LMFDB - Newform orbit 1169.1.f.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1169,1,Mod(333,1169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1169, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 3]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1169.333");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1169 = 7 \cdot 167 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1169.f (of order $6$, 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:	$0.583406999768$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{33}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{33} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1169\mathbb{Z}\right)^\times$.

$n$	$673$	$836$
$\chi(n)$	$-1$	$\zeta_{66}^{22}$

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} $

333.1

−0.995472	+	0.0950560i
0.580057	−	0.814576i
0.981929	+	0.189251i
−0.327068	+	0.945001i
0.723734	+	0.690079i
0.235759	+	0.971812i
0.928368	−	0.371662i
−0.786053	+	0.618159i
−0.888835	−	0.458227i
0.0475819	−	0.998867i
−0.995472	−	0.0950560i
0.580057	+	0.814576i
0.981929	−	0.189251i
−0.327068	−	0.945001i
0.723734	−	0.690079i
0.235759	−	0.971812i
0.928368	+	0.371662i
−0.786053	−	0.618159i
−0.888835	+	0.458227i
0.0475819	+	0.998867i

−0.841254

−

1.45709i

−0.981929

1.70075i

−0.915415

1.58555i

3.30420

0.723734

0.690079i

1.39788

−1.42837

−

2.47401i

333.2

−0.841254

−

1.45709i

0.327068

−

0.566498i

−0.915415

1.58555i

−1.10059

0.235759

0.971812i

1.39788

0.286053

0.495458i

333.3

−0.415415

−

0.719520i

−0.928368

1.60798i

0.154861

−

0.268227i

1.54263

0.0475819

−

0.998867i

−1.08816

−1.22373

−

2.11957i

333.4

−0.415415

−

0.719520i

0.786053

−

1.36148i

0.154861

−

0.268227i

−1.30615

−0.888835

−

0.458227i

−1.08816

−0.735759

−

1.27437i

333.5

0.142315

0.246497i

−0.0475819

0.0824143i

0.459493

−

0.795865i

−0.0270865

0.981929

0.189251i

0.546200

0.495472

0.858183i

333.6

0.142315

0.246497i

0.888835

−

1.53951i

0.459493

−

0.795865i

0.505978

−0.327068

0.945001i

0.546200

−1.08006

−

1.87071i

333.7

0.654861

1.13425i

−0.723734

1.25354i

−0.357685

0.619529i

−1.89578

−0.995472

0.0950560i

0.372786

−0.547582

−

0.948440i

333.8

0.654861

1.13425i

−0.235759

0.408346i

−0.357685

0.619529i

−0.617557

0.580057

−

0.814576i

0.372786

0.388835

0.673483i

333.9

0.959493

1.66189i

−0.580057

1.00469i

−1.34125

2.32312i

−2.22624

−0.786053

0.618159i

−3.22871

−0.172932

−

0.299527i

333.10

0.959493

1.66189i

0.995472

−

1.72421i

−1.34125

2.32312i

3.82059

0.928368

−

0.371662i

−3.22871

−1.48193

−

2.56678i

667.1

−0.841254

1.45709i

−0.981929

−

1.70075i

−0.915415

−

1.58555i

3.30420

0.723734

−

0.690079i

1.39788

−1.42837

2.47401i

667.2

−0.841254

1.45709i

0.327068

0.566498i

