# Properties

 Label 9.828986949632.16t1294.a.a Dimension $9$ Group $S_4\wr C_2$ Conductor $828986949632$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $9$ Group: $S_4\wr C_2$ Conductor: $$828986949632$$$$\medspace = 2^{16} \cdot 233^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 8.2.754507653376.2 Galois orbit size: $1$ Smallest permutation container: 16T1294 Parity: odd Determinant: 1.932.2t1.a.a Projective image: $S_4\wr C_2$ Projective field: Galois closure of 8.2.754507653376.2

## Defining polynomial

 $f(x)$ $=$ $x^{8} - 4 x^{6} - 8 x^{5} + 9 x^{4} + 16 x^{3} + 6 x^{2} - 20 x - 52$.

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 110.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $x^{3} + 2 x + 27$

Roots:
 $r_{ 1 }$ $=$ $7 a^{2} + 19 a + 22 + \left(4 a^{2} + 11 a\right)\cdot 29 + \left(25 a^{2} + 19 a + 26\right)\cdot 29^{2} + \left(7 a^{2} + 4 a + 18\right)\cdot 29^{3} + \left(22 a^{2} + 12 a + 11\right)\cdot 29^{4} + \left(5 a^{2} + 23 a + 13\right)\cdot 29^{5} + \left(13 a^{2} + 7 a + 7\right)\cdot 29^{6} + \left(9 a^{2} + 22\right)\cdot 29^{7} + \left(25 a^{2} + 11 a\right)\cdot 29^{8} + \left(16 a + 5\right)\cdot 29^{9} + \left(25 a^{2} + 17 a + 17\right)\cdot 29^{10} + \left(a^{2} + 28 a + 25\right)\cdot 29^{11} + \left(3 a^{2} + 2 a + 19\right)\cdot 29^{12} + \left(2 a^{2} + 2 a + 27\right)\cdot 29^{13} + \left(4 a^{2} + 24 a + 6\right)\cdot 29^{14} + \left(7 a^{2} + a + 2\right)\cdot 29^{15} + \left(15 a^{2} + 2 a + 16\right)\cdot 29^{16} + \left(22 a^{2} + 3 a\right)\cdot 29^{17} + \left(3 a^{2} + 19 a + 6\right)\cdot 29^{18} + \left(26 a^{2} + 28 a + 19\right)\cdot 29^{19} + \left(23 a^{2} + 10 a + 23\right)\cdot 29^{20} + \left(8 a + 16\right)\cdot 29^{21} + \left(21 a^{2} + 16 a + 5\right)\cdot 29^{22} + \left(20 a^{2} + 13 a + 20\right)\cdot 29^{23} + \left(14 a^{2} + 24 a + 24\right)\cdot 29^{24} + \left(15 a^{2} + 26 a + 11\right)\cdot 29^{25} + \left(21 a^{2} + 9 a + 27\right)\cdot 29^{26} + \left(4 a^{2} + 11 a + 22\right)\cdot 29^{27} + \left(8 a + 2\right)\cdot 29^{28} + \left(11 a^{2} + 3 a + 12\right)\cdot 29^{29} + \left(25 a^{2} + 20 a + 13\right)\cdot 29^{30} + \left(10 a^{2} + 22 a + 20\right)\cdot 29^{31} + \left(a^{2} + 7 a + 7\right)\cdot 29^{32} + \left(3 a^{2} + 12 a + 28\right)\cdot 29^{33} + \left(5 a^{2} + 12 a + 8\right)\cdot 29^{34} + \left(15 a^{2} + 21 a + 11\right)\cdot 29^{35} + \left(3 a^{2} + 8 a + 5\right)\cdot 29^{36} + \left(25 a^{2} + 15 a + 17\right)\cdot 29^{37} + \left(4 a^{2} + 20 a + 19\right)\cdot 29^{38} + \left(28 a^{2} + 14 a + 13\right)\cdot 29^{39} + \left(27 a^{2} + 14 a + 9\right)\cdot 29^{40} + \left(6 a^{2} + 10 a + 24\right)\cdot 29^{41} + \left(22 a^{2} + 28 a + 6\right)\cdot 29^{42} + \left(13 a^{2} + 8 a + 23\right)\cdot 29^{43} + \left(6 a^{2} + 17 a + 13\right)\cdot 29^{44} + \left(a^{2} + 14 a + 7\right)\cdot 29^{45} + \left(2 a^{2} + 2\right)\cdot 29^{46} + \left(12 a^{2} + 11 a + 2\right)\cdot 29^{47} + \left(23 a^{2} + 27 a\right)\cdot 29^{48} + \left(17 a^{2} + 5 a + 6\right)\cdot 29^{49} + \left(21 a^{2} + 14 a + 4\right)\cdot 29^{50} + \left(23 a^{2} + 12 a + 3\right)\cdot 29^{51} + \left(27 a^{2} + a + 27\right)\cdot 29^{52} + \left(10 a^{2} + 11 a + 3\right)\cdot 29^{53} + \left(20 a^{2} + 11 a + 6\right)\cdot 29^{54} + \left(21 a^{2} + 15 a + 18\right)\cdot 29^{55} + \left(21 a^{2} + 22 a + 26\right)\cdot 29^{56} + \left(2 a^{2} + 22 a + 2\right)\cdot 29^{57} + \left(26 a^{2} + 12\right)\cdot 29^{58} + \left(14 a^{2} + 11\right)\cdot 29^{59} + \left(11 a^{2} + 19 a + 10\right)\cdot 29^{60} + \left(5 a^{2} + 8 a + 27\right)\cdot 29^{61} + \left(18 a^{2} + 8 a + 9\right)\cdot 29^{62} + \left(10 a^{2} + 19 a\right)\cdot 29^{63} + \left(15 a^{2} + 24 a + 20\right)\cdot 29^{64} + \left(23 a^{2} + 10 a + 3\right)\cdot 29^{65} + \left(3 a^{2} + 14 a + 3\right)\cdot 29^{66} + \left(9 a^{2} + 24 a + 23\right)\cdot 29^{67} + \left(27 a^{2} + 24 a + 5\right)\cdot 29^{68} + \left(18 a^{2} + 13 a + 12\right)\cdot 29^{69} + \left(27 a^{2} + 7 a + 14\right)\cdot 29^{70} + \left(11 a^{2} + 25 a + 28\right)\cdot 29^{71} + \left(27 a^{2} + 19 a + 27\right)\cdot 29^{72} + \left(8 a^{2} + 22 a + 14\right)\cdot 29^{73} + \left(18 a^{2} + 19 a + 16\right)\cdot 29^{74} + \left(26 a^{2} + 12 a + 3\right)\cdot 29^{75} + \left(21 a^{2} + 24 a + 17\right)\cdot 29^{76} + \left(8 a^{2} + 21 a + 21\right)\cdot 29^{77} + \left(6 a^{2} + 17 a + 1\right)\cdot 29^{78} + \left(16 a^{2} + 18\right)\cdot 29^{79} + \left(4 a^{2} + 12 a + 23\right)\cdot 29^{80} + \left(6 a^{2} + 27 a + 9\right)\cdot 29^{81} + \left(5 a^{2} + 15 a + 14\right)\cdot 29^{82} + \left(26 a^{2} + 4 a + 16\right)\cdot 29^{83} + \left(17 a^{2} + 5 a + 24\right)\cdot 29^{84} + \left(24 a^{2} + 11 a + 19\right)\cdot 29^{85} + \left(16 a^{2} + 5 a + 15\right)\cdot 29^{86} + \left(18 a^{2} + 17 a + 23\right)\cdot 29^{87} + \left(20 a^{2} + 17 a + 2\right)\cdot 29^{88} + \left(8 a^{2} + 15 a + 13\right)\cdot 29^{89} + \left(21 a^{2} + 7 a + 2\right)\cdot 29^{90} + \left(19 a^{2} + 5 a + 21\right)\cdot 29^{91} + \left(11 a^{2} + 6 a + 17\right)\cdot 29^{92} + \left(7 a^{2} + 22\right)\cdot 29^{93} + \left(24 a^{2} + 18 a + 28\right)\cdot 29^{94} + \left(3 a^{2} + 10 a + 2\right)\cdot 29^{95} + \left(2 a^{2} + 10 a\right)\cdot 29^{96} + \left(25 a^{2} + 14 a + 5\right)\cdot 29^{97} + \left(a^{2} + 11 a + 21\right)\cdot 29^{98} + \left(6 a^{2} + 1\right)\cdot 29^{99} + \left(8 a^{2} + 18 a\right)\cdot 29^{100} + \left(25 a^{2} + 24 a + 25\right)\cdot 29^{101} + \left(5 a^{2} + 10 a + 14\right)\cdot 29^{102} + \left(7 a^{2} + 17 a\right)\cdot 29^{103} + \left(2 a + 18\right)\cdot 29^{104} + \left(9 a^{2} + 13 a + 27\right)\cdot 29^{105} + \left(a^{2} + 19 a + 22\right)\cdot 29^{106} + \left(a^{2} + 7 a + 11\right)\cdot 29^{107} + \left(3 a^{2} + 11 a + 20\right)\cdot 29^{108} + \left(9 a^{2} + 28 a + 1\right)\cdot 29^{109} +O\left(29^{ 110 }\right)$ $r_{ 2 }$ $=$ $21 + 26\cdot 29 + 17\cdot 29^{2} + 4\cdot 29^{3} + 4\cdot 29^{4} + 7\cdot 29^{5} + 27\cdot 29^{6} + 4\cdot 29^{7} + 25\cdot 29^{8} + 16\cdot 29^{9} + 12\cdot 29^{10} + 14\cdot 29^{11} + 9\cdot 29^{12} + 19\cdot 29^{13} + 24\cdot 29^{14} + 2\cdot 29^{15} + 14\cdot 29^{16} + 5\cdot 29^{17} + 7\cdot 29^{18} + 15\cdot 29^{19} + 4\cdot 29^{20} + 29^{21} + 8\cdot 29^{22} + 16\cdot 29^{23} + 16\cdot 29^{24} + 18\cdot 29^{25} + 11\cdot 29^{26} + 10\cdot 29^{27} + 20\cdot 29^{28} + 14\cdot 29^{29} + 16\cdot 29^{30} + 15\cdot 29^{31} + 9\cdot 29^{32} + 4\cdot 29^{33} + 6\cdot 29^{34} + 5\cdot 29^{35} + 19\cdot 29^{36} + 29^{37} + 18\cdot 29^{38} + 8\cdot 29^{39} + 14\cdot 29^{40} + 27\cdot 29^{42} + 12\cdot 29^{43} + 18\cdot 29^{44} + 28\cdot 29^{45} + 25\cdot 29^{46} + 14\cdot 29^{47} + 14\cdot 29^{48} + 3\cdot 29^{49} + 23\cdot 29^{50} + 6\cdot 29^{51} + 18\cdot 29^{52} + 16\cdot 29^{53} + 28\cdot 29^{54} + 19\cdot 29^{55} + 7\cdot 29^{56} + 27\cdot 29^{57} + 13\cdot 29^{58} + 5\cdot 29^{59} + 20\cdot 29^{60} + 24\cdot 29^{61} + 9\cdot 29^{62} + 22\cdot 29^{63} + 15\cdot 29^{64} + 24\cdot 29^{65} + 23\cdot 29^{66} + 22\cdot 29^{67} + 8\cdot 29^{68} + 26\cdot 29^{69} + 17\cdot 29^{70} + 19\cdot 29^{71} + 24\cdot 29^{72} + 21\cdot 29^{73} + 14\cdot 29^{74} + 26\cdot 29^{75} + 28\cdot 29^{76} + 11\cdot 29^{77} + 11\cdot 29^{78} + 25\cdot 29^{79} + 7\cdot 29^{80} + 28\cdot 29^{81} + 18\cdot 29^{82} + 12\cdot 29^{83} + 18\cdot 29^{84} + 14\cdot 29^{85} + 19\cdot 29^{86} + 14\cdot 29^{87} + 10\cdot 29^{88} + 23\cdot 29^{89} + 28\cdot 29^{90} + 26\cdot 29^{91} + 25\cdot 29^{92} + 11\cdot 29^{93} + 3\cdot 29^{94} + 15\cdot 29^{95} + 21\cdot 29^{96} + 29^{97} + 21\cdot 29^{99} + 8\cdot 29^{100} + 15\cdot 29^{101} + 2\cdot 29^{102} + 26\cdot 29^{103} + 13\cdot 29^{104} + 8\cdot 29^{105} + 19\cdot 29^{106} + 9\cdot 29^{107} + 26\cdot 29^{108} + 28\cdot 29^{109} +O\left(29^{ 110 }\right)$ $r_{ 3 }$ $=$ $5 a^{2} + 10 a + 19 + \left(25 a^{2} + 28 a + 24\right)\cdot 29 + \left(23 a^{2} + 13 a + 25\right)\cdot 29^{2} + \left(20 a^{2} + 7 a + 6\right)\cdot 29^{3} + \left(16 a^{2} + 19 a + 11\right)\cdot 29^{4} + \left(15 a^{2} + 19 a + 18\right)\cdot 29^{5} + \left(21 a^{2} + a + 19\right)\cdot 29^{6} + \left(5 a^{2} + 28 a + 15\right)\cdot 29^{7} + \left(13 a^{2} + 21 a + 28\right)\cdot 29^{8} + \left(6 a^{2} + 6 a + 2\right)\cdot 29^{9} + \left(7 a^{2} + 14 a + 15\right)\cdot 29^{10} + \left(20 a + 24\right)\cdot 29^{11} + \left(24 a + 25\right)\cdot 29^{12} + \left(14 a^{2} + 28 a + 21\right)\cdot 29^{13} + \left(14 a^{2} + 10 a + 20\right)\cdot 29^{14} + \left(21 a^{2} + 5 a + 27\right)\cdot 29^{15} + \left(2 a^{2} + 27 a + 27\right)\cdot 29^{16} + \left(23 a + 7\right)\cdot 29^{17} + \left(23 a^{2} + 21 a + 28\right)\cdot 29^{18} + \left(2 a^{2} + 7 a + 17\right)\cdot 29^{19} + \left(2 a^{2} + 7 a + 20\right)\cdot 29^{20} + \left(15 a^{2} + 15 a + 19\right)\cdot 29^{21} + \left(6 a^{2} + 20 a + 15\right)\cdot 29^{22} + \left(9 a^{2} + 12 a + 16\right)\cdot 29^{23} + \left(19 a^{2} + 16 a + 10\right)\cdot 29^{24} + \left(21 a^{2} + 13 a + 3\right)\cdot 29^{25} + \left(9 a^{2} + 10 a + 9\right)\cdot 29^{26} + \left(16 a^{2} + a + 18\right)\cdot 29^{27} + \left(23 a^{2} + 11 a + 24\right)\cdot 29^{28} + \left(21 a^{2} + 3 a + 4\right)\cdot 29^{29} + \left(23 a^{2} + 28 a + 26\right)\cdot 29^{30} + \left(6 a^{2} + 22 a + 3\right)\cdot 29^{31} + \left(26 a^{2} + 13 a + 22\right)\cdot 29^{32} + \left(27 a^{2} + 18 a + 6\right)\cdot 29^{33} + \left(4 a^{2} + 8 a + 14\right)\cdot 29^{34} + \left(4 a^{2} + 17 a + 13\right)\cdot 29^{35} + \left(20 a^{2} + 26 a + 20\right)\cdot 29^{36} + \left(2 a^{2} + 6 a + 12\right)\cdot 29^{37} + \left(20 a^{2} + 15 a + 1\right)\cdot 29^{38} + \left(2 a^{2} + 6 a + 20\right)\cdot 29^{39} + \left(18 a^{2} + 26 a + 9\right)\cdot 29^{40} + \left(9 a^{2} + a + 22\right)\cdot 29^{41} + \left(15 a^{2} + 16 a + 1\right)\cdot 29^{42} + \left(4 a^{2} + 24 a + 21\right)\cdot 29^{43} + \left(20 a^{2} + 19 a + 20\right)\cdot 29^{44} + \left(27 a^{2} + 12 a + 17\right)\cdot 29^{45} + \left(14 a^{2} + 27 a + 1\right)\cdot 29^{46} + \left(14 a^{2} + 3 a + 24\right)\cdot 29^{47} + \left(18 a^{2} + 13 a + 19\right)\cdot 29^{48} + \left(13 a^{2} + 10 a + 26\right)\cdot 29^{49} + \left(28 a^{2} + 12 a + 10\right)\cdot 29^{50} + \left(4 a^{2} + 4 a + 4\right)\cdot 29^{51} + \left(13 a^{2} + 13 a + 21\right)\cdot 29^{52} + \left(26 a^{2} + 14 a\right)\cdot 29^{53} + \left(24 a^{2} + 18 a + 14\right)\cdot 29^{54} + \left(7 a^{2} + 27 a + 13\right)\cdot 29^{55} + \left(18 a^{2} + 24 a + 2\right)\cdot 29^{56} + \left(7 a^{2} + 21 a + 1\right)\cdot 29^{57} + \left(18 a^{2} + 19 a + 10\right)\cdot 29^{58} + \left(12 a^{2} + 25 a + 5\right)\cdot 29^{59} + \left(6 a^{2} + 21\right)\cdot 29^{60} + \left(23 a^{2} + 25 a + 22\right)\cdot 29^{61} + \left(26 a^{2} + a + 22\right)\cdot 29^{62} + \left(9 a^{2} + 25 a + 5\right)\cdot 29^{63} + \left(27 a^{2} + 6 a + 2\right)\cdot 29^{64} + \left(12 a^{2} + 8 a + 9\right)\cdot 29^{65} + \left(4 a^{2} + 8 a + 17\right)\cdot 29^{66} + \left(14 a^{2} + 10 a + 1\right)\cdot 29^{67} + \left(15 a^{2} + 3 a + 8\right)\cdot 29^{68} + \left(20 a^{2} + 22 a + 28\right)\cdot 29^{69} + \left(20 a^{2} + 11\right)\cdot 29^{70} + \left(14 a^{2} + 25 a + 3\right)\cdot 29^{71} + \left(25 a^{2} + 20 a + 16\right)\cdot 29^{72} + \left(21 a^{2} + 12 a + 2\right)\cdot 29^{73} + \left(21 a^{2} + 21 a + 24\right)\cdot 29^{74} + \left(18 a^{2} + 6 a + 25\right)\cdot 29^{75} + \left(27 a^{2} + 11 a + 7\right)\cdot 29^{76} + \left(2 a^{2} + 4 a + 19\right)\cdot 29^{77} + \left(2 a^{2} + 2 a + 8\right)\cdot 29^{78} + \left(8 a^{2} + 20 a + 2\right)\cdot 29^{79} + \left(a^{2} + 3 a + 28\right)\cdot 29^{80} + \left(9 a^{2} + 12 a + 21\right)\cdot 29^{81} + \left(28 a^{2} + 4 a + 21\right)\cdot 29^{82} + \left(12 a^{2} + 3 a + 22\right)\cdot 29^{83} + \left(3 a^{2} + 4 a + 17\right)\cdot 29^{84} + \left(16 a + 14\right)\cdot 29^{85} + \left(15 a^{2} + 16 a + 13\right)\cdot 29^{86} + \left(10 a^{2} + 26 a + 28\right)\cdot 29^{87} + \left(28 a^{2} + 28 a + 14\right)\cdot 29^{88} + \left(a^{2} + 3 a + 4\right)\cdot 29^{89} + \left(27 a^{2} + 22 a + 7\right)\cdot 29^{90} + \left(28 a^{2} + 12 a + 10\right)\cdot 29^{91} + \left(5 a^{2} + 26 a + 28\right)\cdot 29^{92} + \left(9 a^{2} + 10 a + 17\right)\cdot 29^{93} + \left(10 a^{2} + 12 a + 12\right)\cdot 29^{94} + \left(8 a^{2} + 15 a + 25\right)\cdot 29^{95} + \left(23 a^{2} + 21 a + 23\right)\cdot 29^{96} + \left(19 a^{2} + 28 a + 25\right)\cdot 29^{97} + \left(22 a^{2} + 4 a + 10\right)\cdot 29^{98} + \left(6 a^{2} + 10 a + 21\right)\cdot 29^{99} + \left(8 a^{2} + 21 a + 17\right)\cdot 29^{100} + \left(2 a^{2} + 3 a + 7\right)\cdot 29^{101} + \left(27 a^{2} + 16 a + 6\right)\cdot 29^{102} + \left(21 a^{2} + 8 a + 1\right)\cdot 29^{103} + \left(25 a^{2} + 5 a + 20\right)\cdot 29^{104} + \left(15 a^{2} + 25 a + 8\right)\cdot 29^{105} + \left(15 a^{2} + 22 a + 14\right)\cdot 29^{106} + \left(4 a^{2} + 11 a + 12\right)\cdot 29^{107} + \left(14 a^{2} + 13 a\right)\cdot 29^{108} + \left(19 a^{2} + 3 a + 26\right)\cdot 29^{109} +O\left(29^{ 110 }\right)$ $r_{ 4 }$ $=$ $8 a^{2} + 14 a + 23 + \left(9 a^{2} + 6 a + 22\right)\cdot 29 + \left(7 a + 13\right)\cdot 29^{2} + \left(6 a^{2} + 22 a + 6\right)\cdot 29^{3} + \left(4 a^{2} + 21 a + 4\right)\cdot 29^{4} + \left(18 a^{2} + 7 a + 12\right)\cdot 29^{5} + \left(13 a^{2} + 20 a + 28\right)\cdot 29^{6} + \left(20 a^{2} + 11 a + 15\right)\cdot 29^{7} + \left(8 a^{2} + 6 a + 22\right)\cdot 29^{8} + \left(14 a + 23\right)\cdot 29^{9} + \left(23 a^{2} + 6 a + 16\right)\cdot 29^{10} + \left(18 a^{2} + 28 a + 10\right)\cdot 29^{11} + \left(6 a^{2} + 19 a + 15\right)\cdot 29^{12} + \left(11 