LMFDB - Artin representation 4.63423.12t175.a.a

Basic invariants

Dimension:	$4$
Group:	$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Conductor:	$63423$$\medspace = 3^{7} \cdot 29 $
Artin stem field:	Galois closure of 9.5.9448798306221.1
Galois orbit size:	$2$
Smallest permutation container:	12T175
Parity:	even
Determinant:	1.29.2t1.a.a
Projective image:	$C_3^3:S_4$
Projective stem field:	Galois closure of 9.5.9448798306221.1

Defining polynomial

$f(x)$

$=$

$ x^{9} + 9x^{7} - 6x^{6} + 9x^{5} - 27x^{4} - 36x^{3} + 81x^{2} - 36x + 3 $

The roots of $f$ are computed in an extension of $\Q_{ 139 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 139 }$: $ x^{3} + 6x + 137 $ x^3 + 6*x + 137

Roots:

$r_{ 1 }$	$=$	$ 16 a^{2} + 112 a + 42 + \left(109 a^{2} + 110 a + 79\right)\cdot 139 + \left(117 a^{2} + 90 a + 82\right)\cdot 139^{2} + \left(108 a^{2} + 33 a\right)\cdot 139^{3} + \left(73 a^{2} + 28 a + 40\right)\cdot 139^{4} + \left(53 a^{2} + 136 a + 105\right)\cdot 139^{5} + \left(46 a^{2} + 102 a + 110\right)\cdot 139^{6} + \left(90 a^{2} + 36 a + 15\right)\cdot 139^{7} + \left(103 a^{2} + 106 a + 117\right)\cdot 139^{8} + \left(10 a^{2} + 59 a + 39\right)\cdot 139^{9} +O(139^{10})$ 16a^2 + 112a + 42 + (109a^2 + 110a + 79)139 + (117a^2 + 90a + 82)139^2 + (108a^2 + 33a)139^3 + (73a^2 + 28a + 40)139^4 + (53a^2 + 136a + 105)139^5 + (46a^2 + 102a + 110)139^6 + (90a^2 + 36a + 15)139^7 + (103a^2 + 106a + 117)139^8 + (10a^2 + 59a + 39)*139^9+O(139^10)
$r_{ 2 }$	$=$	$ 57 a^{2} + 138 a + 67 + \left(110 a^{2} + 128 a + 84\right)\cdot 139 + \left(107 a^{2} + 102 a + 42\right)\cdot 139^{2} + \left(19 a^{2} + 70 a + 61\right)\cdot 139^{3} + \left(114 a^{2} + 109 a + 62\right)\cdot 139^{4} + \left(127 a^{2} + 9 a + 124\right)\cdot 139^{5} + \left(130 a^{2} + 14 a + 31\right)\cdot 139^{6} + \left(103 a^{2} + 11 a + 70\right)\cdot 139^{7} + \left(58 a^{2} + 45 a + 76\right)\cdot 139^{8} + \left(127 a^{2} + 83 a + 89\right)\cdot 139^{9} +O(139^{10})$ 57a^2 + 138a + 67 + (110a^2 + 128a + 84)139 + (107a^2 + 102a + 42)139^2 + (19a^2 + 70a + 61)139^3 + (114a^2 + 109a + 62)139^4 + (127a^2 + 9a + 124)139^5 + (130a^2 + 14a + 31)139^6 + (103a^2 + 11a + 70)139^7 + (58a^2 + 45a + 76)139^8 + (127a^2 + 83a + 89)*139^9+O(139^10)
