Properties

Label 9.3e7_619e3.18t272.1
Dimension 9
Group $S_4\wr C_2$
Conductor $ 3^{7} \cdot 619^{3}$
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$9$
Group:$S_4\wr C_2$
Conductor:$518705353233= 3^{7} \cdot 619^{3} $
Artin number field: Splitting field of $f= x^{8} - 4 x^{7} - 14 x^{6} + 17 x^{5} + 43 x^{4} + 323 x^{3} + 310 x^{2} - 1135 x - 413 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 18T272
Parity: Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 159.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 25 a + 28 + \left(8 a + 27\right)\cdot 29 + 25\cdot 29^{2} + \left(27 a + 14\right)\cdot 29^{3} + \left(19 a + 22\right)\cdot 29^{4} + \left(21 a + 2\right)\cdot 29^{5} + \left(25 a + 14\right)\cdot 29^{6} + 7 a\cdot 29^{7} + \left(4 a + 15\right)\cdot 29^{8} + \left(11 a + 10\right)\cdot 29^{9} + \left(a + 3\right)\cdot 29^{10} + \left(20 a + 14\right)\cdot 29^{11} + 28 a\cdot 29^{12} + \left(13 a + 11\right)\cdot 29^{13} + \left(2 a + 15\right)\cdot 29^{14} + \left(12 a + 19\right)\cdot 29^{15} + \left(16 a + 26\right)\cdot 29^{16} + \left(10 a + 9\right)\cdot 29^{17} + \left(20 a + 21\right)\cdot 29^{18} + \left(23 a + 16\right)\cdot 29^{19} + \left(16 a + 26\right)\cdot 29^{20} + \left(12 a + 28\right)\cdot 29^{21} + \left(28 a + 7\right)\cdot 29^{22} + \left(17 a + 14\right)\cdot 29^{23} + \left(26 a + 16\right)\cdot 29^{24} + \left(16 a + 16\right)\cdot 29^{25} + \left(14 a + 5\right)\cdot 29^{26} + \left(8 a + 3\right)\cdot 29^{27} + \left(25 a + 4\right)\cdot 29^{28} + \left(25 a + 14\right)\cdot 29^{29} + \left(4 a + 19\right)\cdot 29^{30} + \left(7 a + 26\right)\cdot 29^{31} + \left(20 a + 17\right)\cdot 29^{32} + 15\cdot 29^{33} + \left(9 a + 18\right)\cdot 29^{34} + \left(19 a + 22\right)\cdot 29^{35} + \left(26 a + 24\right)\cdot 29^{36} + \left(25 a + 4\right)\cdot 29^{37} + \left(16 a + 15\right)\cdot 29^{38} + \left(26 a + 28\right)\cdot 29^{39} + \left(4 a + 23\right)\cdot 29^{40} + \left(21 a + 10\right)\cdot 29^{41} + \left(a + 3\right)\cdot 29^{42} + 28 a\cdot 29^{43} + \left(16 a + 26\right)\cdot 29^{44} + \left(a + 21\right)\cdot 29^{45} + \left(13 a + 19\right)\cdot 29^{46} + \left(a + 9\right)\cdot 29^{47} + \left(24 a + 13\right)\cdot 29^{48} + \left(13 a + 16\right)\cdot 29^{49} + \left(12 a + 27\right)\cdot 29^{50} + \left(25 a + 26\right)\cdot 29^{51} + \left(5 a + 17\right)\cdot 29^{52} + \left(24 a + 14\right)\cdot 29^{53} + \left(12 a + 3\right)\cdot 29^{54} + \left(23 a + 18\right)\cdot 29^{55} + \left(5 a + 26\right)\cdot 29^{56} + \left(8 a + 25\right)\cdot 29^{57} + \left(24 a + 10\right)\cdot 29^{58} + \left(14 a + 27\right)\cdot 29^{59} + \left(3 a + 1\right)\cdot 29^{60} + \left(3 a + 11\right)\cdot 29^{61} + \left(16 a + 6\right)\cdot 29^{62} + \left(13 a + 24\right)\cdot 29^{63} + \left(20 a + 18\right)\cdot 29^{64} + 20\cdot 29^{65} + \left(8 a + 22\right)\cdot 29^{66} + \left(14 a + 20\right)\cdot 29^{67} + \left(18 a + 19\right)\cdot 29^{68} + \left(21 a + 8\right)\cdot 29^{69} + \left(6 a + 23\right)\cdot 29^{70} + \left(5 a + 23\right)\cdot 29^{71} + \left(24 a + 24\right)\cdot 29^{72} + \left(10 a + 24\right)\cdot 29^{73} + \left(25 a + 23\right)\cdot 29^{74} + 18\cdot 29^{75} + \left(8 a + 11\right)\cdot 29^{76} + \left(24 a + 18\right)\cdot 29^{77} + \left(23 a + 14\right)\cdot 29^{78} + \left(28 a + 16\right)\cdot 29^{79} + \left(7 a + 15\right)\cdot 29^{80} + \left(21 a + 27\right)\cdot 29^{81} + \left(19 a + 28\right)\cdot 29^{82} + \left(4 a + 12\right)\cdot 29^{83} + \left(18 a + 25\right)\cdot 29^{84} + \left(15 a + 22\right)\cdot 29^{85} + \left(4 a + 10\right)\cdot 29^{86} + \left(18 a + 2\right)\cdot 29^{87} + \left(11 a + 16\right)\cdot 29^{88} + \left(23 a + 1\right)\cdot 29^{89} + \left(19 a + 13\right)\cdot 29^{90} + \left(2 a + 11\right)\cdot 29^{91} + \left(19 a + 20\right)\cdot 29^{92} + \left(17 a + 18\right)\cdot 29^{93} + \left(19 a + 27\right)\cdot 29^{94} + \left(21 a + 11\right)\cdot 29^{95} + \left(24 a + 11\right)\cdot 29^{96} + \left(2 a + 22\right)\cdot 29^{97} + \left(17 a + 6\right)\cdot 29^{98} + 24 a\cdot 29^{99} + \left(24 a + 13\right)\cdot 29^{100} + \left(7 a + 1\right)\cdot 29^{101} + \left(10 a + 27\right)\cdot 29^{102} + \left(17 a + 23\right)\cdot 29^{103} + \left(a + 24\right)\cdot 29^{104} + \left(22 a + 19\right)\cdot 29^{105} + \left(16 a + 22\right)\cdot 29^{106} + \left(15 a + 16\right)\cdot 29^{107} + \left(26 a + 6\right)\cdot 29^{108} + \left(24 a + 13\right)\cdot 29^{109} + \left(28 a + 28\right)\cdot 29^{110} + \left(28 a + 23\right)\cdot 29^{111} + \left(15 a + 12\right)\cdot 29^{112} + \left(16 a + 16\right)\cdot 29^{113} + \left(a + 17\right)\cdot 29^{114} + \left(7 a + 24\right)\cdot 29^{115} + \left(3 a + 8\right)\cdot 29^{116} + \left(22 a + 12\right)\cdot 29^{117} + \left(a + 2\right)\cdot 29^{118} + \left(27 a + 23\right)\cdot 29^{119} + \left(9 a + 20\right)\cdot 29^{120} + \left(10 a + 16\right)\cdot 29^{121} + \left(13 a + 24\right)\cdot 29^{122} + \left(13 a + 9\right)\cdot 29^{123} + \left(8 a + 18\right)\cdot 29^{124} + 22\cdot 29^{125} + \left(21 a + 21\right)\cdot 29^{126} + 6 a\cdot 29^{127} + \left(12 a + 7\right)\cdot 29^{128} + \left(16 a + 