LMFDB - Artin representation 2.2541.15t4.b.d

Basic invariants

Dimension:	$2$
Group:	$S_3 \times C_5$
Conductor:	$2541$$\medspace = 3 \cdot 7 \cdot 11^{2} $
Artin stem field:	Galois closure of 15.5.140994243189740741031.1
Galois orbit size:	$4$
Smallest permutation container:	$S_3 \times C_5$
Parity:	odd
Determinant:	1.231.10t1.b.d
Projective image:	$S_3$
Projective stem field:	Galois closure of 3.1.231.1

Defining polynomial

$f(x)$

$=$

$ x^{15} - 4 x^{14} + 2 x^{13} + 25 x^{12} - 77 x^{11} + 56 x^{10} + 46 x^{9} + 26 x^{8} - 157 x^{7} + \cdots + 1 $

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{5} + 12x + 59 $ x^5 + 12*x + 59

Roots:

$r_{ 1 }$	$=$	$ 37 a^{3} + 13 a^{2} + 30 a + 44 + \left(32 a^{4} + 41 a^{3} + 2 a^{2} + 29 a\right)\cdot 61 + \left(8 a^{4} + 51 a^{3} + 44 a^{2} + 12 a + 7\right)\cdot 61^{2} + \left(17 a^{4} + 3 a^{3} + 38 a^{2} + 49 a + 5\right)\cdot 61^{3} + \left(49 a^{4} + 25 a^{3} + 33 a^{2} + 14 a + 40\right)\cdot 61^{4} + \left(60 a^{4} + 26 a^{3} + 32 a^{2} + 20\right)\cdot 61^{5} +O(61^{6})$ 37a^3 + 13a^2 + 30a + 44 + (32a^4 + 41a^3 + 2a^2 + 29a)61 + (8a^4 + 51a^3 + 44a^2 + 12a + 7)61^2 + (17a^4 + 3a^3 + 38a^2 + 49a + 5)61^3 + (49a^4 + 25a^3 + 33a^2 + 14a + 40)61^4 + (60a^4 + 26a^3 + 32a^2 + 20)*61^5+O(61^6)
$r_{ 2 }$	$=$	$ a^{4} + 51 a^{3} + 6 a^{2} + 48 a + 17 + \left(11 a^{4} + 52 a^{3} + 40 a^{2} + 11 a + 43\right)\cdot 61 + \left(44 a^{4} + 57 a^{3} + 24 a^{2} + 17 a + 7\right)\cdot 61^{2} + \left(23 a^{4} + 39 a^{3} + 40 a^{2} + 24 a + 56\right)\cdot 61^{3} + \left(34 a^{4} + 47 a^{3} + 45 a^{2} + 31 a + 6\right)\cdot 61^{4} + \left(55 a^{4} + 43 a^{3} + 51 a^{2} + 43 a + 19\right)\cdot 61^{5} +O(61^{6})$ a^4 + 51a^3 + 6a^2 + 48a + 17 + (11a^4 + 52a^3 + 40a^2 + 11a + 43)61 + (44a^4 + 57a^3 + 24a^2 + 17a + 7)61^2 + (23a^4 + 39a^3 + 40a^2 + 24a + 56)61^3 + (34a^4 + 47a^3 + 45a^2 + 31a + 6)61^4 + (55a^4 + 43a^3 + 51a^2 + 43a + 19)61^5+O(61^6)
$r_{ 3 }$	$=$	$ 3 a^{4} + 10 a^{3} + 21 a^{2} + 12 a + 7 + \left(59 a^{4} + 14 a^{3} + 37 a^{2} + 20 a + 52\right)\cdot 61 + \left(28 a^{4} + 55 a^{3} + 35 a^{2} + 17 a + 56\right)\cdot 61^{2} + \left(47 a^{4} + 48 a^{3} + 26 a^{2} + 27 a + 29\right)\cdot 61^{3} + \left(26 a^{4} + 30 a^{3} + 25 a^{2} + 33 a + 44\right)\cdot 61^{4} + \left(28 a^{4} + 46 a^{3} + 43 a^{2} + 54 a + 18\right)\cdot 61^{5} +O(61^{6})$ 3a^4 + 10a^3 + 21a^2 + 12a + 7 + (59a^4 + 14a^3 + 37a^2 + 20a + 52)61 + (28a^4 + 55a^3 + 35a^2 + 17a + 56)61^2 + (47a^4 + 48a^3 + 26a^2 + 27a + 29)61^3 + (26a^4 + 30a^3 + 25a^2 + 33a + 44)61^4 + (28a^4 + 46a^3 + 43a^2 + 54a + 18)*61^5+O(61^6)
