LMFDB - Galois orbit of Artin representations 16.1119186718586212624367616.24t1334.a

Basic invariants

Dimension:	$16$
Group:	$((C_3^2:Q_8):C_3):C_2$
Conductor:	$111\!\cdots\!616$$\medspace = 2^{26} \cdot 3^{34} $
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 9.3.4760622968832.3
Galois orbit size:	$1$
Smallest permutation container:	24T1334
Parity:	even
Projective image:	$C_3^2:\GL(2,3)$
Projective field:	Galois closure of 9.3.4760622968832.3

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{4} + 6x^{2} + 24x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 15 + 25\cdot 37 + 33\cdot 37^{2} + 4\cdot 37^{3} + 25\cdot 37^{4} + 8\cdot 37^{5} + 18\cdot 37^{6} + 4\cdot 37^{7} + 6\cdot 37^{8} + 37^{9} +O(37^{10})$ 15 + 2537 + 3337^2 + 437^3 + 2537^4 + 837^5 + 1837^6 + 437^7 + 637^8 + 37^9+O(37^10)
$r_{ 2 }$	$=$	$ 3 a^{3} + 21 a^{2} + 19 a + 19 + \left(18 a^{3} + 31 a^{2} + 10 a + 30\right)\cdot 37 + \left(34 a^{3} + 15 a^{2} + 4 a + 12\right)\cdot 37^{2} + \left(34 a^{3} + 4 a^{2} + 33 a + 10\right)\cdot 37^{3} + \left(33 a^{3} + 11 a^{2} + 32 a + 7\right)\cdot 37^{4} + \left(14 a^{3} + 18 a^{2} + 7 a + 11\right)\cdot 37^{5} + \left(18 a^{3} + 2 a^{2} + 13 a + 15\right)\cdot 37^{6} + \left(29 a^{3} + 16 a^{2} + 10 a + 4\right)\cdot 37^{7} + \left(2 a^{3} + 22 a^{2} + 36 a + 16\right)\cdot 37^{8} + \left(11 a^{3} + 17 a^{2} + 16 a + 18\right)\cdot 37^{9} +O(37^{10})$ 3a^3 + 21a^2 + 19a + 19 + (18a^3 + 31a^2 + 10a + 30)37 + (34a^3 + 15a^2 + 4a + 12)37^2 + (34a^3 + 4a^2 + 33a + 10)37^3 + (33a^3 + 11a^2 + 32a + 7)37^4 + (14a^3 + 18a^2 + 7a + 11)37^5 + (18a^3 + 2a^2 + 13a + 15)37^6 + (29a^3 + 16a^2 + 10a + 4)37^7 + (2a^3 + 22a^2 + 36a + 16)37^8 + (11a^3 + 17a^2 + 16a + 18)*37^9+O(37^10)
$r_{ 3 }$	$=$	$ 9 a^{3} + 18 a^{2} + 35 a + 5 + \left(8 a^{3} + 15 a^{2} + 32 a + 23\right)\cdot 37 + \left(2 a^{3} + 3 a^{2} + a + 3\right)\cdot 37^{2} + \left(35 a^{3} + 15 a^{2} + 14 a + 21\right)\cdot 37^{3} + \left(19 a^{3} + 16 a + 10\right)\cdot 37^{4} + \left(31 a^{3} + 28 a^{2} + 30 a + 18\right)\cdot 37^{5} + \left(9 a^{3} + 21 a^{2} + 26 a + 6\right)\cdot 37^{6} + \left(30 a^{3} + 18 a^{2} + 33 a + 27\right)\cdot 37^{7} + \left(30 a^{3} + 17 a^{2} + 32 a + 22\right)\cdot 37^{8} + \left(4 a^{3} + 31 a^{2} + 8 a + 35\right)\cdot 37^{9} +O(37^{10})$ 9a^3 + 18a^2 + 35a + 5 + (8a^3 + 15a^2 + 32a + 23)37 + (2a^3 + 3a^2 + a + 3)37^2 + (35a^3 + 15a^2 + 14a + 21)37^3 + (19a^3 + 16a + 10)37^4 + (31a^3 + 28a^2 + 30a + 18)37^5 + (9a^3 + 21a^2 + 26a + 6)37^6 + (30a^3 + 18a^2 + 33a + 27)37^7 + (30a^3 + 17a^2 + 32a + 22)37^8 + (4a^3 + 31a^2 + 8a + 35)*37^9+O(37^10)
