# Properties

 Label 8.40575253489.12t213.a.a Dimension 8 Group $S_3\wr S_3$ Conductor $17^{6} \cdot 41^{2}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $8$ Group: $S_3\wr S_3$ Conductor: $40575253489= 17^{6} \cdot 41^{2}$ Artin number field: Splitting field of 9.5.5756350841.1 defined by $f= x^{9} - 2 x^{8} - 4 x^{7} + 11 x^{6} + 2 x^{5} - 19 x^{4} + 7 x^{3} + 11 x^{2} - 5 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 12T213 Parity: Even Determinant: 1.1.1t1.a.a Projective image: $S_3\wr S_3$ Projective field: Galois closure of 9.5.5756350841.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 127 }$ to precision 100.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$: $x^{3} + 3 x + 124$
Roots:
 $r_{ 1 }$ $=$ $22 + 45\cdot 127 + 97\cdot 127^{2} + 39\cdot 127^{3} + 18\cdot 127^{4} + 24\cdot 127^{5} + 25\cdot 127^{6} + 19\cdot 127^{7} + 64\cdot 127^{8} + 99\cdot 127^{9} + 108\cdot 127^{10} + 101\cdot 127^{11} + 98\cdot 127^{12} + 49\cdot 127^{13} + 71\cdot 127^{14} + 95\cdot 127^{15} + 68\cdot 127^{16} + 77\cdot 127^{17} + 102\cdot 127^{18} + 71\cdot 127^{19} + 86\cdot 127^{20} + 118\cdot 127^{21} + 27\cdot 127^{22} + 97\cdot 127^{23} + 42\cdot 127^{24} + 34\cdot 127^{25} + 17\cdot 127^{26} + 51\cdot 127^{27} + 10\cdot 127^{28} + 39\cdot 127^{29} + 110\cdot 127^{30} + 63\cdot 127^{31} + 39\cdot 127^{32} + 24\cdot 127^{33} + 29\cdot 127^{34} + 118\cdot 127^{35} + 33\cdot 127^{36} + 107\cdot 127^{37} + 2\cdot 127^{38} + 123\cdot 127^{39} + 67\cdot 127^{40} + 14\cdot 127^{41} + 49\cdot 127^{42} + 17\cdot 127^{43} + 75\cdot 127^{44} + 26\cdot 127^{45} + 119\cdot 127^{46} + 14\cdot 127^{47} + 20\cdot 127^{48} + 4\cdot 127^{49} + 54\cdot 127^{50} + 6\cdot 127^{51} + 68\cdot 127^{52} + 56\cdot 127^{53} + 69\cdot 127^{54} + 125\cdot 127^{55} + 108\cdot 127^{56} + 105\cdot 127^{57} + 83\cdot 127^{58} + 14\cdot 127^{59} + 51\cdot 127^{60} + 32\cdot 127^{61} + 33\cdot 127^{62} + 17\cdot 127^{63} + 118\cdot 127^{64} + 30\cdot 127^{65} + 112\cdot 127^{66} + 68\cdot 127^{67} + 86\cdot 127^{68} + 36\cdot 127^{69} + 104\cdot 127^{70} + 126\cdot 127^{71} + 94\cdot 127^{72} + 45\cdot 127^{73} + 55\cdot 127^{74} + 54\cdot 127^{75} + 26\cdot 127^{76} + 116\cdot 127^{77} + 114\cdot 127^{78} + 32\cdot 127^{79} + 69\cdot 127^{80} + 3\cdot 127^{81} + 112\cdot 127^{82} + 89\cdot 127^{83} + 22\cdot 127^{84} + 72\cdot 127^{85} + 59\cdot 127^{86} + 80\cdot 127^{87} + 59\cdot 127^{88} + 35\cdot 127^{89} + 27\cdot 127^{90} + 12\cdot 127^{91} + 38\cdot 127^{92} + 81\cdot 127^{93} + 86\cdot 127^{94} + 58\cdot 127^{95} + 92\cdot 127^{96} + 11\cdot 127^{97} + 111\cdot 127^{98} + 40\cdot 127^{99} +O\left(127^{ 100 }\right)$ $r_{ 2 }$ $=$ $66 + 41\cdot 127 + 91\cdot 127^{2} + 124\cdot 127^{3} + 27\cdot 127^{4} + 2\cdot 127^{5} + 81\cdot 127^{6} + 86\cdot 127^{7} + 96\cdot 127^{8} + 17\cdot 127^{9} + 81\cdot 127^{10} + 56\cdot 127^{11} + 19\cdot 127^{12} + 55\cdot 127^{13} + 79\cdot 127^{14} + 86\cdot 127^{15} + 94\cdot 127^{16} + 39\cdot 127^{17} + 61\cdot 127^{18} + 105\cdot 127^{19} + 112\cdot 127^{20} + 123\cdot 127^{21} + 114\cdot 127^{22} + 82\cdot 127^{23} + 127^{24} + 86\cdot 127^{25} + 10\cdot 127^{26} + 63\cdot 127^{27} + 92\cdot 127^{28} + 9\cdot 127^{29} + 17\cdot 127^{30} + 30\cdot 127^{31} + 127^{32} + 99\cdot 127^{33} + 18\cdot 127^{34} + 74\cdot 127^{35} + 52\cdot 127^{36} + 7\cdot 127^{37} + 26\cdot 127^{38} + 24\cdot 127^{39} + 62\cdot 127^{40} + 50\cdot 127^{41} + 121\cdot 127^{42} + 94\cdot 127^{43} + 15\cdot 127^{44} + 13\cdot 127^{45} + 7\cdot 127^{46} + 102\cdot 127^{47} + 90\cdot 127^{48} + 92\cdot 127^{49} + 96\cdot 127^{50} + 71\cdot 127^{51} + 91\cdot 127^{52} + 10\cdot 127^{53} + 47\cdot 127^{54} + 36\cdot 127^{55} + 48\cdot 127^{56} + 87\cdot 127^{57} + 105\cdot 127^{58} + 87\cdot 127^{59} + 94\cdot 127^{60} + 84\cdot 127^{61} + 98\cdot 127^{62} + 58\cdot 127^{63} + 15\cdot 127^{64} + 15\cdot 127^{65} + 63\cdot 127^{66} + 110\cdot 127^{67} + 14\cdot 127^{68} + 87\cdot 127^{69} + 13\cdot 127^{70} + 84\cdot 127^{71} + 57\cdot 127^{72} + 49\cdot 127^{73} + 75\cdot 127^{74} + 112\cdot 127^{75} + 89\cdot 127^{76} + 68\cdot 127^{77} + 32\cdot 127^{78} + 119\cdot 127^{79} + 23\cdot 127^{80} + 41\cdot 127^{81} + 57\cdot 127^{82} + 9\cdot 127^{83} + 64\cdot 127^{84} + 117\cdot 127^{85} + 90\cdot 127^{86} + 62\cdot 127^{87} + 83\cdot 127^{88} + 86\cdot 127^{89} + 114\cdot 127^{90} + 48\cdot 127^{91} + 92\cdot 127^{92} + 126\cdot 127^{93} + 113\cdot 127^{94} + 101\cdot 127^{95} + 97\cdot 127^{96} + 122\cdot 127^{97} + 92\cdot 127^{98} + 101\cdot 127^{99} +O\left(127^{ 100 }\right)$ $r_{ 3 }$ $=$ $110 + 19\cdot 127 + 124\cdot 127^{2} + 70\cdot 127^{3} + 51\cdot 127^{4} + 31\cdot 127^{6} + 64\cdot 127^{7} + 