LMFDB - Newform orbit 3484.1.dz.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3484,1,Mod(103,3484)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3484, base_ring=CyclotomicField(66))

chi = DirichletCharacter(H, H._module([33, 33, 14]))

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

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

chi := DirichletCharacter("3484.103");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3484 = 2^{2} \cdot 13 \cdot 67 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3484.dz (of order $66$, 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:	$1.73874250401$
Analytic rank:	$0$
Dimension:	$20$
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:	$\Z[a_1, a_2]$
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 $\left(\mathbb{Z}/3484\mathbb{Z}\right)^\times$.

$n$	$1341$	$1743$	$3017$
$\chi(n)$	$-1$	$-1$	$-\zeta_{66}^{5}$

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} $

103.1

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

−0.786053

−

0.618159i

0.235759

0.971812i

1.07701

−

0.431171i

0.415415

−

0.909632i

−0.142315

0.989821i

155.1

0.981929

−

0.189251i

0.928368

−

0.371662i

−0.0311250

0.0899299i

0.841254

−

0.540641i

−0.654861

−

0.755750i

207.1

−0.327068

0.945001i

−0.786053

−

0.618159i

−1.74555

0.336426i

0.841254

−

0.540641i

−0.654861

−

0.755750i

467.1

0.0475819

−

0.998867i

−0.995472

−

0.0950560i

−1.65033

0.850806i

−0.142315

0.989821i

0.841254

0.540641i

571.1

0.235759

−

0.971812i

−0.888835

−

0.458227i

−0.473420

−

0.451405i

−0.654861

0.755750i

−0.959493

0.281733i

987.1

0.580057

−

0.814576i

−0.327068

−

0.945001i

−0.469383

−

0.0448206i

−0.959493

−

0.281733i

0.415415

0.909632i

1091.1

−0.888835

−

0.458227i

0.580057

0.814576i

−0.0748038

−

1.57033i

−0.142315

−

0.989821i

0.841254

−

0.540641i

1143.1

−0.995472

−

0.0950560i

0.981929

0.189251i

0.839614

−

1.17907i

−0.959493

−

0.281733i

0.415415

0.909632i

1195.1

−0.327068

−

0.945001i

−0.786053

0.618159i

−1.74555

−

0.336426i

0.841254

0.540641i

−0.654861

0.755750i

1299.1

0.580057

0.814576i

−0.327068

0.945001i

−0.469383

0.0448206i

−0.959493

0.281733i

0.415415

−

0.909632i

1507.1

0.0475819

0.998867i

−0.995472

0.0950560i

−1.65033

−

0.850806i

−0.142315

−

0.989821i

0.841254

−

0.540641i

1663.1

0.928368

0.371662i

0.723734

0.690079i

1.56499

−

1.23072i

0.415415

0.909632i

−0.142315

−

0.989821i

1819.1

0.723734

0.690079i

0.0475819

0.998867i

0.462997

−

1.90850i

−0.654861

0.755750i

−0.959493

0.281733i

1923.1

0.723734

−

0.690079i

0.0475819

−

0.998867i

0.462997

1.90850i

−0.654861

−

0.755750i

−0.959493

−

0.281733i

2027.1

−0.995472

0.0950560i

0.981929

−

0.189251i

0.839614

1.17907i

−0.959493

0.281733i

0.415415

−

0.909632i

2131.1

−0.786053

0.618159i

0.235759

−

0.971812i

1.07701

0.431171i

0.415415

0.909632i

−0.142315

−

0.989821i

2183.1

0.928368

−

0.371662i

0.723734

−

0.690079i

1.56499

1.23072i

0.415415

−

0.909632i

−0.142315

0.989821i

2495.1

0.981929

0.189251i

0.928368

0.371662i

−0.0311250

−

0.0899299i

0.841254

0.540641i

−0.654861

0.755750i

2703.1

0.235759

0.971812i

−0.888835

0.458227i

−0.473420

0.451405i

−0.654861

−

0.755750i

−0.959493

−

0.281733i

2807.1

−0.888835

0.458227i

0.580057

−

0.814576i

−0.0748038

1.57033i

−0.142315

0.989821i

0.841254

0.540641i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
52.b	odd	2	1	CM by $\Q(\sqrt{-13}) $
67.g	even	33	1	inner
