LMFDB - Newform orbit 297.2.j.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [297,2,Mod(34,297)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(297, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([16, 0]))

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

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

chi := DirichletCharacter("297.34");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 297 = 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	297.j (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:	$2.37155694003$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/297\mathbb{Z}\right)^\times$.

$n$	$56$	$244$
$\chi(n)$	$-\zeta_{18}^{5}$	$1$

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} $

34.1

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

−1.43969

0.524005i

−1.50000

−

0.866025i

0.266044

−

0.223238i

−0.560307

−

3.17766i

2.61334

0.460802i

−0.500000

−

0.419550i

1.26604

−

2.19285i

1.50000

2.59808i

2.47178

4.28125i

67.1

−0.326352

−

1.85083i

−1.50000

0.866025i

−1.43969

0.524005i

−1.67365

−

1.40436i

2.09240

2.49362i

−0.500000

−

0.181985i

−0.439693

−

0.761570i

1.50000

−

2.59808i

−2.05303

3.55596i

133.1

−0.326352

1.85083i

−1.50000

−

0.866025i

−1.43969

−

0.524005i

−1.67365

1.40436i

2.09240

−

2.49362i

−0.500000

0.181985i

−0.439693

0.761570i

1.50000

2.59808i

−2.05303

−

3.55596i

166.1

−1.43969

−

0.524005i

−1.50000

0.866025i

0.266044

0.223238i

−0.560307

3.17766i

2.61334

−

0.460802i

−0.500000

0.419550i

1.26604

2.19285i

1.50000

−

2.59808i

2.47178

−

4.28125i

232.1

0.266044

0.223238i

−1.50000

−

0.866025i

−0.326352

−

1.85083i

−2.26604

−

0.824773i

−0.205737

−

0.565258i

−0.500000

2.83564i

0.673648

−

1.16679i

1.50000

2.59808i

−0.418748

−

0.725293i

265.1

0.266044

−

0.223238i

−1.50000

0.866025i

−0.326352

1.85083i

−2.26604

0.824773i

−0.205737

0.565258i

−0.500000

−

2.83564i

0.673648

1.16679i

1.50000

−

2.59808i

−0.418748

0.725293i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	297.2.j.a	✓	6
3.b	odd	2	1		891.2.j.a		6
27.e	even	9	1	inner	297.2.j.a	✓	6
27.e	even	9	1		8019.2.a.a		3
27.f	odd	18	1		891.2.j.a		6
27.f	odd	18	1		8019.2.a.b		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
