LMFDB - Newform orbit 171.2.u.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,2,Mod(28,171)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(171, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 8]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("171.28");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 171 = 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	171.u (of order $9$, degree $6$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.36544187456$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/171\mathbb{Z}\right)^\times$.

$n$	$20$	$154$
$\chi(n)$	$1$	$\zeta_{18}^{2}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

28.1

−0.173648	−	0.984808i
−0.173648	+	0.984808i
−0.766044	−	0.642788i
−0.766044	+	0.642788i
0.939693	−	0.342020i
0.939693	+	0.342020i

0.233956

1.32683i

0.173648

−

0.0632028i

0.826352

0.300767i

−0.173648

0.300767i

1.47178

2.54920i

−0.205737

1.16679i

55.1

0.233956

−

1.32683i

0.173648

0.0632028i

0.826352

−

0.300767i

−0.173648

−

0.300767i

1.47178

−

2.54920i

−0.205737

−

1.16679i

73.1

1.93969

1.62760i

0.766044

4.34445i

0.233956

−

1.32683i

−0.766044

−

1.32683i

−3.05303

5.28801i

2.61334

−

2.19285i

82.1

1.93969

−

1.62760i

0.766044

−

4.34445i

0.233956

1.32683i

−0.766044

1.32683i

−3.05303

−

5.28801i

2.61334

2.19285i

100.1

0.826352

−

0.300767i

−0.939693

0.788496i

1.93969

1.62760i

0.939693

1.62760i

−1.41875

2.45734i

2.09240

0.761570i

118.1

0.826352

0.300767i

−0.939693

−

0.788496i

1.93969

−

1.62760i

0.939693

−

1.62760i

−1.41875

−

2.45734i

2.09240

−

0.761570i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.e	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	171.2.u.c		6
3.b	odd	2	1		19.2.e.a	✓	6
12.b	even	2	1		304.2.u.b		6
15.d	odd	2	1		475.2.l.a		6
15.e	even	4	2		475.2.u.a		12
19.e	even	9	1	inner	171.2.u.c		6
19.e	even	9	1		3249.2.a.z		3
19.f	odd	18	1		3249.2.a.s		3
21.c	even	2	1		931.2.w.a		6
21.g	even	6	1		931.2.v.a		6
21.g	even	6	1		931.2.x.b		6
21.h	odd	6	1		931.2.v.b		6
21.h	odd	6	1		931.2.x.a		6
57.d	even	2	1		361.2.e.h		6
57.f	even	6	1		361.2.e.a		6
57.f	even	6	1		361.2.e.b		6