121\cdot 127^{8} + 87\cdot 127^{9} + 80\cdot 127^{10} + 123\cdot 127^{11} + 86\cdot 127^{12} + 86\cdot 127^{13} + 93\cdot 127^{14} + 58\cdot 127^{15} + 7\cdot 127^{16} + 62\cdot 127^{17} + 58\cdot 127^{18} + 109\cdot 127^{19} + 83\cdot 127^{20} + 93\cdot 127^{21} + 88\cdot 127^{22} + 63\cdot 127^{23} + 32\cdot 127^{24} + 4\cdot 127^{25} + 18\cdot 127^{26} + 59\cdot 127^{27} + 64\cdot 127^{28} + 6\cdot 127^{29} + 85\cdot 127^{30} + 122\cdot 127^{31} + 51\cdot 127^{32} + 21\cdot 127^{33} + 76\cdot 127^{34} + 95\cdot 127^{35} + 4\cdot 127^{36} + 37\cdot 127^{37} + 77\cdot 127^{38} + 108\cdot 127^{39} + 12\cdot 127^{40} + 59\cdot 127^{41} + 105\cdot 127^{42} + 73\cdot 127^{43} + 84\cdot 127^{44} + 69\cdot 127^{45} + 67\cdot 127^{46} + 3\cdot 127^{47} + 57\cdot 127^{48} + 24\cdot 127^{49} + 83\cdot 127^{50} + 85\cdot 127^{51} + 32\cdot 127^{52} + 87\cdot 127^{53} + 105\cdot 127^{54} + 19\cdot 127^{55} + 78\cdot 127^{56} + 53\cdot 127^{57} + 112\cdot 127^{58} + 123\cdot 127^{59} + 118\cdot 127^{60} + 21\cdot 127^{61} + 24\cdot 127^{62} + 43\cdot 127^{63} + 63\cdot 127^{64} + 22\cdot 127^{65} + 6\cdot 127^{66} + 106\cdot 127^{67} + 35\cdot 127^{68} + 12\cdot 127^{69} + 47\cdot 127^{70} + 85\cdot 127^{71} + 53\cdot 127^{72} + 86\cdot 127^{73} + 84\cdot 127^{74} + 76\cdot 127^{75} + 69\cdot 127^{76} + 72\cdot 127^{77} + 26\cdot 127^{78} + 74\cdot 127^{79} + 104\cdot 127^{80} + 62\cdot 127^{81} + 12\cdot 127^{82} + 101\cdot 127^{83} + 100\cdot 127^{84} + 6\cdot 127^{85} + 79\cdot 127^{86} + 93\cdot 127^{87} + 124\cdot 127^{88} + 35\cdot 127^{89} + 6\cdot 127^{90} + 45\cdot 127^{91} + 37\cdot 127^{92} + 53\cdot 127^{93} + 4\cdot 127^{94} + 126\cdot 127^{95} + 61\cdot 127^{96} + 52\cdot 127^{97} + 79\cdot 127^{98} + 94\cdot 127^{99} +O\left(127^{ 100 }\right)$ $r_{ 4 }$ $=$ $79 a + 83 + \left(14 a^{2} + 125 a + 74\right)\cdot 127 + \left(101 a^{2} + 19 a + 124\right)\cdot 127^{2} + \left(15 a^{2} + 39 a + 33\right)\cdot 127^{3} + \left(37 a^{2} + 94 a + 10\right)\cdot 127^{4} + \left(105 a^{2} + 94 a + 106\right)\cdot 127^{5} + \left(87 a^{2} + 64 a + 60\right)\cdot 127^{6} + \left(105 a^{2} + 4 a + 62\right)\cdot 127^{7} + \left(41 a^{2} + 20 a + 24\right)\cdot 127^{8} + \left(97 a^{2} + 84 a + 59\right)\cdot 127^{9} + \left(104 a^{2} + 93 a + 50\right)\cdot 127^{10} + \left(12 a^{2} + 100 a + 98\right)\cdot 127^{11} + \left(101 a^{2} + 55 a + 102\right)\cdot 127^{12} + \left(47 a^{2} + 5 a + 23\right)\cdot 127^{13} + \left(73 a^{2} + 87 a + 100\right)\cdot 127^{14} + \left(32 a^{2} + 73 a + 117\right)\cdot 127^{15} + \left(50 a^{2} + 104 a + 106\right)\cdot 127^{16} + \left(98 a^{2} + 105 a + 99\right)\cdot 127^{17} + \left(9 a^{2} + 72 a + 86\right)\cdot 127^{18} + \left(45 a^{2} + 108 a + 93\right)\cdot 127^{19} + \left(112 a^{2} + 106 a + 99\right)\cdot 127^{20} + \left(23 a^{2} + 97 a + 121\right)\cdot 127^{21} + \left(76 a^{2} + 30 a + 49\right)\cdot 127^{22} + \left(40 a^{2} + 20 a + 115\right)\cdot 127^{23} + \left(92 a^{2} + 82 a + 64\right)\cdot 127^{24} + \left(97 a^{2} + 9 a + 2\right)\cdot 127^{25} + \left(104 a^{2} + 50 a + 69\right)\cdot 127^{26} + \left(121 a^{2} + 83 a + 101\right)\cdot 127^{27} + \left(106 a^{2} + 80 a + 81\right)\cdot 127^{28} + \left(65 a^{2} + 108 a + 91\right)\cdot 127^{29} + \left(67 a^{2} + 69 a + 39\right)\cdot 127^{30} + \left(110 a^{2} + 73 a + 46\right)\cdot 127^{31} + \left(34 a^{2} + 107 a + 17\right)\cdot 127^{32} + \left(9 a^{2} + 2 a + 87\right)\cdot 127^{33} + \left(5 a^{2} + 5 a + 2\right)\cdot 127^{34} + \left(105 a^{2} + 13 a + 93\right)\cdot 127^{35} + \left(52 a^{2} + 123 a + 68\right)\cdot 127^{36} + \left(35 a^{2} + 45 a + 15\right)\cdot 127^{37} + \left(44 a^{2} + 97 a + 2\right)\cdot 127^{38} + \left(104 a^{2} + 106 a + 34\right)\cdot 127^{39} + \left(82 a^{2} + 20 a + 40\right)\cdot 127^{40} + \left(48 a^{2} + 13 a + 64\right)\cdot 127^{41} + \left(119 a^{2} + 54 a + 80\right)\cdot 127^{42} + \left(66 a^{2} + 56 a + 12\right)\cdot 127^{43} + \left(38 a^{2} + 71 a + 62\right)\cdot 127^{44} + \left(59 a^{2} + 66 a + 91\right)\cdot 127^{45} + \left(111 a^{2} + 69 a + 20\right)\cdot 127^{46} + \left(78 a^{2} + 126 a + 124\right)\cdot 127^{47} + \left(80 a^{2} + 48 a + 16\right)\cdot 127^{48} + \left(103 a^{2} + 8 a + 119\right)\cdot 127^{49} + \left(59 a^{2} + 72 a + 42\right)\cdot 127^{50} + \left(64 a^{2} + 33 a + 16\right)\cdot 127^{51} + \left(80 a^{2} + 8 a + 19\right)\cdot 127^{52} + \left(93 a^{2} + 27 a + 125\right)\cdot 127^{53} + \left(106 a^{2} + 105 a + 26\right)\cdot 127^{54} + \left(14 a^{2} + 86 a + 27\right)\cdot 127^{55} + \left(88 a^{2} + 69 a + 94\right)\cdot 127^{56} + \left(20 a^{2} + 124 a + 33\right)\cdot 127^{57} + \left(100 a^{2} + 