$r_{ 4 }$	$=$	$ 3 a^{4} + 15 a^{3} + 42 a^{2} + 4 a + 44 + \left(9 a^{4} + 11 a^{3} + 12 a^{2} + 16 a + 29\right)\cdot 61 + \left(2 a^{4} + 50 a^{3} + 46 a^{2} + 2 a + 60\right)\cdot 61^{2} + \left(11 a^{4} + 33 a^{3} + 24 a^{2} + 9 a + 7\right)\cdot 61^{3} + \left(35 a^{4} + 13 a^{3} + 21 a^{2} + 56\right)\cdot 61^{4} + \left(24 a^{4} + 56 a^{3} + 6 a^{2} + 36 a + 52\right)\cdot 61^{5} +O(61^{6})$ 3a^4 + 15a^3 + 42a^2 + 4a + 44 + (9a^4 + 11a^3 + 12a^2 + 16a + 29)61 + (2a^4 + 50a^3 + 46a^2 + 2a + 60)61^2 + (11a^4 + 33a^3 + 24a^2 + 9a + 7)61^3 + (35a^4 + 13a^3 + 21a^2 + 56)61^4 + (24a^4 + 56a^3 + 6a^2 + 36a + 52)61^5+O(61^6)
$r_{ 5 }$	$=$	$ 5 a^{4} + 31 a^{3} + 8 a^{2} + 8 a + 14 + \left(49 a^{4} + 30 a^{3} + 32 a^{2} + 35 a + 5\right)\cdot 61 + \left(4 a^{4} + 54 a^{3} + 56 a^{2} + 31 a + 20\right)\cdot 61^{2} + \left(52 a^{4} + 45 a^{3} + 16 a^{2} + 12 a + 25\right)\cdot 61^{3} + \left(59 a^{4} + 58 a^{3} + 58 a^{2} + 32 a + 20\right)\cdot 61^{4} + \left(52 a^{4} + 36 a^{3} + 55 a^{2} + 3 a + 10\right)\cdot 61^{5} +O(61^{6})$ 5a^4 + 31a^3 + 8a^2 + 8a + 14 + (49a^4 + 30a^3 + 32a^2 + 35a + 5)61 + (4a^4 + 54a^3 + 56a^2 + 31a + 20)61^2 + (52a^4 + 45a^3 + 16a^2 + 12a + 25)61^3 + (59a^4 + 58a^3 + 58a^2 + 32a + 20)61^4 + (52a^4 + 36a^3 + 55a^2 + 3a + 10)*61^5+O(61^6)
$r_{ 6 }$	$=$	$ 10 a^{4} + 3 a^{3} + 21 a^{2} + 21 a + 38 + \left(45 a^{4} + 46 a^{3} + 24 a^{2} + 10 a + 22\right)\cdot 61 + \left(30 a^{4} + 17 a^{3} + 12 a^{2} + 41 a + 54\right)\cdot 61^{2} + \left(32 a^{4} + 28 a^{3} + 36 a^{2} + 34 a + 18\right)\cdot 61^{3} + \left(33 a^{4} + 33 a^{3} + 10 a^{2} + 59 a + 40\right)\cdot 61^{4} + \left(a^{4} + 56 a^{3} + 16 a^{2} + 9 a + 2\right)\cdot 61^{5} +O(61^{6})$ 10a^4 + 3a^3 + 21a^2 + 21a + 38 + (45a^4 + 46a^3 + 24a^2 + 10a + 22)61 + (30a^4 + 17a^3 + 12a^2 + 41a + 54)61^2 + (32a^4 + 28a^3 + 36a^2 + 34a + 18)61^3 + (33a^4 + 33a^3 + 10a^2 + 59a + 40)61^4 + (a^4 + 56a^3 + 16a^2 + 9a + 2)61^5+O(61^6)
$r_{ 7 }$	$=$	$ 13 a^{4} + 29 a^{3} + 26 a^{2} + 26 a + 59 + \left(5 a^{4} + 13 a^{3} + 11 a^{2} + 59 a + 11\right)\cdot 61 + \left(a^{4} + 55 a^{3} + 41 a^{2} + 38 a + 33\right)\cdot 61^{2} + \left(55 a^{4} + 28 a^{3} + 5 a^{2} + 12 a + 51\right)\cdot 61^{3} + \left(41 a^{4} + 39 a^{3} + 48 a^{2} + 51 a + 5\right)\cdot 61^{4} + \left(7 a^{4} + 26 a^{3} + 10 a^{2} + 8 a + 23\right)\cdot 61^{5} +O(61^{6})$ 13a^4 + 29a^3 + 26a^2 + 26a + 59 + (5a^4 + 13a^3 + 11a^2 + 59a + 11)61 + (a^4 + 55a^3 + 41a^2 + 38a + 33)61^2 + (55a^4 + 28a^3 + 5a^2 + 12a + 51)61^3 + (41a^4 + 39a^3 + 48a^2 + 51a + 5)61^4 + (7a^4 + 26a^3 + 10a^2 + 8a + 23)61^5+O(61^6)