57.h	odd	6	1		361.2.e.f		6
57.h	odd	6	1		361.2.e.g		6
57.j	even	18	1		361.2.a.h		3
57.j	even	18	2		361.2.c.h		6
57.j	even	18	1		361.2.e.a		6
57.j	even	18	1		361.2.e.b		6
57.j	even	18	1		361.2.e.h		6
57.l	odd	18	1		19.2.e.a	✓	6
57.l	odd	18	1		361.2.a.g		3
57.l	odd	18	2		361.2.c.i		6
57.l	odd	18	1		361.2.e.f		6
57.l	odd	18	1		361.2.e.g		6
228.u	odd	18	1		5776.2.a.bi		3
228.v	even	18	1		304.2.u.b		6
228.v	even	18	1		5776.2.a.br		3
285.bd	odd	18	1		475.2.l.a		6
285.bd	odd	18	1		9025.2.a.bd		3
285.bf	even	18	1		9025.2.a.x		3
285.bi	even	36	2		475.2.u.a		12
399.ca	odd	18	1		931.2.v.b		6
399.cb	even	18	1		931.2.v.a		6
399.ch	odd	18	1		931.2.x.a		6
399.ci	even	18	1		931.2.x.b		6
399.cj	even	18	1		931.2.w.a		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
19.2.e.a	✓	6	3.b	odd	2	1
19.2.e.a	✓	6	57.l	odd	18	1
171.2.u.c		6	1.a	even	1	1	trivial
171.2.u.c		6	19.e	even	9	1	inner
304.2.u.b		6	12.b	even	2	1
304.2.u.b		6	228.v	even	18	1
361.2.a.g		3	57.l	odd	18	1
361.2.a.h		3	57.j	even	18	1
361.2.c.h		6	57.j	even	18	2
361.2.c.i		6	57.l	odd	18	2
361.2.e.a		6	57.f	even	6	1
361.2.e.a		6	57.j	even	18	1
361.2.e.b		6	57.f	even	6	1
361.2.e.b		6	57.j	even	18	1
361.2.e.f		6	57.h	odd	6	1
361.2.e.f		6	57.l	odd	18	1
361.2.e.g		6	57.h	odd	6	1
361.2.e.g		6	57.l	odd	18	1
361.2.e.h		6	57.d	even	2	1
361.2.e.h		6	57.j	even	18	1
475.2.l.a		6	15.d	odd	2	1
475.2.l.a		6	285.bd	odd	18	1
475.2.u.a		12	15.e	even	4	2
475.2.u.a		12	285.bi	even	36	2
931.2.v.a		6	21.g	even	6	1
931.2.v.a		6	399.cb	even	18	1
931.2.v.b		6	21.h	odd	6	1
931.2.v.b		6	399.ca	odd	18	1
931.2.w.a		6	21.c	even	2	1
931.2.w.a		6	399.cj	even	18	1
931.2.x.a		6	21.h	odd	6	1
931.2.x.a		6	399.ch	odd	18	1
931.2.x.b		6	21.g	even	6	1
931.2.x.b		6	399.ci	even	18	1
3249.2.a.s		3	19.f	odd	18	1
3249.2.a.z		3	19.e	even	9	1
5776.2.a.bi		3	228.u	odd	18	1
5776.2.a.br		3	228.v	even	18	1
9025.2.a.x		3	285.bf	even	18	1
9025.2.a.bd		3	285.bd	odd	18	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{6} - 6T_{2}^{5} + 18T_{2}^{4} - 30T_{2}^{3} + 36T_{2}^{2} - 27T_{2} + 9 $ T2^6 - 6*T2^5 + 18*T2^4 - 30*T2^3 + 36*T2^2 - 27*T2 + 9 acting on $S_{2}^{\mathrm{new}}(171, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} - 6 T^{5} + \cdots + 9 $ T^6 - 6T^5 + 18T^4 - 30T^3 + 36T^2 - 27*T + 9
