LMFDB - Newform orbit 2008.1.w.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2008,1,Mod(51,2008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2008, base_ring=CyclotomicField(50))

chi = DirichletCharacter(H, H._module([25, 25, 18]))

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

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

chi := DirichletCharacter("2008.51");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2008 = 2^{3} \cdot 251 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2008.w (of order $50$, degree $20$, 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:	$1.00212254537$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{50})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{15} + x^{10} - x^{5} + 1 $ x^20 - x^15 + x^10 - x^5 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{25}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{25} - \cdots)$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q - \zeta_{50}^{5} q^{2} + (\zeta_{50}^{4} + \zeta_{50}^{2}) q^{3} + \zeta_{50}^{10} q^{4} + ( - \zeta_{50}^{9} - \zeta_{50}^{7}) q^{6} - \zeta_{50}^{15} q^{8} + (\zeta_{50}^{8} + \cdots + \zeta_{50}^{4}) q^{9} +O(q^{10}) $ q - z^5 * q^2 + (z^4 + z^2) * q^3 + z^10 * q^4 + (-z^9 - z^7) * q^6 - z^15 * q^8 + (z^8 + z^6 + z^4) * q^9	$ q - \zeta_{50}^{5} q^{2} + (\zeta_{50}^{4} + \zeta_{50}^{2}) q^{3} + \zeta_{50}^{10} q^{4} + ( - \zeta_{50}^{9} - \zeta_{50}^{7}) q^{6} - \zeta_{50}^{15} q^{8} + (\zeta_{50}^{8} + \cdots + \zeta_{50}^{4}) q^{9} + \cdots + (\zeta_{50}^{24} + \zeta_{50}^{14} + \cdots - \zeta_{50}) q^{99} +O(q^{100}) $ q - z^5 * q^2 + (z^4 + z^2) * q^3 + z^10 * q^4 + (-z^9 - z^7) * q^6 - z^15 * q^8 + (z^8 + z^6 + z^4) * q^9 + (z^20 + z^6) * q^11 + (z^14 + z^12) * q^12 + z^20 * q^16 - z^17 * q^17 + (-z^13 - z^11 - z^9) * q^18 + (z^24 - z^9) * q^19 + (-z^11 + 1) * q^22 + (-z^19 - z^17) * q^24 - z^5 * q^25 + (z^12 + z^10 + z^8 + z^6) * q^27 + q^32 + (z^24 + z^22 + z^10 + z^8) * q^33 + 2z^22 q^34 + (z^18 + z^16 + z^14) * q^36 + (z^14 + z^4) * q^38 + (-z^23 + z^16) * q^41 + (z^18 - z^9) * q^43 + (z^16 - z^5) * q^44 + (z^24 + z^22) * q^48 + z^18 * q^49 + z^10 * q^50 + (-2z^21 - 2z^19) * q^51 + (-z^17 - z^15 - z^13 - z^11) * q^54 + (-z^13 - z^11 - z^3 - z) * q^57 + (-z^21 + z^20) * q^59 - z^5 * q^64 + (-z^15 - z^13 + z^4 + z^2) * q^66 + (-z^11 - z^3) * q^67 + 2z^2 q^68 + (-z^23 - z^21 - z^19) * q^72 + (-z^17 + z^2) * q^73 + (-z^9 - z^7) * q^75 + (-z^19 - z^9) * q^76 + (z^16 + z^14 + z^12 + z^10 + z^8) * q^81 + (-z^21 - z^3) * q^82 + (z^24 - z^5) * q^83 + (-z^23 + z^14) * q^86 + (-z^21 + z^10) * q^88 + (z^6 - z^3) * q^89 + (z^4 + z^2) * q^96 + (z^18 - z^11) * q^97 - z^23 * q^98 + (z^24 + z^14 + z^12 + z^10 - z^3 - z) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 5 q^{2} - 5 q^{4} - 5 q^{8}+O(q^{10}) $ 20 * q - 5 * q^2 - 5 * q^4 - 5 * q^8	$ 20 q - 5 q^{2} - 5 q^{4} - 5 q^{8} - 5 q^{11} - 5 q^{16} + 20 q^{22} - 5 q^{25} - 5 q^{27} + 20 q^{32} - 5 q^{33} - 5 q^{44} - 5 q^{50} - 5 q^{54} - 5 q^{59} - 5 q^{64} - 5 q^{66} - 5 q^{81} - 5 q^{83} - 5 q^{88} - 5 q^{99}+O(q^{100}) $ 20 * q - 5 * q^2 - 5 * q^4 - 5 * q^8 - 5 * q^11 - 5 * q^16 + 20 * q^22 - 5 * q^25 - 5 * q^27 + 20 * q^32 - 5 * q^33 - 5 * q^44 - 5 * q^50 - 5 * q^54 - 5 * q^59 - 5 * q^64 - 5 * q^66 - 5 * q^81 - 5 * q^83 - 5 * q^88 - 5 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2008\mathbb{Z}\right)^\times$.

