Properties

Label 8.2e12_3e4_11e2_13e2.12t213.1c1
Dimension 8
Group $S_3\wr S_3$
Conductor $ 2^{12} \cdot 3^{4} \cdot 11^{2} \cdot 13^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$8$
Group:$S_3\wr S_3$
Conductor:$6784487424= 2^{12} \cdot 3^{4} \cdot 11^{2} \cdot 13^{2} $
Artin number field: Splitting field of $f= x^{9} - x^{8} + 3 x^{7} + x^{6} - x^{5} - x^{4} - 4 x^{2} - 3 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 12T213
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 211 }$ to precision 98.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 211 }$: $ x^{3} + 2 x + 209 $
Roots:
$r_{ 1 }$ $=$ $ 2 + 91\cdot 211 + 54\cdot 211^{2} + 145\cdot 211^{3} + 134\cdot 211^{4} + 142\cdot 211^{5} + 135\cdot 211^{6} + 157\cdot 211^{7} + 11\cdot 211^{8} + 68\cdot 211^{9} + 126\cdot 211^{10} + 135\cdot 211^{11} + 134\cdot 211^{12} + 144\cdot 211^{13} + 7\cdot 211^{14} + 145\cdot 211^{15} + 201\cdot 211^{16} + 121\cdot 211^{17} + 114\cdot 211^{18} + 22\cdot 211^{19} + 23\cdot 211^{20} + 83\cdot 211^{21} + 132\cdot 211^{22} + 31\cdot 211^{23} + 168\cdot 211^{24} + 46\cdot 211^{25} + 18\cdot 211^{26} + 58\cdot 211^{27} + 118\cdot 211^{28} + 82\cdot 211^{29} + 28\cdot 211^{30} + 112\cdot 211^{31} + 148\cdot 211^{32} + 116\cdot 211^{33} + 33\cdot 211^{34} + 43\cdot 211^{35} + 153\cdot 211^{36} + 89\cdot 211^{37} + 122\cdot 211^{38} + 207\cdot 211^{39} + 93\cdot 211^{40} + 31\cdot 211^{41} + 171\cdot 211^{42} + 7\cdot 211^{43} + 80\cdot 211^{44} + 93\cdot 211^{45} + 64\cdot 211^{46} + 47\cdot 211^{47} + 151\cdot 211^{48} + 89\cdot 211^{49} + 156\cdot 211^{50} + 57\cdot 211^{51} + 182\cdot 211^{52} + 77\cdot 211^{53} + 141\cdot 211^{54} + 80\cdot 211^{55} + 69\cdot 211^{56} + 22\cdot 211^{57} + 161\cdot 211^{59} + 76\cdot 211^{60} + 45\cdot 211^{61} + 80\cdot 211^{62} + 5\cdot 211^{63} + 148\cdot 211^{64} + 161\cdot 211^{65} + 108\cdot 211^{66} + 210\cdot 211^{67} + 38\cdot 211^{68} + 127\cdot 211^{69} + 98\cdot 211^{70} + 134\cdot 211^{71} + 8\cdot 211^{72} + 29\cdot 211^{73} + 72\cdot 211^{74} + 208\cdot 211^{75} + 119\cdot 211^{76} + 163\cdot 211^{77} + 105\cdot 211^{78} + 145\cdot 211^{79} + 77\cdot 211^{80} + 46\cdot 211^{81} + 116\cdot 211^{82} + 209\cdot 211^{83} + 176\cdot 211^{84} + 25\cdot 211^{85} + 64\cdot 211^{86} + 125\cdot 211^{87} + 19\cdot 211^{88} + 95\cdot 211^{89} + 101\cdot 211^{90} + 122\cdot 211^{91} + 102\cdot 211^{92} + 116\cdot 211^{93} + 69\cdot 211^{94} + 60\cdot 211^{95} + 165\cdot 211^{96} + 155\cdot 211^{97} +O\left(211^{ 98 }\right)$
$r_{ 2 }$ $=$ $ 114 + 186\cdot 211 + 82\cdot 211^{2} + 133\cdot 211^{3} + 191\cdot 211^{4} + 23\cdot 211^{5} + 180\cdot 211^{6} + 164\cdot 211^{7} + 3\cdot 211^{8} + 6\cdot 211^{9} + 91\cdot 211^{10} + 123\cdot 211^{11} + 164\cdot 211^{12} + 43\cdot 211^{13} + 157\cdot 211^{14} + 181\cdot 211^{15} + 122\cdot 211^{16} + 2\cdot 211^{17} + 4\cdot 211^{18} + 104\cdot 211^{19} + 5\cdot 211^{20} + 59\cdot 211^{21} + 6\cdot 211^{22} + 36\cdot 211^{23} + 5\cdot 211^{24} + 94\cdot 211^{25} + 133\cdot 211^{26} + 69\cdot 211^{27} + 50\cdot 211^{28} + 21\cdot 211^{29} + 133\cdot 211^{30} + 172\cdot 211^{31} + 63\cdot 211^{32} + 192\cdot 211^{33} + 130\cdot 211^{34} + 36\cdot 211^{35} + 210\cdot 211^{36} + 41\cdot 211^{38} + 21\cdot 211^{39} + 209\cdot 211^{40} + 17\cdot 211^{41} + 84\cdot 211^{42} + 16\cdot 211^{43} + 136\cdot 211^{44} + 13\cdot 211^{45} + 22\cdot 211^{46} + 47\cdot 211^{47} + 52\cdot 211^{48} + 150\cdot 211^{49} + 157\cdot 211^{50} + 23\cdot 211^{51} + 50\cdot 211^{52} + 129\cdot 211^{53} + 103\cdot 211^{54} + 167\cdot 211^{55} + 36\cdot 211^{56} + 43\cdot 211^{57} + 109\cdot 211^{58} + 96\cdot 211^{59} + 176\cdot 211^{60} + 63\cdot 211^{61} + 172\cdot 211^{62} + 50\cdot 211^{63} + 67\cdot 211^{64} + 100\cdot 211^{65} + 89\cdot 211^{66} + 121\cdot 211^{67} + 99\cdot 211^{68} + 17\cdot 211^{69} + 129\cdot 211^{70} + 104\cdot 211^{71} + 165\cdot 211^{72} + 28\cdot 211^{73} + 136\cdot 211^{74} + 41\cdot 211^{75} + 203\cdot 211^{76} + 140\cdot 211^{77} + 175\cdot 211^{78} + 175\cdot 211^{79} + 38\cdot 211^{80} + 61\cdot 211^{81} + 94\cdot 211^{82} + 81\cdot 211^{83} + 188\cdot 211^{84} + 197\cdot 211^{85} + 2\cdot 211^{86} + 181\cdot 211^{87} + 28\cdot 211^{88} + 188\cdot 211^{89} + 179\cdot 211^{90} + 136\cdot 211^{91} + 25\cdot 211^{92} + 90\cdot 211^{93} + 207\cdot 211^{94} + 105\cdot 211^{95} + 114\cdot 211^{96} + 72\cdot 211^{97} +O\left(211^{ 98 }\right)$
