LMFDB - Artin representation 2.961.15t4.a.c

Basic invariants

Dimension:	$2$
Group:	$S_3 \times C_5$
Conductor:	$961$$\medspace = 31^{2} $
Artin stem field:	Galois closure of 15.5.24417546297445042591.1
Galois orbit size:	$4$
Smallest permutation container:	$S_3 \times C_5$
Parity:	odd
Determinant:	1.31.10t1.a.b
Projective image:	$S_3$
Projective stem field:	Galois closure of 3.1.31.1

Defining polynomial

$f(x)$

$=$

$ x^{15} - 4 x^{14} + 7 x^{13} - 17 x^{12} + 51 x^{11} - 87 x^{10} - 7 x^{9} + 172 x^{8} - 100 x^{7} + \cdots - 1 $

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{5} + x + 42 $ x^5 + x + 42

Roots:

$r_{ 1 }$	$=$	$ 26 a^{3} + 21 a^{2} + 40 a + 17 + \left(13 a^{4} + 37 a^{3} + 40 a^{2} + 45 a + 13\right)\cdot 47 + \left(23 a^{4} + 43 a^{3} + 36 a^{2} + 42 a + 31\right)\cdot 47^{2} + \left(29 a^{4} + 23 a^{3} + 12 a^{2} + 42 a + 43\right)\cdot 47^{3} + \left(33 a^{4} + 39 a^{3} + 11 a^{2} + 19 a + 31\right)\cdot 47^{4} + \left(5 a^{4} + 35 a^{3} + 17 a^{2} + 22 a + 23\right)\cdot 47^{5} + \left(18 a^{4} + 36 a^{2} + 10 a + 19\right)\cdot 47^{6} +O(47^{7})$ 26a^3 + 21a^2 + 40a + 17 + (13a^4 + 37a^3 + 40a^2 + 45a + 13)47 + (23a^4 + 43a^3 + 36a^2 + 42a + 31)47^2 + (29a^4 + 23a^3 + 12a^2 + 42a + 43)47^3 + (33a^4 + 39a^3 + 11a^2 + 19a + 31)47^4 + (5a^4 + 35a^3 + 17a^2 + 22a + 23)47^5 + (18a^4 + 36a^2 + 10a + 19)47^6+O(47^7)
$r_{ 2 }$	$=$	$ 4 a^{4} + 33 a^{3} + 38 a^{2} + 40 a + 16 + \left(42 a^{4} + 42 a^{3} + 24 a^{2} + 4 a + 39\right)\cdot 47 + \left(27 a^{4} + 19 a^{3} + 27 a^{2} + 24 a + 27\right)\cdot 47^{2} + \left(14 a^{4} + 9 a^{3} + 18 a^{2} + 22 a + 31\right)\cdot 47^{3} + \left(19 a^{4} + 33 a^{2} + 11 a + 14\right)\cdot 47^{4} + \left(a^{4} + 33 a^{3} + 23 a^{2} + 16 a + 18\right)\cdot 47^{5} + \left(34 a^{4} + 44 a^{3} + 6 a^{2} + 31 a + 1\right)\cdot 47^{6} +O(47^{7})$ 4a^4 + 33a^3 + 38a^2 + 40a + 16 + (42a^4 + 42a^3 + 24a^2 + 4a + 39)47 + (27a^4 + 19a^3 + 27a^2 + 24a + 27)47^2 + (14a^4 + 9a^3 + 18a^2 + 22a + 31)47^3 + (19a^4 + 33a^2 + 11a + 14)47^4 + (a^4 + 33a^3 + 23a^2 + 16a + 18)47^5 + (34a^4 + 44a^3 + 6a^2 + 31a + 1)47^6+O(47^7)
$r_{ 3 }$	$=$	$ 5 a^{4} + 5 a^{3} + 46 a^{2} + 43 a + 22 + \left(23 a^{4} + 15 a^{3} + 19 a^{2} + 17 a\right)\cdot 47 + \left(34 a^{4} + 34 a^{3} + 40 a^{2} + 46 a\right)\cdot 47^{2} + \left(13 a^{4} + 36 a^{3} + 33 a^{2} + 42 a + 18\right)\cdot 47^{3} + \left(43 a^{4} + 4 a^{3} + 43 a^{2} + 15 a + 30\right)\cdot 47^{4} + \left(44 a^{4} + 43 a^{3} + 8 a^{2} + 17 a + 46\right)\cdot 47^{5} + \left(28 a^{4} + 46 a^{3} + 5 a^{2} + 45 a + 43\right)\cdot 47^{6} +O(47^{7})$ 5a^4 + 5a^3 + 46a^2 + 43a + 22 + (23a^4 + 15a^3 + 19a^2 + 17a)47 + (34a^4 + 34a^3 + 40a^2 + 46a)47^2 + (13a^4 + 36a^3 + 33a^2 + 42a + 18)47^3 + (43a^4 + 4a^3 + 43a^2 + 15a + 30)47^4 + (44a^4 + 43a^3 + 8a^2 + 17a + 46)47^5 + (28a^4 + 46a^3 + 5a^2 + 45a + 43)47^6+O(47^7)
