LMFDB - Artin representation 1.116.14t1.b.f

Basic invariants

Dimension:	$1$
Group:	$C_{14}$
Conductor:	$116$$\medspace = 2^{2} \cdot 29 $
Artin field:	Galois closure of 14.0.5796901408038404767744.1
Galois orbit size:	$6$
Smallest permutation container:	$C_{14}$
Parity:	odd
Dirichlet character:	$\chi_{116}(111,\cdot)$
Projective image:	$C_1$
Projective field:	Galois closure of $\Q$

Defining polynomial

$f(x)$

$=$

$ x^{14} + 25x^{12} + 214x^{10} + 767x^{8} + 1194x^{6} + 686x^{4} + 53x^{2} + 1 $

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{7} + 7x + 35 $ x^7 + 7*x + 35

Roots:

$r_{ 1 }$	$=$	$ 9 a^{6} + 22 a^{4} + 11 a^{3} + 18 a^{2} + 5 a + 32 + \left(15 a^{6} + 24 a^{5} + 36 a^{4} + 8 a^{3} + 7 a^{2} + 27 a + 32\right)\cdot 37 + \left(9 a^{6} + 17 a^{5} + 22 a^{4} + 34 a^{3} + 25 a^{2} + 31 a + 9\right)\cdot 37^{2} + \left(13 a^{6} + 22 a^{5} + 17 a^{4} + 36 a^{3} + 20 a^{2} + 3\right)\cdot 37^{3} + \left(10 a^{6} + 29 a^{5} + 26 a^{4} + 4 a^{3} + a^{2} + 23 a + 14\right)\cdot 37^{4} + \left(34 a^{6} + 4 a^{5} + 34 a^{4} + 5 a^{3} + 36 a^{2} + 21 a + 6\right)\cdot 37^{5} + \left(10 a^{6} + 2 a^{5} + 18 a^{4} + a^{3} + 32 a^{2} + 25 a + 27\right)\cdot 37^{6} + \left(3 a^{6} + 26 a^{5} + 17 a^{4} + 5 a^{3} + 30 a^{2} + 28 a + 10\right)\cdot 37^{7} +O(37^{8})$ 9a^6 + 22a^4 + 11a^3 + 18a^2 + 5a + 32 + (15a^6 + 24a^5 + 36a^4 + 8a^3 + 7a^2 + 27a + 32)37 + (9a^6 + 17a^5 + 22a^4 + 34a^3 + 25a^2 + 31a + 9)37^2 + (13a^6 + 22a^5 + 17a^4 + 36a^3 + 20a^2 + 3)37^3 + (10a^6 + 29a^5 + 26a^4 + 4a^3 + a^2 + 23a + 14)37^4 + (34a^6 + 4a^5 + 34a^4 + 5a^3 + 36a^2 + 21a + 6)37^5 + (10a^6 + 2a^5 + 18a^4 + a^3 + 32a^2 + 25a + 27)37^6 + (3a^6 + 26a^5 + 17a^4 + 5a^3 + 30a^2 + 28a + 10)*37^7+O(37^8)
$r_{ 2 }$	$=$	$ 10 a^{6} + 5 a^{5} + 11 a^{4} + 7 a^{3} + 29 a^{2} + 5 a + 8 + \left(11 a^{6} + 5 a^{5} + 20 a^{4} + 17 a^{3} + 20 a^{2} + 7 a + 15\right)\cdot 37 + \left(24 a^{6} + 20 a^{5} + 26 a^{4} + 13 a^{3} + 22 a^{2} + 4 a + 7\right)\cdot 37^{2} + \left(2 a^{6} + 17 a^{5} + 15 a^{4} + 21 a^{3} + 21 a^{2} + 4 a + 18\right)\cdot 37^{3} + \left(3 a^{6} + 36 a^{5} + 18 a^{4} + 5 a^{3} + 8 a^{2} + 19 a + 29\right)\cdot 37^{4} + \left(27 a^{6} + a^{5} + 7 a^{4} + 28 a^{3} + 14 a^{2} + 8 a + 28\right)\cdot 37^{5} + \left(34 a^{5} + 12 a^{4} + 19 a^{3} + 8 a^{2} + 21 a + 5\right)\cdot 37^{6} + \left(23 a^{6} + 10 a^{5} + 6 a^{4} + 3 a^{3} + 15 a^{2} + 31 a + 36\right)\cdot 37^{7} +O(37^{8})$ 10a^6 + 5a^5 + 11a^4 + 7a^3 + 29a^2 + 5a + 8 + (11a^6 + 5a^5 + 20a^4 + 17a^3 + 20a^2 + 7a + 15)37 + (24a^6 + 20a^5 + 26a^4 + 13a^3 + 22a^2 + 4a + 7)37^2 + (2a^6 + 17a^5 + 15a^4 + 21a^3 + 21a^2 + 4a + 18)37^3 + (3a^6 + 36a^5 + 18a^4 + 5a^3 + 8a^2 + 19a + 29)37^4 + (27a^6 + a^5 + 7a^4 + 28a^3 + 14a^2 + 8a + 28)37^5 + (34a^5 + 12a^4 + 19a^3 + 8a^2 + 21a + 5)37^6 + (23a^6 + 10a^5 + 6a^4 + 3a^3 + 15a^2 + 31a + 36)*37^7+O(37^8)
$r_{ 3 }$	$=$	$ 11 a^{6} + 18 a^{5} + 21 a^{4} + 20 a^{3} + 2 a^{2} + 19 a + 14 + \left(33 a^{6} + 5 a^{5} + 28 a^{4} + 21 a^{3} + 2 a^{2} + 10 a + 36\right)\cdot 37 + \left(22 a^{6} + 31 a^{5} + 15 a^{4} + 21 a^{3} + 18 a^{2} + 18 a + 35\right)\cdot 37^{2} + \left(29 a^{6} + 34 a^{5} + 22 a^{4} + 33 a^{3} + 36 a^{2} + 31\right)\cdot 37^{3} + \left(15 a^{6} + 25 a^{5} + 2 a^{4} + 11 a^{3} + 17 a^{2} + 30 a + 31\right)\cdot 37^{4} + \left(22 a^{6} + 2 a^{5} + 23 a^{4} + 24 a^{3} + 24 a^{2} + 27 a\right)\cdot 37^{5} + \left(16 a^{6} + 13 a^{5} + 28 a^{4} + 36 a^{3} + 22 a^{2} + 35 a + 27\right)\cdot 37^{6} + \left(22 a^{6} + 7 a^{5} + 22 a^{4} + 32 a^{3} + 12 a^{2} + 4 a + 32\right)\cdot 37^{7} +O(37^{8})$ 11a^6 + 18a^5 + 21a^4 + 20a^3 + 2a^2 + 19a + 14 + (33a^6 + 5a^5 + 28a^4 + 21a^3 + 2a^2 + 10a + 36)37 + (22a^6 + 31a^5 + 15a^4 + 21a^3 + 18a^2 + 18a + 35)37^2 + (29a^6 + 34a^5 + 22a^4 + 33a^3 + 36a^2 + 31)37^3 + (15a^6 + 25a^5 + 2a^4 + 11a^3 + 17a^2 + 30a + 31)37^4 + (22a^6 + 2a^5 + 23a^4 + 24a^3 + 24a^2 + 27a)37^5 + (16a^6 + 13a^5 + 28a^4 + 36a^3 + 22a^2 + 35a + 27)37^6 + (22a^6 + 7a^5 + 22a^4 + 32a^3 + 12a^2 + 4a + 32)37^7+O(37^8)
