LMFDB - Newform orbit 847.2.e.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,2,Mod(485,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("847.485");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	847.e (of order $3$, degree $2$, 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:	$6.76332905120$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/847\mathbb{Z}\right)^\times$.

$n$	$122$	$365$
$\chi(n)$	$-\zeta_{18}^{3}$	$1$

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} $

485.1

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

−0.766044

1.32683i

−1.43969

−

2.49362i

−0.173648

−

0.300767i

−1.17365

2.03282i

4.41147

−2.05303

1.66885i

−2.53209

−2.64543

4.58202i

−1.79813

−

3.11446i

485.2

−0.173648

0.300767i

0.266044

0.460802i

0.939693

1.62760i

−0.0603074

0.104455i

−0.184793

2.47178

0.943555i

−1.34730

1.35844

−

2.35289i

−0.0209445

−

0.0362770i

485.3

0.939693

−

1.62760i

−0.326352

−

0.565258i

−0.766044

−

1.32683i

−1.76604

3.05888i

−1.22668

−0.418748

−

2.61240i

0.879385

1.28699

−

2.22913i

3.31908

5.74881i

606.1

−0.766044

−

1.32683i

−1.43969

2.49362i

−0.173648

0.300767i

−1.17365

−

2.03282i

4.41147

−2.05303

−

1.66885i

−2.53209

−2.64543

−

4.58202i

−1.79813

3.11446i

606.2

−0.173648

−

0.300767i

0.266044

−

0.460802i

0.939693

−

1.62760i

−0.0603074

−

0.104455i

−0.184793

2.47178

−

0.943555i

−1.34730

1.35844

2.35289i

−0.0209445

0.0362770i

606.3

0.939693

1.62760i

−0.326352

0.565258i

−0.766044

1.32683i

−1.76604

−

3.05888i

−1.22668

−0.418748

2.61240i

0.879385

1.28699

2.22913i

3.31908

−

5.74881i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	847.2.e.c		6
7.c	even	3	1	inner	847.2.e.c		6
7.c	even	3	1		5929.2.a.x		3
7.d	odd	6	1		5929.2.a.u		3
11.b	odd	2	1		77.2.e.a	✓	6
11.c	even	5	4		847.2.n.f		24
11.d	odd	10	4		847.2.n.g		24
33.d	even	2	1		693.2.i.h		6
44.c	even	2	1		1232.2.q.m		6
77.b	even	2	1		539.2.e.m		6
77.h	odd	6	1		77.2.e.a	✓	6
77.h	odd	6	1		539.2.a.j		3
77.i	even	6	1		539.2.a.g		3
77.i	even	6	1		539.2.e.m		6
77.m	even	15	4		847.2.n.f		24
77.o	odd	30	4		847.2.n.g		24
231.k	odd	6	1		4851.2.a.bk		3
231.l	even	6	1		693.2.i.h		6