$r_{ 4 }$	$=$	$ 9 a^{4} + 15 a^{3} + 21 a^{2} + 35 a + 43 + \left(11 a^{4} + 21 a^{3} + 37 a^{2} + 21 a + 11\right)\cdot 47 + \left(38 a^{4} + 30 a^{3} + 39 a^{2} + 4 a + 43\right)\cdot 47^{2} + \left(16 a^{4} + 16 a^{3} + 9 a^{2} + 15 a + 42\right)\cdot 47^{3} + \left(28 a^{4} + 42 a^{3} + 42 a^{2} + 34 a + 8\right)\cdot 47^{4} + \left(37 a^{4} + 19 a^{3} + 34 a^{2} + 29 a + 2\right)\cdot 47^{5} + \left(11 a^{4} + 12 a^{3} + 19 a^{2} + 46 a + 5\right)\cdot 47^{6} +O(47^{7})$ 9a^4 + 15a^3 + 21a^2 + 35a + 43 + (11a^4 + 21a^3 + 37a^2 + 21a + 11)47 + (38a^4 + 30a^3 + 39a^2 + 4a + 43)47^2 + (16a^4 + 16a^3 + 9a^2 + 15a + 42)47^3 + (28a^4 + 42a^3 + 42a^2 + 34a + 8)47^4 + (37a^4 + 19a^3 + 34a^2 + 29a + 2)47^5 + (11a^4 + 12a^3 + 19a^2 + 46a + 5)47^6+O(47^7)
$r_{ 5 }$	$=$	$ 13 a^{4} + 42 a^{3} + a^{2} + 43 a + 19 + \left(24 a^{4} + 32 a^{3} + 35 a^{2} + 21 a + 1\right)\cdot 47 + \left(19 a^{4} + 37 a^{3} + 46 a^{2} + 23 a + 35\right)\cdot 47^{2} + \left(13 a^{4} + 5 a^{3} + 38 a^{2} + 41 a + 17\right)\cdot 47^{3} + \left(13 a^{4} + 45 a^{3} + 28 a^{2} + 27 a + 6\right)\cdot 47^{4} + \left(11 a^{4} + 39 a^{3} + 36 a^{2} + 18 a + 29\right)\cdot 47^{5} + \left(6 a^{4} + 24 a^{3} + 17 a^{2} + 30 a + 44\right)\cdot 47^{6} +O(47^{7})$ 13a^4 + 42a^3 + a^2 + 43a + 19 + (24a^4 + 32a^3 + 35a^2 + 21a + 1)47 + (19a^4 + 37a^3 + 46a^2 + 23a + 35)47^2 + (13a^4 + 5a^3 + 38a^2 + 41a + 17)47^3 + (13a^4 + 45a^3 + 28a^2 + 27a + 6)47^4 + (11a^4 + 39a^3 + 36a^2 + 18a + 29)47^5 + (6a^4 + 24a^3 + 17a^2 + 30a + 44)*47^6+O(47^7)
$r_{ 6 }$	$=$	$ 16 a^{4} + 36 a^{3} + 28 a^{2} + 4 a + 12 + \left(6 a^{4} + 38 a^{3} + 41 a^{2} + 6 a + 15\right)\cdot 47 + \left(39 a^{4} + 27 a^{3} + 45 a^{2} + 17 a + 41\right)\cdot 47^{2} + \left(37 a^{4} + 14 a^{3} + 36 a^{2} + 27\right)\cdot 47^{3} + \left(19 a^{4} + 3 a^{3} + 20 a^{2} + 19 a + 11\right)\cdot 47^{4} + \left(31 a^{4} + 17 a^{3} + 13 a^{2} + 13 a + 45\right)\cdot 47^{5} + \left(11 a^{4} + 40 a^{3} + 22 a^{2} + 10 a + 1\right)\cdot 47^{6} +O(47^{7})$ 16a^4 + 36a^3 + 28a^2 + 4a + 12 + (6a^4 + 38a^3 + 41a^2 + 6a + 15)47 + (39a^4 + 27a^3 + 45a^2 + 17a + 41)47^2 + (37a^4 + 14a^3 + 36a^2 + 27)47^3 + (19a^4 + 3a^3 + 20a^2 + 19a + 11)47^4 + (31a^4 + 17a^3 + 13a^2 + 13a + 45)47^5 + (11a^4 + 40a^3 + 22a^2 + 10a + 1)*47^6+O(47^7)
$r_{ 7 }$	$=$	$ 22 a^{4} + 28 a^{3} + 37 a^{2} + 22 a + 44 + \left(39 a^{4} + 33 a^{3} + 18 a^{2} + 35 a + 43\right)\cdot 47 + \left(19 a^{4} + 7 a^{3} + 34 a^{2} + 29 a + 37\right)\cdot 47^{2} + \left(25 a^{4} + 38 a^{3} + 9 a^{2} + 18 a + 2\right)\cdot 47^{3} + \left(31 a^{4} + 21 a^{3} + 19 a^{2} + 34 a + 2\right)\cdot 47^{4} + \left(44 a^{4} + 23 a^{3} + 19 a^{2} + 21 a + 36\right)\cdot 47^{5} + \left(19 a^{4} + 40 a^{3} + 21 a^{2} + 28 a + 39\right)\cdot 47^{6} +O(47^{7})$ 22a^4 + 28a^3 + 37a^2 + 22a + 44 + (39a^4 + 33a^3 + 18a^2 + 35a + 43)47 + (19a^4 + 7a^3 + 34a^2 + 29a + 37)47^2 + (25a^4 + 38a^3 + 9a^2 + 18a + 2)47^3 + (31a^4 + 21a^3 + 19a^2 + 34a + 2)47^4 + (44a^4 + 23a^3 + 19a^2 + 21a + 36)47^5 + (19a^4 + 40a^3 + 21a^2 + 28a + 39)47^6+O(47^7)