$r_{ 4 }$	$=$	$ 11 a^{6} + 19 a^{5} + 15 a^{4} + 11 a^{3} + 29 a^{2} + 29 a + 14 + \left(23 a^{6} + 7 a^{5} + 11 a^{4} + a^{3} + 10 a^{2} + 20 a + 13\right)\cdot 37 + \left(18 a^{6} + 12 a^{5} + 27 a^{4} + 6 a^{3} + 29 a^{2} + a + 10\right)\cdot 37^{2} + \left(3 a^{6} + 11 a^{5} + a^{4} + 26 a^{3} + 7 a^{2} + 12 a + 23\right)\cdot 37^{3} + \left(19 a^{5} + 21 a^{4} + 23 a^{3} + 27 a^{2} + 8 a + 11\right)\cdot 37^{4} + \left(18 a^{6} + 24 a^{5} + 17 a^{4} + 11 a^{3} + 29 a^{2} + 32 a + 11\right)\cdot 37^{5} + \left(11 a^{6} + 20 a^{5} + 31 a^{4} + 14 a^{3} + 24 a^{2} + 36 a + 33\right)\cdot 37^{6} + \left(20 a^{6} + 20 a^{5} + 23 a^{4} + 14 a^{3} + 4 a^{2} + 27 a + 19\right)\cdot 37^{7} +O(37^{8})$ 11a^6 + 19a^5 + 15a^4 + 11a^3 + 29a^2 + 29a + 14 + (23a^6 + 7a^5 + 11a^4 + a^3 + 10a^2 + 20a + 13)37 + (18a^6 + 12a^5 + 27a^4 + 6a^3 + 29a^2 + a + 10)37^2 + (3a^6 + 11a^5 + a^4 + 26a^3 + 7a^2 + 12a + 23)37^3 + (19a^5 + 21a^4 + 23a^3 + 27a^2 + 8a + 11)37^4 + (18a^6 + 24a^5 + 17a^4 + 11a^3 + 29a^2 + 32a + 11)37^5 + (11a^6 + 20a^5 + 31a^4 + 14a^3 + 24a^2 + 36a + 33)37^6 + (20a^6 + 20a^5 + 23a^4 + 14a^3 + 4a^2 + 27a + 19)*37^7+O(37^8)
$r_{ 5 }$	$=$	$ 13 a^{6} + 20 a^{5} + 29 a^{4} + 10 a^{3} + 23 a^{2} + 35 a + 26 + \left(20 a^{6} + 20 a^{5} + 5 a^{4} + 3 a^{3} + 2 a^{2} + 11 a + 32\right)\cdot 37 + \left(9 a^{6} + 26 a^{5} + 16 a^{4} + a^{3} + 5 a^{2} + 27 a + 29\right)\cdot 37^{2} + \left(34 a^{6} + 12 a^{5} + 4 a^{4} + 22 a^{3} + 32 a + 22\right)\cdot 37^{3} + \left(26 a^{6} + 36 a^{5} + 31 a^{4} + 26 a^{3} + 14 a^{2} + 15 a + 24\right)\cdot 37^{4} + \left(18 a^{6} + 30 a^{5} + 5 a^{4} + 29 a^{3} + 14 a^{2} + 4 a + 15\right)\cdot 37^{5} + \left(24 a^{6} + 3 a^{5} + 29 a^{4} + 13 a^{3} + 35 a^{2} + 27 a\right)\cdot 37^{6} + \left(26 a^{6} + 13 a^{5} + 18 a^{4} + 16 a^{3} + 3 a^{2} + 18 a + 21\right)\cdot 37^{7} +O(37^{8})$ 13a^6 + 20a^5 + 29a^4 + 10a^3 + 23a^2 + 35a + 26 + (20a^6 + 20a^5 + 5a^4 + 3a^3 + 2a^2 + 11a + 32)37 + (9a^6 + 26a^5 + 16a^4 + a^3 + 5a^2 + 27a + 29)37^2 + (34a^6 + 12a^5 + 4a^4 + 22a^3 + 32a + 22)37^3 + (26a^6 + 36a^5 + 31a^4 + 26a^3 + 14a^2 + 15a + 24)37^4 + (18a^6 + 30a^5 + 5a^4 + 29a^3 + 14a^2 + 4a + 15)37^5 + (24a^6 + 3a^5 + 29a^4 + 13a^3 + 35a^2 + 27a)37^6 + (26a^6 + 13a^5 + 18a^4 + 16a^3 + 3a^2 + 18a + 21)*37^7+O(37^8)
$r_{ 6 }$	$=$	$ 17 a^{6} + 19 a^{5} + 4 a^{4} + a^{3} + 15 a^{2} + 32 a + 6 + \left(33 a^{6} + 4 a^{5} + 21 a^{4} + 3 a^{3} + 20 a^{2} + 2 a + 31\right)\cdot 37 + \left(23 a^{6} + 7 a^{5} + 23 a^{4} + 29 a^{3} + 16 a^{2} + 35 a + 22\right)\cdot 37^{2} + \left(31 a^{6} + 18 a^{5} + 15 a^{4} + 20 a^{3} + 16 a^{2} + 10 a + 2\right)\cdot 37^{3} + \left(6 a^{6} + 4 a^{5} + 14 a^{4} + 14 a^{3} + 13 a^{2} + 14 a + 30\right)\cdot 37^{4} + \left(13 a^{6} + 17 a^{5} + 12 a^{4} + 5 a^{3} + 36 a^{2} + 6 a + 27\right)\cdot 37^{5} + \left(22 a^{6} + 18 a^{5} + 36 a^{4} + 16 a^{3} + 10 a^{2} + 9 a + 21\right)\cdot 37^{6} + \left(14 a^{6} + 26 a^{5} + 25 a^{4} + 17 a^{3} + 28 a^{2} + 25 a + 4\right)\cdot 37^{7} +O(37^{8})$ 17a^6 + 19a^5 + 4a^4 + a^3 + 15a^2 + 32a + 6 + (33a^6 + 4a^5 + 21a^4 + 3a^3 + 20a^2 + 2a + 31)37 + (23a^6 + 7a^5 + 23a^4 + 29a^3 + 16a^2 + 35a + 22)37^2 + (31a^6 + 18a^5 + 15a^4 + 20a^3 + 16a^2 + 10a + 2)37^3 + (6a^6 + 4a^5 + 14a^4 + 14a^3 + 13a^2 + 14a + 30)37^4 + (13a^6 + 17a^5 + 12a^4 + 5a^3 + 36a^2 + 6a + 27)37^5 + (22a^6 + 18a^5 + 36a^4 + 16a^3 + 10a^2 + 9a + 21)37^6 + (14a^6 + 26a^5 + 25a^4 + 17a^3 + 28a^2 + 25a + 4)37^7+O(37^8)
$r_{ 7 }$	$=$	$ 18 a^{6} + 31 a^{5} + 24 a^{4} + a^{3} + 24 a^{2} + 23 a + 19 + \left(34 a^{6} + 26 a^{5} + 28 a^{4} + 5 a^{3} + 28 a^{2} + 16 a + 6\right)\cdot 37 + \left(31 a^{6} + 8 a^{5} + 34 a^{4} + 21 a^{3} + 3 a^{2} + 15 a + 16\right)\cdot 37^{2} + \left(11 a^{6} + a^{5} + 25 a^{4} + 28 a^{3} + 8 a^{2} + 36 a + 36\right)\cdot 37^{3} + \left(8 a^{6} + 27 a^{5} + 4 a^{4} + 25 a^{3} + 21 a^{2} + 23\right)\cdot 37^{4} + \left(35 a^{6} + 35 a^{5} + 30 a^{4} + 27 a^{3} + 26 a^{2} + 29 a + 3\right)\cdot 37^{5} + \left(16 a^{6} + 22 a^{5} + 27 a^{4} + 6 a^{3} + 26 a^{2} + 24 a + 29\right)\cdot 37^{6} + \left(36 a^{6} + 8 a^{4} + 29 a^{3} + 22 a^{2} + 7 a + 5\right)\cdot 37^{7} +O(37^{8})$ 18a^6 + 31a^5 + 24a^4 + a^3 + 24a^2 + 23a + 19 + (34a^6 + 26a^5 + 28a^4 + 5a^3 + 28a^2 + 16a + 6)37 + (31a^6 + 8a^5 + 34a^4 + 21a^3 + 3a^2 + 15a + 16)37^2 + (11a^6 + a^5 + 25a^4 + 28a^3 + 8a^2 + 36a + 36)37^3 + (8a^6 + 27a^5 + 4a^4 + 25a^3 + 21a^2 + 23)37^4 + (35a^6 + 35a^5 + 30a^4 + 27a^3 + 26a^2 + 29a + 3)37^5 + (16a^6 + 22a^5 + 27a^4 + 6a^3 + 26a^2 + 24a + 29)37^6 + (36a^6 + 8a^4 + 29a^3 + 22a^2 + 7a + 5)*37^7+O(37^8)
