LMFDB - Galois orbit of Artin representations 8.13254456384.12t213.a

Basic invariants

Dimension:	$8$
Group:	$S_3\wr S_3$
Conductor:	$13254456384$$\medspace = 2^{6} \cdot 3^{6} \cdot 13^{2} \cdot 41^{2} $
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 9.1.784958361408.1
Galois orbit size:	$1$
Smallest permutation container:	12T213
Parity:	even
Projective image:	$S_3\wr S_3$
Projective field:	Galois closure of 9.1.784958361408.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{3} + 6x + 65 $

Roots:

$r_{ 1 }$	$=$	$ 36 a^{2} + 12 a + 61 + \left(22 a^{2} + 18 a + 7\right)\cdot 67 + \left(59 a^{2} + 14 a + 51\right)\cdot 67^{2} + \left(26 a^{2} + 52 a + 23\right)\cdot 67^{3} + \left(3 a^{2} + 32 a + 30\right)\cdot 67^{4} + \left(57 a^{2} + 19 a + 54\right)\cdot 67^{5} + \left(58 a^{2} + 6 a + 14\right)\cdot 67^{6} + \left(61 a^{2} + 4 a + 2\right)\cdot 67^{7} + \left(31 a^{2} + 33 a + 54\right)\cdot 67^{8} + \left(56 a^{2} + 2 a + 16\right)\cdot 67^{9} +O(67^{10})$ 36a^2 + 12a + 61 + (22a^2 + 18a + 7)67 + (59a^2 + 14a + 51)67^2 + (26a^2 + 52a + 23)67^3 + (3a^2 + 32a + 30)67^4 + (57a^2 + 19a + 54)67^5 + (58a^2 + 6a + 14)67^6 + (61a^2 + 4a + 2)67^7 + (31a^2 + 33a + 54)67^8 + (56a^2 + 2a + 16)*67^9+O(67^10)
$r_{ 2 }$	$=$	$ 59 a^{2} + 52 a + 56 + \left(37 a^{2} + 50 a + 51\right)\cdot 67 + \left(6 a^{2} + 13 a + 20\right)\cdot 67^{2} + \left(32 a^{2} + 51 a + 23\right)\cdot 67^{3} + \left(39 a^{2} + 32\right)\cdot 67^{4} + \left(54 a^{2} + 66 a + 36\right)\cdot 67^{5} + \left(50 a^{2} + 20 a + 17\right)\cdot 67^{6} + \left(53 a^{2} + 42 a + 56\right)\cdot 67^{7} + \left(47 a^{2} + a + 2\right)\cdot 67^{8} + \left(52 a^{2} + 19 a + 52\right)\cdot 67^{9} +O(67^{10})$ 59a^2 + 52a + 56 + (37a^2 + 50a + 51)67 + (6a^2 + 13a + 20)67^2 + (32a^2 + 51a + 23)67^3 + (39a^2 + 32)67^4 + (54a^2 + 66a + 36)67^5 + (50a^2 + 20a + 17)67^6 + (53a^2 + 42a + 56)67^7 + (47a^2 + a + 2)67^8 + (52a^2 + 19a + 52)*67^9+O(67^10)
$r_{ 3 }$	$=$	$ 11 a^{2} + 59 a + 65 + \left(22 a^{2} + 33 a + 55\right)\cdot 67 + \left(26 a^{2} + 55 a + 32\right)\cdot 67^{2} + \left(30 a^{2} + 48 a + 16\right)\cdot 67^{3} + \left(47 a^{2} + 31 a + 64\right)\cdot 67^{4} + \left(15 a^{2} + 38 a + 14\right)\cdot 67^{5} + \left(52 a^{2} + 14 a + 23\right)\cdot 67^{6} + \left(44 a^{2} + 38 a + 20\right)\cdot 67^{7} + \left(26 