$r_{ 3 }$ $=$ $ 149 + 122\cdot 211 + 33\cdot 211^{2} + 77\cdot 211^{3} + 41\cdot 211^{4} + 69\cdot 211^{5} + 34\cdot 211^{6} + 202\cdot 211^{7} + 93\cdot 211^{8} + 60\cdot 211^{9} + 68\cdot 211^{10} + 137\cdot 211^{11} + 25\cdot 211^{12} + 29\cdot 211^{13} + 94\cdot 211^{14} + 141\cdot 211^{15} + 27\cdot 211^{16} + 57\cdot 211^{17} + 82\cdot 211^{18} + 107\cdot 211^{19} + 60\cdot 211^{20} + 204\cdot 211^{21} + 26\cdot 211^{22} + 169\cdot 211^{23} + 186\cdot 211^{24} + 38\cdot 211^{25} + 71\cdot 211^{26} + 119\cdot 211^{27} + 199\cdot 211^{28} + 117\cdot 211^{29} + 116\cdot 211^{30} + 109\cdot 211^{31} + 95\cdot 211^{32} + 77\cdot 211^{33} + 127\cdot 211^{35} + 183\cdot 211^{36} + 202\cdot 211^{37} + 107\cdot 211^{38} + 125\cdot 211^{39} + 64\cdot 211^{40} + 147\cdot 211^{41} + 13\cdot 211^{42} + 147\cdot 211^{43} + 63\cdot 211^{44} + 206\cdot 211^{45} + 208\cdot 211^{46} + 203\cdot 211^{47} + 208\cdot 211^{48} + 201\cdot 211^{49} + 123\cdot 211^{50} + 136\cdot 211^{51} + 139\cdot 211^{52} + 170\cdot 211^{53} + 56\cdot 211^{54} + 181\cdot 211^{55} + 70\cdot 211^{56} + 146\cdot 211^{57} + 20\cdot 211^{58} + 19\cdot 211^{59} + 206\cdot 211^{60} + 146\cdot 211^{61} + 96\cdot 211^{62} + 209\cdot 211^{63} + 192\cdot 211^{64} + 92\cdot 211^{65} + 62\cdot 211^{66} + 58\cdot 211^{67} + 55\cdot 211^{68} + 43\cdot 211^{69} + 93\cdot 211^{70} + 73\cdot 211^{71} + 103\cdot 211^{72} + 118\cdot 211^{73} + 32\cdot 211^{74} + 188\cdot 211^{75} + 145\cdot 211^{76} + 51\cdot 211^{77} + 65\cdot 211^{78} + 120\cdot 211^{79} + 198\cdot 211^{80} + 131\cdot 211^{81} + 32\cdot 211^{82} + 49\cdot 211^{83} + 84\cdot 211^{84} + 99\cdot 211^{85} + 5\cdot 211^{86} + 41\cdot 211^{87} + 14\cdot 211^{88} + 34\cdot 211^{89} + 11\cdot 211^{90} + 52\cdot 211^{91} + 126\cdot 211^{92} + 152\cdot 211^{93} + 131\cdot 211^{94} + 84\cdot 211^{95} + 165\cdot 211^{96} + 9\cdot 211^{97} +O\left(211^{ 98 }\right)$
$r_{ 4 }$ $=$ $ 21 a^{2} + 119 a + 115 + \left(207 a^{2} + 25 a + 176\right)\cdot 211 + \left(152 a^{2} + 90 a + 123\right)\cdot 211^{2} + \left(190 a^{2} + 172 a + 4\right)\cdot 211^{3} + \left(12 a^{2} + 107 a + 173\right)\cdot 211^{4} + \left(170 a^{2} + 133 a + 21\right)\cdot 211^{5} + \left(209 a^{2} + 138 a + 198\right)\cdot 211^{6} + \left(121 a^{2} + 7 a + 33\right)\cdot 211^{7} + \left(35 a^{2} + 180 a + 86\right)\cdot 211^{8} + \left(137 a^{2} + 100 a + 198\right)\cdot 211^{9} + \left(73 a^{2} + 34 a + 137\right)\cdot 211^{10} + \left(189 a^{2} + 162 a + 97\right)\cdot 211^{11} + \left(80 a^{2} + 10 a + 131\right)\cdot 211^{12} + \left(25 a^{2} + 25 a + 86\right)\cdot 211^{13} + \left(80 a^{2} + 65 a + 172\right)\cdot 211^{14} + \left(100 a^{2} + 134 a + 33\right)\cdot 211^{15} + \left(86 a^{2} + 127 a + 152\right)\cdot 211^{16} + \left(30 a^{2} + 119 a + 46\right)\cdot 211^{17} + \left(10 a^{2} + 127 a + 38\right)\cdot 211^{18} + \left(13 a^{2} + 85 a + 135\right)\cdot 211^{19} + \left(188 a^{2} + 43 a + 25\right)\cdot 211^{20} + \left(105 a^{2} + 22 a + 76\right)\cdot 211^{21} + \left(173 a^{2} + 122 a + 199\right)\cdot 211^{22} + \left(164 a^{2} + 70 a + 49\right)\cdot 211^{23} + \left(13 a^{2} + 41 a + 57\right)\cdot 211^{24} + \left(195 a^{2} + 101 a + 125\right)\cdot 211^{25} + \left(91 a^{2} + 95 a + 3\right)\cdot 211^{26} + \left(19 a^{2} + 89 a + 10\right)\cdot 211^{27} + \left(151 a^{2} + 199 a + 127\right)\cdot 211^{28} + \left(106 a^{2} + 140 a + 126\right)\cdot 211^{29} + \left(198 a^{2} + 206 a + 78\right)\cdot 211^{30} + \left(174 a + 162\right)\cdot 211^{31} + \left(60 a^{2} + 29 a + 44\right)\cdot 211^{32} + \left(88 a^{2} + 183 a + 32\right)\cdot 211^{33} + \left(201 a^{2} + 168 a + 126\right)\cdot 211^{34} + \left(146 a^{2} + 199 a + 95\right)\cdot 211^{35} + \left(a^{2} + 63 a + 122\right)\cdot 211^{36} + \left(54 a^{2} + 176 a + 201\right)\cdot 211^{37} + \left(87 a^{2} + 183 a + 35\right)\cdot 211^{38} + \left(140 a^{2} + 138 a + 97\right)\cdot 211^{39} + \left(14 a^{2} + 47 a + 80\right)\cdot 211^{40} + \left(167 a^{2} + 179 a + 157\right)\cdot 211^{41} + \left(150 a^{2} + 12 a + 109\right)\cdot 211^{42} + \left(17 a^{2} + 17 a + 62\right)\cdot 211^{43} + \left(120 a^{2} + 97 a + 149\right)\cdot 211^{44} + \left(34 a^{2} + 206 a + 71\right)\cdot 211^{45} + \left(39 a^{2} + 125 a + 96\right)\cdot 211^{46} + \left(169 a^{2} + 133 a + 116\right)\cdot 211^{47} + \left(148 a^{2} + 203 a + 74\right)\cdot 211^{48} + \left(164 a^{2} + 83 a + 202\right)\cdot 211^{49} + \left(16 a^{2} + 109 a + 23\right)\cdot 211^{50} + \left(28 a^{2} + 7 a + 198\right)\cdot 211^{51} + \left(35 a^{2} + 139 a + 90\right)\cdot 211^{52} + \left(105 a^{2} + 199 a + 9\right)\cdot 211^{53} + \left(162 a^{2} + 98 a + 68\right)\cdot 211^{54} + \left(71 a^{2} + 44 a + 46\right)\cdot 211^{55} + \left(195 a^{2} + 