a^{2} + 25 a + 8\right)\cdot 29^{13} + \left(3 a^{2} + 16 a + 25\right)\cdot 29^{14} + \left(21 a^{2} + 23 a + 7\right)\cdot 29^{15} + \left(17 a^{2} + 26 a + 9\right)\cdot 29^{16} + \left(4 a^{2} + 23 a + 4\right)\cdot 29^{17} + \left(4 a^{2} + 18 a + 3\right)\cdot 29^{18} + \left(11 a^{2} + 21 a\right)\cdot 29^{19} + \left(20 a^{2} + 7 a + 16\right)\cdot 29^{20} + a\cdot 29^{21} + \left(20 a^{2} + 15 a + 24\right)\cdot 29^{22} + \left(21 a^{2} + 17 a + 13\right)\cdot 29^{23} + \left(18 a^{2} + 13 a + 19\right)\cdot 29^{24} + \left(16 a^{2} + 19 a + 25\right)\cdot 29^{25} + \left(24 a^{2} + a + 28\right)\cdot 29^{26} + \left(9 a^{2} + 23 a + 28\right)\cdot 29^{27} + \left(18 a^{2} + 11 a + 7\right)\cdot 29^{28} + \left(18 a^{2} + 2 a + 10\right)\cdot 29^{29} + \left(3 a + 24\right)\cdot 29^{30} + \left(22 a^{2} + 25 a + 4\right)\cdot 29^{31} + \left(24 a^{2} + 17 a + 20\right)\cdot 29^{32} + \left(23 a^{2} + 3 a + 20\right)\cdot 29^{33} + \left(24 a^{2} + 20 a + 11\right)\cdot 29^{34} + \left(20 a^{2} + 3 a + 16\right)\cdot 29^{35} + \left(19 a^{2} + 2 a\right)\cdot 29^{36} + \left(9 a^{2} + 26 a + 22\right)\cdot 29^{37} + \left(9 a^{2} + 15 a + 25\right)\cdot 29^{38} + \left(13 a^{2} + 3 a + 14\right)\cdot 29^{39} + \left(26 a^{2} + 11 a + 1\right)\cdot 29^{40} + \left(19 a^{2} + 21 a + 7\right)\cdot 29^{41} + \left(3 a^{2} + 13 a + 15\right)\cdot 29^{42} + \left(16 a^{2} + 21 a + 7\right)\cdot 29^{43} + \left(21 a^{2} + 15 a + 3\right)\cdot 29^{44} + \left(8 a^{2} + 11 a + 2\right)\cdot 29^{45} + \left(23 a^{2} + 19 a + 3\right)\cdot 29^{46} + \left(21 a^{2} + 12 a + 24\right)\cdot 29^{47} + \left(3 a^{2} + 5 a + 9\right)\cdot 29^{48} + \left(10 a^{2} + 20 a + 12\right)\cdot 29^{49} + \left(18 a^{2} + 19 a + 26\right)\cdot 29^{50} + \left(7 a^{2} + 12 a + 7\right)\cdot 29^{51} + \left(3 a^{2} + 15 a + 27\right)\cdot 29^{52} + \left(24 a^{2} + 7 a + 16\right)\cdot 29^{53} + \left(a^{2} + 8 a + 2\right)\cdot 29^{54} + \left(25 a + 3\right)\cdot 29^{55} + \left(15 a^{2} + 16 a + 27\right)\cdot 29^{56} + \left(21 a^{2} + 9\right)\cdot 29^{57} + \left(20 a^{2} + 18 a + 13\right)\cdot 29^{58} + \left(13 a^{2} + 27 a + 16\right)\cdot 29^{59} + \left(10 a^{2} + 3 a + 26\right)\cdot 29^{60} + \left(3 a^{2} + 8 a + 5\right)\cdot 29^{61} + \left(8 a^{2} + 7 a + 17\right)\cdot 29^{62} + \left(12 a^{2} + 13 a + 18\right)\cdot 29^{63} + \left(20 a^{2} + 8 a + 2\right)\cdot 29^{64} + \left(3 a^{2} + 17 a + 16\right)\cdot 29^{65} + \left(27 a^{2} + 28 a + 18\right)\cdot 29^{66} + \left(8 a^{2} + 4 a + 23\right)\cdot 29^{67} + \left(3 a^{2} + 22 a + 20\right)\cdot 29^{68} + \left(11 a^{2} + 6 a + 15\right)\cdot 29^{69} + \left(22 a^{2} + 8 a + 4\right)\cdot 29^{70} + \left(24 a^{2} + 26 a + 7\right)\cdot 29^{71} + \left(3 a^{2} + a + 16\right)\cdot 29^{72} + \left(3 a^{2} + 9 a + 6\right)\cdot 29^{73} + \left(4 a^{2} + 10\right)\cdot 29^{74} + \left(5 a^{2} + 13 a + 17\right)\cdot 29^{75} + \left(16 a^{2} + a + 21\right)\cdot 29^{76} + \left(10 a^{2} + 24 a + 19\right)\cdot 29^{77} + \left(24 a^{2} + 20 a + 28\right)\cdot 29^{78} + \left(13 a^{2} + 8 a + 9\right)\cdot 29^{79} + \left(4 a^{2} + 27 a + 3\right)\cdot 29^{80} + \left(7 a^{2} + 11 a\right)\cdot 29^{81} + \left(13 a^{2} + a + 21\right)\cdot 29^{82} + \left(27 a^{2} + 6 a + 22\right)\cdot 29^{83} + \left(11 a^{2} + 18 a + 9\right)\cdot 29^{84} + \left(8 a^{2} + 18 a + 6\right)\cdot 29^{85} + \left(8 a^{2} + 12 a + 14\right)\cdot 29^{86} + \left(8 a^{2} + 25 a + 25\right)\cdot 29^{87} + \left(16 a^{2} + 23 a + 27\right)\cdot 29^{88} + \left(6 a^{2} + 27 a\right)\cdot 29^{89} + \left(6 a^{2} + 11 a + 18\right)\cdot 29^{90} + \left(27 a^{2} + 4 a + 17\right)\cdot 29^{91} + \left(19 a^{2} + 18 a + 27\right)\cdot 29^{92} + \left(12 a^{2} + 25 a + 12\right)\cdot 29^{93} + \left(4 a^{2} + 5 a + 14\right)\cdot 29^{94} + \left(a^{2} + 2 a + 25\right)\cdot 29^{95} + \left(26 a^{2} + 13 a + 17\right)\cdot 29^{96} + \left(24 a^{2} + 7 a + 3\right)\cdot 29^{97} + \left(9 a^{2} + 2 a + 13\right)\cdot 29^{98} + \left(11 a^{2} + 18 a + 27\right)\cdot 29^{99} + \left(17 a^{2} + 21 a\right)\cdot 29^{100} + \left(21 a^{2} + 3 a + 14\right)\cdot 29^{101} + \left(28 a^{2} + 11 a + 8\right)\cdot 29^{102} + \left(28 a^{2} + 3 a + 20\right)\cdot 29^{103} + \left(9 a^{2} + 25 a + 8\right)\cdot 29^{104} + \left(19 a^{2} + 2 a + 13\right)\cdot 29^{105} + \left(18 a^{2} + 2 a + 18\right)\cdot 29^{106} + \left(7 a^{2} + 28 a + 16\right)\cdot 29^{107} + \left(17 a^{2} + a + 4\right)\cdot 29^{108} + \left(12 a^{2} + 18 a + 