231.l	even	6	1		4851.2.a.bj		3
308.m	odd	6	1		8624.2.a.co		3
308.n	even	6	1		1232.2.q.m		6
308.n	even	6	1		8624.2.a.ch		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.2.e.a	✓	6	11.b	odd	2	1
77.2.e.a	✓	6	77.h	odd	6	1
539.2.a.g		3	77.i	even	6	1
539.2.a.j		3	77.h	odd	6	1
539.2.e.m		6	77.b	even	2	1
539.2.e.m		6	77.i	even	6	1
693.2.i.h		6	33.d	even	2	1
693.2.i.h		6	231.l	even	6	1
847.2.e.c		6	1.a	even	1	1	trivial
847.2.e.c		6	7.c	even	3	1	inner
847.2.n.f		24	11.c	even	5	4
847.2.n.f		24	77.m	even	15	4
847.2.n.g		24	11.d	odd	10	4
847.2.n.g		24	77.o	odd	30	4
1232.2.q.m		6	44.c	even	2	1
1232.2.q.m		6	308.n	even	6	1
4851.2.a.bj		3	231.l	even	6	1
4851.2.a.bk		3	231.k	odd	6	1
5929.2.a.u		3	7.d	odd	6	1
5929.2.a.x		3	7.c	even	3	1
8624.2.a.ch		3	308.n	even	6	1
8624.2.a.co		3	308.m	odd	6	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{6} + 3T_{2}^{4} + 2T_{2}^{3} + 9T_{2}^{2} + 3T_{2} + 1 $ T2^6 + 3*T2^4 + 2*T2^3 + 9*T2^2 + 3*T2 + 1 acting on $S_{2}^{\mathrm{new}}(847, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$3$	$ T^{6} + 3 T^{5} + \cdots + 1 $ T^6 + 3T^5 + 9T^4 + 2T^3 + 3T^2 + 1
$5$	$ T^{6} + 6 T^{5} + \cdots + 1 $ T^6 + 6T^5 + 27T^4 + 52T^3 + 75T^2 + 9*T + 1
$7$	$ T^{6} - 17T^{3} + 343 $ T^6 - 17*T^3 + 343
$11$	$ T^{6} $ T^6
$13$	$ (T^{3} + 3 T^{2} - 6 T + 1)^{2} $ (T^3 + 3T^2 - 6T + 1)^2
$17$	$ T^{6} + 3 T^{5} + \cdots + 16129 $ T^6 + 3T^5 + 45T^4 + 146T^3 + 1677T^2 + 4572*T + 16129
$19$	$ T^{6} - 9 T^{5} + \cdots + 81 $ T^6 - 9T^5 + 63T^4 - 144T^3 + 243T^2 - 162*T + 81
$23$	$ T^{6} + 57 T^{4} + \cdots + 11449 $ T^6 + 57T^4 - 214T^3 + 3249T^2 - 6099T + 11449
$29$	$ (T^{3} - 3 T^{2} - 36 T - 51)^{2} $ (T^3 - 3T^2 - 36T - 51)^2
$31$	$ T^{6} + 9 T^{5} + \cdots + 361 $ T^6 + 9T^5 + 57T^4 + 178T^3 + 405T^2 + 456*T + 361
$37$	$ T^{6} + 36 T^{4} + \cdots + 5184 $ T^6 + 36T^4 + 144T^3 + 1296T^2 + 2592T + 5184
$41$	$ (T^{3} + 9 T^{2} + 6 T + 1)^{2} $ (T^3 + 9T^2 + 6T + 1)^2
$43$	$ (T^{3} - 9 T - 9)^{2} $ (T^3 - 9*T - 9)^2
$47$	$ T^{6} - 3 T^{5} + \cdots + 104329 $ T^6 - 3T^5 + 87T^4 - 412T^3 + 7053T^2 - 25194*T + 104329
$53$	$ T^{6} + 9 T^{5} + \cdots + 210681 $ T^6 + 9T^5 + 135T^4 + 432T^3 + 7047T^2 + 24786*T + 210681