114 a + 32\right)\cdot 127^{58} + \left(53 a^{2} + 112 a + 74\right)\cdot 127^{59} + \left(71 a^{2} + 72 a + 20\right)\cdot 127^{60} + \left(16 a^{2} + 87 a + 108\right)\cdot 127^{61} + \left(26 a^{2} + 17 a + 34\right)\cdot 127^{62} + \left(86 a^{2} + 56 a + 69\right)\cdot 127^{63} + \left(117 a^{2} + 61 a + 97\right)\cdot 127^{64} + \left(73 a^{2} + 9 a + 59\right)\cdot 127^{65} + \left(83 a^{2} + 18 a + 100\right)\cdot 127^{66} + \left(87 a^{2} + 85 a + 20\right)\cdot 127^{67} + \left(47 a^{2} + 6 a + 9\right)\cdot 127^{68} + \left(102 a^{2} + 115 a\right)\cdot 127^{69} + \left(44 a^{2} + 12 a + 57\right)\cdot 127^{70} + \left(a^{2} + 52 a + 98\right)\cdot 127^{71} + \left(59 a^{2} + 78 a + 11\right)\cdot 127^{72} + \left(75 a^{2} + 49 a + 16\right)\cdot 127^{73} + \left(92 a^{2} + 12 a + 21\right)\cdot 127^{74} + \left(44 a^{2} + 121 a + 124\right)\cdot 127^{75} + \left(124 a^{2} + 60 a + 48\right)\cdot 127^{76} + \left(49 a^{2} + 84 a + 21\right)\cdot 127^{77} + \left(97 a^{2} + 39 a + 6\right)\cdot 127^{78} + \left(7 a^{2} + 109 a + 86\right)\cdot 127^{79} + \left(74 a^{2} + 11 a + 78\right)\cdot 127^{80} + \left(90 a^{2} + 31 a + 19\right)\cdot 127^{81} + \left(35 a^{2} + 2 a + 66\right)\cdot 127^{82} + \left(22 a^{2} + 108 a + 26\right)\cdot 127^{83} + \left(44 a^{2} + 52 a + 83\right)\cdot 127^{84} + \left(59 a^{2} + 45 a + 98\right)\cdot 127^{85} + \left(84 a^{2} + 80 a + 109\right)\cdot 127^{86} + \left(4 a^{2} + 12 a + 37\right)\cdot 127^{87} + \left(46 a^{2} + 117 a + 99\right)\cdot 127^{88} + \left(89 a^{2} + 12 a + 114\right)\cdot 127^{89} + \left(42 a + 21\right)\cdot 127^{90} + \left(51 a^{2} + 19 a + 50\right)\cdot 127^{91} + \left(36 a^{2} + 69 a + 122\right)\cdot 127^{92} + \left(98 a^{2} + 126 a + 16\right)\cdot 127^{93} + \left(62 a^{2} + 43 a + 74\right)\cdot 127^{94} + \left(92 a^{2} + 91 a + 95\right)\cdot 127^{95} + \left(91 a^{2} + 114 a + 24\right)\cdot 127^{96} + \left(111 a^{2} + 87 a\right)\cdot 127^{97} + \left(81 a^{2} + 118 a + 45\right)\cdot 127^{98} + \left(13 a^{2} + 35 a + 28\right)\cdot 127^{99} +O\left(127^{ 100 }\right)$ $r_{ 5 }$ $=$ $11 a^{2} + 35 a + 43 + \left(93 a^{2} + 42 a + 19\right)\cdot 127 + \left(118 a^{2} + 62 a + 126\right)\cdot 127^{2} + \left(122 a^{2} + 15 a + 37\right)\cdot 127^{3} + \left(124 a^{2} + 63 a + 27\right)\cdot 127^{4} + \left(77 a^{2} + 118 a + 82\right)\cdot 127^{5} + \left(44 a^{2} + 90 a + 73\right)\cdot 127^{6} + \left(34 a^{2} + 47 a + 118\right)\cdot 127^{7} + \left(34 a^{2} + 92 a + 75\right)\cdot 127^{8} + \left(8 a^{2} + 24 a + 83\right)\cdot 127^{9} + \left(19 a^{2} + 34 a + 64\right)\cdot 127^{10} + \left(55 a^{2} + 100 a + 70\right)\cdot 127^{11} + \left(90 a + 116\right)\cdot 127^{12} + \left(126 a^{2} + 23 a + 5\right)\cdot 127^{13} + \left(122 a^{2} + 110 a + 84\right)\cdot 127^{14} + \left(34 a^{2} + 37 a + 21\right)\cdot 127^{15} + \left(20 a^{2} + 23 a + 104\right)\cdot 127^{16} + \left(124 a^{2} + 72 a + 73\right)\cdot 127^{17} + \left(122 a^{2} + 100 a + 104\right)\cdot 127^{18} + \left(5 a^{2} + 101 a + 39\right)\cdot 127^{19} + \left(25 a^{2} + 39 a + 38\right)\cdot 127^{20} + \left(33 a^{2} + 84 a + 7\right)\cdot 127^{21} + \left(106 a^{2} + 10 a + 26\right)\cdot 127^{22} + \left(95 a^{2} + 74 a + 76\right)\cdot 127^{23} + \left(44 a^{2} + 2 a + 56\right)\cdot 127^{24} + \left(10 a^{2} + 61 a + 87\right)\cdot 127^{25} + \left(8 a^{2} + 19 a + 56\right)\cdot 127^{26} + \left(115 a^{2} + 88 a + 60\right)\cdot 127^{27} + \left(71 a^{2} + 84 a + 8\right)\cdot 127^{28} + \left(50 a^{2} + 86 a + 38\right)\cdot 127^{29} + \left(58 a^{2} + 122 a + 14\right)\cdot 127^{30} + \left(48 a^{2} + 69 a + 30\right)\cdot 127^{31} + \left(42 a^{2} + 62 a + 106\right)\cdot 127^{32} + \left(104 a^{2} + 65 a + 91\right)\cdot 127^{33} + \left(94 a^{2} + 119 a + 28\right)\cdot 127^{34} + \left(77 a^{2} + 2 a + 7\right)\cdot 127^{35} + \left(75 a^{2} + 72 a + 73\right)\cdot 127^{36} + \left(67 a^{2} + 29 a + 97\right)\cdot 127^{37} + \left(97 a^{2} + 88 a + 76\right)\cdot 127^{38} + \left(36 a^{2} + 95 a + 120\right)\cdot 127^{39} + \left(5 a^{2} + 77 a + 45\right)\cdot 127^{40} + \left(40 a^{2} + 73 a + 29\right)\cdot 127^{41} + \left(90 a^{2} + 51 a + 35\right)\cdot 127^{42} + \left(82 a^{2} + 123 a + 55\right)\cdot 127^{43} + \left(9 a^{2} + 33 a + 60\right)\cdot 127^{44} + \left(47 a^{2} + 93 a + 42\right)\cdot 127^{45} + \left(33 a^{2} + 123 a + 77\right)\cdot 127^{46} + \left(89 a^{2} + 18 a + 2\right)\cdot 127^{47} + \left(87 a^{2} + 113 a + 52\right)\cdot 127^{48} + \left(33 a^{2} + 57 a + 30\right)\cdot 127^{49} + \left(80 a^{2} + 46 a + 32\right)\cdot 127^{50} + \left(12 a^{2} + 11 a + 83\right)\cdot 127^{51} + \left(95 a^{2} + 