$n$	$257$	$503$	$1005$
$\chi(n)$	$\zeta_{50}^{16}$	$-1$	$-1$

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} $

51.1

−0.968583	+	0.248690i
−0.876307	+	0.481754i
−0.728969	+	0.684547i
−0.0627905	−	0.998027i
0.992115	+	0.125333i
0.637424	−	0.770513i
−0.968583	−	0.248690i
−0.876307	−	0.481754i
0.187381	+	0.982287i
−0.0627905	+	0.998027i
−0.535827	−	0.844328i
0.425779	−	0.904827i
0.425779	+	0.904827i
−0.728969	−	0.684547i
0.637424	+	0.770513i
0.929776	−	0.368125i
0.187381	−	0.982287i
0.929776	+	0.368125i
−0.535827	+	0.844328i
0.992115	−	0.125333i

0.309017

−

0.951057i

1.41213

−

1.32608i

−0.809017

−

0.587785i

−0.824805

−

1.75280i

−0.809017

0.587785i

0.172838

−

2.74718i

91.1

−0.809017

−

0.587785i

0.110048

−

1.74915i

0.309017

0.951057i

−1.11716

1.35041i

0.309017

−

0.951057i

−2.05532

−

0.259647i

123.1

−0.809017

0.587785i

−0.929324

−

1.12336i

0.309017

−

0.951057i

1.41213

0.362574i

0.309017

0.951057i

−0.210913

1.10564i

211.1

0.309017

0.951057i

−0.0235315

−

0.123357i

−0.809017

0.587785i

0.110048

−

0.0604991i

−0.809017

−

0.587785i

0.915113

−

0.362319i

243.1

−0.809017

−

0.587785i

1.84489

0.730444i

0.309017

0.951057i

−1.06320

−

1.67534i

0.309017

−

0.951057i

2.14110

2.01063i

267.1

0.309017

−

0.951057i

−1.11716

−

0.614163i

−0.809017

−

0.587785i

−0.929324

0.872693i

−0.809017

0.587785i

0.335019

0.527905i

315.1

0.309017

0.951057i

1.41213

1.32608i

−0.809017

0.587785i

−0.824805

1.75280i

−0.809017

−

0.587785i

0.172838

2.74718i

331.1

−0.809017

0.587785i

0.110048

1.74915i

0.309017

−

0.951057i

−1.11716

−

1.35041i

0.309017

0.951057i

−2.05532

0.259647i

507.1

−0.809017

−

0.587785i

−0.200808

−

0.316423i

0.309017

0.951057i

−0.0235315

0.374023i

0.309017

−

0.951057i

0.365980

−

0.777747i

571.1

0.309017

−

0.951057i

−0.0235315

0.123357i

−0.809017

−

0.587785i

0.110048

0.0604991i

−0.809017

0.587785i

0.915113

0.362319i

627.1

0.309017

−

0.951057i

−1.06320

0.134314i

−0.809017

−

0.587785i

−0.200808

1.05267i

−0.809017

0.587785i

0.143778

−

0.0369159i

1067.1

−0.809017

−

0.587785i

−0.824805

0.211774i

0.309017

0.951057i

0.791759

0.313480i

0.309017

−

0.951057i

−0.240851

0.132409i

1259.1

−0.809017

0.587785i

−0.824805

−

0.211774i

0.309017

−

0.951057i

0.791759

−

0.313480i

0.309017

0.951057i

−0.240851

−

0.132409i

1355.1

−0.809017

−

0.587785i

−0.929324

1.12336i

0.309017

0.951057i

1.41213

−

0.362574i

0.309017

−

0.951057i

−0.210913

−

1.10564i

1459.1

0.309017

0.951057i

−1.11716

0.614163i

−0.809017

0.587785i

−0.929324

−

0.872693i

−0.809017

−

0.587785i

0.335019

−

0.527905i

1531.1

0.309017

0.951057i

0.791759

−

1.68257i

−0.809017

0.587785i

1.84489

0.233064i

−0.809017

−

0.587785i

−1.56675

−

1.89387i

1707.1

−0.809017

