LMFDB - Newform orbit 1503.1.f.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,1,Mod(166,1503)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("1503.166");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1503.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:	$0.750094713987$
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:	$\Z[a_1, a_2, a_3]$
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 $\left(\mathbb{Z}/1503\mathbb{Z}\right)^\times$.

$n$	$172$	$335$
$\chi(n)$	$-1$	$\zeta_{66}^{22}$

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} $

166.1

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

−0.981929

−

1.70075i

0.235759

0.971812i

−1.42837

2.47401i

1.42131

−

1.35522i

−0.580057

−

1.00469i

3.64636

−0.888835

0.458227i

166.2

−0.928368

−

1.60798i

−0.888835

−

0.458227i

−1.22373

2.11957i

0.0883470

1.85463i

0.327068

0.566498i

2.68757

0.580057

0.814576i

166.3

−0.723734

−

1.25354i

0.580057

−

0.814576i

−0.547582

0.948440i

−1.44091

−

0.137591i

0.786053

1.36148i

0.137747

−0.327068

−

0.945001i

166.4

−0.580057

−

1.00469i

0.928368

−

0.371662i

−0.172932

0.299527i

−0.911911

−

0.717135i

−0.0475819

−

0.0824143i

−0.758872

0.723734

−

0.690079i

166.5

−0.235759

−

0.408346i

−0.995472

0.0950560i

0.388835

−

0.673483i

0.273507

0.384087i

−0.928368

−

1.60798i

−0.838204

0.981929

−

0.189251i

166.6

−0.0475819

−

0.0824143i

−0.327068

0.945001i

0.495472

−

0.858183i

0.0934441

−

0.0180099i

−0.235759

−

0.408346i

−0.189466

−0.786053

−

0.618159i

166.7

0.327068

0.566498i

0.723734

0.690079i

0.286053

−

0.495458i

−0.154218

0.635697i

0.995472

1.72421i

1.02837

0.0475819

0.998867i

166.8

0.786053

1.36148i

0.0475819

−

0.998867i

−0.735759

1.27437i

1.39734

−

0.720381i

−0.981929

−

1.70075i

−0.741276

−0.995472

−

0.0950560i

166.9

0.888835

1.53951i

0.981929

0.189251i

−1.08006

1.87071i

0.581419

1.67990i

−0.723734

−

1.25354i

−2.06230

0.928368

0.371662i

166.10

0.995472

1.72421i

−0.786053

0.618159i

−1.48193

2.56678i

−1.84833

−

0.739959i

0.888835

1.53951i

−3.90993

0.235759

−

0.971812i

1168.1

−0.981929

1.70075i

0.235759

−

0.971812i

−1.42837

−

2.47401i

1.42131

1.35522i

−0.580057

1.00469i

3.64636

−0.888835

−

0.458227i

1168.2

−0.928368

1.60798i

−0.888835

0.458227i

−1.22373

−

2.11957i

0.0883470

−

1.85463i

0.327068

−

0.566498i

2.68757

0.580057

−

0.814576i

1168.3

−0.723734

1.25354i

0.580057

0.814576i

−0.547582

−

0.948440i

−1.44091

0.137591i

0.786053

−

1.36148i

0.137747

−0.327068

0.945001i

1168.4

−0.580057

1.00469i

0.928368

0.371662i

−0.172932

−

0.299527i

−0.911911

0.717135i

−0.0475819

0.0824143i

−0.758872

0.723734

0.690079i

1168.5

−0.235759

0.408346i

−0.995472

−

0.0950560i

0.388835

0.673483i

0.273507

−

0.384087i

−0.928368

1.60798i

−0.838204

0.981929

0.189251i

1168.6

−0.0475819

0.0824143i

−0.327068

−

0.945001i

0.495472

0.858183i

0.0934441

0.0180099i

−0.235759

0.408346i

−0.189466

−0.786053

0.618159i

1168.7

0.327068

−

0.566498i

0.723734

−

0.690079i

0.286053

0.495458i

−0.154218

−

0.635697i

0.995472

−

1.72421i

1.02837

0.0475819

−

0.998867i

1168.8

0.786053

−

1.36148i

0.0475819

0.998867i

−0.735759

−

1.27437i

1.39734

0.720381i

−0.981929

1.70075i

−0.741276

−0.995472

0.0950560i

1168.9

0.888835

−

1.53951i

0.981929

−

0.189251i

−1.08006

−

1.87071i

0.581419

−

1.67990i

−0.723734

1.25354i

−2.06230

0.928368

−

0.371662i

1168.10

0.995472

−

1.72421i

−0.786053

