{34} + \left(24 a^{2} + 9 a + 30\right)\cdot 97^{35} + \left(45 a^{2} + 9 a + 37\right)\cdot 97^{36} + \left(2 a^{2} + 46 a + 82\right)\cdot 97^{37} + \left(43 a^{2} + 79 a + 21\right)\cdot 97^{38} + \left(77 a^{2} + 52 a + 67\right)\cdot 97^{39} + \left(78 a^{2} + 4 a + 56\right)\cdot 97^{40} + \left(59 a^{2} + 51 a + 73\right)\cdot 97^{41} + \left(19 a^{2} + 91 a + 12\right)\cdot 97^{42} + \left(59 a^{2} + 75 a + 6\right)\cdot 97^{43} + \left(56 a^{2} + 47 a + 75\right)\cdot 97^{44} + \left(25 a^{2} + 25 a + 8\right)\cdot 97^{45} + \left(59 a^{2} + 20 a + 82\right)\cdot 97^{46} + \left(a^{2} + 67 a + 74\right)\cdot 97^{47} + \left(31 a^{2} + 11 a + 14\right)\cdot 97^{48} + \left(89 a^{2} + 68 a + 17\right)\cdot 97^{49} + \left(91 a^{2} + 53 a + 49\right)\cdot 97^{50} + \left(22 a^{2} + 55 a + 66\right)\cdot 97^{51} + \left(94 a^{2} + 4 a + 90\right)\cdot 97^{52} + \left(96 a^{2} + 57 a + 51\right)\cdot 97^{53} + \left(41 a^{2} + 78 a + 45\right)\cdot 97^{54} + \left(42 a^{2} + 44 a + 41\right)\cdot 97^{55} + \left(17 a^{2} + 70 a + 70\right)\cdot 97^{56} + \left(14 a + 85\right)\cdot 97^{57} + \left(63 a^{2} + 18 a + 21\right)\cdot 97^{58} + \left(73 a^{2} + 67 a + 33\right)\cdot 97^{59} + \left(18 a^{2} + 46 a + 87\right)\cdot 97^{60} + \left(95 a^{2} + 47 a + 22\right)\cdot 97^{61} + \left(15 a^{2} + 9 a + 68\right)\cdot 97^{62} + \left(89 a^{2} + 61 a + 67\right)\cdot 97^{63} + \left(23 a^{2} + 45 a + 3\right)\cdot 97^{64} + \left(72 a^{2} + 96 a + 83\right)\cdot 97^{65} + \left(56 a^{2} + 21 a + 81\right)\cdot 97^{66} + \left(23 a^{2} + 55 a + 48\right)\cdot 97^{67} + \left(4 a^{2} + 26 a + 45\right)\cdot 97^{68} + \left(a^{2} + 65 a + 55\right)\cdot 97^{69} + \left(44 a^{2} + 71 a + 68\right)\cdot 97^{70} + \left(21 a^{2} + 62 a + 74\right)\cdot 97^{71} + \left(12 a^{2} + 19 a + 71\right)\cdot 97^{72} + \left(26 a^{2} + 9 a + 50\right)\cdot 97^{73} + \left(84 a^{2} + 89 a + 35\right)\cdot 97^{74} + \left(53 a^{2} + 47 a + 67\right)\cdot 97^{75} + \left(31 a^{2} + 25 a + 82\right)\cdot 97^{76} + \left(63 a^{2} + 19 a + 89\right)\cdot 97^{77} + \left(21 a^{2} + 74 a + 43\right)\cdot 97^{78} + \left(83 a^{2} + 96 a + 51\right)\cdot 97^{79} + \left(54 a^{2} + 30 a + 41\right)\cdot 97^{80} + \left(69 a^{2} + 35 a + 92\right)\cdot 97^{81} + \left(55 a^{2} + 86 a + 61\right)\cdot 97^{82} + \left(14 a^{2} + 69 a + 38\right)\cdot 97^{83} + \left(32 a^{2} + 88 a + 3\right)\cdot 97^{84} + \left(38 a^{2} + 16 a + 19\right)\cdot 97^{85} + \left(66 a^{2} + 28 a + 61\right)\cdot 97^{86} + \left(5 a^{2} + 40 a + 33\right)\cdot 97^{87} + \left(65 a^{2} + 42 a + 20\right)\cdot 