$r_{ 8 }$	$=$	$ 27 a^{4} + 12 a^{3} + 26 a^{2} + 44 a + 2 + \left(32 a^{4} + 16 a^{3} + 37 a^{2} + 10 a + 8\right)\cdot 47 + \left(38 a^{4} + 13 a^{3} + 40 a^{2} + 6 a + 22\right)\cdot 47^{2} + \left(10 a^{4} + 4 a^{3} + 2 a^{2} + 25 a + 34\right)\cdot 47^{3} + \left(10 a^{4} + 2 a^{3} + 17 a^{2} + 19 a + 22\right)\cdot 47^{4} + \left(27 a^{4} + a^{3} + 27 a^{2} + 42 a + 32\right)\cdot 47^{5} + \left(42 a^{4} + 44 a^{3} + 36 a^{2} + 11 a + 26\right)\cdot 47^{6} +O(47^{7})$ 27a^4 + 12a^3 + 26a^2 + 44a + 2 + (32a^4 + 16a^3 + 37a^2 + 10a + 8)47 + (38a^4 + 13a^3 + 40a^2 + 6a + 22)47^2 + (10a^4 + 4a^3 + 2a^2 + 25a + 34)47^3 + (10a^4 + 2a^3 + 17a^2 + 19a + 22)47^4 + (27a^4 + a^3 + 27a^2 + 42a + 32)47^5 + (42a^4 + 44a^3 + 36a^2 + 11a + 26)*47^6+O(47^7)
$r_{ 9 }$	$=$	$ 27 a^{4} + 26 a^{3} + 24 a^{2} + 30 a + 25 + \left(20 a^{4} + 35 a^{3} + 12 a^{2} + 5 a + 31\right)\cdot 47 + \left(42 a^{4} + 34 a^{3} + 30 a^{2} + 30 a + 20\right)\cdot 47^{2} + \left(16 a^{3} + 34 a^{2} + 46 a + 39\right)\cdot 47^{3} + \left(3 a^{4} + 35 a^{3} + 36 a^{2} + 27 a + 29\right)\cdot 47^{4} + \left(37 a^{4} + 15 a^{3} + 7 a^{2} + 8 a + 18\right)\cdot 47^{5} + \left(25 a^{4} + 46 a^{3} + 46 a^{2} + 24 a + 32\right)\cdot 47^{6} +O(47^{7})$ 27a^4 + 26a^3 + 24a^2 + 30a + 25 + (20a^4 + 35a^3 + 12a^2 + 5a + 31)47 + (42a^4 + 34a^3 + 30a^2 + 30a + 20)47^2 + (16a^3 + 34a^2 + 46a + 39)47^3 + (3a^4 + 35a^3 + 36a^2 + 27a + 29)47^4 + (37a^4 + 15a^3 + 7a^2 + 8a + 18)47^5 + (25a^4 + 46a^3 + 46a^2 + 24a + 32)*47^6+O(47^7)
$r_{ 10 }$	$=$	$ 28 a^{4} + 36 a^{3} + 41 a^{2} + a + 7 + \left(36 a^{4} + 28 a^{3} + 39 a^{2} + 40 a + 16\right)\cdot 47 + \left(29 a^{4} + 13 a^{3} + 24 a^{2} + 44 a + 29\right)\cdot 47^{2} + \left(19 a^{4} + 9 a^{3} + 26 a^{2} + 4 a + 35\right)\cdot 47^{3} + \left(11 a^{4} + 29 a^{3} + 4 a^{2} + 46 a + 17\right)\cdot 47^{4} + \left(25 a^{4} + 12 a^{3} + 10 a^{2} + 39 a + 37\right)\cdot 47^{5} + \left(4 a^{4} + 42 a^{3} + 15 a^{2} + 36 a + 24\right)\cdot 47^{6} +O(47^{7})$ 28a^4 + 36a^3 + 41a^2 + a + 7 + (36a^4 + 28a^3 + 39a^2 + 40a + 16)47 + (29a^4 + 13a^3 + 24a^2 + 44a + 29)47^2 + (19a^4 + 9a^3 + 26a^2 + 4a + 35)47^3 + (11a^4 + 29a^3 + 4a^2 + 46a + 17)47^4 + (25a^4 + 12a^3 + 10a^2 + 39a + 37)47^5 + (4a^4 + 42a^3 + 15a^2 + 36a + 24)*47^6+O(47^7)
$r_{ 11 }$	$=$	$ 29 a^{4} + 27 a^{3} + 7 a^{2} + 45 a + 12 + \left(16 a^{4} + 22 a^{3} + 2 a^{2} + a + 16\right)\cdot 47 + \left(19 a^{4} + 21 a^{3} + 40 a^{2} + 30 a + 9\right)\cdot 47^{2} + \left(17 a^{4} + 11 a^{3} + 10 a^{2} + 15\right)\cdot 47^{3} + \left(40 a^{4} + 36 a^{3} + 22 a^{2} + 37 a + 37\right)\cdot 47^{4} + \left(2 a^{4} + 31 a^{3} + 10 a^{2} + 7 a + 30\right)\cdot 47^{5} + \left(13 a^{4} + 23 a^{3} + 7 a + 43\right)\cdot 47^{6} +O(47^{7})$ 29a^4 + 27a^3 + 7a^2 + 45a + 12 + (16a^4 + 22a^3 + 2a^2 + a + 16)47 + (19a^4 + 21a^3 + 40a^2 + 30a + 9)47^2 + (17a^4 + 11a^3 + 10a^2 + 15)47^3 + (40a^4 + 36a^3 + 22a^2 + 37a + 37)47^4 + (2a^4 + 31a^3 + 10a^2 + 7a + 30)47^5 + (13a^4 + 23a^3 + 7a + 43)*47^6+O(47^7)