85 a + 98\right)\cdot 127^{52} + \left(92 a^{2} + 105 a + 26\right)\cdot 127^{53} + \left(101 a^{2} + 113 a + 104\right)\cdot 127^{54} + \left(43 a^{2} + 63 a + 71\right)\cdot 127^{55} + \left(105 a^{2} + 39 a + 2\right)\cdot 127^{56} + \left(91 a^{2} + 56 a + 109\right)\cdot 127^{57} + \left(27 a^{2} + 45 a + 122\right)\cdot 127^{58} + \left(23 a^{2} + 79 a + 88\right)\cdot 127^{59} + \left(75 a^{2} + 40 a + 99\right)\cdot 127^{60} + \left(86 a^{2} + 72 a + 51\right)\cdot 127^{61} + \left(103 a^{2} + 31 a + 3\right)\cdot 127^{62} + \left(82 a^{2} + 12 a + 102\right)\cdot 127^{63} + \left(85 a^{2} + 13 a + 31\right)\cdot 127^{64} + \left(14 a^{2} + 119 a + 52\right)\cdot 127^{65} + \left(46 a^{2} + 100 a + 98\right)\cdot 127^{66} + \left(31 a^{2} + 34 a + 79\right)\cdot 127^{67} + \left(75 a^{2} + 59 a + 21\right)\cdot 127^{68} + \left(104 a^{2} + 112 a + 72\right)\cdot 127^{69} + \left(100 a^{2} + 2 a + 94\right)\cdot 127^{70} + \left(126 a^{2} + 9 a + 16\right)\cdot 127^{71} + \left(104 a^{2} + 77 a + 78\right)\cdot 127^{72} + \left(106 a^{2} + 19 a + 118\right)\cdot 127^{73} + \left(35 a^{2} + 65 a + 36\right)\cdot 127^{74} + \left(105 a^{2} + 4 a + 52\right)\cdot 127^{75} + \left(14 a^{2} + 51 a + 40\right)\cdot 127^{76} + \left(42 a^{2} + 31 a + 119\right)\cdot 127^{77} + \left(77 a^{2} + 46 a + 115\right)\cdot 127^{78} + \left(111 a^{2} + 66 a + 34\right)\cdot 127^{79} + \left(116 a^{2} + 21 a + 110\right)\cdot 127^{80} + \left(9 a^{2} + 101 a + 60\right)\cdot 127^{81} + \left(88 a^{2} + 38 a + 78\right)\cdot 127^{82} + \left(80 a^{2} + 27 a + 112\right)\cdot 127^{83} + \left(87 a^{2} + 43 a + 117\right)\cdot 127^{84} + \left(48 a^{2} + 74 a + 51\right)\cdot 127^{85} + \left(48 a^{2} + 19 a + 79\right)\cdot 127^{86} + \left(87 a^{2} + 92 a + 109\right)\cdot 127^{87} + \left(106 a^{2} + 60 a + 116\right)\cdot 127^{88} + \left(119 a^{2} + 103 a + 123\right)\cdot 127^{89} + \left(92 a^{2} + 89 a + 115\right)\cdot 127^{90} + \left(85 a^{2} + 105 a + 60\right)\cdot 127^{91} + \left(7 a^{2} + 59 a + 121\right)\cdot 127^{92} + \left(94 a^{2} + 26 a + 68\right)\cdot 127^{93} + \left(28 a^{2} + 116 a + 40\right)\cdot 127^{94} + \left(46 a^{2} + 46 a + 86\right)\cdot 127^{95} + \left(56 a^{2} + 28 a + 102\right)\cdot 127^{96} + \left(97 a^{2} + 96 a + 101\right)\cdot 127^{97} + \left(12 a^{2} + 95 a + 49\right)\cdot 127^{98} + \left(75 a^{2} + 73 a + 112\right)\cdot 127^{99} +O\left(127^{ 100 }\right)$ $r_{ 6 }$ $=$ $40 a^{2} + 46 a + 101 + \left(36 a^{2} + 123 a + 32\right)\cdot 127 + \left(125 a^{2} + 103 a + 12\right)\cdot 127^{2} + \left(110 a^{2} + 9 a + 14\right)\cdot 127^{3} + \left(3 a^{2} + 111 a + 39\right)\cdot 127^{4} + \left(77 a^{2} + 11 a + 80\right)\cdot 127^{5} + \left(25 a^{2} + 45 a + 35\right)\cdot 127^{6} + \left(45 a^{2} + 25 a + 13\right)\cdot 127^{7} + \left(74 a^{2} + 62 a + 29\right)\cdot 127^{8} + \left(40 a^{2} + 40 a + 21\right)\cdot 127^{9} + \left(44 a^{2} + 89 a + 115\right)\cdot 127^{10} + \left(27 a + 87\right)\cdot 127^{11} + \left(10 a^{2} + 30 a + 8\right)\cdot 127^{12} + \left(51 a^{2} + 54 a + 110\right)\cdot 127^{13} + \left(44 a^{2} + 93 a + 53\right)\cdot 127^{14} + \left(98 a^{2} + 36 a + 21\right)\cdot 127^{15} + \left(110 a^{2} + 77 a + 31\right)\cdot 127^{16} + \left(70 a^{2} + 125 a + 94\right)\cdot 127^{17} + \left(10 a^{2} + 65 a + 6\right)\cdot 127^{18} + \left(46 a^{2} + 113 a + 120\right)\cdot 127^{19} + \left(34 a^{2} + 11 a + 56\right)\cdot 127^{20} + \left(99 a^{2} + 9 a + 12\right)\cdot 127^{21} + \left(105 a^{2} + 18 a + 25\right)\cdot 127^{22} + \left(80 a^{2} + 110 a + 46\right)\cdot 127^{23} + \left(61 a^{2} + 19 a + 90\right)\cdot 127^{24} + \left(67 a^{2} + 90 a + 74\right)\cdot 127^{25} + \left(43 a^{2} + 62 a\right)\cdot 127^{26} + \left(99 a^{2} + 86 a + 29\right)\cdot 127^{27} + \left(51 a^{2} + 56 a + 95\right)\cdot 127^{28} + \left(57 a^{2} + 36 a + 51\right)\cdot 127^{29} + \left(60 a^{2} + 47 a + 18\right)\cdot 127^{30} + \left(8 a^{2} + 102 a + 77\right)\cdot 127^{31} + \left(120 a^{2} + 34 a + 7\right)\cdot 127^{32} + \left(118 a^{2} + 60 a + 121\right)\cdot 127^{33} + \left(81 a^{2} + 10 a + 2\right)\cdot 127^{34} + \left(34 a^{2} + 15 a + 48\right)\cdot 127^{35} + \left(7 a^{2} + 82 a + 63\right)\cdot 127^{36} + \left(98 a^{2} + 86 a + 31\right)\cdot 127^{37} + \left(8 a^{2} + 72 a + 26\right)\cdot 127^{38} + \left(48 a^{2} + 103 a + 16\right)\cdot 127^{39} + \left(a + 36\right)\cdot 127^{40} + \left(87 a^{2} + 117 a + 123\right)\cdot 127^{41} + \left(43 a^{2} + 11 a + 68\right)\cdot 127^{42} + \left(86 a^{2} + 124 a + 62\right)\cdot 127^{43} + \left(43 a^{2} + 61 a + 1\right)\cdot 127^{44} + \left(117 a^{2} + 14 a + 56\right)\cdot 