Properties

Label 8.2e20_127e2.12t213.1c1
Dimension 8
Group $S_3\wr S_3$
Conductor $ 2^{20} \cdot 127^{2}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$8$
Group:$S_3\wr S_3$
Conductor:$16912482304= 2^{20} \cdot 127^{2} $
Artin number field: Splitting field of $f= x^{9} - 4 x^{8} + 9 x^{7} - 6 x^{6} - 7 x^{5} + 12 x^{4} + 5 x^{3} - 2 x^{2} - 8 x - 4 $ 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 $\Q_{ 29 }$ to precision 175.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{3} + 2 x + 27 $
Roots:
$r_{ 1 }$ $=$ $ 24 + 17\cdot 29 + 17\cdot 29^{2} + 5\cdot 29^{3} + 16\cdot 29^{4} + 19\cdot 29^{5} + 25\cdot 29^{6} + 21\cdot 29^{7} + 26\cdot 29^{8} + 23\cdot 29^{10} + 18\cdot 29^{11} + 19\cdot 29^{12} + 8\cdot 29^{13} + 6\cdot 29^{14} + 8\cdot 29^{15} + 27\cdot 29^{16} + 11\cdot 29^{17} + 5\cdot 29^{18} + 21\cdot 29^{19} + 27\cdot 29^{20} + 22\cdot 29^{21} + 28\cdot 29^{22} + 23\cdot 29^{23} + 14\cdot 29^{24} + 13\cdot 29^{25} + 29^{26} + 28\cdot 29^{27} + 2\cdot 29^{28} + 26\cdot 29^{30} + 4\cdot 29^{31} + 24\cdot 29^{32} + 25\cdot 29^{33} + 17\cdot 29^{35} + 13\cdot 29^{36} + 16\cdot 29^{37} + 6\cdot 29^{39} + 20\cdot 29^{40} + 9\cdot 29^{41} + 28\cdot 29^{42} + 8\cdot 29^{43} + 4\cdot 29^{44} + 7\cdot 29^{45} + 11\cdot 29^{46} + 15\cdot 29^{47} + 24\cdot 29^{48} + 4\cdot 29^{49} + 26\cdot 29^{50} + 26\cdot 29^{51} + 9\cdot 29^{52} + 12\cdot 29^{53} + 12\cdot 29^{54} + 27\cdot 29^{55} + 16\cdot 29^{56} + 4\cdot 29^{57} + 20\cdot 29^{58} + 2\cdot 29^{59} + 3\cdot 29^{60} + 25\cdot 29^{61} + 8\cdot 29^{62} + 29^{63} + 15\cdot 29^{64} + 22\cdot 29^{65} + 15\cdot 29^{66} + 7\cdot 29^{69} + 29^{70} + 7\cdot 29^{71} + 27\cdot 29^{72} + 22\cdot 29^{73} + 3\cdot 29^{74} + 20\cdot 29^{75} + 20\cdot 29^{76} + 14\cdot 29^{77} + 14\cdot 29^{78} + 13\cdot 29^{79} + 29^{80} + 5\cdot 29^{81} + 9\cdot 29^{82} + 2\cdot 29^{83} + 13\cdot 29^{84} + 6\cdot 29^{85} + 23\cdot 29^{86} + 18\cdot 29^{87} + 8\cdot 29^{88} + 20\cdot 29^{89} + 8\cdot 29^{90} + 23\cdot 29^{91} + 9\cdot 29^{92} + 5\cdot 29^{93} + 9\cdot 29^{94} + 20\cdot 29^{95} + 23\cdot 29^{96} + 8\cdot 29^{97} + 12\cdot 29^{98} + 5\cdot 29^{99} + 23\cdot 29^{100} + 11\cdot 29^{102} + 26\cdot 29^{103} + 12\cdot 29^{104} + 6\cdot 29^{105} + 22\cdot 29^{106} + 15\cdot 29^{107} + 20\cdot 29^{108} + 4\cdot 29^{109} + 21\cdot 29^{110} + 6\cdot 29^{111} + 28\cdot 29^{112} + 24\cdot 29^{113} + 23\cdot 29^{114} + 21\cdot 29^{115} + 23\cdot 29^{116} + 23\cdot 29^{117} + 9\cdot 29^{118} + 14\cdot 29^{119} + 15\cdot 29^{120} + 7\cdot 29^{121} + 26\cdot 29^{122} + 24\cdot 29^{123} + 16\cdot 29^{124} + 17\cdot 29^{125} + 28\cdot 29^{126} + 26\cdot 29^{127} + 16\cdot 29^{128} + 27\cdot 29^{129} + 20\cdot 29^{130} + 4\cdot 29^{131} + 8\cdot 29^{132} + 29^{133} + 17\cdot 29^{134} + 27\cdot 29^{135} + 3\cdot 29^{136} + 7\cdot 29^{137} + 15\cdot 29^{138} + 4\cdot 29^{139} + 19\cdot 29^{140} + 24\cdot 29^{141} + 9\cdot 29^{142} + 28\cdot 29^{143} + 12\cdot 29^{144} + 21\cdot 29^{145} + 4\cdot 29^{146} + 26\cdot 29^{147} + 4\cdot 29^{148} + 16\cdot 29^{149} + 20\cdot 29^{150} + 24\cdot 29^{151} + 20\cdot 29^{152} + 4\cdot 29^{153} + 16\cdot 29^{154} + 6\cdot 29^{155} + 24\cdot 29^{156} + 4\cdot 29^{157} + 6\cdot 29^{158} + 3\cdot 29^{159} + 27\cdot 29^{160} + 7\cdot 29^{161} + 5\cdot 29^{162} + 9\cdot 29^{163} + 28\cdot 29^{164} + 26\cdot 29^{165} + 26\cdot 29^{166} + 28\cdot 29^{167} + 21\cdot 29^{168} + 24\cdot 29^{169} + 17\cdot 29^{170} + 29^{171} + 25\cdot 29^{173} + 19\cdot 29^{174} +O\left(29^{ 175 }\right)$
$r_{ 2 }$ $=$ $ 9 + 25\cdot 29 + 17\cdot 29^{2} + 8\cdot 29^{4} + 4\cdot 29^{6} + 19\cdot 29^{7} + 19\cdot 29^{8} + 3\cdot 29^{9} + 2\cdot 29^{10} + 29^{11} + 16\cdot 29^{12} + 3\cdot 29^{13} + 6\cdot 29^{14} + 23\cdot 29^{15} + 20\cdot 29^{16} + 23\cdot 29^{17} + 29^{18} + 2\cdot 29^{19} + 8\cdot 29^{20} + 9\cdot 29^{21} + 18\cdot 29^{22} + 23\cdot 29^{23} + 23\cdot 29^{25} + 21\cdot 29^{26} + 15\cdot 29^{27} + 22\cdot 29^{28} + 14\cdot 29^{29} + 4\cdot 29^{30} + 12\cdot 29^{31} + 11\cdot 29^{32} + 15\cdot 29^{33} + 11\cdot 29^{34} + 9\cdot 29^{35} + 5\cdot 29^{36} + 19\cdot 29^{37} + 25\cdot 29^{38} + 27\cdot 29^{39} + 24\cdot 29^{40} + 19\cdot 29^{41} + 19\cdot 29^{42} + 19\cdot 29^{43} + 19\cdot 29^{44} + 12\cdot 29^{45} + 29^{46} + 24\cdot 29^{47} + 28\cdot 29^{49} + 4\cdot 29^{51} + 6\cdot 29^{52} + 16\cdot 29^{53} + 6\cdot 29^{54} + 5\cdot 29^{55} + 17\cdot 29^{56} + 16\cdot 29^{57} + 6\cdot 29^{58} + 24\cdot 29^{59} + 14\cdot 29^{60} + 26\cdot 29^{61} + 14\cdot 29^{62} + 2\cdot 29^{63} + 20\cdot 29^{64} + 10\cdot 29^{65} + 6\cdot 29^{66} + 15\cdot 29^{67} + 28\cdot 29^{68} + 27\cdot 29^{69} + 14\cdot 29^{70} + 11\cdot 29^{71} + 29^{72} + 3\cdot 29^{73} + 17\cdot 29^{74} + 27\cdot 29^{75} + 23\cdot 29^{76} + 13\cdot 29^{77} + 28\cdot 29^{78} + 28\cdot 29^{79} + 9\cdot 29^{80} + 29^{81} + 11\cdot 29^{82} + 27\cdot 29^{83} + 2\cdot 29^{84} + 13\cdot 29^{85} + 16\cdot 29^{86} + 25\cdot 29^{87} + 12\cdot 29^{88} + 16\cdot 29^{89} + 12\cdot 29^{90} + 25\cdot 29^{91} + 18\cdot 29^{92} + 8\cdot 29^{93} + 9\cdot 29^{94} + 21\cdot 29^{95} + 27\cdot 29^{96} + 10\cdot 29^{97} + 7\cdot 29^{98} + 13\cdot 29^{99} + 12\cdot 29^{100} + 5\cdot 29^{101} + 13\cdot 29^{102} + 10\cdot 29^{103} + 23\cdot 29^{104} + 20\cdot 29^{105} + 10\cdot 29^{106} + 2\cdot 29^{107} + 18\cdot 29^{108} + 9\cdot 29^{109} + 6\cdot 29^{110} + 5\cdot 29^{111} + 20\cdot 29^{113} + 27\cdot 29^{114} + 17\cdot 29^{115} + 8\cdot 29^{116} + 12\cdot 29^{117} + 19\cdot 29^{118} + 19\cdot 29^{119} + 27\cdot 29^{120} + 28\cdot 29^{121} + 25\cdot 29^{122} + 20\cdot 29^{123} + 17\cdot 29^{124} + 23\cdot 29^{125} + 11\cdot 29^{126} + 12\cdot 29^{127} + 28\cdot 29^{128} + 12\cdot 29^{129} + 27\cdot 29^{130} + 15\cdot 29^{131} + 29^{132} + 25\cdot 29^{133} + 7\cdot 29^{134} + 27\cdot 29^{135} + 12\cdot 29^{136} + 4\cdot 29^{137} + 29^{138} + 8\cdot 29^{139} + 5\cdot 29^{140} + 29^{141} + 25\cdot 29^{142} + 15\cdot 29^{143} + 20\cdot 29^{145} + 5\cdot 29^{146} + 8\cdot 29^{147} + 24\cdot 29^{148} + 11\cdot 29^{149} + 18\cdot 29^{151} + 3\cdot 29^{152} + 21\cdot 29^{153} + 27\cdot 29^{154} + 2\cdot 29^{155} + 20\cdot 29^{156} + 17\cdot 29^{157} + 16\cdot 29^{158} + 7\cdot 29^{159} + 7\cdot 29^{160} + 7\cdot 29^{161} + 17\cdot 29^{162} + 8\cdot 29^{163} + 2\cdot 29^{164} + 23\cdot 29^{165} + 15\cdot 29^{166} + 26\cdot 29^{167} + 5\cdot 29^{168} + 29^{169} + 20\cdot 29^{170} + 5\cdot 29^{171} + 17\cdot 29^{172} + 20\cdot 29^{173} + 11\cdot 29^{174} +O\left(29^{ 175 }\right)$