$r_{ 4 }$	$=$	$ 5 a^{3} + 15 a^{2} + 20 a + \left(21 a^{3} + 11 a^{2} + 26 a + 25\right)\cdot 37 + \left(29 a^{3} + 11 a^{2} + 34 a + 21\right)\cdot 37^{2} + \left(7 a^{3} + 4 a + 27\right)\cdot 37^{3} + \left(34 a^{3} + 11 a^{2} + 6 a + 11\right)\cdot 37^{4} + \left(10 a^{3} + 21 a^{2} + 36 a + 22\right)\cdot 37^{5} + \left(8 a^{3} + 32 a^{2} + 7 a + 34\right)\cdot 37^{6} + \left(10 a^{3} + 23 a^{2} + 8 a + 13\right)\cdot 37^{7} + \left(13 a^{3} + 5 a^{2} + 16 a + 6\right)\cdot 37^{8} + \left(19 a^{3} + 28 a^{2} + 31 a + 14\right)\cdot 37^{9} +O(37^{10})$ 5a^3 + 15a^2 + 20a + (21a^3 + 11a^2 + 26a + 25)37 + (29a^3 + 11a^2 + 34a + 21)37^2 + (7a^3 + 4a + 27)37^3 + (34a^3 + 11a^2 + 6a + 11)37^4 + (10a^3 + 21a^2 + 36a + 22)37^5 + (8a^3 + 32a^2 + 7a + 34)37^6 + (10a^3 + 23a^2 + 8a + 13)37^7 + (13a^3 + 5a^2 + 16a + 6)37^8 + (19a^3 + 28a^2 + 31a + 14)37^9+O(37^10)
$r_{ 5 }$	$=$	$ 23 a^{3} + 16 a^{2} + 35 a + 29 + \left(7 a^{2} + 27 a + 9\right)\cdot 37 + \left(31 a^{3} + 3 a^{2} + 33 a + 3\right)\cdot 37^{2} + \left(12 a^{3} + 7 a^{2} + 26 a + 4\right)\cdot 37^{3} + \left(35 a^{3} + 25 a^{2} + 35 a + 29\right)\cdot 37^{4} + \left(36 a^{3} + 13 a^{2} + 36 a + 35\right)\cdot 37^{5} + \left(21 a^{3} + 10 a^{2} + 33 a + 5\right)\cdot 37^{6} + \left(7 a^{3} + 10 a^{2} + 26 a + 1\right)\cdot 37^{7} + \left(7 a^{3} + 15 a^{2} + 33 a + 35\right)\cdot 37^{8} + \left(14 a^{3} + 33 a^{2} + 27 a + 24\right)\cdot 37^{9} +O(37^{10})$ 23a^3 + 16a^2 + 35a + 29 + (7a^2 + 27a + 9)37 + (31a^3 + 3a^2 + 33a + 3)37^2 + (12a^3 + 7a^2 + 26a + 4)37^3 + (35a^3 + 25a^2 + 35a + 29)37^4 + (36a^3 + 13a^2 + 36a + 35)37^5 + (21a^3 + 10a^2 + 33a + 5)37^6 + (7a^3 + 10a^2 + 26a + 1)37^7 + (7a^3 + 15a^2 + 33a + 35)37^8 + (14a^3 + 33a^2 + 27a + 24)37^9+O(37^10)