297.2.j.a	✓	6	1.a	even	1	1	trivial
297.2.j.a	✓	6	27.e	even	9	1	inner
891.2.j.a		6	3.b	odd	2	1
891.2.j.a		6	27.f	odd	18	1
8019.2.a.a		3	27.e	even	9	1
8019.2.a.b		3	27.f	odd	18	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{6} + 3T_{2}^{5} + 6T_{2}^{4} + 8T_{2}^{3} + 3T_{2}^{2} - 3T_{2} + 1 $ T2^6 + 3*T2^5 + 6*T2^4 + 8*T2^3 + 3*T2^2 - 3*T2 + 1 acting on $S_{2}^{\mathrm{new}}(297, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} + 3 T^{5} + \cdots + 1 $ T^6 + 3T^5 + 6T^4 + 8T^3 + 3T^2 - 3*T + 1
$3$	$ (T^{2} + 3 T + 3)^{3} $ (T^2 + 3*T + 3)^3
$5$	$ T^{6} + 9 T^{5} + \cdots + 289 $ T^6 + 9T^5 + 45T^4 + 152T^3 + 342T^2 + 459*T + 289
$7$	$ T^{6} + 3 T^{5} + \cdots + 1 $ T^6 + 3T^5 + 12T^4 + 19T^3 + 15T^2 + 6*T + 1
$11$	$ T^{6} - T^{3} + 1 $ T^6 - T^3 + 1
$13$	$ T^{6} + 15 T^{5} + \cdots + 5329 $ T^6 + 15T^5 + 132T^4 + 739T^3 + 2883T^2 + 6132*T + 5329
$17$	$ T^{6} + 12 T^{5} + \cdots + 2809 $ T^6 + 12T^5 + 99T^4 + 434T^3 + 1389T^2 + 2385*T + 2809
$19$	$ T^{6} + 12 T^{5} + \cdots + 3249 $ T^6 + 12T^5 + 117T^4 + 438T^3 + 1413T^2 - 1539*T + 3249
$23$	$ T^{6} + 6 T^{5} + \cdots + 64 $ T^6 + 6T^5 + 24T^4 + 64T^3 + 48T^2 - 96*T + 64
$29$	$ T^{6} + 6 T^{5} + \cdots + 207936 $ T^6 + 6T^5 + 36T^4 + 192T^3 - 288T^2 + 16416*T + 207936
$31$	$ T^{6} + 18 T^{5} + \cdots + 1369 $ T^6 + 18T^5 + 126T^4 + 469T^3 + 1422T^2 + 1998*T + 1369
$37$	$ T^{6} + 36 T^{4} + \cdots + 5184 $ T^6 + 36T^4 - 144T^3 + 1296T^2 - 2592T + 5184
$41$	$ T^{6} + 6 T^{5} + \cdots + 361 $ T^6 + 6T^5 + 66T^4 + 422T^3 + 1752T^2 + 1425*T + 361
$43$	$ T^{6} + 6 T^{5} + \cdots + 45369 $ T^6 + 6T^5 + 126T^4 + 219T^3 - 1746T^2 + 3834*T + 45369
$47$	$ T^{6} - 15 T^{5} + \cdots + 289 $ T^6 - 15T^5 + 156T^4 - 917T^3 + 3885T^2 - 1938*T + 289
$53$	$ (T^{3} + 21 T^{2} + \cdots + 267)^{2} $ (T^3 + 21T^2 + 135T + 267)^2
$59$	$ T^{6} + 6 T^{5} + \cdots + 289 $ T^6 + 6T^5 - 12T^4 - 53T^3 + 480T^2 + 102*T + 289
$61$	$ T^{6} - 6 T^{5} + \cdots + 576 $ T^6 - 6T^5 + 36T^4 - 192T^3 + 576T^2 - 864*T + 576
$67$	$ T^{6} - 27 T^{5} + \cdots + 683929 $ T^6 - 27T^5 + 432T^4 - 4789T^3 + 50553T^2 - 297720*T + 683929
$71$	$ T^{6} + 21 T^{5} + \cdots + 356409 $ T^6 + 21T^5 + 387T^4 + 2328T^3 + 15453T^2 - 32238*T + 356409
$73$	$ T^{6} + 3 T^{5} + \cdots + 72361 $ T^6 + 3T^5 + 123T^4 + 196T^3 + 13803T^2 + 30666*T + 72361