$59$	$ T^{6} + 93 T^{4} + \cdots + 361 $ T^6 + 93T^4 + 38T^3 + 8649T^2 + 1767T + 361
$61$	$ T^{6} - 12 T^{5} + \cdots + 576 $ T^6 - 12T^5 + 180T^4 + 480T^3 + 1008T^2 + 864*T + 576
$67$	$ T^{6} + 12 T^{4} + \cdots + 64 $ T^6 + 12T^4 - 16T^3 + 144T^2 - 96T + 64
$71$	$ (T^{3} + 9 T^{2} + \cdots - 801)^{2} $ (T^3 + 9T^2 - 90T - 801)^2
$73$	$ T^{6} - 6 T^{5} + \cdots + 1 $ T^6 - 6T^5 + 27T^4 - 52T^3 + 75T^2 - 9*T + 1
$79$	$ T^{6} + 3 T^{5} + \cdots + 1369 $ T^6 + 3T^5 + 123T^4 - 416T^3 + 12885T^2 - 4218*T + 1369
$83$	$ (T^{3} - 15 T^{2} + \cdots + 267)^{2} $ (T^3 - 15T^2 + 18T + 267)^2
$89$	$ T^{6} - 15 T^{5} + \cdots + 12321 $ T^6 - 15T^5 + 153T^4 - 858T^3 + 3519T^2 - 7992*T + 12321
$97$	$ (T^{3} - 45 T^{2} + \cdots - 3329)^{2} $ (T^3 - 45T^2 + 672T - 3329)^2
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Overview	Random
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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:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{18}^{5} - \zeta_{18}^{4} + \cdots + \zeta_{18}) q^{2}+ \cdots + (3 \zeta_{18}^{5} + \cdots + 3 \zeta_{18}) q^{9}+O(q^{10}) \) q + (z^5 - z^4 - z^2 + z) * q^2 + (-z^5 - z^4 + z^3 + z^2 - 1) * q^3 + (-z^5 - z^4 + z) * q^4 + (z^4 - 2z^3 + z^2) q^5 + (-z^5 + 2z^4 - z^2 - z + 1) q^6 + (-2z^4 + 3z) * q^7 + (z^5 - z^4 - 1) * q^8 + (3z^5 - z^4 - z^2 + 3z) * q^9	\( q + (\zeta_{18}^{5} - \zeta_{18}^{4} + \cdots + \zeta_{18}) q^{2}+ \cdots + (8 \zeta_{18}^{4} - 8 \zeta_{18}^{3} + \cdots + 5) q^{98}+O(q^{100}) \) q + (z^5 - z^4 - z^2 + z) * q^2 + (-z^5 - z^4 + z^3 + z^2 - 1) * q^3 + (-z^5 - z^4 + z) * q^4 + (z^4 - 2z^3 + z^2) q^5 + (-z^5 + 2z^4 - z^2 - z + 1) q^6 + (-2z^4 + 3z) * q^7 + (z^5 - z^4 - 1) * q^8 + (3z^5 - z^4 - z^2 + 3z) * q^9 + (-z^3 + 3z^2 - 3z + 1) * q^10 - z^3 * q^12 + (-2z^5 + z^4 + z^2 + z - 1) q^13 + (-3z^5 + 2z^3 + z^2 - 3) * q^14 + (2z^4 - 2z^2 - 2z + 3) q^15 + (-z^5 - 2z^4 + 2z^3 - 2z^2 - z) q^16 + (-4z^5 - 4z^4 + z^3 + z^2 + 3z - 1) q^17 + (-3z^5 - 3z^4 + 4z^3 + 2z^2 + z - 4) * q^18 + (z^5 + z^4 + 3z^3 + z^2 + z) q^19 + (2z^5 - z^4 - z^2 - z + 2) q^20 + (-z^5 + 3z^4 - 2z^3 - 2z^2 - z + 3) q^21 + (5z^5 - 3z^4 - 3z^2 + 5z) * q^23 + (2z^5 + 2z^4 - 2z^3 - 3z^2 + z + 2) * q^24 + (-3z^5 - 3z^4 + z^3 - z^2 + 4z - 1) q^25 + (-2z^5 + 3z^4 - 3z^3 + 3z^2 - 2z) q^26 + (-3z^5 + 3z^4 + 2) * q^27 + (-3z^5 - 3z^3 + z^2 + 1) * q^28 + (-z^5 + 4z^4 - 3z^2 - 3z + 1) q^29 + (5z^5 - 3z^4 - 2z^3 - 3z^2 + 5z) q^30 + (z^5 + z^4 + 3z^3 - z - 3) q^31 + (z^5 + z^4 + 3z^3 - 3z^2 + 2z - 3) q^32 + (-4z^5 + 5z^4 - z^2 - z - 2) * q^34 + (z^5 - 2z^4 + z^3 + 2z^2 - 4z + 2) q^35 + (-3z^5 + z^4 + 2z^2 + 2z + 1) q^36 + (-4z^5 + 2z^4 + 2z^2 - 4z) * q^37 + (-z^5 - z^4 - z^2 + 2z) q^38 + (-z^3 + z^2 - z + 1) * q^39 + (-3z^5 + 2z^4 + z^3 + 2z^2 - 3z) * q^40 + (2z^5 + z^4 - 3z^2 - 3z - 3) q^41 + (4z^5 - 2z^4 - 4z^3 + z^2 + 3z - 1) * q^42 + (2z^5 - z^4 - z^2 - z) q^43 + (-2z^5 - 2z^4 + z^3 + 7z^2 - 5z - 1) * q^45 + (-5z^5 - 5z^4 + 8z^3 + 2z^2 + 3z - 8) q^46 + (-3z^5 - 3z^4 + z^3 - 3z^2 - 3z) * q^47 + (z^5 - 4z^4 + 3z^2 + 3z - 6) q^48 + (-8z^5 + 5z^2) * q^49 + (-5z^5 + 4z^4 + z^2 + z - 5) * q^50 + (3z^5 - z^4 - 6z^3 - z^2 + 3z) q^51 + (-3z^3 + 2z^2 - 2z + 3) q^52 + (3z^5 + 3z^4 + 3z^3 + 3z^2 - 6z - 3) q^53 + (2z^5 + z^4 - 6z^3 + z^2 + 2z) q^54 + (-z^5 + 2z^4 + 3z^3 - 2z^2 - 3z - 1) * q^56 + (-z^5 - z^4 + 2z^2 + 2z) * q^57 + (4z^5 - 5z^3 + 4z) q^58 + (-5z^5 - 5z^4 - z^2 + 6z) q^59 + (-z^5 - z^4 + 2z^3 + z - 2) q^60 + (-6z^5 + 2z^4 + 4z^3 + 2z^2 - 6z) q^61 + (-2z^5 + 2z^4 + 1) * q^62 + (-7z^5 + 8z^3 + 7z^2 - 5) q^63 + (-z^5 + 4z^4 - 3z^2 - 3z - 1) q^64 + (4z^5 - 6z^4 + 5z^3 - 6z^2 + 4z) q^65 + (-2z^5 - 2z^4 + 2z^2) q^67 + (-3z^5 - 7z^3 - 3z) q^68 + (-5z^5 + 10z^4 - 5z^2 - 5z + 7) * q^69 + (6z^5 - 3z^4 + 3z^3 - 9z^2 + z + 6) * q^70 + (7z^5 - 5z^4 - 2z^2 - 2z - 3) * q^71 + (-5z^5 + 4z^3 - 5z) q^72 + (-z^5 - z^4 - 2z^3 + z^2 + 2) q^73 + (4z^5 + 4z^4 - 6z^3 - 2z^2 - 2z + 6) q^74 + (-z^5 + z^4 - 3z^3 + z^2 - z) q^75 + (-4z^5 - z^4 + 5z^2 + 5z + 3) q^76 + (2z^5 - z^4 - z^2 - z + 2) q^78 + (-2z^5 + 7z^4 - z^3 + 7z^2 - 2z) * q^79 + (5z^5 + 5z^4 - 9z^3 - 5z + 9) * q^80 + (z^5 + z^4 + 5z^3 - z^2 - 5) q^81 + (z^4 + z^3 + z^2) * q^82 + (3z^5 - 5z^4 + 2z^2 + 2z + 5) * q^83 + (-z^4 - 2z) q^84 + (6z^5 - z^4 - 5z^2 - 5z + 9) q^85 + (z^5 - 2z^4 + 3z^3 - 2z^2 + z) q^86 + (-5z^5 - 5z^4 + 8z^3 + 9z^2 - 4z - 8) q^87 + (z^4 + 5z^3 + z^2) q^89 + (4z^5 + 3z^4 - 7z^2 - 7z + 12) * q^90 + (-z^5 + 2z^4 - 5z^3 + 5z^2 - 