$r_{ 6 }$	$=$	$ 9 a^{3} + 18 a^{2} + 29 a + 7 + \left(36 a^{3} + 10 a^{2} + 33 a + 35\right)\cdot 37 + \left(28 a^{3} + 8 a^{2} + 26 a + 1\right)\cdot 37^{2} + \left(12 a^{3} + 34 a^{2} + 14 a + 34\right)\cdot 37^{3} + \left(35 a^{3} + 2 a^{2} + 17 a + 7\right)\cdot 37^{4} + \left(9 a^{3} + 7 a^{2} + 31 a + 36\right)\cdot 37^{5} + \left(27 a^{3} + 22 a^{2} + 10 a + 11\right)\cdot 37^{6} + \left(4 a^{3} + 21 a^{2} + 18 a + 19\right)\cdot 37^{7} + \left(16 a^{3} + 26 a^{2} + 27 a + 9\right)\cdot 37^{8} + \left(15 a^{3} + 19 a^{2} + 13 a + 29\right)\cdot 37^{9} +O(37^{10})$ 9a^3 + 18a^2 + 29a + 7 + (36a^3 + 10a^2 + 33a + 35)37 + (28a^3 + 8a^2 + 26a + 1)37^2 + (12a^3 + 34a^2 + 14a + 34)37^3 + (35a^3 + 2a^2 + 17a + 7)37^4 + (9a^3 + 7a^2 + 31a + 36)37^5 + (27a^3 + 22a^2 + 10a + 11)37^6 + (4a^3 + 21a^2 + 18a + 19)37^7 + (16a^3 + 26a^2 + 27a + 9)37^8 + (15a^3 + 19a^2 + 13a + 29)*37^9+O(37^10)
$r_{ 7 }$	$=$	$ 34 a^{3} + 20 a^{2} + 26 a + 17 + \left(33 a^{3} + 26 a^{2} + 30 a\right)\cdot 37 + \left(3 a^{3} + 11 a^{2} + 18 a + 22\right)\cdot 37^{2} + \left(7 a^{3} + 15 a^{2} + 16 a + 36\right)\cdot 37^{3} + \left(14 a^{3} + 18 a^{2} + 27 a + 34\right)\cdot 37^{4} + \left(12 a^{3} + 19 a^{2} + 33 a + 17\right)\cdot 37^{5} + \left(15 a^{3} + 13 a^{2} + 16 a + 6\right)\cdot 37^{6} + \left(26 a^{3} + 32 a^{2} + 15 a + 36\right)\cdot 37^{7} + \left(35 a^{3} + 30 a^{2} + 2\right)\cdot 37^{8} + \left(10 a^{3} + 34 a^{2} + 35 a + 8\right)\cdot 37^{9} +O(37^{10})$ 34a^3 + 20a^2 + 26a + 17 + (33a^3 + 26a^2 + 30a)37 + (3a^3 + 11a^2 + 18a + 22)37^2 + (7a^3 + 15a^2 + 16a + 36)37^3 + (14a^3 + 18a^2 + 27a + 34)37^4 + (12a^3 + 19a^2 + 33a + 17)37^5 + (15a^3 + 13a^2 + 16a + 6)37^6 + (26a^3 + 32a^2 + 15a + 36)37^7 + (35a^3 + 30a^2 + 2)37^8 + (10a^3 + 34a^2 + 35a + 8)37^9+O(37^10)
$r_{ 8 }$	$=$	$ 20 a^{3} + 20 a^{2} + 6 a + 26 + \left(35 a^{3} + 20 a^{2} + 3 a + 15\right)\cdot 37 + \left(17 a^{3} + a^{2} + 8 a + 5\right)\cdot 37^{2} + \left(18 a^{3} + 35 a^{2} + 21 a + 28\right)\cdot 37^{3} + \left(7 a^{3} + 11 a^{2} + 17 a + 14\right)\cdot 37^{4} + \left(a^{3} + 27 a^{2} + 35 a + 13\right)\cdot 37^{5} + \left(20 a^{3} + 16 a^{2} + 4 a + 13\right)\cdot 37^{6} + \left(29 a^{3} + 12 a^{2} + 31\right)\cdot 37^{7} + \left(4 a^{3} + 19 a^{2} + 31 a + 5\right)\cdot 37^{8} + \left(28 a^{3} + 8 a^{2} + 11 a + 2\right)\cdot 37^{9} +O(37^{10})$ 20a^3 + 20a^2 + 6a + 26 + (35a^3 + 20a^2 + 3a + 15)37 + (17a^3 + a^2 + 8a + 5)37^2 + (18a^3 + 35a^2 + 21a + 28)37^3 + (7a^3 + 11a^2 + 17a + 14)37^4 + (a^3 + 27a^2 + 35a + 13)37^5 + (20a^3 + 16a^2 + 4a + 13)37^6 + (29a^3 + 12a^2 + 31)37^7 + (4a^3 + 19a^2 + 31a + 5)37^8 + (28a^3 + 8a^2 + 11a + 2)37^9+O(37^10)