183 a + 121\right)\cdot 211^{56} + \left(125 a^{2} + 43 a + 190\right)\cdot 211^{57} + \left(32 a^{2} + 109 a + 76\right)\cdot 211^{58} + \left(123 a^{2} + a + 197\right)\cdot 211^{59} + \left(85 a^{2} + 151 a + 170\right)\cdot 211^{60} + \left(129 a^{2} + 198 a + 181\right)\cdot 211^{61} + \left(178 a^{2} + 86 a + 136\right)\cdot 211^{62} + \left(72 a^{2} + 123 a + 162\right)\cdot 211^{63} + \left(85 a^{2} + 2 a + 76\right)\cdot 211^{64} + \left(135 a^{2} + 170 a + 87\right)\cdot 211^{65} + \left(33 a^{2} + 11 a + 197\right)\cdot 211^{66} + \left(191 a^{2} + 144 a + 45\right)\cdot 211^{67} + \left(15 a^{2} + 110 a + 169\right)\cdot 211^{68} + \left(179 a^{2} + 160 a + 14\right)\cdot 211^{69} + \left(86 a^{2} + 43 a + 103\right)\cdot 211^{70} + \left(55 a^{2} + 96 a + 33\right)\cdot 211^{71} + \left(82 a^{2} + 163 a + 43\right)\cdot 211^{72} + \left(172 a^{2} + 97 a + 145\right)\cdot 211^{73} + \left(9 a^{2} + 78 a + 92\right)\cdot 211^{74} + \left(90 a^{2} + 126 a + 153\right)\cdot 211^{75} + \left(18 a^{2} + 150 a + 77\right)\cdot 211^{76} + \left(17 a^{2} + 121 a + 65\right)\cdot 211^{77} + \left(108 a^{2} + 65\right)\cdot 211^{78} + \left(a^{2} + 98 a + 138\right)\cdot 211^{79} + \left(89 a^{2} + 164 a + 2\right)\cdot 211^{80} + \left(65 a^{2} + 149 a + 200\right)\cdot 211^{81} + \left(45 a^{2} + 38 a + 103\right)\cdot 211^{82} + \left(37 a^{2} + 204 a + 142\right)\cdot 211^{83} + \left(44 a^{2} + 193 a + 7\right)\cdot 211^{84} + \left(130 a^{2} + 205 a + 138\right)\cdot 211^{85} + \left(31 a^{2} + 174 a + 85\right)\cdot 211^{86} + \left(155 a^{2} + 156 a + 76\right)\cdot 211^{87} + \left(155 a^{2} + 135 a + 160\right)\cdot 211^{88} + \left(58 a^{2} + a + 16\right)\cdot 211^{89} + \left(156 a^{2} + 94 a + 68\right)\cdot 211^{90} + \left(179 a^{2} + 170 a + 90\right)\cdot 211^{91} + \left(58 a^{2} + 7 a + 173\right)\cdot 211^{92} + \left(204 a^{2} + 32 a + 165\right)\cdot 211^{93} + \left(52 a^{2} + 70 a + 135\right)\cdot 211^{94} + \left(16 a^{2} + 210 a + 78\right)\cdot 211^{95} + \left(52 a^{2} + 157 a + 131\right)\cdot 211^{96} + \left(80 a^{2} + 66 a + 118\right)\cdot 211^{97} +O\left(211^{ 98 }\right)$
$r_{ 5 }$ $=$ $ 28 a^{2} + 69 a + 3 + \left(90 a^{2} + 120 a + 16\right)\cdot 211 + \left(72 a^{2} + 5 a + 190\right)\cdot 211^{2} + \left(51 a^{2} + 47 a + 58\right)\cdot 211^{3} + \left(150 a^{2} + 169 a + 203\right)\cdot 211^{4} + \left(192 a^{2} + 180 a + 101\right)\cdot 211^{5} + \left(63 a^{2} + 45 a + 120\right)\cdot 211^{6} + \left(105 a^{2} + 22 a + 164\right)\cdot 211^{7} + \left(71 a^{2} + 104 a + 160\right)\cdot 211^{8} + \left(109 a^{2} + 114 a + 155\right)\cdot 211^{9} + \left(20 a^{2} + 37 a + 173\right)\cdot 211^{10} + \left(165 a^{2} + 66 a + 31\right)\cdot 211^{11} + \left(206 a^{2} + 47 a + 3\right)\cdot 211^{12} + \left(157 a^{2} + 174 a + 15\right)\cdot 211^{13} + \left(63 a^{2} + 155 a + 144\right)\cdot 211^{14} + \left(74 a^{2} + 103 a + 183\right)\cdot 211^{15} + \left(42 a^{2} + 162 a + 42\right)\cdot 211^{16} + \left(38 a^{2} + 8 a + 195\right)\cdot 211^{17} + \left(148 a^{2} + 93 a + 105\right)\cdot 211^{18} + \left(44 a^{2} + 12 a + 4\right)\cdot 211^{19} + \left(145 a^{2} + 103 a + 178\right)\cdot 211^{20} + \left(31 a^{2} + 26 a + 202\right)\cdot 211^{21} + \left(18 a^{2} + 182 a\right)\cdot 211^{22} + \left(202 a^{2} + 64 a + 79\right)\cdot 211^{23} + \left(114 a^{2} + 36 a + 205\right)\cdot 211^{24} + \left(43 a^{2} + 77 a + 132\right)\cdot 211^{25} + \left(207 a^{2} + 186 a + 39\right)\cdot 211^{26} + \left(200 a^{2} + 26 a + 131\right)\cdot 211^{27} + \left(81 a^{2} + 62 a + 201\right)\cdot 211^{28} + \left(162 a^{2} + 200 a + 87\right)\cdot 211^{29} + \left(29 a^{2} + 98 a + 203\right)\cdot 211^{30} + \left(184 a^{2} + 40 a + 163\right)\cdot 211^{31} + \left(184 a^{2} + 140 a + 108\right)\cdot 211^{32} + \left(129 a^{2} + 39 a + 59\right)\cdot 211^{33} + \left(102 a^{2} + 10 a + 13\right)\cdot 211^{34} + \left(13 a^{2} + 48 a + 190\right)\cdot 211^{35} + \left(141 a^{2} + 178 a + 166\right)\cdot 211^{36} + \left(58 a^{2} + 107 a + 61\right)\cdot 211^{37} + \left(139 a^{2} + 115 a + 105\right)\cdot 211^{38} + \left(168 a^{2} + 116 a + 126\right)\cdot 211^{39} + \left(48 a^{2} + 95 a + 92\right)\cdot 211^{40} + \left(134 a^{2} + 127 a + 108\right)\cdot 211^{41} + \left(45 a^{2} + 158 a + 62\right)\cdot 211^{42} + \left(19 a^{2} + 97 a + 70\right)\cdot 211^{43} + \left(3 a^{2} + 5 a + 62\right)\cdot 211^{44} + \left(30 a^{2} + 158 a + 191\right)\cdot 211^{45} + \left(118 a^{2} + 93 a + 14\right)\cdot 211^{46} + \left(160 a^{2} + 42 a + 83\right)\cdot 