10\right)\cdot 29^{129} + \left(12 a + 1\right)\cdot 29^{130} + \left(9 a + 10\right)\cdot 29^{131} + \left(28 a + 21\right)\cdot 29^{132} + \left(14 a + 14\right)\cdot 29^{133} + \left(3 a + 24\right)\cdot 29^{134} + 15\cdot 29^{135} + \left(12 a + 18\right)\cdot 29^{136} + \left(3 a + 4\right)\cdot 29^{137} + \left(3 a + 7\right)\cdot 29^{138} + \left(22 a + 27\right)\cdot 29^{139} + 8\cdot 29^{140} + \left(12 a + 20\right)\cdot 29^{141} + \left(22 a + 27\right)\cdot 29^{142} + \left(23 a + 10\right)\cdot 29^{143} + \left(5 a + 3\right)\cdot 29^{144} + \left(20 a + 13\right)\cdot 29^{145} + \left(12 a + 23\right)\cdot 29^{146} + \left(2 a + 16\right)\cdot 29^{147} + \left(4 a + 18\right)\cdot 29^{148} + \left(18 a + 10\right)\cdot 29^{149} + \left(3 a + 22\right)\cdot 29^{150} + \left(26 a + 24\right)\cdot 29^{151} + \left(16 a + 3\right)\cdot 29^{152} + \left(27 a + 26\right)\cdot 29^{153} + \left(3 a + 13\right)\cdot 29^{154} + \left(27 a + 5\right)\cdot 29^{155} + \left(18 a + 2\right)\cdot 29^{156} + \left(27 a + 1\right)\cdot 29^{157} + \left(21 a + 3\right)\cdot 29^{158} +O\left(29^{ 159 }\right)$
$r_{ 2 }$ $=$ $ 14 + 10\cdot 29 + 11\cdot 29^{2} + 4\cdot 29^{3} + 8\cdot 29^{4} + 15\cdot 29^{5} + 15\cdot 29^{6} + 25\cdot 29^{7} + 21\cdot 29^{8} + 8\cdot 29^{9} + 29^{10} + 11\cdot 29^{11} + 8\cdot 29^{12} + 27\cdot 29^{13} + 24\cdot 29^{14} + 10\cdot 29^{15} + 13\cdot 29^{16} + 16\cdot 29^{17} + 17\cdot 29^{18} + 23\cdot 29^{19} + 17\cdot 29^{20} + 15\cdot 29^{21} + 8\cdot 29^{22} + 11\cdot 29^{23} + 3\cdot 29^{24} + 10\cdot 29^{25} + 8\cdot 29^{26} + 9\cdot 29^{27} + 24\cdot 29^{28} + 20\cdot 29^{29} + 25\cdot 29^{30} + 22\cdot 29^{31} + 14\cdot 29^{32} + 16\cdot 29^{33} + 2\cdot 29^{34} + 8\cdot 29^{35} + 13\cdot 29^{36} + 22\cdot 29^{37} + 23\cdot 29^{38} + 18\cdot 29^{39} + 17\cdot 29^{41} + 12\cdot 29^{42} + 2\cdot 29^{43} + 2\cdot 29^{44} + 17\cdot 29^{45} + 14\cdot 29^{46} + 23\cdot 29^{47} + 22\cdot 29^{48} + 6\cdot 29^{49} + 4\cdot 29^{50} + 6\cdot 29^{51} + 20\cdot 29^{53} + 19\cdot 29^{54} + 18\cdot 29^{55} + 27\cdot 29^{56} + 26\cdot 29^{57} + 20\cdot 29^{58} + 7\cdot 29^{59} + 6\cdot 29^{60} + 8\cdot 29^{61} + 25\cdot 29^{62} + 20\cdot 29^{63} + 5\cdot 29^{64} + 12\cdot 29^{65} + 16\cdot 29^{66} + 16\cdot 29^{67} + 18\cdot 29^{68} + 8\cdot 29^{69} + 4\cdot 29^{70} + 8\cdot 29^{71} + 7\cdot 29^{72} + 25\cdot 29^{73} + 29^{74} + 8\cdot 29^{75} + 29^{76} + 17\cdot 29^{77} + 21\cdot 29^{78} + 8\cdot 29^{79} + 21\cdot 29^{80} + 23\cdot 29^{81} + 14\cdot 29^{82} + 13\cdot 29^{83} + 23\cdot 29^{84} + 13\cdot 29^{85} + 27\cdot 29^{86} + 22\cdot 29^{87} + 22\cdot 29^{88} + 14\cdot 29^{89} + 17\cdot 29^{90} + 5\cdot 29^{91} + 24\cdot 29^{92} + 25\cdot 29^{93} + 8\cdot 29^{94} + 9\cdot 29^{95} + 23\cdot 29^{96} + 7\cdot 29^{97} + 11\cdot 29^{98} + 26\cdot 29^{99} + 12\cdot 29^{100} + 7\cdot 29^{101} + 12\cdot 29^{102} + 19\cdot 29^{103} + 2\cdot 29^{104} + 15\cdot 29^{105} + 24\cdot 29^{106} + 23\cdot 29^{108} + 29^{109} + 6\cdot 29^{110} + 12\cdot 29^{111} + 15\cdot 29^{112} + 4\cdot 29^{113} + 6\cdot 29^{114} + 9\cdot 29^{115} + 10\cdot 29^{116} + 5\cdot 29^{117} + 17\cdot 29^{118} + 26\cdot 29^{119} + 28\cdot 29^{120} + 27\cdot 29^{121} + 12\cdot 29^{122} + 19\cdot 29^{123} + 14\cdot 29^{124} + 26\cdot 29^{125} + 29^{126} + 26\cdot 29^{127} + 10\cdot 29^{128} + 13\cdot 29^{130} + 3\cdot 29^{131} + 25\cdot 29^{132} + 20\cdot 29^{133} + 5\cdot 29^{134} + 2\cdot 29^{135} + 29^{136} + 8\cdot 29^{137} + 22\cdot 29^{138} + 13\cdot 29^{139} + 11\cdot 29^{140} + 18\cdot 29^{141} + 28\cdot 29^{142} + 29^{143} + 23\cdot 29^{144} + 22\cdot 29^{145} + 8\cdot 29^{146} + 6\cdot 29^{147} + 16\cdot 29^{148} + 5\cdot 29^{149} + 25\cdot 29^{150} + 22\cdot 29^{151} + 10\cdot 29^{152} + 3\cdot 29^{153} + 27\cdot 29^{154} + 20\cdot 29^{155} + 15\cdot 29^{156} + 19\cdot 29^{157} + 3\cdot 29^{158} +O\left(29^{ 159 }\right)$
$r_{ 3 }$ $=$ $ 28 a + 7 a\cdot 29 + \left(22 a + 13\right)\cdot 29^{2} + 17 a\cdot 29^{3} + \left(28 a + 9\right)\cdot 29^{4} + \left(3 a + 15\right)\cdot 29^{5} + \left(27 a + 11\right)\cdot 29^{6} + \left(28 a + 6\right)\cdot 29^{7} + \left(19 a + 15\right)\cdot 29^{8} + \left(26 a + 8\right)\cdot 29^{9} + \left(20 a + 3\right)\cdot 29^{10} + \left(24 a + 1\right)\cdot 29^{11} + \left(28 a + 23\right)\cdot 29^{12} + \left(17 a + 24\right)\cdot 29^{13} + \left(5 a + 23\right)\cdot 29^{14} + 11\cdot 29^{15} + \left(16 a + 14\right)\cdot 29^{16} + \left(a + 19\right)\cdot 29^{17} + \left(7 a + 17\right)\cdot 29^{18} + \left(5 a + 11\right)\cdot 29^{19} + \left(11 a + 19\right)\cdot 29^{20} + \left(3 a + 17\right)\cdot 29^{21} + \left(a + 27\right)\cdot 29^{22} + \left(12 a + 26\right)\cdot 29^{23} + \left(15 a + 23\right)\cdot 29^{24} + \left(10 a + 22\right)\cdot 29^{25} + \left(16 a + 17\right)\cdot 29^{26} + \left(3 a + 25\right)\cdot 29^{27} + 4 a\cdot 29^{28} + \left(24 a + 6\right)\cdot 29^{29} + \left(20 a + 28\right)\cdot 29^{30} + \left(2 a + 4\right)\cdot 29^{31} + \left(11 a + 10\right)\cdot 29^{32} + \left(21 a + 17\right)\cdot 29^{33} + \left(11 a + 27\right)\cdot 29^{34} + \left(18 a + 23\right)\cdot 29^{35} + \left(19 a + 8\right)\cdot 29^{36} + \left(25 a + 5\right)\cdot 