LMFDB - Artin representation 2.671.15t2.a.b

Basic invariants

Dimension:	$2$
Group:	$D_{15}$
Conductor:	$671$$\medspace = 11 \cdot 61 $
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 15.1.61243167054566186591.1
Galois orbit size:	$4$
Smallest permutation container:	$D_{15}$
Parity:	odd
Determinant:	1.671.2t1.a.a
Projective image:	$D_{15}$
Projective stem field:	Galois closure of 15.1.61243167054566186591.1

Defining polynomial

$f(x)$

$=$

$ x^{15} - 3 x^{14} + 6 x^{13} - x^{12} - 20 x^{11} + 56 x^{10} - 81 x^{9} + 46 x^{8} + 90 x^{7} + \cdots + 11 $

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{5} + 3x + 27 $ x^5 + 3*x + 27

Roots:

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

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

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

Character values on conjugacy classes

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

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	2.671.15t2.a.b
Dimension	$2$
Group	$D_{15}$
Conductor	$671$
Root number	$1$
Indicator	$1$

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