7\right)\cdot 29^{109} +O\left(29^{ 110 }\right)$ $r_{ 5 }$ $=$ $20 + 14\cdot 29 + 22\cdot 29^{2} + 3\cdot 29^{3} + 25\cdot 29^{4} + 11\cdot 29^{5} + 29^{6} + 12\cdot 29^{8} + 17\cdot 29^{9} + 19\cdot 29^{10} + 17\cdot 29^{11} + 10\cdot 29^{12} + 12\cdot 29^{13} + 24\cdot 29^{14} + 21\cdot 29^{15} + 12\cdot 29^{16} + 29^{17} + 26\cdot 29^{18} + 17\cdot 29^{19} + 24\cdot 29^{20} + 10\cdot 29^{21} + 9\cdot 29^{22} + 22\cdot 29^{23} + 13\cdot 29^{24} + 26\cdot 29^{25} + 3\cdot 29^{26} + 8\cdot 29^{27} + 21\cdot 29^{28} + 7\cdot 29^{29} + 3\cdot 29^{30} + 11\cdot 29^{31} + 11\cdot 29^{32} + 14\cdot 29^{33} + 22\cdot 29^{34} + 26\cdot 29^{35} + 26\cdot 29^{36} + 19\cdot 29^{37} + 18\cdot 29^{38} + 13\cdot 29^{39} + 25\cdot 29^{40} + 12\cdot 29^{41} + 10\cdot 29^{42} + 14\cdot 29^{43} + 13\cdot 29^{44} + 11\cdot 29^{45} + 29^{46} + 13\cdot 29^{47} + 6\cdot 29^{48} + 24\cdot 29^{49} + 15\cdot 29^{50} + 27\cdot 29^{51} + 3\cdot 29^{53} + 5\cdot 29^{54} + 3\cdot 29^{55} + 7\cdot 29^{56} + 2\cdot 29^{57} + 10\cdot 29^{58} + 25\cdot 29^{59} + 14\cdot 29^{60} + 26\cdot 29^{61} + 13\cdot 29^{62} + 12\cdot 29^{63} + 29^{64} + 25\cdot 29^{65} + 5\cdot 29^{66} + 25\cdot 29^{67} + 4\cdot 29^{68} + 10\cdot 29^{69} + 9\cdot 29^{70} + 20\cdot 29^{71} + 25\cdot 29^{72} + 19\cdot 29^{73} + 23\cdot 29^{74} + 8\cdot 29^{75} + 7\cdot 29^{76} + 28\cdot 29^{77} + 19\cdot 29^{78} + 10\cdot 29^{79} + 5\cdot 29^{80} + 24\cdot 29^{81} + 6\cdot 29^{82} + 26\cdot 29^{83} + 26\cdot 29^{84} + 9\cdot 29^{85} + 20\cdot 29^{86} + 3\cdot 29^{87} + 16\cdot 29^{88} + 24\cdot 29^{89} + 19\cdot 29^{90} + 15\cdot 29^{91} + 22\cdot 29^{92} + 19\cdot 29^{93} + 10\cdot 29^{94} + 6\cdot 29^{95} + 8\cdot 29^{96} + 27\cdot 29^{97} + 29^{98} + 19\cdot 29^{99} + 3\cdot 29^{100} + 26\cdot 29^{101} + 7\cdot 29^{102} + 27\cdot 29^{103} + 4\cdot 29^{104} + 11\cdot 29^{105} + 23\cdot 29^{106} + 26\cdot 29^{107} + 8\cdot 29^{108} + 2\cdot 29^{109} +O\left(29^{ 110 }\right)$ $r_{ 6 }$ $=$ $13 a^{2} + 2 a + 1 + \left(5 a^{2} + 25 a + 12\right)\cdot 29 + \left(3 a^{2} + 18 a + 6\right)\cdot 29^{2} + \left(13 a^{2} + 20 a + 16\right)\cdot 29^{3} + \left(17 a^{2} + 16 a + 24\right)\cdot 29^{4} + \left(3 a^{2} + 15 a\right)\cdot 29^{5} + \left(3 a^{2} + 27 a + 23\right)\cdot 29^{6} + \left(19 a^{2} + 3 a + 15\right)\cdot 29^{7} + \left(2 a^{2} + 28\right)\cdot 29^{8} + \left(3 a^{2} + 5 a + 7\right)\cdot 29^{9} + \left(5 a^{2} + 5 a\right)\cdot 29^{10} + \left(21 a^{2} + 9 a + 3\right)\cdot 29^{11} + \left(a^{2} + 25 a + 18\right)\cdot 29^{12} + \left(14 a^{2} + 8 a + 14\right)\cdot 29^{13} + \left(22 a^{2} + 2 a + 2\right)\cdot 29^{14} + \left(24 a^{2} + 22 a + 16\right)\cdot 29^{15} + \left(10 a^{2} + 23 a\right)\cdot 29^{16} + 27 a\cdot 29^{17} + \left(13 a + 1\right)\cdot 29^{18} + \left(25 a + 23\right)\cdot 29^{19} + \left(20 a^{2} + 28 a + 8\right)\cdot 29^{20} + \left(21 a^{2} + 11 a + 25\right)\cdot 29^{21} + \left(18 a^{2} + 23 a + 21\right)\cdot 29^{22} + \left(2 a^{2} + 8 a + 5\right)\cdot 29^{23} + \left(6 a^{2} + 7 a + 13\right)\cdot 29^{24} + \left(18 a^{2} + 23 a + 15\right)\cdot 29^{25} + \left(8 a^{2} + 7 a\right)\cdot 29^{26} + \left(16 a^{2} + 8 a + 19\right)\cdot 29^{27} + \left(3 a^{2} + 26\right)\cdot 29^{28} + \left(4 a^{2} + 27 a + 2\right)\cdot 29^{29} + \left(11 a^{2} + 6 a + 4\right)\cdot 29^{30} + \left(19 a^{2} + 24 a + 22\right)\cdot 29^{31} + \left(a^{2} + 19 a + 17\right)\cdot 29^{32} + \left(15 a^{2} + 27 a + 5\right)\cdot 29^{33} + \left(a^{2} + a + 4\right)\cdot 29^{34} + \left(8 a^{2} + 10 a + 21\right)\cdot 29^{35} + \left(2 a^{2} + 20 a + 3\right)\cdot 29^{36} + \left(2 a + 3\right)\cdot 29^{37} + \left(a^{2} + 28 a + 24\right)\cdot 29^{38} + \left(26 a^{2} + 16 a + 10\right)\cdot 29^{39} + \left(15 a^{2} + 18 a + 22\right)\cdot 29^{40} + \left(15 a^{2} + 27 a + 6\right)\cdot 29^{41} + \left(22 a^{2} + 8 a + 7\right)\cdot 29^{42} + \left(17 a^{2} + 26 a + 9\right)\cdot 29^{43} + \left(25 a^{2} + 8 a + 10\right)\cdot 29^{44} + \left(a^{2} + 8 a + 8\right)\cdot 29^{45} + \left(11 a^{2} + 2 a + 14\right)\cdot 29^{46} + \left(5 a^{2} + 6 a + 12\right)\cdot 29^{47} + \left(23 a^{2} + 18 a + 9\right)\cdot 29^{48} + \left(15 a^{2} + 14 a + 3\right)\cdot 29^{49} + \left(28 a^{2} + 14 a + 23\right)\cdot 29^{50} + \left(11 a^{2} + 18 a + 6\right)\cdot 29^{51} + \left(4 a^{2} + 13 a + 15\right)\cdot 29^{52} + \left(11 a^{2} + a + 23\right)\cdot 29^{53} + \left(22 a^{2} + 19 a + 8\right)\cdot 29^{54} + \left(19 a^{2} + 25 a + 25\right)\cdot 29^{55} + \left(15 a^{2} + 20 a + 8\right)\cdot 