97^{88} + \left(93 a^{2} + 52 a + 20\right)\cdot 97^{89} + \left(71 a^{2} + 92 a + 50\right)\cdot 97^{90} + \left(88 a^{2} + 9 a + 49\right)\cdot 97^{91} + \left(51 a^{2} + 48 a + 67\right)\cdot 97^{92} + \left(a + 3\right)\cdot 97^{93} + \left(9 a^{2} + 71 a + 84\right)\cdot 97^{94} + \left(36 a^{2} + 59 a + 3\right)\cdot 97^{95} + \left(84 a^{2} + 83 a + 57\right)\cdot 97^{96} + \left(62 a^{2} + 6 a + 81\right)\cdot 97^{97} + \left(43 a^{2} + 15 a + 33\right)\cdot 97^{98} +O\left(97^{ 99 }\right)$
$r_{ 7 }$ $=$ $ 24 + 43\cdot 97 + 41\cdot 97^{2} + 75\cdot 97^{3} + 96\cdot 97^{4} + 55\cdot 97^{5} + 71\cdot 97^{6} + 39\cdot 97^{7} + 8\cdot 97^{8} + 58\cdot 97^{9} + 96\cdot 97^{10} + 54\cdot 97^{11} + 32\cdot 97^{12} + 25\cdot 97^{13} + 81\cdot 97^{14} + 71\cdot 97^{15} + 53\cdot 97^{16} + 39\cdot 97^{17} + 25\cdot 97^{18} + 31\cdot 97^{19} + 36\cdot 97^{20} + 69\cdot 97^{21} + 80\cdot 97^{22} + 60\cdot 97^{23} + 2\cdot 97^{24} + 72\cdot 97^{25} + 57\cdot 97^{26} + 44\cdot 97^{27} + 94\cdot 97^{28} + 68\cdot 97^{29} + 57\cdot 97^{30} + 32\cdot 97^{31} + 13\cdot 97^{32} + 76\cdot 97^{33} + 31\cdot 97^{34} + 51\cdot 97^{35} + 23\cdot 97^{36} + 88\cdot 97^{37} + 29\cdot 97^{38} + 28\cdot 97^{39} + 84\cdot 97^{40} + 79\cdot 97^{41} + 23\cdot 97^{42} + 77\cdot 97^{43} + 17\cdot 97^{44} + 46\cdot 97^{45} + 44\cdot 97^{46} + 95\cdot 97^{47} + 28\cdot 97^{48} + 4\cdot 97^{49} + 52\cdot 97^{50} + 19\cdot 97^{51} + 66\cdot 97^{52} + 37\cdot 97^{53} + 37\cdot 97^{54} + 57\cdot 97^{55} + 5\cdot 97^{56} + 37\cdot 97^{57} + 97^{58} + 62\cdot 97^{59} + 75\cdot 97^{60} + 92\cdot 97^{61} + 82\cdot 97^{62} + 43\cdot 97^{63} + 31\cdot 97^{64} + 81\cdot 97^{65} + 96\cdot 97^{66} + 83\cdot 97^{67} + 36\cdot 97^{68} + 46\cdot 97^{69} + 4\cdot 97^{70} + 65\cdot 97^{71} + 4\cdot 97^{72} + 27\cdot 97^{73} + 52\cdot 97^{74} + 88\cdot 97^{75} + 28\cdot 97^{76} + 94\cdot 97^{77} + 63\cdot 97^{78} + 82\cdot 97^{79} + 86\cdot 97^{80} + 4\cdot 97^{81} + 41\cdot 97^{82} + 49\cdot 97^{83} + 83\cdot 97^{84} + 50\cdot 97^{85} + 41\cdot 97^{86} + 97^{87} + 43\cdot 97^{88} + 73\cdot 97^{89} + 77\cdot 97^{90} + 90\cdot 97^{91} + 52\cdot 97^{92} + 95\cdot 97^{93} + 6\cdot 97^{94} + 56\cdot 97^{95} + 86\cdot 97^{96} + 13\cdot 97^{97} + 5\cdot 97^{98} +O\left(97^{ 99 }\right)$