$r_{ 8 }$	$=$	$ 20 a^{4} + 25 a^{3} + 20 a^{2} + 31 a + 12 + \left(20 a^{4} + 17 a^{3} + 30 a^{2} + 14 a + 28\right)\cdot 61 + \left(a^{3} + a^{2} + 21 a + 6\right)\cdot 61^{2} + \left(12 a^{4} + 42 a^{3} + 36 a^{2} + 32 a + 5\right)\cdot 61^{3} + \left(60 a^{4} + 4 a^{3} + 5 a^{2} + 39 a + 40\right)\cdot 61^{4} + \left(41 a^{4} + 4 a^{3} + 46 a^{2} + 12 a + 12\right)\cdot 61^{5} +O(61^{6})$ 20a^4 + 25a^3 + 20a^2 + 31a + 12 + (20a^4 + 17a^3 + 30a^2 + 14a + 28)61 + (a^3 + a^2 + 21a + 6)61^2 + (12a^4 + 42a^3 + 36a^2 + 32a + 5)61^3 + (60a^4 + 4a^3 + 5a^2 + 39a + 40)61^4 + (41a^4 + 4a^3 + 46a^2 + 12a + 12)61^5+O(61^6)
$r_{ 9 }$	$=$	$ 22 a^{4} + 34 a^{3} + 35 a^{2} + 22 a + 43 + \left(36 a^{4} + 46 a^{3} + 41 a^{2} + 3 a + 41\right)\cdot 61 + \left(3 a^{4} + 49 a^{3} + 32 a^{2} + 2 a + 20\right)\cdot 61^{2} + \left(53 a^{4} + 22 a^{3} + 18 a^{2} + 21 a + 10\right)\cdot 61^{3} + \left(53 a^{4} + 26 a^{3} + 37 a^{2} + 22 a + 36\right)\cdot 61^{4} + \left(26 a^{4} + 29 a^{3} + 12 a^{2} + 51 a + 52\right)\cdot 61^{5} +O(61^{6})$ 22a^4 + 34a^3 + 35a^2 + 22a + 43 + (36a^4 + 46a^3 + 41a^2 + 3a + 41)61 + (3a^4 + 49a^3 + 32a^2 + 2a + 20)61^2 + (53a^4 + 22a^3 + 18a^2 + 21a + 10)61^3 + (53a^4 + 26a^3 + 37a^2 + 22a + 36)61^4 + (26a^4 + 29a^3 + 12a^2 + 51a + 52)*61^5+O(61^6)
$r_{ 10 }$	$=$	$ 31 a^{4} + 48 a^{3} + 2 a^{2} + 26 a + 20 + \left(50 a^{4} + 39 a^{3} + 34 a^{2} + 30 a + 37\right)\cdot 61 + \left(28 a^{4} + 18 a^{3} + 20 a^{2} + 21 a + 11\right)\cdot 61^{2} + \left(26 a^{4} + 33 a^{3} + 55 a^{2} + 49 a + 34\right)\cdot 61^{3} + \left(17 a^{4} + 28 a^{3} + 44 a^{2} + 34 a + 56\right)\cdot 61^{4} + \left(23 a^{4} + 21 a^{3} + 35 a^{2} + 43 a + 3\right)\cdot 61^{5} +O(61^{6})$ 31a^4 + 48a^3 + 2a^2 + 26a + 20 + (50a^4 + 39a^3 + 34a^2 + 30a + 37)61 + (28a^4 + 18a^3 + 20a^2 + 21a + 11)61^2 + (26a^4 + 33a^3 + 55a^2 + 49a + 34)61^3 + (17a^4 + 28a^3 + 44a^2 + 34a + 56)61^4 + (23a^4 + 21a^3 + 35a^2 + 43a + 3)*61^5+O(61^6)
$r_{ 11 }$	$=$	$ 37 a^{4} + 34 a^{3} + 56 a^{2} + 35 a + 4 + \left(3 a^{4} + 35 a^{3} + 21 a^{2} + 52 a + 20\right)\cdot 61 + \left(39 a^{4} + 53 a^{3} + a^{2} + 49 a + 7\right)\cdot 61^{2} + \left(38 a^{4} + 42 a^{3} + 38 a^{2} + 58 a + 6\right)\cdot 61^{3} + \left(49 a^{4} + 18 a^{3} + 36 a^{2} + 20 a + 32\right)\cdot 61^{4} + \left(13 a^{4} + 11 a^{3} + 19 a^{2} + 20 a + 12\right)\cdot 61^{5} +O(61^{6})$ 37a^4 + 34a^3 + 56a^2 + 35a + 4 + (3a^4 + 35a^3 + 21a^2 + 52a + 20)61 + (39a^4 + 53a^3 + a^2 + 49a + 7)61^2 + (38a^4 + 42a^3 + 38a^2 + 58a + 6)61^3 + (49a^4 + 18a^3 + 36a^2 + 20a + 32)61^4 + (13a^4 + 11a^3 + 19a^2 + 20a + 12)61^5+O(61^6)