29^{37} + \left(5 a + 26\right)\cdot 29^{38} + \left(15 a + 8\right)\cdot 29^{39} + \left(7 a + 9\right)\cdot 29^{40} + \left(10 a + 18\right)\cdot 29^{41} + \left(7 a + 18\right)\cdot 29^{42} + \left(25 a + 1\right)\cdot 29^{43} + \left(25 a + 24\right)\cdot 29^{44} + \left(2 a + 2\right)\cdot 29^{45} + \left(6 a + 7\right)\cdot 29^{46} + \left(9 a + 2\right)\cdot 29^{47} + \left(5 a + 20\right)\cdot 29^{48} + \left(18 a + 19\right)\cdot 29^{49} + \left(12 a + 12\right)\cdot 29^{50} + \left(21 a + 13\right)\cdot 29^{51} + \left(4 a + 22\right)\cdot 29^{52} + \left(27 a + 21\right)\cdot 29^{53} + 16\cdot 29^{54} + \left(12 a + 1\right)\cdot 29^{55} + \left(6 a + 4\right)\cdot 29^{56} + \left(10 a + 21\right)\cdot 29^{57} + \left(3 a + 1\right)\cdot 29^{58} + \left(22 a + 10\right)\cdot 29^{59} + \left(12 a + 19\right)\cdot 29^{60} + \left(16 a + 20\right)\cdot 29^{61} + \left(7 a + 16\right)\cdot 29^{62} + \left(11 a + 12\right)\cdot 29^{63} + \left(5 a + 1\right)\cdot 29^{64} + \left(12 a + 18\right)\cdot 29^{65} + \left(22 a + 23\right)\cdot 29^{66} + \left(13 a + 25\right)\cdot 29^{67} + \left(13 a + 15\right)\cdot 29^{68} + \left(26 a + 17\right)\cdot 29^{69} + \left(16 a + 28\right)\cdot 29^{70} + \left(26 a + 9\right)\cdot 29^{71} + \left(7 a + 12\right)\cdot 29^{72} + 20 a\cdot 29^{73} + \left(28 a + 16\right)\cdot 29^{74} + \left(6 a + 17\right)\cdot 29^{75} + \left(6 a + 14\right)\cdot 29^{76} + \left(24 a + 12\right)\cdot 29^{77} + \left(18 a + 4\right)\cdot 29^{78} + \left(15 a + 24\right)\cdot 29^{79} + \left(16 a + 17\right)\cdot 29^{80} + \left(6 a + 16\right)\cdot 29^{81} + \left(18 a + 20\right)\cdot 29^{82} + \left(17 a + 22\right)\cdot 29^{83} + \left(18 a + 9\right)\cdot 29^{84} + 27\cdot 29^{85} + \left(28 a + 2\right)\cdot 29^{86} + \left(28 a + 12\right)\cdot 29^{87} + \left(a + 2\right)\cdot 29^{88} + \left(23 a + 5\right)\cdot 29^{89} + \left(23 a + 17\right)\cdot 29^{90} + \left(a + 13\right)\cdot 29^{91} + \left(9 a + 27\right)\cdot 29^{92} + \left(6 a + 7\right)\cdot 29^{93} + \left(28 a + 24\right)\cdot 29^{94} + \left(19 a + 23\right)\cdot 29^{95} + 7 a\cdot 29^{96} + \left(15 a + 21\right)\cdot 29^{97} + \left(16 a + 19\right)\cdot 29^{98} + \left(11 a + 27\right)\cdot 29^{99} + \left(3 a + 6\right)\cdot 29^{100} + \left(a + 19\right)\cdot 29^{101} + \left(11 a + 25\right)\cdot 29^{102} + \left(12 a + 13\right)\cdot 29^{103} + \left(26 a + 21\right)\cdot 29^{104} + \left(11 a + 10\right)\cdot 29^{105} + \left(19 a + 5\right)\cdot 29^{106} + \left(27 a + 9\right)\cdot 29^{107} + \left(2 a + 28\right)\cdot 29^{108} + 25\cdot 29^{109} + \left(8 a + 7\right)\cdot 29^{110} + \left(20 a + 16\right)\cdot 29^{111} + \left(14 a + 7\right)\cdot 29^{112} + \left(22 a + 17\right)\cdot 29^{113} + \left(3 a + 17\right)\cdot 29^{114} + \left(9 a + 24\right)\cdot 29^{115} + \left(22 a + 7\right)\cdot 29^{116} + \left(25 a + 11\right)\cdot 29^{117} + \left(13 a + 11\right)\cdot 29^{118} + \left(5 a + 19\right)\cdot 29^{119} + \left(4 a + 3\right)\cdot 29^{120} + \left(15 a + 28\right)\cdot 29^{121} + \left(5 a + 27\right)\cdot 29^{122} + \left(27 a + 13\right)\cdot 29^{123} + \left(10 a + 11\right)\cdot 29^{124} + \left(19 a + 10\right)\cdot 29^{125} + \left(13 a + 17\right)\cdot 29^{126} + \left(25 a + 8\right)\cdot 29^{127} + \left(a + 17\right)\cdot 29^{128} + \left(9 a + 5\right)\cdot 29^{129} + \left(10 a + 12\right)\cdot 29^{130} + \left(12 a + 19\right)\cdot 29^{131} + \left(16 a + 22\right)\cdot 29^{132} + \left(18 a + 10\right)\cdot 29^{133} + \left(11 a + 12\right)\cdot 29^{134} + \left(2 a + 14\right)\cdot 29^{135} + \left(17 a + 11\right)\cdot 29^{136} + \left(28 a + 2\right)\cdot 29^{137} + \left(12 a + 12\right)\cdot 29^{138} + \left(22 a + 14\right)\cdot 29^{139} + \left(2 a + 4\right)\cdot 29^{140} + \left(6 a + 23\right)\cdot 29^{141} + \left(2 a + 6\right)\cdot 29^{142} + \left(13 a + 25\right)\cdot 29^{143} + \left(25 a + 23\right)\cdot 29^{144} + 26 a\cdot 29^{145} + \left(12 a + 23\right)\cdot 29^{146} + \left(15 a + 23\right)\cdot 29^{147} + \left(24 a + 5\right)\cdot 29^{148} + \left(a + 26\right)\cdot 29^{149} + \left(12 a + 20\right)\cdot 29^{150} + \left(13 a + 14\right)\cdot 29^{151} + \left(16 a + 19\right)\cdot 29^{152} + \left(17 a + 22\right)\cdot 29^{153} + \left(24 a + 9\right)\cdot 29^{154} + \left(9 a + 3\right)\cdot 29^{155} + \left(23 a + 12\right)\cdot 29^{156} + \left(8 a + 1\right)\cdot 29^{157} + \left(4 a + 22\right)\cdot 29^{158} +O\left(29^{ 159 }\right)$