$r_{ 12 }$	$=$	$ 33 a^{4} + 46 a^{3} + 40 a^{2} + 7 a + 35 + \left(7 a^{4} + 37 a^{3} + 6 a^{2} + 37 a + 25\right)\cdot 47 + \left(9 a^{4} + 27 a^{3} + 14 a^{2} + 45\right)\cdot 47^{2} + \left(18 a^{4} + 32 a^{3} + 28 a^{2} + 31 a + 30\right)\cdot 47^{3} + \left(7 a^{4} + 38 a^{3} + 30 a^{2} + 11 a + 1\right)\cdot 47^{4} + \left(26 a^{4} + 39 a^{3} + 7 a^{2} + 2 a + 41\right)\cdot 47^{5} + \left(4 a^{4} + 31 a^{3} + 12 a^{2} + 43 a + 33\right)\cdot 47^{6} +O(47^{7})$ 33a^4 + 46a^3 + 40a^2 + 7a + 35 + (7a^4 + 37a^3 + 6a^2 + 37a + 25)47 + (9a^4 + 27a^3 + 14a^2 + 45)47^2 + (18a^4 + 32a^3 + 28a^2 + 31a + 30)47^3 + (7a^4 + 38a^3 + 30a^2 + 11a + 1)47^4 + (26a^4 + 39a^3 + 7a^2 + 2a + 41)47^5 + (4a^4 + 31a^3 + 12a^2 + 43a + 33)*47^6+O(47^7)
$r_{ 13 }$	$=$	$ 34 a^{4} + 45 a^{3} + 8 a^{2} + 46 a + 16 + \left(13 a^{4} + 25 a^{3} + 42 a^{2} + 35 a + 23\right)\cdot 47 + \left(40 a^{4} + 37 a^{3} + 36 a^{2} + 33 a + 35\right)\cdot 47^{2} + \left(4 a^{4} + 3 a^{3} + 3 a^{2} + 16 a + 42\right)\cdot 47^{3} + \left(7 a^{4} + a^{3} + 46 a^{2} + 15 a + 38\right)\cdot 47^{4} + \left(3 a^{4} + 30 a^{3} + 11 a^{2} + 12 a + 2\right)\cdot 47^{5} + \left(31 a^{4} + 16 a^{3} + 16 a^{2} + a + 11\right)\cdot 47^{6} +O(47^{7})$ 34a^4 + 45a^3 + 8a^2 + 46a + 16 + (13a^4 + 25a^3 + 42a^2 + 35a + 23)47 + (40a^4 + 37a^3 + 36a^2 + 33a + 35)47^2 + (4a^4 + 3a^3 + 3a^2 + 16a + 42)47^3 + (7a^4 + a^3 + 46a^2 + 15a + 38)47^4 + (3a^4 + 30a^3 + 11a^2 + 12a + 2)47^5 + (31a^4 + 16a^3 + 16a^2 + a + 11)47^6+O(47^7)
$r_{ 14 }$	$=$	$ 39 a^{4} + 36 a^{3} + 28 a^{2} + 4 a + 44 + \left(22 a^{4} + 18 a^{3} + 39 a^{2} + 2 a + 23\right)\cdot 47 + \left(33 a^{4} + 39 a^{3} + 20 a^{2} + 14 a + 13\right)\cdot 47^{2} + \left(42 a^{4} + 4 a^{3} + 17 a^{2} + 15 a + 35\right)\cdot 47^{3} + \left(40 a^{4} + 7 a^{3} + 14 a^{2} + 29 a + 3\right)\cdot 47^{4} + \left(19 a^{4} + 30 a^{3} + 34 a^{2} + 14 a + 33\right)\cdot 47^{5} + \left(18 a^{4} + 20 a^{2} + 11 a + 7\right)\cdot 47^{6} +O(47^{7})$ 39a^4 + 36a^3 + 28a^2 + 4a + 44 + (22a^4 + 18a^3 + 39a^2 + 2a + 23)47 + (33a^4 + 39a^3 + 20a^2 + 14a + 13)47^2 + (42a^4 + 4a^3 + 17a^2 + 15a + 35)47^3 + (40a^4 + 7a^3 + 14a^2 + 29a + 3)47^4 + (19a^4 + 30a^3 + 34a^2 + 14a + 33)47^5 + (18a^4 + 20a^2 + 11a + 7)*47^6+O(47^7)
$r_{ 15 }$	$=$	$ 43 a^{4} + 10 a^{3} + 10 a^{2} + 19 a + 19 + \left(18 a^{4} + 15 a^{3} + 24 a^{2} + 41 a + 11\right)\cdot 47 + \left(7 a^{4} + 33 a^{3} + 37 a^{2} + 27 a + 30\right)\cdot 47^{2} + \left(16 a^{4} + 6 a^{3} + 43 a^{2} + 4 a + 4\right)\cdot 47^{3} + \left(19 a^{4} + 22 a^{3} + 4 a^{2} + 26 a + 24\right)\cdot 47^{4} + \left(10 a^{4} + 2 a^{3} + 18 a^{2} + 14 a + 25\right)\cdot 47^{5} + \left(11 a^{4} + 7 a^{3} + 5 a^{2} + 37 a + 39\right)\cdot 47^{6} +O(47^{7})$ 43a^4 + 10a^3 + 10a^2 + 19a + 19 + (18a^4 + 15a^3 + 24a^2 + 41a + 11)47 + (7a^4 + 33a^3 + 37a^2 + 27a + 30)47^2 + (16a^4 + 6a^3 + 43a^2 + 4a + 4)47^3 + (19a^4 + 22a^3 + 4a^2 + 26a + 24)47^4 + (10a^4 + 2a^3 + 18a^2 + 14a + 25)47^5 + (11a^4 + 7a^3 + 5a^2 + 37a + 39)47^6+O(47^7)