$r_{ 12 }$	$=$	$ 51 a^{4} + 31 a^{3} + 52 a^{2} + 58 a + 9 + \left(3 a^{4} + 47 a^{3} + 48 a^{2} + 42 a + 23\right)\cdot 61 + \left(37 a^{4} + 24 a^{3} + 39 a^{2} + 9 a\right)\cdot 61^{2} + \left(37 a^{4} + 59 a^{3} + 52 a^{2} + 52 a + 43\right)\cdot 61^{3} + \left(27 a^{4} + 35 a^{3} + 22 a^{2} + 4 a + 51\right)\cdot 61^{4} + \left(18 a^{4} + 10 a^{3} + 39 a^{2} + 13 a + 28\right)\cdot 61^{5} +O(61^{6})$ 51a^4 + 31a^3 + 52a^2 + 58a + 9 + (3a^4 + 47a^3 + 48a^2 + 42a + 23)61 + (37a^4 + 24a^3 + 39a^2 + 9a)61^2 + (37a^4 + 59a^3 + 52a^2 + 52a + 43)61^3 + (27a^4 + 35a^3 + 22a^2 + 4a + 51)61^4 + (18a^4 + 10a^3 + 39a^2 + 13a + 28)*61^5+O(61^6)
$r_{ 13 }$	$=$	$ 55 a^{4} + 13 a^{3} + 2 a^{2} + 45 a + 6 + \left(34 a^{4} + 56 a^{3} + 50 a^{2} + 10 a + 52\right)\cdot 61 + \left(45 a^{4} + 30 a^{3} + 56 a^{2} + 21 a + 8\right)\cdot 61^{2} + \left(52 a^{4} + 22 a^{3} + 21 a^{2} + 2 a + 56\right)\cdot 61^{3} + \left(53 a^{4} + 48 a^{3} + 25 a^{2} + 13 a + 23\right)\cdot 61^{4} + \left(60 a^{4} + 58 a^{3} + 51 a^{2} + 53 a + 37\right)\cdot 61^{5} +O(61^{6})$ 55a^4 + 13a^3 + 2a^2 + 45a + 6 + (34a^4 + 56a^3 + 50a^2 + 10a + 52)61 + (45a^4 + 30a^3 + 56a^2 + 21a + 8)61^2 + (52a^4 + 22a^3 + 21a^2 + 2a + 56)61^3 + (53a^4 + 48a^3 + 25a^2 + 13a + 23)61^4 + (60a^4 + 58a^3 + 51a^2 + 53a + 37)*61^5+O(61^6)
$r_{ 14 }$	$=$	$ 57 a^{4} + 35 a^{3} + 25 a^{2} + 21 a + 30 + \left(8 a^{4} + 27 a^{3} + 19 a^{2} + 39 a + 35\right)\cdot 61 + \left(31 a^{4} + 54 a^{3} + 33 a^{2} + 43 a + 4\right)\cdot 61^{2} + \left(49 a^{4} + 50 a^{3} + 45 a^{2} + 44 a + 23\right)\cdot 61^{3} + \left(29 a^{4} + 34 a^{3} + 32 a^{2} + 19 a + 48\right)\cdot 61^{4} + \left(40 a^{4} + 14 a^{3} + 48 a^{2} + 56 a + 20\right)\cdot 61^{5} +O(61^{6})$ 57a^4 + 35a^3 + 25a^2 + 21a + 30 + (8a^4 + 27a^3 + 19a^2 + 39a + 35)61 + (31a^4 + 54a^3 + 33a^2 + 43a + 4)61^2 + (49a^4 + 50a^3 + 45a^2 + 44a + 23)61^3 + (29a^4 + 34a^3 + 32a^2 + 19a + 48)61^4 + (40a^4 + 14a^3 + 48a^2 + 56a + 20)*61^5+O(61^6)
$r_{ 15 }$	$=$	$ 58 a^{4} + 31 a^{3} + 37 a^{2} + 40 a + 23 + \left(57 a^{4} + 7 a^{3} + 20 a^{2} + 50 a + 23\right)\cdot 61 + \left(59 a^{4} + 34 a^{3} + 41 a^{2} + 35 a + 5\right)\cdot 61^{2} + \left(39 a^{4} + 45 a^{3} + 30 a^{2} + 57 a + 54\right)\cdot 61^{3} + \left(36 a^{4} + 41 a^{3} + 39 a^{2} + 48 a + 45\right)\cdot 61^{4} + \left(30 a^{4} + 44 a^{3} + 17 a^{2} + 19 a + 49\right)\cdot 61^{5} +O(61^{6})$ 58a^4 + 31a^3 + 37a^2 + 40a + 23 + (57a^4 + 7a^3 + 20a^2 + 50a + 23)61 + (59a^4 + 34a^3 + 41a^2 + 35a + 5)61^2 + (39a^4 + 45a^3 + 30a^2 + 57a + 54)61^3 + (36a^4 + 41a^3 + 39a^2 + 48a + 45)61^4 + (30a^4 + 44a^3 + 17a^2 + 19a + 49)*61^5+O(61^6)