$r_{ 4 }$ $=$ $ a + 24 + \left(21 a + 11\right)\cdot 29 + 6 a\cdot 29^{2} + \left(11 a + 9\right)\cdot 29^{3} + 18\cdot 29^{4} + \left(25 a + 6\right)\cdot 29^{5} + \left(a + 27\right)\cdot 29^{6} + 7\cdot 29^{7} + \left(9 a + 28\right)\cdot 29^{8} + \left(2 a + 5\right)\cdot 29^{9} + \left(8 a + 23\right)\cdot 29^{10} + \left(4 a + 16\right)\cdot 29^{11} + 26\cdot 29^{12} + \left(11 a + 27\right)\cdot 29^{13} + \left(23 a + 4\right)\cdot 29^{14} + \left(28 a + 7\right)\cdot 29^{15} + \left(12 a + 7\right)\cdot 29^{16} + \left(27 a + 11\right)\cdot 29^{17} + \left(21 a + 22\right)\cdot 29^{18} + \left(23 a + 1\right)\cdot 29^{19} + \left(17 a + 12\right)\cdot 29^{20} + \left(25 a + 23\right)\cdot 29^{21} + 27 a\cdot 29^{22} + \left(16 a + 28\right)\cdot 29^{23} + \left(13 a + 1\right)\cdot 29^{24} + \left(18 a + 2\right)\cdot 29^{25} + \left(12 a + 2\right)\cdot 29^{26} + \left(25 a + 27\right)\cdot 29^{27} + \left(24 a + 17\right)\cdot 29^{28} + \left(4 a + 6\right)\cdot 29^{29} + \left(8 a + 21\right)\cdot 29^{30} + \left(26 a + 26\right)\cdot 29^{31} + \left(17 a + 4\right)\cdot 29^{32} + \left(7 a + 26\right)\cdot 29^{33} + \left(17 a + 6\right)\cdot 29^{34} + \left(10 a + 17\right)\cdot 29^{35} + \left(9 a + 1\right)\cdot 29^{36} + \left(3 a + 27\right)\cdot 29^{37} + 23 a\cdot 29^{38} + \left(13 a + 21\right)\cdot 29^{39} + \left(21 a + 2\right)\cdot 29^{40} + \left(18 a + 4\right)\cdot 29^{41} + \left(21 a + 16\right)\cdot 29^{42} + \left(3 a + 4\right)\cdot 29^{43} + \left(3 a + 12\right)\cdot 29^{44} + \left(26 a + 20\right)\cdot 29^{45} + \left(22 a + 5\right)\cdot 29^{46} + \left(19 a + 13\right)\cdot 29^{47} + \left(23 a + 8\right)\cdot 29^{48} + \left(10 a + 18\right)\cdot 29^{49} + \left(16 a + 28\right)\cdot 29^{50} + \left(7 a + 20\right)\cdot 29^{51} + \left(24 a + 24\right)\cdot 29^{52} + \left(a + 7\right)\cdot 29^{53} + \left(28 a + 23\right)\cdot 29^{54} + \left(16 a + 2\right)\cdot 29^{55} + \left(22 a + 24\right)\cdot 29^{56} + \left(18 a + 7\right)\cdot 29^{57} + \left(25 a + 8\right)\cdot 29^{58} + \left(6 a + 1\right)\cdot 29^{59} + \left(16 a + 3\right)\cdot 29^{60} + \left(12 a + 3\right)\cdot 29^{61} + \left(21 a + 9\right)\cdot 29^{62} + \left(17 a + 3\right)\cdot 29^{63} + \left(23 a + 17\right)\cdot 29^{64} + \left(16 a + 15\right)\cdot 29^{65} + \left(6 a + 7\right)\cdot 29^{66} + \left(15 a + 14\right)\cdot 29^{67} + \left(15 a + 11\right)\cdot 29^{68} + \left(2 a + 20\right)\cdot 29^{69} + \left(12 a + 28\right)\cdot 29^{70} + \left(2 a + 9\right)\cdot 29^{71} + \left(21 a + 25\right)\cdot 29^{72} + \left(8 a + 6\right)\cdot 29^{73} + 23\cdot 29^{74} + \left(22 a + 23\right)\cdot 29^{75} + \left(22 a + 9\right)\cdot 29^{76} + \left(4 a + 11\right)\cdot 29^{77} + \left(10 a + 16\right)\cdot 29^{78} + \left(13 a + 25\right)\cdot 29^{79} + \left(12 a + 26\right)\cdot 29^{80} + \left(22 a + 3\right)\cdot 29^{81} + \left(10 a + 18\right)\cdot 29^{82} + \left(11 a + 5\right)\cdot 29^{83} + \left(10 a + 27\right)\cdot 29^{84} + \left(28 a + 11\right)\cdot 29^{85} + 26\cdot 29^{86} + 12\cdot 29^{87} + \left(27 a + 12\right)\cdot 29^{88} + \left(5 a + 2\right)\cdot 29^{89} + \left(5 a + 26\right)\cdot 29^{90} + \left(27 a + 27\right)\cdot 29^{91} + \left(19 a + 12\right)\cdot 29^{92} + \left(22 a + 1\right)\cdot 29^{93} + 14\cdot 29^{94} + \left(9 a + 8\right)\cdot 29^{95} + \left(21 a + 19\right)\cdot 29^{96} + \left(13 a + 2\right)\cdot 29^{97} + 12 a\cdot 29^{98} + \left(17 a + 11\right)\cdot 29^{99} + \left(25 a + 12\right)\cdot 29^{100} + \left(27 a + 21\right)\cdot 29^{101} + \left(17 a + 21\right)\cdot 29^{102} + \left(16 a + 6\right)\cdot 29^{103} + \left(2 a + 25\right)\cdot 29^{104} + \left(17 a + 14\right)\cdot 29^{105} + \left(9 a + 3\right)\cdot 29^{106} + \left(a + 12\right)\cdot 29^{107} + \left(26 a + 15\right)\cdot 29^{108} + \left(28 a + 23\right)\cdot 29^{109} + \left(20 a + 18\right)\cdot 29^{110} + \left(8 a + 22\right)\cdot 29^{111} + \left(14 a + 2\right)\cdot 29^{112} + \left(6 a + 28\right)\cdot 29^{113} + \left(25 a + 13\right)\cdot 29^{114} + \left(19 a + 8\right)\cdot 29^{115} + \left(6 a + 23\right)\cdot 29^{116} + \left(3 a + 1\right)\cdot 29^{117} + \left(15 a + 26\right)\cdot 29^{118} + \left(23 a + 3\right)\cdot 29^{119} + \left(24 a + 19\right)\cdot 29^{120} + \left(13 a + 12\right)\cdot 29^{121} + \left(23 a + 11\right)\cdot 29^{122} + \left(a + 28\right)\cdot 29^{123} + \left(18 a + 9\right)\cdot 29^{124} + \left(9 a + 9\right)\cdot 29^{125} + \left(15 a + 8\right)\cdot 29^{126} + \left(3 a + 6\right)\cdot 29^{127} + \left(27 a + 1\right)\cdot 29^{128} + \left(19 a + 20\right)\cdot 29^{129} + \left(18 a + 25\right)\cdot 29^{130} + \left(16 a + 12\right)\cdot 29^{131} + \left(12 a + 5\right)\cdot 29^{132} + 10 a\cdot 29^{133} + \left(17 a + 23\right)\cdot 29^{134} + \left(26 a + 14\right)\cdot 29^{135} + \left(11 a + 7\right)\cdot 29^{136} + 12\cdot 29^{137} + \left(16 a + 19\right)\cdot 29^{138} + \left(6 a + 26\right)\cdot 29^{139} + \left(26 a + 24\right)\cdot 29^{140} + \left(22 a + 21\right)\cdot 29^{141} + \left(26 a + 11\right)\cdot 29^{142} + \left(15 a + 1\right)\cdot 29^{143} + \left(3 a + 22\right)\cdot 29^{144} + \left(2 a + 22\right)\cdot 29^{145} + \left(16 a + 2\right)\cdot 29^{146} + \left(13 a + 1\right)\cdot 29^{147} + \left(4 a + 26\right)\cdot 29^{148} + \left(27 a + 10\right)\cdot 29^{149} + \left(16 a + 21\right)\cdot 29^{150} + \left(15 a + 11\right)\cdot 29^{151} + \left(12 a + 1\right)\cdot 29^{152} + \left(11 a + 7\right)\cdot 29^{153} + \left(4 a + 28\right)\cdot 29^{154} + \left(19 a + 27\right)\cdot 29^{155} + \left(5 a + 2\right)\cdot 29^{156} + \left(20 a + 22\right)\cdot 29^{157} + \left(24 a + 5\right)\cdot 29^{158} +O\left(29^{ 159 }\right)$