$r_{ 3 }$ $=$ $ 4 a^{2} + 20 a + 21 + \left(27 a^{2} + a + 9\right)\cdot 29 + \left(13 a^{2} + 10 a + 27\right)\cdot 29^{2} + \left(23 a^{2} + 15 a + 28\right)\cdot 29^{3} + \left(24 a^{2} + 18 a + 15\right)\cdot 29^{4} + \left(4 a^{2} + 23 a + 22\right)\cdot 29^{5} + \left(19 a^{2} + 26 a + 2\right)\cdot 29^{6} + \left(5 a^{2} + 5 a\right)\cdot 29^{7} + \left(16 a^{2} + 21 a + 15\right)\cdot 29^{8} + \left(a^{2} + 10 a + 8\right)\cdot 29^{9} + \left(15 a^{2} + 11 a + 17\right)\cdot 29^{10} + \left(a^{2} + 21 a + 12\right)\cdot 29^{11} + \left(21 a^{2} + 22 a + 2\right)\cdot 29^{12} + \left(5 a^{2} + 14 a + 18\right)\cdot 29^{13} + \left(12 a^{2} + a + 18\right)\cdot 29^{14} + \left(16 a^{2} + 9 a + 19\right)\cdot 29^{15} + \left(21 a^{2} + 4 a + 17\right)\cdot 29^{16} + \left(23 a^{2} + 27 a + 7\right)\cdot 29^{17} + \left(13 a^{2} + 5 a + 21\right)\cdot 29^{18} + \left(20 a^{2} + 28 a + 17\right)\cdot 29^{19} + \left(2 a^{2} + 21 a + 12\right)\cdot 29^{20} + \left(16 a^{2} + 9 a + 26\right)\cdot 29^{21} + \left(20 a^{2} + 10 a + 13\right)\cdot 29^{22} + \left(10 a^{2} + 8 a + 7\right)\cdot 29^{23} + \left(8 a^{2} + 28 a + 27\right)\cdot 29^{24} + \left(8 a^{2} + 15 a + 9\right)\cdot 29^{25} + \left(4 a^{2} + 15\right)\cdot 29^{26} + \left(17 a^{2} + 14 a + 13\right)\cdot 29^{27} + \left(20 a^{2} + 7 a + 24\right)\cdot 29^{28} + \left(17 a^{2} + 11 a + 25\right)\cdot 29^{29} + \left(15 a^{2} + a + 11\right)\cdot 29^{30} + \left(12 a^{2} + 9 a + 8\right)\cdot 29^{31} + \left(8 a^{2} + 4 a + 28\right)\cdot 29^{32} + \left(a^{2} + 20 a + 19\right)\cdot 29^{33} + \left(26 a^{2} + 22 a + 24\right)\cdot 29^{34} + \left(23 a^{2} + 14 a + 25\right)\cdot 29^{35} + \left(23 a^{2} + 15 a + 5\right)\cdot 29^{36} + \left(10 a^{2} + 13 a\right)\cdot 29^{37} + \left(18 a + 21\right)\cdot 29^{38} + \left(7 a^{2} + 7 a + 3\right)\cdot 29^{39} + \left(11 a^{2} + 6 a + 21\right)\cdot 29^{40} + \left(18 a^{2} + 16 a + 3\right)\cdot 29^{41} + \left(24 a^{2} + 17 a + 22\right)\cdot 29^{42} + \left(25 a + 13\right)\cdot 29^{43} + \left(9 a^{2} + 19 a + 3\right)\cdot 29^{44} + \left(10 a^{2} + 10 a + 7\right)\cdot 29^{45} + \left(7 a^{2} + 5 a + 22\right)\cdot 29^{46} + \left(25 a^{2} + 21 a + 2\right)\cdot 29^{47} + \left(6 a^{2} + 27 a + 24\right)\cdot 29^{48} + \left(3 a^{2} + 18 a + 28\right)\cdot 29^{49} + \left(10 a^{2} + 23 a + 28\right)\cdot 29^{50} + \left(10 a^{2} + 18 a + 17\right)\cdot 29^{51} + \left(20 a^{2} + 16 a + 20\right)\cdot 29^{52} + \left(27 a^{2} + 27 a + 19\right)\cdot 29^{53} + \left(17 a^{2} + 4 a + 22\right)\cdot 29^{54} + \left(6 a^{2} + 3 a + 14\right)\cdot 29^{55} + \left(13 a^{2} + 20 a\right)\cdot 29^{56} + \left(26 a^{2} + 5 a + 15\right)\cdot 29^{57} + \left(7 a^{2} + 11 a + 17\right)\cdot 29^{58} + \left(7 a^{2} + 5 a + 18\right)\cdot 29^{59} + \left(28 a^{2} + 16 a + 22\right)\cdot 29^{60} + \left(23 a^{2} + 12 a\right)\cdot 29^{61} + \left(24 a^{2} + 10 a + 23\right)\cdot 29^{62} + \left(27 a^{2} + 6 a + 1\right)\cdot 29^{63} + \left(10 a^{2} + 26 a + 21\right)\cdot 29^{64} + \left(25 a^{2} + 21 a + 13\right)\cdot 29^{65} + \left(28 a^{2} + 9 a + 2\right)\cdot 29^{66} + \left(15 a^{2} + 19 a + 8\right)\cdot 29^{67} + \left(21 a^{2} + 10 a + 25\right)\cdot 29^{68} + \left(2 a^{2} + 18 a + 21\right)\cdot 29^{69} + \left(18 a^{2} + a + 27\right)\cdot 29^{70} + \left(8 a^{2} + 8 a + 7\right)\cdot 29^{71} + \left(10 a^{2} + 13 a + 10\right)\cdot 29^{72} + \left(26 a^{2} + 21 a + 5\right)\cdot 29^{73} + \left(26 a^{2} + 27 a + 4\right)\cdot 29^{74} + \left(13 a^{2} + 7 a + 5\right)\cdot 29^{75} + \left(26 a^{2} + 3 a + 27\right)\cdot 29^{76} + \left(27 a^{2} + 22 a + 21\right)\cdot 29^{77} + \left(2 a^{2} + 17 a + 12\right)\cdot 29^{78} + \left(8 a^{2} + 23 a + 5\right)\cdot 29^{79} + \left(21 a^{2} + 6 a + 22\right)\cdot 29^{80} + \left(2 a^{2} + 4 a + 21\right)\cdot 29^{81} + \left(24 a^{2} + 15 a + 21\right)\cdot 29^{82} + \left(18 a^{2} + 12 a + 13\right)\cdot 29^{83} + \left(16 a^{2} + 6 a + 13\right)\cdot 29^{84} + \left(6 a^{2} + 26 a + 24\right)\cdot 29^{85} + \left(5 a^{2} + 3 a + 22\right)\cdot 29^{86} + \left(21 a^{2} + 2 a + 1\right)\cdot 29^{87} + \left(13 a^{2} + 5 a + 23\right)\cdot 29^{88} + \left(18 a^{2} + 20 a + 24\right)\cdot 29^{89} + \left(24 a^{2} + 15 a + 27\right)\cdot 29^{90} + \left(28 a^{2} + 15 a + 10\right)\cdot 29^{91} + \left(17 a^{2} + 20 a + 11\right)\cdot 29^{92} + \left(6 a^{2} + 28 a + 19\right)\cdot 29^{93} + \left(4 a^{2} + 6 a + 2\right)\cdot 29^{94} + \left(17 a^{2} + 6 a + 23\right)\cdot 29^{95} + \left(2 a^{2} + 15 a + 22\right)\cdot 29^{96} + \left(13 a^{2} + 17\right)\cdot 29^{97} + \left(2 a^{2} + 27 a + 20\right)\cdot 29^{98} + \left(16 a^{2} + a + 28\right)\cdot 29^{99} + \left(26 a^{2} + 28 a + 28\right)\cdot 29^{100} + \left(16 a^{2} + 18 a + 5\right)\cdot 29^{101} + \left(27 a^{2} + 27 a + 3\right)\cdot 29^{102} + \left(25 a^{2} + 25 a + 14\right)\cdot 29^{103} + \left(28 a^{2} + 18 a + 15\right)\cdot 29^{104} + \left(18 a^{2} + 8 a + 27\right)\cdot 29^{105} + \left(8 a^{2} + 20 a + 28\right)\cdot 29^{106} + \left(27 a^{2} + 24 a + 4\right)\cdot 29^{107} + \left(17 a^{2} + 15 a + 4\right)\cdot 29^{108} + \left(11 a^{2} + 12 a + 12\right)\cdot 29^{109} + \left(25 a^{2} + 5 a + 25\right)\cdot 29^{110} + \left(26 a^{2} + 9 a + 5\right)\cdot 29^{111} + \left(10 a^{2} + 23 a + 4\right)\cdot 29^{112} + \left(12 a^{2} + 25 a + 28\right)\cdot 29^{113} + \left(15 a^{2} + 16 a + 25\right)\cdot 29^{114} + \left(7 a^{2} + 4 a + 5\right)\cdot 29^{115} + \left(16 a^{2} + 22 a + 1\right)\cdot 29^{116} + \left(9 a^{2} + 6 a + 16\right)\cdot 29^{117} + \left(27 a^{2} + 5 a + 17\right)\cdot 29^{118} + \left(16 a^{2} + 26 a + 14\right)\cdot 29^{119} + \left(22 a^{2} + 10 a + 5\right)\cdot 29^{120} + \left(28 a^{2} + 21 a + 22\right)\cdot 29^{121} + \left(23 a^{2} + 4 a + 1\right)\cdot 29^{122} + \left(a^{2} + 21 a + 9\right)\cdot 29^{123} + \left(a^{2} + 28 a + 24\right)\cdot 29^{124} + \left(26 a^{2} + 28 a + 13\right)\cdot 29^{125} + \left(27 a^{2} + 13 a + 20\right)\cdot 29^{126} + \left(11 a^{2} + 24 a + 25\right)\cdot 29^{127} + \left(6 a^{2} + 21 a + 16\right)\cdot 29^{128} + \left(18 a^{2} + 2\right)\cdot 29^{129} + \left(9 a^{2} + 6 a + 20\right)\cdot 29^{130} + \left(11 a^{2} + 8 a + 23\right)\cdot 29^{131} + \left(19 a^{2} + 17 a + 23\right)\cdot 29^{132} + \left(21 a^{2} + 11 a + 6\right)\cdot 29^{133} + \left(13 a^{2} + 19 a + 11\right)\cdot 29^{134} + \left(7 a^{2} + 25 a + 4\right)\cdot 29^{135} + \left(16 a^{2} + 23 a + 13\right)\cdot 29^{136} + \left(2 a^{2} + 4 a + 5\right)\cdot 29^{137} + \left(20 a^{2} + 24 a + 9\right)\cdot 29^{138} + \left(19 a^{2} + 16 a + 8\right)\cdot 29^{139} + \left(2 a^{2} + 20 a + 28\right)\cdot 29^{140} + \left(28 a^{2} + 28 a + 1\right)\cdot 29^{141} + \left(9 a^{2} + 21 a + 22\right)\cdot 29^{142} + \left(8 a^{2} + 19\right)\cdot 29^{143} + \left(15 a^{2} + 27 a + 11\right)\cdot 29^{144} + \left(19 a^{2} + 21 a + 18\right)\cdot 29^{145} + \left(21 a^{2} + 17 a + 1\right)\cdot 29^{146} + \left(18 a^{2} + 22 a + 19\right)\cdot 29^{147} + \left(9 a^{2} + 22 a + 11\right)\cdot 29^{148} + \left(18 a^{2} + 20 a + 15\right)\cdot 29^{149} + \left(27 a^{2} + 9 a + 9\right)\cdot 29^{150} + \left(14 a^{2} + 15 a + 4\right)\cdot 29^{151} + \left(10 a^{2} + 6 a + 25\right)\cdot 29^{152} + \left(14 a^{2} + 10 a + 15\right)\cdot 29^{153} + \left(28 a^{2} + 25 a + 13\right)\cdot 29^{154} + \left(11 a^{2} + 26 a + 7\right)\cdot 29^{155} + \left(18 a^{2} + 25 a + 4\right)\cdot 29^{156} + \left(11 a^{2} + 3 a + 8\right)\cdot 29^{157} + \left(6 a^{2} + 15 a + 9\right)\cdot 29^{158} + \left(24 a^{2} + 16 a + 26\right)\cdot 29^{159} + \left(23 a^{2} + 23 a + 5\right)\cdot 29^{160} + \left(10 a^{2} + 7 a + 19\right)\cdot 29^{161} + \left(25 a^{2} + 25 a + 4\right)\cdot 29^{162} + \left(7 a^{2} + 27 a + 28\right)\cdot 29^{163} + \left(12 a^{2} + 3 a + 3\right)\cdot 29^{164} + \left(19 a^{2} + 2 a + 25\right)\cdot 29^{165} + \left(16 a^{2} + 18 a + 2\right)\cdot 29^{166} + \left(5 a + 7\right)\cdot 29^{167} + \left(25 a^{2} + 16 a + 23\right)\cdot 29^{168} + \left(12 a^{2} + 18 a + 11\right)\cdot 29^{169} + \left(27 a^{2} + a + 6\right)\cdot 29^{170} + \left(6 a^{2} + 9 a + 13\right)\cdot 29^{171} + \left(21 a^{2} + 5 a + 5\right)\cdot 29^{172} + \left(2 a^{2} + 28 a + 1\right)\cdot 29^{173} + \left(11 a^{2} + 22 a\right)\cdot 29^{174} +O\left(29^{ 175 }\right)$