a^{2} + 66 a + 52\right)\cdot 67^{8} + \left(23 a^{2} + 38 a + 1\right)\cdot 67^{9} +O(67^{10})$ 11a^2 + 59a + 65 + (22a^2 + 33a + 55)67 + (26a^2 + 55a + 32)67^2 + (30a^2 + 48a + 16)67^3 + (47a^2 + 31a + 64)67^4 + (15a^2 + 38a + 14)67^5 + (52a^2 + 14a + 23)67^6 + (44a^2 + 38a + 20)67^7 + (26a^2 + 66a + 52)67^8 + (23a^2 + 38a + 1)*67^9+O(67^10)
$r_{ 4 }$	$=$	$ 2 a^{2} + 57 a + 4 + \left(44 a^{2} + 53 a + 23\right)\cdot 67 + \left(31 a^{2} + 59 a + 50\right)\cdot 67^{2} + \left(61 a^{2} + 43 a + 32\right)\cdot 67^{3} + \left(17 a^{2} + 62 a + 46\right)\cdot 67^{4} + \left(21 a^{2} + 3 a + 38\right)\cdot 67^{5} + \left(30 a^{2} + 37 a + 58\right)\cdot 67^{6} + \left(36 a^{2} + 45 a + 13\right)\cdot 67^{7} + \left(16 a^{2} + 50 a + 60\right)\cdot 67^{8} + \left(6 a^{2} + 32 a + 57\right)\cdot 67^{9} +O(67^{10})$ 2a^2 + 57a + 4 + (44a^2 + 53a + 23)67 + (31a^2 + 59a + 50)67^2 + (61a^2 + 43a + 32)67^3 + (17a^2 + 62a + 46)67^4 + (21a^2 + 3a + 38)67^5 + (30a^2 + 37a + 58)67^6 + (36a^2 + 45a + 13)67^7 + (16a^2 + 50a + 60)67^8 + (6a^2 + 32a + 57)*67^9+O(67^10)
$r_{ 5 }$	$=$	$ 33 a^{2} + a + 49 + \left(42 a^{2} + 47 a + 20\right)\cdot 67 + \left(43 a^{2} + 22 a + 55\right)\cdot 67^{2} + \left(34 a^{2} + 52 a + 54\right)\cdot 67^{3} + \left(20 a^{2} + 10 a + 31\right)\cdot 67^{4} + \left(34 a^{2} + 31 a + 30\right)\cdot 67^{5} + \left(64 a^{2} + 21 a + 37\right)\cdot 67^{6} + \left(21 a^{2} + 9 a + 43\right)\cdot 67^{7} + \left(47 a^{2} + 19 a + 48\right)\cdot 67^{8} + \left(12 a^{2} + 33 a + 42\right)\cdot 67^{9} +O(67^{10})$ 33a^2 + a + 49 + (42a^2 + 47a + 20)67 + (43a^2 + 22a + 55)67^2 + (34a^2 + 52a + 54)67^3 + (20a^2 + 10a + 31)67^4 + (34a^2 + 31a + 30)67^5 + (64a^2 + 21a + 37)67^6 + (21a^2 + 9a + 43)67^7 + (47a^2 + 19a + 48)67^8 + (12a^2 + 33a + 42)67^9+O(67^10)
$r_{ 6 }$	$=$	$ 64 a^{2} + 23 a + 9 + \left(6 a^{2} + 49 a + 62\right)\cdot 67 + \left(34 a^{2} + 64 a + 63\right)\cdot 67^{2} + \left(4 a^{2} + 33 a + 46\right)\cdot 67^{3} + \left(47 a^{2} + 34 a + 62\right)\cdot 67^{4} + \left(63 a^{2} + 29 a + 5\right)\cdot 67^{5} + \left(30 a^{2} + 31 a + 5\right)\cdot 67^{6} + \left(35 a^{2} + 53 a + 50\right)\cdot 67^{7} + \left(59 a^{2} + 65 a + 49\right)\cdot 67^{8} + \left(57 a^{2} + 8 a + 5\right)\cdot 67^{9} +O(67^{10})$ 64a^2 + 23a + 9 + (6a^2 + 49a + 62)67 + (34a^2 + 64a + 63)67^2 + (4a^2 + 33a + 46)67^3 + (47a^2 + 34a + 62)67^4 + (63a^2 + 29a + 5)67^5 + (30a^2 + 31a + 5)67^6 + (35a^2 + 53a + 50)67^7 + (59a^2 + 65a + 49)67^8 + (57a^2 + 8a + 5)*67^9+O(67^10)