$r_{ 8 }$	$=$	$ 19 a^{6} + 6 a^{5} + 13 a^{4} + 36 a^{3} + 13 a^{2} + 14 a + 18 + \left(2 a^{6} + 10 a^{5} + 8 a^{4} + 31 a^{3} + 8 a^{2} + 20 a + 30\right)\cdot 37 + \left(5 a^{6} + 28 a^{5} + 2 a^{4} + 15 a^{3} + 33 a^{2} + 21 a + 20\right)\cdot 37^{2} + \left(25 a^{6} + 35 a^{5} + 11 a^{4} + 8 a^{3} + 28 a^{2}\right)\cdot 37^{3} + \left(28 a^{6} + 9 a^{5} + 32 a^{4} + 11 a^{3} + 15 a^{2} + 36 a + 13\right)\cdot 37^{4} + \left(a^{6} + a^{5} + 6 a^{4} + 9 a^{3} + 10 a^{2} + 7 a + 33\right)\cdot 37^{5} + \left(20 a^{6} + 14 a^{5} + 9 a^{4} + 30 a^{3} + 10 a^{2} + 12 a + 7\right)\cdot 37^{6} + \left(36 a^{5} + 28 a^{4} + 7 a^{3} + 14 a^{2} + 29 a + 31\right)\cdot 37^{7} +O(37^{8})$ 19a^6 + 6a^5 + 13a^4 + 36a^3 + 13a^2 + 14a + 18 + (2a^6 + 10a^5 + 8a^4 + 31a^3 + 8a^2 + 20a + 30)37 + (5a^6 + 28a^5 + 2a^4 + 15a^3 + 33a^2 + 21a + 20)37^2 + (25a^6 + 35a^5 + 11a^4 + 8a^3 + 28a^2)37^3 + (28a^6 + 9a^5 + 32a^4 + 11a^3 + 15a^2 + 36a + 13)37^4 + (a^6 + a^5 + 6a^4 + 9a^3 + 10a^2 + 7a + 33)37^5 + (20a^6 + 14a^5 + 9a^4 + 30a^3 + 10a^2 + 12a + 7)37^6 + (36a^5 + 28a^4 + 7a^3 + 14a^2 + 29a + 31)*37^7+O(37^8)
$r_{ 9 }$	$=$	$ 20 a^{6} + 18 a^{5} + 33 a^{4} + 36 a^{3} + 22 a^{2} + 5 a + 31 + \left(3 a^{6} + 32 a^{5} + 15 a^{4} + 33 a^{3} + 16 a^{2} + 34 a + 5\right)\cdot 37 + \left(13 a^{6} + 29 a^{5} + 13 a^{4} + 7 a^{3} + 20 a^{2} + a + 14\right)\cdot 37^{2} + \left(5 a^{6} + 18 a^{5} + 21 a^{4} + 16 a^{3} + 20 a^{2} + 26 a + 34\right)\cdot 37^{3} + \left(30 a^{6} + 32 a^{5} + 22 a^{4} + 22 a^{3} + 23 a^{2} + 22 a + 6\right)\cdot 37^{4} + \left(23 a^{6} + 19 a^{5} + 24 a^{4} + 31 a^{3} + 30 a + 9\right)\cdot 37^{5} + \left(14 a^{6} + 18 a^{5} + 20 a^{3} + 26 a^{2} + 27 a + 15\right)\cdot 37^{6} + \left(22 a^{6} + 10 a^{5} + 11 a^{4} + 19 a^{3} + 8 a^{2} + 11 a + 32\right)\cdot 37^{7} +O(37^{8})$ 20a^6 + 18a^5 + 33a^4 + 36a^3 + 22a^2 + 5a + 31 + (3a^6 + 32a^5 + 15a^4 + 33a^3 + 16a^2 + 34a + 5)37 + (13a^6 + 29a^5 + 13a^4 + 7a^3 + 20a^2 + a + 14)37^2 + (5a^6 + 18a^5 + 21a^4 + 16a^3 + 20a^2 + 26a + 34)37^3 + (30a^6 + 32a^5 + 22a^4 + 22a^3 + 23a^2 + 22a + 6)37^4 + (23a^6 + 19a^5 + 24a^4 + 31a^3 + 30a + 9)37^5 + (14a^6 + 18a^5 + 20a^3 + 26a^2 + 27a + 15)37^6 + (22a^6 + 10a^5 + 11a^4 + 19a^3 + 8a^2 + 11a + 32)37^7+O(37^8)
$r_{ 10 }$	$=$	$ 24 a^{6} + 17 a^{5} + 8 a^{4} + 27 a^{3} + 14 a^{2} + 2 a + 11 + \left(16 a^{6} + 16 a^{5} + 31 a^{4} + 33 a^{3} + 34 a^{2} + 25 a + 4\right)\cdot 37 + \left(27 a^{6} + 10 a^{5} + 20 a^{4} + 35 a^{3} + 31 a^{2} + 9 a + 7\right)\cdot 37^{2} + \left(2 a^{6} + 24 a^{5} + 32 a^{4} + 14 a^{3} + 36 a^{2} + 4 a + 14\right)\cdot 37^{3} + \left(10 a^{6} + 5 a^{4} + 10 a^{3} + 22 a^{2} + 21 a + 12\right)\cdot 37^{4} + \left(18 a^{6} + 6 a^{5} + 31 a^{4} + 7 a^{3} + 22 a^{2} + 32 a + 21\right)\cdot 37^{5} + \left(12 a^{6} + 33 a^{5} + 7 a^{4} + 23 a^{3} + a^{2} + 9 a + 36\right)\cdot 37^{6} + \left(10 a^{6} + 23 a^{5} + 18 a^{4} + 20 a^{3} + 33 a^{2} + 18 a + 15\right)\cdot 37^{7} +O(37^{8})$ 24a^6 + 17a^5 + 8a^4 + 27a^3 + 14a^2 + 2a + 11 + (16a^6 + 16a^5 + 31a^4 + 33a^3 + 34a^2 + 25a + 4)37 + (27a^6 + 10a^5 + 20a^4 + 35a^3 + 31a^2 + 9a + 7)37^2 + (2a^6 + 24a^5 + 32a^4 + 14a^3 + 36a^2 + 4a + 14)37^3 + (10a^6 + 5a^4 + 10a^3 + 22a^2 + 21a + 12)37^4 + (18a^6 + 6a^5 + 31a^4 + 7a^3 + 22a^2 + 32a + 21)37^5 + (12a^6 + 33a^5 + 7a^4 + 23a^3 + a^2 + 9a + 36)37^6 + (10a^6 + 23a^5 + 18a^4 + 20a^3 + 33a^2 + 18a + 15)*37^7+O(37^8)
$r_{ 11 }$	$=$	$ 26 a^{6} + 18 a^{5} + 22 a^{4} + 26 a^{3} + 8 a^{2} + 8 a + 23 + \left(13 a^{6} + 29 a^{5} + 25 a^{4} + 35 a^{3} + 26 a^{2} + 16 a + 23\right)\cdot 37 + \left(18 a^{6} + 24 a^{5} + 9 a^{4} + 30 a^{3} + 7 a^{2} + 35 a + 26\right)\cdot 37^{2} + \left(33 a^{6} + 25 a^{5} + 35 a^{4} + 10 a^{3} + 29 a^{2} + 24 a + 13\right)\cdot 37^{3} + \left(36 a^{6} + 17 a^{5} + 15 a^{4} + 13 a^{3} + 9 a^{2} + 28 a + 25\right)\cdot 37^{4} + \left(18 a^{6} + 12 a^{5} + 19 a^{4} + 25 a^{3} + 7 a^{2} + 4 a + 25\right)\cdot 37^{5} + \left(25 a^{6} + 16 a^{5} + 5 a^{4} + 22 a^{3} + 12 a^{2} + 3\right)\cdot 37^{6} + \left(16 a^{6} + 16 a^{5} + 13 a^{4} + 22 a^{3} + 32 a^{2} + 9 a + 17\right)\cdot 37^{7} +O(37^{8})$ 26a^6 + 18a^5 + 22a^4 + 26a^3 + 8a^2 + 8a + 23 + (13a^6 + 29a^5 + 25a^4 + 35a^3 + 26a^2 + 16a + 23)37 + (18a^6 + 24a^5 + 9a^4 + 30a^3 + 7a^2 + 35a + 26)37^2 + (33a^6 + 25a^5 + 35a^4 + 10a^3 + 29a^2 + 24a + 13)37^3 + (36a^6 + 17a^5 + 15a^4 + 13a^3 + 9a^2 + 28a + 25)37^4 + (18a^6 + 12a^5 + 19a^4 + 25a^3 + 7a^2 + 4a + 25)37^5 + (25a^6 + 16a^5 + 5a^4 + 22a^3 + 12a^2 + 3)37^6 + (16a^6 + 16a^5 + 13a^4 + 22a^3 + 32a^2 + 9a + 17)37^7+O(37^8)