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

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

Character values on conjugacy classes

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

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 $r_1, \ldots, r_{ 15 }$

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	2.2541.15t4.b.d
Dimension	$2$
Group	$S_3 \times C_5$
Conductor	$2541$
Root number	not computed
Indicator	$0$

$r_{ 1 }$	$=$	\( 37 a^{3} + 13 a^{2} + 30 a + 44 + \left(32 a^{4} + 41 a^{3} + 2 a^{2} + 29 a\right)\cdot 61 + \left(8 a^{4} + 51 a^{3} + 44 a^{2} + 12 a + 7\right)\cdot 61^{2} + \left(17 a^{4} + 3 a^{3} + 38 a^{2} + 49 a + 5\right)\cdot 61^{3} + \left(49 a^{4} + 25 a^{3} + 33 a^{2} + 14 a + 40\right)\cdot 61^{4} + \left(60 a^{4} + 26 a^{3} + 32 a^{2} + 20\right)\cdot 61^{5} +O(61^{6})\) 37a^3 + 13a^2 + 30a + 44 + (32a^4 + 41a^3 + 2a^2 + 29a)61 + (8a^4 + 51a^3 + 44a^2 + 12a + 7)61^2 + (17a^4 + 3a^3 + 38a^2 + 49a + 5)61^3 + (49a^4 + 25a^3 + 33a^2 + 14a + 40)61^4 + (60a^4 + 26a^3 + 32a^2 + 20)*61^5+O(61^6)
$r_{ 2 }$	$=$	\( a^{4} + 51 a^{3} + 6 a^{2} + 48 a + 17 + \left(11 a^{4} + 52 a^{3} + 40 a^{2} + 11 a + 43\right)\cdot 61 + \left(44 a^{4} + 57 a^{3} + 24 a^{2} + 17 a + 7\right)\cdot 61^{2} + \left(23 a^{4} + 39 a^{3} + 40 a^{2} + 24 a + 56\right)\cdot 61^{3} + \left(34 a^{4} + 47 a^{3} + 45 a^{2} + 31 a + 6\right)\cdot 61^{4} + \left(55 a^{4} + 43 a^{3} + 51 a^{2} + 43 a + 19\right)\cdot 61^{5} +O(61^{6})\) a^4 + 51a^3 + 6a^2 + 48a + 17 + (11a^4 + 52a^3 + 40a^2 + 11a + 43)61 + (44a^4 + 57a^3 + 24a^2 + 17a + 7)61^2 + (23a^4 + 39a^3 + 40a^2 + 24a + 56)61^3 + (34a^4 + 47a^3 + 45a^2 + 31a + 6)61^4 + (55a^4 + 43a^3 + 51a^2 + 43a + 19)61^5+O(61^6)
$r_{ 3 }$	$=$	\( 3 a^{4} + 10 a^{3} + 21 a^{2} + 12 a + 7 + \left(59 a^{4} + 14 a^{3} + 37 a^{2} + 20 a + 52\right)\cdot 61 + \left(28 a^{4} + 55 a^{3} + 35 a^{2} + 17 a + 56\right)\cdot 61^{2} + \left(47 a^{4} + 48 a^{3} + 26 a^{2} + 27 a + 29\right)\cdot 61^{3} + \left(26 a^{4} + 30 a^{3} + 25 a^{2} + 33 a + 44\right)\cdot 61^{4} + \left(28 a^{4} + 46 a^{3} + 43 a^{2} + 54 a + 18\right)\cdot 61^{5} +O(61^{6})\) 3a^4 + 10a^3 + 21a^2 + 12a + 7 + (59a^4 + 14a^3 + 37a^2 + 20a + 52)61 + (28a^4 + 55a^3 + 35a^2 + 17a + 56)61^2 + (47a^4 + 48a^3 + 26a^2 + 27a + 29)61^3 + (26a^4 + 30a^3 + 25a^2 + 33a + 44)61^4 + (28a^4 + 46a^3 + 43a^2 + 54a + 18)*61^5+O(61^6)