$r_{ 7 }$	$=$	$ 6 a^{2} + 30 a + 20 + \left(57 a^{2} + 52 a + 8\right)\cdot 67 + \left(21 a^{2} + 6 a + 11\right)\cdot 67^{2} + \left(40 a^{2} + 41 a + 15\right)\cdot 67^{3} + \left(33 a^{2} + 52 a + 42\right)\cdot 67^{4} + \left(25 a^{2} + 4 a + 55\right)\cdot 67^{5} + \left(36 a^{2} + 19 a + 15\right)\cdot 67^{6} + \left(39 a^{2} + 26 a + 26\right)\cdot 67^{7} + \left(23 a^{2} + 32 a + 21\right)\cdot 67^{8} + \left(34 a^{2} + 62 a + 36\right)\cdot 67^{9} +O(67^{10})$ 6a^2 + 30a + 20 + (57a^2 + 52a + 8)67 + (21a^2 + 6a + 11)67^2 + (40a^2 + 41a + 15)67^3 + (33a^2 + 52a + 42)67^4 + (25a^2 + 4a + 55)67^5 + (36a^2 + 19a + 15)67^6 + (39a^2 + 26a + 26)67^7 + (23a^2 + 32a + 21)67^8 + (34a^2 + 62a + 36)*67^9+O(67^10)
$r_{ 8 }$	$=$	$ 59 a^{2} + 47 a + 31 + \left(32 a^{2} + 27 a + 45\right)\cdot 67 + \left(13 a^{2} + 44\right)\cdot 67^{2} + \left(32 a^{2} + 49 a + 49\right)\cdot 67^{3} + \left(15 a^{2} + 18 a + 36\right)\cdot 67^{4} + \left(20 a^{2} + 58 a + 34\right)\cdot 67^{5} + \left(10 a + 5\right)\cdot 67^{6} + \left(58 a^{2} + 62 a + 33\right)\cdot 67^{7} + \left(26 a^{2} + 50 a + 34\right)\cdot 67^{8} + \left(26 a^{2} + 38 a + 4\right)\cdot 67^{9} +O(67^{10})$ 59a^2 + 47a + 31 + (32a^2 + 27a + 45)67 + (13a^2 + 44)67^2 + (32a^2 + 49a + 49)67^3 + (15a^2 + 18a + 36)67^4 + (20a^2 + 58a + 34)67^5 + (10a + 5)67^6 + (58a^2 + 62a + 33)67^7 + (26a^2 + 50a + 34)67^8 + (26a^2 + 38a + 4)*67^9+O(67^10)
$r_{ 9 }$	$=$	$ 65 a^{2} + 54 a + 43 + \left(a^{2} + a + 59\right)\cdot 67 + \left(31 a^{2} + 30 a + 4\right)\cdot 67^{2} + \left(5 a^{2} + 29 a + 5\right)\cdot 67^{3} + \left(43 a^{2} + 23 a + 55\right)\cdot 67^{4} + \left(42 a^{2} + 16 a + 63\right)\cdot 67^{5} + \left(10 a^{2} + 39 a + 22\right)\cdot 67^{6} + \left(50 a^{2} + 53 a + 22\right)\cdot 67^{7} + \left(54 a^{2} + 14 a + 11\right)\cdot 67^{8} + \left(64 a^{2} + 31 a + 50\right)\cdot 67^{9} +O(67^{10})$ 65a^2 + 54a + 43 + (a^2 + a + 59)67 + (31a^2 + 30a + 4)67^2 + (5a^2 + 29a + 5)67^3 + (43a^2 + 23a + 55)67^4 + (42a^2 + 16a + 63)67^5 + (10a^2 + 39a + 22)67^6 + (50a^2 + 53a + 22)67^7 + (54a^2 + 14a + 11)67^8 + (64a^2 + 31a + 50)*67^9+O(67^10)