$r_{ 5 }$ $=$ $ 26 a + 19 + \left(18 a + 2\right)\cdot 29 + \left(19 a + 15\right)\cdot 29^{2} + \left(18 a + 18\right)\cdot 29^{3} + \left(5 a + 4\right)\cdot 29^{4} + \left(7 a + 21\right)\cdot 29^{5} + \left(6 a + 2\right)\cdot 29^{6} + \left(24 a + 10\right)\cdot 29^{7} + \left(8 a + 17\right)\cdot 29^{8} + \left(a + 10\right)\cdot 29^{9} + \left(23 a + 5\right)\cdot 29^{10} + \left(3 a + 15\right)\cdot 29^{11} + \left(8 a + 22\right)\cdot 29^{12} + \left(7 a + 8\right)\cdot 29^{13} + \left(20 a + 13\right)\cdot 29^{14} + \left(27 a + 23\right)\cdot 29^{15} + \left(4 a + 20\right)\cdot 29^{16} + \left(12 a + 13\right)\cdot 29^{17} + \left(13 a + 14\right)\cdot 29^{18} + \left(4 a + 23\right)\cdot 29^{19} + \left(27 a + 10\right)\cdot 29^{20} + \left(23 a + 6\right)\cdot 29^{21} + \left(2 a + 23\right)\cdot 29^{22} + \left(12 a + 22\right)\cdot 29^{23} + \left(6 a + 17\right)\cdot 29^{24} + \left(11 a + 5\right)\cdot 29^{25} + \left(11 a + 13\right)\cdot 29^{26} + \left(4 a + 24\right)\cdot 29^{27} + \left(4 a + 19\right)\cdot 29^{28} + \left(14 a + 24\right)\cdot 29^{29} + \left(a + 16\right)\cdot 29^{30} + \left(25 a + 12\right)\cdot 29^{31} + \left(3 a + 27\right)\cdot 29^{32} + 19 a\cdot 29^{33} + \left(4 a + 6\right)\cdot 29^{34} + \left(15 a + 12\right)\cdot 29^{35} + \left(10 a + 9\right)\cdot 29^{36} + 2 a\cdot 29^{37} + \left(5 a + 17\right)\cdot 29^{38} + \left(18 a + 19\right)\cdot 29^{39} + \left(16 a + 27\right)\cdot 29^{40} + \left(26 a + 18\right)\cdot 29^{41} + \left(12 a + 19\right)\cdot 29^{42} + \left(a + 13\right)\cdot 29^{43} + \left(7 a + 24\right)\cdot 29^{44} + \left(9 a + 28\right)\cdot 29^{45} + \left(14 a + 28\right)\cdot 29^{46} + \left(12 a + 26\right)\cdot 29^{47} + \left(28 a + 2\right)\cdot 29^{48} + \left(9 a + 25\right)\cdot 29^{49} + \left(9 a + 26\right)\cdot 29^{50} + \left(16 a + 15\right)\cdot 29^{51} + \left(26 a + 14\right)\cdot 29^{52} + \left(18 a + 16\right)\cdot 29^{53} + \left(17 a + 18\right)\cdot 29^{54} + \left(6 a + 12\right)\cdot 29^{55} + \left(23 a + 9\right)\cdot 29^{56} + \left(10 a + 28\right)\cdot 29^{57} + \left(4 a + 13\right)\cdot 29^{58} + \left(8 a + 7\right)\cdot 29^{59} + \left(17 a + 13\right)\cdot 29^{60} + \left(6 a + 21\right)\cdot 29^{61} + \left(24 a + 7\right)\cdot 29^{62} + \left(a + 1\right)\cdot 29^{63} + \left(3 a + 22\right)\cdot 29^{64} + \left(8 a + 2\right)\cdot 29^{65} + \left(25 a + 20\right)\cdot 29^{66} + \left(19 a + 6\right)\cdot 29^{67} + \left(22 a + 27\right)\cdot 29^{68} + \left(10 a + 26\right)\cdot 29^{69} + \left(5 a + 17\right)\cdot 29^{70} + \left(16 a + 27\right)\cdot 29^{71} + \left(22 a + 8\right)\cdot 29^{72} + \left(6 a + 12\right)\cdot 29^{73} + \left(22 a + 12\right)\cdot 29^{74} + \left(23 a + 19\right)\cdot 29^{75} + \left(18 a + 25\right)\cdot 29^{76} + \left(22 a + 5\right)\cdot 29^{77} + \left(6 a + 12\right)\cdot 29^{78} + \left(22 a + 14\right)\cdot 29^{79} + \left(6 a + 19\right)\cdot 29^{80} + \left(17 a + 11\right)\cdot 29^{81} + \left(10 a + 8\right)\cdot 29^{82} + \left(a + 26\right)\cdot 29^{83} + \left(17 a + 6\right)\cdot 29^{84} + \left(14 a + 24\right)\cdot 29^{85} + \left(24 a + 19\right)\cdot 29^{86} + \left(26 a + 27\right)\cdot 29^{87} + \left(11 a + 7\right)\cdot 29^{88} + \left(25 a + 16\right)\cdot 29^{89} + \left(15 a + 23\right)\cdot 29^{90} + \left(9 a + 1\right)\cdot 29^{91} + \left(2 a + 17\right)\cdot 29^{92} + \left(15 a + 22\right)\cdot 29^{93} + \left(2 a + 20\right)\cdot 29^{94} + \left(7 a + 21\right)\cdot 29^{95} + \left(15 a + 14\right)\cdot 29^{96} + \left(23 a + 23\right)\cdot 29^{97} + \left(16 a + 9\right)\cdot 29^{98} + \left(22 a + 7\right)\cdot 29^{99} + \left(4 a + 28\right)\cdot 29^{100} + 22 a\cdot 29^{101} + \left(26 a + 14\right)\cdot 29^{102} + \left(21 a + 16\right)\cdot 29^{103} + \left(9 a + 14\right)\cdot 29^{104} + \left(11 a + 1\right)\cdot 29^{105} + \left(11 a + 3\right)\cdot 29^{106} + \left(15 a + 20\right)\cdot 29^{107} + \left(5 a + 13\right)\cdot 29^{108} + \left(12 a + 27\right)\cdot 29^{109} + \left(24 a + 9\right)\cdot 29^{110} + \left(17 a + 15\right)\cdot 29^{111} + \left(15 a + 25\right)\cdot 29^{112} + \left(4 a + 9\right)\cdot 29^{113} + \left(10 a + 15\right)\cdot 29^{114} + \left(16 a + 2\right)\cdot 29^{115} + \left(26 a + 5\right)\cdot 29^{116} + \left(23 a + 4\right)\cdot 29^{117} + \left(8 a + 5\right)\cdot 29^{118} + \left(7 a + 2\right)\cdot 29^{119} + \left(27 a + 11\right)\cdot 29^{120} + \left(25 a + 7\right)\cdot 29^{121} + 17\cdot 29^{122} + \left(14 a + 25\right)\cdot 29^{123} + \left(27 a + 18\right)\cdot 29^{124} + \left(18 a + 9\right)\cdot 29^{125} + \left(3 a + 20\right)\cdot 29^{126} + \left(20 a + 26\right)\cdot 29^{127} + \left(10 a + 15\right)\cdot 29^{128} + \left(27 a + 10\right)\cdot 29^{129} + \left(8 a + 21\right)\cdot 29^{130} + \left(12 a + 19\right)\cdot 29^{131} + \left(27 a + 5\right)\cdot 29^{132} + \left(9 a + 1\right)\cdot 29^{133} + \left(27 a + 15\right)\cdot 29^{134} + 22\cdot 29^{135} + \left(4 a + 20\right)\cdot 29^{136} + \left(13 a + 20\right)\cdot 29^{137} + \left(14 a + 25\right)\cdot 29^{138} + \left(17 a + 14\right)\cdot 29^{139} + \left(a + 28\right)\cdot 29^{140} + \left(27 a + 21\right)\cdot 29^{141} + \left(24 a + 8\right)\cdot 29^{142} + \left(28 a + 19\right)\cdot 29^{143} + \left(22 a + 28\right)\cdot 29^{144} + \left(28 a + 13\right)\cdot 29^{145} + \left(27 a + 27\right)\cdot 29^{146} + \left(12 a + 5\right)\cdot 29^{147} + \left(16 a + 15\right)\cdot 29^{148} + \left(20 a + 14\right)\cdot 29^{149} + 27 a\cdot 29^{150} + \left(13 a + 24\right)\cdot 29^{151} + \left(28 a + 17\right)\cdot 29^{152} + \left(5 a + 10\right)\cdot 29^{153} + \left(7 a + 11\right)\cdot 29^{154} + \left(19 a + 20\right)\cdot 29^{155} + \left(20 a + 1\right)\cdot 29^{156} + \left(24 a + 23\right)\cdot 29^{157} + \left(2 a + 12\right)\cdot 29^{158} +O\left(29^{ 159 }\right)$