127^{45} + \left(28 a^{2} + 42 a + 68\right)\cdot 127^{46} + \left(103 a^{2} + 24 a + 30\right)\cdot 127^{47} + \left(8 a^{2} + 22 a + 21\right)\cdot 127^{48} + \left(39 a^{2} + 11 a + 41\right)\cdot 127^{49} + \left(53 a^{2} + 112 a + 105\right)\cdot 127^{50} + \left(9 a^{2} + 73 a + 76\right)\cdot 127^{51} + \left(36 a^{2} + 82 a + 107\right)\cdot 127^{52} + \left(86 a^{2} + 73 a + 13\right)\cdot 127^{53} + \left(110 a^{2} + 9 a + 122\right)\cdot 127^{54} + \left(20 a^{2} + 64 a + 25\right)\cdot 127^{55} + \left(25 a^{2} + 13 a + 96\right)\cdot 127^{56} + \left(3 a^{2} + 84 a + 58\right)\cdot 127^{57} + \left(31 a^{2} + 95 a + 2\right)\cdot 127^{58} + \left(60 a^{2} + 46 a + 36\right)\cdot 127^{59} + \left(101 a^{2} + 117 a + 25\right)\cdot 127^{60} + \left(15 a^{2} + 103 a + 37\right)\cdot 127^{61} + \left(46 a^{2} + 14 a + 15\right)\cdot 127^{62} + \left(20 a^{2} + 106 a + 104\right)\cdot 127^{63} + \left(105 a^{2} + 100 a + 70\right)\cdot 127^{64} + \left(77 a^{2} + 6 a + 51\right)\cdot 127^{65} + \left(99 a^{2} + 121 a + 78\right)\cdot 127^{66} + \left(38 a^{2} + 53 a + 94\right)\cdot 127^{67} + \left(43 a^{2} + 78 a + 84\right)\cdot 127^{68} + \left(44 a^{2} + 24 a + 78\right)\cdot 127^{69} + \left(10 a^{2} + 88 a + 40\right)\cdot 127^{70} + \left(88 a^{2} + 85 a + 66\right)\cdot 127^{71} + \left(19 a^{2} + 67 a + 34\right)\cdot 127^{72} + \left(45 a^{2} + 5 a + 122\right)\cdot 127^{73} + \left(21 a^{2} + 43 a + 7\right)\cdot 127^{74} + \left(103 a^{2} + 105 a + 48\right)\cdot 127^{75} + \left(121 a^{2} + 8 a\right)\cdot 127^{76} + \left(73 a^{2} + 93 a + 56\right)\cdot 127^{77} + \left(52 a^{2} + 99 a + 66\right)\cdot 127^{78} + \left(126 a^{2} + 3 a + 64\right)\cdot 127^{79} + \left(98 a^{2} + 3 a + 74\right)\cdot 127^{80} + \left(78 a^{2} + 105 a + 71\right)\cdot 127^{81} + \left(21 a^{2} + 19 a + 72\right)\cdot 127^{82} + \left(25 a^{2} + 34 a + 1\right)\cdot 127^{83} + \left(49 a^{2} + 99 a + 41\right)\cdot 127^{84} + \left(100 a^{2} + 111 a + 28\right)\cdot 127^{85} + \left(61 a^{2} + 10 a + 106\right)\cdot 127^{86} + \left(70 a^{2} + 23 a + 75\right)\cdot 127^{87} + \left(2 a^{2} + 117 a + 35\right)\cdot 127^{88} + \left(92 a^{2} + 48 a + 68\right)\cdot 127^{89} + \left(68 a^{2} + 80 a + 67\right)\cdot 127^{90} + \left(99 a^{2} + 30 a + 88\right)\cdot 127^{91} + \left(43 a^{2} + 42 a + 66\right)\cdot 127^{92} + \left(67 a^{2} + 36 a + 15\right)\cdot 127^{93} + \left(11 a^{2} + 90 a + 6\right)\cdot 127^{94} + \left(57 a^{2} + 96 a + 108\right)\cdot 127^{95} + \left(49 a^{2} + 30 a + 88\right)\cdot 127^{96} + \left(44 a^{2} + 62 a + 122\right)\cdot 127^{97} + \left(100 a^{2} + 75 a + 97\right)\cdot 127^{98} + \left(32 a^{2} + 101 a + 27\right)\cdot 127^{99} +O\left(127^{ 100 }\right)$ $r_{ 7 }$ $=$ $57 a^{2} + 46 a + 70 + \left(79 a^{2} + 107 a + 78\right)\cdot 127 + \left(8 a^{2} + 57 a + 66\right)\cdot 127^{2} + \left(30 a^{2} + 72 a + 62\right)\cdot 127^{3} + \left(51 a^{2} + 10 a + 38\right)\cdot 127^{4} + \left(67 a^{2} + 124 a + 30\right)\cdot 127^{5} + \left(61 a^{2} + 101 a + 8\right)\cdot 127^{6} + \left(31 a^{2} + 28 a + 41\right)\cdot 127^{7} + \left(92 a^{2} + 108 a + 125\right)\cdot 127^{8} + \left(26 a^{2} + 16 a + 44\right)\cdot 127^{9} + \left(95 a^{2} + 55 a + 31\right)\cdot 127^{10} + \left(49 a^{2} + 57 a + 45\right)\cdot 127^{11} + \left(126 a^{2} + 26 a + 26\right)\cdot 127^{12} + \left(25 a^{2} + 2 a + 107\right)\cdot 127^{13} + \left(3 a^{2} + 80 a + 86\right)\cdot 127^{14} + \left(121 a^{2} + 60 a + 40\right)\cdot 127^{15} + \left(58 a^{2} + 4 a + 124\right)\cdot 127^{16} + \left(27 a^{2} + 99 a + 84\right)\cdot 127^{17} + \left(55 a^{2} + 50\right)\cdot 127^{18} + \left(97 a^{2} + 77 a + 71\right)\cdot 127^{19} + \left(34 a^{2} + 85 a + 71\right)\cdot 127^{20} + \left(27 a^{2} + 60 a + 1\right)\cdot 127^{21} + \left(58 a^{2} + 20 a + 14\right)\cdot 127^{22} + \left(67 a^{2} + 120 a + 42\right)\cdot 127^{23} + \left(91 a + 8\right)\cdot 127^{24} + \left(91 a^{2} + 118 a + 116\right)\cdot 127^{25} + \left(93 a^{2} + 5 a + 46\right)\cdot 127^{26} + \left(35 a^{2} + 90 a + 56\right)\cdot 127^{27} + \left(108 a^{2} + 40 a + 84\right)\cdot 127^{28} + \left(90 a^{2} + 118 a + 14\right)\cdot 127^{29} + \left(78 a^{2} + 64 a + 62\right)\cdot 127^{30} + \left(20 a^{2} + 98 a + 120\right)\cdot 127^{31} + \left(78 a^{2} + 120 a + 103\right)\cdot 127^{32} + \left(94 a^{2} + 87 a + 3\right)\cdot 127^{33} + \left(25 a^{2} + 12 a + 44\right)\cdot 127^{34} + \left(9 a^{2} + 95 a + 28\right)\cdot 127^{35} + \left(26 a^{2} + 98 a + 15\right)\cdot 127^{36} + \left(70 a^{2} + 63 a + 85\right)\cdot 127^{37} + \left(4 a^{2} + 112 a + 49\right)\cdot 127^{38} + \left(91 a^{2} + 115 a + 