$r_{ 12 }$	$=$	$ 26 a^{6} + 19 a^{5} + 16 a^{4} + 17 a^{3} + 35 a^{2} + 18 a + 23 + \left(3 a^{6} + 31 a^{5} + 8 a^{4} + 15 a^{3} + 34 a^{2} + 26 a\right)\cdot 37 + \left(14 a^{6} + 5 a^{5} + 21 a^{4} + 15 a^{3} + 18 a^{2} + 18 a + 1\right)\cdot 37^{2} + \left(7 a^{6} + 2 a^{5} + 14 a^{4} + 3 a^{3} + 36 a + 5\right)\cdot 37^{3} + \left(21 a^{6} + 11 a^{5} + 34 a^{4} + 25 a^{3} + 19 a^{2} + 6 a + 5\right)\cdot 37^{4} + \left(14 a^{6} + 34 a^{5} + 13 a^{4} + 12 a^{3} + 12 a^{2} + 9 a + 36\right)\cdot 37^{5} + \left(20 a^{6} + 23 a^{5} + 8 a^{4} + 14 a^{2} + a + 9\right)\cdot 37^{6} + \left(14 a^{6} + 29 a^{5} + 14 a^{4} + 4 a^{3} + 24 a^{2} + 32 a + 4\right)\cdot 37^{7} +O(37^{8})$ 26a^6 + 19a^5 + 16a^4 + 17a^3 + 35a^2 + 18a + 23 + (3a^6 + 31a^5 + 8a^4 + 15a^3 + 34a^2 + 26a)37 + (14a^6 + 5a^5 + 21a^4 + 15a^3 + 18a^2 + 18a + 1)37^2 + (7a^6 + 2a^5 + 14a^4 + 3a^3 + 36a + 5)37^3 + (21a^6 + 11a^5 + 34a^4 + 25a^3 + 19a^2 + 6a + 5)37^4 + (14a^6 + 34a^5 + 13a^4 + 12a^3 + 12a^2 + 9a + 36)37^5 + (20a^6 + 23a^5 + 8a^4 + 14a^2 + a + 9)37^6 + (14a^6 + 29a^5 + 14a^4 + 4a^3 + 24a^2 + 32a + 4)37^7+O(37^8)
$r_{ 13 }$	$=$	$ 27 a^{6} + 32 a^{5} + 26 a^{4} + 30 a^{3} + 8 a^{2} + 32 a + 29 + \left(25 a^{6} + 31 a^{5} + 16 a^{4} + 19 a^{3} + 16 a^{2} + 29 a + 21\right)\cdot 37 + \left(12 a^{6} + 16 a^{5} + 10 a^{4} + 23 a^{3} + 14 a^{2} + 32 a + 29\right)\cdot 37^{2} + \left(34 a^{6} + 19 a^{5} + 21 a^{4} + 15 a^{3} + 15 a^{2} + 32 a + 18\right)\cdot 37^{3} + \left(33 a^{6} + 18 a^{4} + 31 a^{3} + 28 a^{2} + 17 a + 7\right)\cdot 37^{4} + \left(9 a^{6} + 35 a^{5} + 29 a^{4} + 8 a^{3} + 22 a^{2} + 28 a + 8\right)\cdot 37^{5} + \left(36 a^{6} + 2 a^{5} + 24 a^{4} + 17 a^{3} + 28 a^{2} + 15 a + 31\right)\cdot 37^{6} + \left(13 a^{6} + 26 a^{5} + 30 a^{4} + 33 a^{3} + 21 a^{2} + 5 a\right)\cdot 37^{7} +O(37^{8})$ 27a^6 + 32a^5 + 26a^4 + 30a^3 + 8a^2 + 32a + 29 + (25a^6 + 31a^5 + 16a^4 + 19a^3 + 16a^2 + 29a + 21)37 + (12a^6 + 16a^5 + 10a^4 + 23a^3 + 14a^2 + 32a + 29)37^2 + (34a^6 + 19a^5 + 21a^4 + 15a^3 + 15a^2 + 32a + 18)37^3 + (33a^6 + 18a^4 + 31a^3 + 28a^2 + 17a + 7)37^4 + (9a^6 + 35a^5 + 29a^4 + 8a^3 + 22a^2 + 28a + 8)37^5 + (36a^6 + 2a^5 + 24a^4 + 17a^3 + 28a^2 + 15a + 31)37^6 + (13a^6 + 26a^5 + 30a^4 + 33a^3 + 21a^2 + 5a)37^7+O(37^8)
$r_{ 14 }$	$=$	$ 28 a^{6} + 15 a^{4} + 26 a^{3} + 19 a^{2} + 32 a + 5 + \left(21 a^{6} + 13 a^{5} + 28 a^{3} + 29 a^{2} + 9 a + 4\right)\cdot 37 + \left(27 a^{6} + 19 a^{5} + 14 a^{4} + 2 a^{3} + 11 a^{2} + 5 a + 27\right)\cdot 37^{2} + \left(23 a^{6} + 14 a^{5} + 19 a^{4} + 16 a^{2} + 36 a + 33\right)\cdot 37^{3} + \left(26 a^{6} + 7 a^{5} + 10 a^{4} + 32 a^{3} + 35 a^{2} + 13 a + 22\right)\cdot 37^{4} + \left(2 a^{6} + 32 a^{5} + 2 a^{4} + 31 a^{3} + 15 a + 30\right)\cdot 37^{5} + \left(26 a^{6} + 34 a^{5} + 18 a^{4} + 35 a^{3} + 4 a^{2} + 11 a + 9\right)\cdot 37^{6} + \left(33 a^{6} + 10 a^{5} + 19 a^{4} + 31 a^{3} + 6 a^{2} + 8 a + 26\right)\cdot 37^{7} +O(37^{8})$ 28a^6 + 15a^4 + 26a^3 + 19a^2 + 32a + 5 + (21a^6 + 13a^5 + 28a^3 + 29a^2 + 9a + 4)37 + (27a^6 + 19a^5 + 14a^4 + 2a^3 + 11a^2 + 5a + 27)37^2 + (23a^6 + 14a^5 + 19a^4 + 16a^2 + 36a + 33)37^3 + (26a^6 + 7a^5 + 10a^4 + 32a^3 + 35a^2 + 13a + 22)37^4 + (2a^6 + 32a^5 + 2a^4 + 31a^3 + 15a + 30)37^5 + (26a^6 + 34a^5 + 18a^4 + 35a^3 + 4a^2 + 11a + 9)37^6 + (33a^6 + 10a^5 + 19a^4 + 31a^3 + 6a^2 + 8a + 26)*37^7+O(37^8)

Generators of the action on the roots $r_1, \ldots, r_{ 14 }$

Cycle notation
$(1,4,8,5,6,3,13,14,11,7,10,9,12,2)$
$(1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 14 }$	Character value
$1$	$1$	$()$	$1$
$1$	$2$	$(1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)$	$-1$
$1$	$7$	$(1,8,6,13,11,10,12)(2,4,5,3,14,7,9)$	$\zeta_{7}$
$1$	$7$	$(1,6,11,12,8,13,10)(2,5,14,9,4,3,7)$	$\zeta_{7}^{2}$