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

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 9 }$	Character values
			$c1$
$1$	$1$	$()$	$8$
$9$	$2$	$(1,3)$	$0$
$18$	$2$	$(1,6)(3,8)(7,9)$	$4$
$27$	$2$	$(1,3)(2,4)(6,8)$	$0$
$27$	$2$	$(1,3)(6,8)$	$0$
$54$	$2$	$(1,3)(2,6)(4,8)(5,9)$	$0$
$6$	$3$	$(2,4,5)$	$-4$
$8$	$3$	$(1,3,7)(2,4,5)(6,8,9)$	$-1$
$12$	$3$	$(2,4,5)(6,8,9)$	$2$
$72$	$3$	$(1,6,2)(3,8,4)(5,7,9)$	$2$
$54$	$4$	$(1,8,3,6)(7,9)$	$0$
$162$	$4$	$(1,4,3,2)(5,7)(8,9)$	$0$
$36$	$6$	$(1,6)(2,4,5)(3,8)(7,9)$	$-2$
$36$	$6$	$(1,2,3,4,7,5)$	$-2$
$36$	$6$	$(1,3)(2,4,5)$	$0$
$36$	$6$	$(1,3)(2,4,5)(6,8,9)$	$0$
$54$	$6$	$(1,3)(2,5,4)(6,8)$	$0$
$72$	$6$	$(1,6,3,8,7,9)(2,4,5)$	$1$
$108$	$6$	$(1,3)(2,8,4,9,5,6)$	$0$
$216$	$6$	$(1,8,4,3,6,2)(5,7,9)$	$0$
$144$	$9$	$(1,6,2,3,8,4,7,9,5)$	$-1$
$108$	$12$	$(1,8,3,6)(2,4,5)(7,9)$	$0$

The blue line marks the conjugacy class containing complex conjugation.

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Basic invariants

Galois action

Roots of 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	8.13254456384.12t213.a
Dimension	$8$
Group	$S_3\wr S_3$
Conductor	$13254456384$
Indicator	$1$