211^{47} + \left(79 a^{2} + 83 a + 22\right)\cdot 211^{48} + \left(10 a^{2} + 118 a + 165\right)\cdot 211^{49} + \left(145 a^{2} + 167 a + 45\right)\cdot 211^{50} + \left(67 a^{2} + 74 a + 138\right)\cdot 211^{51} + \left(104 a^{2} + 161 a + 111\right)\cdot 211^{52} + \left(29 a^{2} + 68 a + 114\right)\cdot 211^{53} + \left(154 a^{2} + 86 a + 42\right)\cdot 211^{54} + \left(196 a^{2} + 63 a + 98\right)\cdot 211^{55} + \left(188 a^{2} + 111 a + 191\right)\cdot 211^{56} + \left(137 a^{2} + 91 a + 160\right)\cdot 211^{57} + \left(146 a^{2} + 83 a + 118\right)\cdot 211^{58} + \left(128 a^{2} + 75 a + 116\right)\cdot 211^{59} + \left(130 a^{2} + 63 a + 34\right)\cdot 211^{60} + \left(204 a^{2} + 133 a + 178\right)\cdot 211^{61} + \left(152 a^{2} + 106 a + 188\right)\cdot 211^{62} + \left(166 a^{2} + 123 a + 208\right)\cdot 211^{63} + \left(115 a^{2} + 163 a + 195\right)\cdot 211^{64} + \left(40 a^{2} + 83 a + 28\right)\cdot 211^{65} + \left(137 a^{2} + 23 a + 84\right)\cdot 211^{66} + \left(202 a^{2} + 102 a + 208\right)\cdot 211^{67} + \left(147 a^{2} + 37 a + 54\right)\cdot 211^{68} + \left(94 a^{2} + 108 a + 6\right)\cdot 211^{69} + \left(209 a^{2} + 173 a + 185\right)\cdot 211^{70} + \left(38 a^{2} + 14 a + 128\right)\cdot 211^{71} + \left(179 a^{2} + 165 a + 142\right)\cdot 211^{72} + \left(41 a^{2} + 66 a + 81\right)\cdot 211^{73} + \left(168 a^{2} + 55 a + 64\right)\cdot 211^{74} + \left(84 a^{2} + 82 a + 74\right)\cdot 211^{75} + \left(19 a^{2} + 48 a + 27\right)\cdot 211^{76} + \left(86 a^{2} + 158 a + 94\right)\cdot 211^{77} + \left(99 a^{2} + 151 a + 25\right)\cdot 211^{78} + \left(11 a^{2} + 92 a + 13\right)\cdot 211^{79} + \left(52 a^{2} + 89 a + 10\right)\cdot 211^{80} + \left(30 a^{2} + 22 a + 129\right)\cdot 211^{81} + \left(101 a^{2} + 139 a + 80\right)\cdot 211^{82} + \left(131 a^{2} + 126 a + 39\right)\cdot 211^{83} + \left(158 a^{2} + 99 a + 183\right)\cdot 211^{84} + \left(7 a^{2} + 114 a + 78\right)\cdot 211^{85} + \left(65 a^{2} + 87 a + 89\right)\cdot 211^{86} + \left(92 a^{2} + 144 a + 67\right)\cdot 211^{87} + \left(69 a^{2} + 7 a + 189\right)\cdot 211^{88} + \left(165 a^{2} + 95 a + 105\right)\cdot 211^{89} + \left(43 a^{2} + 158 a + 171\right)\cdot 211^{90} + \left(63 a^{2} + 157 a + 129\right)\cdot 211^{91} + \left(72 a^{2} + 175 a + 127\right)\cdot 211^{92} + \left(126 a^{2} + 166 a + 14\right)\cdot 211^{93} + \left(169 a^{2} + 118 a + 95\right)\cdot 211^{94} + \left(123 a^{2} + 24\right)\cdot 211^{95} + \left(106 a^{2} + 189 a + 2\right)\cdot 211^{96} + \left(180 a^{2} + 191 a + 9\right)\cdot 211^{97} +O\left(211^{ 98 }\right)$
$r_{ 6 }$ $=$ $ 47 a^{2} + 67 a + 9 + \left(140 a^{2} + 71 a + 17\right)\cdot 211 + \left(149 a^{2} + 153 a + 49\right)\cdot 211^{2} + \left(163 a^{2} + 140 a + 109\right)\cdot 211^{3} + \left(133 a^{2} + 46 a + 193\right)\cdot 211^{4} + \left(201 a^{2} + 207 a + 63\right)\cdot 211^{5} + \left(128 a^{2} + 80 a + 90\right)\cdot 211^{6} + \left(66 a^{2} + 197 a + 30\right)\cdot 211^{7} + \left(111 a^{2} + 195 a + 187\right)\cdot 211^{8} + \left(132 a^{2} + 122 a + 121\right)\cdot 211^{9} + \left(39 a^{2} + 134 a + 92\right)\cdot 211^{10} + \left(52 a^{2} + 163 a + 55\right)\cdot 211^{11} + \left(52 a^{2} + 93\right)\cdot 211^{12} + \left(167 a^{2} + 168 a + 205\right)\cdot 211^{13} + \left(168 a^{2} + 81 a + 149\right)\cdot 211^{14} + \left(126 a^{2} + 5 a + 209\right)\cdot 211^{15} + \left(100 a^{2} + 102 a + 170\right)\cdot 211^{16} + \left(96 a^{2} + 61 a + 134\right)\cdot 211^{17} + \left(182 a^{2} + 149 a + 197\right)\cdot 211^{18} + \left(196 a^{2} + 40 a + 98\right)\cdot 211^{19} + \left(83 a^{2} + 197 a + 27\right)\cdot 211^{20} + \left(145 a^{2} + 21 a + 199\right)\cdot 211^{21} + \left(155 a^{2} + 34 a + 34\right)\cdot 211^{22} + \left(49 a^{2} + 153 a + 37\right)\cdot 211^{23} + \left(143 a^{2} + 151 a + 89\right)\cdot 211^{24} + \left(105 a^{2} + 38 a + 76\right)\cdot 211^{25} + \left(115 a^{2} + 193 a + 105\right)\cdot 211^{26} + \left(179 a^{2} + 144 a + 12\right)\cdot 211^{27} + \left(122 a^{2} + 185 a + 19\right)\cdot 211^{28} + \left(101 a^{2} + 168 a + 190\right)\cdot 211^{29} + \left(159 a^{2} + 24 a + 96\right)\cdot 211^{30} + \left(38 a^{2} + 6 a + 142\right)\cdot 211^{31} + \left(149 a^{2} + 88 a + 163\right)\cdot 211^{32} + \left(152 a^{2} + 141 a + 47\right)\cdot 211^{33} + \left(175 a^{2} + 40 a + 162\right)\cdot 211^{34} + \left(112 a^{2} + 7 a + 190\right)\cdot 211^{35} + \left(72 a^{2} + 114 a + 5\right)\cdot 211^{36} + \left(110 a^{2} + 21 a + 136\right)\cdot 211^{37} + \left(210 a^{2} + 67 a + 59\right)\cdot 211^{38} + \left(134 