$r_{ 3 }$	$=$	$ 58 a^{2} + 136 a + 88 + \left(127 a^{2} + 2 a + 17\right)\cdot 139 + \left(91 a^{2} + 92 a + 138\right)\cdot 139^{2} + \left(5 a^{2} + 80 a + 112\right)\cdot 139^{3} + \left(10 a^{2} + 83 a + 47\right)\cdot 139^{4} + \left(59 a^{2} + 79 a + 34\right)\cdot 139^{5} + \left(80 a^{2} + 136 a + 33\right)\cdot 139^{6} + \left(74 a^{2} + 104 a + 32\right)\cdot 139^{7} + \left(101 a^{2} + 111 a + 54\right)\cdot 139^{8} + \left(72 a^{2} + 86 a + 79\right)\cdot 139^{9} +O(139^{10})$ 58a^2 + 136a + 88 + (127a^2 + 2a + 17)139 + (91a^2 + 92a + 138)139^2 + (5a^2 + 80a + 112)139^3 + (10a^2 + 83a + 47)139^4 + (59a^2 + 79a + 34)139^5 + (80a^2 + 136a + 33)139^6 + (74a^2 + 104a + 32)139^7 + (101a^2 + 111a + 54)139^8 + (72a^2 + 86a + 79)*139^9+O(139^10)
$r_{ 4 }$	$=$	$ 65 a^{2} + 30 a + 9 + \left(41 a^{2} + 25 a + 42\right)\cdot 139 + \left(68 a^{2} + 95 a + 57\right)\cdot 139^{2} + \left(24 a^{2} + 24 a + 25\right)\cdot 139^{3} + \left(55 a^{2} + 27 a + 51\right)\cdot 139^{4} + \left(26 a^{2} + 62 a + 138\right)\cdot 139^{5} + \left(12 a^{2} + 38 a + 133\right)\cdot 139^{6} + \left(113 a^{2} + 136 a + 90\right)\cdot 139^{7} + \left(72 a^{2} + 59 a + 106\right)\cdot 139^{8} + \left(55 a^{2} + 131 a + 19\right)\cdot 139^{9} +O(139^{10})$ 65a^2 + 30a + 9 + (41a^2 + 25a + 42)139 + (68a^2 + 95a + 57)139^2 + (24a^2 + 24a + 25)139^3 + (55a^2 + 27a + 51)139^4 + (26a^2 + 62a + 138)139^5 + (12a^2 + 38a + 133)139^6 + (113a^2 + 136a + 90)139^7 + (72a^2 + 59a + 106)139^8 + (55a^2 + 131a + 19)*139^9+O(139^10)
$r_{ 5 }$	$=$	$ 66 a^{2} + 28 a + 103 + \left(58 a^{2} + 38 a + 15\right)\cdot 139 + \left(52 a^{2} + 84 a + 99\right)\cdot 139^{2} + \left(10 a^{2} + 34 a + 23\right)\cdot 139^{3} + \left(90 a^{2} + a + 105\right)\cdot 139^{4} + \left(96 a^{2} + 132 a + 138\right)\cdot 139^{5} + \left(100 a^{2} + 21 a + 49\right)\cdot 139^{6} + \left(83 a^{2} + 91 a + 128\right)\cdot 139^{7} + \left(115 a^{2} + 126 a + 25\right)\cdot 139^{8} + 134 a\cdot 139^{9} +O(139^{10})$ 66a^2 + 28a + 103 + (58a^2 + 38a + 15)139 + (52a^2 + 84a + 99)139^2 + (10a^2 + 34a + 23)139^3 + (90a^2 + a + 105)139^4 + (96a^2 + 132a + 138)139^5 + (100a^2 + 21a + 49)139^6 + (83a^2 + 91a + 128)139^7 + (115a^2 + 126a + 25)139^8 + 134a*139^9+O(139^10)
$r_{ 6 }$	$=$	$ 82 a^{2} + 16 a + 77 + \left(113 a^{2} + 34 a + 52\right)\cdot 139 + \left(a^{2} + 92 a + 69\right)\cdot 139^{2} + \left(32 a^{2} + 124 a + 55\right)\cdot 139^{3} + \left(119 a^{2} + 3 a + 29\right)\cdot 139^{4} + \left(7 a^{2} + 98 a + 64\right)\cdot 139^{5} + \left(29 a^{2} + 32 a + 62\right)\cdot 139^{6} + \left(31 a^{2} + 97 a + 41\right)\cdot 139^{7} + \left(26 a^{2} + 95 a + 59\right)\cdot 139^{8} + \left(76 a^{2} + 3 a + 102\right)\cdot 139^{9} +O(139^{10})$ 82a^2 + 16a + 77 + (113a^2 + 34a + 52)139 + (a^2 + 92a + 69)139^2 + (32a^2 + 124a + 55)139^3 + (119a^2 + 3a + 29)139^4 + (7a^2 + 98a + 64)139^5 + (29a^2 + 32a + 62)139^6 + (31a^2 + 97a + 41)139^7 + (26a^2 + 95a + 59)139^8 + (76a^2 + 3a + 102)139^9+O(139^10)