0.587785i

−0.200808

0.316423i

0.309017

−

0.951057i

−0.0235315

−

0.374023i

0.309017

0.951057i

0.365980

0.777747i

1747.1

0.309017

−

0.951057i

0.791759

1.68257i

−0.809017

−

0.587785i

1.84489

−

0.233064i

−0.809017

0.587785i

−1.56675

1.89387i

1755.1

0.309017

0.951057i

−1.06320

−

0.134314i

−0.809017

0.587785i

−0.200808

−

1.05267i

−0.809017

−

0.587785i

0.143778

0.0369159i

1851.1

−0.809017

0.587785i

1.84489

−

0.730444i

0.309017

−

0.951057i

−1.06320

1.67534i

0.309017

0.951057i

2.14110

−

2.01063i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
251.e	even	25	1	inner
2008.w	odd	50	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2008.1.w.a	✓	20
8.d	odd	2	1	CM	2008.1.w.a	✓	20
251.e	even	25	1	inner	2008.1.w.a	✓	20
2008.w	odd	50	1	inner	2008.1.w.a	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2008.1.w.a	✓	20	1.a	even	1	1	trivial
2008.1.w.a	✓	20	8.d	odd	2	1	CM
2008.1.w.a	✓	20	251.e	even	25	1	inner
2008.1.w.a	✓	20	2008.w	odd	50	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2008, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{5} $ (T^4 + T^3 + T^2 + T + 1)^5
$3$	$ T^{20} + 5 T^{17} + \cdots + 1 $ T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
$5$	$ T^{20} $ T^20
$7$	$ T^{20} $ T^20
$11$	$ T^{20} + 5 T^{19} + \cdots + 1 $ T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 170T^14 + 185T^13 + 165T^12 + 205T^11 + 629T^10 + 1235T^9 + 1655T^8 + 1520T^7 + 665T^6 - 252T^5 - 340T^4 - 45T^3 + 70T^2 - 10*T + 1
$13$	$ T^{20} $ T^20
$17$	$ T^{20} + 32 T^{15} + \cdots + 1048576 $ T^20 + 32T^15 + 1024T^10 + 32768*T^5 + 1048576
$19$	$ T^{20} - 18 T^{15} + \cdots + 1 $ T^20 - 18T^15 + 124T^10 - 7*T^5 + 1
$23$	$ T^{20} $ T^20
$29$	$ T^{20} $ T^20
$31$	$ T^{20} $ T^20
$37$	$ T^{20} $ T^20
$41$	$ T^{20} + 5 T^{17} + \cdots + 1 $ T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 120T^11 + 379T^10 - 225T^9 + 925T^8 - 715T^7 + 1010T^6 - 747T^5 + 600T^4 - 175T^3 - 15T^2 + 10*T + 1
$43$	$ T^{20} + 5 T^{17} + \cdots + 1 $ T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
$47$	$ T^{20} $ T^20
$53$	$ T^{20} $ T^20
$59$	$ T^{20} + 5 T^{19} + \cdots + 1 $ T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 170T^14 + 185T^13 + 165T^12 + 205T^11 + 629T^10 + 1235T^9 + 1655T^8 + 1520T^7 + 665T^6 - 252T^5 - 340T^4 - 45T^3 + 70T^2 - 10*T + 1
$61$	$ T^{20} $ T^20
$67$	$ T^{20} - 20 T^{17} + \cdots + 1 $ T^20 - 20T^17 + 5T^16 + 2T^15 + 150T^14 - 75T^13 - 5T^12 - 505T^11 + 379T^10 - 100T^9 + 800T^8 - 640T^7 + 360T^6 - 747T^5 + 600T^4 - 175T^3 + 85T^2 - 15*T + 1
$71$	$ T^{20} $ T^20
$73$	$ T^{20} + 7 T^{15} + \cdots + 1 $ T^20 + 7T^15 + 124T^10 + 18*T^5 + 1
$79$	$ T^{20} $ T^20
$83$	$ T^{20} + 5 T^{19} + \cdots + 1 $ T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 170T^14 + 185T^13 + 165T^12 + 205T^11 + 629T^10 + 1235T^9 + 1655T^8 + 1520T^7 + 665T^6 - 252T^5 - 340T^4 - 45T^3 + 70T^2 - 10*T + 1