−0.915415

−

1.58555i

−1.10059

0.235759

−

0.971812i

1.39788

0.286053

−

0.495458i

667.3

−0.415415

0.719520i

−0.928368

−

1.60798i

0.154861

0.268227i

1.54263

0.0475819

0.998867i

−1.08816

−1.22373

2.11957i

667.4

−0.415415

0.719520i

0.786053

1.36148i

0.154861

0.268227i

−1.30615

−0.888835

0.458227i

−1.08816

−0.735759

1.27437i

667.5

0.142315

−

0.246497i

−0.0475819

−

0.0824143i

0.459493

0.795865i

−0.0270865

0.981929

−

0.189251i

0.546200

0.495472

−

0.858183i

667.6

0.142315

−

0.246497i

0.888835

1.53951i

0.459493

0.795865i

0.505978

−0.327068

−

0.945001i

0.546200

−1.08006

1.87071i

667.7

0.654861

−

1.13425i

−0.723734

−

1.25354i

−0.357685

−

0.619529i

−1.89578

−0.995472

−

0.0950560i

0.372786

−0.547582

0.948440i

667.8

0.654861

−

1.13425i

−0.235759

−

0.408346i

−0.357685

−

0.619529i

−0.617557

0.580057

0.814576i

0.372786

0.388835

−

0.673483i

667.9

0.959493

−

1.66189i

−0.580057

−

1.00469i

−1.34125

−

2.32312i

−2.22624

−0.786053

−

0.618159i

−3.22871

−0.172932

0.299527i

667.10

0.959493

−

1.66189i

0.995472

1.72421i

−1.34125

−

2.32312i

3.82059

0.928368

0.371662i

−3.22871

−1.48193

2.56678i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
167.b	odd	2	1	CM by $\Q(\sqrt{-167}) $
7.c	even	3	1	inner
1169.f	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1169.1.f.c	✓	20
7.c	even	3	1	inner	1169.1.f.c	✓	20
167.b	odd	2	1	CM	1169.1.f.c	✓	20
1169.f	odd	6	1	inner	1169.1.f.c	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1169.1.f.c	✓	20	1.a	even	1	1	trivial
1169.1.f.c	✓	20	7.c	even	3	1	inner
1169.1.f.c	✓	20	167.b	odd	2	1	CM
1169.1.f.c	✓	20	1169.f	odd	6	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{10} - T_{2}^{9} + 5T_{2}^{8} - 2T_{2}^{7} + 16T_{2}^{6} - 7T_{2}^{5} + 20T_{2}^{4} + T_{2}^{3} + 12T_{2}^{2} - 3T_{2} + 1 $ T2^10 - T2^9 + 5*T2^8 - 2*T2^7 + 16*T2^6 - 7*T2^5 + 20*T2^4 + T2^3 + 12*T2^2 - 3*T2 + 1 acting on $S_{1}^{\mathrm{new}}(1169, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{10} - T^{9} + 5 T^{8} + \cdots + 1)^{2} $ (T^10 - T^9 + 5T^8 - 2T^7 + 16T^6 - 7T^5 + 20T^4 + T^3 + 12T^2 - 3*T + 1)^2
$3$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 + 11T^18 + 10T^17 + 76T^16 + 66T^15 + 320T^14 + 254T^13 + 968T^12 + 714T^11 + 1935T^10 + 1220T^9 + 2673T^8 + 1431T^7 + 2025T^6 + 550T^5 + 703T^4 + 230T^3 + 132T^2 + 12T + 1
$5$	$ T^{20} $ T^20
$7$	$ T^{20} - T^{19} + \cdots + 1 $ T^20 - T^19 + T^17 - T^16 + T^14 - T^13 + T^11 - T^10 + T^9 - T^7 + T^6 - T^4 + T^3 - T + 1
$11$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 + 11T^18 + 10T^17 + 76T^16 + 66T^15 + 320T^14 + 254T^13 + 968T^12 + 714T^11 + 1935T^10 + 1220T^9 + 2673T^8 + 1431T^7 + 2025T^6 + 550T^5 + 703T^4 + 230T^3 + 132T^2 + 12T + 1
$13$	$ T^{20} $ T^20
$17$	$ T^{20} $ T^20