$r_{ 4 }$	$=$	\( 3 a^{4} + 15 a^{3} + 42 a^{2} + 4 a + 44 + \left(9 a^{4} + 11 a^{3} + 12 a^{2} + 16 a + 29\right)\cdot 61 + \left(2 a^{4} + 50 a^{3} + 46 a^{2} + 2 a + 60\right)\cdot 61^{2} + \left(11 a^{4} + 33 a^{3} + 24 a^{2} + 9 a + 7\right)\cdot 61^{3} + \left(35 a^{4} + 13 a^{3} + 21 a^{2} + 56\right)\cdot 61^{4} + \left(24 a^{4} + 56 a^{3} + 6 a^{2} + 36 a + 52\right)\cdot 61^{5} +O(61^{6})\) 3a^4 + 15a^3 + 42a^2 + 4a + 44 + (9a^4 + 11a^3 + 12a^2 + 16a + 29)61 + (2a^4 + 50a^3 + 46a^2 + 2a + 60)61^2 + (11a^4 + 33a^3 + 24a^2 + 9a + 7)61^3 + (35a^4 + 13a^3 + 21a^2 + 56)61^4 + (24a^4 + 56a^3 + 6a^2 + 36a + 52)61^5+O(61^6)
$r_{ 5 }$	$=$	\( 5 a^{4} + 31 a^{3} + 8 a^{2} + 8 a + 14 + \left(49 a^{4} + 30 a^{3} + 32 a^{2} + 35 a + 5\right)\cdot 61 + \left(4 a^{4} + 54 a^{3} + 56 a^{2} + 31 a + 20\right)\cdot 61^{2} + \left(52 a^{4} + 45 a^{3} + 16 a^{2} + 12 a + 25\right)\cdot 61^{3} + \left(59 a^{4} + 58 a^{3} + 58 a^{2} + 32 a + 20\right)\cdot 61^{4} + \left(52 a^{4} + 36 a^{3} + 55 a^{2} + 3 a + 10\right)\cdot 61^{5} +O(61^{6})\) 5a^4 + 31a^3 + 8a^2 + 8a + 14 + (49a^4 + 30a^3 + 32a^2 + 35a + 5)61 + (4a^4 + 54a^3 + 56a^2 + 31a + 20)61^2 + (52a^4 + 45a^3 + 16a^2 + 12a + 25)61^3 + (59a^4 + 58a^3 + 58a^2 + 32a + 20)61^4 + (52a^4 + 36a^3 + 55a^2 + 3a + 10)*61^5+O(61^6)
$r_{ 6 }$	$=$	\( 10 a^{4} + 3 a^{3} + 21 a^{2} + 21 a + 38 + \left(45 a^{4} + 46 a^{3} + 24 a^{2} + 10 a + 22\right)\cdot 61 + \left(30 a^{4} + 17 a^{3} + 12 a^{2} + 41 a + 54\right)\cdot 61^{2} + \left(32 a^{4} + 28 a^{3} + 36 a^{2} + 34 a + 18\right)\cdot 61^{3} + \left(33 a^{4} + 33 a^{3} + 10 a^{2} + 59 a + 40\right)\cdot 61^{4} + \left(a^{4} + 56 a^{3} + 16 a^{2} + 9 a + 2\right)\cdot 61^{5} +O(61^{6})\) 10a^4 + 3a^3 + 21a^2 + 21a + 38 + (45a^4 + 46a^3 + 24a^2 + 10a + 22)61 + (30a^4 + 17a^3 + 12a^2 + 41a + 54)61^2 + (32a^4 + 28a^3 + 36a^2 + 34a + 18)61^3 + (33a^4 + 33a^3 + 10a^2 + 59a + 40)61^4 + (a^4 + 56a^3 + 16a^2 + 9a + 2)61^5+O(61^6)
$r_{ 7 }$	$=$	\( 13 a^{4} + 29 a^{3} + 26 a^{2} + 26 a + 59 + \left(5 a^{4} + 13 a^{3} + 11 a^{2} + 59 a + 11\right)\cdot 61 + \left(a^{4} + 55 a^{3} + 41 a^{2} + 38 a + 33\right)\cdot 61^{2} + \left(55 a^{4} + 28 a^{3} + 5 a^{2} + 12 a + 51\right)\cdot 61^{3} + \left(41 a^{4} + 39 a^{3} + 48 a^{2} + 51 a + 5\right)\cdot 61^{4} + \left(7 a^{4} + 26 a^{3} + 10 a^{2} + 8 a + 23\right)\cdot 61^{5} +O(61^{6})\) 13a^4 + 29a^3 + 26a^2 + 26a + 59 + (5a^4 + 13a^3 + 11a^2 + 59a + 11)61 + (a^4 + 55a^3 + 41a^2 + 38a + 33)61^2 + (55a^4 + 28a^3 + 5a^2 + 12a + 51)61^3 + (41a^4 + 39a^3 + 48a^2 + 51a + 5)61^4 + (7a^4 + 26a^3 + 10a^2 + 8a + 23)61^5+O(61^6)