Generators of the action on the roots $r_1, \ldots, r_{ 15 }$

Cycle notation
$(2,6)(3,14)(5,10)(8,9)(12,15)$
$(1,2)(4,9)(7,15)(10,13)(11,14)$
$(1,11,7,13,4)(2,14,15,10,9)(3,12,5,8,6)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 15 }$	Character value	Complex conjugation
$1$	$1$	$()$	$2$
$3$	$2$	$(1,2)(4,9)(7,15)(10,13)(11,14)$	$0$	✓
$2$	$3$	$(1,6,2)(3,14,11)(4,8,9)(5,10,13)(7,12,15)$	$-1$
$1$	$5$	$(1,11,7,13,4)(2,14,15,10,9)(3,12,5,8,6)$	$2 \zeta_{5}^{3}$
$1$	$5$	$(1,7,4,11,13)(2,15,9,14,10)(3,5,6,12,8)$	$2 \zeta_{5}$
$1$	$5$	$(1,13,11,4,7)(2,10,14,9,15)(3,8,12,6,5)$	$-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$
$1$	$5$	$(1,4,13,7,11)(2,9,10,15,14)(3,6,8,5,12)$	$2 \zeta_{5}^{2}$
$3$	$10$	$(1,14,7,10,4,2,11,15,13,9)(3,12,5,8,6)$	$0$
$3$	$10$	$(1,10,11,9,7,2,13,14,4,15)(3,8,12,6,5)$	$0$
$3$	$10$	$(1,15,4,14,13,2,7,9,11,10)(3,5,6,12,8)$	$0$
$3$	$10$	$(1,9,13,15,11,2,4,10,7,14)(3,6,8,5,12)$	$0$
$2$	$15$	$(1,3,15,13,8,2,11,12,10,4,6,14,7,5,9)$	$-\zeta_{5}^{3}$
$2$	$15$	$(1,15,8,11,10,6,7,9,3,13,2,12,4,14,5)$	$-\zeta_{5}$
$2$	$15$	$(1,8,10,7,3,2,4,5,15,11,6,9,13,12,14)$	$-\zeta_{5}^{2}$
$2$	$15$	$(1,5,14,4,12,2,13,3,9,7,6,10,11,8,15)$	$\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$

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Basic invariants

Defining polynomial

Generators of the action on the roots $r_1, \ldots, r_{ 15 }$

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	2.961.15t4.a.c
Dimension	$2$
Group	$S_3 \times C_5$
Conductor	$961$
Root number	not computed
Indicator	$0$