$19$	$ (T^{10} - T^{9} + 5 T^{8} + \cdots + 1)^{2} $ (T^10 - T^9 + 5T^8 - 2T^7 + 16T^6 - 7T^5 + 20T^4 + T^3 + 12T^2 - 3*T + 1)^2
$23$	$ T^{20} $ T^20
$29$	$ (T^{10} - T^{9} - 10 T^{8} + \cdots + 1)^{2} $ (T^10 - T^9 - 10T^8 + 10T^7 + 34T^6 - 34T^5 - 43T^4 + 43T^3 + 12T^2 - 12T + 1)^2
$31$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 + 11T^18 + 10T^17 + 76T^16 + 66T^15 + 320T^14 + 254T^13 + 968T^12 + 714T^11 + 1935T^10 + 1220T^9 + 2673T^8 + 1431T^7 + 2025T^6 + 550T^5 + 703T^4 + 230T^3 + 132T^2 + 12T + 1
$37$	$ T^{20} $ T^20
$41$	$ T^{20} $ T^20
$43$	$ T^{20} $ T^20
$47$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 + 11T^18 + 10T^17 + 76T^16 + 66T^15 + 320T^14 + 254T^13 + 968T^12 + 714T^11 + 1935T^10 + 1220T^9 + 2673T^8 + 1431T^7 + 2025T^6 + 550T^5 + 703T^4 + 230T^3 + 132T^2 + 12T + 1
$53$	$ T^{20} $ T^20
$59$	$ T^{20} $ T^20
$61$	$ (T^{10} - T^{9} + 5 T^{8} + \cdots + 1)^{2} $ (T^10 - T^9 + 5T^8 - 2T^7 + 16T^6 - 7T^5 + 20T^4 + T^3 + 12T^2 - 3*T + 1)^2
$67$	$ T^{20} $ T^20
$71$	$ T^{20} $ T^20
$73$	$ T^{20} $ T^20
$79$	$ T^{20} $ T^20
$83$	$ T^{20} $ T^20
$89$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 + 11T^18 + 10T^17 + 76T^16 + 66T^15 + 320T^14 + 254T^13 + 968T^12 + 714T^11 + 1935T^10 + 1220T^9 + 2673T^8 + 1431T^7 + 2025T^6 + 550T^5 + 703T^4 + 230T^3 + 132T^2 + 12T + 1
$97$	$ (T^{5} + T^{4} - 4 T^{3} + \cdots + 1)^{4} $ (T^5 + T^4 - 4T^3 - 3T^2 + 3*T + 1)^4
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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:	\( N \)	\(=\)	\( 1169 = 7 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1169.f (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{66}^{28} + \zeta_{66}^{16}) q^{2} + ( - \zeta_{66}^{13} - \zeta_{66}^{9}) q^{3} + (\zeta_{66}^{32} + \cdots - \zeta_{66}^{11}) q^{4}+ \cdots + (\zeta_{66}^{26} + \cdots + \zeta_{66}^{18}) q^{9}+O(q^{10}) \) q + (z^28 + z^16) * q^2 + (-z^13 - z^9) * q^3 + (z^32 - z^23 - z^11) * q^4 + (-z^29 - z^25 + z^8 + z^4) * q^6 - z^25 * q^7 + (-z^27 + z^18 - z^15 - z^6) * q^8 + (z^26 + z^22 + z^18) * q^9	\( q + (\zeta_{66}^{28} + \zeta_{66}^{16}) q^{2} + ( - \zeta_{66}^{13} - \zeta_{66}^{9}) q^{3} + (\zeta_{66}^{32} + \cdots - \zeta_{66}^{11}) q^{4}+ \cdots + (\zeta_{66}^{32} + \zeta_{66}^{30} + \cdots - \zeta_{66}) q^{99}+O(q^{100}) \) q + (z^28 + z^16) * q^2 + (-z^13 - z^9) * q^3 + (z^32 - z^23 - z^11) * q^4 + (-z^29 - z^25 + z^8 + z^4) * q^6 - z^25 * q^7 + (-z^27 + z^18 - z^15 - z^6) * q^8 + (z^26 + z^22 + z^18) * q^9 + (z^12 + z^10) * q^11 + (z^32 + z^24 + z^20 + z^12 + z^8 - z^3) * q^12 + (z^20 + z^8) * q^14 + (-z^31 + z^22 - z^13 - z^10 + z) * q^16 + (-z^21 - z^17 - z^13 - z^9 - z^5 - z) * q^18 + (-z^7 + z^4) * q^19 + (-z^5 - z) * q^21 + (z^28 + z^26 - z^7 - z^5) * q^22 + (-z^31 + z^28 - z^27 + z^24 - z^19 - z^15 - z^7 - z^3) * q^24 - z^11 * q^25 + (-z^31 - z^27 + z^6 - z^2) * q^27 + (z^24 - z^15 - z^3) * q^28 + (-z^19 + z^14) * q^29 + (z^16 + z^6) * q^31 + (-z^29 + z^26 - z^17 + z^14 + z^8 + z^5) * q^32 + (-z^25 - z^23 - z^21 - z^19) * q^33 + (-z^29 - z^25 - z^21 - z^17 + z^16 + z^12 + z^8 + z^4 + 1) * q^36 + (z^32 - z^23 + z^20 + z^2) * q^38 + (-z^29 - z^21 - z^17 + 1) * q^42 + (-z^23 - z^21 - z^11 - z^9 + z^2 + 1) * q^44 + (-z^29 - z^15) * q^47 + (-z^31 + z^26 - z^23 + z^22 - z^19 + z^14 - z^11 + z^10 - z^7 + z^2) * q^48 - z^17 * q^49 + (-z^27 + z^6) * q^50 + (z^30 + z^26 + z^22 + z^18 + z^14 + z^10 - z) * q^54 + (-z^31 - z^19 + z^10 - z^7) * q^56 + (z^20 - z^17 + z^16 - z^13) * q^57 + (z^30 + z^14 - z^9 + z^2) * q^58 + (-z^31 - z^13) * q^61 + (z^32 + z^22 - z^11 - z) * q^62 + (z^18 + z^14 + z^10) * q^63 + (z^30 + z^24 - z^21 + z^12 + z^9 - z^3 - 1) * q^64 + (z^20 + z^18 + z^16 + z^14 + z^8 + z^6 + z^4 + z^2) * q^66 + (z^32 + z^28 + z^24 + z^20 + z^16 + z^12 - z^11 + z^8 - z^7 + z^4 - z^3 + 1) * q^72 + (z^24 + z^20) * q^75 + (z^30 - z^27 + z^18 - z^15 + z^6 - z^3) * q^76 + (z^4 + z^2) * q^77 + (-z^19 + z^15 + z^11 + z^7 - z^3) * q^81 + (z^28 + z^24 + z^16 + z^12 + z^4 + 1) * q^84 + (z^32 + z^28 - z^27 - z^23) * q^87 + (z^30 + z^28 - z^27 - z^25 + z^18 + z^16 + z^6 + z^4) * q^88 + (-z^9 + z^2) * q^89 + (-z^29 - z^25 - z^19 - z^15) * q^93 + (-z^31 + z^24 + z^12 + z^10) * q^94 + (z^30 - z^27 + z^26 - z^23 - z^21 + z^18 - z^17 + z^14 - z^9 + z^6 - z^5 + z^2) * q^96 + (-z^21 + z^12) * q^97 + (z^12 + 1) * q^98 + (z^32 + z^30 + z^28 - z^5 - z^3 - z) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} - q^{3} - 8 q^{4} + 4 q^{6} + q^{7} - 8 q^{8} - 11 q^{9}+O(q^{10}) \) 20 * q + 2 * q^2 - q^3 - 8 * q^4 + 4 * q^6 + q^7 - 8 * q^8 - 11 * q^9	\( 20 q + 2 q^{2} - q^{3} - 8 q^{4} + 4 q^{6} + q^{7} - 8 q^{8} - 11 q^{9} - q^{11} - 3 q^{12} + 2 q^{14} - 6 q^{16} + 2 q^{19} + 2 q^{21} + 4 q^{22} - 4 q^{24} - 10 q^{25} - 2 q^{27} - 6 q^{28} + 2 q^{29} - q^{31} + 6 q^{32} + q^{33} + 22 q^{36} + 4 q^{38} + 20 q^{42} + 8 q^{44} - q^{47} - 12 q^{48} + q^{49} - 4 q^{50} - 20 q^{54} + 4 q^{56} + 4 q^{57} - 2 q^{58} + 2 q^{61} - 18 q^{62} + 8 q^{64} + 2 q^{66} + 11 q^{72} - q^{75} - 12 q^{76} + 2 q^{77} - 12 q^{81} + 19 q^{84} + q^{87} - 4 q^{88} - q^{89} + q^{93} - 2 q^{94} - 6 q^{96} - 4 q^{97} + 18 q^{98}+O(q^{100}) \) 20 * q + 2 * q^2 - q^3 - 8 * q^4 + 4 * q^6 + q^7 - 8 * q^8 - 11 * q^9 - q^11 - 3 * q^12 + 2 * q^14 - 6 * q^16 + 2 * q^19 + 2 * q^21 + 4 * q^22 - 4 * q^24 - 10 * q^25 - 2 * q^27 - 6 * q^28 + 2 * q^29 - q^31 + 6 * q^32 + q^33 + 22 * q^36 + 4 * q^38 + 20 * q^42 + 8 * q^44 - q^47 - 12 * q^48 + q^49 - 4 * q^50 - 20 * q^54 + 4 * q^56 + 4 * q^57 - 2 * q^58 + 2 * q^61 - 18 * q^62 + 8 * q^64 + 2 * q^66 + 11 * q^72 - q^75 - 12 * q^76 + 2 * q^77 - 12 * q^81 + 19 * q^84 + q^87 - 4 * q^88 - q^89 + q^93 - 2 * q^94 - 6 * q^96 - 4 * q^97 + 18 * q^98