29^{56} + \left(27 a^{2} + 25 a + 26\right)\cdot 29^{57} + \left(7 a^{2} + 9 a + 16\right)\cdot 29^{58} + \left(a^{2} + 21 a + 12\right)\cdot 29^{59} + \left(12 a^{2} + a + 1\right)\cdot 29^{60} + \left(19 a^{2} + 15 a + 17\right)\cdot 29^{61} + \left(9 a^{2} + 27 a + 27\right)\cdot 29^{62} + \left(23 a^{2} + 10 a + 26\right)\cdot 29^{63} + \left(2 a^{2} + 14 a + 12\right)\cdot 29^{64} + \left(24 a^{2} + 9 a + 4\right)\cdot 29^{65} + \left(15 a^{2} + 20 a + 19\right)\cdot 29^{66} + \left(23 a^{2} + 3\right)\cdot 29^{67} + \left(25 a^{2} + 23 a + 23\right)\cdot 29^{68} + \left(15 a^{2} + 26 a + 17\right)\cdot 29^{69} + \left(21 a^{2} + 15 a + 25\right)\cdot 29^{70} + \left(12 a^{2} + 24 a + 19\right)\cdot 29^{71} + \left(5 a^{2} + 14 a + 27\right)\cdot 29^{72} + \left(18 a^{2} + 9 a + 7\right)\cdot 29^{73} + \left(10 a^{2} + 6\right)\cdot 29^{74} + \left(10 a^{2} + 27 a + 1\right)\cdot 29^{75} + \left(13 a^{2} + 10 a + 25\right)\cdot 29^{76} + \left(25 a^{2} + 18 a + 14\right)\cdot 29^{77} + \left(12 a^{2} + 20 a + 10\right)\cdot 29^{78} + \left(20 a^{2} + 5 a + 4\right)\cdot 29^{79} + \left(4 a^{2} + 11 a + 14\right)\cdot 29^{80} + \left(20 a^{2} + 6 a + 28\right)\cdot 29^{81} + \left(22 a^{2} + 27\right)\cdot 29^{82} + \left(28 a^{2} + 10 a + 19\right)\cdot 29^{83} + \left(12 a^{2} + 6 a + 27\right)\cdot 29^{84} + \left(18 a^{2} + 1\right)\cdot 29^{85} + \left(8 a^{2} + 12 a + 24\right)\cdot 29^{86} + \left(19 a^{2} + 14\right)\cdot 29^{87} + \left(20 a^{2} + 15 a + 12\right)\cdot 29^{88} + \left(6 a^{2} + 21 a + 10\right)\cdot 29^{89} + \left(9 a^{2} + 12 a + 15\right)\cdot 29^{90} + \left(8 a^{2} + 4 a + 15\right)\cdot 29^{91} + \left(14 a^{2} + 5 a + 11\right)\cdot 29^{92} + \left(10 a^{2} + 2 a + 7\right)\cdot 29^{93} + \left(22 a^{2} + 10 a + 26\right)\cdot 29^{94} + \left(8 a^{2} + 21 a + 28\right)\cdot 29^{95} + \left(28 a^{2} + 27 a + 5\right)\cdot 29^{96} + \left(18 a^{2} + 20 a + 16\right)\cdot 29^{97} + \left(15 a^{2} + 22 a + 10\right)\cdot 29^{98} + \left(9 a^{2} + 14 a + 6\right)\cdot 29^{99} + \left(16 a^{2} + 12 a + 1\right)\cdot 29^{100} + \left(10 a^{2} + 9 a + 15\right)\cdot 29^{101} + \left(19 a^{2} + 24 a + 13\right)\cdot 29^{102} + \left(6 a^{2} + 6 a + 9\right)\cdot 29^{103} + \left(20 a^{2} + 7 a + 25\right)\cdot 29^{104} + \left(16 a^{2} + 14 a + 8\right)\cdot 29^{105} + \left(13 a^{2} + 24 a + 10\right)\cdot 29^{106} + \left(27 a^{2} + 20 a + 8\right)\cdot 29^{107} + \left(16 a^{2} + 13 a\right)\cdot 29^{108} + \left(20 a^{2} + 28 a + 17\right)\cdot 29^{109} +O\left(29^{ 110 }\right)$ $r_{ 7 }$ $=$ $16 a^{2} + 5 a + 24 + \left(23 a^{2} + 23 a + 12\right)\cdot 29 + \left(4 a^{2} + 7 a\right)\cdot 29^{2} + \left(2 a^{2} + 28 a + 11\right)\cdot 29^{3} + \left(8 a^{2} + 16 a + 9\right)\cdot 29^{4} + \left(24 a^{2} + a + 20\right)\cdot 29^{5} + \left(22 a^{2} + 7 a + 11\right)\cdot 29^{6} + \left(2 a^{2} + 18 a + 21\right)\cdot 29^{7} + \left(7 a^{2} + 10\right)\cdot 29^{8} + \left(22 a^{2} + 8 a + 14\right)\cdot 29^{9} + \left(27 a^{2} + 8 a + 13\right)\cdot 29^{10} + \left(9 a^{2} + 9 a + 8\right)\cdot 29^{11} + \left(22 a^{2} + 13 a + 7\right)\cdot 29^{12} + \left(3 a^{2} + 3 a + 8\right)\cdot 29^{13} + \left(11 a^{2} + a + 16\right)\cdot 29^{14} + \left(15 a^{2} + 19\right)\cdot 29^{15} + \left(8 a^{2} + 4 a + 6\right)\cdot 29^{16} + \left(24 a^{2} + 10 a + 11\right)\cdot 29^{17} + \left(a^{2} + 17 a + 19\right)\cdot 29^{18} + \left(15 a^{2} + 28 a + 24\right)\cdot 29^{19} + \left(6 a^{2} + 13 a + 16\right)\cdot 29^{20} + \left(13 a^{2} + 12 a + 7\right)\cdot 29^{21} + \left(2 a^{2} + 22 a + 10\right)\cdot 29^{22} + \left(27 a^{2} + 27 a + 11\right)\cdot 29^{23} + \left(19 a^{2} + 27 a + 11\right)\cdot 29^{24} + \left(19 a^{2} + 24 a + 10\right)\cdot 29^{25} + \left(23 a^{2} + 16 a + 8\right)\cdot 29^{26} + \left(2 a^{2} + 4 a\right)\cdot 29^{27} + \left(16 a^{2} + 6 a + 5\right)\cdot 29^{28} + \left(17 a^{2} + 23 a + 28\right)\cdot 29^{29} + \left(4 a^{2} + 26 a + 19\right)\cdot 29^{30} + \left(9 a + 4\right)\cdot 29^{31} + \left(7 a^{2} + 26 a + 6\right)\cdot 29^{32} + \left(6 a^{2} + 6 a + 26\right)\cdot 29^{33} + \left(28 a^{2} + 25\right)\cdot 29^{34} + \left(3 a^{2} + 8 a + 22\right)\cdot 29^{35} + \left(18 a^{2} + 17\right)\cdot 29^{36} + \left(16 a^{2} + 25 a + 21\right)\cdot 29^{37} + \left(28 a^{2} + 26 a + 12\right)\cdot 29^{38} + \left(12 a^{2} + 18 a + 14\right)\cdot 29^{39} + \left(13 a^{2} + 20 a + 3\right)\cdot 29^{40} + \left(28 a^{2} + 5 a + 28\right)\cdot 29^{41} + \left(9 a^{2} + 28 a + 13\right)\cdot 29^{42} + \left(8 