$r_{ 7 }$	$=$	$ 90 a^{2} + 47 a + 77 + \left(44 a^{2} + 69 a + 103\right)\cdot 139 + \left(101 a^{2} + 84 a + 36\right)\cdot 139^{2} + \left(36 a^{2} + 78 a + 98\right)\cdot 139^{3} + \left(60 a^{2} + 60 a + 109\right)\cdot 139^{4} + \left(45 a^{2} + 11 a + 118\right)\cdot 139^{5} + \left(49 a^{2} + 57 a + 47\right)\cdot 139^{6} + \left(40 a^{2} + 83 a + 34\right)\cdot 139^{7} + \left(40 a^{2} + 110 a + 87\right)\cdot 139^{8} + \left(4 a^{2} + 51 a + 83\right)\cdot 139^{9} +O(139^{10})$ 90a^2 + 47a + 77 + (44a^2 + 69a + 103)139 + (101a^2 + 84a + 36)139^2 + (36a^2 + 78a + 98)139^3 + (60a^2 + 60a + 109)139^4 + (45a^2 + 11a + 118)139^5 + (49a^2 + 57a + 47)139^6 + (40a^2 + 83a + 34)139^7 + (40a^2 + 110a + 87)139^8 + (4a^2 + 51a + 83)*139^9+O(139^10)
$r_{ 8 }$	$=$	$ 130 a^{2} + 95 a + 98 + \left(105 a^{2} + 66 a + 70\right)\cdot 139 + \left(84 a^{2} + 101 a + 109\right)\cdot 139^{2} + \left(96 a^{2} + 118 a + 59\right)\cdot 139^{3} + \left(68 a^{2} + 133 a + 4\right)\cdot 139^{4} + \left(34 a^{2} + 47 a + 75\right)\cdot 139^{5} + \left(9 a^{2} + 84 a + 26\right)\cdot 139^{6} + \left(24 a^{2} + 89 a + 108\right)\cdot 139^{7} + \left(136 a^{2} + 55 a + 53\right)\cdot 139^{8} + \left(61 a^{2} + 36\right)\cdot 139^{9} +O(139^{10})$ 130a^2 + 95a + 98 + (105a^2 + 66a + 70)139 + (84a^2 + 101a + 109)139^2 + (96a^2 + 118a + 59)139^3 + (68a^2 + 133a + 4)139^4 + (34a^2 + 47a + 75)139^5 + (9a^2 + 84a + 26)139^6 + (24a^2 + 89a + 108)139^7 + (136a^2 + 55a + 53)139^8 + (61a^2 + 36)139^9+O(139^10)
$r_{ 9 }$	$=$	$ 131 a^{2} + 93 a + 134 + \left(122 a^{2} + 79 a + 89\right)\cdot 139 + \left(68 a^{2} + 90 a + 59\right)\cdot 139^{2} + \left(82 a^{2} + 128 a + 118\right)\cdot 139^{3} + \left(103 a^{2} + 107 a + 105\right)\cdot 139^{4} + \left(104 a^{2} + 117 a + 34\right)\cdot 139^{5} + \left(97 a^{2} + 67 a + 59\right)\cdot 139^{6} + \left(133 a^{2} + 44 a + 34\right)\cdot 139^{7} + \left(39 a^{2} + 122 a + 114\right)\cdot 139^{8} + \left(7 a^{2} + 3 a + 104\right)\cdot 139^{9} +O(139^{10})$ 131a^2 + 93a + 134 + (122a^2 + 79a + 89)139 + (68a^2 + 90a + 59)139^2 + (82a^2 + 128a + 118)139^3 + (103a^2 + 107a + 105)139^4 + (104a^2 + 117a + 34)139^5 + (97a^2 + 67a + 59)139^6 + (133a^2 + 44a + 34)139^7 + (39a^2 + 122a + 114)139^8 + (7a^2 + 3a + 104)*139^9+O(139^10)