$r_{ 8 }$	$=$	\( 20 a^{4} + 25 a^{3} + 20 a^{2} + 31 a + 12 + \left(20 a^{4} + 17 a^{3} + 30 a^{2} + 14 a + 28\right)\cdot 61 + \left(a^{3} + a^{2} + 21 a + 6\right)\cdot 61^{2} + \left(12 a^{4} + 42 a^{3} + 36 a^{2} + 32 a + 5\right)\cdot 61^{3} + \left(60 a^{4} + 4 a^{3} + 5 a^{2} + 39 a + 40\right)\cdot 61^{4} + \left(41 a^{4} + 4 a^{3} + 46 a^{2} + 12 a + 12\right)\cdot 61^{5} +O(61^{6})\) 20a^4 + 25a^3 + 20a^2 + 31a + 12 + (20a^4 + 17a^3 + 30a^2 + 14a + 28)61 + (a^3 + a^2 + 21a + 6)61^2 + (12a^4 + 42a^3 + 36a^2 + 32a + 5)61^3 + (60a^4 + 4a^3 + 5a^2 + 39a + 40)61^4 + (41a^4 + 4a^3 + 46a^2 + 12a + 12)61^5+O(61^6)
$r_{ 9 }$	$=$	\( 22 a^{4} + 34 a^{3} + 35 a^{2} + 22 a + 43 + \left(36 a^{4} + 46 a^{3} + 41 a^{2} + 3 a + 41\right)\cdot 61 + \left(3 a^{4} + 49 a^{3} + 32 a^{2} + 2 a + 20\right)\cdot 61^{2} + \left(53 a^{4} + 22 a^{3} + 18 a^{2} + 21 a + 10\right)\cdot 61^{3} + \left(53 a^{4} + 26 a^{3} + 37 a^{2} + 22 a + 36\right)\cdot 61^{4} + \left(26 a^{4} + 29 a^{3} + 12 a^{2} + 51 a + 52\right)\cdot 61^{5} +O(61^{6})\) 22a^4 + 34a^3 + 35a^2 + 22a + 43 + (36a^4 + 46a^3 + 41a^2 + 3a + 41)61 + (3a^4 + 49a^3 + 32a^2 + 2a + 20)61^2 + (53a^4 + 22a^3 + 18a^2 + 21a + 10)61^3 + (53a^4 + 26a^3 + 37a^2 + 22a + 36)61^4 + (26a^4 + 29a^3 + 12a^2 + 51a + 52)*61^5+O(61^6)
$r_{ 10 }$	$=$	\( 31 a^{4} + 48 a^{3} + 2 a^{2} + 26 a + 20 + \left(50 a^{4} + 39 a^{3} + 34 a^{2} + 30 a + 37\right)\cdot 61 + \left(28 a^{4} + 18 a^{3} + 20 a^{2} + 21 a + 11\right)\cdot 61^{2} + \left(26 a^{4} + 33 a^{3} + 55 a^{2} + 49 a + 34\right)\cdot 61^{3} + \left(17 a^{4} + 28 a^{3} + 44 a^{2} + 34 a + 56\right)\cdot 61^{4} + \left(23 a^{4} + 21 a^{3} + 35 a^{2} + 43 a + 3\right)\cdot 61^{5} +O(61^{6})\) 31a^4 + 48a^3 + 2a^2 + 26a + 20 + (50a^4 + 39a^3 + 34a^2 + 30a + 37)61 + (28a^4 + 18a^3 + 20a^2 + 21a + 11)61^2 + (26a^4 + 33a^3 + 55a^2 + 49a + 34)61^3 + (17a^4 + 28a^3 + 44a^2 + 34a + 56)61^4 + (23a^4 + 21a^3 + 35a^2 + 43a + 3)*61^5+O(61^6)
$r_{ 11 }$	$=$	\( 37 a^{4} + 34 a^{3} + 56 a^{2} + 35 a + 4 + \left(3 a^{4} + 35 a^{3} + 21 a^{2} + 52 a + 20\right)\cdot 61 + \left(39 a^{4} + 53 a^{3} + a^{2} + 49 a + 7\right)\cdot 61^{2} + \left(38 a^{4} + 42 a^{3} + 38 a^{2} + 58 a + 6\right)\cdot 61^{3} + \left(49 a^{4} + 18 a^{3} + 36 a^{2} + 20 a + 32\right)\cdot 61^{4} + \left(13 a^{4} + 11 a^{3} + 19 a^{2} + 20 a + 12\right)\cdot 61^{5} +O(61^{6})\) 37a^4 + 34a^3 + 56a^2 + 35a + 4 + (3a^4 + 35a^3 + 21a^2 + 52a + 20)61 + (39a^4 + 53a^3 + a^2 + 49a + 7)61^2 + (38a^4 + 42a^3 + 38a^2 + 58a + 6)61^3 + (49a^4 + 18a^3 + 36a^2 + 20a + 32)61^4 + (13a^4 + 11a^3 + 19a^2 + 20a + 12)61^5+O(61^6)