$79$	$ T^{6} - 18 T^{5} + \cdots + 72361 $ T^6 - 18T^5 + 234T^4 - 3322T^3 + 26424T^2 - 65367*T + 72361
$83$	$ T^{6} + 39 T^{5} + \cdots + 1172889 $ T^6 + 39T^5 + 693T^4 + 6720T^3 + 45396T^2 + 282663*T + 1172889
$89$	$ T^{6} + 18 T^{5} + \cdots + 1896129 $ T^6 + 18T^5 + 405T^4 + 1296T^3 + 31347T^2 + 111537*T + 1896129
$97$	$ T^{6} + 30 T^{5} + \cdots + 26569 $ T^6 + 30T^5 + 330T^4 + 3005T^3 + 37506T^2 + 47922*T + 26569
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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:	\( N \)	\(=\)	\( 297 = 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	297.j (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{2} + (\zeta_{18}^{3} - 2) q^{3} + (\zeta_{18}^{4} + \zeta_{18}^{3} - 1) q^{4} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + \cdots - 2) q^{5}+ \cdots + ( - 3 \zeta_{18}^{3} + 3) q^{9}+O(q^{10}) \) q + (-z^3 + z^2) * q^2 + (z^3 - 2) * q^3 + (z^4 + z^3 - 1) * q^4 + (-z^5 + z^4 + z^3 + z^2 - 2z - 2) q^5 + (z^5 + z^3 - 2z^2 + 1) q^6 + (-z^5 - z^4 - z^3) * q^7 + (-z^5 + z^4 + z^3 + z^2 - z) * q^8 + (-3z^3 + 3) q^9	\( q + ( - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{2} + (\zeta_{18}^{3} - 2) q^{3} + (\zeta_{18}^{4} + \zeta_{18}^{3} - 1) q^{4} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + \cdots - 2) q^{5}+ \cdots + ( - 3 \zeta_{18}^{4} + 3 \zeta_{18}) q^{99}+O(q^{100}) \) q + (-z^3 + z^2) * q^2 + (z^3 - 2) * q^3 + (z^4 + z^3 - 1) * q^4 + (-z^5 + z^4 + z^3 + z^2 - 2z - 2) q^5 + (z^5 + z^3 - 2z^2 + 1) q^6 + (-z^5 - z^4 - z^3) * q^7 + (-z^5 + z^4 + z^3 + z^2 - z) * q^8 + (-3z^3 + 3) q^9 + (z^5 + z^4 - 3z^2 + 2z) * q^10 + z * q^11 + (-z^4 - 2z^3 - z + 1) q^12 + (-3z^4 + z^3 - 3z^2 + z - 3) * q^13 - z^2 * q^14 + (2z^5 - 3z^4 - 3z^3 - z^2 + 3z + 3) * q^15 + (z^5 - z^3 - 3z^2 + 2z - 2) * q^16 + (4z^3 - z^2 + z - 4) q^17 + (-3z^5 + 3z^2 - 3) * q^18 + (2z^5 - 3z^4 - 4z^3 - 3z^2 + 2z) q^19 + (-3z^4 - z^3 - z^2 + 1) q^20 + (z^5 + z^4 + z^3 + z^2 + z + 1) * q^21 + (-z^4 + z^3) * q^22 + (2z^4 + 2z^3 - 2) * q^23 + (2z^5 - 2z^4 - z^3 - z^2 + z - 1) * q^24 + (z^5 - 2z^4 - z^3 + z^2 + 2z - 1) * q^25 + (4z^5 - z^4 - 3z^2 - 3z + 4) q^26 + (6z^3 - 3) q^27 + (-z^5 - z^4 + 2z^2 + 2z + 2) * q^28 + (-2z^5 - 2z^4 - 6z^3 + 6z^2 + 2z + 2) q^29 + (-4z^5 + z^4 + 5z^2 - 5z) q^30 + (-z^5 + 2z^4 + 2z^3 + 2z - 4) q^31 + (3z^5 - 2z^4 + 3z^3 - 3z^2 - z - 1) * q^32 + (z^4 - 2z) q^33 + (5z^5 - 2z^4 + z^3 - 4z^2 + 4) q^34 + (2z^5 + z^4 + 2z^3 + z^2 + 2z) q^35 + (3z^3 + 3z) * q^36 + (4z^5 + 4z^4 - 2z^2 - 2z) * q^37 + (-3z^5 + 3z^3 + 2z^2 - 5z - 1) * q^38 + (-3z^5 + 4z^4 - 4z^3 + 6z^2 + z + 5) * q^39 + (-4z^4 + z^3 + z^2 + z - 4) q^40 + (z^4 - 4z^3 + z^2 - 4z + 1) * q^41 + (-z^5 + 2z^2) q^42 + (6z^5 - 6z^3 - 4z^2 + z + 2) q^43 + (z^5 + z^4 - z) * q^44 + (-3z^5 + 6z^4 + 6z^3 - 3z - 3) * q^45 + (2z^5 - 2z^4 + 2z^3 - 2z^2 + 2z) q^46 + (3z^5 + 5z^4 + z^3 - 2z^2 + 2) q^47 + (-4z^5 + 2z^4 - z^3 + 5z^2 - 4z + 5) * q^48 + (-4z^5 + 2z^4 + z^3 + 4z^2 - 3z - 3) * q^49 + (-3z^5 + 2z^4 + 2z^3 - 3z + 1) * q^50 + (-z^5 + z^4 - 8z^3 + 2z^2 - 2z + 4) q^51 + (-5z^5 - z^4 - 6z^3 + 6z^2 + z + 5) q^52 + (2z^5 - 2z^2 - 2z - 7) q^53 + (6z^5 - 3z^3 - 3z^2 + 6) q^54 + (z^5 + z^4 - 2z^2 - 2z + 1) * q^55 + (-z^5 - 2z^4 - z^3 + z^2 + 2z + 1) * q^56 + (-5z^5 + 5z^4 + 4z^3 + 4z^2 - z + 4) * q^57 + (-10z^5 + 4z^4 + 4z^3 - 4) q^58 + (3z^5 + 2z^3 - 3z^2 - 2z - 2) * q^59 + (-z^5 + 3z^4 + 2z^3 + 2z^2 + 3z - 1) * q^60 + (-2z^4 - 2z^3 - 2z^2 + 2) q^61 + (3z^5 - 5z^4 + 6z^3 - 5z^2 + 3z) q^62 + (-3z^2 - 3z - 3) * q^63 + (-3z^5 - 3z^4 - z^3 + 4z^2 - z + 1) q^64 + (4z^5 - 4z^3 + 5z^2 + 3z + 9) * q^65 + (z^4 - z^3 + z - 1) * q^66 + (z^4 + 7z^3 - z^2 + 7z + 1) * q^67 + (z^4 - 5z^3 + z^2 - 5z + 1) * q^68 + (-2z^4 - 4z^3 - 2z + 2) q^69 + (-z^5 + z^3 + 2z^2 - z + 1) q^70 + (-6z^5 - 6z^4 + 7z^3 + z^2 + 5z - 7) * q^71 + (-3z^5 + 3z^4 + 3) * q^72 + (-7z^5 + 5z^4 - z^3 + 5z^2 - 7z) * q^73 + (-2z^5 + 2z^3 + 4z^2 - 4) q^74 + (4z^4 - 3z^2 - 2z + 3) q^75 + (-4z^5 - 2z^4 - 3z^3 + 4z^2 + 5z + 5) q^76 + (-z^5 - z^4 - z^3 + 1) * q^77 + (-7z^5 - 2z^4 + 4z^3 + 2z^2 + 7z - 8) q^78 + (-5z^5 + 7z^4 - 4z^3 + 4z^2 - 7z + 5) q^79 + (2z^4 - 2z^2 - 2z + 11) q^80 - 9z^3 q^81 + (-5z^5 + 4z^4 + z^2 + z - 5) * q^82 + (9z^5 - z^4 + 5z^3 - 5z^2 + z - 9) q^83 + (3z^5 + 3z^4 + 2z^3 - 3z^2 - 3z - 4) q^84 + (4z^5 - 7z^4 - 7z^3 + z + 6) q^85 + (-8z^5 + z^4 + 5z^3 + 8z^2 - 6z - 6) * q^86 + (8z^5 + 4z^4 + 8z^3 - 10z^2 - 2z + 2) q^87 + (z^5 + z^4 - z^2 + 1) * q^88 + (6z^5 + 3z^4 - 6z^3 + 3z^2 + 6z) q^89 + (9z^5 - 6z^4 - 6z^2 + 9z) * q^90 + (7z^5 + 7z^4 + 4z^3 - 2z^2 - 5z - 4) q^91 + (2z^5 - 2z^3 - 2z^2 - 4z) * q^92 + (z^5 - 6z^3 + z^2 - 6z + 6) * q^93 + (-4z^4 + 2z^3 + 5z^2 + 2z - 4) * q^94 + (11z^4 - z^2 + 11) q^95 + (-6z^5 + z^4 - 4z^3 + 3z^2 + 4z - 1) * q^96 + (-2z^5 + 2z^3 - 4z^2 - 9z - 6) * q^97 + (z^5 + z^4 + z^3 - 7z^2 + 6z - 1) * q^98 + (-3z^4 + 3z) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} + 9 q^{6} - 3 q^{7} + 3 q^{8} + 9 q^{9}+O(q^{10}) \) 6 * q - 3 * q^2 - 9 * q^3 - 3 * q^4 - 9 * q^5 + 9 * q^6 - 3 * q^7 + 3 * q^8 + 9 * q^9	\( 6 q - 3 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} + 9 q^{6} - 3 q^{7} + 3 q^{8} + 9 q^{9} - 15 q^{13} + 9 q^{15} - 15 q^{16} - 12 q^{17} - 18 q^{18} - 12 q^{19} + 3 q^{20} + 9 q^{21} + 3 q^{22} - 6 q^{23} - 9 q^{24} - 9 q^{25} + 24 q^{26} + 12 q^{28} - 6 q^{29} - 18 q^{31} + 3 q^{32} + 27 q^{34} + 6 q^{35} + 9 q^{36} + 3 q^{38} + 18 q^{39} - 21 q^{40} - 6 q^{41} - 6 q^{43} + 6 q^{46} + 15 q^{47} + 27 q^{48} - 15 q^{49} + 12 q^{50} + 12 q^{52} - 42 q^{53} + 27 q^{54} + 6 q^{55} + 3 q^{56} + 36 q^{57} - 12 q^{58} - 6 q^{59} + 6 q^{61} + 18 q^{62} - 18 q^{63} + 3 q^{64} + 42 q^{65} - 9 q^{66} + 27 q^{67} - 9 q^{68} + 9 q^{70} - 21 q^{71} + 18 q^{72} - 3 q^{73} - 18 q^{74} + 18 q^{75} + 21 q^{76} + 3 q^{77} - 36 q^{78} + 18 q^{79} + 66 q^{80} - 27 q^{81} - 30 q^{82} - 39 q^{83} - 18 q^{84} + 15 q^{85} - 21 q^{86} + 36 q^{87} + 6 q^{88} - 18 q^{89} - 12 q^{91} - 6 q^{92} + 18 q^{93} - 18 q^{94} + 66 q^{95} - 18 q^{96} - 30 q^{97} - 3 q^{98}+O(q^{100}) \) 6 * q - 3 * q^2 - 9 * q^3 - 3 * q^4 - 9 * q^5 + 9 * q^6 - 3 * q^7 + 3 * q^8 + 9 * q^9 - 15 * q^13 + 9 * q^15 - 15 * q^16 - 12 * q^17 - 18 * q^18 - 12 * q^19 + 3 * q^20 + 9 * q^21 + 3 * q^22 - 6 * q^23 - 9 * q^24 - 9 * q^25 + 24 * q^26 + 12 * q^28 - 6 * q^29 - 18 * q^31 + 3 * q^32 + 27 * q^34 + 6 * q^35 + 9 * q^36 + 3 * q^38 + 18 * q^39 - 21 * q^40 - 6 * q^41 - 6 * q^43 + 6 * q^46 + 15 * q^47 + 27 * q^48 - 15 * q^49 + 12 * q^50 + 12 * q^52 - 42 * q^53 + 27 * q^54 + 6 * q^55 + 3 * q^56 + 36 * q^57 - 12 * q^58 - 6 * q^59 + 6 * q^61 + 18 * q^62 - 18 * q^63 + 3 * q^64 + 42 * q^65 - 9 * q^66 + 27 * q^67 - 9 * q^68 + 9 * q^70 - 21 * q^71 + 18 * q^72 - 3 * q^73 - 18 * q^74 + 18 * q^75 + 21 * q^76 + 3 * q^77 - 36 * q^78 + 18 * q^79 + 66 * q^80 - 27 * q^81 - 30 * q^82 - 39 * q^83 - 18 * q^84 + 15 * q^85 - 21 * q^86 + 36 * q^87 + 6 * q^88 - 18 * q^89 - 12 * q^91 - 6 * q^92 + 18 * q^93 - 18 * q^94 + 66 * q^95 - 18 * q^96 - 30 * q^97 - 3 * 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	297.2.j.a