3z + 4) * q^91 + (-5z^5 + 3z^4 + 2z^2 + 2z - 1) * q^92 + (3z^5 - 2z^3 + 3z) q^93 + (3z^5 + 3z^4 - 7z^2 + 4z) * q^94 + (z^5 + z^4 - 3z^3 + z^2 - 2z + 3) * q^95 + (-3z^5 + 3z^4 + z^3 + 3z^2 - 3z) * q^96 + (-z^4 + z^2 + z + 15) * q^97 + (8z^4 - 8z^3 - 5z + 5) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - 6 q^{5} + 6 q^{6} - 6 q^{8}+O(q^{10}) \) 6 * q - 3 * q^3 - 6 * q^5 + 6 * q^6 - 6 * q^8	\( 6 q - 3 q^{3} - 6 q^{5} + 6 q^{6} - 6 q^{8} + 3 q^{10} - 3 q^{12} - 6 q^{13} - 12 q^{14} + 18 q^{15} + 6 q^{16} - 3 q^{17} - 12 q^{18} + 9 q^{19} + 12 q^{20} + 12 q^{21} + 6 q^{24} - 3 q^{25} - 9 q^{26} + 12 q^{27} - 3 q^{28} + 6 q^{29} - 6 q^{30} - 9 q^{31} - 9 q^{32} - 12 q^{34} + 15 q^{35} + 6 q^{36} + 3 q^{39} + 3 q^{40} - 18 q^{41} - 18 q^{42} - 3 q^{45} - 24 q^{46} + 3 q^{47} - 36 q^{48} - 30 q^{50} - 18 q^{51} + 9 q^{52} - 9 q^{53} - 18 q^{54} + 3 q^{56} - 15 q^{58} - 6 q^{60} + 12 q^{61} + 6 q^{62} - 6 q^{63} - 6 q^{64} + 15 q^{65} - 21 q^{68} + 42 q^{69} + 45 q^{70} - 18 q^{71} + 12 q^{72} + 6 q^{73} + 18 q^{74} - 9 q^{75} + 18 q^{76} + 12 q^{78} - 3 q^{79} + 27 q^{80} - 15 q^{81} + 3 q^{82} + 30 q^{83} + 54 q^{85} + 9 q^{86} - 24 q^{87} + 15 q^{89} + 72 q^{90} + 9 q^{91} - 6 q^{92} - 6 q^{93} + 9 q^{95} + 3 q^{96} + 90 q^{97} + 6 q^{98}+O(q^{100}) \) 6 * q - 3 * q^3 - 6 * q^5 + 6 * q^6 - 6 * q^8 + 3 * q^10 - 3 * q^12 - 6 * q^13 - 12 * q^14 + 18 * q^15 + 6 * q^16 - 3 * q^17 - 12 * q^18 + 9 * q^19 + 12 * q^20 + 12 * q^21 + 6 * q^24 - 3 * q^25 - 9 * q^26 + 12 * q^27 - 3 * q^28 + 6 * q^29 - 6 * q^30 - 9 * q^31 - 9 * q^32 - 12 * q^34 + 15 * q^35 + 6 * q^36 + 3 * q^39 + 3 * q^40 - 18 * q^41 - 18 * q^42 - 3 * q^45 - 24 * q^46 + 3 * q^47 - 36 * q^48 - 30 * q^50 - 18 * q^51 + 9 * q^52 - 9 * q^53 - 18 * q^54 + 3 * q^56 - 15 * q^58 - 6 * q^60 + 12 * q^61 + 6 * q^62 - 6 * q^63 - 6 * q^64 + 15 * q^65 - 21 * q^68 + 42 * q^69 + 45 * q^70 - 18 * q^71 + 12 * q^72 + 6 * q^73 + 18 * q^74 - 9 * q^75 + 18 * q^76 + 12 * q^78 - 3 * q^79 + 27 * q^80 - 15 * q^81 + 3 * q^82 + 30 * q^83 + 54 * q^85 + 9 * q^86 - 24 * q^87 + 15 * q^89 + 72 * q^90 + 9 * q^91 - 6 * q^92 - 6 * q^93 + 9 * q^95 + 3 * q^96 + 90 * q^97 + 6 * 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	847.2.e.c