$r_{ 9 }$	$=$	$ 8 a^{3} + 20 a^{2} + 15 a + 30 + \left(31 a^{3} + 24 a^{2} + 19 a + 19\right)\cdot 37 + \left(36 a^{3} + 18 a^{2} + 19 a + 6\right)\cdot 37^{2} + \left(18 a^{3} + 36 a^{2} + 16 a + 18\right)\cdot 37^{3} + \left(4 a^{3} + 29 a^{2} + 31 a + 6\right)\cdot 37^{4} + \left(30 a^{3} + 12 a^{2} + 9 a + 21\right)\cdot 37^{5} + \left(26 a^{3} + 28 a^{2} + 33 a + 35\right)\cdot 37^{6} + \left(9 a^{3} + 12 a^{2} + 34 a + 9\right)\cdot 37^{7} + \left(10 a^{2} + 6 a + 6\right)\cdot 37^{8} + \left(7 a^{3} + 11 a^{2} + 2 a + 14\right)\cdot 37^{9} +O(37^{10})$ 8a^3 + 20a^2 + 15a + 30 + (31a^3 + 24a^2 + 19a + 19)37 + (36a^3 + 18a^2 + 19a + 6)37^2 + (18a^3 + 36a^2 + 16a + 18)37^3 + (4a^3 + 29a^2 + 31a + 6)37^4 + (30a^3 + 12a^2 + 9a + 21)37^5 + (26a^3 + 28a^2 + 33a + 35)37^6 + (9a^3 + 12a^2 + 34a + 9)37^7 + (10a^2 + 6a + 6)37^8 + (7a^3 + 11a^2 + 2a + 14)37^9+O(37^10)

Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

Cycle notation
$(3,5,4)(7,9,8)$
$(1,9,5)(2,3,8)$
$(1,7,5,3,2,4,8,9)$
$(1,6,2)(3,5,4)(7,8,9)$
$(1,5,9)(2,3,8)(4,7,6)$
$(1,9,2,3)(4,8,7,5)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 9 }$	Character values
			$c1$
$1$	$1$	$()$	$16$
$9$	$2$	$(1,3)(2,5)(4,6)(8,9)$	$0$
$36$	$2$	$(1,5)(2,8)(4,7)$	$0$
$8$	$3$	$(1,6,2)(3,5,4)(7,8,9)$	$-2$
$24$	$3$	$(2,9,4)(5,8,6)$	$-2$
$48$	$3$	$(1,3,9)(2,4,8)(5,7,6)$	$1$
$54$	$4$	$(1,6,3,4)(2,9,5,8)$	$0$
$72$	$6$	$(1,7,8,5,4,2)(3,9,6)$	$0$
$72$	$6$	$(1,9,3,4,7,2)(5,6)$	$0$
$54$	$8$	$(1,7,5,3,2,4,8,9)$	$0$
$54$	$8$	$(1,4,5,9,2,7,8,3)$	$0$

The blue line marks the conjugacy class containing complex conjugation.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Basic invariants

Galois action

Roots of defining polynomial

Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

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	16.111...616.24t1334.a
Dimension	$16$
Group	$((C_3^2:Q_8):C_3):C_2$
Conductor	$1.119\times 10^{24}$
Indicator	$1$

$r_{ 1 }$	$=$	\( 15 + 25\cdot 37 + 33\cdot 37^{2} + 4\cdot 37^{3} + 25\cdot 37^{4} + 8\cdot 37^{5} + 18\cdot 37^{6} + 4\cdot 37^{7} + 6\cdot 37^{8} + 37^{9} +O(37^{10})\) 15 + 2537 + 3337^2 + 437^3 + 2537^4 + 837^5 + 1837^6 + 437^7 + 637^8 + 37^9+O(37^10)
