Properties

Label 7.2e16_1423e2.8t48.1c1
Dimension 7
Group $C_2^3:\GL(3,2)$
Conductor $ 2^{16} \cdot 1423^{2}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$7$
Group:$C_2^3:\GL(3,2)$
Conductor:$132705746944= 2^{16} \cdot 1423^{2} $
Artin number field: Splitting field of $f= x^{8} - 10 x^{6} - 4 x^{5} + 27 x^{4} + 20 x^{3} - 10 x^{2} - 8 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_2^3:\GL(3,2)$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$7$
$7$$2$$(1,5)(2,6)(3,4)(7,8)$$-1$
$42$$2$$(1,8)(2,4)(3,7)(5,6)$$-1$
$42$$2$$(3,5)(6,7)$$3$
$224$$3$$(2,6,7)(3,5,4)$$1$
$84$$4$$(1,2,5,6)(3,8,4,7)$$-1$
$168$$4$$(1,6,8,5)(2,3,4,7)$$-1$
$168$$4$$(1,4)(3,6,5,7)$$1$
$224$$6$$(1,8)(2,5,6,4,7,3)$$-1$
$192$$7$$(1,6,3,7,2,8,5)$$0$
$192$$7$$(1,7,5,3,8,6,2)$$0$
The blue line marks the conjugacy class containing complex conjugation.