3484.dz	odd	66	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3484.1.dz.b	yes	20
4.b	odd	2	1		3484.1.dz.a	✓	20
13.b	even	2	1		3484.1.dz.a	✓	20
52.b	odd	2	1	CM	3484.1.dz.b	yes	20
67.g	even	33	1	inner	3484.1.dz.b	yes	20
268.o	odd	66	1		3484.1.dz.a	✓	20
871.bp	even	66	1		3484.1.dz.a	✓	20
3484.dz	odd	66	1	inner	3484.1.dz.b	yes	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3484.1.dz.a	✓	20	4.b	odd	2	1
3484.1.dz.a	✓	20	13.b	even	2	1
3484.1.dz.a	✓	20	268.o	odd	66	1
3484.1.dz.a	✓	20	871.bp	even	66	1
3484.1.dz.b	yes	20	1.a	even	1	1	trivial
3484.1.dz.b	yes	20	52.b	odd	2	1	CM
3484.1.dz.b	yes	20	67.g	even	33	1	inner
3484.1.dz.b	yes	20	3484.dz	odd	66	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{20} + T_{7}^{19} + 10 T_{7}^{17} + 10 T_{7}^{16} + 78 T_{7}^{14} + 89 T_{7}^{13} + 11 T_{7}^{12} + \cdots + 1 $ T7^20 + T7^19 + 10*T7^17 + 10*T7^16 + 78*T7^14 + 89*T7^13 + 11*T7^12 + 241*T7^11 + 87*T7^10 - 155*T7^9 + 605*T7^8 - 87*T7^7 - 560*T7^6 + 253*T7^5 + 934*T7^4 + 626*T7^3 + 154*T7^2 + 12*T7 + 1 acting on $S_{1}^{\mathrm{new}}(3484, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ 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
$3$	$ T^{20} $ T^20
$5$	$ T^{20} $ T^20
$7$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 + 10T^17 + 10T^16 + 78T^14 + 89T^13 + 11T^12 + 241T^11 + 87T^10 - 155T^9 + 605T^8 - 87T^7 - 560T^6 + 253T^5 + 934T^4 + 626T^3 + 154T^2 + 12T + 1
$11$	$ T^{20} - 2 T^{19} + \cdots + 1 $ T^20 - 2T^19 - 14T^17 + 28T^16 + 75T^14 - 139T^13 - 22T^12 - 170T^11 + 318T^10 + 46T^9 + 209T^8 - 360T^7 - 17T^6 + 22T^5 + 145T^4 - 70T^3 + 22T^2 - 6*T + 1
$13$	$ 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
$17$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 - T^17 + 10T^16 + 22T^15 + 12T^14 - 10T^13 + 77T^12 + 362T^11 + 395T^10 + 32T^9 - 121T^8 - 274T^7 - 395T^6 - 22T^5 + 835T^4 - 595T^3 + 143T^2 - 10T + 1
$19$	$ T^{20} - 2 T^{19} + \cdots + 1 $ T^20 - 2T^19 - 14T^17 + 28T^16 + 75T^14 - 139T^13 - 22T^12 - 170T^11 + 318T^10 + 46T^9 + 209T^8 - 360T^7 - 17T^6 + 22T^5 + 145T^4 - 70T^3 + 22T^2 - 6*T + 1
$23$	$ 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 - T^17 - T^16 + T^14 + T^13 - T^11 - T^10 - T^9 + T^7 + T^6 - T^4 - T^3 + T + 1
$37$	$ T^{20} $ T^20
$41$	$ T^{20} $ T^20
$43$	$ T^{20} $ T^20
$47$	$ T^{20} - 21 T^{19} + \cdots + 1 $ T^20 - 21T^19 + 209T^18 - 1310T^17 + 5796T^16 - 19228T^15 + 49589T^14 - 101727T^13 + 168454T^12 - 227239T^11 + 250953T^10 - 227239T^9 + 168443T^8 - 101683T^7 + 49479T^6 - 19052T^5 + 5620T^4 - 1200T^3 + 165T^2 - 10*T + 1
$53$	$ T^{20} - 2 T^{19} + \cdots + 1 $ T^20 - 2T^19 + 3T^18 - 4T^17 + 5T^16 + 16T^15 - 37T^14 - 8T^13 + 31T^12 - 43T^11 + 528T^10 - 1011T^9 + 768T^8 - 767T^7 + 1613T^6 - 1491T^5 + 1149T^4 - 565T^3 + 157T^2 - 13*T + 1
$59$	$ (T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} $ (T^10 + 2T^9 + 4T^8 + 8T^7 + 5T^6 - T^5 - 2T^4 - 4T^3 + 14T^2 - 5T + 1)^2
$61$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 - T^17 - T^16 + 22T^15 + 23T^14 + 34T^13 + 22T^12 - T^11 + 450T^10 + 450T^9 + 1452T^8 + 760T^7 + 881T^6 - 12T^4 + 230T^3 + 154T^2 + 23*T + 1
$67$	$ 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
$71$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 - T^17 - T^16 + 12T^14 - 98T^13 - 33T^12 + 109T^11 + 120T^10 + 10T^9 + 11T^8 - 604T^7 + 1563T^6 - 1100T^5 + 725T^4 - 375T^3 + 44T^2 + 12T + 1
$73$	$ T^{20} $ T^20
$79$	$ T^{20} $ T^20
$83$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 - T^17 - T^16 + 22T^15 + 23T^14 + 34T^13 + 22T^12 - T^11 + 450T^10 + 450T^9 + 1452T^8 + 760T^7 + 881T^6 - 12T^4 + 230T^3 + 154T^2 + 23*T + 1
$89$	$ T^{20} $ T^20
$97$	$ T^{20} $ T^20
show more
show less