$3$	$ T^{6} $ T^6
$5$	$ T^{6} - 6 T^{5} + \cdots + 9 $ T^6 - 6T^5 + 18T^4 - 30T^3 + 36T^2 - 27*T + 9
$7$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$11$	$ T^{6} + 9 T^{4} + \cdots + 81 $ T^6 + 9T^4 + 18T^3 + 81T^2 + 81T + 81
$13$	$ T^{6} + 3 T^{5} + \cdots + 1369 $ T^6 + 3T^5 + 24T^4 + 26T^3 - 114T^2 + 222*T + 1369
$17$	$ T^{6} + 3 T^{5} + \cdots + 9 $ T^6 + 3T^5 - 30T^3 + 36*T^2 + 9
$19$	$ T^{6} + 12 T^{5} + \cdots + 6859 $ T^6 + 12T^5 + 78T^4 + 385T^3 + 1482T^2 + 4332*T + 6859
$23$	$ T^{6} + 6 T^{5} + \cdots + 576 $ T^6 + 6T^5 + 36T^4 + 192T^3 + 576T^2 + 864*T + 576
$29$	$ T^{6} - 3 T^{5} + \cdots + 12321 $ T^6 - 3T^5 + 36T^4 + 57T^3 - 477T^2 + 1998*T + 12321
$31$	$ T^{6} - 9 T^{5} + \cdots + 2809 $ T^6 - 9T^5 + 75T^4 - 160T^3 + 513T^2 + 318*T + 2809
$37$	$ (T^{3} - 21 T - 17)^{2} $ (T^3 - 21*T - 17)^2
$41$	$ T^{6} + 21 T^{5} + \cdots + 12321 $ T^6 + 21T^5 + 162T^4 + 672T^3 + 3411T^2 + 8991*T + 12321
$43$	$ T^{6} + 3 T^{5} + \cdots + 26569 $ T^6 + 3T^5 + 60T^4 + 8T^3 - 663T^2 + 5379*T + 26569
$47$	$ T^{6} - 3 T^{5} + \cdots + 9 $ T^6 - 3T^5 + 54T^4 - 24T^3 - 18T^2 + 9
$53$	$ T^{6} - 3 T^{5} + \cdots + 2601 $ T^6 - 3T^5 + 84T^3 + 387T^2 + 1377T + 2601
$59$	$ T^{6} + 12 T^{5} + \cdots + 71289 $ T^6 + 12T^5 + 18T^4 + 159T^3 + 3006T^2 - 19224*T + 71289
$61$	$ T^{6} + 12 T^{5} + \cdots + 32761 $ T^6 + 12T^5 + 24T^4 - 37T^3 + 984T^2 - 2172*T + 32761
$67$	$ T^{6} + 30 T^{5} + \cdots + 179776 $ T^6 + 30T^5 + 348T^4 + 2528T^3 + 20928T^2 + 86496*T + 179776
$71$	$ T^{6} - 6 T^{5} + \cdots + 788544 $ T^6 - 6T^5 - 36T^4 + 1536T^3 + 8352T^2 + 31968*T + 788544
$73$	$ T^{6} + 12 T^{5} + \cdots + 4096 $ T^6 + 12T^5 + 96T^4 + 512T^3 + 768T^2 - 3072*T + 4096
$79$	$ T^{6} + 39 T^{5} + \cdots + 654481 $ T^6 + 39T^5 + 708T^4 + 7487T^3 + 51663T^2 + 242700*T + 654481
$83$	$ T^{6} + 189 T^{4} + \cdots + 210681 $ T^6 + 189T^4 - 918T^3 + 35721T^2 - 86751T + 210681
$89$	$ T^{6} - 12 T^{5} + \cdots + 3249 $ T^6 - 12T^5 + 54T^4 + 300T^3 + 522T^2 + 1539*T + 3249
$97$	$ T^{6} - 18 T^{5} + \cdots + 16129 $ T^6 - 18T^5 + 234T^4 - 1855T^3 + 9522T^2 - 20574*T + 16129