$r_{ 1 }$	$=$	\( 26 a^{3} + 21 a^{2} + 40 a + 17 + \left(13 a^{4} + 37 a^{3} + 40 a^{2} + 45 a + 13\right)\cdot 47 + \left(23 a^{4} + 43 a^{3} + 36 a^{2} + 42 a + 31\right)\cdot 47^{2} + \left(29 a^{4} + 23 a^{3} + 12 a^{2} + 42 a + 43\right)\cdot 47^{3} + \left(33 a^{4} + 39 a^{3} + 11 a^{2} + 19 a + 31\right)\cdot 47^{4} + \left(5 a^{4} + 35 a^{3} + 17 a^{2} + 22 a + 23\right)\cdot 47^{5} + \left(18 a^{4} + 36 a^{2} + 10 a + 19\right)\cdot 47^{6} +O(47^{7})\) 26a^3 + 21a^2 + 40a + 17 + (13a^4 + 37a^3 + 40a^2 + 45a + 13)47 + (23a^4 + 43a^3 + 36a^2 + 42a + 31)47^2 + (29a^4 + 23a^3 + 12a^2 + 42a + 43)47^3 + (33a^4 + 39a^3 + 11a^2 + 19a + 31)47^4 + (5a^4 + 35a^3 + 17a^2 + 22a + 23)47^5 + (18a^4 + 36a^2 + 10a + 19)47^6+O(47^7)
$r_{ 2 }$	$=$	\( 4 a^{4} + 33 a^{3} + 38 a^{2} + 40 a + 16 + \left(42 a^{4} + 42 a^{3} + 24 a^{2} + 4 a + 39\right)\cdot 47 + \left(27 a^{4} + 19 a^{3} + 27 a^{2} + 24 a + 27\right)\cdot 47^{2} + \left(14 a^{4} + 9 a^{3} + 18 a^{2} + 22 a + 31\right)\cdot 47^{3} + \left(19 a^{4} + 33 a^{2} + 11 a + 14\right)\cdot 47^{4} + \left(a^{4} + 33 a^{3} + 23 a^{2} + 16 a + 18\right)\cdot 47^{5} + \left(34 a^{4} + 44 a^{3} + 6 a^{2} + 31 a + 1\right)\cdot 47^{6} +O(47^{7})\) 4a^4 + 33a^3 + 38a^2 + 40a + 16 + (42a^4 + 42a^3 + 24a^2 + 4a + 39)47 + (27a^4 + 19a^3 + 27a^2 + 24a + 27)47^2 + (14a^4 + 9a^3 + 18a^2 + 22a + 31)47^3 + (19a^4 + 33a^2 + 11a + 14)47^4 + (a^4 + 33a^3 + 23a^2 + 16a + 18)47^5 + (34a^4 + 44a^3 + 6a^2 + 31a + 1)47^6+O(47^7)
$r_{ 3 }$	$=$	\( 5 a^{4} + 5 a^{3} + 46 a^{2} + 43 a + 22 + \left(23 a^{4} + 15 a^{3} + 19 a^{2} + 17 a\right)\cdot 47 + \left(34 a^{4} + 34 a^{3} + 40 a^{2} + 46 a\right)\cdot 47^{2} + \left(13 a^{4} + 36 a^{3} + 33 a^{2} + 42 a + 18\right)\cdot 47^{3} + \left(43 a^{4} + 4 a^{3} + 43 a^{2} + 15 a + 30\right)\cdot 47^{4} + \left(44 a^{4} + 43 a^{3} + 8 a^{2} + 17 a + 46\right)\cdot 47^{5} + \left(28 a^{4} + 46 a^{3} + 5 a^{2} + 45 a + 43\right)\cdot 47^{6} +O(47^{7})\) 5a^4 + 5a^3 + 46a^2 + 43a + 22 + (23a^4 + 15a^3 + 19a^2 + 17a)47 + (34a^4 + 34a^3 + 40a^2 + 46a)47^2 + (13a^4 + 36a^3 + 33a^2 + 42a + 18)47^3 + (43a^4 + 4a^3 + 43a^2 + 15a + 30)47^4 + (44a^4 + 43a^3 + 8a^2 + 17a + 46)47^5 + (28a^4 + 46a^3 + 5a^2 + 45a + 43)47^6+O(47^7)
$r_{ 4 }$	$=$	\( 9 a^{4} + 15 a^{3} + 21 a^{2} + 35 a + 43 + \left(11 a^{4} + 21 a^{3} + 37 a^{2} + 21 a + 11\right)\cdot 47 + \left(38 a^{4} + 30 a^{3} + 39 a^{2} + 4 a + 43\right)\cdot 47^{2} + \left(16 a^{4} + 16 a^{3} + 9 a^{2} + 15 a + 42\right)\cdot 47^{3} + \left(28 a^{4} + 42 a^{3} + 42 a^{2} + 34 a + 8\right)\cdot 47^{4} + \left(37 a^{4} + 19 a^{3} + 34 a^{2} + 29 a + 2\right)\cdot 47^{5} + \left(11 a^{4} + 12 a^{3} + 19 a^{2} + 46 a + 5\right)\cdot 47^{6} +O(47^{7})\) 9a^4 + 15a^3 + 21a^2 + 35a + 43 + (11a^4 + 21a^3 + 37a^2 + 21a + 11)47 + (38a^4 + 30a^3 + 39a^2 + 4a + 43)47^2 + (16a^4 + 16a^3 + 9a^2 + 15a + 42)47^3 + (28a^4 + 42a^3 + 42a^2 + 34a + 8)47^4 + (37a^4 + 19a^3 + 34a^2 + 29a + 2)47^5 + (11a^4 + 12a^3 + 19a^2 + 46a + 5)47^6+O(47^7)
$r_{ 5 }$	$=$	\( 13 a^{4} + 42 a^{3} + a^{2} + 43 a + 19 + \left(24 a^{4} + 32 a^{3} + 35 a^{2} + 21 a + 1\right)\cdot 47 + \left(19 a^{4} + 37 a^{3} + 46 a^{2} + 23 a + 35\right)\cdot 47^{2} + \left(13 a^{4} + 5 a^{3} + 38 a^{2} + 41 a + 17\right)\cdot 47^{3} + \left(13 a^{4} + 45 a^{3} + 28 a^{2} + 27 a + 6\right)\cdot 47^{4} + \left(11 a^{4} + 39 a^{3} + 36 a^{2} + 18 a + 29\right)\cdot 47^{5} + \left(6 a^{4} + 24 a^{3} + 17 a^{2} + 30 a + 44\right)\cdot 47^{6} +O(47^{7})\) 13a^4 + 42a^3 + a^2 + 43a + 19 + (24a^4 + 32a^3 + 35a^2 + 21a + 1)47 + (19a^4 + 37a^3 + 46a^2 + 23a + 35)47^2 + (13a^4 + 5a^3 + 38a^2 + 41a + 17)47^3 + (13a^4 + 45a^3 + 28a^2 + 27a + 6)47^4 + (11a^4 + 39a^3 + 36a^2 + 18a + 29)47^5 + (6a^4 + 24a^3 + 17a^2 + 30a + 44)*47^6+O(47^7)