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

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 9 }$	Character value
$1$	$1$	$()$	$4$
$18$	$2$	$(1,6)(3,5)(4,8)$	$2$
$27$	$2$	$(1,3)(5,8)$	$0$
$4$	$3$	$(1,4,3)(2,9,7)(5,6,8)$	$3 \zeta_{3} + 1$
$4$	$3$	$(1,3,4)(2,7,9)(5,8,6)$	$-3 \zeta_{3} - 2$
$6$	$3$	$(1,3,4)$	$-2$
$12$	$3$	$(2,7,9)(5,8,6)$	$1$
$72$	$3$	$(1,8,9)(2,3,6)(4,5,7)$	$1$
$162$	$4$	$(1,5,3,8)(2,9)(4,6)$	$0$
$18$	$6$	$(1,6)(2,9,7)(3,5)(4,8)$	$-2 \zeta_{3} - 2$
$18$	$6$	$(1,6)(2,7,9)(3,5)(4,8)$	$2 \zeta_{3}$
$36$	$6$	$(1,8,4,5,3,6)(2,9,7)$	$\zeta_{3} + 1$
$36$	$6$	$(1,6,3,5,4,8)(2,7,9)$	$-\zeta_{3}$
$36$	$6$	$(1,7,4,9,3,2)$	$-1$
$54$	$6$	$(1,4,3)(6,8)(7,9)$	$0$
$72$	$9$	$(1,2,5,4,9,6,3,7,8)$	$-\zeta_{3} - 1$
$72$	$9$	$(1,5,9,3,8,2,4,6,7)$	$\zeta_{3}$

The blue line marks the conjugacy class containing complex conjugation.

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Basic invariants

Defining polynomial

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

Character values on conjugacy classes

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4.63423.12t175.a.a
Dimension	$4$
Group	$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Conductor	$63423$
Root number	not computed
Indicator	$0$