Level	$297$
Weight	$2$
Character orbit	297.j
Analytic conductor	$2.372$
Analytic rank	$1$
Dimension	$6$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(2.37155694003\)
Analytic rank:	\(1\)
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:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 34.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} + 3 T^{5} + \cdots + 1 \) T^6 + 3T^5 + 6T^4 + 8T^3 + 3T^2 - 3*T + 1
$3$	\( (T^{2} + 3 T + 3)^{3} \) (T^2 + 3*T + 3)^3
$5$	\( T^{6} + 9 T^{5} + \cdots + 289 \) T^6 + 9T^5 + 45T^4 + 152T^3 + 342T^2 + 459*T + 289
$7$	\( T^{6} + 3 T^{5} + \cdots + 1 \) T^6 + 3T^5 + 12T^4 + 19T^3 + 15T^2 + 6*T + 1
$11$	\( T^{6} - T^{3} + 1 \) T^6 - T^3 + 1
$13$	\( T^{6} + 15 T^{5} + \cdots + 5329 \) T^6 + 15T^5 + 132T^4 + 739T^3 + 2883T^2 + 6132*T + 5329
$17$	\( T^{6} + 12 T^{5} + \cdots + 2809 \) T^6 + 12T^5 + 99T^4 + 434T^3 + 1389T^2 + 2385*T + 2809
$19$	\( T^{6} + 12 T^{5} + \cdots + 3249 \) T^6 + 12T^5 + 117T^4 + 438T^3 + 1413T^2 - 1539*T + 3249
$23$	\( T^{6} + 6 T^{5} + \cdots + 64 \) T^6 + 6T^5 + 24T^4 + 64T^3 + 48T^2 - 96*T + 64
$29$	\( T^{6} + 6 T^{5} + \cdots + 207936 \) T^6 + 6T^5 + 36T^4 + 192T^3 - 288T^2 + 16416*T + 207936
$31$	\( T^{6} + 18 T^{5} + \cdots + 1369 \) T^6 + 18T^5 + 126T^4 + 469T^3 + 1422T^2 + 1998*T + 1369
$37$	\( T^{6} + 36 T^{4} + \cdots + 5184 \) T^6 + 36T^4 - 144T^3 + 1296T^2 - 2592T + 5184
$41$	\( T^{6} + 6 T^{5} + \cdots + 361 \) T^6 + 6T^5 + 66T^4 + 422T^3 + 1752T^2 + 1425*T + 361
$43$	\( T^{6} + 6 T^{5} + \cdots + 45369 \) T^6 + 6T^5 + 126T^4 + 219T^3 - 1746T^2 + 3834*T + 45369
$47$	\( T^{6} - 15 T^{5} + \cdots + 289 \) T^6 - 15T^5 + 156T^4 - 917T^3 + 3885T^2 - 1938*T + 289
$53$	\( (T^{3} + 21 T^{2} + \cdots + 267)^{2} \) (T^3 + 21T^2 + 135T + 267)^2
$59$	\( T^{6} + 6 T^{5} + \cdots + 289 \) T^6 + 6T^5 - 12T^4 - 53T^3 + 480T^2 + 102*T + 289
$61$	\( T^{6} - 6 T^{5} + \cdots + 576 \) T^6 - 6T^5 + 36T^4 - 192T^3 + 576T^2 - 864*T + 576
$67$	\( T^{6} - 27 T^{5} + \cdots + 683929 \) T^6 - 27T^5 + 432T^4 - 4789T^3 + 50553T^2 - 297720*T + 683929
$71$	\( T^{6} + 21 T^{5} + \cdots + 356409 \) T^6 + 21T^5 + 387T^4 + 2328T^3 + 15453T^2 - 32238*T + 356409
$73$	\( T^{6} + 3 T^{5} + \cdots + 72361 \) T^6 + 3T^5 + 123T^4 + 196T^3 + 13803T^2 + 30666*T + 72361
$79$	\( T^{6} - 18 T^{5} + \cdots + 72361 \) T^6 - 18T^5 + 234T^4 - 3322T^3 + 26424T^2 - 65367*T + 72361
$83$	\( T^{6} + 39 T^{5} + \cdots + 1172889 \) T^6 + 39T^5 + 693T^4 + 6720T^3 + 45396T^2 + 282663*T + 1172889
$89$	\( T^{6} + 18 T^{5} + \cdots + 1896129 \) T^6 + 18T^5 + 405T^4 + 1296T^3 + 31347T^2 + 111537*T + 1896129
$97$	\( T^{6} + 30 T^{5} + \cdots + 26569 \) T^6 + 30T^5 + 330T^4 + 3005T^3 + 37506T^2 + 47922*T + 26569
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\(n\)	\(56\)	\(244\)
\(\chi(n)\)	\(-\zeta_{18}^{5}\)	\(1\)