Level	$847$
Weight	$2$
Character orbit	847.e
Analytic conductor	$6.763$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(6.76332905120\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
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:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 77)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 485.3
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$3$	\( T^{6} + 3 T^{5} + \cdots + 1 \) T^6 + 3T^5 + 9T^4 + 2T^3 + 3T^2 + 1
$5$	\( T^{6} + 6 T^{5} + \cdots + 1 \) T^6 + 6T^5 + 27T^4 + 52T^3 + 75T^2 + 9*T + 1
$7$	\( T^{6} - 17T^{3} + 343 \) T^6 - 17*T^3 + 343
$11$	\( T^{6} \) T^6
$13$	\( (T^{3} + 3 T^{2} - 6 T + 1)^{2} \) (T^3 + 3T^2 - 6T + 1)^2
$17$	\( T^{6} + 3 T^{5} + \cdots + 16129 \) T^6 + 3T^5 + 45T^4 + 146T^3 + 1677T^2 + 4572*T + 16129
$19$	\( T^{6} - 9 T^{5} + \cdots + 81 \) T^6 - 9T^5 + 63T^4 - 144T^3 + 243T^2 - 162*T + 81
$23$	\( T^{6} + 57 T^{4} + \cdots + 11449 \) T^6 + 57T^4 - 214T^3 + 3249T^2 - 6099T + 11449
$29$	\( (T^{3} - 3 T^{2} - 36 T - 51)^{2} \) (T^3 - 3T^2 - 36T - 51)^2
$31$	\( T^{6} + 9 T^{5} + \cdots + 361 \) T^6 + 9T^5 + 57T^4 + 178T^3 + 405T^2 + 456*T + 361
$37$	\( T^{6} + 36 T^{4} + \cdots + 5184 \) T^6 + 36T^4 + 144T^3 + 1296T^2 + 2592T + 5184
$41$	\( (T^{3} + 9 T^{2} + 6 T + 1)^{2} \) (T^3 + 9T^2 + 6T + 1)^2
$43$	\( (T^{3} - 9 T - 9)^{2} \) (T^3 - 9*T - 9)^2
$47$	\( T^{6} - 3 T^{5} + \cdots + 104329 \) T^6 - 3T^5 + 87T^4 - 412T^3 + 7053T^2 - 25194*T + 104329
$53$	\( T^{6} + 9 T^{5} + \cdots + 210681 \) T^6 + 9T^5 + 135T^4 + 432T^3 + 7047T^2 + 24786*T + 210681
$59$	\( T^{6} + 93 T^{4} + \cdots + 361 \) T^6 + 93T^4 + 38T^3 + 8649T^2 + 1767T + 361
$61$	\( T^{6} - 12 T^{5} + \cdots + 576 \) T^6 - 12T^5 + 180T^4 + 480T^3 + 1008T^2 + 864*T + 576
$67$	\( T^{6} + 12 T^{4} + \cdots + 64 \) T^6 + 12T^4 - 16T^3 + 144T^2 - 96T + 64
$71$	\( (T^{3} + 9 T^{2} + \cdots - 801)^{2} \) (T^3 + 9T^2 - 90T - 801)^2
$73$	\( T^{6} - 6 T^{5} + \cdots + 1 \) T^6 - 6T^5 + 27T^4 - 52T^3 + 75T^2 - 9*T + 1
$79$	\( T^{6} + 3 T^{5} + \cdots + 1369 \) T^6 + 3T^5 + 123T^4 - 416T^3 + 12885T^2 - 4218*T + 1369
$83$	\( (T^{3} - 15 T^{2} + \cdots + 267)^{2} \) (T^3 - 15T^2 + 18T + 267)^2
$89$	\( T^{6} - 15 T^{5} + \cdots + 12321 \) T^6 - 15T^5 + 153T^4 - 858T^3 + 3519T^2 - 7992*T + 12321
$97$	\( (T^{3} - 45 T^{2} + \cdots - 3329)^{2} \) (T^3 - 45T^2 + 672T - 3329)^2
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\(n\)	\(122\)	\(365\)
\(\chi(n)\)	\(-\zeta_{18}^{3}\)	\(1\)