$r_{ 4 }$ $=$ $ 13 a^{2} + 7 a + 25 + \left(a^{2} + 2 a + 13\right)\cdot 29 + \left(27 a^{2} + 6 a + 23\right)\cdot 29^{2} + \left(23 a^{2} + 8 a + 16\right)\cdot 29^{3} + \left(22 a^{2} + 26 a + 5\right)\cdot 29^{4} + \left(21 a^{2} + 23 a + 12\right)\cdot 29^{5} + \left(11 a^{2} + 13 a + 6\right)\cdot 29^{6} + \left(21 a^{2} + 2 a + 5\right)\cdot 29^{7} + \left(20 a^{2} + 3 a + 10\right)\cdot 29^{8} + \left(22 a^{2} + a + 14\right)\cdot 29^{9} + \left(24 a^{2} + 8 a + 19\right)\cdot 29^{10} + \left(2 a^{2} + 21 a + 4\right)\cdot 29^{11} + \left(17 a^{2} + 27 a + 11\right)\cdot 29^{12} + \left(2 a^{2} + 12 a + 7\right)\cdot 29^{13} + \left(27 a^{2} + 22 a + 25\right)\cdot 29^{14} + \left(14 a^{2} + 15 a + 21\right)\cdot 29^{15} + \left(14 a^{2} + 10 a + 14\right)\cdot 29^{16} + \left(10 a^{2} + 11 a + 27\right)\cdot 29^{17} + \left(19 a^{2} + 6 a + 11\right)\cdot 29^{18} + \left(12 a^{2} + 4 a + 6\right)\cdot 29^{19} + \left(4 a^{2} + 8 a + 21\right)\cdot 29^{20} + \left(19 a^{2} + 18 a + 18\right)\cdot 29^{21} + \left(9 a^{2} + 25 a + 8\right)\cdot 29^{22} + \left(10 a^{2} + 3 a + 2\right)\cdot 29^{23} + \left(2 a^{2} + 20 a + 28\right)\cdot 29^{24} + \left(28 a^{2} + 22 a + 14\right)\cdot 29^{25} + \left(4 a^{2} + 17\right)\cdot 29^{26} + \left(21 a^{2} + 5 a + 14\right)\cdot 29^{27} + \left(27 a^{2} + 28 a + 25\right)\cdot 29^{28} + \left(15 a^{2} + 22 a + 13\right)\cdot 29^{29} + \left(22 a^{2} + 14 a + 26\right)\cdot 29^{30} + \left(21 a^{2} + 19 a + 11\right)\cdot 29^{31} + \left(19 a^{2} + 21 a + 26\right)\cdot 29^{32} + \left(16 a^{2} + 26 a + 21\right)\cdot 29^{33} + \left(7 a^{2} + 8 a + 7\right)\cdot 29^{34} + \left(19 a^{2} + 21 a + 16\right)\cdot 29^{35} + \left(28 a^{2} + 8 a\right)\cdot 29^{36} + 19\cdot 29^{37} + \left(12 a^{2} + 10 a + 19\right)\cdot 29^{38} + \left(19 a^{2} + 9 a + 2\right)\cdot 29^{39} + \left(19 a^{2} + 20 a + 21\right)\cdot 29^{40} + \left(28 a^{2} + 17 a + 16\right)\cdot 29^{41} + \left(25 a^{2} + 28 a + 3\right)\cdot 29^{42} + \left(10 a^{2} + 8 a + 9\right)\cdot 29^{43} + \left(7 a^{2} + 6 a + 10\right)\cdot 29^{44} + \left(18 a^{2} + 13 a + 22\right)\cdot 29^{45} + \left(13 a^{2} + 6 a + 20\right)\cdot 29^{46} + \left(14 a^{2} + 24 a + 10\right)\cdot 29^{47} + \left(17 a^{2} + 13 a + 14\right)\cdot 29^{48} + \left(24 a^{2} + 13 a + 23\right)\cdot 29^{49} + \left(3 a^{2} + 2 a + 26\right)\cdot 29^{50} + \left(14 a^{2} + 9 a + 26\right)\cdot 29^{51} + \left(26 a^{2} + 18 a + 4\right)\cdot 29^{52} + \left(11 a^{2} + 8 a + 25\right)\cdot 29^{53} + \left(15 a^{2} + 24 a + 25\right)\cdot 29^{54} + \left(6 a^{2} + 5 a + 2\right)\cdot 29^{55} + \left(10 a^{2} + 14 a + 19\right)\cdot 29^{56} + \left(25 a^{2} + 13 a + 7\right)\cdot 29^{57} + \left(22 a^{2} + 10 a + 13\right)\cdot 29^{58} + \left(6 a^{2} + 19 a + 25\right)\cdot 29^{59} + \left(18 a^{2} + 19 a + 10\right)\cdot 29^{60} + \left(23 a^{2} + 9 a + 25\right)\cdot 29^{61} + \left(9 a^{2} + 10 a + 23\right)\cdot 29^{62} + \left(7 a^{2} + 21 a + 2\right)\cdot 29^{63} + \left(5 a^{2} + 13 a + 17\right)\cdot 29^{64} + \left(25 a^{2} + 20 a + 2\right)\cdot 29^{65} + \left(19 a^{2} + 28 a\right)\cdot 29^{66} + \left(19 a^{2} + 20 a + 9\right)\cdot 29^{67} + \left(13 a^{2} + 21 a + 18\right)\cdot 29^{68} + \left(25 a^{2} + 14 a + 20\right)\cdot 29^{69} + \left(10 a^{2} + 7 a + 1\right)\cdot 29^{70} + \left(7 a^{2} + 19 a + 25\right)\cdot 29^{71} + \left(a^{2} + 17\right)\cdot 29^{72} + \left(23 a + 27\right)\cdot 29^{73} + \left(4 a^{2} + 12 a + 11\right)\cdot 29^{74} + \left(21 a^{2} + 18 a + 23\right)\cdot 29^{75} + \left(19 a^{2} + 19 a + 8\right)\cdot 29^{76} + \left(15 a^{2} + 23 a + 3\right)\cdot 29^{77} + \left(22 a^{2} + 23 a + 14\right)\cdot 29^{78} + \left(13 a^{2} + 5 a + 18\right)\cdot 29^{79} + \left(20 a^{2} + 28 a + 13\right)\cdot 29^{80} + \left(8 a^{2} + 9 a + 12\right)\cdot 29^{81} + \left(25 a^{2} + 2 a + 8\right)\cdot 29^{82} + \left(28 a^{2} + 7 a + 22\right)\cdot 29^{83} + \left(24 a^{2} + 15 a + 17\right)\cdot 29^{84} + \left(26 a^{2} + 27 a + 15\right)\cdot 29^{85} + \left(27 a^{2} + 9 a + 7\right)\cdot 29^{86} + \left(15 a^{2} + 26 a + 27\right)\cdot 29^{87} + \left(22 a^{2} + a + 15\right)\cdot 29^{88} + \left(19 a^{2} + 6 a + 22\right)\cdot 29^{89} + \left(25 a^{2} + 2 a + 22\right)\cdot 29^{90} + \left(3 a^{2} + 5 a + 13\right)\cdot 29^{91} + \left(17 a^{2} + 23 a + 16\right)\cdot 29^{92} + \left(20 a^{2} + a + 20\right)\cdot 29^{93} + \left(28 a^{2} + 7 a + 5\right)\cdot 29^{94} + \left(11 a^{2} + 11 a + 23\right)\cdot 29^{95} + \left(11 a^{2} + 15 a + 8\right)\cdot 29^{96} + \left(18 a^{2} + 17 a + 19\right)\cdot 29^{97} + \left(14 a^{2} + 24\right)\cdot 29^{98} + \left(15 a^{2} + 10 a + 15\right)\cdot 29^{99} + \left(24 a^{2} + 10 a + 11\right)\cdot 29^{100} + \left(23 a^{2} + 12 a + 15\right)\cdot 29^{101} + \left(11 a^{2} + 25 a + 14\right)\cdot 29^{102} + \left(17 a^{2} + 11 a + 26\right)\cdot 29^{103} + \left(23 a^{2} + 22 a + 10\right)\cdot 29^{104} + \left(8 a^{2} + 14 a + 1\right)\cdot 29^{105} + \left(15 a^{2} + 6 a + 24\right)\cdot 29^{106} + \left(4 a^{2} + 8 a + 12\right)\cdot 29^{107} + \left(15 a^{2} + 25 a + 11\right)\cdot 29^{108} + \left(13 a^{2} + 23 a + 24\right)\cdot 29^{109} + \left(14 a^{2} + a + 5\right)\cdot 29^{110} + \left(14 a^{2} + 19 a + 11\right)\cdot 29^{111} + \left(26 a^{2} + 5 a + 22\right)\cdot 29^{112} + \left(13 a^{2} + 28 a + 23\right)\cdot 29^{113} + \left(4 a^{2} + 15 a + 24\right)\cdot 29^{114} + \left(11 a^{2} + 25 a + 13\right)\cdot 29^{115} + \left(5 a^{2} + 24 a + 5\right)\cdot 29^{116} + \left(24 a^{2} + 19 a + 3\right)\cdot 29^{117} + \left(27 a + 1\right)\cdot 29^{118} + \left(13 a^{2} + 7 a + 3\right)\cdot 29^{119} + \left(a^{2} + 9 a + 12\right)\cdot 29^{120} + \left(15 a^{2} + 2 a + 8\right)\cdot 29^{121} + \left(5 a^{2} + 9 a + 15\right)\cdot 29^{122} + \left(20 a^{2} + 4 a + 28\right)\cdot 29^{123} + \left(18 a^{2} + 2 a + 1\right)\cdot 29^{124} + \left(7 a^{2} + 25 a + 12\right)\cdot 29^{125} + \left(17 a^{2} + 20 a + 17\right)\cdot 29^{126} + \left(25 a^{2} + 8 a + 3\right)\cdot 29^{127} + \left(11 a^{2} + 9 a + 8\right)\cdot 29^{128} + \left(3 a^{2} + 27 a + 21\right)\cdot 29^{129} + \left(3 a^{2} + 21 a + 24\right)\cdot 29^{130} + \left(16 a^{2} + 28 a + 10\right)\cdot 29^{131} + \left(11 a^{2} + 14 a + 11\right)\cdot 29^{132} + \left(20 a^{2} + 25 a + 6\right)\cdot 29^{133} + \left(14 a^{2} + 7 a + 24\right)\cdot 29^{134} + \left(28 a^{2} + 4 a + 8\right)\cdot 29^{135} + \left(7 a^{2} + 15 a + 10\right)\cdot 29^{136} + \left(25 a^{2} + 25 a + 27\right)\cdot 29^{137} + \left(28 a^{2} + 13 a + 8\right)\cdot 29^{138} + \left(13 a^{2} + 12 a + 22\right)\cdot 29^{139} + \left(7 a^{2} + 5 a + 16\right)\cdot 29^{140} + \left(a^{2} + 20 a + 14\right)\cdot 29^{141} + \left(18 a^{2} + 23 a + 19\right)\cdot 29^{142} + \left(9 a^{2} + 14 a + 24\right)\cdot 29^{143} + \left(8 a^{2} + 17 a + 2\right)\cdot 29^{144} + \left(26 a^{2} + 10 a + 5\right)\cdot 29^{145} + \left(26 a^{2} + 3 a + 21\right)\cdot 29^{146} + \left(8 a^{2} + 6 a\right)\cdot 29^{147} + \left(17 a^{2} + 17 a + 18\right)\cdot 29^{148} + \left(17 a^{2} + 4 a + 26\right)\cdot 29^{149} + \left(12 a^{2} + 11 a + 28\right)\cdot 29^{150} + \left(10 a^{2} + 6 a + 7\right)\cdot 29^{151} + \left(9 a^{2} + 2 a + 23\right)\cdot 29^{152} + \left(27 a^{2} + 12 a + 9\right)\cdot 29^{153} + \left(14 a^{2} + 22 a + 26\right)\cdot 29^{154} + \left(18 a^{2} + 21 a + 5\right)\cdot 29^{155} + \left(6 a^{2} + 15 a + 21\right)\cdot 29^{156} + \left(a^{2} + 2 a + 19\right)\cdot 29^{157} + \left(21 a^{2} + 5 a + 1\right)\cdot 29^{158} + \left(4 a^{2} + 23 a + 19\right)\cdot 29^{159} + \left(28 a^{2} + 26 a + 21\right)\cdot 29^{160} + \left(25 a^{2} + 18 a + 14\right)\cdot 29^{161} + \left(18 a^{2} + 18 a + 1\right)\cdot 29^{162} + \left(27 a^{2} + 6\right)\cdot 29^{163} + \left(6 a^{2} + 17 a + 26\right)\cdot 29^{164} + \left(24 a^{2} + 7 a + 15\right)\cdot 29^{165} + \left(21 a^{2} + 5 a + 25\right)\cdot 29^{166} + \left(23 a^{2} + 26 a + 4\right)\cdot 29^{167} + \left(13 a^{2} + 11 a + 19\right)\cdot 29^{168} + \left(14 a^{2} + 6 a + 27\right)\cdot 29^{169} + \left(28 a^{2} + 11 a + 2\right)\cdot 29^{170} + \left(19 a^{2} + 19 a\right)\cdot 29^{171} + \left(8 a^{2} + a + 14\right)\cdot 29^{172} + \left(12 a^{2} + 17 a + 20\right)\cdot 29^{173} + \left(15 a^{2} + 9 a + 7\right)\cdot 29^{174} +O\left(29^{ 175 }\right)$