$r_{ 1 }$	$=$	\( 36 a^{2} + 12 a + 61 + \left(22 a^{2} + 18 a + 7\right)\cdot 67 + \left(59 a^{2} + 14 a + 51\right)\cdot 67^{2} + \left(26 a^{2} + 52 a + 23\right)\cdot 67^{3} + \left(3 a^{2} + 32 a + 30\right)\cdot 67^{4} + \left(57 a^{2} + 19 a + 54\right)\cdot 67^{5} + \left(58 a^{2} + 6 a + 14\right)\cdot 67^{6} + \left(61 a^{2} + 4 a + 2\right)\cdot 67^{7} + \left(31 a^{2} + 33 a + 54\right)\cdot 67^{8} + \left(56 a^{2} + 2 a + 16\right)\cdot 67^{9} +O(67^{10})\) 36a^2 + 12a + 61 + (22a^2 + 18a + 7)67 + (59a^2 + 14a + 51)67^2 + (26a^2 + 52a + 23)67^3 + (3a^2 + 32a + 30)67^4 + (57a^2 + 19a + 54)67^5 + (58a^2 + 6a + 14)67^6 + (61a^2 + 4a + 2)67^7 + (31a^2 + 33a + 54)67^8 + (56a^2 + 2a + 16)*67^9+O(67^10)
$r_{ 2 }$	$=$	\( 59 a^{2} + 52 a + 56 + \left(37 a^{2} + 50 a + 51\right)\cdot 67 + \left(6 a^{2} + 13 a + 20\right)\cdot 67^{2} + \left(32 a^{2} + 51 a + 23\right)\cdot 67^{3} + \left(39 a^{2} + 32\right)\cdot 67^{4} + \left(54 a^{2} + 66 a + 36\right)\cdot 67^{5} + \left(50 a^{2} + 20 a + 17\right)\cdot 67^{6} + \left(53 a^{2} + 42 a + 56\right)\cdot 67^{7} + \left(47 a^{2} + a + 2\right)\cdot 67^{8} + \left(52 a^{2} + 19 a + 52\right)\cdot 67^{9} +O(67^{10})\) 59a^2 + 52a + 56 + (37a^2 + 50a + 51)67 + (6a^2 + 13a + 20)67^2 + (32a^2 + 51a + 23)67^3 + (39a^2 + 32)67^4 + (54a^2 + 66a + 36)67^5 + (50a^2 + 20a + 17)67^6 + (53a^2 + 42a + 56)67^7 + (47a^2 + a + 2)67^8 + (52a^2 + 19a + 52)*67^9+O(67^10)
$r_{ 3 }$	$=$	\( 11 a^{2} + 59 a + 65 + \left(22 a^{2} + 33 a + 55\right)\cdot 67 + \left(26 a^{2} + 55 a + 32\right)\cdot 67^{2} + \left(30 a^{2} + 48 a + 16\right)\cdot 67^{3} + \left(47 a^{2} + 31 a + 64\right)\cdot 67^{4} + \left(15 a^{2} + 38 a + 14\right)\cdot 67^{5} + \left(52 a^{2} + 14 a + 23\right)\cdot 67^{6} + \left(44 a^{2} + 38 a + 20\right)\cdot 67^{7} + \left(26 a^{2} + 66 a + 52\right)\cdot 67^{8} + \left(23 a^{2} + 38 a + 1\right)\cdot 67^{9} +O(67^{10})\) 11a^2 + 59a + 65 + (22a^2 + 33a + 55)67 + (26a^2 + 55a + 32)67^2 + (30a^2 + 48a + 16)67^3 + (47a^2 + 31a + 64)67^4 + (15a^2 + 38a + 14)67^5 + (52a^2 + 14a + 23)67^6 + (44a^2 + 38a + 20)67^7 + (26a^2 + 66a + 52)67^8 + (23a^2 + 38a + 1)*67^9+O(67^10)
$r_{ 4 }$	$=$	\( 2 a^{2} + 57 a + 4 + \left(44 a^{2} + 53 a + 23\right)\cdot 67 + \left(31 a^{2} + 59 a + 50\right)\cdot 67^{2} + \left(61 a^{2} + 43 a + 32\right)\cdot 67^{3} + \left(17 a^{2} + 62 a + 46\right)\cdot 67^{4} + \left(21 a^{2} + 3 a + 38\right)\cdot 67^{5} + \left(30 a^{2} + 37 a + 58\right)\cdot 67^{6} + \left(36 a^{2} + 45 a + 13\right)\cdot 67^{7} + \left(16 a^{2} + 50 a + 60\right)\cdot 67^{8} + \left(6 a^{2} + 32 a + 57\right)\cdot 67^{9} +O(67^{10})\) 2a^2 + 57a + 4 + (44a^2 + 53a + 23)67 + (31a^2 + 59a + 50)67^2 + (61a^2 + 43a + 32)67^3 + (17a^2 + 62a + 46)67^4 + (21a^2 + 3a + 38)67^5 + (30a^2 + 37a + 58)67^6 + (36a^2 + 45a + 13)67^7 + (16a^2 + 50a + 60)67^8 + (6a^2 + 32a + 57)*67^9+O(67^10)