7\right)\cdot 127^{39} + \left(22 a^{2} + 113 a + 47\right)\cdot 127^{40} + \left(4 a^{2} + 33 a + 102\right)\cdot 127^{41} + \left(100 a^{2} + 3 a + 41\right)\cdot 127^{42} + \left(68 a^{2} + 57 a + 16\right)\cdot 127^{43} + \left(49 a^{2} + 78 a + 84\right)\cdot 127^{44} + \left(77 a^{2} + 102 a\right)\cdot 127^{45} + \left(11 a^{2} + 110 a + 75\right)\cdot 127^{46} + \left(69 a^{2} + 94 a + 104\right)\cdot 127^{47} + \left(65 a^{2} + 25 a + 113\right)\cdot 127^{48} + \left(93 a^{2} + 37 a + 98\right)\cdot 127^{49} + \left(121 a^{2} + 18 a + 39\right)\cdot 127^{50} + \left(56 a^{2} + 18 a + 1\right)\cdot 127^{51} + \left(30 a^{2} + 4 a + 46\right)\cdot 127^{52} + \left(6 a^{2} + 75 a + 77\right)\cdot 127^{53} + \left(98 a^{2} + 50 a + 9\right)\cdot 127^{54} + \left(89 a^{2} + 56 a + 50\right)\cdot 127^{55} + \left(95 a^{2} + 53 a + 109\right)\cdot 127^{56} + \left(9 a^{2} + 63 a + 11\right)\cdot 127^{57} + \left(21 a^{2} + 49 a + 1\right)\cdot 127^{58} + \left(94 a^{2} + 29 a + 28\right)\cdot 127^{59} + \left(58 a^{2} + 85 a + 122\right)\cdot 127^{60} + \left(58 a^{2} + 94 a + 64\right)\cdot 127^{61} + \left(48 a^{2} + 85 a + 79\right)\cdot 127^{62} + \left(15 a^{2} + 72 a + 54\right)\cdot 127^{63} + \left(73 a^{2} + 123 a + 8\right)\cdot 127^{64} + \left(50 a^{2} + 109 a + 13\right)\cdot 127^{65} + \left(56 a^{2} + 75 a + 46\right)\cdot 127^{66} + \left(101 a^{2} + 114 a + 48\right)\cdot 127^{67} + \left(63 a^{2} + 87 a + 41\right)\cdot 127^{68} + \left(90 a^{2} + 51 a + 103\right)\cdot 127^{69} + \left(54 a^{2} + 81 a + 76\right)\cdot 127^{70} + \left(107 a^{2} + 125 a + 56\right)\cdot 127^{71} + \left(98 a^{2} + 119 a + 91\right)\cdot 127^{72} + \left(38 a^{2} + 116 a + 69\right)\cdot 127^{73} + \left(119 a^{2} + 30 a + 74\right)\cdot 127^{74} + \left(10 a^{2} + 59 a + 56\right)\cdot 127^{75} + \left(124 a^{2} + 38 a + 48\right)\cdot 127^{76} + \left(57 a^{2} + 107 a + 37\right)\cdot 127^{77} + \left(85 a^{2} + 86 a + 109\right)\cdot 127^{78} + \left(26 a^{2} + 55 a + 123\right)\cdot 127^{79} + \left(100 a^{2} + 110 a + 3\right)\cdot 127^{80} + \left(126 a^{2} + 7 a + 92\right)\cdot 127^{81} + \left(48 a^{2} + 92 a + 92\right)\cdot 127^{82} + \left(54 a^{2} + 117 a + 90\right)\cdot 127^{83} + \left(122 a^{2} + 123 a + 112\right)\cdot 127^{84} + \left(55 a^{2} + 81 a + 91\right)\cdot 127^{85} + \left(86 a^{2} + 107 a + 113\right)\cdot 127^{86} + \left(86 a^{2} + 72 a + 74\right)\cdot 127^{87} + \left(91 a^{2} + 16 a + 63\right)\cdot 127^{88} + \left(18 a^{2} + 83 a + 100\right)\cdot 127^{89} + \left(112 a^{2} + 123 a + 117\right)\cdot 127^{90} + \left(60 a^{2} + 71 a + 69\right)\cdot 127^{91} + \left(11 a^{2} + 34 a + 72\right)\cdot 127^{92} + \left(115 a^{2} + 50\right)\cdot 127^{93} + \left(63 a^{2} + 125 a + 76\right)\cdot 127^{94} + \left(40 a^{2} + 69 a + 118\right)\cdot 127^{95} + \left(126 a^{2} + 98 a + 93\right)\cdot 127^{96} + \left(47 a^{2} + 105 a + 126\right)\cdot 127^{97} + \left(68 a^{2} + 91 a + 17\right)\cdot 127^{98} + \left(111 a^{2} + 79 a + 97\right)\cdot 127^{99} +O\left(127^{ 100 }\right)$ $r_{ 8 }$ $=$ $70 a^{2} + 2 a + 96 + \left(33 a^{2} + 21 a + 113\right)\cdot 127 + \left(17 a^{2} + 49 a + 83\right)\cdot 127^{2} + \left(81 a^{2} + 15 a + 37\right)\cdot 127^{3} + \left(38 a^{2} + 22 a + 13\right)\cdot 127^{4} + \left(81 a^{2} + 35 a + 58\right)\cdot 127^{5} + \left(104 a^{2} + 87 a + 94\right)\cdot 127^{6} + \left(116 a^{2} + 93 a + 84\right)\cdot 127^{7} + \left(119 a^{2} + 125 a + 53\right)\cdot 127^{8} + \left(2 a^{2} + 25 a + 124\right)\cdot 127^{9} + \left(54 a^{2} + 105 a + 75\right)\cdot 127^{10} + \left(64 a^{2} + 95 a + 74\right)\cdot 127^{11} + \left(26 a^{2} + 44 a + 80\right)\cdot 127^{12} + \left(53 a^{2} + 119 a + 34\right)\cdot 127^{13} + \left(50 a^{2} + 86 a + 54\right)\cdot 127^{14} + \left(100 a^{2} + 119 a + 126\right)\cdot 127^{15} + \left(17 a^{2} + 17 a + 41\right)\cdot 127^{16} + \left(a^{2} + 49 a + 32\right)\cdot 127^{17} + \left(62 a^{2} + 53 a + 64\right)\cdot 127^{18} + \left(111 a^{2} + 68 a + 99\right)\cdot 127^{19} + \left(106 a^{2} + 61 a + 88\right)\cdot 127^{20} + \left(75 a^{2} + 95 a + 98\right)\cdot 127^{21} + \left(119 a^{2} + 75 a + 9\right)\cdot 127^{22} + \left(18 a^{2} + 113 a + 72\right)\cdot 127^{23} + \left(34 a^{2} + 79 a + 75\right)\cdot 127^{24} + \left(65 a^{2} + 125 a + 64\right)\cdot 127^{25} + \left(55 a^{2} + 70 a + 97\right)\cdot 127^{26} + \left(96 a^{2} + 80 a + 50\right)\cdot 127^{27} + \left(38 a^{2} + 5 a + 72\right)\cdot 127^{28} + \left(97 a^{2} + 27 a + 27\right)\cdot 127^{29} + \left(107 a^{2} + 119 a + 120\right)\cdot 127^{30} + \left(122 a^{2} + 81 a + 70\right)\cdot 127^{31} + \left(13 a^{2} + 25 a + 102\right)\cdot 