LMFDB - Artin representation 2.484.15t4.b.a

Basic invariants

Dimension:	$2$
Group:	$S_3 \times C_5$
Conductor:	$484$$\medspace = 2^{2} \cdot 11^{2} $
Artin stem field:	Galois closure of 15.5.388863829589238784.1
Galois orbit size:	$4$
Smallest permutation container:	$S_3 \times C_5$
Parity:	odd
Determinant:	1.44.10t1.a.b
Projective image:	$S_3$
Projective stem field:	Galois closure of 3.1.484.1

Defining polynomial

$f(x)$

$=$

$ x^{15} - 3 x^{14} + 2 x^{13} - x^{12} + 5 x^{11} - 22 x^{9} + 33 x^{8} - 22 x^{7} + 22 x^{6} - 22 x^{5} + \cdots - 1 $

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.

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

Roots:

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

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

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

Character values on conjugacy classes

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

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 $r_1, \ldots, r_{ 15 }$

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	2.484.15t4.b.a
Dimension	$2$
Group	$S_3 \times C_5$
Conductor	$484$
Root number	not computed
Indicator	$0$

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