$1$	$7$	$(1,13,12,6,10,8,11)(2,3,9,5,7,4,14)$	$\zeta_{7}^{3}$
$1$	$7$	$(1,11,8,10,6,12,13)(2,14,4,7,5,9,3)$	$\zeta_{7}^{4}$
$1$	$7$	$(1,10,13,8,12,11,6)(2,7,3,4,9,14,5)$	$\zeta_{7}^{5}$
$1$	$7$	$(1,12,10,11,13,6,8)(2,9,7,14,3,5,4)$	$-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$
$1$	$14$	$(1,4,8,5,6,3,13,14,11,7,10,9,12,2)$	$-\zeta_{7}^{4}$
$1$	$14$	$(1,5,13,7,12,4,6,14,10,2,8,3,11,9)$	$-\zeta_{7}^{5}$
$1$	$14$	$(1,3,10,4,13,9,8,14,12,5,11,2,6,7)$	$\zeta_{7}^{5} + \zeta_{7}^{4} + \zeta_{7}^{3} + \zeta_{7}^{2} + \zeta_{7} + 1$
$1$	$14$	$(1,7,6,2,11,5,12,14,8,9,13,4,10,3)$	$-\zeta_{7}$
$1$	$14$	$(1,9,11,3,8,2,10,14,6,4,12,7,13,5)$	$-\zeta_{7}^{2}$
$1$	$14$	$(1,2,12,9,10,7,11,14,13,3,6,5,8,4)$	$-\zeta_{7}^{3}$

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 $r_1, \ldots, r_{ 14 }$

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	1.116.14t1.b.f
Dimension	$1$
Group	$C_{14}$
Conductor	$116$
Root number	not computed
Indicator	$0$

$r_{ 1 }$	$=$	\( 9 a^{6} + 22 a^{4} + 11 a^{3} + 18 a^{2} + 5 a + 32 + \left(15 a^{6} + 24 a^{5} + 36 a^{4} + 8 a^{3} + 7 a^{2} + 27 a + 32\right)\cdot 37 + \left(9 a^{6} + 17 a^{5} + 22 a^{4} + 34 a^{3} + 25 a^{2} + 31 a + 9\right)\cdot 37^{2} + \left(13 a^{6} + 22 a^{5} + 17 a^{4} + 36 a^{3} + 20 a^{2} + 3\right)\cdot 37^{3} + \left(10 a^{6} + 29 a^{5} + 26 a^{4} + 4 a^{3} + a^{2} + 23 a + 14\right)\cdot 37^{4} + \left(34 a^{6} + 4 a^{5} + 34 a^{4} + 5 a^{3} + 36 a^{2} + 21 a + 6\right)\cdot 37^{5} + \left(10 a^{6} + 2 a^{5} + 18 a^{4} + a^{3} + 32 a^{2} + 25 a + 27\right)\cdot 37^{6} + \left(3 a^{6} + 26 a^{5} + 17 a^{4} + 5 a^{3} + 30 a^{2} + 28 a + 10\right)\cdot 37^{7} +O(37^{8})\) 9a^6 + 22a^4 + 11a^3 + 18a^2 + 5a + 32 + (15a^6 + 24a^5 + 36a^4 + 8a^3 + 7a^2 + 27a + 32)37 + (9a^6 + 17a^5 + 22a^4 + 34a^3 + 25a^2 + 31a + 9)37^2 + (13a^6 + 22a^5 + 17a^4 + 36a^3 + 20a^2 + 3)37^3 + (10a^6 + 29a^5 + 26a^4 + 4a^3 + a^2 + 23a + 14)37^4 + (34a^6 + 4a^5 + 34a^4 + 5a^3 + 36a^2 + 21a + 6)37^5 + (10a^6 + 2a^5 + 18a^4 + a^3 + 32a^2 + 25a + 27)37^6 + (3a^6 + 26a^5 + 17a^4 + 5a^3 + 30a^2 + 28a + 10)*37^7+O(37^8)
$r_{ 2 }$	$=$	\( 10 a^{6} + 5 a^{5} + 11 a^{4} + 7 a^{3} + 29 a^{2} + 5 a + 8 + \left(11 a^{6} + 5 a^{5} + 20 a^{4} + 17 a^{3} + 20 a^{2} + 7 a + 15\right)\cdot 37 + \left(24 a^{6} + 20 a^{5} + 26 a^{4} + 13 a^{3} + 22 a^{2} + 4 a + 7\right)\cdot 37^{2} + \left(2 a^{6} + 17 a^{5} + 15 a^{4} + 21 a^{3} + 21 a^{2} + 4 a + 18\right)\cdot 37^{3} + \left(3 a^{6} + 36 a^{5} + 18 a^{4} + 5 a^{3} + 8 a^{2} + 19 a + 29\right)\cdot 37^{4} + \left(27 a^{6} + a^{5} + 7 a^{4} + 28 a^{3} + 14 a^{2} + 8 a + 28\right)\cdot 37^{5} + \left(34 a^{5} + 12 a^{4} + 19 a^{3} + 8 a^{2} + 21 a + 5\right)\cdot 37^{6} + \left(23 a^{6} + 10 a^{5} + 6 a^{4} + 3 a^{3} + 15 a^{2} + 31 a + 36\right)\cdot 37^{7} +O(37^{8})\) 10a^6 + 5a^5 + 11a^4 + 7a^3 + 29a^2 + 5a + 8 + (11a^6 + 5a^5 + 20a^4 + 17a^3 + 20a^2 + 7a + 15)37 + (24a^6 + 20a^5 + 26a^4 + 13a^3 + 22a^2 + 4a + 7)37^2 + (2a^6 + 17a^5 + 15a^4 + 21a^3 + 21a^2 + 4a + 18)37^3 + (3a^6 + 36a^5 + 18a^4 + 5a^3 + 8a^2 + 19a + 29)37^4 + (27a^6 + a^5 + 7a^4 + 28a^3 + 14a^2 + 8a + 28)37^5 + (34a^5 + 12a^4 + 19a^3 + 8a^2 + 21a + 5)37^6 + (23a^6 + 10a^5 + 6a^4 + 3a^3 + 15a^2 + 31a + 36)*37^7+O(37^8)
$r_{ 3 }$	$=$	\( 11 a^{6} + 18 a^{5} + 21 a^{4} + 20 a^{3} + 2 a^{2} + 19 a + 14 + \left(33 a^{6} + 5 a^{5} + 28 a^{4} + 21 a^{3} + 2 a^{2} + 10 a + 36\right)\cdot 37 + \left(22 a^{6} + 31 a^{5} + 15 a^{4} + 21 a^{3} + 18 a^{2} + 18 a + 35\right)\cdot 37^{2} + \left(29 a^{6} + 34 a^{5} + 22 a^{4} + 33 a^{3} + 36 a^{2} + 31\right)\cdot 37^{3} + \left(15 a^{6} + 25 a^{5} + 2 a^{4} + 11 a^{3} + 17 a^{2} + 30 a + 31\right)\cdot 37^{4} + \left(22 a^{6} + 2 a^{5} + 23 a^{4} + 24 a^{3} + 24 a^{2} + 27 a\right)\cdot 37^{5} + \left(16 a^{6} + 13 a^{5} + 28 a^{4} + 36 a^{3} + 22 a^{2} + 35 a + 27\right)\cdot 37^{6} + \left(22 a^{6} + 7 a^{5} + 22 a^{4} + 32 a^{3} + 12 a^{2} + 4 a + 32\right)\cdot 37^{7} +O(37^{8})\) 11a^6 + 18a^5 + 21a^4 + 20a^3 + 2a^2 + 19a + 14 + (33a^6 + 5a^5 + 28a^4 + 21a^3 + 2a^2 + 10a + 36)37 + (22a^6 + 31a^5 + 15a^4 + 21a^3 + 18a^2 + 18a + 35)37^2 + (29a^6 + 34a^5 + 22a^4 + 33a^3 + 36a^2 + 31)37^3 + (15a^6 + 25a^5 + 2a^4 + 11a^3 + 17a^2 + 30a + 31)37^4 + (22a^6 + 2a^5 + 23a^4 + 24a^3 + 24a^2 + 27a)37^5 + (16a^6 + 13a^5 + 28a^4 + 36a^3 + 22a^2 + 35a + 27)37^6 + (22a^6 + 7a^5 + 22a^4 + 32a^3 + 12a^2 + 4a + 32)37^7+O(37^8)