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:	\( N \)	\(=\)	\( 3484 = 2^{2} \cdot 13 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3484.dz (of order \(66\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{66}^{19} q^{2} - \zeta_{66}^{5} q^{4} + (\zeta_{66}^{28} - \zeta_{66}^{21}) q^{7} + \zeta_{66}^{24} q^{8} + \zeta_{66}^{30} q^{9} +O(q^{10}) \) q - z^19 * q^2 - z^5 * q^4 + (z^28 - z^21) * q^7 + z^24 * q^8 + z^30 * q^9	\( q - \zeta_{66}^{19} q^{2} - \zeta_{66}^{5} q^{4} + (\zeta_{66}^{28} - \zeta_{66}^{21}) q^{7} + \zeta_{66}^{24} q^{8} + \zeta_{66}^{30} q^{9} + ( - \zeta_{66}^{29} + \zeta_{66}^{2}) q^{11} - \zeta_{66}^{31} q^{13} + (\zeta_{66}^{14} - \zeta_{66}^{7}) q^{14} + \zeta_{66}^{10} q^{16} + ( - \zeta_{66}^{29} - \zeta_{66}^{27}) q^{17} + \zeta_{66}^{16} q^{18} + (\zeta_{66}^{16} - \zeta_{66}) q^{19} + ( - \zeta_{66}^{21} - \zeta_{66}^{15}) q^{22} - \zeta_{66}^{9} q^{25} - \zeta_{66}^{17} q^{26} + (\zeta_{66}^{26} + 1) q^{28} + (\zeta_{66}^{16} + \zeta_{66}^{6}) q^{29} - \zeta_{66}^{2} q^{31} - \zeta_{66}^{29} q^{32} + ( - \zeta_{66}^{15} - \zeta_{66}^{13}) q^{34} + \zeta_{66}^{2} q^{36} + (\zeta_{66}^{20} + \zeta_{66}^{2}) q^{38} + ( - \zeta_{66}^{7} - \zeta_{66}) q^{44} + ( - \zeta_{66}^{19} + 1) q^{47} + ( - \zeta_{66}^{23} + \cdots - \zeta_{66}^{9}) q^{49} + \cdots + (\zeta_{66}^{32} + \zeta_{66}^{26}) q^{99} +O(q^{100}) \) q - z^19 * q^2 - z^5 * q^4 + (z^28 - z^21) * q^7 + z^24 * q^8 + z^30 * q^9 + (-z^29 + z^2) * q^11 - z^31 * q^13 + (z^14 - z^7) * q^14 + z^10 * q^16 + (-z^29 - z^27) * q^17 + z^16 * q^18 + (z^16 - z) * q^19 + (-z^21 - z^15) * q^22 - z^9 * q^25 - z^17 * q^26 + (z^26 + 1) * q^28 + (z^16 + z^6) * q^29 - z^2 * q^31 - z^29 * q^32 + (-z^15 - z^13) * q^34 + z^2 * q^36 + (z^20 + z^2) * q^38 + (-z^7 - z) * q^44 + (-z^19 + 1) * q^47 + (-z^23 + z^16 - z^9) * q^49 + z^28 * q^50 - z^3 * q^52 + (-z^13 + z^8) * q^53 + (-z^19 + z^12) * q^56 + (-z^25 + z^2) * q^58 + (-z^9 + z^6) * q^59 + (z^18 - z^17) * q^61 + z^21 * q^62 + (-z^25 + z^18) * q^63 - z^15 * q^64 - z^17 * q^67 + (z^32 - z) * q^68 + (-z^7 - z^3) * q^71 - z^21 * q^72 + (-z^21 + z^6) * q^76 + (z^30 + z^24 - z^23 - z^17) * q^77 - z^27 * q^81 + (-z^15 - z^5) * q^83 + (z^26 + z^20) * q^88 + (z^26 - z^19) * q^91 + (-z^19 - z^5) * q^94 + (z^28 - z^9 + z^2) * q^98 + (z^32 + z^26) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + q^{2} + q^{4} - q^{7} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) 20 * q + q^2 + q^4 - q^7 - 2 * q^8 - 2 * q^9	\( 20 q + q^{2} + q^{4} - q^{7} - 2 q^{8} - 2 q^{9} + 2 q^{11} + q^{13} + 2 q^{14} + q^{16} - q^{17} + q^{18} + 2 q^{19} - 4 q^{22} - 2 q^{25} + q^{26} + 21 q^{28} - q^{29} - q^{31} + q^{32} - q^{34} + q^{36} + 2 q^{38} + 2 q^{44} + 21 q^{47} + q^{50} - 2 q^{52} + 2 q^{53} - q^{56} + 2 q^{58} - 4 q^{59} - q^{61} + 2 q^{62} - q^{63} - 2 q^{64} + q^{67} + 2 q^{68} - q^{71} - 2 q^{72} - 4 q^{76} - 2 q^{77} - 2 q^{81} - q^{83} + 2 q^{88} + 2 q^{91} + 2 q^{94} + 2 q^{99}+O(q^{100}) \) 20 * q + q^2 + q^4 - q^7 - 2 * q^8 - 2 * q^9 + 2 * q^11 + q^13 + 2 * q^14 + q^16 - q^17 + q^18 + 2 * q^19 - 4 * q^22 - 2 * q^25 + q^26 + 21 * q^28 - q^29 - q^31 + q^32 - q^34 + q^36 + 2 * q^38 + 2 * q^44 + 21 * q^47 + q^50 - 2 * q^52 + 2 * q^53 - q^56 + 2 * q^58 - 4 * q^59 - q^61 + 2 * q^62 - q^63 - 2 * q^64 + q^67 + 2 * q^68 - q^71 - 2 * q^72 - 4 * q^76 - 2 * q^77 - 2 * q^81 - q^83 + 2 * q^88 + 2 * q^91 + 2 * q^94 + 2 * 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	3484.1.dz.b