$89$	$ T^{20} + 5 T^{17} + \cdots + 1 $ T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
$97$	$ T^{20} + 5 T^{17} + \cdots + 1 $ T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
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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:	\( N \)	\(=\)	\( 2008 = 2^{3} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2008.w (of order \(50\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{50}^{5} q^{2} + (\zeta_{50}^{4} + \zeta_{50}^{2}) q^{3} + \zeta_{50}^{10} q^{4} + ( - \zeta_{50}^{9} - \zeta_{50}^{7}) q^{6} - \zeta_{50}^{15} q^{8} + (\zeta_{50}^{8} + \cdots + \zeta_{50}^{4}) q^{9} +O(q^{10}) \) q - z^5 * q^2 + (z^4 + z^2) * q^3 + z^10 * q^4 + (-z^9 - z^7) * q^6 - z^15 * q^8 + (z^8 + z^6 + z^4) * q^9	\( q - \zeta_{50}^{5} q^{2} + (\zeta_{50}^{4} + \zeta_{50}^{2}) q^{3} + \zeta_{50}^{10} q^{4} + ( - \zeta_{50}^{9} - \zeta_{50}^{7}) q^{6} - \zeta_{50}^{15} q^{8} + (\zeta_{50}^{8} + \cdots + \zeta_{50}^{4}) q^{9} + \cdots + (\zeta_{50}^{24} + \zeta_{50}^{14} + \cdots - \zeta_{50}) q^{99} +O(q^{100}) \) q - z^5 * q^2 + (z^4 + z^2) * q^3 + z^10 * q^4 + (-z^9 - z^7) * q^6 - z^15 * q^8 + (z^8 + z^6 + z^4) * q^9 + (z^20 + z^6) * q^11 + (z^14 + z^12) * q^12 + z^20 * q^16 - z^17 * q^17 + (-z^13 - z^11 - z^9) * q^18 + (z^24 - z^9) * q^19 + (-z^11 + 1) * q^22 + (-z^19 - z^17) * q^24 - z^5 * q^25 + (z^12 + z^10 + z^8 + z^6) * q^27 + q^32 + (z^24 + z^22 + z^10 + z^8) * q^33 + 2z^22 q^34 + (z^18 + z^16 + z^14) * q^36 + (z^14 + z^4) * q^38 + (-z^23 + z^16) * q^41 + (z^18 - z^9) * q^43 + (z^16 - z^5) * q^44 + (z^24 + z^22) * q^48 + z^18 * q^49 + z^10 * q^50 + (-2z^21 - 2z^19) * q^51 + (-z^17 - z^15 - z^13 - z^11) * q^54 + (-z^13 - z^11 - z^3 - z) * q^57 + (-z^21 + z^20) * q^59 - z^5 * q^64 + (-z^15 - z^13 + z^4 + z^2) * q^66 + (-z^11 - z^3) * q^67 + 2z^2 q^68 + (-z^23 - z^21 - z^19) * q^72 + (-z^17 + z^2) * q^73 + (-z^9 - z^7) * q^75 + (-z^19 - z^9) * q^76 + (z^16 + z^14 + z^12 + z^10 + z^8) * q^81 + (-z^21 - z^3) * q^82 + (z^24 - z^5) * q^83 + (-z^23 + z^14) * q^86 + (-z^21 + z^10) * q^88 + (z^6 - z^3) * q^89 + (z^4 + z^2) * q^96 + (z^18 - z^11) * q^97 - z^23 * q^98 + (z^24 + z^14 + z^12 + z^10 - z^3 - z) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{2} - 5 q^{4} - 5 q^{8}+O(q^{10}) \) 20 * q - 5 * q^2 - 5 * q^4 - 5 * q^8	\( 20 q - 5 q^{2} - 5 q^{4} - 5 q^{8} - 5 q^{11} - 5 q^{16} + 20 q^{22} - 5 q^{25} - 5 q^{27} + 20 q^{32} - 5 q^{33} - 5 q^{44} - 5 q^{50} - 5 q^{54} - 5 q^{59} - 5 q^{64} - 5 q^{66} - 5 q^{81} - 5 q^{83} - 5 q^{88} - 5 q^{99}+O(q^{100}) \) 20 * q - 5 * q^2 - 5 * q^4 - 5 * q^8 - 5 * q^11 - 5 * q^16 + 20 * q^22 - 5 * q^25 - 5 * q^27 + 20 * q^32 - 5 * q^33 - 5 * q^44 - 5 * q^50 - 5 * q^54 - 5 * q^59 - 5 * q^64 - 5 * q^66 - 5 * q^81 - 5 * q^83 - 5 * q^88 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	2008.1.w.a