$r_{ 6 }$ $=$ $ 23 + 2\cdot 29 + 24\cdot 29^{2} + 2\cdot 29^{4} + 28\cdot 29^{5} + 12\cdot 29^{6} + 13\cdot 29^{7} + 10\cdot 29^{8} + 29^{9} + 19\cdot 29^{10} + 20\cdot 29^{11} + 25\cdot 29^{12} + 13\cdot 29^{13} + 28\cdot 29^{14} + 26\cdot 29^{15} + 5\cdot 29^{16} + 16\cdot 29^{17} + 14\cdot 29^{18} + 7\cdot 29^{19} + 3\cdot 29^{20} + 24\cdot 29^{21} + 12\cdot 29^{22} + 29^{23} + 28\cdot 29^{24} + 15\cdot 29^{25} + 6\cdot 29^{26} + 18\cdot 29^{27} + 6\cdot 29^{28} + 8\cdot 29^{29} + 5\cdot 29^{30} + 2\cdot 29^{31} + 23\cdot 29^{32} + 5\cdot 29^{33} + 10\cdot 29^{34} + 12\cdot 29^{35} + 17\cdot 29^{36} + 4\cdot 29^{37} + 6\cdot 29^{38} + 29^{39} + 24\cdot 29^{40} + 2\cdot 29^{41} + 26\cdot 29^{42} + 4\cdot 29^{43} + 2\cdot 29^{44} + 2\cdot 29^{45} + 10\cdot 29^{46} + 19\cdot 29^{47} + 15\cdot 29^{48} + 8\cdot 29^{49} + 21\cdot 29^{50} + 5\cdot 29^{51} + 28\cdot 29^{52} + 23\cdot 29^{53} + 18\cdot 29^{54} + 27\cdot 29^{55} + 17\cdot 29^{56} + 29^{57} + 27\cdot 29^{58} + 27\cdot 29^{59} + 4\cdot 29^{60} + 20\cdot 29^{61} + 18\cdot 29^{62} + 20\cdot 29^{63} + 23\cdot 29^{64} + 2\cdot 29^{65} + 28\cdot 29^{66} + 11\cdot 29^{67} + 7\cdot 29^{68} + 22\cdot 29^{69} + 29^{70} + 6\cdot 29^{71} + 23\cdot 29^{72} + 25\cdot 29^{73} + 13\cdot 29^{74} + 29^{75} + 22\cdot 29^{76} + 21\cdot 29^{77} + 3\cdot 29^{79} + 15\cdot 29^{80} + 18\cdot 29^{81} + 19\cdot 29^{82} + 24\cdot 29^{83} + 23\cdot 29^{84} + 26\cdot 29^{85} + 27\cdot 29^{86} + 14\cdot 29^{87} + 15\cdot 29^{88} + 11\cdot 29^{89} + 26\cdot 29^{90} + 16\cdot 29^{91} + 26\cdot 29^{92} + 10\cdot 29^{94} + 29^{95} + 23\cdot 29^{96} + 16\cdot 29^{97} + 24\cdot 29^{98} + 7\cdot 29^{99} + 16\cdot 29^{100} + 22\cdot 29^{102} + 9\cdot 29^{103} + 28\cdot 29^{104} + 21\cdot 29^{105} + 10\cdot 29^{106} + 9\cdot 29^{107} + 24\cdot 29^{108} + 3\cdot 29^{109} + 9\cdot 29^{110} + 8\cdot 29^{111} + 18\cdot 29^{112} + 26\cdot 29^{113} + 3\cdot 29^{114} + 29^{115} + 8\cdot 29^{116} + 9\cdot 29^{117} + 10\cdot 29^{118} + 28\cdot 29^{119} + 22\cdot 29^{120} + 28\cdot 29^{121} + 2\cdot 29^{122} + 5\cdot 29^{123} + 27\cdot 29^{124} + 2\cdot 29^{125} + 16\cdot 29^{126} + 26\cdot 29^{127} + 10\cdot 29^{128} + 26\cdot 29^{129} + 13\cdot 29^{130} + 20\cdot 29^{131} + 12\cdot 29^{132} + 12\cdot 29^{133} + 11\cdot 29^{134} + 4\cdot 29^{135} + 25\cdot 29^{136} + 4\cdot 29^{137} + 12\cdot 29^{138} + 28\cdot 29^{139} + 27\cdot 29^{140} + 6\cdot 29^{141} + 29^{142} + 14\cdot 29^{143} + 7\cdot 29^{144} + 2\cdot 29^{145} + 28\cdot 29^{146} + 2\cdot 29^{147} + 24\cdot 29^{149} + 3\cdot 29^{151} + 28\cdot 29^{152} + 2\cdot 29^{153} + 7\cdot 29^{154} + 23\cdot 29^{155} + 12\cdot 29^{156} + 5\cdot 29^{157} + 10\cdot 29^{158} +O\left(29^{ 159 }\right)$
$r_{ 7 }$ $=$ $ 4 a + 8 + \left(20 a + 18\right)\cdot 29 + \left(28 a + 18\right)\cdot 29^{2} + \left(a + 4\right)\cdot 29^{3} + \left(9 a + 8\right)\cdot 29^{4} + \left(7 a + 4\right)\cdot 29^{5} + \left(3 a + 5\right)\cdot 29^{6} + \left(21 a + 14\right)\cdot 29^{7} + \left(24 a + 28\right)\cdot 29^{8} + \left(17 a + 3\right)\cdot 29^{9} + \left(27 a + 28\right)\cdot 29^{10} + \left(8 a + 25\right)\cdot 29^{11} + 7\cdot 29^{12} + \left(15 a + 23\right)\cdot 29^{13} + \left(26 a + 13\right)\cdot 29^{14} + \left(16 a + 19\right)\cdot 29^{15} + \left(12 a + 9\right)\cdot 29^{16} + \left(18 a + 17\right)\cdot 29^{17} + \left(8 a + 25\right)\cdot 29^{18} + \left(5 a + 27\right)\cdot 29^{19} + \left(12 a + 28\right)\cdot 29^{20} + \left(16 a + 16\right)\cdot 29^{21} + 21\cdot 29^{22} + \left(11 a + 17\right)\cdot 29^{23} + \left(2 a + 15\right)\cdot 29^{24} + \left(12 a + 16\right)\cdot 29^{25} + \left(14 a + 3\right)\cdot 29^{26} + \left(20 a + 2\right)\cdot 29^{27} + \left(3 a + 6\right)\cdot 29^{28} + \left(3 a + 2\right)\cdot 29^{29} + \left(24 a + 18\right)\cdot 29^{30} + \left(21 a + 28\right)\cdot 29^{31} + \left(8 a + 24\right)\cdot 29^{32} + \left(28 a + 27\right)\cdot 29^{33} + \left(19 a + 4\right)\cdot 29^{34} + \left(9 a + 23\right)\cdot 29^{35} + \left(2 a + 22\right)\cdot 29^{36} + \left(3 a + 20\right)\cdot 29^{37} + \left(12 a + 15\right)\cdot 29^{38} + \left(2 a + 28\right)\cdot 29^{39} + \left(24 a + 21\right)\cdot 29^{40} + \left(7 a + 24\right)\cdot 29^{41} + \left(27 a + 19\right)\cdot 29^{42} + 22\cdot 29^{43} + \left(12 a + 24\right)\cdot 29^{44} + \left(27 a + 12\right)\cdot 29^{45} + \left(15 a + 25\right)\cdot 29^{46} + \left(27 a + 3\right)\cdot 29^{47} + \left(4 a + 16\right)\cdot 29^{48} + \left(15 a + 3\right)\cdot 29^{49} + \left(16 a + 18\right)\cdot 29^{50} + \left(3 a + 25\right)\cdot 29^{51} + \left(23 a + 21\right)\cdot 29^{52} + \left(4 a + 