$r_{ 4 }$	$=$	\( 11 a^{6} + 19 a^{5} + 15 a^{4} + 11 a^{3} + 29 a^{2} + 29 a + 14 + \left(23 a^{6} + 7 a^{5} + 11 a^{4} + a^{3} + 10 a^{2} + 20 a + 13\right)\cdot 37 + \left(18 a^{6} + 12 a^{5} + 27 a^{4} + 6 a^{3} + 29 a^{2} + a + 10\right)\cdot 37^{2} + \left(3 a^{6} + 11 a^{5} + a^{4} + 26 a^{3} + 7 a^{2} + 12 a + 23\right)\cdot 37^{3} + \left(19 a^{5} + 21 a^{4} + 23 a^{3} + 27 a^{2} + 8 a + 11\right)\cdot 37^{4} + \left(18 a^{6} + 24 a^{5} + 17 a^{4} + 11 a^{3} + 29 a^{2} + 32 a + 11\right)\cdot 37^{5} + \left(11 a^{6} + 20 a^{5} + 31 a^{4} + 14 a^{3} + 24 a^{2} + 36 a + 33\right)\cdot 37^{6} + \left(20 a^{6} + 20 a^{5} + 23 a^{4} + 14 a^{3} + 4 a^{2} + 27 a + 19\right)\cdot 37^{7} +O(37^{8})\) 11a^6 + 19a^5 + 15a^4 + 11a^3 + 29a^2 + 29a + 14 + (23a^6 + 7a^5 + 11a^4 + a^3 + 10a^2 + 20a + 13)37 + (18a^6 + 12a^5 + 27a^4 + 6a^3 + 29a^2 + a + 10)37^2 + (3a^6 + 11a^5 + a^4 + 26a^3 + 7a^2 + 12a + 23)37^3 + (19a^5 + 21a^4 + 23a^3 + 27a^2 + 8a + 11)37^4 + (18a^6 + 24a^5 + 17a^4 + 11a^3 + 29a^2 + 32a + 11)37^5 + (11a^6 + 20a^5 + 31a^4 + 14a^3 + 24a^2 + 36a + 33)37^6 + (20a^6 + 20a^5 + 23a^4 + 14a^3 + 4a^2 + 27a + 19)*37^7+O(37^8)
$r_{ 5 }$	$=$	\( 13 a^{6} + 20 a^{5} + 29 a^{4} + 10 a^{3} + 23 a^{2} + 35 a + 26 + \left(20 a^{6} + 20 a^{5} + 5 a^{4} + 3 a^{3} + 2 a^{2} + 11 a + 32\right)\cdot 37 + \left(9 a^{6} + 26 a^{5} + 16 a^{4} + a^{3} + 5 a^{2} + 27 a + 29\right)\cdot 37^{2} + \left(34 a^{6} + 12 a^{5} + 4 a^{4} + 22 a^{3} + 32 a + 22\right)\cdot 37^{3} + \left(26 a^{6} + 36 a^{5} + 31 a^{4} + 26 a^{3} + 14 a^{2} + 15 a + 24\right)\cdot 37^{4} + \left(18 a^{6} + 30 a^{5} + 5 a^{4} + 29 a^{3} + 14 a^{2} + 4 a + 15\right)\cdot 37^{5} + \left(24 a^{6} + 3 a^{5} + 29 a^{4} + 13 a^{3} + 35 a^{2} + 27 a\right)\cdot 37^{6} + \left(26 a^{6} + 13 a^{5} + 18 a^{4} + 16 a^{3} + 3 a^{2} + 18 a + 21\right)\cdot 37^{7} +O(37^{8})\) 13a^6 + 20a^5 + 29a^4 + 10a^3 + 23a^2 + 35a + 26 + (20a^6 + 20a^5 + 5a^4 + 3a^3 + 2a^2 + 11a + 32)37 + (9a^6 + 26a^5 + 16a^4 + a^3 + 5a^2 + 27a + 29)37^2 + (34a^6 + 12a^5 + 4a^4 + 22a^3 + 32a + 22)37^3 + (26a^6 + 36a^5 + 31a^4 + 26a^3 + 14a^2 + 15a + 24)37^4 + (18a^6 + 30a^5 + 5a^4 + 29a^3 + 14a^2 + 4a + 15)37^5 + (24a^6 + 3a^5 + 29a^4 + 13a^3 + 35a^2 + 27a)37^6 + (26a^6 + 13a^5 + 18a^4 + 16a^3 + 3a^2 + 18a + 21)*37^7+O(37^8)
$r_{ 6 }$	$=$	\( 17 a^{6} + 19 a^{5} + 4 a^{4} + a^{3} + 15 a^{2} + 32 a + 6 + \left(33 a^{6} + 4 a^{5} + 21 a^{4} + 3 a^{3} + 20 a^{2} + 2 a + 31\right)\cdot 37 + \left(23 a^{6} + 7 a^{5} + 23 a^{4} + 29 a^{3} + 16 a^{2} + 35 a + 22\right)\cdot 37^{2} + \left(31 a^{6} + 18 a^{5} + 15 a^{4} + 20 a^{3} + 16 a^{2} + 10 a + 2\right)\cdot 37^{3} + \left(6 a^{6} + 4 a^{5} + 14 a^{4} + 14 a^{3} + 13 a^{2} + 14 a + 30\right)\cdot 37^{4} + \left(13 a^{6} + 17 a^{5} + 12 a^{4} + 5 a^{3} + 36 a^{2} + 6 a + 27\right)\cdot 37^{5} + \left(22 a^{6} + 18 a^{5} + 36 a^{4} + 16 a^{3} + 10 a^{2} + 9 a + 21\right)\cdot 37^{6} + \left(14 a^{6} + 26 a^{5} + 25 a^{4} + 17 a^{3} + 28 a^{2} + 25 a + 4\right)\cdot 37^{7} +O(37^{8})\) 17a^6 + 19a^5 + 4a^4 + a^3 + 15a^2 + 32a + 6 + (33a^6 + 4a^5 + 21a^4 + 3a^3 + 20a^2 + 2a + 31)37 + (23a^6 + 7a^5 + 23a^4 + 29a^3 + 16a^2 + 35a + 22)37^2 + (31a^6 + 18a^5 + 15a^4 + 20a^3 + 16a^2 + 10a + 2)37^3 + (6a^6 + 4a^5 + 14a^4 + 14a^3 + 13a^2 + 14a + 30)37^4 + (13a^6 + 17a^5 + 12a^4 + 5a^3 + 36a^2 + 6a + 27)37^5 + (22a^6 + 18a^5 + 36a^4 + 16a^3 + 10a^2 + 9a + 21)37^6 + (14a^6 + 26a^5 + 25a^4 + 17a^3 + 28a^2 + 25a + 4)37^7+O(37^8)
$r_{ 7 }$	$=$	\( 18 a^{6} + 31 a^{5} + 24 a^{4} + a^{3} + 24 a^{2} + 23 a + 19 + \left(34 a^{6} + 26 a^{5} + 28 a^{4} + 5 a^{3} + 28 a^{2} + 16 a + 6\right)\cdot 37 + \left(31 a^{6} + 8 a^{5} + 34 a^{4} + 21 a^{3} + 3 a^{2} + 15 a + 16\right)\cdot 37^{2} + \left(11 a^{6} + a^{5} + 25 a^{4} + 28 a^{3} + 8 a^{2} + 36 a + 36\right)\cdot 37^{3} + \left(8 a^{6} + 27 a^{5} + 4 a^{4} + 25 a^{3} + 21 a^{2} + 23\right)\cdot 37^{4} + \left(35 a^{6} + 35 a^{5} + 30 a^{4} + 27 a^{3} + 26 a^{2} + 29 a + 3\right)\cdot 37^{5} + \left(16 a^{6} + 22 a^{5} + 27 a^{4} + 6 a^{3} + 26 a^{2} + 24 a + 29\right)\cdot 37^{6} + \left(36 a^{6} + 8 a^{4} + 29 a^{3} + 22 a^{2} + 7 a + 5\right)\cdot 37^{7} +O(37^{8})\) 18a^6 + 31a^5 + 24a^4 + a^3 + 24a^2 + 23a + 19 + (34a^6 + 26a^5 + 28a^4 + 5a^3 + 28a^2 + 16a + 6)37 + (31a^6 + 8a^5 + 34a^4 + 21a^3 + 3a^2 + 15a + 16)37^2 + (11a^6 + a^5 + 25a^4 + 28a^3 + 8a^2 + 36a + 36)37^3 + (8a^6 + 27a^5 + 4a^4 + 25a^3 + 21a^2 + 23)37^4 + (35a^6 + 35a^5 + 30a^4 + 27a^3 + 26a^2 + 29a + 3)37^5 + (16a^6 + 22a^5 + 27a^4 + 6a^3 + 26a^2 + 24a + 29)37^6 + (36a^6 + 8a^4 + 29a^3 + 22a^2 + 7a + 5)*37^7+O(37^8)