a^{2} + 143 a + 160\right)\cdot 211^{39} + \left(206 a^{2} + 63 a + 195\right)\cdot 211^{40} + \left(18 a^{2} + 106 a + 170\right)\cdot 211^{41} + \left(57 a^{2} + 24 a + 195\right)\cdot 211^{42} + \left(3 a^{2} + 22 a + 183\right)\cdot 211^{43} + \left(139 a^{2} + 2 a + 33\right)\cdot 211^{44} + \left(17 a^{2} + 29 a + 49\right)\cdot 211^{45} + \left(203 a^{2} + 30 a + 174\right)\cdot 211^{46} + \left(187 a^{2} + 129 a\right)\cdot 211^{47} + \left(170 a^{2} + 117 a + 104\right)\cdot 211^{48} + \left(83 a^{2} + 154 a + 94\right)\cdot 211^{49} + \left(127 a^{2} + 90 a + 171\right)\cdot 211^{50} + \left(158 a^{2} + 97 a + 90\right)\cdot 211^{51} + \left(172 a^{2} + 153 a + 63\right)\cdot 211^{52} + \left(35 a^{2} + 66 a + 198\right)\cdot 211^{53} + \left(208 a^{2} + 206 a + 128\right)\cdot 211^{54} + \left(135 a^{2} + 200 a + 61\right)\cdot 211^{55} + \left(58 a^{2} + 9\right)\cdot 211^{56} + \left(119 a^{2} + 173 a + 41\right)\cdot 211^{57} + \left(158 a^{2} + 74 a + 104\right)\cdot 211^{58} + \left(111 a^{2} + 4 a + 41\right)\cdot 211^{59} + \left(193 a^{2} + 84 a + 174\right)\cdot 211^{60} + \left(65 a^{2} + 104 a + 26\right)\cdot 211^{61} + \left(158 a^{2} + 72 a + 180\right)\cdot 211^{62} + \left(55 a^{2} + 117 a + 139\right)\cdot 211^{63} + \left(171 a^{2} + 205 a + 50\right)\cdot 211^{64} + \left(37 a^{2} + 7 a + 168\right)\cdot 211^{65} + \left(104 a^{2} + 179 a + 150\right)\cdot 211^{66} + \left(188 a^{2} + 129 a + 112\right)\cdot 211^{67} + \left(194 a^{2} + 140 a + 126\right)\cdot 211^{68} + \left(163 a^{2} + 121 a + 205\right)\cdot 211^{69} + \left(25 a^{2} + 51 a + 91\right)\cdot 211^{70} + \left(43 a^{2} + 146 a + 87\right)\cdot 211^{71} + \left(169 a^{2} + 73 a + 18\right)\cdot 211^{72} + \left(198 a^{2} + 127 a + 110\right)\cdot 211^{73} + \left(210 a^{2} + 21 a + 79\right)\cdot 211^{74} + \left(95 a^{2} + 134 a + 161\right)\cdot 211^{75} + \left(131 a^{2} + 116 a + 87\right)\cdot 211^{76} + \left(155 a^{2} + 168 a + 109\right)\cdot 211^{77} + \left(112 a^{2} + 82 a + 71\right)\cdot 211^{78} + \left(79 a^{2} + 181 a + 31\right)\cdot 211^{79} + \left(20 a^{2} + 103 a + 122\right)\cdot 211^{80} + \left(66 a^{2} + 11 a + 130\right)\cdot 211^{81} + \left(134 a^{2} + 177 a + 11\right)\cdot 211^{82} + \left(4 a^{2} + 121 a + 99\right)\cdot 211^{83} + \left(204 a^{2} + 76 a + 150\right)\cdot 211^{84} + \left(60 a^{2} + 157 a + 45\right)\cdot 211^{85} + \left(27 a^{2} + 40 a + 150\right)\cdot 211^{86} + \left(134 a^{2} + 71 a + 118\right)\cdot 211^{87} + \left(117 a^{2} + 131 a + 109\right)\cdot 211^{88} + \left(155 a^{2} + 178 a + 75\right)\cdot 211^{89} + \left(82 a^{2} + 165 a + 40\right)\cdot 211^{90} + \left(41 a^{2} + 120 a + 187\right)\cdot 211^{91} + \left(95 a^{2} + 64 a + 10\right)\cdot 211^{92} + \left(204 a^{2} + 73 a + 166\right)\cdot 211^{93} + \left(190 a^{2} + 70 a + 108\right)\cdot 211^{94} + \left(203 a^{2} + 103 a + 47\right)\cdot 211^{95} + \left(169 a^{2} + 54 a + 7\right)\cdot 211^{96} + \left(81 a^{2} + 206 a + 191\right)\cdot 211^{97} +O\left(211^{ 98 }\right)$
$r_{ 7 }$ $=$ $ 58 a^{2} + 92 a + 43 + \left(5 a^{2} + 25 a + 184\right)\cdot 211 + \left(13 a^{2} + 85 a + 110\right)\cdot 211^{2} + \left(178 a^{2} + 8 a + 157\right)\cdot 211^{3} + \left(121 a^{2} + 147 a + 24\right)\cdot 211^{4} + \left(198 a^{2} + 6 a + 180\right)\cdot 211^{5} + \left(92 a^{2} + 83 a + 88\right)\cdot 211^{6} + \left(92 a^{2} + 205 a + 147\right)\cdot 211^{7} + \left(158 a^{2} + 80 a + 65\right)\cdot 211^{8} + \left(68 a^{2} + 117 a + 31\right)\cdot 211^{9} + \left(31 a^{2} + 183 a + 188\right)\cdot 211^{10} + \left(39 a^{2} + 202 a + 74\right)\cdot 211^{11} + \left(101 a^{2} + 168 a + 73\right)\cdot 211^{12} + \left(177 a^{2} + 56 a + 111\right)\cdot 211^{13} + \left(156 a^{2} + 201 a + 127\right)\cdot 211^{14} + \left(92 a^{2} + 155 a + 67\right)\cdot 211^{15} + \left(124 a^{2} + 152\right)\cdot 211^{16} + \left(21 a^{2} + 83 a + 102\right)\cdot 211^{17} + \left(148 a^{2} + 26 a + 35\right)\cdot 211^{18} + \left(187 a^{2} + 105 a + 195\right)\cdot 211^{19} + \left(55 a^{2} + 177 a + 58\right)\cdot 211^{20} + \left(96 a^{2} + 174 a + 148\right)\cdot 211^{21} + \left(54 a^{2} + 210 a + 119\right)\cdot 211^{22} + \left(167 a^{2} + 62 a + 32\right)\cdot 211^{23} + \left(157 a^{2} + 89 a + 192\right)\cdot 211^{24} + \left(110 a^{2} + 76 a + 81\right)\cdot 211^{25} + \left(136 a^{2} + 121 a + 156\right)\cdot 211^{26} + \left(152 a^{2} + 4 a + 66\right)\cdot 211^{27} + \left(80 a^{2} + 129 a + 59\right)\cdot 211^{28} + \left(86 a^{2} + 161 a + 127\right)\cdot 211^{29} + \left(100 a^{2} + 109 a + 