$r_{ 6 }$	$=$	\( 16 a^{4} + 36 a^{3} + 28 a^{2} + 4 a + 12 + \left(6 a^{4} + 38 a^{3} + 41 a^{2} + 6 a + 15\right)\cdot 47 + \left(39 a^{4} + 27 a^{3} + 45 a^{2} + 17 a + 41\right)\cdot 47^{2} + \left(37 a^{4} + 14 a^{3} + 36 a^{2} + 27\right)\cdot 47^{3} + \left(19 a^{4} + 3 a^{3} + 20 a^{2} + 19 a + 11\right)\cdot 47^{4} + \left(31 a^{4} + 17 a^{3} + 13 a^{2} + 13 a + 45\right)\cdot 47^{5} + \left(11 a^{4} + 40 a^{3} + 22 a^{2} + 10 a + 1\right)\cdot 47^{6} +O(47^{7})\) 16a^4 + 36a^3 + 28a^2 + 4a + 12 + (6a^4 + 38a^3 + 41a^2 + 6a + 15)47 + (39a^4 + 27a^3 + 45a^2 + 17a + 41)47^2 + (37a^4 + 14a^3 + 36a^2 + 27)47^3 + (19a^4 + 3a^3 + 20a^2 + 19a + 11)47^4 + (31a^4 + 17a^3 + 13a^2 + 13a + 45)47^5 + (11a^4 + 40a^3 + 22a^2 + 10a + 1)*47^6+O(47^7)
$r_{ 7 }$	$=$	\( 22 a^{4} + 28 a^{3} + 37 a^{2} + 22 a + 44 + \left(39 a^{4} + 33 a^{3} + 18 a^{2} + 35 a + 43\right)\cdot 47 + \left(19 a^{4} + 7 a^{3} + 34 a^{2} + 29 a + 37\right)\cdot 47^{2} + \left(25 a^{4} + 38 a^{3} + 9 a^{2} + 18 a + 2\right)\cdot 47^{3} + \left(31 a^{4} + 21 a^{3} + 19 a^{2} + 34 a + 2\right)\cdot 47^{4} + \left(44 a^{4} + 23 a^{3} + 19 a^{2} + 21 a + 36\right)\cdot 47^{5} + \left(19 a^{4} + 40 a^{3} + 21 a^{2} + 28 a + 39\right)\cdot 47^{6} +O(47^{7})\) 22a^4 + 28a^3 + 37a^2 + 22a + 44 + (39a^4 + 33a^3 + 18a^2 + 35a + 43)47 + (19a^4 + 7a^3 + 34a^2 + 29a + 37)47^2 + (25a^4 + 38a^3 + 9a^2 + 18a + 2)47^3 + (31a^4 + 21a^3 + 19a^2 + 34a + 2)47^4 + (44a^4 + 23a^3 + 19a^2 + 21a + 36)47^5 + (19a^4 + 40a^3 + 21a^2 + 28a + 39)47^6+O(47^7)
$r_{ 8 }$	$=$	\( 27 a^{4} + 12 a^{3} + 26 a^{2} + 44 a + 2 + \left(32 a^{4} + 16 a^{3} + 37 a^{2} + 10 a + 8\right)\cdot 47 + \left(38 a^{4} + 13 a^{3} + 40 a^{2} + 6 a + 22\right)\cdot 47^{2} + \left(10 a^{4} + 4 a^{3} + 2 a^{2} + 25 a + 34\right)\cdot 47^{3} + \left(10 a^{4} + 2 a^{3} + 17 a^{2} + 19 a + 22\right)\cdot 47^{4} + \left(27 a^{4} + a^{3} + 27 a^{2} + 42 a + 32\right)\cdot 47^{5} + \left(42 a^{4} + 44 a^{3} + 36 a^{2} + 11 a + 26\right)\cdot 47^{6} +O(47^{7})\) 27a^4 + 12a^3 + 26a^2 + 44a + 2 + (32a^4 + 16a^3 + 37a^2 + 10a + 8)47 + (38a^4 + 13a^3 + 40a^2 + 6a + 22)47^2 + (10a^4 + 4a^3 + 2a^2 + 25a + 34)47^3 + (10a^4 + 2a^3 + 17a^2 + 19a + 22)47^4 + (27a^4 + a^3 + 27a^2 + 42a + 32)47^5 + (42a^4 + 44a^3 + 36a^2 + 11a + 26)*47^6+O(47^7)
$r_{ 9 }$	$=$	\( 27 a^{4} + 26 a^{3} + 24 a^{2} + 30 a + 25 + \left(20 a^{4} + 35 a^{3} + 12 a^{2} + 5 a + 31\right)\cdot 47 + \left(42 a^{4} + 34 a^{3} + 30 a^{2} + 30 a + 20\right)\cdot 47^{2} + \left(16 a^{3} + 34 a^{2} + 46 a + 39\right)\cdot 47^{3} + \left(3 a^{4} + 35 a^{3} + 36 a^{2} + 27 a + 29\right)\cdot 47^{4} + \left(37 a^{4} + 15 a^{3} + 7 a^{2} + 8 a + 18\right)\cdot 47^{5} + \left(25 a^{4} + 46 a^{3} + 46 a^{2} + 24 a + 32\right)\cdot 47^{6} +O(47^{7})\) 27a^4 + 26a^3 + 24a^2 + 30a + 25 + (20a^4 + 35a^3 + 12a^2 + 5a + 31)47 + (42a^4 + 34a^3 + 30a^2 + 30a + 20)47^2 + (16a^3 + 34a^2 + 46a + 39)47^3 + (3a^4 + 35a^3 + 36a^2 + 27a + 29)47^4 + (37a^4 + 15a^3 + 7a^2 + 8a + 18)47^5 + (25a^4 + 46a^3 + 46a^2 + 24a + 32)*47^6+O(47^7)
$r_{ 10 }$	$=$	\( 28 a^{4} + 36 a^{3} + 41 a^{2} + a + 7 + \left(36 a^{4} + 28 a^{3} + 39 a^{2} + 40 a + 16\right)\cdot 47 + \left(29 a^{4} + 13 a^{3} + 24 a^{2} + 44 a + 29\right)\cdot 47^{2} + \left(19 a^{4} + 9 a^{3} + 26 a^{2} + 4 a + 35\right)\cdot 47^{3} + \left(11 a^{4} + 29 a^{3} + 4 a^{2} + 46 a + 17\right)\cdot 47^{4} + \left(25 a^{4} + 12 a^{3} + 10 a^{2} + 39 a + 37\right)\cdot 47^{5} + \left(4 a^{4} + 42 a^{3} + 15 a^{2} + 36 a + 24\right)\cdot 47^{6} +O(47^{7})\) 28a^4 + 36a^3 + 41a^2 + a + 7 + (36a^4 + 28a^3 + 39a^2 + 40a + 16)47 + (29a^4 + 13a^3 + 24a^2 + 44a + 29)47^2 + (19a^4 + 9a^3 + 26a^2 + 4a + 35)47^3 + (11a^4 + 29a^3 + 4a^2 + 46a + 17)47^4 + (25a^4 + 12a^3 + 10a^2 + 39a + 37)47^5 + (4a^4 + 42a^3 + 15a^2 + 36a + 24)*47^6+O(47^7)
