LMFDB - Newform orbit 58.2.d.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [58,2,Mod(7,58)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(58, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([6]))

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

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

chi := DirichletCharacter("58.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 58 = 2 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	58.d (of order $7$, 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:	$0.463132331723$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/58\mathbb{Z}\right)^\times$.

$n$	$31$
$\chi(n)$	$-\zeta_{14}$

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} $

7.1

0.900969	−	0.433884i
0.222521	+	0.974928i
0.900969	+	0.433884i
−0.623490	−	0.781831i
−0.623490	+	0.781831i
0.222521	−	0.974928i

0.222521

0.974928i

0.500000

0.240787i

−0.900969

0.433884i

−0.0440730

−

0.193096i

−0.123490

0.541044i

−0.0990311

−

0.0476909i

−0.623490

−

0.781831i

−1.67845

−

2.10471i

0.178448

−

0.0859360i

23.1

−0.623490

0.781831i

0.500000

−

2.19064i

−0.222521

−

0.974928i

0.969501

−

1.21572i

1.40097

1.75676i

−0.777479

3.40636i

0.900969

0.433884i

−1.84601

−

0.888992i

0.346011

1.51597i

25.1

0.222521

−

0.974928i

0.500000

−

0.240787i

−0.900969

−

0.433884i

−0.0440730

0.193096i

−0.123490

−

0.541044i

−0.0990311

0.0476909i

−0.623490

0.781831i

−1.67845

2.10471i

0.178448

0.0859360i

45.1

0.900969

0.433884i

0.500000

−

0.626980i

0.623490

0.781831i

−2.92543

−

1.40881i

0.722521

−

0.347948i

−1.62349

2.03579i

0.222521

0.974928i

0.524459

2.29780i

−2.02446

−

2.53859i

49.1

0.900969

−

0.433884i

0.500000

0.626980i

0.623490

−

0.781831i

−2.92543

1.40881i

0.722521

0.347948i

−1.62349

−

2.03579i

0.222521

−

0.974928i

0.524459

−

2.29780i

−2.02446

2.53859i

53.1

−0.623490

−

0.781831i

0.500000

2.19064i

−0.222521

0.974928i

0.969501

1.21572i

1.40097

−

1.75676i

−0.777479

−

3.40636i

0.900969

−

0.433884i

−1.84601

0.888992i

0.346011

−

1.51597i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
29.d	even	7	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	58.2.d.a	✓	6
3.b	odd	2	1		522.2.k.c		6
4.b	odd	2	1		464.2.u.b		6
29.d	even	7	1	inner	58.2.d.a	✓	6
29.d	even	7	1		1682.2.a.m		3
29.e	even	14	1		1682.2.a.n		3
29.f	odd	28	2		1682.2.b.g		6
87.j	odd	14	1		522.2.k.c		6
116.j	odd	14	1		464.2.u.b		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
58.2.d.a	✓	6	1.a	even	1	1	trivial
58.2.d.a	✓	6	29.d	even	7	1	inner
464.2.u.b		6	4.b	odd	2	1
464.2.u.b		6	116.j	odd	14	1
522.2.k.c		6	3.b	odd	2	1
522.2.k.c		6	87.j	odd	14	1
1682.2.a.m		3	29.d	even	7	1
1682.2.a.n		3	29.e	even	14	1
1682.2.b.g		6	29.f	odd	28	2