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	171.u (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{18}^{2} - \zeta_{18} + 1) q^{2} + (\zeta_{18}^{4} - 2 \zeta_{18}^{3} + \cdots + 1) q^{4}+ \cdots + ( - 3 \zeta_{18}^{5} + \cdots - 3 \zeta_{18}) q^{8}+O(q^{10}) \) q + (z^2 - z + 1) * q^2 + (z^4 - 2z^3 + z^2 - 2z + 1) * q^4 + (-z^5 - z^4 + z + 1) * q^5 + (-z^5 - z^4 + z) * q^7 + (-3z^5 + 2z^4 - 2z^3 + 2z^2 - 3z) q^8	\( q + (\zeta_{18}^{2} - \zeta_{18} + 1) q^{2} + (\zeta_{18}^{4} - 2 \zeta_{18}^{3} + \cdots + 1) q^{4}+ \cdots + (4 \zeta_{18}^{5} - 6 \zeta_{18}^{4} + \cdots - 1) q^{98}+O(q^{100}) \) q + (z^2 - z + 1) * q^2 + (z^4 - 2z^3 + z^2 - 2z + 1) * q^4 + (-z^5 - z^4 + z + 1) * q^5 + (-z^5 - z^4 + z) * q^7 + (-3z^5 + 2z^4 - 2z^3 + 2z^2 - 3z) q^8 + (-2z^4 + z^3 + z + 1) q^10 + (z^5 + z^4 + z^2 + z) * q^11 + (3z^4 + 3z^3 - z - 2) * q^13 + (-2z^4 + z^3 - z^2 + 2z) * q^14 + (-3z^5 + z^4 + 3z^2 - 3) * q^16 + (-z^5 + z^3 + 2z - 1) q^17 + (z^5 - 4z^4 - 2z^2 + 2z - 2) q^19 + (3z^5 - z^4 - 2z^2 - 2z + 1) q^20 + 3z^4 q^22 + (-2z^3 + 2z^2 - 2z) q^23 + (2z^5 - 2z^4 - 2z^3 + z + 1) q^25 + (5z^3 - z^2 + z - 5) q^26 + (3z^5 - 2z^4 + 2z^3 - 3z^2) * q^28 + (5z^5 - z^3 - 5z^2 + z + 1) * q^29 + (3z^5 + 3z^4 - 3z^3 - 2z^2 - z + 3) * q^31 + (3z^4 + 3z^3 - 3) * q^32 + (-2z^4 + 4z^3 - 3z^2 + 4z - 2) * q^34 + (-2z^5 - z^4 - 2z^3) * q^35 + (-2z^5 + 3z^4 - z^2 - z) * q^37 + (5z^5 - 5z^4 - z^3 - 6z^2 + 3z + 3) * q^38 + (2z^5 - 2z^3 - z^2 - 6z + 1) q^40 + (-3z^5 + 4z^4 + z^3 + 4z^2 - 4) q^41 + (-2z^5 - 5z^3 + 5z^2 + 2) q^43 + (-3z^5 - z^4 - z^3 + 2z - 1) * q^44 + (-2z^5 + 4z^4 - 6z^3 + 4z^2 - 2z) q^46 + (2z^5 + z^4 - 3z^3 - 2z^2 + 2z + 2) * q^47 + (-z^5 + 5z^3 - z) q^49 + (2z^5 + 2z^4 - 5z^3 - 2z + 5) * q^50 + (-z^5 - 6z^4 + 3z^3 - 3z^2 + 6z + 1) * q^52 + (-z^4 + 3z^3 + 2z^2 + 3z - 1) q^53 + (3z^2 + 3z + 3) * q^55 + (5z^5 - 2z^4 - 3z^2 - 3z + 1) * q^56 + (4z^5 + z^4 - 5z^2 - 5z + 6) q^58 + (-2z^2 - 7z - 2) * q^59 + (-4z^4 + 4z^3 + 3z^2 + 4z - 4) * q^61 + (-3z^5 + 7z^4 - 2z^3 + 2z^2 - 7z + 3) q^62 + (4z^3 + 3z^2 - 3z - 4) q^64 + (5z^4 + 4z^3 + 5z^2) q^65 + (6z^5 + 6z^4 - 2z^3 - 6z^2 - 4z - 4) q^67 + (4z^5 - 7z^4 + 5z^3 - 7z^2 + 4z) q^68 + (-3z^5 - z^4 - z^3 + 2z - 1) * q^70 + (-2z^5 - 10z^4 - 2z^3 + 2z^2 + 10z + 2) q^71 + (-4z^5 - 4z^3) * q^73 + (-5z^5 + 5z^3 + z - 5) * q^74 + (9z^5 - 3z^4 + 2z^3 - 2z^2 - 3z + 5) q^76 + (-z^5 - z^4 + 2z^2 + 2z + 3) * q^77 + (-7z^5 - 3z^4 - z^3 + 6z^2 - 6) q^79 + (z^4 - 3z^3 + 4z^2 - 3z + 1) q^80 + (-6z^5 + 4z^4 + 4z^3 + 7z - 11) * q^82 + (-3z^5 - 3z^4 + 9z^2 - 6z) * q^83 + (-3z^5 + z^4 - 2z^3 + 3z^2 + z + 1) q^85 + (-7z^5 + 8z^4 - 8z^3 + 7z^2) * q^86 + (-3z^5 - 3z^4 - 3z^3 - 3z^2 + 6z + 3) q^88 + (-5z^5 + 2z^4 + 2z^3 - 3z + 1) * q^89 + (2z^4 + z^3 + 5z^2 + z + 2) * q^91 + (-8z^5 + 8z^4 - 2z^3 + 6z^2 - 6) * q^92 + (-2z^5 + 4z^4 - 2z^2 - 2z + 3) * q^94 + (z^5 - z^3 - 4z^2 - z - 6) q^95 + (-2z^5 + 2z^3 + 4z^2 - 5z + 2) * q^97 + (4z^5 - 6z^4 + 5z^3 + z^2 - 1) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{5} - 6 q^{8}+O(q^{10}) \) 6 * q + 6 * q^2 + 6 * q^5 - 6 * q^8	\( 6 q + 6 q^{2} + 6 q^{5} - 6 q^{8} + 9 q^{10} - 3 q^{13} + 3 q^{14} - 18 q^{16} - 3 q^{17} - 12 q^{19} + 6 q^{20} - 6 q^{23} - 15 q^{26} + 6 q^{28} + 3 q^{29} + 9 q^{31} - 9 q^{32} - 6 q^{35} + 15 q^{38} - 21 q^{41} - 3 q^{43} - 9 q^{44} - 18 q^{46} + 3 q^{47} + 15 q^{49} + 15 q^{50} + 15 q^{52} + 3 q^{53} + 18 q^{55} + 6 q^{56} + 36 q^{58} - 12 q^{59} - 12 q^{61} + 12 q^{62} - 12 q^{64} + 12 q^{65} - 30 q^{67} + 15 q^{68} - 9 q^{70} + 6 q^{71} - 12 q^{73} - 15 q^{74} + 36 q^{76} + 18 q^{77} - 39 q^{79} - 3 q^{80} - 54 q^{82} - 24 q^{86} + 9 q^{88} + 12 q^{89} + 15 q^{91} - 42 q^{92} + 18 q^{94} - 39 q^{95} + 18 q^{97} + 9 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^5 - 6 * q^8 + 9 * q^10 - 3 * q^13 + 3 * q^14 - 18 * q^16 - 3 * q^17 - 12 * q^19 + 6 * q^20 - 6 * q^23 - 15 * q^26 + 6 * q^28 + 3 * q^29 + 9 * q^31 - 9 * q^32 - 6 * q^35 + 15 * q^38 - 21 * q^41 - 3 * q^43 - 9 * q^44 - 18 * q^46 + 3 * q^47 + 15 * q^49 + 15 * q^50 + 15 * q^52 + 3 * q^53 + 18 * q^55 + 6 * q^56 + 36 * q^58 - 12 * q^59 - 12 * q^61 + 12 * q^62 - 12 * q^64 + 12 * q^65 - 30 * q^67 + 15 * q^68 - 9 * q^70 + 6 * q^71 - 12 * q^73 - 15 * q^74 + 36 * q^76 + 18 * q^77 - 39 * q^79 - 3 * q^80 - 54 * q^82 - 24 * q^86 + 9 * q^88 + 12 * q^89 + 15 * q^91 - 42 * q^92 + 18 * q^94 - 39 * q^95 + 18 * q^97 + 9 * q^98

