LMFDB - Artin representation 2.1991.14t3.a.a

Basic invariants

Dimension:	$2$
Group:	$D_{14}$
Conductor:	$1991$$\medspace = 11 \cdot 181 $
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 14.2.11274729599284301762821.1
Galois orbit size:	$3$
Smallest permutation container:	$D_{14}$
Parity:	odd
Determinant:	1.1991.2t1.a.a
Projective image:	$D_7$
Projective stem field:	Galois closure of 7.1.7892485271.1

Defining polynomial

$f(x)$

$=$

$ x^{14} - 2 x^{13} + 7 x^{12} - 10 x^{11} - 52 x^{10} - 90 x^{9} - 340 x^{8} - 179 x^{7} + 955 x^{6} + \cdots - 1111 $

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.

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

Roots:

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

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

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 14 }$	Character value
$1$	$1$	$()$	$2$
$1$	$2$	$(1,4)(2,14)(3,7)(5,10)(6,8)(9,13)(11,12)$	$-2$
$7$	$2$	$(1,6)(2,7)(3,14)(4,8)(5,13)(9,10)$	$0$
$7$	$2$	$(1,8)(2,3)(4,6)(5,9)(7,14)(10,13)(11,12)$	$0$
$2$	$7$	$(1,2,9,10,7,6,11)(3,8,12,4,14,13,5)$	$\zeta_{7}^{4} + \zeta_{7}^{3}$
$2$	$7$	$(1,9,7,11,2,10,6)(3,12,14,5,8,4,13)$	$-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$
$2$	$7$	$(1,10,11,9,6,2,7)(3,4,5,12,13,8,14)$	$\zeta_{7}^{5} + \zeta_{7}^{2}$
$2$	$14$	$(1,14,9,5,7,8,11,4,2,13,10,3,6,12)$	$-\zeta_{7}^{4} - \zeta_{7}^{3}$
$2$	$14$	$(1,5,11,13,6,14,7,4,10,12,9,8,2,3)$	$-\zeta_{7}^{5} - \zeta_{7}^{2}$
$2$	$14$	$(1,8,10,14,11,3,9,4,6,5,2,12,7,13)$	$\zeta_{7}^{5} + \zeta_{7}^{4} + \zeta_{7}^{3} + \zeta_{7}^{2} + 1$

The blue line marks the conjugacy class containing complex conjugation.

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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	2.1991.14t3.a.a
Dimension	$2$
Group	$D_{14}$
Conductor	$1991$
Root number	$1$
Indicator	$1$