$r_{ 8 }$ $=$ $ 47 a^{2} + 32 a + 22 + \left(21 a^{2} + 46 a + 69\right)\cdot 97 + \left(59 a^{2} + 73 a + 20\right)\cdot 97^{2} + \left(43 a^{2} + 81 a + 46\right)\cdot 97^{3} + \left(16 a^{2} + 57 a + 45\right)\cdot 97^{4} + \left(51 a^{2} + 45 a + 13\right)\cdot 97^{5} + \left(88 a^{2} + 2 a + 74\right)\cdot 97^{6} + \left(18 a^{2} + 6 a + 7\right)\cdot 97^{7} + \left(48 a^{2} + 28 a + 26\right)\cdot 97^{8} + \left(45 a^{2} + 24 a + 62\right)\cdot 97^{9} + \left(29 a^{2} + 27 a + 9\right)\cdot 97^{10} + \left(73 a^{2} + 43 a + 30\right)\cdot 97^{11} + \left(76 a^{2} + 70 a + 8\right)\cdot 97^{12} + \left(8 a^{2} + 34 a + 94\right)\cdot 97^{13} + \left(11 a^{2} + 9 a + 51\right)\cdot 97^{14} + \left(52 a^{2} + 63 a + 58\right)\cdot 97^{15} + \left(30 a^{2} + 48 a + 73\right)\cdot 97^{16} + \left(28 a^{2} + 94 a + 73\right)\cdot 97^{17} + \left(54 a^{2} + 77 a + 16\right)\cdot 97^{18} + \left(79 a^{2} + 73 a + 10\right)\cdot 97^{19} + \left(33 a^{2} + 5 a + 91\right)\cdot 97^{20} + \left(92 a^{2} + 52 a + 73\right)\cdot 97^{21} + \left(60 a^{2} + 20 a + 34\right)\cdot 97^{22} + \left(23 a^{2} + 37 a + 60\right)\cdot 97^{23} + \left(68 a^{2} + 50 a + 56\right)\cdot 97^{24} + \left(49 a^{2} + 13 a + 92\right)\cdot 97^{25} + \left(34 a^{2} + 42 a + 1\right)\cdot 97^{26} + \left(91 a^{2} + 50 a + 30\right)\cdot 97^{27} + \left(60 a^{2} + 58 a + 40\right)\cdot 97^{28} + \left(37 a^{2} + 92 a + 78\right)\cdot 97^{29} + \left(63 a^{2} + 83 a + 11\right)\cdot 97^{30} + \left(8 a^{2} + 75 a + 64\right)\cdot 97^{31} + \left(70 a^{2} + 5 a + 47\right)\cdot 97^{32} + \left(63 a^{2} + 73 a + 89\right)\cdot 97^{33} + \left(22 a^{2} + 87 a + 46\right)\cdot 97^{34} + \left(96 a^{2} + 50 a + 94\right)\cdot 97^{35} + \left(93 a^{2} + 40 a + 58\right)\cdot 97^{36} + \left(62 a^{2} + 90 a + 69\right)\cdot 97^{37} + \left(5 a^{2} + 96 a + 94\right)\cdot 97^{38} + \left(50 a^{2} + 31\right)\cdot 97^{39} + \left(57 a^{2} + 45 a + 25\right)\cdot 97^{40} + \left(45 a^{2} + 56 a + 52\right)\cdot 97^{41} + \left(75 a^{2} + 86 a + 68\right)\cdot 97^{42} + \left(33 a^{2} + 49 a + 35\right)\cdot 97^{43} + \left(92 a^{2} + 66 a + 30\right)\cdot 97^{44} + \left(4 a^{2} + 42 a + 12\right)\cdot 97^{45} + \left(14 a^{2} + 52 a + 67\right)\cdot 97^{46} + \left(38 a^{2} + 41 a + 91\right)\cdot 97^{47} + \left(78 a^{2} + 65 a + 71\right)\cdot 97^{48} + \left(23 a^{2} + 27 a + 27\right)\cdot 97^{49} + \left(17 a^{2} + 26 a + 45\right)\cdot 97^{50} + \left(38 a^{2} + 13 a + 50\right)\cdot 97^{51} + \left(35 a^{2} + 83 a + 93\right)\cdot 97^{52} + \left(60 a^{2} + 49 a + 34\right)\cdot 97^{53} + \left(52 a^{2} + 78 a + 16\right)\cdot 97^{54} + \left(11 a^{2} + 79\right)\cdot 97^{55} + \left(14 a^{2} + 59 a\right)\cdot 97^{56} + \left(89 a^{2} + 41 a + 34\right)\cdot 97^{57} + \left(52 a^{2} + 23 a + 20\right)\cdot 97^{58} + \left(34 a^{2} + 48 a + 65\right)\cdot 97^{59} + \left(96 a^{2} + 94 a + 36\right)\cdot 97^{60} + \left(7 