86\right)\cdot 211^{30} + \left(163 a^{2} + 209 a + 136\right)\cdot 211^{31} + \left(176 a^{2} + 35 a + 27\right)\cdot 211^{32} + \left(22 a^{2} + 45 a + 198\right)\cdot 211^{33} + \left(31 a^{2} + 101 a + 128\right)\cdot 211^{34} + \left(155 a^{2} + 123 a + 97\right)\cdot 211^{35} + \left(37 a^{2} + 159 a + 99\right)\cdot 211^{36} + \left(147 a^{2} + 50 a + 109\right)\cdot 211^{37} + \left(184 a^{2} + 119 a + 95\right)\cdot 211^{38} + \left(62 a^{2} + 59 a + 196\right)\cdot 211^{39} + \left(207 a^{2} + 160 a + 92\right)\cdot 211^{40} + \left(177 a^{2} + 80 a + 96\right)\cdot 211^{41} + \left(33 a^{2} + 9 a + 187\right)\cdot 211^{42} + \left(208 a^{2} + 53 a + 40\right)\cdot 211^{43} + \left(172 a^{2} + 195 a + 148\right)\cdot 211^{44} + \left(111 a^{2} + 132 a + 159\right)\cdot 211^{45} + \left(131 a^{2} + 15 a + 32\right)\cdot 211^{46} + \left(169 a^{2} + 180 a + 95\right)\cdot 211^{47} + \left(11 a^{2} + 161 a + 72\right)\cdot 211^{48} + \left(51 a^{2} + 102 a + 8\right)\cdot 211^{49} + \left(4 a^{2} + 87 a + 69\right)\cdot 211^{50} + \left(4 a^{2} + 148 a + 53\right)\cdot 211^{51} + \left(2 a^{2} + 168 a + 186\right)\cdot 211^{52} + \left(60 a^{2} + 128 a + 84\right)\cdot 211^{53} + \left(91 a^{2} + 130 a + 99\right)\cdot 211^{54} + \left(123 a^{2} + 25 a\right)\cdot 211^{55} + \left(149 a^{2} + 184 a + 139\right)\cdot 211^{56} + \left(113 a^{2} + 22 a + 128\right)\cdot 211^{57} + \left(94 a^{2} + 105 a + 119\right)\cdot 211^{58} + \left(24 a^{2} + 11 a + 188\right)\cdot 211^{59} + \left(64 a^{2} + 22 a + 156\right)\cdot 211^{60} + \left(200 a^{2} + 37 a + 31\right)\cdot 211^{61} + \left(139 a^{2} + 121 a + 101\right)\cdot 211^{62} + \left(35 a^{2} + 61 a + 104\right)\cdot 211^{63} + \left(43 a^{2} + 49 a + 169\right)\cdot 211^{64} + \left(75 a^{2} + 116 a + 4\right)\cdot 211^{65} + \left(61 a^{2} + 52 a + 194\right)\cdot 211^{66} + \left(55 a^{2} + 207 a + 11\right)\cdot 211^{67} + \left(174 a^{2} + 18 a + 90\right)\cdot 211^{68} + \left(207 a^{2} + 185 a + 16\right)\cdot 211^{69} + \left(182 a^{2} + 131 a + 9\right)\cdot 211^{70} + \left(199 a^{2} + 40 a + 62\right)\cdot 211^{71} + \left(156 a^{2} + 132 a + 183\right)\cdot 211^{72} + \left(82 a^{2} + 178 a + 65\right)\cdot 211^{73} + \left(156 a^{2} + 159 a + 189\right)\cdot 211^{74} + \left(189 a^{2} + 152 a + 143\right)\cdot 211^{75} + \left(115 a^{2} + 65 a + 85\right)\cdot 211^{76} + \left(114 a^{2} + 17 a + 202\right)\cdot 211^{77} + \left(6 a^{2} + 18 a + 182\right)\cdot 211^{78} + \left(135 a^{2} + 19 a + 177\right)\cdot 211^{79} + \left(202 a^{2} + 6 a + 210\right)\cdot 211^{80} + \left(95 a^{2} + 191 a + 75\right)\cdot 211^{81} + \left(185 a^{2} + 73 a + 52\right)\cdot 211^{82} + \left(190 a^{2} + 27 a + 48\right)\cdot 211^{83} + \left(173 a^{2} + 205 a + 133\right)\cdot 211^{84} + \left(66 a^{2} + 59 a + 157\right)\cdot 211^{85} + \left(199 a^{2} + 62 a + 127\right)\cdot 211^{86} + \left(163 a^{2} + 130 a + 92\right)\cdot 211^{87} + \left(190 a^{2} + 18 a + 210\right)\cdot 211^{88} + \left(145 a^{2} + 63 a + 79\right)\cdot 211^{89} + \left(98 a^{2} + 83 a + 174\right)\cdot 211^{90} + \left(56 a^{2} + 166 a + 120\right)\cdot 211^{91} + \left(108 a^{2} + 67 a + 175\right)\cdot 211^{92} + \left(154 a^{2} + 128 a + 192\right)\cdot 211^{93} + \left(25 a^{2} + 64 a + 43\right)\cdot 211^{94} + \left(150 a^{2} + 112 a + 200\right)\cdot 211^{95} + \left(208 a^{2} + 89 a + 67\right)\cdot 211^{96} + \left(154 a^{2} + 41 a + 45\right)\cdot 211^{97} +O\left(211^{ 98 }\right)$
$r_{ 8 }$ $=$ $ 125 a^{2} + 50 a + 62 + \left(115 a^{2} + 65 a + 120\right)\cdot 211 + \left(125 a^{2} + 120 a + 190\right)\cdot 211^{2} + \left(192 a^{2} + 155 a + 176\right)\cdot 211^{3} + \left(149 a^{2} + 105 a + 202\right)\cdot 211^{4} + \left(30 a^{2} + 23 a + 96\right)\cdot 211^{5} + \left(54 a^{2} + 82 a + 107\right)\cdot 211^{6} + \left(13 a^{2} + 194 a + 182\right)\cdot 211^{7} + \left(192 a^{2} + 25 a + 180\right)\cdot 211^{8} + \left(32 a^{2} + 190 a + 53\right)\cdot 211^{9} + \left(159 a^{2} + 200 a + 77\right)\cdot 211^{10} + \left(6 a^{2} + 152 a + 172\right)\cdot 211^{11} + \left(114 a^{2} + 205 a + 160\right)\cdot 211^{12} + \left(86 a^{2} + 190 a + 130\right)\cdot 211^{13} + \left(201 a^{2} + 64 a + 116\right)\cdot 211^{14} + \left(43 a^{2} + 162 a + 2\right)\cdot 211^{15} + \left(44 a^{2} + 47 a + 186\right)\cdot 211^{16} + \left(151 a^{2} + 119 a + 134\right)\cdot 211^{17} + \left(125 a^{2} + 91 a + 5\right)\cdot 211^{18} + \left(189 a^{2} + 93 a + 57\right)\cdot 211^{19} + \left(9 a^{2} + 141 a + 138\right)\cdot 211^{20} + \left(83 a^{2} + 9 a + 130\right)\cdot 