−

0.618159i

−1.48193

−

2.56678i

−1.84833

0.739959i

0.888835

−

1.53951i

−3.90993

0.235759

0.971812i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
167.b	odd	2	1	CM by $\Q(\sqrt{-167}) $
9.c	even	3	1	inner
1503.f	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1503.1.f.b	✓	20
9.c	even	3	1	inner	1503.1.f.b	✓	20
167.b	odd	2	1	CM	1503.1.f.b	✓	20
1503.f	odd	6	1	inner	1503.1.f.b	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1503.1.f.b	✓	20	1.a	even	1	1	trivial
1503.1.f.b	✓	20	9.c	even	3	1	inner
1503.1.f.b	✓	20	167.b	odd	2	1	CM
1503.1.f.b	✓	20	1503.f	odd	6	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{20} + T_{2}^{19} + 11 T_{2}^{18} + 10 T_{2}^{17} + 76 T_{2}^{16} + 66 T_{2}^{15} + 320 T_{2}^{14} + \cdots + 1 $ T2^20 + T2^19 + 11*T2^18 + 10*T2^17 + 76*T2^16 + 66*T2^15 + 320*T2^14 + 254*T2^13 + 968*T2^12 + 714*T2^11 + 1935*T2^10 + 1220*T2^9 + 2673*T2^8 + 1431*T2^7 + 2025*T2^6 + 550*T2^5 + 703*T2^4 + 230*T2^3 + 132*T2^2 + 12*T2 + 1 acting on $S_{1}^{\mathrm{new}}(1503, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ 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
$3$	$ 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
$5$	$ T^{20} $ T^20
$7$	$ 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
$11$	$ 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} - 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
$23$	$ T^{20} $ T^20
$29$	$ 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
$31$	$ 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^{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
$97$	$ (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
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1503.f (of order \(6\), degree \(2\), minimal)

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

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Newspace parameters

Newform invariants

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Character values

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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	1503.1.f.b

Level	$1503$
Weight	$1$
Character orbit	1503.f
Analytic conductor	$0.750$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{33}$
CM discriminant	-167
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.750094713987\)
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:	\(\Z[a_1, a_2, a_3]\)
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 166.10
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{20} + T^{19} + \cdots + 1 \) 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
$3$	\( T^{20} - T^{19} + \cdots + 1 \) 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
$5$	\( T^{20} \) T^20
$7$	\( T^{20} + T^{19} + \cdots + 1 \) 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
$11$	\( T^{20} + T^{19} + \cdots + 1 \) 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
$19$	\( (T^{10} - T^{9} - 10 T^{8} + \cdots + 1)^{2} \) (T^10 - T^9 - 10T^8 + 10T^7 + 34T^6 - 34T^5 - 43T^4 + 43T^3 + 12T^2 - 12T + 1)^2
$23$	\( T^{20} \) T^20
$29$	\( T^{20} + T^{19} + \cdots + 1 \) 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
$31$	\( T^{20} + T^{19} + \cdots + 1 \) 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
$89$	\( (T^{10} - T^{9} - 10 T^{8} + \cdots + 1)^{2} \) (T^10 - T^9 - 10T^8 + 10T^7 + 34T^6 - 34T^5 - 43T^4 + 43T^3 + 12T^2 - 12T + 1)^2
$97$	\( (T^{10} - T^{9} + 5 T^{8} + \cdots + 1)^{2} \) (T^10 - T^9 + 5T^8 - 2T^7 + 16T^6 - 7T^5 + 20T^4 + T^3 + 12T^2 - 3*T + 1)^2
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\(n\)	\(172\)	\(335\)
\(\chi(n)\)	\(-1\)	\(\zeta_{66}^{22}\)