$r_{ 5 }$ $=$ $ 3 a^{2} + 21 a + 10 + \left(20 a^{2} + 2 a\right)\cdot 29 + \left(11 a^{2} + 16 a + 5\right)\cdot 29^{2} + \left(3 a^{2} + 19 a + 2\right)\cdot 29^{3} + \left(27 a^{2} + 22 a + 19\right)\cdot 29^{4} + \left(27 a^{2} + 25 a + 14\right)\cdot 29^{5} + \left(7 a^{2} + 24 a + 26\right)\cdot 29^{6} + \left(19 a^{2} + 11 a + 27\right)\cdot 29^{7} + \left(17 a^{2} + 7 a + 16\right)\cdot 29^{8} + \left(7 a^{2} + 6 a + 16\right)\cdot 29^{9} + \left(2 a^{2} + 13 a + 19\right)\cdot 29^{10} + \left(28 a + 10\right)\cdot 29^{11} + \left(14 a^{2} + 23 a + 12\right)\cdot 29^{12} + \left(13 a^{2} + 26 a + 28\right)\cdot 29^{13} + \left(25 a^{2} + 3 a + 16\right)\cdot 29^{14} + \left(17 a + 8\right)\cdot 29^{15} + \left(10 a^{2} + 10 a + 2\right)\cdot 29^{16} + \left(23 a^{2} + 9 a + 7\right)\cdot 29^{17} + \left(2 a^{2} + 8 a + 16\right)\cdot 29^{18} + \left(17 a^{2} + 14 a + 3\right)\cdot 29^{19} + 12 a\cdot 29^{20} + \left(20 a^{2} + 6 a + 22\right)\cdot 29^{21} + \left(26 a^{2} + 18 a + 2\right)\cdot 29^{22} + \left(17 a^{2} + 21 a + 17\right)\cdot 29^{23} + \left(27 a + 26\right)\cdot 29^{24} + \left(20 a^{2} + 28 a + 15\right)\cdot 29^{25} + \left(27 a^{2} + 17\right)\cdot 29^{26} + \left(27 a^{2} + 8\right)\cdot 29^{27} + \left(21 a^{2} + 7 a + 26\right)\cdot 29^{28} + \left(13 a^{2} + 13 a + 10\right)\cdot 29^{29} + \left(8 a^{2} + 8 a + 2\right)\cdot 29^{30} + \left(8 a^{2} + 21 a + 22\right)\cdot 29^{31} + \left(7 a^{2} + 26\right)\cdot 29^{32} + \left(18 a^{2} + 16 a + 3\right)\cdot 29^{33} + \left(9 a^{2} + 22\right)\cdot 29^{34} + \left(13 a^{2} + a + 11\right)\cdot 29^{35} + \left(8 a^{2} + 16 a + 14\right)\cdot 29^{36} + \left(15 a^{2} + 25 a + 25\right)\cdot 29^{37} + \left(26 a^{2} + 14 a + 26\right)\cdot 29^{38} + \left(8 a^{2} + 22 a + 15\right)\cdot 29^{39} + \left(4 a^{2} + 28 a + 21\right)\cdot 29^{40} + \left(18 a^{2} + 14 a + 22\right)\cdot 29^{41} + \left(6 a^{2} + 5 a + 7\right)\cdot 29^{42} + \left(a^{2} + 5 a + 14\right)\cdot 29^{43} + \left(5 a^{2} + 7 a + 17\right)\cdot 29^{44} + \left(17 a^{2} + 25 a + 6\right)\cdot 29^{45} + \left(24 a^{2} + 18 a + 16\right)\cdot 29^{46} + \left(24 a^{2} + 22 a + 21\right)\cdot 29^{47} + \left(28 a^{2} + 4 a + 14\right)\cdot 29^{48} + \left(2 a^{2} + 4 a + 28\right)\cdot 29^{49} + \left(15 a^{2} + 4 a + 25\right)\cdot 29^{50} + \left(14 a^{2} + 13 a + 13\right)\cdot 29^{51} + \left(16 a^{2} + 17 a + 15\right)\cdot 29^{52} + \left(22 a^{2} + 11 a + 22\right)\cdot 29^{53} + \left(3 a^{2} + 16 a + 3\right)\cdot 29^{54} + \left(2 a^{2} + 2 a + 28\right)\cdot 29^{55} + \left(3 a^{2} + 10 a + 15\right)\cdot 29^{56} + \left(4 a^{2} + 27 a + 4\right)\cdot 29^{57} + \left(4 a^{2} + 14 a + 22\right)\cdot 29^{58} + \left(18 a^{2} + 25 a + 13\right)\cdot 29^{59} + \left(24 a + 24\right)\cdot 29^{60} + \left(24 a^{2} + 19 a\right)\cdot 29^{61} + \left(5 a^{2} + 20 a + 17\right)\cdot 29^{62} + \left(3 a^{2} + 7\right)\cdot 29^{63} + \left(27 a^{2} + 5 a + 23\right)\cdot 29^{64} + \left(12 a^{2} + 8 a + 6\right)\cdot 29^{65} + \left(15 a^{2} + 20 a + 23\right)\cdot 29^{66} + \left(a^{2} + 3 a + 17\right)\cdot 29^{67} + \left(22 a^{2} + 27 a + 6\right)\cdot 29^{68} + \left(21 a^{2} + 18\right)\cdot 29^{69} + \left(15 a^{2} + 15 a + 24\right)\cdot 29^{70} + \left(3 a + 6\right)\cdot 29^{71} + \left(4 a^{2} + 14 a + 21\right)\cdot 29^{72} + \left(16 a^{2} + 2 a + 20\right)\cdot 29^{73} + \left(5 a^{2} + 26 a + 4\right)\cdot 29^{74} + \left(4 a^{2} + 10 a + 21\right)\cdot 29^{75} + \left(4 a^{2} + 19 a + 16\right)\cdot 29^{76} + \left(17 a^{2} + 9 a + 7\right)\cdot 29^{77} + \left(3 a^{2} + 18 a + 23\right)\cdot 29^{78} + \left(21 a^{2} + 8 a + 22\right)\cdot 29^{79} + \left(14 a^{2} + 9 a + 3\right)\cdot 29^{80} + \left(9 a^{2} + 20 a + 21\right)\cdot 29^{81} + \left(18 a^{2} + 17 a + 23\right)\cdot 29^{82} + \left(23 a^{2} + 2 a\right)\cdot 29^{83} + \left(18 a^{2} + 9 a + 26\right)\cdot 29^{84} + \left(22 a^{2} + 24 a + 16\right)\cdot 29^{85} + \left(15 a^{2} + 27 a + 17\right)\cdot 29^{86} + \left(a^{2} + 26 a + 4\right)\cdot 29^{87} + \left(5 a^{2} + 13 a + 21\right)\cdot 29^{88} + \left(6 a^{2} + a + 27\right)\cdot 29^{89} + \left(17 a^{2} + 18 a + 17\right)\cdot 29^{90} + \left(18 a^{2} + 4 a + 16\right)\cdot 29^{91} + \left(2 a^{2} + 26 a\right)\cdot 29^{92} + \left(28 a^{2} + 24 a + 19\right)\cdot 29^{93} + \left(26 a^{2} + 12 a + 13\right)\cdot 29^{94} + \left(3 a^{2} + 18 a + 5\right)\cdot 29^{95} + \left(18 a^{2} + 18 a + 24\right)\cdot 29^{96} + \left(13 a^{2} + 7 a + 8\right)\cdot 29^{97} + \left(24 a^{2} + 24 a + 11\right)\cdot 29^{98} + \left(21 a^{2} + 7\right)\cdot 29^{99} + \left(5 a^{2} + 17 a + 1\right)\cdot 29^{100} + \left(5 a^{2} + 22 a\right)\cdot 29^{101} + \left(4 a^{2} + 26 a + 1\right)\cdot 29^{102} + \left(4 a^{2} + 7 a + 14\right)\cdot 29^{103} + \left(22 a^{2} + 3 a + 6\right)\cdot 29^{104} + \left(18 a^{2} + 25 a + 27\right)\cdot 29^{105} + \left(11 a^{2} + 14 a + 3\right)\cdot 29^{106} + \left(7 a^{2} + 26 a + 17\right)\cdot 29^{107} + \left(25 a^{2} + 21 a + 23\right)\cdot 29^{108} + \left(23 a^{2} + 16 a + 18\right)\cdot 29^{109} + \left(8 a^{2} + 14 a + 22\right)\cdot 29^{110} + \left(17 a^{2} + 10 a + 2\right)\cdot 29^{111} + \left(13 a^{2} + 22 a + 27\right)\cdot 29^{112} + \left(21 a^{2} + 7 a + 20\right)\cdot 29^{113} + \left(19 a^{2} + 10 a + 2\right)\cdot 29^{114} + \left(a^{2} + 16 a + 27\right)\cdot 29^{115} + \left(4 a^{2} + 21 a + 13\right)\cdot 29^{116} + \left(20 a^{2} + 22 a + 20\right)\cdot 29^{117} + \left(2 a^{2} + 5 a + 13\right)\cdot 29^{118} + \left(27 a^{2} + 18 a + 18\right)\cdot 29^{119} + \left(12 a^{2} + 16 a + 21\right)\cdot 29^{120} + \left(12 a^{2} + 24 a + 19\right)\cdot 29^{121} + \left(24 a^{2} + 28 a + 21\right)\cdot 29^{122} + \left(26 a^{2} + 20 a + 3\right)\cdot 29^{123} + \left(a^{2} + 27 a + 6\right)\cdot 29^{124} + \left(13 a^{2} + 20 a + 6\right)\cdot 29^{125} + \left(27 a^{2} + a + 10\right)\cdot 29^{126} + \left(7 a^{2} + 21 a + 20\right)\cdot 29^{127} + \left(4 a^{2} + 23 a + 23\right)\cdot 29^{128} + \left(7 a^{2} + 13 a + 16\right)\cdot 29^{129} + \left(8 a^{2} + 4 a + 8\right)\cdot 29^{130} + \left(3 a^{2} + 9 a + 3\right)\cdot 29^{131} + \left(18 a^{2} + 27 a + 22\right)\cdot 29^{132} + \left(9 a + 7\right)\cdot 29^{133} + \left(7 a^{2} + 28 a + 2\right)\cdot 29^{134} + \left(21 a^{2} + 2 a + 13\right)\cdot 29^{135} + \left(11 a^{2} + 23 a + 26\right)\cdot 29^{136} + \left(12 a^{2} + 14 a + 8\right)\cdot 29^{137} + \left(13 a^{2} + 9 a\right)\cdot 29^{138} + \left(10 a + 2\right)\cdot 29^{139} + \left(4 a^{2} + 8 a + 1\right)\cdot 29^{140} + \left(24 a^{2} + 28 a + 16\right)\cdot 29^{141} + \left(2 a^{2} + 18 a + 12\right)\cdot 29^{142} + \left(6 a^{2} + 9 a + 26\right)\cdot 29^{143} + \left(8 a^{2} + 18 a + 11\right)\cdot 29^{144} + \left(15 a^{2} + 24 a + 22\right)\cdot 29^{145} + \left(18 a^{2} + 20 a + 16\right)\cdot 29^{146} + \left(6 a^{2} + 