13\right)\cdot 29^{53} + \left(16 a + 14\right)\cdot 29^{54} + \left(5 a + 6\right)\cdot 29^{55} + \left(23 a + 3\right)\cdot 29^{56} + \left(20 a + 3\right)\cdot 29^{57} + \left(4 a + 8\right)\cdot 29^{58} + \left(14 a + 19\right)\cdot 29^{59} + \left(25 a + 4\right)\cdot 29^{60} + \left(25 a + 23\right)\cdot 29^{61} + \left(12 a + 25\right)\cdot 29^{62} + \left(15 a + 17\right)\cdot 29^{63} + \left(8 a + 20\right)\cdot 29^{64} + \left(28 a + 3\right)\cdot 29^{65} + \left(20 a + 4\right)\cdot 29^{66} + \left(14 a + 26\right)\cdot 29^{67} + \left(10 a + 10\right)\cdot 29^{68} + \left(7 a + 11\right)\cdot 29^{69} + \left(22 a + 6\right)\cdot 29^{70} + \left(23 a + 14\right)\cdot 29^{71} + \left(4 a + 24\right)\cdot 29^{72} + \left(18 a + 25\right)\cdot 29^{73} + \left(3 a + 23\right)\cdot 29^{74} + \left(28 a + 26\right)\cdot 29^{75} + \left(20 a + 21\right)\cdot 29^{76} + \left(4 a + 15\right)\cdot 29^{77} + \left(5 a + 22\right)\cdot 29^{78} + 20\cdot 29^{79} + \left(21 a + 26\right)\cdot 29^{80} + \left(7 a + 9\right)\cdot 29^{81} + \left(9 a + 19\right)\cdot 29^{82} + \left(24 a + 16\right)\cdot 29^{83} + \left(10 a + 24\right)\cdot 29^{84} + \left(13 a + 24\right)\cdot 29^{85} + \left(24 a + 17\right)\cdot 29^{86} + \left(10 a + 1\right)\cdot 29^{87} + \left(17 a + 27\right)\cdot 29^{88} + \left(5 a + 19\right)\cdot 29^{89} + \left(9 a + 1\right)\cdot 29^{90} + \left(26 a + 5\right)\cdot 29^{91} + \left(9 a + 26\right)\cdot 29^{92} + 11 a\cdot 29^{93} + \left(9 a + 21\right)\cdot 29^{94} + \left(7 a + 13\right)\cdot 29^{95} + \left(4 a + 26\right)\cdot 29^{96} + \left(26 a + 11\right)\cdot 29^{97} + \left(11 a + 2\right)\cdot 29^{98} + \left(4 a + 19\right)\cdot 29^{99} + \left(4 a + 25\right)\cdot 29^{100} + \left(21 a + 15\right)\cdot 29^{101} + \left(18 a + 12\right)\cdot 29^{102} + \left(11 a + 13\right)\cdot 29^{103} + \left(27 a + 15\right)\cdot 29^{104} + \left(6 a + 12\right)\cdot 29^{105} + \left(12 a + 26\right)\cdot 29^{106} + \left(13 a + 19\right)\cdot 29^{107} + \left(2 a + 7\right)\cdot 29^{108} + \left(4 a + 24\right)\cdot 29^{109} + 2\cdot 29^{110} + 24\cdot 29^{111} + \left(13 a + 5\right)\cdot 29^{112} + \left(12 a + 25\right)\cdot 29^{113} + \left(27 a + 8\right)\cdot 29^{114} + 21 a\cdot 29^{115} + \left(25 a + 18\right)\cdot 29^{116} + \left(6 a + 3\right)\cdot 29^{117} + \left(27 a + 18\right)\cdot 29^{118} + \left(a + 11\right)\cdot 29^{119} + \left(19 a + 14\right)\cdot 29^{120} + 18 a\cdot 29^{121} + \left(15 a + 23\right)\cdot 29^{122} + \left(15 a + 5\right)\cdot 29^{123} + \left(20 a + 18\right)\cdot 29^{124} + \left(28 a + 15\right)\cdot 29^{125} + \left(7 a + 10\right)\cdot 29^{126} + \left(22 a + 13\right)\cdot 29^{127} + \left(16 a + 3\right)\cdot 29^{128} + \left(12 a + 22\right)\cdot 29^{129} + \left(16 a + 18\right)\cdot 29^{130} + \left(19 a + 15\right)\cdot 29^{131} + 8\cdot 29^{132} + \left(14 a + 3\right)\cdot 29^{133} + \left(25 a + 27\right)\cdot 29^{134} + \left(28 a + 12\right)\cdot 29^{135} + \left(16 a + 20\right)\cdot 29^{136} + \left(25 a + 9\right)\cdot 29^{137} + \left(25 a + 19\right)\cdot 29^{138} + \left(6 a + 18\right)\cdot 29^{139} + \left(28 a + 19\right)\cdot 29^{140} + \left(16 a + 21\right)\cdot 29^{141} + \left(6 a + 11\right)\cdot 29^{142} + \left(5 a + 20\right)\cdot 29^{143} + \left(23 a + 8\right)\cdot 29^{144} + \left(8 a + 21\right)\cdot 29^{145} + \left(16 a + 8\right)\cdot 29^{146} + \left(26 a + 16\right)\cdot 29^{147} + \left(24 a + 7\right)\cdot 29^{148} + \left(10 a + 10\right)\cdot 29^{149} + \left(25 a + 22\right)\cdot 29^{150} + \left(2 a + 6\right)\cdot 29^{151} + \left(12 a + 4\right)\cdot 29^{152} + \left(a + 2\right)\cdot 29^{153} + \left(25 a + 6\right)\cdot 29^{154} + \left(a + 21\right)\cdot 29^{155} + \left(10 a + 11\right)\cdot 29^{156} + \left(a + 4\right)\cdot 29^{157} + \left(7 a + 27\right)\cdot 29^{158} +O\left(29^{ 159 }\right)$
$r_{ 8 }$ $=$ $ 3 a + 4 + \left(10 a + 13\right)\cdot 29 + \left(9 a + 7\right)\cdot 29^{2} + \left(10 a + 5\right)\cdot 29^{3} + \left(23 a + 14\right)\cdot 29^{4} + \left(21 a + 22\right)\cdot 29^{5} + \left(22 a + 26\right)\cdot 29^{6} + \left(4 a + 8\right)\cdot 29^{7} + \left(20 a + 8\right)\cdot 29^{8} + \left(27 a + 8\right)\cdot 29^{9} + \left(5 a + 3\right)\cdot 29^{10} + \left(25 a + 11\right)\cdot 29^{11} + \left(20 a + 1\right)\cdot 29^{12} + \left(21 a + 8\right)\cdot 29^{13} + \left(8 a + 20\right)\cdot 29^{14} + \left(a + 25\right)\cdot 29^{15} + \left(24 a + 17\right)\cdot 29^{16} + \left(16 a + 11\right)\cdot 29^{17} + \left(15 a + 11\right)\cdot 29^{18} + \left(24 a + 3\right)\cdot 29^{19} + \left(a + 26\right)\cdot 29^{20} + \left(5 a + 11\right)\cdot 29^{21} + \left(26 a + 13\right)\cdot 29^{22} + \left(16 a + 22\right)\cdot 29^{23} + \left(22 a + 8\right)\cdot 29^{24} + \left(17 a + 26\right)\cdot 29^{25} + 17 a\cdot 29^{26} + \left(24 a + 6\right)\cdot 29^{27} + \left(24 a + 7\right)\cdot 29^{28} + \left(14 a + 4\right)\cdot 29^{29} + \left(27 a + 10\right)\cdot 29^{30} + \left(3 a + 20\right)\cdot 29^{31} + \left(25 a + 21\right)\cdot 29^{32} + \left(9 a + 5\right)\cdot 29^{33} + \left(24 a + 10\right)\cdot 29^{34} + \left(13 a + 25\right)\cdot 