$r_{ 8 }$	$=$	\( 19 a^{6} + 6 a^{5} + 13 a^{4} + 36 a^{3} + 13 a^{2} + 14 a + 18 + \left(2 a^{6} + 10 a^{5} + 8 a^{4} + 31 a^{3} + 8 a^{2} + 20 a + 30\right)\cdot 37 + \left(5 a^{6} + 28 a^{5} + 2 a^{4} + 15 a^{3} + 33 a^{2} + 21 a + 20\right)\cdot 37^{2} + \left(25 a^{6} + 35 a^{5} + 11 a^{4} + 8 a^{3} + 28 a^{2}\right)\cdot 37^{3} + \left(28 a^{6} + 9 a^{5} + 32 a^{4} + 11 a^{3} + 15 a^{2} + 36 a + 13\right)\cdot 37^{4} + \left(a^{6} + a^{5} + 6 a^{4} + 9 a^{3} + 10 a^{2} + 7 a + 33\right)\cdot 37^{5} + \left(20 a^{6} + 14 a^{5} + 9 a^{4} + 30 a^{3} + 10 a^{2} + 12 a + 7\right)\cdot 37^{6} + \left(36 a^{5} + 28 a^{4} + 7 a^{3} + 14 a^{2} + 29 a + 31\right)\cdot 37^{7} +O(37^{8})\) 19a^6 + 6a^5 + 13a^4 + 36a^3 + 13a^2 + 14a + 18 + (2a^6 + 10a^5 + 8a^4 + 31a^3 + 8a^2 + 20a + 30)37 + (5a^6 + 28a^5 + 2a^4 + 15a^3 + 33a^2 + 21a + 20)37^2 + (25a^6 + 35a^5 + 11a^4 + 8a^3 + 28a^2)37^3 + (28a^6 + 9a^5 + 32a^4 + 11a^3 + 15a^2 + 36a + 13)37^4 + (a^6 + a^5 + 6a^4 + 9a^3 + 10a^2 + 7a + 33)37^5 + (20a^6 + 14a^5 + 9a^4 + 30a^3 + 10a^2 + 12a + 7)37^6 + (36a^5 + 28a^4 + 7a^3 + 14a^2 + 29a + 31)*37^7+O(37^8)
$r_{ 9 }$	$=$	\( 20 a^{6} + 18 a^{5} + 33 a^{4} + 36 a^{3} + 22 a^{2} + 5 a + 31 + \left(3 a^{6} + 32 a^{5} + 15 a^{4} + 33 a^{3} + 16 a^{2} + 34 a + 5\right)\cdot 37 + \left(13 a^{6} + 29 a^{5} + 13 a^{4} + 7 a^{3} + 20 a^{2} + a + 14\right)\cdot 37^{2} + \left(5 a^{6} + 18 a^{5} + 21 a^{4} + 16 a^{3} + 20 a^{2} + 26 a + 34\right)\cdot 37^{3} + \left(30 a^{6} + 32 a^{5} + 22 a^{4} + 22 a^{3} + 23 a^{2} + 22 a + 6\right)\cdot 37^{4} + \left(23 a^{6} + 19 a^{5} + 24 a^{4} + 31 a^{3} + 30 a + 9\right)\cdot 37^{5} + \left(14 a^{6} + 18 a^{5} + 20 a^{3} + 26 a^{2} + 27 a + 15\right)\cdot 37^{6} + \left(22 a^{6} + 10 a^{5} + 11 a^{4} + 19 a^{3} + 8 a^{2} + 11 a + 32\right)\cdot 37^{7} +O(37^{8})\) 20a^6 + 18a^5 + 33a^4 + 36a^3 + 22a^2 + 5a + 31 + (3a^6 + 32a^5 + 15a^4 + 33a^3 + 16a^2 + 34a + 5)37 + (13a^6 + 29a^5 + 13a^4 + 7a^3 + 20a^2 + a + 14)37^2 + (5a^6 + 18a^5 + 21a^4 + 16a^3 + 20a^2 + 26a + 34)37^3 + (30a^6 + 32a^5 + 22a^4 + 22a^3 + 23a^2 + 22a + 6)37^4 + (23a^6 + 19a^5 + 24a^4 + 31a^3 + 30a + 9)37^5 + (14a^6 + 18a^5 + 20a^3 + 26a^2 + 27a + 15)37^6 + (22a^6 + 10a^5 + 11a^4 + 19a^3 + 8a^2 + 11a + 32)37^7+O(37^8)
$r_{ 10 }$	$=$	\( 24 a^{6} + 17 a^{5} + 8 a^{4} + 27 a^{3} + 14 a^{2} + 2 a + 11 + \left(16 a^{6} + 16 a^{5} + 31 a^{4} + 33 a^{3} + 34 a^{2} + 25 a + 4\right)\cdot 37 + \left(27 a^{6} + 10 a^{5} + 20 a^{4} + 35 a^{3} + 31 a^{2} + 9 a + 7\right)\cdot 37^{2} + \left(2 a^{6} + 24 a^{5} + 32 a^{4} + 14 a^{3} + 36 a^{2} + 4 a + 14\right)\cdot 37^{3} + \left(10 a^{6} + 5 a^{4} + 10 a^{3} + 22 a^{2} + 21 a + 12\right)\cdot 37^{4} + \left(18 a^{6} + 6 a^{5} + 31 a^{4} + 7 a^{3} + 22 a^{2} + 32 a + 21\right)\cdot 37^{5} + \left(12 a^{6} + 33 a^{5} + 7 a^{4} + 23 a^{3} + a^{2} + 9 a + 36\right)\cdot 37^{6} + \left(10 a^{6} + 23 a^{5} + 18 a^{4} + 20 a^{3} + 33 a^{2} + 18 a + 15\right)\cdot 37^{7} +O(37^{8})\) 24a^6 + 17a^5 + 8a^4 + 27a^3 + 14a^2 + 2a + 11 + (16a^6 + 16a^5 + 31a^4 + 33a^3 + 34a^2 + 25a + 4)37 + (27a^6 + 10a^5 + 20a^4 + 35a^3 + 31a^2 + 9a + 7)37^2 + (2a^6 + 24a^5 + 32a^4 + 14a^3 + 36a^2 + 4a + 14)37^3 + (10a^6 + 5a^4 + 10a^3 + 22a^2 + 21a + 12)37^4 + (18a^6 + 6a^5 + 31a^4 + 7a^3 + 22a^2 + 32a + 21)37^5 + (12a^6 + 33a^5 + 7a^4 + 23a^3 + a^2 + 9a + 36)37^6 + (10a^6 + 23a^5 + 18a^4 + 20a^3 + 33a^2 + 18a + 15)*37^7+O(37^8)