Level	$3484$
Weight	$1$
Character orbit	3484.dz
Analytic conductor	$1.739$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{33}$
CM discriminant	-52
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.73874250401\)
Analytic rank:	\(0\)
Dimension:	\(20\)
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:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{33}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{33} - \cdots)\)

\(n\)	\(1341\)	\(1743\)	\(3017\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-\zeta_{66}^{5}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 103.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{20} - T^{19} + \cdots + 1 \) 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
$3$	\( T^{20} \) T^20
$5$	\( T^{20} \) T^20
$7$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 + 10T^17 + 10T^16 + 78T^14 + 89T^13 + 11T^12 + 241T^11 + 87T^10 - 155T^9 + 605T^8 - 87T^7 - 560T^6 + 253T^5 + 934T^4 + 626T^3 + 154T^2 + 12T + 1
$11$	\( T^{20} - 2 T^{19} + \cdots + 1 \) T^20 - 2T^19 - 14T^17 + 28T^16 + 75T^14 - 139T^13 - 22T^12 - 170T^11 + 318T^10 + 46T^9 + 209T^8 - 360T^7 - 17T^6 + 22T^5 + 145T^4 - 70T^3 + 22T^2 - 6*T + 1
$13$	\( T^{20} - T^{19} + \cdots + 1 \) 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
$17$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 - T^17 + 10T^16 + 22T^15 + 12T^14 - 10T^13 + 77T^12 + 362T^11 + 395T^10 + 32T^9 - 121T^8 - 274T^7 - 395T^6 - 22T^5 + 835T^4 - 595T^3 + 143T^2 - 10T + 1
$19$	\( T^{20} - 2 T^{19} + \cdots + 1 \) T^20 - 2T^19 - 14T^17 + 28T^16 + 75T^14 - 139T^13 - 22T^12 - 170T^11 + 318T^10 + 46T^9 + 209T^8 - 360T^7 - 17T^6 + 22T^5 + 145T^4 - 70T^3 + 22T^2 - 6*T + 1
$23$	\( T^{20} \) 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 - T^17 - T^16 + T^14 + T^13 - T^11 - T^10 - T^9 + T^7 + T^6 - T^4 - T^3 + T + 1
$37$	\( T^{20} \) T^20
$41$	\( T^{20} \) T^20
$43$	\( T^{20} \) T^20
$47$	\( T^{20} - 21 T^{19} + \cdots + 1 \) T^20 - 21T^19 + 209T^18 - 1310T^17 + 5796T^16 - 19228T^15 + 49589T^14 - 101727T^13 + 168454T^12 - 227239T^11 + 250953T^10 - 227239T^9 + 168443T^8 - 101683T^7 + 49479T^6 - 19052T^5 + 5620T^4 - 1200T^3 + 165T^2 - 10*T + 1
$53$	\( T^{20} - 2 T^{19} + \cdots + 1 \) T^20 - 2T^19 + 3T^18 - 4T^17 + 5T^16 + 16T^15 - 37T^14 - 8T^13 + 31T^12 - 43T^11 + 528T^10 - 1011T^9 + 768T^8 - 767T^7 + 1613T^6 - 1491T^5 + 1149T^4 - 565T^3 + 157T^2 - 13*T + 1
$59$	\( (T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) (T^10 + 2T^9 + 4T^8 + 8T^7 + 5T^6 - T^5 - 2T^4 - 4T^3 + 14T^2 - 5T + 1)^2
$61$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 - T^17 - T^16 + 22T^15 + 23T^14 + 34T^13 + 22T^12 - T^11 + 450T^10 + 450T^9 + 1452T^8 + 760T^7 + 881T^6 - 12T^4 + 230T^3 + 154T^2 + 23*T + 1
$67$	\( T^{20} - T^{19} + \cdots + 1 \) 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
$71$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 - T^17 - T^16 + 12T^14 - 98T^13 - 33T^12 + 109T^11 + 120T^10 + 10T^9 + 11T^8 - 604T^7 + 1563T^6 - 1100T^5 + 725T^4 - 375T^3 + 44T^2 + 12T + 1
$73$	\( T^{20} \) T^20
$79$	\( T^{20} \) T^20
$83$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 - T^17 - T^16 + 22T^15 + 23T^14 + 34T^13 + 22T^12 - T^11 + 450T^10 + 450T^9 + 1452T^8 + 760T^7 + 881T^6 - 12T^4 + 230T^3 + 154T^2 + 23*T + 1
$89$	\( T^{20} \) T^20
$97$	\( T^{20} \) T^20
show more
show less