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{6} - 3T_{3}^{5} + 9T_{3}^{4} - 13T_{3}^{3} + 11T_{3}^{2} - 5T_{3} + 1 $ T3^6 - 3*T3^5 + 9*T3^4 - 13*T3^3 + 11*T3^2 - 5*T3 + 1 acting on $S_{2}^{\mathrm{new}}(58, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} - T^{5} + T^{4} + \cdots + 1 $ T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$3$	$ T^{6} - 3 T^{5} + \cdots + 1 $ T^6 - 3T^5 + 9T^4 - 13T^3 + 11T^2 - 5*T + 1
$5$	$ T^{6} + 4 T^{5} + \cdots + 1 $ T^6 + 4T^5 + 2T^4 - 6T^3 + 25T^2 + 2*T + 1
$7$	$ T^{6} + 5 T^{5} + \cdots + 1 $ T^6 + 5T^5 + 25T^4 + 55T^3 + 93T^2 + 17*T + 1
$11$	$ T^{6} + 6 T^{5} + \cdots + 729 $ T^6 + 6T^5 + 36T^4 + 216T^3 + 729T^2 + 972*T + 729
$13$	$ T^{6} - 11 T^{5} + \cdots + 841 $ T^6 - 11T^5 + 58T^4 - 169T^3 + 312T^2 - 464*T + 841
$17$	$ (T^{3} + 5 T^{2} - 8 T - 41)^{2} $ (T^3 + 5T^2 - 8T - 41)^2
$19$	$ T^{6} + 6 T^{5} + \cdots + 1681 $ T^6 + 6T^5 + 57T^4 + 286T^3 + 1478T^2 + 2624*T + 1681
$23$	$ T^{6} + 7 T^{5} + \cdots + 8281 $ T^6 + 7T^5 + 84T^4 + 161T^3 + 490T^2 - 5096*T + 8281
$29$	$ T^{6} - 15 T^{5} + \cdots + 24389 $ T^6 - 15T^5 + 113T^4 - 659T^3 + 3277T^2 - 12615*T + 24389
$31$	$ T^{6} - 4 T^{5} + \cdots + 1849 $ T^6 - 4T^5 + 86T^4 + 27T^3 - 80T^2 + 817*T + 1849
$37$	$ T^{6} + 13 T^{5} + \cdots + 19321 $ T^6 + 13T^5 + 99T^4 + 713T^3 + 4201T^2 + 11815*T + 19321
$41$	$ (T^{3} + 6 T^{2} - 37 T + 1)^{2} $ (T^3 + 6T^2 - 37T + 1)^2
$43$	$ T^{6} + 7 T^{5} + \cdots + 41209 $ T^6 + 7T^5 + 91T^4 - 329T^3 + 5929T^2 - 26999*T + 41209
$47$	$ T^{6} - 3 T^{5} + \cdots + 1 $ T^6 - 3T^5 + 9T^4 - 13T^3 + 11T^2 - 5*T + 1
$53$	$ T^{6} + 22 T^{5} + \cdots + 113569 $ T^6 + 22T^5 + 211T^4 + 1156T^3 + 5314T^2 - 20894*T + 113569
$59$	$ (T^{3} + 19 T^{2} + \cdots + 169)^{2} $ (T^3 + 19T^2 + 104T + 169)^2
$61$	$ T^{6} - 31 T^{5} + \cdots + 169 $ T^6 - 31T^5 + 422T^4 - 2925T^3 + 9580T^2 + 520*T + 169
$67$	$ T^{6} - 11 T^{5} + \cdots + 1771561 $ T^6 - 11T^5 + 121T^4 - 1331T^3 + 14641T^2 - 161051*T + 1771561
$71$	$ T^{6} - 23 T^{5} + \cdots + 63001 $ T^6 - 23T^5 + 319T^4 - 2395T^3 + 13169T^2 - 42921*T + 63001
$73$	$ T^{6} - 5 T^{5} + \cdots + 241081 $ T^6 - 5T^5 + 25T^4 - 587T^3 + 3243T^2 + 12275*T + 241081
$79$	$ T^{6} + 56 T^{4} + \cdots + 3136 $ T^6 + 56T^4 - 112T^3 + 784T^2 + 3136T + 3136
$83$	$ T^{6} - 35 T^{5} + \cdots + 790321 $ T^6 - 35T^5 + 630T^4 - 7315T^3 + 58408T^2 - 286258*T + 790321
$89$	$ T^{6} - 13 T^{5} + \cdots + 253009 $ T^6 - 13T^5 + 169T^4 - 1007T^3 + 1373T^2 - 7545*T + 253009
$97$	$ T^{6} - 20 T^{5} + \cdots + 10816 $ T^6 - 20T^5 + 148T^4 - 496T^3 + 2976T^2 - 6656*T + 10816
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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:	\( N \)	\(=\)	\( 58 = 2 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	58.d (of order \(7\), degree \(6\), minimal)

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

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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	58.2.d.a