$r_{ 11 }$	$=$	\( 29 a^{4} + 27 a^{3} + 7 a^{2} + 45 a + 12 + \left(16 a^{4} + 22 a^{3} + 2 a^{2} + a + 16\right)\cdot 47 + \left(19 a^{4} + 21 a^{3} + 40 a^{2} + 30 a + 9\right)\cdot 47^{2} + \left(17 a^{4} + 11 a^{3} + 10 a^{2} + 15\right)\cdot 47^{3} + \left(40 a^{4} + 36 a^{3} + 22 a^{2} + 37 a + 37\right)\cdot 47^{4} + \left(2 a^{4} + 31 a^{3} + 10 a^{2} + 7 a + 30\right)\cdot 47^{5} + \left(13 a^{4} + 23 a^{3} + 7 a + 43\right)\cdot 47^{6} +O(47^{7})\) 29a^4 + 27a^3 + 7a^2 + 45a + 12 + (16a^4 + 22a^3 + 2a^2 + a + 16)47 + (19a^4 + 21a^3 + 40a^2 + 30a + 9)47^2 + (17a^4 + 11a^3 + 10a^2 + 15)47^3 + (40a^4 + 36a^3 + 22a^2 + 37a + 37)47^4 + (2a^4 + 31a^3 + 10a^2 + 7a + 30)47^5 + (13a^4 + 23a^3 + 7a + 43)*47^6+O(47^7)
$r_{ 12 }$	$=$	\( 33 a^{4} + 46 a^{3} + 40 a^{2} + 7 a + 35 + \left(7 a^{4} + 37 a^{3} + 6 a^{2} + 37 a + 25\right)\cdot 47 + \left(9 a^{4} + 27 a^{3} + 14 a^{2} + 45\right)\cdot 47^{2} + \left(18 a^{4} + 32 a^{3} + 28 a^{2} + 31 a + 30\right)\cdot 47^{3} + \left(7 a^{4} + 38 a^{3} + 30 a^{2} + 11 a + 1\right)\cdot 47^{4} + \left(26 a^{4} + 39 a^{3} + 7 a^{2} + 2 a + 41\right)\cdot 47^{5} + \left(4 a^{4} + 31 a^{3} + 12 a^{2} + 43 a + 33\right)\cdot 47^{6} +O(47^{7})\) 33a^4 + 46a^3 + 40a^2 + 7a + 35 + (7a^4 + 37a^3 + 6a^2 + 37a + 25)47 + (9a^4 + 27a^3 + 14a^2 + 45)47^2 + (18a^4 + 32a^3 + 28a^2 + 31a + 30)47^3 + (7a^4 + 38a^3 + 30a^2 + 11a + 1)47^4 + (26a^4 + 39a^3 + 7a^2 + 2a + 41)47^5 + (4a^4 + 31a^3 + 12a^2 + 43a + 33)*47^6+O(47^7)
$r_{ 13 }$	$=$	\( 34 a^{4} + 45 a^{3} + 8 a^{2} + 46 a + 16 + \left(13 a^{4} + 25 a^{3} + 42 a^{2} + 35 a + 23\right)\cdot 47 + \left(40 a^{4} + 37 a^{3} + 36 a^{2} + 33 a + 35\right)\cdot 47^{2} + \left(4 a^{4} + 3 a^{3} + 3 a^{2} + 16 a + 42\right)\cdot 47^{3} + \left(7 a^{4} + a^{3} + 46 a^{2} + 15 a + 38\right)\cdot 47^{4} + \left(3 a^{4} + 30 a^{3} + 11 a^{2} + 12 a + 2\right)\cdot 47^{5} + \left(31 a^{4} + 16 a^{3} + 16 a^{2} + a + 11\right)\cdot 47^{6} +O(47^{7})\) 34a^4 + 45a^3 + 8a^2 + 46a + 16 + (13a^4 + 25a^3 + 42a^2 + 35a + 23)47 + (40a^4 + 37a^3 + 36a^2 + 33a + 35)47^2 + (4a^4 + 3a^3 + 3a^2 + 16a + 42)47^3 + (7a^4 + a^3 + 46a^2 + 15a + 38)47^4 + (3a^4 + 30a^3 + 11a^2 + 12a + 2)47^5 + (31a^4 + 16a^3 + 16a^2 + a + 11)47^6+O(47^7)
$r_{ 14 }$	$=$	\( 39 a^{4} + 36 a^{3} + 28 a^{2} + 4 a + 44 + \left(22 a^{4} + 18 a^{3} + 39 a^{2} + 2 a + 23\right)\cdot 47 + \left(33 a^{4} + 39 a^{3} + 20 a^{2} + 14 a + 13\right)\cdot 47^{2} + \left(42 a^{4} + 4 a^{3} + 17 a^{2} + 15 a + 35\right)\cdot 47^{3} + \left(40 a^{4} + 7 a^{3} + 14 a^{2} + 29 a + 3\right)\cdot 47^{4} + \left(19 a^{4} + 30 a^{3} + 34 a^{2} + 14 a + 33\right)\cdot 47^{5} + \left(18 a^{4} + 20 a^{2} + 11 a + 7\right)\cdot 47^{6} +O(47^{7})\) 39a^4 + 36a^3 + 28a^2 + 4a + 44 + (22a^4 + 18a^3 + 39a^2 + 2a + 23)47 + (33a^4 + 39a^3 + 20a^2 + 14a + 13)47^2 + (42a^4 + 4a^3 + 17a^2 + 15a + 35)47^3 + (40a^4 + 7a^3 + 14a^2 + 29a + 3)47^4 + (19a^4 + 30a^3 + 34a^2 + 14a + 33)47^5 + (18a^4 + 20a^2 + 11a + 7)*47^6+O(47^7)
$r_{ 15 }$	$=$	\( 43 a^{4} + 10 a^{3} + 10 a^{2} + 19 a + 19 + \left(18 a^{4} + 15 a^{3} + 24 a^{2} + 41 a + 11\right)\cdot 47 + \left(7 a^{4} + 33 a^{3} + 37 a^{2} + 27 a + 30\right)\cdot 47^{2} + \left(16 a^{4} + 6 a^{3} + 43 a^{2} + 4 a + 4\right)\cdot 47^{3} + \left(19 a^{4} + 22 a^{3} + 4 a^{2} + 26 a + 24\right)\cdot 47^{4} + \left(10 a^{4} + 2 a^{3} + 18 a^{2} + 14 a + 25\right)\cdot 47^{5} + \left(11 a^{4} + 7 a^{3} + 5 a^{2} + 37 a + 39\right)\cdot 47^{6} +O(47^{7})\) 43a^4 + 10a^3 + 10a^2 + 19a + 19 + (18a^4 + 15a^3 + 24a^2 + 41a + 11)47 + (7a^4 + 33a^3 + 37a^2 + 27a + 30)47^2 + (16a^4 + 6a^3 + 43a^2 + 4a + 4)47^3 + (19a^4 + 22a^3 + 4a^2 + 26a + 24)47^4 + (10a^4 + 2a^3 + 18a^2 + 14a + 25)47^5 + (11a^4 + 7a^3 + 5a^2 + 37a + 39)47^6+O(47^7)