29^{35} + \left(18 a + 17\right)\cdot 29^{36} + \left(26 a + 1\right)\cdot 29^{37} + \left(23 a + 11\right)\cdot 29^{38} + \left(10 a + 18\right)\cdot 29^{39} + \left(12 a + 5\right)\cdot 29^{40} + \left(2 a + 19\right)\cdot 29^{41} + \left(16 a + 28\right)\cdot 29^{42} + \left(27 a + 7\right)\cdot 29^{43} + 21 a\cdot 29^{44} + \left(19 a + 10\right)\cdot 29^{45} + \left(14 a + 4\right)\cdot 29^{46} + \left(16 a + 17\right)\cdot 29^{47} + 16\cdot 29^{48} + \left(19 a + 17\right)\cdot 29^{49} + \left(19 a + 5\right)\cdot 29^{50} + \left(12 a + 1\right)\cdot 29^{51} + \left(2 a + 15\right)\cdot 29^{52} + \left(10 a + 26\right)\cdot 29^{53} + 11 a\cdot 29^{54} + \left(22 a + 28\right)\cdot 29^{55} + \left(5 a + 2\right)\cdot 29^{56} + \left(18 a + 1\right)\cdot 29^{57} + \left(24 a + 25\right)\cdot 29^{58} + \left(20 a + 14\right)\cdot 29^{59} + \left(11 a + 4\right)\cdot 29^{60} + \left(22 a + 8\right)\cdot 29^{61} + \left(4 a + 6\right)\cdot 29^{62} + \left(27 a + 15\right)\cdot 29^{63} + \left(25 a + 6\right)\cdot 29^{64} + \left(20 a + 11\right)\cdot 29^{65} + \left(3 a + 22\right)\cdot 29^{66} + \left(9 a + 22\right)\cdot 29^{67} + \left(6 a + 4\right)\cdot 29^{68} + 18 a\cdot 29^{69} + \left(23 a + 5\right)\cdot 29^{70} + \left(12 a + 16\right)\cdot 29^{71} + \left(6 a + 18\right)\cdot 29^{72} + \left(22 a + 23\right)\cdot 29^{73} + 6 a\cdot 29^{74} + 5 a\cdot 29^{75} + \left(10 a + 9\right)\cdot 29^{76} + \left(6 a + 13\right)\cdot 29^{77} + \left(22 a + 23\right)\cdot 29^{78} + \left(6 a + 2\right)\cdot 29^{79} + \left(22 a + 2\right)\cdot 29^{80} + \left(11 a + 4\right)\cdot 29^{81} + \left(18 a + 15\right)\cdot 29^{82} + \left(27 a + 22\right)\cdot 29^{83} + \left(11 a + 3\right)\cdot 29^{84} + \left(14 a + 22\right)\cdot 29^{85} + \left(4 a + 11\right)\cdot 29^{86} + \left(2 a + 21\right)\cdot 29^{87} + \left(17 a + 11\right)\cdot 29^{88} + \left(3 a + 15\right)\cdot 29^{89} + \left(13 a + 19\right)\cdot 29^{90} + \left(19 a + 4\right)\cdot 29^{91} + \left(26 a + 19\right)\cdot 29^{92} + \left(13 a + 8\right)\cdot 29^{93} + \left(26 a + 18\right)\cdot 29^{94} + \left(21 a + 25\right)\cdot 29^{95} + \left(13 a + 25\right)\cdot 29^{96} + \left(5 a + 9\right)\cdot 29^{97} + \left(12 a + 12\right)\cdot 29^{98} + \left(6 a + 16\right)\cdot 29^{99} + 24 a\cdot 29^{100} + \left(6 a + 20\right)\cdot 29^{101} + \left(2 a + 9\right)\cdot 29^{102} + \left(7 a + 12\right)\cdot 29^{103} + \left(19 a + 12\right)\cdot 29^{104} + \left(17 a + 19\right)\cdot 29^{105} + \left(17 a + 19\right)\cdot 29^{106} + \left(13 a + 27\right)\cdot 29^{107} + \left(23 a + 25\right)\cdot 29^{108} + \left(16 a + 24\right)\cdot 29^{109} + \left(4 a + 3\right)\cdot 29^{110} + \left(11 a + 22\right)\cdot 29^{111} + \left(13 a + 27\right)\cdot 29^{112} + \left(24 a + 16\right)\cdot 29^{113} + \left(18 a + 3\right)\cdot 29^{114} + \left(12 a + 16\right)\cdot 29^{115} + \left(2 a + 5\right)\cdot 29^{116} + \left(5 a + 10\right)\cdot 29^{117} + \left(20 a + 25\right)\cdot 29^{118} + 21 a\cdot 29^{119} + \left(a + 24\right)\cdot 29^{120} + \left(3 a + 22\right)\cdot 29^{121} + \left(28 a + 24\right)\cdot 29^{122} + \left(14 a + 7\right)\cdot 29^{123} + \left(a + 26\right)\cdot 29^{124} + \left(10 a + 18\right)\cdot 29^{125} + \left(25 a + 19\right)\cdot 29^{126} + \left(8 a + 7\right)\cdot 29^{127} + \left(18 a + 20\right)\cdot 29^{128} + \left(a + 20\right)\cdot 29^{129} + \left(20 a + 9\right)\cdot 29^{130} + \left(16 a + 14\right)\cdot 29^{131} + \left(a + 14\right)\cdot 29^{132} + \left(19 a + 23\right)\cdot 29^{133} + \left(a + 25\right)\cdot 29^{134} + \left(28 a + 28\right)\cdot 29^{135} + \left(24 a + 10\right)\cdot 29^{136} + \left(15 a + 24\right)\cdot 29^{137} + \left(14 a + 26\right)\cdot 29^{138} + 11 a\cdot 29^{139} + \left(27 a + 19\right)\cdot 29^{140} + \left(a + 10\right)\cdot 29^{141} + \left(4 a + 19\right)\cdot 29^{142} + 22\cdot 29^{143} + \left(6 a + 27\right)\cdot 29^{144} + 18\cdot 29^{145} + \left(a + 22\right)\cdot 29^{146} + \left(16 a + 13\right)\cdot 29^{147} + \left(12 a + 26\right)\cdot 29^{148} + \left(8 a + 13\right)\cdot 29^{149} + \left(a + 2\right)\cdot 29^{150} + \left(15 a + 8\right)\cdot 29^{151} + 29^{152} + \left(23 a + 12\right)\cdot 29^{153} + \left(21 a + 12\right)\cdot 29^{154} + \left(9 a + 22\right)\cdot 29^{155} + \left(8 a + 27\right)\cdot 29^{156} + \left(4 a + 9\right)\cdot 29^{157} + \left(26 a + 2\right)\cdot 29^{158} +O\left(29^{ 159 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character values
$c1$
$1$ $1$ $()$ $9$
$6$ $2$ $(1,4)(3,7)$ $-3$
$9$ $2$ $(1,4)(2,6)(3,7)(5,8)$ $1$
$12$ $2$ $(1,3)$ $-3$
$24$ $2$ $(1,2)(3,5)(4,6)(7,8)$ $3$
$36$ $2$ $(1,3)(2,5)$ $1$
$36$ $2$ $(1,3)(2,6)(5,8)$ $1$
$16$ $3$ $(3,4,7)$ $0$
$64$ $3$ $(3,4,7)(5,6,8)$ $0$
$12$ $4$ $(1,3,4,7)$ $3$
$36$ $4$ $(1,3,4,7)(2,5,6,8)$ $1$
$36$ $4$ $(1,4)(2,5,6,8)(3,7)$ $-1$
$72$ $4$ $(1,6,4,2)(3,8,7,5)$ $-1$
$72$ $4$ $(1,3)(2,5,6,8)$ $-1$
$144$ $4$ $(1,5,3,2)(4,6)(7,8)$ $-1$
$48$ $6$ $(2,6)(3,7,4)(5,8)$ $0$
$96$ $6$ $(1,3)(2,6,8)$ $0$
$192$ $6$ $(1,2)(3,6,4,8,7,5)$ $0$
$144$ $8$ $(1,5,3,6,4,8,7,2)$ $1$
$96$ $12$ $(2,5,6,8)(3,4,7)$ $0$
The blue line marks the conjugacy class containing complex conjugation.