Level	$58$
Weight	$2$
Character orbit	58.d
Analytic conductor	$0.463$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(0.463132331723\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\Q(\zeta_{14})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 7.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} - T^{5} + T^{4} + \cdots + 1 \) T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$3$	\( T^{6} - 3 T^{5} + \cdots + 1 \) T^6 - 3T^5 + 9T^4 - 13T^3 + 11T^2 - 5*T + 1
$5$	\( T^{6} + 4 T^{5} + \cdots + 1 \) T^6 + 4T^5 + 2T^4 - 6T^3 + 25T^2 + 2*T + 1
$7$	\( T^{6} + 5 T^{5} + \cdots + 1 \) T^6 + 5T^5 + 25T^4 + 55T^3 + 93T^2 + 17*T + 1
$11$	\( T^{6} + 6 T^{5} + \cdots + 729 \) T^6 + 6T^5 + 36T^4 + 216T^3 + 729T^2 + 972*T + 729
$13$	\( T^{6} - 11 T^{5} + \cdots + 841 \) T^6 - 11T^5 + 58T^4 - 169T^3 + 312T^2 - 464*T + 841
$17$	\( (T^{3} + 5 T^{2} - 8 T - 41)^{2} \) (T^3 + 5T^2 - 8T - 41)^2
$19$	\( T^{6} + 6 T^{5} + \cdots + 1681 \) T^6 + 6T^5 + 57T^4 + 286T^3 + 1478T^2 + 2624*T + 1681
$23$	\( T^{6} + 7 T^{5} + \cdots + 8281 \) T^6 + 7T^5 + 84T^4 + 161T^3 + 490T^2 - 5096*T + 8281
$29$	\( T^{6} - 15 T^{5} + \cdots + 24389 \) T^6 - 15T^5 + 113T^4 - 659T^3 + 3277T^2 - 12615*T + 24389
$31$	\( T^{6} - 4 T^{5} + \cdots + 1849 \) T^6 - 4T^5 + 86T^4 + 27T^3 - 80T^2 + 817*T + 1849
$37$	\( T^{6} + 13 T^{5} + \cdots + 19321 \) T^6 + 13T^5 + 99T^4 + 713T^3 + 4201T^2 + 11815*T + 19321
$41$	\( (T^{3} + 6 T^{2} - 37 T + 1)^{2} \) (T^3 + 6T^2 - 37T + 1)^2
$43$	\( T^{6} + 7 T^{5} + \cdots + 41209 \) T^6 + 7T^5 + 91T^4 - 329T^3 + 5929T^2 - 26999*T + 41209
$47$	\( T^{6} - 3 T^{5} + \cdots + 1 \) T^6 - 3T^5 + 9T^4 - 13T^3 + 11T^2 - 5*T + 1
$53$	\( T^{6} + 22 T^{5} + \cdots + 113569 \) T^6 + 22T^5 + 211T^4 + 1156T^3 + 5314T^2 - 20894*T + 113569
$59$	\( (T^{3} + 19 T^{2} + \cdots + 169)^{2} \) (T^3 + 19T^2 + 104T + 169)^2
$61$	\( T^{6} - 31 T^{5} + \cdots + 169 \) T^6 - 31T^5 + 422T^4 - 2925T^3 + 9580T^2 + 520*T + 169
$67$	\( T^{6} - 11 T^{5} + \cdots + 1771561 \) T^6 - 11T^5 + 121T^4 - 1331T^3 + 14641T^2 - 161051*T + 1771561
$71$	\( T^{6} - 23 T^{5} + \cdots + 63001 \) T^6 - 23T^5 + 319T^4 - 2395T^3 + 13169T^2 - 42921*T + 63001
$73$	\( T^{6} - 5 T^{5} + \cdots + 241081 \) T^6 - 5T^5 + 25T^4 - 587T^3 + 3243T^2 + 12275*T + 241081
$79$	\( T^{6} + 56 T^{4} + \cdots + 3136 \) T^6 + 56T^4 - 112T^3 + 784T^2 + 3136T + 3136
$83$	\( T^{6} - 35 T^{5} + \cdots + 790321 \) T^6 - 35T^5 + 630T^4 - 7315T^3 + 58408T^2 - 286258*T + 790321
$89$	\( T^{6} - 13 T^{5} + \cdots + 253009 \) T^6 - 13T^5 + 169T^4 - 1007T^3 + 1373T^2 - 7545*T + 253009
$97$	\( T^{6} - 20 T^{5} + \cdots + 10816 \) T^6 - 20T^5 + 148T^4 - 496T^3 + 2976T^2 - 6656*T + 10816
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\(n\)	\(31\)
\(\chi(n)\)	\(-\zeta_{14}\)