26 a + 12\right)\cdot 29^{147} + \left(7 a^{2} + 18 a + 8\right)\cdot 29^{148} + \left(4 a^{2} + 12 a + 6\right)\cdot 29^{149} + \left(6 a^{2} + 28 a\right)\cdot 29^{150} + \left(2 a^{2} + 3 a + 26\right)\cdot 29^{151} + \left(18 a^{2} + 6 a + 15\right)\cdot 29^{152} + \left(17 a^{2} + 9 a + 10\right)\cdot 29^{153} + \left(16 a^{2} + 7\right)\cdot 29^{154} + \left(20 a^{2} + 20 a + 9\right)\cdot 29^{155} + \left(22 a^{2} + 14 a\right)\cdot 29^{156} + \left(21 a^{2} + 3 a + 12\right)\cdot 29^{157} + \left(13 a^{2} + 9 a + 9\right)\cdot 29^{158} + \left(20 a^{2} + 28 a + 21\right)\cdot 29^{159} + \left(13 a^{2} + 25 a + 11\right)\cdot 29^{160} + \left(15 a^{2} + a + 25\right)\cdot 29^{161} + \left(10 a^{2} + 5 a + 13\right)\cdot 29^{162} + \left(18 a^{2} + 10 a + 3\right)\cdot 29^{163} + \left(6 a^{2} + 10 a + 6\right)\cdot 29^{164} + \left(19 a^{2} + 20 a + 15\right)\cdot 29^{165} + \left(10 a^{2} + 16 a + 4\right)\cdot 29^{166} + \left(27 a^{2} + 8 a + 4\right)\cdot 29^{167} + \left(8 a^{2} + 27 a + 21\right)\cdot 29^{168} + \left(17 a^{2} + 26 a + 17\right)\cdot 29^{169} + \left(7 a^{2} + 15 a + 18\right)\cdot 29^{170} + \left(12 a^{2} + 5 a + 10\right)\cdot 29^{171} + \left(a^{2} + a + 27\right)\cdot 29^{172} + \left(6 a^{2} + 10 a + 24\right)\cdot 29^{173} + \left(3 a^{2} + 2 a + 8\right)\cdot 29^{174} +O\left(29^{ 175 }\right)$
$r_{ 6 }$ $=$ $ 17 a^{2} + 3 a + 11 + \left(3 a^{2} + 20 a + 26\right)\cdot 29 + \left(5 a^{2} + 11 a + 3\right)\cdot 29^{2} + \left(2 a^{2} + 26 a + 7\right)\cdot 29^{3} + \left(14 a^{2} + 16 a + 13\right)\cdot 29^{4} + \left(7 a^{2} + 26 a + 12\right)\cdot 29^{5} + \left(14 a^{2} + 18 a + 19\right)\cdot 29^{6} + \left(24 a^{2} + 20 a + 28\right)\cdot 29^{7} + \left(19 a^{2} + 13 a + 8\right)\cdot 29^{8} + \left(16 a^{2} + 2 a + 6\right)\cdot 29^{9} + \left(a^{2} + 15 a + 27\right)\cdot 29^{10} + \left(25 a^{2} + 2 a + 14\right)\cdot 29^{11} + \left(19 a^{2} + 15 a + 24\right)\cdot 29^{12} + \left(2 a^{2} + 26\right)\cdot 29^{13} + \left(17 a^{2} + 14 a + 11\right)\cdot 29^{14} + \left(25 a^{2} + 18 a + 26\right)\cdot 29^{15} + \left(10 a^{2} + 28 a + 9\right)\cdot 29^{16} + \left(3 a^{2} + 13 a + 8\right)\cdot 29^{17} + \left(12 a^{2} + 18 a + 2\right)\cdot 29^{18} + \left(25 a^{2} + 8 a + 4\right)\cdot 29^{19} + \left(11 a^{2} + 10 a + 2\right)\cdot 29^{20} + \left(7 a^{2} + 16 a + 3\right)\cdot 29^{21} + \left(16 a^{2} + 27\right)\cdot 29^{22} + \left(23 a^{2} + 9 a + 19\right)\cdot 29^{23} + \left(11 a^{2} + 16 a + 11\right)\cdot 29^{24} + \left(13 a^{2} + 18 a + 24\right)\cdot 29^{25} + \left(11 a^{2} + 7 a + 6\right)\cdot 29^{26} + \left(17 a^{2} + 12 a + 19\right)\cdot 29^{27} + \left(28 a^{2} + 4 a + 26\right)\cdot 29^{28} + \left(25 a^{2} + 25 a + 7\right)\cdot 29^{29} + \left(17 a^{2} + 15 a + 20\right)\cdot 29^{30} + \left(4 a^{2} + 20 a + 27\right)\cdot 29^{31} + \left(6 a^{2} + 13 a + 17\right)\cdot 29^{32} + \left(9 a^{2} + 2 a + 21\right)\cdot 29^{33} + \left(7 a^{2} + 21 a + 26\right)\cdot 29^{34} + \left(8 a^{2} + a + 20\right)\cdot 29^{35} + \left(27 a^{2} + 4 a + 27\right)\cdot 29^{36} + \left(8 a^{2} + 10 a + 19\right)\cdot 29^{37} + \left(11 a^{2} + 28 a + 18\right)\cdot 29^{38} + \left(15 a^{2} + 17 a + 16\right)\cdot 29^{39} + \left(15 a^{2} + 22 a + 15\right)\cdot 29^{40} + \left(3 a^{2} + 4 a + 2\right)\cdot 29^{41} + \left(13 a^{2} + 12 a + 25\right)\cdot 29^{42} + \left(22 a^{2} + 24 a + 14\right)\cdot 29^{43} + \left(12 a^{2} + 12 a + 17\right)\cdot 29^{44} + \left(8 a^{2} + 20 a + 28\right)\cdot 29^{45} + \left(22 a^{2} + 18 a + 12\right)\cdot 29^{46} + \left(25 a^{2} + 9 a + 6\right)\cdot 29^{47} + \left(28 a^{2} + 24 a + 10\right)\cdot 29^{48} + \left(27 a^{2} + 11 a + 18\right)\cdot 29^{49} + \left(23 a^{2} + a + 24\right)\cdot 29^{50} + \left(11 a^{2} + 25 a + 23\right)\cdot 29^{51} + \left(2 a^{2} + 22 a + 1\right)\cdot 29^{52} + \left(18 a^{2} + 28 a + 14\right)\cdot 29^{53} + \left(8 a^{2} + 15 a + 26\right)\cdot 29^{54} + \left(24 a^{2} + 25 a + 16\right)\cdot 29^{55} + \left(7 a^{2} + 26 a + 25\right)\cdot 29^{56} + \left(26 a^{2} + 21 a + 8\right)\cdot 29^{57} + \left(20 a^{2} + 9 a + 20\right)\cdot 29^{58} + \left(4 a^{2} + 24 a + 22\right)\cdot 29^{59} + \left(11 a^{2} + 10 a + 20\right)\cdot 29^{60} + \left(23 a^{2} + 13 a + 5\right)\cdot 29^{61} + \left(8 a^{2} + 18 a + 3\right)\cdot 29^{62} + \left(18 a^{2} + 14 a + 27\right)\cdot 29^{63} + \left(18 a^{2} + a + 5\right)\cdot 29^{64} + \left(7 a + 18\right)\cdot 29^{65} + \left(26 a^{2} + 10 a + 27\right)\cdot 29^{66} + \left(10 a^{2} + 25 a + 6\right)\cdot 29^{67} + \left(18 a^{2} + 26 a + 5\right)\cdot 29^{68} + \left(3 a^{2} + 27 a + 1\right)\cdot 29^{69} + \left(2 a^{2} + 5 a + 19\right)\cdot 29^{70} + \left(4 a^{2} + 26 a + 20\right)\cdot 29^{71} + \left(5 a^{2} + 13 a + 3\right)\cdot 29^{72} + \left(26 a^{2} + 13 a + 14\right)\cdot 29^{73} + \left(17 a^{2} + 12 a + 1\right)\cdot 29^{74} + \left(18 a^{2} + 11 a + 20\right)\cdot 29^{75} + \left(3 a^{2} + 10 a + 6\right)\cdot 29^{76} + \left(13 a^{2} + 2 a + 19\right)\cdot 29^{77} + \left(10 a^{2} + a + 7\right)\cdot 29^{78} + \left(4 a^{2} + 23 a + 25\right)\cdot 29^{79} + \left(14 a^{2} + 19 a + 14\right)\cdot 29^{80} + \left(6 a^{2} + 17 a + 9\right)\cdot 29^{81} + \left(15 a^{2} + 25 a + 14\right)\cdot 29^{82} + \left(25 a^{2} + 16 a + 27\right)\cdot 29^{83} + \left(23 a^{2} + 21 a + 25\right)\cdot 29^{84} + \left(20 a^{2} + 23 a + 26\right)\cdot 29^{85} + \left(19 a^{2} + 20 a + 15\right)\cdot 29^{86} + \left(10 a^{2} + 6 a + 10\right)\cdot 29^{87} + \left(13 a^{2} + 28 a + 13\right)\cdot 29^{88} + \left(16 a^{2} + 5 a + 8\right)\cdot 29^{89} + \left(14 a^{2} + 26 a + 27\right)\cdot 29^{90} + \left(5 a^{2} + 21 a + 15\right)\cdot 29^{91} + \left(8 a^{2} + 12 a + 4\right)\cdot 29^{92} + \left(25 a^{2} + 18 a + 17\right)\cdot 29^{93} + \left(15 a^{2} + 4 a + 17\right)\cdot 29^{94} + \left(12 a^{2} + 4\right)\cdot 29^{95} + \left(4 a^{2} + 27 a + 9\right)\cdot 29^{96} + \left(3 a^{2} + 14 a + 18\right)\cdot 29^{97} + \left(21 a^{2} + 3 a + 23\right)\cdot 29^{98} + \left(9 a^{2} + 11 a + 17\right)\cdot 29^{99} + \left(17 a^{2} + 13 a + 11\right)\cdot 29^{100} + \left(4 a^{2} + 11 a + 28\right)\cdot 29^{101} + \left(7 a^{2} + 19 a + 17\right)\cdot 29^{102} + \left(16 a^{2} + 3 a + 5\right)\cdot 29^{103} + \left(9 a^{2} + 25 a + 21\right)\cdot 29^{104} + \left(7 a^{2} + 6 a + 18\right)\cdot 29^{105} + \left(3 a^{2} + 18 a + 17\right)\cdot 29^{106} + \left(9 a + 16\right)\cdot 29^{107} + \left(15 a^{2} + 9 a + 1\right)\cdot 29^{108} + \left(17 a^{2} + 9 a + 20\right)\cdot 29^{109} + \left(22 a^{2} + 5 a + 16\right)\cdot 29^{110} + \left(23 a^{2} + 15 a + 23\right)\cdot 29^{111} + \left(7 a^{2} + 7 a + 16\right)\cdot 29^{112} + \left(15 a^{2} + 12 a + 25\right)\cdot 29^{113} + \left(13 a^{2} + 16 a + 7\right)\cdot 29^{114} + \left(2 a^{2} + 10 a + 2\right)\cdot 29^{115} + \left(3 a^{2} + 7 a + 12\right)\cdot 29^{116} + \left(27 a^{2} + 8 a + 26\right)\cdot 29^{117} + \left(6 a^{2} + 13 a + 18\right)\cdot 29^{118} + \left(19 a^{2} + a + 1\right)\cdot 29^{119} + \left(10 a^{2} + 11 a + 5\right)\cdot 29^{120} + \left(6 a^{2} + 11 a + 16\right)\cdot 29^{121} + \left(4 a^{2} + 9 a + 13\right)\cdot 29^{122} + \left(24 a^{2} + 7 a + 14\right)\cdot 29^{123} + \left(15 a^{2} + 14 a + 17\right)\cdot 29^{124} + \left(5 a^{2} + 27 a + 28\right)\cdot 29^{125} + \left(21 a^{2} + 12\right)\cdot 29^{126} + \left(a^{2} + 10 a + 10\right)\cdot 29^{127} + \left(6 a^{2} + 5 a\right)\cdot 29^{128} + \left(26 a^{2} + 16 a + 13\right)\cdot 