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

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	1169.1.f.c

Level	$1169$
Weight	$1$
Character orbit	1169.f
Analytic conductor	$0.583$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{33}$
CM discriminant	-167
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.583406999768\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{33})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{33}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{33} - \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 333.10
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{10} - T^{9} + 5 T^{8} + \cdots + 1)^{2} \) (T^10 - T^9 + 5T^8 - 2T^7 + 16T^6 - 7T^5 + 20T^4 + T^3 + 12T^2 - 3*T + 1)^2
$3$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 + 11T^18 + 10T^17 + 76T^16 + 66T^15 + 320T^14 + 254T^13 + 968T^12 + 714T^11 + 1935T^10 + 1220T^9 + 2673T^8 + 1431T^7 + 2025T^6 + 550T^5 + 703T^4 + 230T^3 + 132T^2 + 12T + 1
$5$	\( T^{20} \) T^20
$7$	\( T^{20} - T^{19} + \cdots + 1 \) T^20 - T^19 + T^17 - T^16 + T^14 - T^13 + T^11 - T^10 + T^9 - T^7 + T^6 - T^4 + T^3 - T + 1
$11$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 + 11T^18 + 10T^17 + 76T^16 + 66T^15 + 320T^14 + 254T^13 + 968T^12 + 714T^11 + 1935T^10 + 1220T^9 + 2673T^8 + 1431T^7 + 2025T^6 + 550T^5 + 703T^4 + 230T^3 + 132T^2 + 12T + 1
$13$	\( T^{20} \) T^20
$17$	\( T^{20} \) T^20
$19$	\( (T^{10} - T^{9} + 5 T^{8} + \cdots + 1)^{2} \) (T^10 - T^9 + 5T^8 - 2T^7 + 16T^6 - 7T^5 + 20T^4 + T^3 + 12T^2 - 3*T + 1)^2
$23$	\( T^{20} \) T^20
$29$	\( (T^{10} - T^{9} - 10 T^{8} + \cdots + 1)^{2} \) (T^10 - T^9 - 10T^8 + 10T^7 + 34T^6 - 34T^5 - 43T^4 + 43T^3 + 12T^2 - 12T + 1)^2
$31$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 + 11T^18 + 10T^17 + 76T^16 + 66T^15 + 320T^14 + 254T^13 + 968T^12 + 714T^11 + 1935T^10 + 1220T^9 + 2673T^8 + 1431T^7 + 2025T^6 + 550T^5 + 703T^4 + 230T^3 + 132T^2 + 12T + 1
$37$	\( T^{20} \) T^20
$41$	\( T^{20} \) T^20
$43$	\( T^{20} \) T^20
$47$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 + 11T^18 + 10T^17 + 76T^16 + 66T^15 + 320T^14 + 254T^13 + 968T^12 + 714T^11 + 1935T^10 + 1220T^9 + 2673T^8 + 1431T^7 + 2025T^6 + 550T^5 + 703T^4 + 230T^3 + 132T^2 + 12T + 1
$53$	\( T^{20} \) T^20
$59$	\( T^{20} \) T^20
$61$	\( (T^{10} - T^{9} + 5 T^{8} + \cdots + 1)^{2} \) (T^10 - T^9 + 5T^8 - 2T^7 + 16T^6 - 7T^5 + 20T^4 + T^3 + 12T^2 - 3*T + 1)^2
$67$	\( T^{20} \) T^20
$71$	\( T^{20} \) T^20
$73$	\( T^{20} \) T^20
$79$	\( T^{20} \) T^20
$83$	\( T^{20} \) T^20
$89$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 + 11T^18 + 10T^17 + 76T^16 + 66T^15 + 320T^14 + 254T^13 + 968T^12 + 714T^11 + 1935T^10 + 1220T^9 + 2673T^8 + 1431T^7 + 2025T^6 + 550T^5 + 703T^4 + 230T^3 + 132T^2 + 12T + 1
$97$	\( (T^{5} + T^{4} - 4 T^{3} + \cdots + 1)^{4} \) (T^5 + T^4 - 4T^3 - 3T^2 + 3*T + 1)^4
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\(n\)	\(673\)	\(836\)
\(\chi(n)\)	\(-1\)	\(\zeta_{66}^{22}\)