LMFDB - Artin representation 2.716.15t2.a.c

Basic invariants

Dimension:	$2$
Group:	$D_{15}$
Conductor:	$716$$\medspace = 2^{2} \cdot 179 $
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 15.1.6029359418000239616.1
Galois orbit size:	$4$
Smallest permutation container:	$D_{15}$
Parity:	odd
Determinant:	1.179.2t1.a.a
Projective image:	$D_{15}$
Projective stem field:	Galois closure of 15.1.6029359418000239616.1

Defining polynomial

$f(x)$

$=$

$ x^{15} - 3 x^{14} + 10 x^{13} - 18 x^{12} + 41 x^{11} - 61 x^{10} + 57 x^{9} - 103 x^{8} + 29 x^{7} + \cdots - 1 $

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{5} + 7x + 28 $ x^5 + 7*x + 28

Roots:

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

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

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

Character values on conjugacy classes

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

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	2.716.15t2.a.c
Dimension	$2$
Group	$D_{15}$
Conductor	$716$
Root number	$1$
Indicator	$1$

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