Properties

Label 12.5e5_1069e6.24t2821.1c1
Dimension 12
Group $S_4\wr C_2$
Conductor $ 5^{5} \cdot 1069^{6}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$12$
Group:$S_4\wr C_2$
Conductor:$4663545871628928378125= 5^{5} \cdot 1069^{6} $
Artin number field: Splitting field of $f= x^{8} - 2 x^{7} + 11 x^{6} - 39 x^{5} + 87 x^{4} - 178 x^{3} + 108 x^{2} + 56 x + 5 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 24T2821
Parity: Even
Determinant: 1.5.2t1.1c1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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