$r_{ 11 }$	$=$	\( 26 a^{6} + 18 a^{5} + 22 a^{4} + 26 a^{3} + 8 a^{2} + 8 a + 23 + \left(13 a^{6} + 29 a^{5} + 25 a^{4} + 35 a^{3} + 26 a^{2} + 16 a + 23\right)\cdot 37 + \left(18 a^{6} + 24 a^{5} + 9 a^{4} + 30 a^{3} + 7 a^{2} + 35 a + 26\right)\cdot 37^{2} + \left(33 a^{6} + 25 a^{5} + 35 a^{4} + 10 a^{3} + 29 a^{2} + 24 a + 13\right)\cdot 37^{3} + \left(36 a^{6} + 17 a^{5} + 15 a^{4} + 13 a^{3} + 9 a^{2} + 28 a + 25\right)\cdot 37^{4} + \left(18 a^{6} + 12 a^{5} + 19 a^{4} + 25 a^{3} + 7 a^{2} + 4 a + 25\right)\cdot 37^{5} + \left(25 a^{6} + 16 a^{5} + 5 a^{4} + 22 a^{3} + 12 a^{2} + 3\right)\cdot 37^{6} + \left(16 a^{6} + 16 a^{5} + 13 a^{4} + 22 a^{3} + 32 a^{2} + 9 a + 17\right)\cdot 37^{7} +O(37^{8})\) 26a^6 + 18a^5 + 22a^4 + 26a^3 + 8a^2 + 8a + 23 + (13a^6 + 29a^5 + 25a^4 + 35a^3 + 26a^2 + 16a + 23)37 + (18a^6 + 24a^5 + 9a^4 + 30a^3 + 7a^2 + 35a + 26)37^2 + (33a^6 + 25a^5 + 35a^4 + 10a^3 + 29a^2 + 24a + 13)37^3 + (36a^6 + 17a^5 + 15a^4 + 13a^3 + 9a^2 + 28a + 25)37^4 + (18a^6 + 12a^5 + 19a^4 + 25a^3 + 7a^2 + 4a + 25)37^5 + (25a^6 + 16a^5 + 5a^4 + 22a^3 + 12a^2 + 3)37^6 + (16a^6 + 16a^5 + 13a^4 + 22a^3 + 32a^2 + 9a + 17)37^7+O(37^8)
$r_{ 12 }$	$=$	\( 26 a^{6} + 19 a^{5} + 16 a^{4} + 17 a^{3} + 35 a^{2} + 18 a + 23 + \left(3 a^{6} + 31 a^{5} + 8 a^{4} + 15 a^{3} + 34 a^{2} + 26 a\right)\cdot 37 + \left(14 a^{6} + 5 a^{5} + 21 a^{4} + 15 a^{3} + 18 a^{2} + 18 a + 1\right)\cdot 37^{2} + \left(7 a^{6} + 2 a^{5} + 14 a^{4} + 3 a^{3} + 36 a + 5\right)\cdot 37^{3} + \left(21 a^{6} + 11 a^{5} + 34 a^{4} + 25 a^{3} + 19 a^{2} + 6 a + 5\right)\cdot 37^{4} + \left(14 a^{6} + 34 a^{5} + 13 a^{4} + 12 a^{3} + 12 a^{2} + 9 a + 36\right)\cdot 37^{5} + \left(20 a^{6} + 23 a^{5} + 8 a^{4} + 14 a^{2} + a + 9\right)\cdot 37^{6} + \left(14 a^{6} + 29 a^{5} + 14 a^{4} + 4 a^{3} + 24 a^{2} + 32 a + 4\right)\cdot 37^{7} +O(37^{8})\) 26a^6 + 19a^5 + 16a^4 + 17a^3 + 35a^2 + 18a + 23 + (3a^6 + 31a^5 + 8a^4 + 15a^3 + 34a^2 + 26a)37 + (14a^6 + 5a^5 + 21a^4 + 15a^3 + 18a^2 + 18a + 1)37^2 + (7a^6 + 2a^5 + 14a^4 + 3a^3 + 36a + 5)37^3 + (21a^6 + 11a^5 + 34a^4 + 25a^3 + 19a^2 + 6a + 5)37^4 + (14a^6 + 34a^5 + 13a^4 + 12a^3 + 12a^2 + 9a + 36)37^5 + (20a^6 + 23a^5 + 8a^4 + 14a^2 + a + 9)37^6 + (14a^6 + 29a^5 + 14a^4 + 4a^3 + 24a^2 + 32a + 4)37^7+O(37^8)
$r_{ 13 }$	$=$	\( 27 a^{6} + 32 a^{5} + 26 a^{4} + 30 a^{3} + 8 a^{2} + 32 a + 29 + \left(25 a^{6} + 31 a^{5} + 16 a^{4} + 19 a^{3} + 16 a^{2} + 29 a + 21\right)\cdot 37 + \left(12 a^{6} + 16 a^{5} + 10 a^{4} + 23 a^{3} + 14 a^{2} + 32 a + 29\right)\cdot 37^{2} + \left(34 a^{6} + 19 a^{5} + 21 a^{4} + 15 a^{3} + 15 a^{2} + 32 a + 18\right)\cdot 37^{3} + \left(33 a^{6} + 18 a^{4} + 31 a^{3} + 28 a^{2} + 17 a + 7\right)\cdot 37^{4} + \left(9 a^{6} + 35 a^{5} + 29 a^{4} + 8 a^{3} + 22 a^{2} + 28 a + 8\right)\cdot 37^{5} + \left(36 a^{6} + 2 a^{5} + 24 a^{4} + 17 a^{3} + 28 a^{2} + 15 a + 31\right)\cdot 37^{6} + \left(13 a^{6} + 26 a^{5} + 30 a^{4} + 33 a^{3} + 21 a^{2} + 5 a\right)\cdot 37^{7} +O(37^{8})\) 27a^6 + 32a^5 + 26a^4 + 30a^3 + 8a^2 + 32a + 29 + (25a^6 + 31a^5 + 16a^4 + 19a^3 + 16a^2 + 29a + 21)37 + (12a^6 + 16a^5 + 10a^4 + 23a^3 + 14a^2 + 32a + 29)37^2 + (34a^6 + 19a^5 + 21a^4 + 15a^3 + 15a^2 + 32a + 18)37^3 + (33a^6 + 18a^4 + 31a^3 + 28a^2 + 17a + 7)37^4 + (9a^6 + 35a^5 + 29a^4 + 8a^3 + 22a^2 + 28a + 8)37^5 + (36a^6 + 2a^5 + 24a^4 + 17a^3 + 28a^2 + 15a + 31)37^6 + (13a^6 + 26a^5 + 30a^4 + 33a^3 + 21a^2 + 5a)37^7+O(37^8)
$r_{ 14 }$	$=$	\( 28 a^{6} + 15 a^{4} + 26 a^{3} + 19 a^{2} + 32 a + 5 + \left(21 a^{6} + 13 a^{5} + 28 a^{3} + 29 a^{2} + 9 a + 4\right)\cdot 37 + \left(27 a^{6} + 19 a^{5} + 14 a^{4} + 2 a^{3} + 11 a^{2} + 5 a + 27\right)\cdot 37^{2} + \left(23 a^{6} + 14 a^{5} + 19 a^{4} + 16 a^{2} + 36 a + 33\right)\cdot 37^{3} + \left(26 a^{6} + 7 a^{5} + 10 a^{4} + 32 a^{3} + 35 a^{2} + 13 a + 22\right)\cdot 37^{4} + \left(2 a^{6} + 32 a^{5} + 2 a^{4} + 31 a^{3} + 15 a + 30\right)\cdot 37^{5} + \left(26 a^{6} + 34 a^{5} + 18 a^{4} + 35 a^{3} + 4 a^{2} + 11 a + 9\right)\cdot 37^{6} + \left(33 a^{6} + 10 a^{5} + 19 a^{4} + 31 a^{3} + 6 a^{2} + 8 a + 26\right)\cdot 37^{7} +O(37^{8})\) 28a^6 + 15a^4 + 26a^3 + 19a^2 + 32a + 5 + (21a^6 + 13a^5 + 28a^3 + 29a^2 + 9a + 4)37 + (27a^6 + 19a^5 + 14a^4 + 2a^3 + 11a^2 + 5a + 27)37^2 + (23a^6 + 14a^5 + 19a^4 + 16a^2 + 36a + 33)37^3 + (26a^6 + 7a^5 + 10a^4 + 32a^3 + 35a^2 + 13a + 22)37^4 + (2a^6 + 32a^5 + 2a^4 + 31a^3 + 15a + 30)37^5 + (26a^6 + 34a^5 + 18a^4 + 35a^3 + 4a^2 + 11a + 9)37^6 + (33a^6 + 10a^5 + 19a^4 + 31a^3 + 6a^2 + 8a + 26)*37^7+O(37^8)