$r_{ 2 }$	$=$	\( 3 a^{3} + 21 a^{2} + 19 a + 19 + \left(18 a^{3} + 31 a^{2} + 10 a + 30\right)\cdot 37 + \left(34 a^{3} + 15 a^{2} + 4 a + 12\right)\cdot 37^{2} + \left(34 a^{3} + 4 a^{2} + 33 a + 10\right)\cdot 37^{3} + \left(33 a^{3} + 11 a^{2} + 32 a + 7\right)\cdot 37^{4} + \left(14 a^{3} + 18 a^{2} + 7 a + 11\right)\cdot 37^{5} + \left(18 a^{3} + 2 a^{2} + 13 a + 15\right)\cdot 37^{6} + \left(29 a^{3} + 16 a^{2} + 10 a + 4\right)\cdot 37^{7} + \left(2 a^{3} + 22 a^{2} + 36 a + 16\right)\cdot 37^{8} + \left(11 a^{3} + 17 a^{2} + 16 a + 18\right)\cdot 37^{9} +O(37^{10})\) 3a^3 + 21a^2 + 19a + 19 + (18a^3 + 31a^2 + 10a + 30)37 + (34a^3 + 15a^2 + 4a + 12)37^2 + (34a^3 + 4a^2 + 33a + 10)37^3 + (33a^3 + 11a^2 + 32a + 7)37^4 + (14a^3 + 18a^2 + 7a + 11)37^5 + (18a^3 + 2a^2 + 13a + 15)37^6 + (29a^3 + 16a^2 + 10a + 4)37^7 + (2a^3 + 22a^2 + 36a + 16)37^8 + (11a^3 + 17a^2 + 16a + 18)*37^9+O(37^10)
$r_{ 3 }$	$=$	\( 9 a^{3} + 18 a^{2} + 35 a + 5 + \left(8 a^{3} + 15 a^{2} + 32 a + 23\right)\cdot 37 + \left(2 a^{3} + 3 a^{2} + a + 3\right)\cdot 37^{2} + \left(35 a^{3} + 15 a^{2} + 14 a + 21\right)\cdot 37^{3} + \left(19 a^{3} + 16 a + 10\right)\cdot 37^{4} + \left(31 a^{3} + 28 a^{2} + 30 a + 18\right)\cdot 37^{5} + \left(9 a^{3} + 21 a^{2} + 26 a + 6\right)\cdot 37^{6} + \left(30 a^{3} + 18 a^{2} + 33 a + 27\right)\cdot 37^{7} + \left(30 a^{3} + 17 a^{2} + 32 a + 22\right)\cdot 37^{8} + \left(4 a^{3} + 31 a^{2} + 8 a + 35\right)\cdot 37^{9} +O(37^{10})\) 9a^3 + 18a^2 + 35a + 5 + (8a^3 + 15a^2 + 32a + 23)37 + (2a^3 + 3a^2 + a + 3)37^2 + (35a^3 + 15a^2 + 14a + 21)37^3 + (19a^3 + 16a + 10)37^4 + (31a^3 + 28a^2 + 30a + 18)37^5 + (9a^3 + 21a^2 + 26a + 6)37^6 + (30a^3 + 18a^2 + 33a + 27)37^7 + (30a^3 + 17a^2 + 32a + 22)37^8 + (4a^3 + 31a^2 + 8a + 35)*37^9+O(37^10)
$r_{ 4 }$	$=$	\( 5 a^{3} + 15 a^{2} + 20 a + \left(21 a^{3} + 11 a^{2} + 26 a + 25\right)\cdot 37 + \left(29 a^{3} + 11 a^{2} + 34 a + 21\right)\cdot 37^{2} + \left(7 a^{3} + 4 a + 27\right)\cdot 37^{3} + \left(34 a^{3} + 11 a^{2} + 6 a + 11\right)\cdot 37^{4} + \left(10 a^{3} + 21 a^{2} + 36 a + 22\right)\cdot 37^{5} + \left(8 a^{3} + 32 a^{2} + 7 a + 34\right)\cdot 37^{6} + \left(10 a^{3} + 23 a^{2} + 8 a + 13\right)\cdot 37^{7} + \left(13 a^{3} + 5 a^{2} + 16 a + 6\right)\cdot 37^{8} + \left(19 a^{3} + 28 a^{2} + 31 a + 14\right)\cdot 37^{9} +O(37^{10})\) 5a^3 + 15a^2 + 20a + (21a^3 + 11a^2 + 26a + 25)37 + (29a^3 + 11a^2 + 34a + 21)37^2 + (7a^3 + 4a + 27)37^3 + (34a^3 + 11a^2 + 6a + 11)37^4 + (10a^3 + 21a^2 + 36a + 22)37^5 + (8a^3 + 32a^2 + 7a + 34)37^6 + (10a^3 + 23a^2 + 8a + 13)37^7 + (13a^3 + 5a^2 + 16a + 6)37^8 + (19a^3 + 28a^2 + 31a + 14)37^9+O(37^10)