$r_{ 5 }$	$=$	\( 33 a^{2} + a + 49 + \left(42 a^{2} + 47 a + 20\right)\cdot 67 + \left(43 a^{2} + 22 a + 55\right)\cdot 67^{2} + \left(34 a^{2} + 52 a + 54\right)\cdot 67^{3} + \left(20 a^{2} + 10 a + 31\right)\cdot 67^{4} + \left(34 a^{2} + 31 a + 30\right)\cdot 67^{5} + \left(64 a^{2} + 21 a + 37\right)\cdot 67^{6} + \left(21 a^{2} + 9 a + 43\right)\cdot 67^{7} + \left(47 a^{2} + 19 a + 48\right)\cdot 67^{8} + \left(12 a^{2} + 33 a + 42\right)\cdot 67^{9} +O(67^{10})\) 33a^2 + a + 49 + (42a^2 + 47a + 20)67 + (43a^2 + 22a + 55)67^2 + (34a^2 + 52a + 54)67^3 + (20a^2 + 10a + 31)67^4 + (34a^2 + 31a + 30)67^5 + (64a^2 + 21a + 37)67^6 + (21a^2 + 9a + 43)67^7 + (47a^2 + 19a + 48)67^8 + (12a^2 + 33a + 42)67^9+O(67^10)
$r_{ 6 }$	$=$	\( 64 a^{2} + 23 a + 9 + \left(6 a^{2} + 49 a + 62\right)\cdot 67 + \left(34 a^{2} + 64 a + 63\right)\cdot 67^{2} + \left(4 a^{2} + 33 a + 46\right)\cdot 67^{3} + \left(47 a^{2} + 34 a + 62\right)\cdot 67^{4} + \left(63 a^{2} + 29 a + 5\right)\cdot 67^{5} + \left(30 a^{2} + 31 a + 5\right)\cdot 67^{6} + \left(35 a^{2} + 53 a + 50\right)\cdot 67^{7} + \left(59 a^{2} + 65 a + 49\right)\cdot 67^{8} + \left(57 a^{2} + 8 a + 5\right)\cdot 67^{9} +O(67^{10})\) 64a^2 + 23a + 9 + (6a^2 + 49a + 62)67 + (34a^2 + 64a + 63)67^2 + (4a^2 + 33a + 46)67^3 + (47a^2 + 34a + 62)67^4 + (63a^2 + 29a + 5)67^5 + (30a^2 + 31a + 5)67^6 + (35a^2 + 53a + 50)67^7 + (59a^2 + 65a + 49)67^8 + (57a^2 + 8a + 5)*67^9+O(67^10)
$r_{ 7 }$	$=$	\( 6 a^{2} + 30 a + 20 + \left(57 a^{2} + 52 a + 8\right)\cdot 67 + \left(21 a^{2} + 6 a + 11\right)\cdot 67^{2} + \left(40 a^{2} + 41 a + 15\right)\cdot 67^{3} + \left(33 a^{2} + 52 a + 42\right)\cdot 67^{4} + \left(25 a^{2} + 4 a + 55\right)\cdot 67^{5} + \left(36 a^{2} + 19 a + 15\right)\cdot 67^{6} + \left(39 a^{2} + 26 a + 26\right)\cdot 67^{7} + \left(23 a^{2} + 32 a + 21\right)\cdot 67^{8} + \left(34 a^{2} + 62 a + 36\right)\cdot 67^{9} +O(67^{10})\) 6a^2 + 30a + 20 + (57a^2 + 52a + 8)67 + (21a^2 + 6a + 11)67^2 + (40a^2 + 41a + 15)67^3 + (33a^2 + 52a + 42)67^4 + (25a^2 + 4a + 55)67^5 + (36a^2 + 19a + 15)67^6 + (39a^2 + 26a + 26)67^7 + (23a^2 + 32a + 21)67^8 + (34a^2 + 62a + 36)*67^9+O(67^10)
$r_{ 8 }$	$=$	\( 59 a^{2} + 47 a + 31 + \left(32 a^{2} + 27 a + 45\right)\cdot 67 + \left(13 a^{2} + 44\right)\cdot 67^{2} + \left(32 a^{2} + 49 a + 49\right)\cdot 67^{3} + \left(15 a^{2} + 18 a + 36\right)\cdot 67^{4} + \left(20 a^{2} + 58 a + 34\right)\cdot 67^{5} + \left(10 a + 5\right)\cdot 67^{6} + \left(58 a^{2} + 62 a + 33\right)\cdot 67^{7} + \left(26 a^{2} + 50 a + 34\right)\cdot 67^{8} + \left(26 a^{2} + 38 a + 4\right)\cdot 67^{9} +O(67^{10})\) 59a^2 + 47a + 31 + (32a^2 + 27a + 45)67 + (13a^2 + 44)67^2 + (32a^2 + 49a + 49)67^3 + (15a^2 + 18a + 36)67^4 + (20a^2 + 58a + 34)67^5 + (10a + 5)67^6 + (58a^2 + 62a + 33)67^7 + (26a^2 + 50a + 34)67^8 + (26a^2 + 38a + 4)*67^9+O(67^10)
$r_{ 9 }$	$=$	\( 65 a^{2} + 54 a + 43 + \left(a^{2} + a + 59\right)\cdot 67 + \left(31 a^{2} + 30 a + 4\right)\cdot 67^{2} + \left(5 a^{2} + 29 a + 5\right)\cdot 67^{3} + \left(43 a^{2} + 23 a + 55\right)\cdot 67^{4} + \left(42 a^{2} + 16 a + 63\right)\cdot 67^{5} + \left(10 a^{2} + 39 a + 22\right)\cdot 67^{6} + \left(50 a^{2} + 53 a + 22\right)\cdot 67^{7} + \left(54 a^{2} + 14 a + 11\right)\cdot 67^{8} + \left(64 a^{2} + 31 a + 50\right)\cdot 67^{9} +O(67^{10})\) 65a^2 + 54a + 43 + (a^2 + a + 59)67 + (31a^2 + 30a + 4)67^2 + (5a^2 + 29a + 5)67^3 + (43a^2 + 23a + 55)67^4 + (42a^2 + 16a + 63)67^5 + (10a^2 + 39a + 22)67^6 + (50a^2 + 53a + 22)67^7 + (54a^2 + 14a + 11)67^8 + (64a^2 + 31a + 50)*67^9+O(67^10)