127^{32} + \left(23 a^{2} + 36 a + 114\right)\cdot 127^{33} + \left(96 a^{2} + 109 a + 57\right)\cdot 127^{34} + \left(12 a^{2} + 18 a + 35\right)\cdot 127^{35} + \left(48 a^{2} + 32 a + 59\right)\cdot 127^{36} + \left(21 a^{2} + 17 a + 114\right)\cdot 127^{37} + \left(78 a^{2} + 44 a + 69\right)\cdot 127^{38} + \left(58 a^{2} + 31 a + 69\right)\cdot 127^{39} + \left(21 a^{2} + 119 a + 44\right)\cdot 127^{40} + \left(74 a^{2} + 79 a + 115\right)\cdot 127^{41} + \left(34 a^{2} + 69 a + 37\right)\cdot 127^{42} + \left(118 a^{2} + 13 a + 115\right)\cdot 127^{43} + \left(38 a^{2} + 104 a + 62\right)\cdot 127^{44} + \left(117 a^{2} + 84 a + 80\right)\cdot 127^{45} + \left(3 a^{2} + 73 a + 59\right)\cdot 127^{46} + \left(106 a^{2} + 32 a + 51\right)\cdot 127^{47} + \left(107 a^{2} + 52 a + 71\right)\cdot 127^{48} + \left(56 a^{2} + 81 a + 25\right)\cdot 127^{49} + \left(72 a^{2} + 36 a + 68\right)\cdot 127^{50} + \left(5 a^{2} + 75 a + 25\right)\cdot 127^{51} + \left(16 a^{2} + 114 a + 17\right)\cdot 127^{52} + \left(27 a^{2} + 24 a + 119\right)\cdot 127^{53} + \left(49 a^{2} + 98 a + 38\right)\cdot 127^{54} + \left(22 a^{2} + 110 a + 42\right)\cdot 127^{55} + \left(70 a^{2} + 3 a + 58\right)\cdot 127^{56} + \left(96 a^{2} + 66 a + 58\right)\cdot 127^{57} + \left(5 a^{2} + 89 a + 97\right)\cdot 127^{58} + \left(106 a^{2} + 111 a + 51\right)\cdot 127^{59} + \left(123 a^{2} + 95 a + 125\right)\cdot 127^{60} + \left(51 a^{2} + 71 a + 51\right)\cdot 127^{61} + \left(52 a^{2} + 23 a + 87\right)\cdot 127^{62} + \left(25 a^{2} + 125 a + 74\right)\cdot 127^{63} + \left(63 a^{2} + 68 a + 115\right)\cdot 127^{64} + \left(2 a^{2} + 7 a + 43\right)\cdot 127^{65} + \left(114 a^{2} + 33 a + 34\right)\cdot 127^{66} + \left(64 a^{2} + 54 a + 102\right)\cdot 127^{67} + \left(15 a^{2} + 32 a + 71\right)\cdot 127^{68} + \left(61 a^{2} + 87 a + 44\right)\cdot 127^{69} + \left(27 a^{2} + 32 a + 22\right)\cdot 127^{70} + \left(18 a^{2} + 76 a + 5\right)\cdot 127^{71} + \left(96 a^{2} + 55 a + 86\right)\cdot 127^{72} + \left(12 a^{2} + 87 a + 17\right)\cdot 127^{73} + \left(42 a^{2} + 83 a + 47\right)\cdot 127^{74} + \left(71 a^{2} + 73 a + 50\right)\cdot 127^{75} + \left(5 a^{2} + 27 a + 65\right)\cdot 127^{76} + \left(19 a^{2} + 62 a + 86\right)\cdot 127^{77} + \left(71 a^{2} + 80\right)\cdot 127^{78} + \left(92 a^{2} + 89 a + 1\right)\cdot 127^{79} + \left(79 a^{2} + 4 a + 90\right)\cdot 127^{80} + \left(36 a^{2} + 88 a + 38\right)\cdot 127^{81} + \left(42 a^{2} + 32 a + 79\right)\cdot 127^{82} + \left(50 a^{2} + 28 a + 82\right)\cdot 127^{83} + \left(87 a^{2} + 77 a + 42\right)\cdot 127^{84} + \left(11 a^{2} + 126 a + 3\right)\cdot 127^{85} + \left(83 a^{2} + 65 a + 107\right)\cdot 127^{86} + \left(35 a^{2} + 41 a + 99\right)\cdot 127^{87} + \left(116 a^{2} + 120 a + 112\right)\cdot 127^{88} + \left(18 a^{2} + 30 a + 100\right)\cdot 127^{89} + \left(14 a^{2} + 88 a + 48\right)\cdot 127^{90} + \left(15 a^{2} + 35 a + 105\right)\cdot 127^{91} + \left(79 a^{2} + 23 a + 80\right)\cdot 127^{92} + \left(40 a^{2} + 28\right)\cdot 127^{93} + \left(85 a + 76\right)\cdot 127^{94} + \left(121 a^{2} + 92 a + 25\right)\cdot 127^{95} + \left(35 a^{2} + 40 a + 40\right)\cdot 127^{96} + \left(94 a^{2} + 60 a + 92\right)\cdot 127^{97} + \left(103 a^{2} + 43 a + 88\right)\cdot 127^{98} + \left(a^{2} + 11 a + 4\right)\cdot 127^{99} +O\left(127^{ 100 }\right)$ $r_{ 9 }$ $=$ $76 a^{2} + 46 a + 46 + \left(124 a^{2} + 88 a + 82\right)\cdot 127 + \left(9 a^{2} + 87 a + 35\right)\cdot 127^{2} + \left(20 a^{2} + 101 a + 86\right)\cdot 127^{3} + \left(125 a^{2} + 79 a + 27\right)\cdot 127^{4} + \left(98 a^{2} + 123 a + 124\right)\cdot 127^{5} + \left(56 a^{2} + 117 a + 97\right)\cdot 127^{6} + \left(47 a^{2} + 53 a + 17\right)\cdot 127^{7} + \left(18 a^{2} + 99 a + 44\right)\cdot 127^{8} + \left(78 a^{2} + 61 a + 96\right)\cdot 127^{9} + \left(63 a^{2} + 3 a + 26\right)\cdot 127^{10} + \left(71 a^{2} + 126 a + 103\right)\cdot 127^{11} + \left(116 a^{2} + 5 a + 94\right)\cdot 127^{12} + \left(76 a^{2} + 49 a + 34\right)\cdot 127^{13} + \left(86 a^{2} + 50 a + 11\right)\cdot 127^{14} + \left(120 a^{2} + 52 a + 66\right)\cdot 127^{15} + \left(122 a^{2} + 26 a + 55\right)\cdot 127^{16} + \left(58 a^{2} + 56 a + 70\right)\cdot 127^{17} + \left(120 a^{2} + 87 a + 99\right)\cdot 127^{18} + \left(74 a^{2} + 38 a + 50\right)\cdot 127^{19} + \left(67 a^{2} + 75 a + 123\right)\cdot 127^{20} + \left(121 a^{2} + 33 a + 56\right)\cdot 127^{21} + \left(41 a^{2} + 98 a + 24\right)\cdot 127^{22} + \left(77 a^{2} + 69 a + 39\right)\cdot 127^{23} + \left(20 a^{2} + 104 a + 8\right)\cdot 127^{24} + \left(49 a^{2} + 102 a + 38\right)\cdot 127^{25} + \left(75 a^{2} + 44 a + 