29^{129} + \left(2 a^{2} + 15 a + 24\right)\cdot 29^{130} + \left(12 a^{2} + 2 a + 24\right)\cdot 29^{131} + \left(7 a + 15\right)\cdot 29^{132} + \left(15 a^{2} + 10 a + 18\right)\cdot 29^{133} + \left(4 a^{2} + 28 a + 10\right)\cdot 29^{134} + \left(28 a^{2} + 26 a + 8\right)\cdot 29^{135} + \left(a^{2} + 2\right)\cdot 29^{136} + \left(9 a^{2} + 13 a + 25\right)\cdot 29^{137} + \left(24 a^{2} + 19 a + 2\right)\cdot 29^{138} + \left(7 a^{2} + a + 14\right)\cdot 29^{139} + \left(19 a^{2} + 13 a + 3\right)\cdot 29^{140} + \left(21 a^{2} + 16 a + 3\right)\cdot 29^{141} + \left(21 a^{2} + 20 a + 5\right)\cdot 29^{142} + \left(26 a + 3\right)\cdot 29^{143} + \left(5 a^{2} + 10 a + 8\right)\cdot 29^{144} + \left(16 a^{2} + 3 a + 1\right)\cdot 29^{145} + \left(22 a^{2} + 26 a + 25\right)\cdot 29^{146} + \left(24 a^{2} + a + 21\right)\cdot 29^{147} + \left(3 a^{2} + 11 a + 9\right)\cdot 29^{148} + \left(18 a^{2} + 22 a + 27\right)\cdot 29^{149} + \left(5 a^{2} + 4 a + 9\right)\cdot 29^{150} + \left(22 a^{2} + a + 4\right)\cdot 29^{151} + \left(17 a^{2} + 26 a + 15\right)\cdot 29^{152} + \left(14 a^{2} + 21 a + 2\right)\cdot 29^{153} + \left(6 a^{2} + 9 a + 15\right)\cdot 29^{154} + \left(12 a^{2} + 15 a + 26\right)\cdot 29^{155} + \left(8 a^{2} + 4 a + 13\right)\cdot 29^{156} + \left(22 a^{2} + 22 a + 28\right)\cdot 29^{157} + \left(3 a^{2} + 15 a + 26\right)\cdot 29^{158} + \left(20 a^{2} + 28 a\right)\cdot 29^{159} + \left(13 a^{2} + 3 a + 12\right)\cdot 29^{160} + \left(7 a^{2} + 22 a + 9\right)\cdot 29^{161} + \left(23 a^{2} + 16 a + 7\right)\cdot 29^{162} + \left(24 a^{2} + 11 a + 2\right)\cdot 29^{163} + \left(5 a^{2} + 28 a + 15\right)\cdot 29^{164} + \left(17 a^{2} + 18 a + 6\right)\cdot 29^{165} + \left(a^{2} + 26 a + 8\right)\cdot 29^{166} + \left(20 a^{2} + 19 a + 19\right)\cdot 29^{167} + \left(18 a^{2} + 11 a + 25\right)\cdot 29^{168} + \left(27 a^{2} + 25\right)\cdot 29^{169} + \left(27 a^{2} + 7 a + 11\right)\cdot 29^{170} + \left(26 a^{2} + 8 a + 9\right)\cdot 29^{171} + \left(19 a^{2} + 15 a + 19\right)\cdot 29^{172} + \left(23 a^{2} + 2 a + 6\right)\cdot 29^{173} + \left(20 a^{2} + 2 a + 5\right)\cdot 29^{174} +O\left(29^{ 175 }\right)$
$r_{ 7 }$ $=$ $ 22 a^{2} + 17 a + 16 + \left(10 a^{2} + 24 a + 26\right)\cdot 29 + \left(3 a^{2} + 2 a + 22\right)\cdot 29^{2} + \left(2 a^{2} + 23 a + 19\right)\cdot 29^{3} + \left(6 a^{2} + 16 a\right)\cdot 29^{4} + \left(25 a^{2} + 8 a + 11\right)\cdot 29^{5} + \left(a^{2} + 6 a + 18\right)\cdot 29^{6} + \left(4 a^{2} + 11 a + 7\right)\cdot 29^{7} + \left(24 a^{2} + 6\right)\cdot 29^{8} + \left(19 a^{2} + 12 a + 23\right)\cdot 29^{9} + \left(11 a^{2} + 4 a + 12\right)\cdot 29^{10} + \left(27 a^{2} + 8 a + 8\right)\cdot 29^{11} + \left(22 a^{2} + 11 a + 24\right)\cdot 29^{12} + \left(9 a^{2} + 16 a + 13\right)\cdot 29^{13} + \left(20 a^{2} + 23 a\right)\cdot 29^{14} + \left(11 a^{2} + 2 a + 23\right)\cdot 29^{15} + \left(26 a^{2} + 14 a + 4\right)\cdot 29^{16} + \left(10 a^{2} + 21 a\right)\cdot 29^{17} + \left(12 a^{2} + 14 a\right)\cdot 29^{18} + \left(20 a^{2} + 15 a + 8\right)\cdot 29^{19} + \left(25 a^{2} + 23 a + 14\right)\cdot 29^{20} + \left(21 a^{2} + 12 a + 24\right)\cdot 29^{21} + 10 a^{2}29^{22} + \left(28 a + 13\right)\cdot 29^{23} + \left(20 a^{2} + a + 23\right)\cdot 29^{24} + \left(13 a + 28\right)\cdot 29^{25} + \left(26 a^{2} + 27 a + 24\right)\cdot 29^{26} + \left(12 a^{2} + 14 a + 7\right)\cdot 29^{27} + \left(15 a^{2} + 14 a + 27\right)\cdot 29^{28} + \left(26 a^{2} + 4 a + 27\right)\cdot 29^{29} + \left(4 a^{2} + 19 a + 16\right)\cdot 29^{30} + \left(8 a^{2} + 27 a + 2\right)\cdot 29^{31} + \left(13 a^{2} + 23 a + 25\right)\cdot 29^{32} + \left(9 a^{2} + 21 a + 1\right)\cdot 29^{33} + \left(22 a^{2} + 5 a + 10\right)\cdot 29^{34} + \left(20 a^{2} + 13 a + 2\right)\cdot 29^{35} + \left(25 a^{2} + 26 a + 18\right)\cdot 29^{36} + \left(2 a^{2} + 18 a + 18\right)\cdot 29^{37} + \left(2 a^{2} + 24 a + 13\right)\cdot 29^{38} + \left(13 a^{2} + 27 a + 21\right)\cdot 29^{39} + \left(13 a^{2} + 22 a + 4\right)\cdot 29^{40} + \left(21 a^{2} + 26 a + 27\right)\cdot 29^{41} + \left(26 a^{2} + 5 a + 24\right)\cdot 29^{42} + \left(26 a^{2} + 27 a + 9\right)\cdot 29^{43} + \left(14 a^{2} + a + 11\right)\cdot 29^{44} + \left(a^{2} + 22 a + 24\right)\cdot 29^{45} + \left(26 a^{2} + 4 a + 27\right)\cdot 29^{46} + \left(7 a^{2} + 14 a + 27\right)\cdot 29^{47} + \left(22 a^{2} + 25 a + 5\right)\cdot 29^{48} + \left(22 a^{2} + 5 a + 16\right)\cdot 29^{49} + \left(3 a^{2} + a + 20\right)\cdot 29^{50} + \left(4 a^{2} + 26 a + 9\right)\cdot 29^{51} + \left(21 a^{2} + 23 a + 2\right)\cdot 29^{52} + \left(7 a^{2} + 18 a + 22\right)\cdot 29^{53} + \left(7 a^{2} + 7 a + 27\right)\cdot 29^{54} + \left(20 a^{2} + 23 a + 3\right)\cdot 29^{55} + \left(12 a^{2} + 27 a + 19\right)\cdot 29^{56} + \left(27 a^{2} + 24 a + 6\right)\cdot 29^{57} + \left(16 a^{2} + 2 a + 10\right)\cdot 29^{58} + \left(3 a^{2} + 27 a + 23\right)\cdot 29^{59} + \left(16 a + 23\right)\cdot 29^{60} + \left(10 a^{2} + 25 a + 20\right)\cdot 29^{61} + \left(27 a^{2} + 26 a + 16\right)\cdot 29^{62} + \left(26 a^{2} + 21 a\right)\cdot 29^{63} + \left(19 a^{2} + 26 a + 4\right)\cdot 29^{64} + \left(19 a^{2} + 27 a + 6\right)\cdot 29^{65} + \left(13 a^{2} + 27 a + 11\right)\cdot 29^{66} + \left(11 a^{2} + 5 a + 21\right)\cdot 29^{67} + \left(14 a^{2} + 20 a + 15\right)\cdot 29^{68} + \left(4 a^{2} + 9 a + 14\right)\cdot 29^{69} + \left(24 a^{2} + 12 a + 16\right)\cdot 29^{70} + \left(19 a^{2} + 17 a + 3\right)\cdot 29^{71} + \left(14 a^{2} + a + 16\right)\cdot 29^{72} + \left(15 a^{2} + 5 a\right)\cdot 29^{73} + \left(25 a^{2} + 4 a + 12\right)\cdot 29^{74} + \left(10 a^{2} + 10 a + 20\right)\cdot 29^{75} + \left(27 a^{2} + 6 a + 18\right)\cdot 29^{76} + \left(12 a^{2} + 26 a + 11\right)\cdot 29^{77} + \left(22 a^{2} + 21 a + 19\right)\cdot 29^{78} + \left(28 a^{2} + 25 a + 13\right)\cdot 29^{79} + \left(21 a^{2} + 12 a + 13\right)\cdot 29^{80} + \left(16 a^{2} + 4 a + 11\right)\cdot 29^{81} + \left(15 a^{2} + 25 a + 10\right)\cdot 29^{82} + \left(15 a^{2} + 13 a + 9\right)\cdot 29^{83} + \left(22 a^{2} + 13 a + 21\right)\cdot 29^{84} + \left(28 a^{2} + 7 a + 5\right)\cdot 29^{85} + \left(7 a^{2} + 26 a + 7\right)\cdot 29^{86} + \left(6 a^{2} + 28 a + 1\right)\cdot 29^{87} + \left(10 a^{2} + 9 a + 28\right)\cdot 29^{88} + \left(4 a^{2} + 7 a + 5\right)\cdot 29^{89} + \left(16 a^{2} + 24 a + 26\right)\cdot 29^{90} + \left(10 a^{2} + 8 a + 5\right)\cdot 29^{91} + \left(8 a^{2} + 11 a + 8\right)\cdot 29^{92} + \left(23 a^{2} + 4 a + 22\right)\cdot 29^{93} + \left(26 a^{2} + 9 a + 3\right)\cdot 29^{94} + \left(7 a^{2} + 4 a + 1\right)\cdot 29^{95} + \left(8 a^{2} + 24 a + 11\right)\cdot 29^{96} + \left(2 a^{2} + 20 a + 3\right)\cdot 29^{97} + \left(2 a^{2} + 6 a + 20\right)\cdot 29^{98} + \left(20 a^{2} + 26 a + 14\right)\cdot 29^{99} + \left(25 a^{2} + 12 a + 8\right)\cdot 29^{100} + \left(6 a^{2} + 16 a + 2\right)\cdot 29^{101} + \left(26 a^{2} + 3 a + 11\right)\cdot 29^{102} + \left(27 a^{2} + 24 a + 26\right)\cdot 29^{103} + \left(6 a^{2} + 6 a + 24\right)\cdot 29^{104} + \left(20 a^{2} + 24 a + 9\right)\cdot 29^{105} + \left(8 a^{2} + 22 a + 19\right)\cdot 29^{106} + \left(23 a^{2} + 6 a + 28\right)\cdot 29^{107} + \left(14 a^{2} + 20 a + 28\right)\cdot 29^{108} + \left(22 a^{2} + 28 a + 16\right)\cdot 29^{109} + \left(23 a^{2} + 8 a + 13\right)\cdot 29^{110} + \left(13 a^{2} + 9 a + 17\right)\cdot 29^{111} + \left(4 a^{2} + 12 a + 24\right)\cdot 29^{112} + \left(24 a^{2} + 24 a + 14\right)\cdot 