a^{2} + 4 a + 3\right)\cdot 97^{61} + \left(53 a^{2} + 29 a + 87\right)\cdot 97^{62} + \left(66 a^{2} + 93 a + 94\right)\cdot 97^{63} + \left(27 a^{2} + 48 a + 51\right)\cdot 97^{64} + \left(62 a^{2} + 63 a + 86\right)\cdot 97^{65} + \left(28 a^{2} + 84 a + 68\right)\cdot 97^{66} + \left(12 a^{2} + 60 a + 89\right)\cdot 97^{67} + \left(83 a^{2} + 49 a + 56\right)\cdot 97^{68} + \left(44 a^{2} + 84 a + 87\right)\cdot 97^{69} + \left(2 a^{2} + 34 a + 44\right)\cdot 97^{70} + \left(86 a^{2} + 40 a + 36\right)\cdot 97^{71} + \left(20 a^{2} + 46 a + 4\right)\cdot 97^{72} + \left(39 a^{2} + 31 a + 19\right)\cdot 97^{73} + \left(34 a^{2} + 80 a + 95\right)\cdot 97^{74} + \left(38 a^{2} + 74 a + 52\right)\cdot 97^{75} + \left(37 a^{2} + 92 a + 52\right)\cdot 97^{76} + \left(19 a^{2} + 31 a + 67\right)\cdot 97^{77} + \left(24 a^{2} + 10 a + 66\right)\cdot 97^{78} + \left(2 a^{2} + 25 a + 30\right)\cdot 97^{79} + \left(16 a^{2} + 10 a + 39\right)\cdot 97^{80} + \left(2 a^{2} + 34 a + 95\right)\cdot 97^{81} + \left(54 a^{2} + 10 a + 48\right)\cdot 97^{82} + \left(14 a^{2} + 12 a + 68\right)\cdot 97^{83} + \left(68 a^{2} + 27 a + 81\right)\cdot 97^{84} + \left(7 a^{2} + 30 a + 86\right)\cdot 97^{85} + \left(63 a^{2} + 62 a + 63\right)\cdot 97^{86} + \left(48 a^{2} + 7 a + 25\right)\cdot 97^{87} + \left(13 a^{2} + 34 a + 71\right)\cdot 97^{88} + \left(7 a^{2} + 12 a + 78\right)\cdot 97^{89} + \left(69 a^{2} + 34 a + 32\right)\cdot 97^{90} + \left(19 a^{2} + 78\right)\cdot 97^{91} + \left(30 a^{2} + 2 a + 52\right)\cdot 97^{92} + \left(13 a^{2} + 86 a + 23\right)\cdot 97^{93} + \left(39 a^{2} + 17 a + 84\right)\cdot 97^{94} + \left(36 a^{2} + 55 a + 34\right)\cdot 97^{95} + \left(29 a^{2} + 68 a + 55\right)\cdot 97^{96} + \left(79 a^{2} + 22 a + 85\right)\cdot 97^{97} + \left(24 a^{2} + 84 a + 29\right)\cdot 97^{98} +O\left(97^{ 99 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(1,2,3,8)$
$(1,2)$
$(1,4)(2,5)(3,6)(7,8)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$9$
$6$$2$$(4,6)(5,7)$$-3$
$9$$2$$(1,3)(2,8)(4,6)(5,7)$$1$
$12$$2$$(1,2)$$3$
$24$$2$$(1,4)(2,5)(3,6)(7,8)$$3$
$36$$2$$(1,2)(4,5)$$1$
$36$$2$$(1,2)(4,6)(5,7)$$-1$
$16$$3$$(1,3,8)$$0$
$64$$3$$(1,3,8)(5,6,7)$$0$
$12$$4$$(4,5,6,7)$$-3$
$36$$4$$(1,2,3,8)(4,5,6,7)$$1$
$36$$4$$(1,2,3,8)(4,6)(5,7)$$1$
$72$$4$$(1,4,3,6)(2,5,8,7)$$-1$
$72$$4$$(1,2)(4,5,6,7)$$-1$
$144$$4$$(1,5,2,4)(3,6)(7,8)$$1$
$48$$6$$(1,8,3)(4,6)(5,7)$$0$
$96$$6$$(1,2)(5,7,6)$$0$
$192$$6$$(1,5,3,6,8,7)(2,4)$$0$
$144$$8$$(1,4,2,5,3,6,8,7)$$-1$
$96$$12$$(1,3,8)(4,5,6,7)$$0$
The blue line marks the conjugacy class containing complex conjugation.