211^{21} + \left(138 a^{2} + 29 a + 20\right)\cdot 211^{22} + \left(52 a^{2} + 83 a + 161\right)\cdot 211^{23} + \left(149 a^{2} + 85 a + 180\right)\cdot 211^{24} + \left(56 a^{2} + 57 a + 9\right)\cdot 211^{25} + \left(78 a^{2} + 114 a + 149\right)\cdot 211^{26} + \left(68 a^{2} + 179 a + 24\right)\cdot 211^{27} + \left(48 a^{2} + 19 a + 16\right)\cdot 211^{28} + \left(173 a^{2} + 60 a + 32\right)\cdot 211^{29} + \left(80 a^{2} + 2 a + 201\right)\cdot 211^{30} + \left(74 a^{2} + 172 a + 87\right)\cdot 211^{31} + \left(60 a^{2} + 34 a + 83\right)\cdot 211^{32} + \left(58 a^{2} + 126 a + 34\right)\cdot 211^{33} + \left(77 a^{2} + 99 a + 120\right)\cdot 211^{34} + \left(42 a^{2} + 39 a + 17\right)\cdot 211^{35} + \left(32 a^{2} + 84 a + 92\right)\cdot 211^{36} + \left(5 a^{2} + 52 a + 201\right)\cdot 211^{37} + \left(98 a^{2} + 187 a + 190\right)\cdot 211^{38} + \left(190 a^{2} + 34 a + 14\right)\cdot 211^{39} + \left(165 a^{2} + 166 a + 108\right)\cdot 211^{40} + \left(109 a^{2} + 2 a + 5\right)\cdot 211^{41} + \left(131 a^{2} + 43 a + 177\right)\cdot 211^{42} + \left(194 a^{2} + 60 a + 22\right)\cdot 211^{43} + \left(34 a^{2} + 10 a + 175\right)\cdot 211^{44} + \left(69 a^{2} + 131 a + 102\right)\cdot 211^{45} + \left(172 a^{2} + 101 a + 157\right)\cdot 211^{46} + \left(91 a^{2} + 199 a + 61\right)\cdot 211^{47} + \left(119 a^{2} + 176 a + 75\right)\cdot 211^{48} + \left(149 a^{2} + 200 a + 69\right)\cdot 211^{49} + \left(61 a^{2} + 166 a + 75\right)\cdot 211^{50} + \left(139 a^{2} + 198 a + 163\right)\cdot 211^{51} + \left(104 a^{2} + 91 a + 41\right)\cdot 211^{52} + \left(121 a^{2} + 13 a + 26\right)\cdot 211^{53} + \left(176 a^{2} + 205 a + 2\right)\cdot 211^{54} + \left(101 a^{2} + 121 a + 42\right)\cdot 211^{55} + \left(83 a^{2} + 126 a + 121\right)\cdot 211^{56} + \left(170 a^{2} + 96 a + 63\right)\cdot 211^{57} + \left(180 a^{2} + 22 a + 164\right)\cdot 211^{58} + \left(57 a^{2} + 124 a + 162\right)\cdot 211^{59} + \left(16 a^{2} + 125 a + 22\right)\cdot 211^{60} + \left(17 a^{2} + 40 a + 139\right)\cdot 211^{61} + \left(129 a^{2} + 194 a + 86\right)\cdot 211^{62} + \left(8 a^{2} + 25 a + 68\right)\cdot 211^{63} + \left(52 a^{2} + 209 a + 181\right)\cdot 211^{64} + \left(95 a^{2} + 10 a + 101\right)\cdot 211^{65} + \left(12 a^{2} + 135 a + 58\right)\cdot 211^{66} + \left(164 a^{2} + 112 a + 16\right)\cdot 211^{67} + \left(99 a^{2} + 154 a + 61\right)\cdot 211^{68} + \left(119 a^{2} + 128 a + 39\right)\cdot 211^{69} + \left(29 a^{2} + 116 a + 156\right)\cdot 211^{70} + \left(183 a^{2} + 155 a + 39\right)\cdot 211^{71} + \left(85 a^{2} + 124 a + 18\right)\cdot 211^{72} + \left(86 a^{2} + 176 a + 141\right)\cdot 211^{73} + \left(97 a^{2} + 206 a + 110\right)\cdot 211^{74} + \left(147 a^{2} + 186 a + 87\right)\cdot 211^{75} + \left(75 a^{2} + 96 a + 102\right)\cdot 211^{76} + \left(10 a^{2} + 35 a + 63\right)\cdot 211^{77} + \left(105 a^{2} + 41 a + 103\right)\cdot 211^{78} + \left(64 a^{2} + 99 a + 13\right)\cdot 211^{79} + \left(167 a^{2} + 115 a + 23\right)\cdot 211^{80} + \left(84 a^{2} + 208 a + 61\right)\cdot 211^{81} + \left(135 a^{2} + 208 a + 126\right)\cdot 211^{82} + \left(99 a^{2} + 56 a + 137\right)\cdot 211^{83} + \left(89 a^{2} + 117 a + 20\right)\cdot 211^{84} + \left(136 a^{2} + 36 a + 180\right)\cdot 211^{85} + \left(157 a^{2} + 61 a + 1\right)\cdot 211^{86} + \left(165 a^{2} + 147 a + 95\right)\cdot 211^{87} + \left(161 a^{2} + 184 a + 101\right)\cdot 211^{88} + \left(110 a^{2} + 52 a + 103\right)\cdot 211^{89} + \left(68 a^{2} + 180 a + 204\right)\cdot 211^{90} + \left(91 a^{2} + 97 a + 96\right)\cdot 211^{91} + \left(30 a^{2} + 178 a + 1\right)\cdot 211^{92} + \left(141 a^{2} + 126 a + 175\right)\cdot 211^{93} + \left(15 a^{2} + 27 a + 100\right)\cdot 211^{94} + \left(148 a^{2} + 98 a + 197\right)\cdot 211^{95} + \left(106 a^{2} + 143 a + 142\right)\cdot 211^{96} + \left(86 a^{2} + 188 a + 94\right)\cdot 211^{97} +O\left(211^{ 98 }\right)$