$r_{ 1 }$	$=$	\( 16 a^{2} + 112 a + 42 + \left(109 a^{2} + 110 a + 79\right)\cdot 139 + \left(117 a^{2} + 90 a + 82\right)\cdot 139^{2} + \left(108 a^{2} + 33 a\right)\cdot 139^{3} + \left(73 a^{2} + 28 a + 40\right)\cdot 139^{4} + \left(53 a^{2} + 136 a + 105\right)\cdot 139^{5} + \left(46 a^{2} + 102 a + 110\right)\cdot 139^{6} + \left(90 a^{2} + 36 a + 15\right)\cdot 139^{7} + \left(103 a^{2} + 106 a + 117\right)\cdot 139^{8} + \left(10 a^{2} + 59 a + 39\right)\cdot 139^{9} +O(139^{10})\) 16a^2 + 112a + 42 + (109a^2 + 110a + 79)139 + (117a^2 + 90a + 82)139^2 + (108a^2 + 33a)139^3 + (73a^2 + 28a + 40)139^4 + (53a^2 + 136a + 105)139^5 + (46a^2 + 102a + 110)139^6 + (90a^2 + 36a + 15)139^7 + (103a^2 + 106a + 117)139^8 + (10a^2 + 59a + 39)*139^9+O(139^10)
$r_{ 2 }$	$=$	\( 57 a^{2} + 138 a + 67 + \left(110 a^{2} + 128 a + 84\right)\cdot 139 + \left(107 a^{2} + 102 a + 42\right)\cdot 139^{2} + \left(19 a^{2} + 70 a + 61\right)\cdot 139^{3} + \left(114 a^{2} + 109 a + 62\right)\cdot 139^{4} + \left(127 a^{2} + 9 a + 124\right)\cdot 139^{5} + \left(130 a^{2} + 14 a + 31\right)\cdot 139^{6} + \left(103 a^{2} + 11 a + 70\right)\cdot 139^{7} + \left(58 a^{2} + 45 a + 76\right)\cdot 139^{8} + \left(127 a^{2} + 83 a + 89\right)\cdot 139^{9} +O(139^{10})\) 57a^2 + 138a + 67 + (110a^2 + 128a + 84)139 + (107a^2 + 102a + 42)139^2 + (19a^2 + 70a + 61)139^3 + (114a^2 + 109a + 62)139^4 + (127a^2 + 9a + 124)139^5 + (130a^2 + 14a + 31)139^6 + (103a^2 + 11a + 70)139^7 + (58a^2 + 45a + 76)139^8 + (127a^2 + 83a + 89)*139^9+O(139^10)
$r_{ 3 }$	$=$	\( 58 a^{2} + 136 a + 88 + \left(127 a^{2} + 2 a + 17\right)\cdot 139 + \left(91 a^{2} + 92 a + 138\right)\cdot 139^{2} + \left(5 a^{2} + 80 a + 112\right)\cdot 139^{3} + \left(10 a^{2} + 83 a + 47\right)\cdot 139^{4} + \left(59 a^{2} + 79 a + 34\right)\cdot 139^{5} + \left(80 a^{2} + 136 a + 33\right)\cdot 139^{6} + \left(74 a^{2} + 104 a + 32\right)\cdot 139^{7} + \left(101 a^{2} + 111 a + 54\right)\cdot 139^{8} + \left(72 a^{2} + 86 a + 79\right)\cdot 139^{9} +O(139^{10})\) 58a^2 + 136a + 88 + (127a^2 + 2a + 17)139 + (91a^2 + 92a + 138)139^2 + (5a^2 + 80a + 112)139^3 + (10a^2 + 83a + 47)139^4 + (59a^2 + 79a + 34)139^5 + (80a^2 + 136a + 33)139^6 + (74a^2 + 104a + 32)139^7 + (101a^2 + 111a + 54)139^8 + (72a^2 + 86a + 79)*139^9+O(139^10)
$r_{ 4 }$	$=$	\( 65 a^{2} + 30 a + 9 + \left(41 a^{2} + 25 a + 42\right)\cdot 139 + \left(68 a^{2} + 95 a + 57\right)\cdot 139^{2} + \left(24 a^{2} + 24 a + 25\right)\cdot 139^{3} + \left(55 a^{2} + 27 a + 51\right)\cdot 139^{4} + \left(26 a^{2} + 62 a + 138\right)\cdot 139^{5} + \left(12 a^{2} + 38 a + 133\right)\cdot 139^{6} + \left(113 a^{2} + 136 a + 90\right)\cdot 139^{7} + \left(72 a^{2} + 59 a + 106\right)\cdot 139^{8} + \left(55 a^{2} + 131 a + 19\right)\cdot 139^{9} +O(139^{10})\) 65a^2 + 30a + 9 + (41a^2 + 25a + 42)139 + (68a^2 + 95a + 57)139^2 + (24a^2 + 24a + 25)139^3 + (55a^2 + 27a + 51)139^4 + (26a^2 + 62a + 138)139^5 + (12a^2 + 38a + 133)139^6 + (113a^2 + 136a + 90)139^7 + (72a^2 + 59a + 106)139^8 + (55a^2 + 131a + 19)*139^9+O(139^10)