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Newspace parameters

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Hecke characteristic polynomials

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	171.2.u.c

Level	$171$
Weight	$2$
Character orbit	171.u
Analytic conductor	$1.365$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(1.36544187456\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\Q(\zeta_{18})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} - x^{3} + 1 \) x^6 - x^3 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 28.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} - 6 T^{5} + \cdots + 9 \) T^6 - 6T^5 + 18T^4 - 30T^3 + 36T^2 - 27*T + 9
$3$	\( T^{6} \) T^6
$5$	\( T^{6} - 6 T^{5} + \cdots + 9 \) T^6 - 6T^5 + 18T^4 - 30T^3 + 36T^2 - 27*T + 9
$7$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$11$	\( T^{6} + 9 T^{4} + \cdots + 81 \) T^6 + 9T^4 + 18T^3 + 81T^2 + 81T + 81
$13$	\( T^{6} + 3 T^{5} + \cdots + 1369 \) T^6 + 3T^5 + 24T^4 + 26T^3 - 114T^2 + 222*T + 1369
$17$	\( T^{6} + 3 T^{5} + \cdots + 9 \) T^6 + 3T^5 - 30T^3 + 36*T^2 + 9
$19$	\( T^{6} + 12 T^{5} + \cdots + 6859 \) T^6 + 12T^5 + 78T^4 + 385T^3 + 1482T^2 + 4332*T + 6859
$23$	\( T^{6} + 6 T^{5} + \cdots + 576 \) T^6 + 6T^5 + 36T^4 + 192T^3 + 576T^2 + 864*T + 576
$29$	\( T^{6} - 3 T^{5} + \cdots + 12321 \) T^6 - 3T^5 + 36T^4 + 57T^3 - 477T^2 + 1998*T + 12321
$31$	\( T^{6} - 9 T^{5} + \cdots + 2809 \) T^6 - 9T^5 + 75T^4 - 160T^3 + 513T^2 + 318*T + 2809
$37$	\( (T^{3} - 21 T - 17)^{2} \) (T^3 - 21*T - 17)^2
$41$	\( T^{6} + 21 T^{5} + \cdots + 12321 \) T^6 + 21T^5 + 162T^4 + 672T^3 + 3411T^2 + 8991*T + 12321
$43$	\( T^{6} + 3 T^{5} + \cdots + 26569 \) T^6 + 3T^5 + 60T^4 + 8T^3 - 663T^2 + 5379*T + 26569
$47$	\( T^{6} - 3 T^{5} + \cdots + 9 \) T^6 - 3T^5 + 54T^4 - 24T^3 - 18T^2 + 9
$53$	\( T^{6} - 3 T^{5} + \cdots + 2601 \) T^6 - 3T^5 + 84T^3 + 387T^2 + 1377T + 2601
$59$	\( T^{6} + 12 T^{5} + \cdots + 71289 \) T^6 + 12T^5 + 18T^4 + 159T^3 + 3006T^2 - 19224*T + 71289
$61$	\( T^{6} + 12 T^{5} + \cdots + 32761 \) T^6 + 12T^5 + 24T^4 - 37T^3 + 984T^2 - 2172*T + 32761
$67$	\( T^{6} + 30 T^{5} + \cdots + 179776 \) T^6 + 30T^5 + 348T^4 + 2528T^3 + 20928T^2 + 86496*T + 179776
$71$	\( T^{6} - 6 T^{5} + \cdots + 788544 \) T^6 - 6T^5 - 36T^4 + 1536T^3 + 8352T^2 + 31968*T + 788544
$73$	\( T^{6} + 12 T^{5} + \cdots + 4096 \) T^6 + 12T^5 + 96T^4 + 512T^3 + 768T^2 - 3072*T + 4096
$79$	\( T^{6} + 39 T^{5} + \cdots + 654481 \) T^6 + 39T^5 + 708T^4 + 7487T^3 + 51663T^2 + 242700*T + 654481
$83$	\( T^{6} + 189 T^{4} + \cdots + 210681 \) T^6 + 189T^4 - 918T^3 + 35721T^2 - 86751T + 210681
$89$	\( T^{6} - 12 T^{5} + \cdots + 3249 \) T^6 - 12T^5 + 54T^4 + 300T^3 + 522T^2 + 1539*T + 3249
$97$	\( T^{6} - 18 T^{5} + \cdots + 16129 \) T^6 - 18T^5 + 234T^4 - 1855T^3 + 9522T^2 - 20574*T + 16129
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\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(1\)	\(\zeta_{18}^{2}\)