29^{113} + \left(22 a^{2} + a + 16\right)\cdot 29^{114} + \left(19 a^{2} + 8 a + 12\right)\cdot 29^{115} + \left(8 a^{2} + 14 a + 10\right)\cdot 29^{116} + \left(28 a^{2} + 28 a + 2\right)\cdot 29^{117} + \left(27 a^{2} + 17 a + 28\right)\cdot 29^{118} + \left(13 a^{2} + 13 a\right)\cdot 29^{119} + \left(22 a^{2} + a + 15\right)\cdot 29^{120} + \left(16 a^{2} + 12 a + 25\right)\cdot 29^{121} + \left(9 a^{2} + 24 a + 1\right)\cdot 29^{122} + \left(15 a + 7\right)\cdot 29^{123} + \left(26 a^{2} + a + 9\right)\cdot 29^{124} + \left(18 a^{2} + 8 a + 4\right)\cdot 29^{125} + \left(2 a^{2} + 13 a + 6\right)\cdot 29^{126} + \left(9 a^{2} + 12 a + 12\right)\cdot 29^{127} + \left(18 a^{2} + 12 a + 13\right)\cdot 29^{128} + \left(3 a^{2} + 14 a + 2\right)\cdot 29^{129} + \left(11 a^{2} + 18 a + 22\right)\cdot 29^{130} + \left(14 a^{2} + 11 a + 27\right)\cdot 29^{131} + \left(20 a^{2} + 13 a + 5\right)\cdot 29^{132} + \left(6 a^{2} + 7 a + 6\right)\cdot 29^{133} + \left(8 a^{2} + 10 a + 23\right)\cdot 29^{134} + 23\cdot 29^{135} + \left(a^{2} + 11 a + 21\right)\cdot 29^{136} + \left(14 a^{2} + 9 a + 20\right)\cdot 29^{137} + \left(24 a^{2} + 24 a + 24\right)\cdot 29^{138} + \left(8 a^{2} + a + 22\right)\cdot 29^{139} + \left(22 a^{2} + 15\right)\cdot 29^{140} + \left(5 a^{2} + a + 20\right)\cdot 29^{141} + \left(16 a^{2} + 17 a + 20\right)\cdot 29^{142} + \left(14 a^{2} + 18 a + 8\right)\cdot 29^{143} + \left(5 a^{2} + 12 a + 8\right)\cdot 29^{144} + \left(23 a^{2} + 11 a + 23\right)\cdot 29^{145} + \left(17 a^{2} + 19 a + 15\right)\cdot 29^{146} + \left(3 a^{2} + 8 a + 8\right)\cdot 29^{147} + \left(12 a^{2} + 16 a + 5\right)\cdot 29^{148} + \left(6 a^{2} + 24 a + 9\right)\cdot 29^{149} + \left(24 a^{2} + 19 a + 24\right)\cdot 29^{150} + \left(11 a^{2} + 9 a + 9\right)\cdot 29^{151} + \left(16 a + 21\right)\cdot 29^{152} + \left(26 a^{2} + 9 a + 21\right)\cdot 29^{153} + \left(12 a^{2} + 3 a + 21\right)\cdot 29^{154} + \left(25 a^{2} + 11 a + 15\right)\cdot 29^{155} + \left(16 a^{2} + 17 a + 21\right)\cdot 29^{156} + \left(24 a^{2} + 21 a + 15\right)\cdot 29^{157} + \left(8 a^{2} + 4 a + 12\right)\cdot 29^{158} + \left(13 a^{2} + 13 a + 21\right)\cdot 29^{159} + \left(20 a^{2} + 8 a + 20\right)\cdot 29^{160} + \left(2 a^{2} + 19 a + 27\right)\cdot 29^{161} + \left(22 a^{2} + 27 a + 9\right)\cdot 29^{162} + \left(2 a^{2} + 19 a + 21\right)\cdot 29^{163} + \left(10 a^{2} + 14 a + 10\right)\cdot 29^{164} + \left(19 a^{2} + 6 a + 15\right)\cdot 29^{165} + \left(a^{2} + 23 a + 21\right)\cdot 29^{166} + \left(a^{2} + 14 a + 7\right)\cdot 29^{167} + \left(24 a^{2} + 14 a + 12\right)\cdot 29^{168} + \left(27 a^{2} + 12 a + 12\right)\cdot 29^{169} + \left(22 a^{2} + 11 a\right)\cdot 29^{170} + \left(9 a^{2} + 14 a + 17\right)\cdot 29^{171} + \left(6 a^{2} + 22 a + 14\right)\cdot 29^{172} + \left(20 a^{2} + 19 a + 24\right)\cdot 29^{173} + \left(14 a^{2} + 3 a + 4\right)\cdot 29^{174} +O\left(29^{ 175 }\right)$
$r_{ 8 }$ $=$ $ 17 + 5\cdot 29^{2} + 17\cdot 29^{3} + 14\cdot 29^{4} + 11\cdot 29^{5} + 8\cdot 29^{6} + 22\cdot 29^{7} + 25\cdot 29^{8} + 23\cdot 29^{9} + 23\cdot 29^{10} + 3\cdot 29^{11} + 18\cdot 29^{12} + 2\cdot 29^{13} + 13\cdot 29^{14} + 27\cdot 29^{15} + 27\cdot 29^{16} + 24\cdot 29^{17} + 25\cdot 29^{18} + 7\cdot 29^{19} + 7\cdot 29^{20} + 2\cdot 29^{21} + 6\cdot 29^{22} + 7\cdot 29^{23} + 6\cdot 29^{24} + 5\cdot 29^{25} + 2\cdot 29^{26} + 25\cdot 29^{27} + 17\cdot 29^{28} + 7\cdot 29^{30} + 29^{31} + 21\cdot 29^{33} + 24\cdot 29^{34} + 19\cdot 29^{35} + 26\cdot 29^{36} + 11\cdot 29^{37} + 17\cdot 29^{38} + 23\cdot 29^{39} + 9\cdot 29^{40} + 10\cdot 29^{41} + 19\cdot 29^{42} + 7\cdot 29^{43} + 6\cdot 29^{45} + 21\cdot 29^{47} + 14\cdot 29^{48} + 8\cdot 29^{49} + 6\cdot 29^{50} + 19\cdot 29^{51} + 7\cdot 29^{52} + 24\cdot 29^{53} + 26\cdot 29^{54} + 24\cdot 29^{55} + 2\cdot 29^{57} + 4\cdot 29^{58} + 13\cdot 29^{59} + 9\cdot 29^{60} + 2\cdot 29^{61} + 3\cdot 29^{62} + 7\cdot 29^{63} + 2\cdot 29^{64} + 4\cdot 29^{65} + 20\cdot 29^{66} + 17\cdot 29^{67} + 10\cdot 29^{68} + 8\cdot 29^{69} + 11\cdot 29^{70} + 4\cdot 29^{71} + 20\cdot 29^{72} + 9\cdot 29^{73} + 25\cdot 29^{74} + 6\cdot 29^{75} + 3\cdot 29^{76} + 12\cdot 29^{77} + 7\cdot 29^{78} + 2\cdot 29^{79} + 19\cdot 29^{80} + 23\cdot 29^{81} + 28\cdot 29^{82} + 23\cdot 29^{83} + 27\cdot 29^{84} + 22\cdot 29^{85} + 29^{86} + 17\cdot 29^{87} + 6\cdot 29^{88} + 3\cdot 29^{89} + 28\cdot 29^{90} + 7\cdot 29^{91} + 28\cdot 29^{92} + 3\cdot 29^{93} + 29^{94} + 23\cdot 29^{95} + 25\cdot 29^{96} + 23\cdot 29^{97} + 28\cdot 29^{98} + 2\cdot 29^{99} + 18\cdot 29^{100} + 5\cdot 29^{101} + 22\cdot 29^{102} + 14\cdot 29^{103} + 7\cdot 29^{104} + 26\cdot 29^{105} + 19\cdot 29^{106} + 26\cdot 29^{107} + 17\cdot 29^{108} + 5\cdot 29^{109} + 9\cdot 29^{110} + 15\cdot 29^{111} + 12\cdot 29^{112} + 20\cdot 29^{113} + 20\cdot 29^{114} + 4\cdot 29^{115} + 5\cdot 29^{116} + 12\cdot 29^{117} + 27\cdot 29^{118} + 3\cdot 29^{119} + 18\cdot 29^{121} + 14\cdot 29^{122} + 16\cdot 29^{123} + 23\cdot 29^{124} + 15\cdot 29^{125} + 26\cdot 29^{126} + 22\cdot 29^{127} + 10\cdot 29^{128} + 3\cdot 29^{129} + 13\cdot 29^{130} + 14\cdot 29^{131} + 8\cdot 29^{132} + 15\cdot 29^{133} + 11\cdot 29^{134} + 20\cdot 29^{135} + 9\cdot 29^{136} + 29^{137} + 9\cdot 29^{138} + 29^{139} + 26\cdot 29^{140} + 12\cdot 29^{141} + 10\cdot 29^{142} + 10\cdot 29^{143} + 8\cdot 29^{144} + 13\cdot 29^{145} + 28\cdot 29^{146} + 16\cdot 29^{147} + 18\cdot 29^{148} + 18\cdot 29^{149} + 24\cdot 29^{150} + 21\cdot 29^{151} + 25\cdot 29^{152} + 5\cdot 29^{153} + 10\cdot 29^{154} + 14\cdot 29^{155} + 8\cdot 29^{156} + 3\cdot 29^{157} + 25\cdot 29^{158} + 26\cdot 29^{159} + 3\cdot 29^{160} + 29^{161} + 20\cdot 29^{162} + 21\cdot 29^{163} + 13\cdot 29^{164} + 29^{165} + 26\cdot 29^{166} + 5\cdot 29^{167} + 23\cdot 29^{169} + 12\cdot 29^{170} + 2\cdot 29^{171} + 15\cdot 29^{172} + 7\cdot 29^{173} + 22\cdot 29^{174} +O\left(29^{ 175 }\right)$
$r_{ 9 }$ $=$ $ 28 a^{2} + 19 a + 16 + \left(23 a^{2} + 6 a + 24\right)\cdot 29 + \left(25 a^{2} + 11 a + 21\right)\cdot 29^{2} + \left(2 a^{2} + 23 a + 17\right)\cdot 29^{3} + \left(21 a^{2} + 14 a + 22\right)\cdot 29^{4} + \left(28 a^{2} + 7 a + 11\right)\cdot 29^{5} + \left(2 a^{2} + 25 a + 4\right)\cdot 29^{6} + \left(12 a^{2} + 5 a + 12\right)\cdot 29^{7} + \left(17 a^{2} + 12 a + 15\right)\cdot 29^{8} + \left(18 a^{2} + 25 a + 18\right)\cdot 29^{9} + \left(2 a^{2} + 5 a + 28\right)\cdot 29^{10} + \left(a^{2} + 5 a + 11\right)\cdot 29^{11} + \left(21 a^{2} + 15 a + 16\right)\cdot 29^{12} + \left(23 a^{2} + 15 a + 6\right)\cdot 29^{13} + \left(13 a^{2} + 21 a + 17\right)\cdot 29^{14} + \left(17 a^{2} + 23 a + 15\right)\cdot 29^{15} + \left(3 a^{2} + 18 a + 19\right)\cdot 29^{16} + \left(15 a^{2} + 3 a + 4\right)\cdot 29^{17} + \left(26 a^{2} + 4 a + 2\right)\cdot 29^{18} + \left(19 a^{2} + 16 a + 16\right)\cdot 29^{19} + \left(12 a^{2} + 10 a + 22\right)\cdot 29^{20} + \left(2 a^{2} + 23 a + 15\right)\cdot 29^{21} + \left(3 a^{2} + 2 a + 9\right)\cdot 29^{22} + \left(24 a^{2} + 16 a + 1\right)\cdot 29^{23} + \left(14 a^{2} + 21 a + 6\right)\cdot 29^{24} + \left(16 a^{2} + 16 a + 9\right)\cdot 29^{25} + \left(12 a^{2} + 20 a + 8\right)\cdot 29^{26} + \left(19 a^{2} + 11 a + 12\right)\cdot 29^{27} + \left(a^{2} + 25 a\right)\cdot 29^{28} + \left(16 a^{2} + 9 a + 14\right)\cdot 29^{29} + \left(17 a^{2} + 27 a\right)\cdot 29^{30} + \left(2 a^{2} + 17 a + 25\right)\cdot 29^{31} + \left(3 a^{2} + 22 a + 13\right)\cdot 29^{32} + \left(3 a^{2} + 28 a + 13\right)\cdot 29^{33} + \left(14 a^{2} + 