a^{2} + 11 a + 16\right)\cdot 29^{43} + \left(16 a^{2} + 22 a + 15\right)\cdot 29^{44} + \left(21 a^{2} + 4 a + 9\right)\cdot 29^{45} + \left(19 a^{2} + 11 a + 27\right)\cdot 29^{46} + \left(21 a^{2} + 12 a + 23\right)\cdot 29^{47} + \left(6 a^{2} + 10 a + 13\right)\cdot 29^{48} + \left(5 a^{2} + 27 a + 15\right)\cdot 29^{49} + \left(11 a^{2} + 25 a + 26\right)\cdot 29^{50} + \left(16 a^{2} + 11 a + 9\right)\cdot 29^{51} + \left(12 a^{2} + 20\right)\cdot 29^{52} + \left(7 a^{2} + 7 a + 23\right)\cdot 29^{53} + \left(2 a^{2} + 2 a + 12\right)\cdot 29^{54} + \left(21 a^{2} + 5 a + 21\right)\cdot 29^{55} + \left(24 a^{2} + 16 a + 20\right)\cdot 29^{56} + \left(28 a^{2} + 6 a + 19\right)\cdot 29^{57} + \left(18 a^{2} + 20 a + 20\right)\cdot 29^{58} + \left(2 a^{2} + 4 a + 1\right)\cdot 29^{59} + \left(12 a^{2} + 24 a + 19\right)\cdot 29^{60} + \left(2 a^{2} + 24 a + 4\right)\cdot 29^{61} + \left(23 a^{2} + 19 a + 8\right)\cdot 29^{62} + \left(6 a^{2} + 19 a + 11\right)\cdot 29^{63} + \left(10 a^{2} + 13 a + 8\right)\cdot 29^{64} + \left(12 a^{2} + 3 a + 8\right)\cdot 29^{65} + \left(26 a^{2} + 21 a + 27\right)\cdot 29^{66} + \left(5 a^{2} + 13 a + 9\right)\cdot 29^{67} + \left(10 a^{2} + 3 a + 20\right)\cdot 29^{68} + \left(26 a^{2} + 16\right)\cdot 29^{69} + \left(14 a^{2} + 20 a + 23\right)\cdot 29^{70} + \left(18 a^{2} + 6 a + 27\right)\cdot 29^{71} + \left(28 a^{2} + 6 a\right)\cdot 29^{72} + \left(3 a^{2} + 7 a + 27\right)\cdot 29^{73} + \left(3 a^{2} + 7 a + 8\right)\cdot 29^{74} + \left(5 a^{2} + 9 a + 17\right)\cdot 29^{75} + \left(14 a^{2} + 16 a + 28\right)\cdot 29^{76} + \left(15 a^{2} + 6\right)\cdot 29^{77} + \left(2 a^{2} + 6 a + 9\right)\cdot 29^{78} + \left(7 a^{2} + 20\right)\cdot 29^{79} + \left(23 a^{2} + 27 a + 18\right)\cdot 29^{80} + \left(12 a^{2} + 4 a + 7\right)\cdot 29^{81} + \left(16 a^{2} + 23 a + 25\right)\cdot 29^{82} + \left(17 a^{2} + 19 a + 28\right)\cdot 29^{83} + \left(13 a^{2} + 6 a + 11\right)\cdot 29^{84} + \left(20 a^{2} + 23 a + 22\right)\cdot 29^{85} + \left(5 a^{2} + 28 a + 10\right)\cdot 29^{86} + \left(10 a^{2} + 5 a + 18\right)\cdot 29^{87} + \left(13 a^{2} + 5 a + 4\right)\cdot 29^{88} + \left(20 a^{2} + 26 a\right)\cdot 29^{89} + \left(24 a^{2} + 23 a + 4\right)\cdot 29^{90} + \left(a^{2} + 11 a + 3\right)\cdot 29^{91} + \left(3 a^{2} + 13 a + 5\right)\cdot 29^{92} + \left(7 a^{2} + 21 a + 15\right)\cdot 29^{93} + \left(14 a^{2} + 10 a + 27\right)\cdot 29^{94} + \left(19 a^{2} + 11 a + 20\right)\cdot 29^{95} + \left(8 a^{2} + 23 a + 23\right)\cdot 29^{96} + \left(13 a^{2} + 21 a + 26\right)\cdot 29^{97} + \left(25 a^{2} + 21 a + 4\right)\cdot 29^{98} + \left(10 a^{2} + 17\right)\cdot 29^{99} + \left(3 a^{2} + 15 a + 1\right)\cdot 29^{100} + \left(5 a^{2} + 21 a + 21\right)\cdot 29^{101} + \left(2 a^{2} + a + 11\right)\cdot 29^{102} + \left(7 a^{2} + 17 a + 10\right)\cdot 29^{103} + \left(22 a^{2} + 27 a + 15\right)\cdot 29^{104} + \left(22 a^{2} + 27\right)\cdot 29^{105} + \left(23 a^{2} + 4 a + 5\right)\cdot 29^{106} + \left(16 a^{2} + 18 a + 19\right)\cdot 29^{107} + \left(26 a^{2} + 13 a + 26\right)\cdot 29^{108} + \left(25 a^{2} + 7 a + 24\right)\cdot 29^{109} +O\left(29^{ 110 }\right)$ $r_{ 8 }$ $=$ $9 a^{2} + 8 a + 15 + \left(19 a^{2} + 21 a + 1\right)\cdot 29 + \left(19 a + 3\right)\cdot 29^{2} + \left(8 a^{2} + 3 a + 19\right)\cdot 29^{3} + \left(18 a^{2} + 25\right)\cdot 29^{4} + \left(19 a^{2} + 19 a + 2\right)\cdot 29^{5} + \left(12 a^{2} + 22 a + 26\right)\cdot 29^{6} + \left(24 a + 19\right)\cdot 29^{7} + \left(a^{2} + 17 a + 16\right)\cdot 29^{8} + \left(25 a^{2} + 7 a + 27\right)\cdot 29^{9} + \left(27 a^{2} + 6 a + 20\right)\cdot 29^{10} + \left(5 a^{2} + 20 a + 11\right)\cdot 29^{11} + \left(24 a^{2} + 9\right)\cdot 29^{12} + \left(12 a^{2} + 18 a + 3\right)\cdot 29^{13} + \left(2 a^{2} + 2 a + 24\right)\cdot 29^{14} + \left(26 a^{2} + 5 a + 17\right)\cdot 29^{15} + \left(2 a^{2} + 3 a + 28\right)\cdot 29^{16} + \left(6 a^{2} + 27 a + 26\right)\cdot 29^{17} + \left(25 a^{2} + 24 a + 24\right)\cdot 29^{18} + \left(2 a^{2} + 3 a + 26\right)\cdot 29^{19} + \left(14 a^{2} + 18 a\right)\cdot 29^{20} + \left(6 a^{2} + 8 a + 5\right)\cdot 29^{21} + \left(18 a^{2} + 18 a + 21\right)\cdot 29^{22} + \left(5 a^{2} + 6 a + 9\right)\cdot 29^{23} + \left(8 a^{2} + 26 a + 6\right)\cdot 29^{24} + \left(24 a^{2} + 7 a + 4\right)\cdot 29^{25} + \left(27 a^{2} + 11 a + 26\right)\cdot 29^{26} + \left(7 a^{2} + 9 a + 7\right)\cdot 29^{27} + \left(25 a^{2} + 20 a + 7\right)\cdot 29^{28} + \left(13 a^{2} + 27 a + 