LMFDB - Artin representation 2.1175.15t2.a.c

Basic invariants

Dimension:	$2$
Group:	$D_{15}$
Conductor:	$1175$$\medspace = 5^{2} \cdot 47 $
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 15.1.4947491410771484375.1
Galois orbit size:	$4$
Smallest permutation container:	$D_{15}$
Parity:	odd
Determinant:	1.47.2t1.a.a
Projective image:	$D_{15}$
Projective stem field:	Galois closure of 15.1.4947491410771484375.1

Defining polynomial

$f(x)$

$=$

$ x^{15} - 6 x^{12} + 9 x^{11} + 20 x^{10} + 57 x^{9} + 29 x^{8} + 27 x^{7} - 53 x^{6} - 44 x^{5} + \cdots - 55 $

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{5} + 10x + 35 $ x^5 + 10*x + 35

Roots:

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

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

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

Character values on conjugacy classes

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

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	2.1175.15t2.a.c
Dimension	$2$
Group	$D_{15}$
Conductor	$1175$
Root number	$1$
Indicator	$1$

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