Level	$2008$
Weight	$1$
Character orbit	2008.w
Analytic conductor	$1.002$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{25}$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.00212254537\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\Q(\zeta_{50})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - x^{15} + x^{10} - x^{5} + 1 \) x^20 - x^15 + x^10 - x^5 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{25}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{25} - \cdots)\)

\(n\)	\(257\)	\(503\)	\(1005\)
\(\chi(n)\)	\(\zeta_{50}^{16}\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 51.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
251.e	even	25	1	inner
2008.w	odd	50	1	inner

$p$	$F_p(T)$
$2$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{5} \) (T^4 + T^3 + T^2 + T + 1)^5
$3$	\( T^{20} + 5 T^{17} + \cdots + 1 \) T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
$5$	\( T^{20} \) T^20
$7$	\( T^{20} \) T^20
$11$	\( T^{20} + 5 T^{19} + \cdots + 1 \) T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 170T^14 + 185T^13 + 165T^12 + 205T^11 + 629T^10 + 1235T^9 + 1655T^8 + 1520T^7 + 665T^6 - 252T^5 - 340T^4 - 45T^3 + 70T^2 - 10*T + 1
$13$	\( T^{20} \) T^20
$17$	\( T^{20} + 32 T^{15} + \cdots + 1048576 \) T^20 + 32T^15 + 1024T^10 + 32768*T^5 + 1048576
$19$	\( T^{20} - 18 T^{15} + \cdots + 1 \) T^20 - 18T^15 + 124T^10 - 7*T^5 + 1
$23$	\( T^{20} \) T^20
$29$	\( T^{20} \) T^20
$31$	\( T^{20} \) T^20
$37$	\( T^{20} \) T^20
$41$	\( T^{20} + 5 T^{17} + \cdots + 1 \) T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 120T^11 + 379T^10 - 225T^9 + 925T^8 - 715T^7 + 1010T^6 - 747T^5 + 600T^4 - 175T^3 - 15T^2 + 10*T + 1
$43$	\( T^{20} + 5 T^{17} + \cdots + 1 \) T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
$47$	\( T^{20} \) T^20
$53$	\( T^{20} \) T^20
$59$	\( T^{20} + 5 T^{19} + \cdots + 1 \) T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 170T^14 + 185T^13 + 165T^12 + 205T^11 + 629T^10 + 1235T^9 + 1655T^8 + 1520T^7 + 665T^6 - 252T^5 - 340T^4 - 45T^3 + 70T^2 - 10*T + 1
$61$	\( T^{20} \) T^20
$67$	\( T^{20} - 20 T^{17} + \cdots + 1 \) T^20 - 20T^17 + 5T^16 + 2T^15 + 150T^14 - 75T^13 - 5T^12 - 505T^11 + 379T^10 - 100T^9 + 800T^8 - 640T^7 + 360T^6 - 747T^5 + 600T^4 - 175T^3 + 85T^2 - 15*T + 1
$71$	\( T^{20} \) T^20
$73$	\( T^{20} + 7 T^{15} + \cdots + 1 \) T^20 + 7T^15 + 124T^10 + 18*T^5 + 1
$79$	\( T^{20} \) T^20
$83$	\( T^{20} + 5 T^{19} + \cdots + 1 \) T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 170T^14 + 185T^13 + 165T^12 + 205T^11 + 629T^10 + 1235T^9 + 1655T^8 + 1520T^7 + 665T^6 - 252T^5 - 340T^4 - 45T^3 + 70T^2 - 10*T + 1
$89$	\( T^{20} + 5 T^{17} + \cdots + 1 \) T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
$97$	\( T^{20} + 5 T^{17} + \cdots + 1 \) T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
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