$r_{ 12 }$	$=$	\( 51 a^{4} + 31 a^{3} + 52 a^{2} + 58 a + 9 + \left(3 a^{4} + 47 a^{3} + 48 a^{2} + 42 a + 23\right)\cdot 61 + \left(37 a^{4} + 24 a^{3} + 39 a^{2} + 9 a\right)\cdot 61^{2} + \left(37 a^{4} + 59 a^{3} + 52 a^{2} + 52 a + 43\right)\cdot 61^{3} + \left(27 a^{4} + 35 a^{3} + 22 a^{2} + 4 a + 51\right)\cdot 61^{4} + \left(18 a^{4} + 10 a^{3} + 39 a^{2} + 13 a + 28\right)\cdot 61^{5} +O(61^{6})\) 51a^4 + 31a^3 + 52a^2 + 58a + 9 + (3a^4 + 47a^3 + 48a^2 + 42a + 23)61 + (37a^4 + 24a^3 + 39a^2 + 9a)61^2 + (37a^4 + 59a^3 + 52a^2 + 52a + 43)61^3 + (27a^4 + 35a^3 + 22a^2 + 4a + 51)61^4 + (18a^4 + 10a^3 + 39a^2 + 13a + 28)*61^5+O(61^6)
$r_{ 13 }$	$=$	\( 55 a^{4} + 13 a^{3} + 2 a^{2} + 45 a + 6 + \left(34 a^{4} + 56 a^{3} + 50 a^{2} + 10 a + 52\right)\cdot 61 + \left(45 a^{4} + 30 a^{3} + 56 a^{2} + 21 a + 8\right)\cdot 61^{2} + \left(52 a^{4} + 22 a^{3} + 21 a^{2} + 2 a + 56\right)\cdot 61^{3} + \left(53 a^{4} + 48 a^{3} + 25 a^{2} + 13 a + 23\right)\cdot 61^{4} + \left(60 a^{4} + 58 a^{3} + 51 a^{2} + 53 a + 37\right)\cdot 61^{5} +O(61^{6})\) 55a^4 + 13a^3 + 2a^2 + 45a + 6 + (34a^4 + 56a^3 + 50a^2 + 10a + 52)61 + (45a^4 + 30a^3 + 56a^2 + 21a + 8)61^2 + (52a^4 + 22a^3 + 21a^2 + 2a + 56)61^3 + (53a^4 + 48a^3 + 25a^2 + 13a + 23)61^4 + (60a^4 + 58a^3 + 51a^2 + 53a + 37)*61^5+O(61^6)
$r_{ 14 }$	$=$	\( 57 a^{4} + 35 a^{3} + 25 a^{2} + 21 a + 30 + \left(8 a^{4} + 27 a^{3} + 19 a^{2} + 39 a + 35\right)\cdot 61 + \left(31 a^{4} + 54 a^{3} + 33 a^{2} + 43 a + 4\right)\cdot 61^{2} + \left(49 a^{4} + 50 a^{3} + 45 a^{2} + 44 a + 23\right)\cdot 61^{3} + \left(29 a^{4} + 34 a^{3} + 32 a^{2} + 19 a + 48\right)\cdot 61^{4} + \left(40 a^{4} + 14 a^{3} + 48 a^{2} + 56 a + 20\right)\cdot 61^{5} +O(61^{6})\) 57a^4 + 35a^3 + 25a^2 + 21a + 30 + (8a^4 + 27a^3 + 19a^2 + 39a + 35)61 + (31a^4 + 54a^3 + 33a^2 + 43a + 4)61^2 + (49a^4 + 50a^3 + 45a^2 + 44a + 23)61^3 + (29a^4 + 34a^3 + 32a^2 + 19a + 48)61^4 + (40a^4 + 14a^3 + 48a^2 + 56a + 20)*61^5+O(61^6)
$r_{ 15 }$	$=$	\( 58 a^{4} + 31 a^{3} + 37 a^{2} + 40 a + 23 + \left(57 a^{4} + 7 a^{3} + 20 a^{2} + 50 a + 23\right)\cdot 61 + \left(59 a^{4} + 34 a^{3} + 41 a^{2} + 35 a + 5\right)\cdot 61^{2} + \left(39 a^{4} + 45 a^{3} + 30 a^{2} + 57 a + 54\right)\cdot 61^{3} + \left(36 a^{4} + 41 a^{3} + 39 a^{2} + 48 a + 45\right)\cdot 61^{4} + \left(30 a^{4} + 44 a^{3} + 17 a^{2} + 19 a + 49\right)\cdot 61^{5} +O(61^{6})\) 58a^4 + 31a^3 + 37a^2 + 40a + 23 + (57a^4 + 7a^3 + 20a^2 + 50a + 23)61 + (59a^4 + 34a^3 + 41a^2 + 35a + 5)61^2 + (39a^4 + 45a^3 + 30a^2 + 57a + 54)61^3 + (36a^4 + 41a^3 + 39a^2 + 48a + 45)61^4 + (30a^4 + 44a^3 + 17a^2 + 19a + 49)*61^5+O(61^6)