6\right)\cdot 29^{29} + \left(21 a^{2} + a + 8\right)\cdot 29^{30} + \left(27 a^{2} + 11 a + 4\right)\cdot 29^{31} + \left(25 a^{2} + a + 21\right)\cdot 29^{32} + \left(10 a^{2} + 18 a + 9\right)\cdot 29^{33} + \left(22 a^{2} + 14 a + 22\right)\cdot 29^{34} + \left(5 a^{2} + 26 a + 27\right)\cdot 29^{35} + \left(23 a^{2} + 28 a + 21\right)\cdot 29^{36} + \left(3 a^{2} + 10 a + 17\right)\cdot 29^{37} + \left(23 a^{2} + 9 a + 24\right)\cdot 29^{38} + \left(3 a^{2} + 26 a + 19\right)\cdot 29^{39} + \left(14 a^{2} + 24 a\right)\cdot 29^{40} + \left(6 a^{2} + 19 a + 14\right)\cdot 29^{41} + \left(13 a^{2} + 20 a + 4\right)\cdot 29^{42} + \left(26 a^{2} + 22 a + 11\right)\cdot 29^{43} + \left(25 a^{2} + 2 a + 20\right)\cdot 29^{44} + \left(25 a^{2} + 6 a + 1\right)\cdot 29^{45} + \left(15 a^{2} + 26 a + 11\right)\cdot 29^{46} + \left(11 a^{2} + 11 a + 1\right)\cdot 29^{47} + \left(11 a^{2} + 12 a + 13\right)\cdot 29^{48} + \left(24 a^{2} + 8 a + 24\right)\cdot 29^{49} + \left(7 a^{2} + 14\right)\cdot 29^{50} + \left(22 a^{2} + 27 a + 20\right)\cdot 29^{51} + \left(25 a^{2} + 13 a + 14\right)\cdot 29^{52} + \left(6 a^{2} + 16 a + 27\right)\cdot 29^{53} + \left(15 a^{2} + 27 a + 8\right)\cdot 29^{54} + \left(16 a^{2} + 16 a + 11\right)\cdot 29^{55} + \left(20 a^{2} + 14 a + 15\right)\cdot 29^{56} + \left(27 a^{2} + 9 a + 26\right)\cdot 29^{57} + \left(23 a^{2} + 18 a + 18\right)\cdot 29^{58} + \left(12 a^{2} + 7 a + 8\right)\cdot 29^{59} + \left(5 a^{2} + 8 a + 2\right)\cdot 29^{60} + \left(4 a^{2} + 5 a + 16\right)\cdot 29^{61} + \left(a^{2} + 22 a + 6\right)\cdot 29^{62} + \left(24 a^{2} + 27 a + 18\right)\cdot 29^{63} + \left(10 a^{2} + 18 a + 23\right)\cdot 29^{64} + \left(10 a^{2} + 8 a + 24\right)\cdot 29^{65} + \left(9 a^{2} + 23 a\right)\cdot 29^{66} + \left(25 a^{2} + 3 a + 6\right)\cdot 29^{67} + \left(4 a^{2} + 10 a + 24\right)\cdot 29^{68} + \left(23 a^{2} + 17 a + 17\right)\cdot 29^{69} + \left(8 a^{2} + 5 a + 8\right)\cdot 29^{70} + \left(4 a^{2} + 8 a + 18\right)\cdot 29^{71} + \left(25 a^{2} + 23 a + 5\right)\cdot 29^{72} + \left(a^{2} + 25 a + 15\right)\cdot 29^{73} + \left(8 a + 11\right)\cdot 29^{74} + \left(21 a^{2} + 18 a + 15\right)\cdot 29^{75} + \left(22 a^{2} + 22 a + 8\right)\cdot 29^{76} + \left(23 a^{2} + 17 a + 22\right)\cdot 29^{77} + \left(9 a^{2} + 19 a + 25\right)\cdot 29^{78} + \left(21 a^{2} + 22 a + 24\right)\cdot 29^{79} + \left(19 a^{2} + 5 a + 14\right)\cdot 29^{80} + \left(2 a^{2} + 24 a + 24\right)\cdot 29^{81} + \left(a^{2} + 12 a + 8\right)\cdot 29^{82} + \left(3 a^{2} + 14 a + 24\right)\cdot 29^{83} + \left(27 a^{2} + 17 a + 7\right)\cdot 29^{84} + \left(14 a^{2} + 17 a + 26\right)\cdot 29^{85} + \left(3 a^{2} + 11 a + 26\right)\cdot 29^{86} + \left(20 a^{2} + 11 a + 15\right)\cdot 29^{87} + \left(16 a^{2} + 25 a + 26\right)\cdot 29^{88} + \left(13 a^{2} + 20 a + 9\right)\cdot 29^{89} + \left(27 a^{2} + 8 a + 20\right)\cdot 29^{90} + \left(19 a + 5\right)\cdot 29^{91} + \left(3 a^{2} + 17 a + 6\right)\cdot 29^{92} + \left(11 a^{2} + 26 a + 8\right)\cdot 29^{93} + \left(11 a^{2} + 21\right)\cdot 29^{94} + \left(16 a^{2} + 26 a + 19\right)\cdot 29^{95} + \left(27 a^{2} + 19 a + 14\right)\cdot 29^{96} + \left(13 a^{2} + 22 a + 9\right)\cdot 29^{97} + \left(11 a^{2} + 23 a + 24\right)\cdot 29^{98} + \left(13 a^{2} + 13 a + 1\right)\cdot 29^{99} + \left(4 a^{2} + 27 a + 24\right)\cdot 29^{100} + \left(22 a^{2} + 23 a + 20\right)\cdot 29^{101} + \left(3 a^{2} + 22 a + 21\right)\cdot 29^{102} + \left(15 a^{2} + 4 a + 20\right)\cdot 29^{103} + \left(8 a^{2} + 19 a + 9\right)\cdot 29^{104} + \left(3 a^{2} + a + 10\right)\cdot 29^{105} + \left(14 a^{2} + 14 a + 1\right)\cdot 29^{106} + 11\cdot 29^{107} + \left(9 a^{2} + 4 a + 28\right)\cdot 29^{108} + \left(28 a^{2} + a + 7\right)\cdot 29^{109} +O\left(29^{ 110 }\right)$

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $9$ $6$ $2$ $(2,4)(3,7)$ $-3$ $9$ $2$ $(1,6)(2,4)(3,7)(5,8)$ $1$ $12$ $2$ $(2,3)$ $3$ $24$ $2$ $(1,2)(3,5)(4,6)(7,8)$ $3$ $36$ $2$ $(1,5)(2,3)$ $1$ $36$ $2$ $(1,6)(2,3)(5,8)$ $-1$ $16$ $3$ $(3,4,7)$ $0$ $64$ $3$ $(2,4,7)(5,6,8)$ $0$ $12$ $4$ $(2,3,4,7)$ $-3$ $36$ $4$ $(1,5,6,8)(2,3,4,7)$ $1$ $36$ $4$ $(1,5,6,8)(2,4)(3,7)$ $1$ $72$ $4$ $(1,2,6,4)(3,8,7,5)$ $-1$ $72$ $4$ $(1,5,6,8)(2,3)$ $-1$ $144$ $4$ $(1,2,5,3)(4,6)(7,8)$ $1$ $48$ $6$ $(1,6)(3,7,4)(5,8)$ $0$ $96$ $6$ $(2,3)(5,8,6)$ $0$ $192$ $6$ $(1,3)(2,5,4,6,7,8)$ $0$ $144$ $8$ $(1,2,5,3,6,4,8,7)$ $-1$ $96$ $12$ $(1,5,6,8)(3,4,7)$ $0$

The blue line marks the conjugacy class containing complex conjugation.