64\right)\cdot 127^{26} + \left(39 a^{2} + 79 a + 36\right)\cdot 127^{27} + \left(3 a^{2} + 112 a + 125\right)\cdot 127^{28} + \left(19 a^{2} + 3 a + 101\right)\cdot 127^{29} + \left(8 a^{2} + 84 a + 40\right)\cdot 127^{30} + \left(70 a^{2} + 81 a + 73\right)\cdot 127^{31} + \left(91 a^{2} + 29 a + 77\right)\cdot 127^{32} + \left(30 a^{2} + a + 71\right)\cdot 127^{33} + \left(77 a^{2} + 124 a + 120\right)\cdot 127^{34} + \left(14 a^{2} + 108 a + 7\right)\cdot 127^{35} + \left(44 a^{2} + 99 a + 10\right)\cdot 127^{36} + \left(88 a^{2} + 10 a + 12\right)\cdot 127^{37} + \left(20 a^{2} + 93 a + 50\right)\cdot 127^{38} + \left(42 a^{2} + 54 a + 4\right)\cdot 127^{39} + \left(121 a^{2} + 47 a + 24\right)\cdot 127^{40} + \left(126 a^{2} + 63 a + 76\right)\cdot 127^{41} + \left(119 a^{2} + 63 a + 94\right)\cdot 127^{42} + \left(84 a^{2} + 6 a + 59\right)\cdot 127^{43} + \left(73 a^{2} + 31 a + 61\right)\cdot 127^{44} + \left(89 a^{2} + 19 a\right)\cdot 127^{45} + \left(64 a^{2} + 88 a + 13\right)\cdot 127^{46} + \left(61 a^{2} + 83 a + 74\right)\cdot 127^{47} + \left(30 a^{2} + 118 a + 64\right)\cdot 127^{48} + \left(54 a^{2} + 57 a + 71\right)\cdot 127^{49} + \left(120 a^{2} + 95 a + 112\right)\cdot 127^{50} + \left(104 a^{2} + 41 a + 13\right)\cdot 127^{51} + \left(122 a^{2} + 86 a + 27\right)\cdot 127^{52} + \left(74 a^{2} + 74 a + 118\right)\cdot 127^{53} + \left(41 a^{2} + 3 a + 110\right)\cdot 127^{54} + \left(62 a^{2} + 126 a + 108\right)\cdot 127^{55} + \left(123 a^{2} + 73 a + 38\right)\cdot 127^{56} + \left(31 a^{2} + 113 a + 116\right)\cdot 127^{57} + \left(68 a^{2} + 112 a + 76\right)\cdot 127^{58} + \left(43 a^{2} + 2\right)\cdot 127^{59} + \left(77 a^{2} + 96 a + 104\right)\cdot 127^{60} + \left(24 a^{2} + 77 a + 54\right)\cdot 127^{61} + \left(104 a^{2} + 80 a + 4\right)\cdot 127^{62} + \left(23 a^{2} + 8 a + 111\right)\cdot 127^{63} + \left(63 a^{2} + 13 a + 113\right)\cdot 127^{64} + \left(34 a^{2} + a + 91\right)\cdot 127^{65} + \left(108 a^{2} + 32 a + 95\right)\cdot 127^{66} + \left(56 a^{2} + 38 a + 3\right)\cdot 127^{67} + \left(8 a^{2} + 116 a + 15\right)\cdot 127^{68} + \left(105 a^{2} + 116 a + 73\right)\cdot 127^{69} + \left(15 a^{2} + 35 a + 51\right)\cdot 127^{70} + \left(39 a^{2} + 32 a + 95\right)\cdot 127^{71} + \left(2 a^{2} + 109 a + 126\right)\cdot 127^{72} + \left(102 a^{2} + 101 a + 108\right)\cdot 127^{73} + \left(69 a^{2} + 18 a + 104\right)\cdot 127^{74} + \left(45 a^{2} + 17 a + 59\right)\cdot 127^{75} + \left(117 a^{2} + 67 a + 118\right)\cdot 127^{76} + \left(10 a^{2} + 2 a + 56\right)\cdot 127^{77} + \left(124 a^{2} + 108 a + 82\right)\cdot 127^{78} + \left(15 a^{2} + 56 a + 97\right)\cdot 127^{79} + \left(38 a^{2} + 102 a + 79\right)\cdot 127^{80} + \left(38 a^{2} + 47 a + 117\right)\cdot 127^{81} + \left(17 a^{2} + 68 a + 63\right)\cdot 127^{82} + \left(21 a^{2} + 65 a + 120\right)\cdot 127^{83} + \left(117 a^{2} + 111 a + 49\right)\cdot 127^{84} + \left(104 a^{2} + 67 a + 37\right)\cdot 127^{85} + \left(16 a^{2} + 96 a + 16\right)\cdot 127^{86} + \left(96 a^{2} + 11 a\right)\cdot 127^{87} + \left(17 a^{2} + 76 a + 66\right)\cdot 127^{88} + \left(42 a^{2} + 101 a + 95\right)\cdot 127^{89} + \left(92 a^{2} + 83 a + 114\right)\cdot 127^{90} + \left(68 a^{2} + 117 a + 26\right)\cdot 127^{91} + \left(75 a^{2} + 24 a + 3\right)\cdot 127^{92} + \left(92 a^{2} + 64 a + 66\right)\cdot 127^{93} + \left(86 a^{2} + 47 a + 29\right)\cdot 127^{94} + \left(23 a^{2} + 110 a + 41\right)\cdot 127^{95} + \left(21 a^{2} + 67 a + 32\right)\cdot 127^{96} + \left(112 a^{2} + 95 a + 4\right)\cdot 127^{97} + \left(13 a^{2} + 82 a + 52\right)\cdot 127^{98} + \left(19 a^{2} + 78 a\right)\cdot 127^{99} +O\left(127^{ 100 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

 Cycle notation $(1,4,5)(2,7,6)(3,8,9)$ $(1,2)$ $(1,4)(2,7)(3,8)$ $(1,2,3)$ $(4,7,8)$ $(5,6,9)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character value $1$ $1$ $()$ $8$ $9$ $2$ $(1,2)$ $0$ $18$ $2$ $(1,4)(2,7)(3,8)$ $4$ $27$ $2$ $(1,2)(4,7)(5,6)$ $0$ $27$ $2$ $(1,2)(5,6)$ $0$ $54$ $2$ $(1,5)(2,6)(3,9)(4,7)$ $0$ $6$ $3$ $(5,6,9)$ $-4$ $8$ $3$ $(1,3,2)(4,8,7)(5,9,6)$ $-1$ $12$ $3$ $(1,3,2)(5,9,6)$ $2$ $72$ $3$ $(1,4,5)(2,7,6)(3,8,9)$ $2$ $54$ $4$ $(1,5,2,6)(3,9)$ $0$ $162$ $4$ $(1,5,2,6)(3,9)(4,7)$ $0$ $36$ $6$ $(1,4)(2,7)(3,8)(5,6,9)$ $-2$ $36$ $6$ $(1,5,3,9,2,6)$ $-2$ $36$ $6$ $(1,2)(5,6,9)$ $0$ $36$ $6$ $(1,2)(4,7,8)(5,6,9)$ $0$ $54$ $6$ $(1,2)(4,7)(5,9,6)$ $0$ $72$ $6$ $(1,4,3,8,2,7)(5,6,9)$ $1$ $108$ $6$ $(1,5,3,9,2,6)(4,7)$ $0$ $216$ $6$ $(1,4,5,2,7,6)(3,8,9)$ $0$ $144$ $9$ $(1,4,5,3,8,9,2,7,6)$ $-1$ $108$ $12$ $(1,4,2,7)(3,8)(5,6,9)$ $0$
The blue line marks the conjugacy class containing complex conjugation.