$r_{ 5 }$	$=$	\( 66 a^{2} + 28 a + 103 + \left(58 a^{2} + 38 a + 15\right)\cdot 139 + \left(52 a^{2} + 84 a + 99\right)\cdot 139^{2} + \left(10 a^{2} + 34 a + 23\right)\cdot 139^{3} + \left(90 a^{2} + a + 105\right)\cdot 139^{4} + \left(96 a^{2} + 132 a + 138\right)\cdot 139^{5} + \left(100 a^{2} + 21 a + 49\right)\cdot 139^{6} + \left(83 a^{2} + 91 a + 128\right)\cdot 139^{7} + \left(115 a^{2} + 126 a + 25\right)\cdot 139^{8} + 134 a\cdot 139^{9} +O(139^{10})\) 66a^2 + 28a + 103 + (58a^2 + 38a + 15)139 + (52a^2 + 84a + 99)139^2 + (10a^2 + 34a + 23)139^3 + (90a^2 + a + 105)139^4 + (96a^2 + 132a + 138)139^5 + (100a^2 + 21a + 49)139^6 + (83a^2 + 91a + 128)139^7 + (115a^2 + 126a + 25)139^8 + 134a*139^9+O(139^10)
$r_{ 6 }$	$=$	\( 82 a^{2} + 16 a + 77 + \left(113 a^{2} + 34 a + 52\right)\cdot 139 + \left(a^{2} + 92 a + 69\right)\cdot 139^{2} + \left(32 a^{2} + 124 a + 55\right)\cdot 139^{3} + \left(119 a^{2} + 3 a + 29\right)\cdot 139^{4} + \left(7 a^{2} + 98 a + 64\right)\cdot 139^{5} + \left(29 a^{2} + 32 a + 62\right)\cdot 139^{6} + \left(31 a^{2} + 97 a + 41\right)\cdot 139^{7} + \left(26 a^{2} + 95 a + 59\right)\cdot 139^{8} + \left(76 a^{2} + 3 a + 102\right)\cdot 139^{9} +O(139^{10})\) 82a^2 + 16a + 77 + (113a^2 + 34a + 52)139 + (a^2 + 92a + 69)139^2 + (32a^2 + 124a + 55)139^3 + (119a^2 + 3a + 29)139^4 + (7a^2 + 98a + 64)139^5 + (29a^2 + 32a + 62)139^6 + (31a^2 + 97a + 41)139^7 + (26a^2 + 95a + 59)139^8 + (76a^2 + 3a + 102)139^9+O(139^10)
$r_{ 7 }$	$=$	\( 90 a^{2} + 47 a + 77 + \left(44 a^{2} + 69 a + 103\right)\cdot 139 + \left(101 a^{2} + 84 a + 36\right)\cdot 139^{2} + \left(36 a^{2} + 78 a + 98\right)\cdot 139^{3} + \left(60 a^{2} + 60 a + 109\right)\cdot 139^{4} + \left(45 a^{2} + 11 a + 118\right)\cdot 139^{5} + \left(49 a^{2} + 57 a + 47\right)\cdot 139^{6} + \left(40 a^{2} + 83 a + 34\right)\cdot 139^{7} + \left(40 a^{2} + 110 a + 87\right)\cdot 139^{8} + \left(4 a^{2} + 51 a + 83\right)\cdot 139^{9} +O(139^{10})\) 90a^2 + 47a + 77 + (44a^2 + 69a + 103)139 + (101a^2 + 84a + 36)139^2 + (36a^2 + 78a + 98)139^3 + (60a^2 + 60a + 109)139^4 + (45a^2 + 11a + 118)139^5 + (49a^2 + 57a + 47)139^6 + (40a^2 + 83a + 34)139^7 + (40a^2 + 110a + 87)139^8 + (4a^2 + 51a + 83)*139^9+O(139^10)
$r_{ 8 }$	$=$	\( 130 a^{2} + 95 a + 98 + \left(105 a^{2} + 66 a + 70\right)\cdot 139 + \left(84 a^{2} + 101 a + 109\right)\cdot 139^{2} + \left(96 a^{2} + 118 a + 59\right)\cdot 139^{3} + \left(68 a^{2} + 133 a + 4\right)\cdot 139^{4} + \left(34 a^{2} + 47 a + 75\right)\cdot 139^{5} + \left(9 a^{2} + 84 a + 26\right)\cdot 139^{6} + \left(24 a^{2} + 89 a + 108\right)\cdot 139^{7} + \left(136 a^{2} + 55 a + 53\right)\cdot 139^{8} + \left(61 a^{2} + 36\right)\cdot 139^{9} +O(139^{10})\) 130a^2 + 95a + 98 + (105a^2 + 66a + 70)139 + (84a^2 + 101a + 109)139^2 + (96a^2 + 118a + 59)139^3 + (68a^2 + 133a + 4)139^4 + (34a^2 + 47a + 75)139^5 + (9a^2 + 84a + 26)139^6 + (24a^2 + 89a + 108)139^7 + (136a^2 + 55a + 53)139^8 + (61a^2 + 36)139^9+O(139^10)
$r_{ 9 }$	$=$	\( 131 a^{2} + 93 a + 134 + \left(122 a^{2} + 79 a + 89\right)\cdot 139 + \left(68 a^{2} + 90 a + 59\right)\cdot 139^{2} + \left(82 a^{2} + 128 a + 118\right)\cdot 139^{3} + \left(103 a^{2} + 107 a + 105\right)\cdot 139^{4} + \left(104 a^{2} + 117 a + 34\right)\cdot 139^{5} + \left(97 a^{2} + 67 a + 59\right)\cdot 139^{6} + \left(133 a^{2} + 44 a + 34\right)\cdot 139^{7} + \left(39 a^{2} + 122 a + 114\right)\cdot 139^{8} + \left(7 a^{2} + 3 a + 104\right)\cdot 139^{9} +O(139^{10})\) 131a^2 + 93a + 134 + (122a^2 + 79a + 89)139 + (68a^2 + 90a + 59)139^2 + (82a^2 + 128a + 118)139^3 + (103a^2 + 107a + 105)139^4 + (104a^2 + 117a + 34)139^5 + (97a^2 + 67a + 59)139^6 + (133a^2 + 44a + 34)139^7 + (39a^2 + 122a + 114)139^8 + (7a^2 + 3a + 104)*139^9+O(139^10)