27 a + 16\right)\cdot 29^{34} + \left(a^{2} + 5 a + 21\right)\cdot 29^{35} + \left(2 a^{2} + 16 a + 3\right)\cdot 29^{36} + \left(19 a^{2} + 18 a + 14\right)\cdot 29^{37} + \left(5 a^{2} + 19 a + 1\right)\cdot 29^{38} + \left(23 a^{2} + a + 27\right)\cdot 29^{39} + \left(22 a^{2} + 15 a + 5\right)\cdot 29^{40} + \left(25 a^{2} + 6 a + 3\right)\cdot 29^{41} + \left(18 a^{2} + 17 a + 23\right)\cdot 29^{42} + \left(24 a^{2} + 24 a + 17\right)\cdot 29^{43} + \left(8 a^{2} + 9 a + 2\right)\cdot 29^{44} + \left(2 a^{2} + 24 a + 1\right)\cdot 29^{45} + \left(22 a^{2} + 3 a + 3\right)\cdot 29^{46} + \left(17 a^{2} + 24 a + 15\right)\cdot 29^{47} + \left(11 a^{2} + 19 a + 6\right)\cdot 29^{48} + \left(5 a^{2} + 3 a + 17\right)\cdot 29^{49} + \left(a^{2} + 25 a + 13\right)\cdot 29^{50} + \left(3 a^{2} + 23 a + 2\right)\cdot 29^{51} + \left(16 a + 18\right)\cdot 29^{52} + \left(28 a^{2} + 20 a + 17\right)\cdot 29^{53} + \left(4 a^{2} + 17 a + 21\right)\cdot 29^{54} + \left(27 a^{2} + 26 a + 20\right)\cdot 29^{55} + \left(10 a^{2} + 16 a\right)\cdot 29^{56} + \left(6 a^{2} + 22 a + 21\right)\cdot 29^{57} + \left(14 a^{2} + 8 a + 1\right)\cdot 29^{58} + \left(17 a^{2} + 14 a + 1\right)\cdot 29^{59} + \left(28 a^{2} + 27 a + 15\right)\cdot 29^{60} + \left(10 a^{2} + 5 a + 8\right)\cdot 29^{61} + \left(10 a^{2} + 5\right)\cdot 29^{62} + \left(3 a^{2} + 22 a + 7\right)\cdot 29^{63} + \left(5 a^{2} + 13 a + 7\right)\cdot 29^{64} + \left(3 a^{2} + a + 2\right)\cdot 29^{65} + \left(12 a^{2} + 19 a + 9\right)\cdot 29^{66} + \left(27 a^{2} + 11 a + 19\right)\cdot 29^{67} + \left(25 a^{2} + 9 a + 5\right)\cdot 29^{68} + \left(28 a^{2} + 15 a + 25\right)\cdot 29^{69} + \left(15 a^{2} + 15 a + 27\right)\cdot 29^{70} + \left(17 a^{2} + 12 a + 28\right)\cdot 29^{71} + \left(22 a^{2} + 14 a + 26\right)\cdot 29^{72} + \left(2 a^{2} + 21 a + 11\right)\cdot 29^{73} + \left(7 a^{2} + 3 a + 6\right)\cdot 29^{74} + \left(18 a^{2} + 28 a\right)\cdot 29^{75} + \left(5 a^{2} + 27 a + 19\right)\cdot 29^{76} + \left(2 a + 11\right)\cdot 29^{77} + \left(25 a^{2} + 4 a + 17\right)\cdot 29^{78} + \left(10 a^{2} + 14\right)\cdot 29^{79} + \left(23 a^{2} + 10 a + 17\right)\cdot 29^{80} + \left(13 a^{2} + a + 9\right)\cdot 29^{81} + \left(17 a^{2} + a + 17\right)\cdot 29^{82} + \left(3 a^{2} + 5 a + 17\right)\cdot 29^{83} + \left(9 a^{2} + 21 a + 25\right)\cdot 29^{84} + \left(10 a^{2} + 6 a + 12\right)\cdot 29^{85} + \left(10 a^{2} + 27 a + 3\right)\cdot 29^{86} + \left(2 a^{2} + 24 a + 9\right)\cdot 29^{87} + \left(22 a^{2} + 27 a + 15\right)\cdot 29^{88} + \left(21 a^{2} + 16 a + 15\right)\cdot 29^{89} + \left(17 a^{2} + 2\right)\cdot 29^{90} + \left(19 a^{2} + 2 a + 25\right)\cdot 29^{91} + \left(3 a^{2} + 22 a + 17\right)\cdot 29^{92} + \left(12 a^{2} + 8 a + 28\right)\cdot 29^{93} + \left(13 a^{2} + 17 a + 23\right)\cdot 29^{94} + \left(4 a^{2} + 17 a + 22\right)\cdot 29^{95} + \left(13 a^{2} + 15 a + 20\right)\cdot 29^{96} + \left(7 a^{2} + 25 a + 4\right)\cdot 29^{97} + \left(22 a^{2} + 24 a + 25\right)\cdot 29^{98} + \left(3 a^{2} + 7 a + 9\right)\cdot 29^{99} + \left(16 a^{2} + 5 a\right)\cdot 29^{100} + \left(5 a + 23\right)\cdot 29^{101} + \left(10 a^{2} + 13 a + 21\right)\cdot 29^{102} + \left(24 a^{2} + 13 a + 6\right)\cdot 29^{103} + \left(24 a^{2} + 10 a + 22\right)\cdot 29^{104} + \left(12 a^{2} + 7 a + 6\right)\cdot 29^{105} + \left(10 a^{2} + 4 a + 27\right)\cdot 29^{106} + \left(24 a^{2} + 11 a + 19\right)\cdot 29^{107} + \left(27 a^{2} + 23 a + 18\right)\cdot 29^{108} + \left(26 a^{2} + 24 a + 3\right)\cdot 29^{109} + \left(20 a^{2} + 21 a + 24\right)\cdot 29^{110} + \left(19 a^{2} + 23 a + 27\right)\cdot 29^{111} + \left(23 a^{2} + 15 a + 8\right)\cdot 29^{112} + \left(28 a^{2} + 17 a + 24\right)\cdot 29^{113} + \left(10 a^{2} + 25 a + 23\right)\cdot 29^{114} + \left(15 a^{2} + 21 a + 9\right)\cdot 29^{115} + \left(20 a^{2} + 25 a + 6\right)\cdot 29^{116} + \left(6 a^{2} + 28\right)\cdot 29^{117} + \left(21 a^{2} + 17 a + 8\right)\cdot 29^{118} + \left(25 a^{2} + 19 a + 10\right)\cdot 29^{119} + \left(16 a^{2} + 8 a + 13\right)\cdot 29^{120} + \left(7 a^{2} + 15 a + 27\right)\cdot 29^{121} + \left(19 a^{2} + 10 a + 23\right)\cdot 29^{122} + \left(13 a^{2} + 17 a + 19\right)\cdot 29^{123} + \left(23 a^{2} + 12 a + 27\right)\cdot 29^{124} + \left(15 a^{2} + 5 a + 22\right)\cdot 29^{125} + \left(19 a^{2} + 7 a + 10\right)\cdot 29^{126} + \left(a^{2} + 10 a + 10\right)\cdot 29^{127} + \left(11 a^{2} + 14 a + 26\right)\cdot 29^{128} + \left(28 a^{2} + 14 a + 15\right)\cdot 29^{129} + \left(22 a^{2} + 20 a + 12\right)\cdot 29^{130} + \left(26 a + 19\right)\cdot 29^{131} + \left(17 a^{2} + 6 a + 18\right)\cdot 29^{132} + \left(22 a^{2} + 22 a + 28\right)\cdot 29^{133} + \left(9 a^{2} + 21 a + 7\right)\cdot 29^{134} + \left(a^{2} + 26 a + 11\right)\cdot 29^{135} + \left(19 a^{2} + 12 a + 15\right)\cdot 29^{136} + \left(23 a^{2} + 19 a + 15\right)\cdot 29^{137} + \left(4 a^{2} + 24 a + 15\right)\cdot 29^{138} + \left(7 a^{2} + 14 a + 3\right)\cdot 29^{139} + \left(2 a^{2} + 10 a\right)\cdot 29^{140} + \left(6 a^{2} + 21 a + 21\right)\cdot 29^{141} + \left(18 a^{2} + 13 a + 19\right)\cdot 29^{142} + \left(18 a^{2} + 16 a + 7\right)\cdot 29^{143} + \left(15 a^{2} + 22\right)\cdot 29^{144} + \left(15 a^{2} + 15 a + 19\right)\cdot 29^{145} + \left(8 a^{2} + 28 a + 25\right)\cdot 29^{146} + \left(24 a^{2} + 20 a + 1\right)\cdot 29^{147} + \left(7 a^{2} + 15\right)\cdot 29^{148} + \left(22 a^{2} + 2 a + 13\right)\cdot 29^{149} + \left(10 a^{2} + 13 a + 26\right)\cdot 29^{150} + \left(25 a^{2} + 21 a + 27\right)\cdot 29^{151} + \left(a^{2} + 22\right)\cdot 29^{152} + \left(16 a^{2} + 24 a + 23\right)\cdot 29^{153} + \left(7 a^{2} + 25 a + 6\right)\cdot 29^{154} + \left(27 a^{2} + 20 a + 27\right)\cdot 29^{155} + \left(13 a^{2} + 8 a + 1\right)\cdot 29^{156} + \left(5 a^{2} + 4 a + 6\right)\cdot 29^{157} + \left(4 a^{2} + 8 a + 8\right)\cdot 29^{158} + \left(4 a^{2} + 6 a + 18\right)\cdot 29^{159} + \left(16 a^{2} + 27 a + 5\right)\cdot 29^{160} + \left(24 a^{2} + 16 a + 3\right)\cdot 29^{161} + \left(15 a^{2} + 22 a + 7\right)\cdot 29^{162} + \left(5 a^{2} + 16 a + 15\right)\cdot 29^{163} + \left(16 a^{2} + 12 a + 9\right)\cdot 29^{164} + \left(16 a^{2} + 2 a + 15\right)\cdot 29^{165} + \left(5 a^{2} + 26 a + 13\right)\cdot 29^{166} + \left(14 a^{2} + 11 a + 11\right)\cdot 29^{167} + \left(25 a^{2} + 5 a + 15\right)\cdot 29^{168} + \left(15 a^{2} + 22 a\right)\cdot 29^{169} + \left(a^{2} + 10 a + 25\right)\cdot 29^{170} + \left(11 a^{2} + a + 26\right)\cdot 29^{171} + \left(12 a + 2\right)\cdot 29^{172} + \left(22 a^{2} + 9 a + 14\right)\cdot 29^{173} + \left(21 a^{2} + 17 a + 6\right)\cdot 29^{174} +O\left(29^{ 175 }\right)$

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

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

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)(8,9)$$4$
$27$$2$$(1,2)(4,6)$$0$
$27$$2$$(1,2)(3,5)(4,6)$$0$
$54$$2$$(1,3)(2,5)(4,6)(7,8)$$0$
$6$$3$$(3,5,7)$$-4$
$8$$3$$(1,2,8)(3,5,7)(4,6,9)$$-1$
$12$$3$$(1,2,8)(3,5,7)$$2$
$72$$3$$(1,4,3)(2,6,5)(7,8,9)$$2$
$54$$4$$(1,4,2,6)(8,9)$$0$
$162$$4$$(1,5,2,3)(4,6)(7,8)$$0$
$36$$6$$(1,4)(2,6)(3,5,7)(8,9)$$-2$
$36$$6$$(1,3,2,5,8,7)$$-2$
$36$$6$$(1,2)(3,5,7)$$0$
$36$$6$$(1,2)(3,5,7)(4,6,9)$$0$
$54$$6$$(1,2)(3,7,5)(4,6)$$0$
$72$$6$$(1,4,8,9,2,6)(3,5,7)$$1$
$108$$6$$(1,3,2,5,8,7)(4,6)$$0$
$216$$6$$(1,6,5,2,4,3)(7,8,9)$$0$
$144$$9$$(1,4,3,2,6,5,8,9,7)$$-1$
$108$$12$$(1,4,2,6)(3,5,7)(8,9)$$0$
The blue line marks the conjugacy class containing complex conjugation.