$r_{ 9 }$ $=$ $ 143 a^{2} + 25 a + 137 + \left(74 a^{2} + 114 a + 140\right)\cdot 211 + \left(119 a^{2} + 178 a + 8\right)\cdot 211^{2} + \left(67 a^{2} + 108 a + 192\right)\cdot 211^{3} + \left(64 a^{2} + 56 a + 100\right)\cdot 211^{4} + \left(50 a^{2} + 81 a + 143\right)\cdot 211^{5} + \left(83 a^{2} + 202 a + 99\right)\cdot 211^{6} + \left(22 a^{2} + 5 a + 182\right)\cdot 211^{7} + \left(64 a^{2} + 46 a + 53\right)\cdot 211^{8} + \left(152 a^{2} + 198 a + 148\right)\cdot 211^{9} + \left(97 a^{2} + 41 a + 99\right)\cdot 211^{10} + \left(180 a^{2} + 96 a + 15\right)\cdot 211^{11} + \left(77 a^{2} + 199 a + 57\right)\cdot 211^{12} + \left(18 a^{2} + 17 a + 77\right)\cdot 211^{13} + \left(173 a^{2} + 64 a + 85\right)\cdot 211^{14} + \left(194 a^{2} + 71 a + 89\right)\cdot 211^{15} + \left(23 a^{2} + 192 a + 209\right)\cdot 211^{16} + \left(84 a^{2} + 29 a + 47\right)\cdot 211^{17} + \left(18 a^{2} + 145 a + 49\right)\cdot 211^{18} + \left(a^{2} + 84 a + 119\right)\cdot 211^{19} + \left(150 a^{2} + 181 a + 115\right)\cdot 211^{20} + \left(170 a^{2} + 166 a + 162\right)\cdot 211^{21} + \left(92 a^{2} + 54 a + 91\right)\cdot 211^{22} + \left(207 a^{2} + 198 a + 36\right)\cdot 211^{23} + \left(53 a^{2} + 17 a + 181\right)\cdot 211^{24} + \left(121 a^{2} + 71 a + 26\right)\cdot 211^{25} + \left(3 a^{2} + 133 a + 167\right)\cdot 211^{26} + \left(12 a^{2} + 187 a + 140\right)\cdot 211^{27} + \left(148 a^{2} + 36 a + 52\right)\cdot 211^{28} + \left(2 a^{2} + 112 a + 58\right)\cdot 211^{29} + \left(64 a^{2} + 190 a + 110\right)\cdot 211^{30} + \left(171 a^{2} + 29 a + 178\right)\cdot 211^{31} + \left(a^{2} + 93 a + 107\right)\cdot 211^{32} + \left(181 a^{2} + 97 a + 85\right)\cdot 211^{33} + \left(44 a^{2} + a + 128\right)\cdot 211^{34} + \left(162 a^{2} + 4 a + 45\right)\cdot 211^{35} + \left(136 a^{2} + 33 a + 21\right)\cdot 211^{36} + \left(46 a^{2} + 13 a + 51\right)\cdot 211^{37} + \left(124 a^{2} + 171 a + 85\right)\cdot 211^{38} + \left(146 a^{2} + 139 a + 105\right)\cdot 211^{39} + \left(200 a^{2} + 99 a + 117\right)\cdot 211^{40} + \left(24 a^{2} + 136 a + 108\right)\cdot 211^{41} + \left(3 a^{2} + 173 a + 53\right)\cdot 211^{42} + \left(190 a^{2} + 171 a + 81\right)\cdot 211^{43} + \left(162 a^{2} + 111 a + 206\right)\cdot 211^{44} + \left(158 a^{2} + 186 a + 166\right)\cdot 211^{45} + \left(179 a^{2} + 54 a + 72\right)\cdot 211^{46} + \left(64 a^{2} + 159 a + 188\right)\cdot 211^{47} + \left(102 a^{2} + 100 a + 82\right)\cdot 211^{48} + \left(173 a^{2} + 183 a + 73\right)\cdot 211^{49} + \left(66 a^{2} + 10 a + 20\right)\cdot 211^{50} + \left(24 a^{2} + 106 a + 193\right)\cdot 211^{51} + \left(3 a^{2} + 129 a + 188\right)\cdot 211^{52} + \left(70 a^{2} + 155 a + 32\right)\cdot 211^{53} + \left(51 a^{2} + 116 a + 201\right)\cdot 211^{54} + \left(3 a^{2} + 176 a + 165\right)\cdot 211^{55} + \left(168 a^{2} + 26 a + 84\right)\cdot 211^{56} + \left(176 a^{2} + 205 a + 47\right)\cdot 211^{57} + \left(19 a^{2} + 26 a + 130\right)\cdot 211^{58} + \left(187 a^{2} + 205 a + 71\right)\cdot 211^{59} + \left(142 a^{2} + 186 a + 36\right)\cdot 211^{60} + \left(15 a^{2} + 118 a + 30\right)\cdot 211^{61} + \left(85 a^{2} + 51 a + 12\right)\cdot 211^{62} + \left(82 a^{2} + 181 a + 105\right)\cdot 211^{63} + \left(165 a^{2} + 2 a + 183\right)\cdot 211^{64} + \left(37 a^{2} + 33 a + 97\right)\cdot 211^{65} + \left(73 a^{2} + 20 a + 109\right)\cdot 211^{66} + \left(42 a^{2} + 148 a + 58\right)\cdot 211^{67} + \left(170 a + 148\right)\cdot 211^{68} + \left(79 a^{2} + 139 a + 162\right)\cdot 211^{69} + \left(98 a^{2} + 115 a + 188\right)\cdot 211^{70} + \left(112 a^{2} + 179 a + 179\right)\cdot 211^{71} + \left(170 a^{2} + 184 a + 160\right)\cdot 211^{72} + \left(50 a^{2} + 196 a + 123\right)\cdot 211^{73} + \left(201 a^{2} + 110 a + 66\right)\cdot 211^{74} + \left(24 a^{2} + 161 a + 207\right)\cdot 211^{75} + \left(61 a^{2} + 154 a + 204\right)\cdot 211^{76} + \left(38 a^{2} + 131 a + 163\right)\cdot 211^{77} + \left(201 a^{2} + 127 a + 48\right)\cdot 211^{78} + \left(129 a^{2} + 142 a + 28\right)\cdot 211^{79} + \left(101 a^{2} + 153 a + 160\right)\cdot 211^{80} + \left(79 a^{2} + 49 a + 7\right)\cdot 211^{81} + \left(31 a^{2} + 206 a + 15\right)\cdot 211^{82} + \left(169 a^{2} + 95 a + 37\right)\cdot 211^{83} + \left(173 a^{2} + 151 a + 110\right)\cdot 211^{84} + \left(19 a^{2} + 58 a + 131\right)\cdot 211^{85} + \left(152 a^{2} + 206 a + 105\right)\cdot 211^{86} + \left(132 a^{2} + 193 a + 46\right)\cdot 211^{87} + \left(148 a^{2} + 154 a + 10\right)\cdot 211^{88} + \left(207 a^{2} + 30 a + 145\right)\cdot 211^{89} + \left(182 a^{2} + 162 a + 103\right)\cdot 211^{90} + \left(200 a^{2} + 130 a + 118\right)\cdot 211^{91} + \left(56 a^{2} + 138 a + 100\right)\cdot 211^{92} + \left(13 a^{2} + 105 a + 192\right)\cdot 211^{93} + \left(178 a^{2} + 70 a + 161\right)\cdot 211^{94} + \left(201 a^{2} + 108 a + 44\right)\cdot 211^{95} + \left(199 a^{2} + 209 a + 47\right)\cdot 211^{96} + \left(48 a^{2} + 148 a + 147\right)\cdot 211^{97} +O\left(211^{ 98 }\right)$

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

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

Character values on conjugacy classes

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