$r_{ 5 }$	$=$	\( 23 a^{3} + 16 a^{2} + 35 a + 29 + \left(7 a^{2} + 27 a + 9\right)\cdot 37 + \left(31 a^{3} + 3 a^{2} + 33 a + 3\right)\cdot 37^{2} + \left(12 a^{3} + 7 a^{2} + 26 a + 4\right)\cdot 37^{3} + \left(35 a^{3} + 25 a^{2} + 35 a + 29\right)\cdot 37^{4} + \left(36 a^{3} + 13 a^{2} + 36 a + 35\right)\cdot 37^{5} + \left(21 a^{3} + 10 a^{2} + 33 a + 5\right)\cdot 37^{6} + \left(7 a^{3} + 10 a^{2} + 26 a + 1\right)\cdot 37^{7} + \left(7 a^{3} + 15 a^{2} + 33 a + 35\right)\cdot 37^{8} + \left(14 a^{3} + 33 a^{2} + 27 a + 24\right)\cdot 37^{9} +O(37^{10})\) 23a^3 + 16a^2 + 35a + 29 + (7a^2 + 27a + 9)37 + (31a^3 + 3a^2 + 33a + 3)37^2 + (12a^3 + 7a^2 + 26a + 4)37^3 + (35a^3 + 25a^2 + 35a + 29)37^4 + (36a^3 + 13a^2 + 36a + 35)37^5 + (21a^3 + 10a^2 + 33a + 5)37^6 + (7a^3 + 10a^2 + 26a + 1)37^7 + (7a^3 + 15a^2 + 33a + 35)37^8 + (14a^3 + 33a^2 + 27a + 24)37^9+O(37^10)
$r_{ 6 }$	$=$	\( 9 a^{3} + 18 a^{2} + 29 a + 7 + \left(36 a^{3} + 10 a^{2} + 33 a + 35\right)\cdot 37 + \left(28 a^{3} + 8 a^{2} + 26 a + 1\right)\cdot 37^{2} + \left(12 a^{3} + 34 a^{2} + 14 a + 34\right)\cdot 37^{3} + \left(35 a^{3} + 2 a^{2} + 17 a + 7\right)\cdot 37^{4} + \left(9 a^{3} + 7 a^{2} + 31 a + 36\right)\cdot 37^{5} + \left(27 a^{3} + 22 a^{2} + 10 a + 11\right)\cdot 37^{6} + \left(4 a^{3} + 21 a^{2} + 18 a + 19\right)\cdot 37^{7} + \left(16 a^{3} + 26 a^{2} + 27 a + 9\right)\cdot 37^{8} + \left(15 a^{3} + 19 a^{2} + 13 a + 29\right)\cdot 37^{9} +O(37^{10})\) 9a^3 + 18a^2 + 29a + 7 + (36a^3 + 10a^2 + 33a + 35)37 + (28a^3 + 8a^2 + 26a + 1)37^2 + (12a^3 + 34a^2 + 14a + 34)37^3 + (35a^3 + 2a^2 + 17a + 7)37^4 + (9a^3 + 7a^2 + 31a + 36)37^5 + (27a^3 + 22a^2 + 10a + 11)37^6 + (4a^3 + 21a^2 + 18a + 19)37^7 + (16a^3 + 26a^2 + 27a + 9)37^8 + (15a^3 + 19a^2 + 13a + 29)*37^9+O(37^10)
$r_{ 7 }$	$=$	\( 34 a^{3} + 20 a^{2} + 26 a + 17 + \left(33 a^{3} + 26 a^{2} + 30 a\right)\cdot 37 + \left(3 a^{3} + 11 a^{2} + 18 a + 22\right)\cdot 37^{2} + \left(7 a^{3} + 15 a^{2} + 16 a + 36\right)\cdot 37^{3} + \left(14 a^{3} + 18 a^{2} + 27 a + 34\right)\cdot 37^{4} + \left(12 a^{3} + 19 a^{2} + 33 a + 17\right)\cdot 37^{5} + \left(15 a^{3} + 13 a^{2} + 16 a + 6\right)\cdot 37^{6} + \left(26 a^{3} + 32 a^{2} + 15 a + 36\right)\cdot 37^{7} + \left(35 a^{3} + 30 a^{2} + 2\right)\cdot 37^{8} + \left(10 a^{3} + 34 a^{2} + 35 a + 8\right)\cdot 37^{9} +O(37^{10})\) 34a^3 + 20a^2 + 26a + 17 + (33a^3 + 26a^2 + 30a)37 + (3a^3 + 11a^2 + 18a + 22)37^2 + (7a^3 + 15a^2 + 16a + 36)37^3 + (14a^3 + 18a^2 + 27a + 34)37^4 + (12a^3 + 19a^2 + 33a + 17)37^5 + (15a^3 + 13a^2 + 16a + 6)37^6 + (26a^3 + 32a^2 + 15a + 36)37^7 + (35a^3 + 30a^2 + 2)37^8 + (10a^3 + 34a^2 + 35a + 8)37^9+O(37^10)
$r_{ 8 }$	$=$	\( 20 a^{3} + 20 a^{2} + 6 a + 26 + \left(35 a^{3} + 20 a^{2} + 3 a + 15\right)\cdot 37 + \left(17 a^{3} + a^{2} + 8 a + 5\right)\cdot 37^{2} + \left(18 a^{3} + 35 a^{2} + 21 a + 28\right)\cdot 37^{3} + \left(7 a^{3} + 11 a^{2} + 17 a + 14\right)\cdot 37^{4} + \left(a^{3} + 27 a^{2} + 35 a + 13\right)\cdot 37^{5} + \left(20 a^{3} + 16 a^{2} + 4 a + 13\right)\cdot 37^{6} + \left(29 a^{3} + 12 a^{2} + 31\right)\cdot 37^{7} + \left(4 a^{3} + 19 a^{2} + 31 a + 5\right)\cdot 37^{8} + \left(28 a^{3} + 8 a^{2} + 11 a + 2\right)\cdot 37^{9} +O(37^{10})\) 20a^3 + 20a^2 + 6a + 26 + (35a^3 + 20a^2 + 3a + 15)37 + (17a^3 + a^2 + 8a + 5)37^2 + (18a^3 + 35a^2 + 21a + 28)37^3 + (7a^3 + 11a^2 + 17a + 14)37^4 + (a^3 + 27a^2 + 35a + 13)37^5 + (20a^3 + 16a^2 + 4a + 13)37^6 + (29a^3 + 12a^2 + 31)37^7 + (4a^3 + 19a^2 + 31a + 5)37^8 + (28a^3 + 8a^2 + 11a + 2)37^9+O(37^10)
$r_{ 9 }$	$=$	\( 8 a^{3} + 20 a^{2} + 15 a + 30 + \left(31 a^{3} + 24 a^{2} + 19 a + 19\right)\cdot 37 + \left(36 a^{3} + 18 a^{2} + 19 a + 6\right)\cdot 37^{2} + \left(18 a^{3} + 36 a^{2} + 16 a + 18\right)\cdot 37^{3} + \left(4 a^{3} + 29 a^{2} + 31 a + 6\right)\cdot 37^{4} + \left(30 a^{3} + 12 a^{2} + 9 a + 21\right)\cdot 37^{5} + \left(26 a^{3} + 28 a^{2} + 33 a + 35\right)\cdot 37^{6} + \left(9 a^{3} + 12 a^{2} + 34 a + 9\right)\cdot 37^{7} + \left(10 a^{2} + 6 a + 6\right)\cdot 37^{8} + \left(7 a^{3} + 11 a^{2} + 2 a + 14\right)\cdot 37^{9} +O(37^{10})\) 8a^3 + 20a^2 + 15a + 30 + (31a^3 + 24a^2 + 19a + 19)37 + (36a^3 + 18a^2 + 19a + 6)37^2 + (18a^3 + 36a^2 + 16a + 18)37^3 + (4a^3 + 29a^2 + 31a + 6)37^4 + (30a^3 + 12a^2 + 9a + 21)37^5 + (26a^3 + 28a^2 + 33a + 35)37^6 + (9a^3 + 12a^2 + 34a + 9)37^7 + (10a^2 + 6a + 6)37^8 + (7a^3 + 11a^2 + 2a + 14)37^9+O(37^10)