LMFDB - Newform orbit 3864.1.cx.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3864,1,Mod(125,3864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3864, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 11, 11, 11, 3]))

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

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

chi := DirichletCharacter("3864.125");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3864.cx (of order $22$, degree $10$, 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:	$1.92838720881$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$4$ over $\Q(\zeta_{22})$
Coefficient field:	$\Q(\zeta_{88})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{40} - x^{36} + x^{32} - x^{28} + x^{24} - x^{20} + x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^40 - x^36 + x^32 - x^28 + x^24 - x^20 + x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{44}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{44} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3864\mathbb{Z}\right)^\times$.

$n$	$967$	$1289$	$1933$	$2761$	$2857$
$\chi(n)$	$1$	$-1$	$-1$	$-1$	$\zeta_{88}^{28}$

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} $

125.1

0.800541	−	0.599278i
−0.800541	+	0.599278i
−0.599278	−	0.800541i
0.599278	+	0.800541i
−0.0713392	+	0.997452i
0.0713392	−	0.997452i
0.997452	+	0.0713392i
−0.997452	−	0.0713392i
0.877679	−	0.479249i
−0.877679	+	0.479249i
0.479249	+	0.877679i
−0.479249	−	0.877679i
0.877679	+	0.479249i
−0.877679	−	0.479249i
0.479249	−	0.877679i
−0.479249	+	0.877679i
−0.936950	+	0.349464i
0.936950	−	0.349464i
0.349464	+	0.936950i
−0.349464	−	0.936950i

−0.755750

0.654861i

−0.707107

0.707107i

0.142315

−

0.989821i

0.729202

−

1.59673i

0.0713392

−

0.997452i

−0.281733

0.959493i

0.540641

0.841254i

−

1.00000i

0.494541

1.68425i

125.2

−0.755750

0.654861i

0.707107

−

0.707107i

0.142315

−

0.989821i

−0.729202

1.59673i

−0.0713392

0.997452i

−0.281733

0.959493i

0.540641

0.841254i

−

1.00000i

−0.494541

−

1.68425i

125.3

0.755750

−

0.654861i

−0.707107

−

0.707107i

0.142315

−

0.989821i

0.398174

−

0.871880i

−0.997452

−

0.0713392i

0.281733

−

0.959493i

−0.540641

−

0.841254i

1.00000i

−0.270040

−

0.919672i

125.4

0.755750

−

0.654861i

0.707107

0.707107i

0.142315

−

0.989821i

−0.398174

0.871880i

0.997452

0.0713392i

0.281733

−

0.959493i

−0.540641

−

0.841254i

1.00000i

0.270040

0.919672i

293.1

−0.909632

−

0.415415i

−0.707107

0.707107i

0.654861

0.755750i

−1.00829

0.647988i

0.936950

−

0.349464i

0.989821

0.142315i

−0.281733

−

0.959493i

−

1.00000i

1.18636

−

0.170572i

293.2

−0.909632

−

0.415415i

0.707107

−

0.707107i

0.654861

0.755750i

1.00829

−

0.647988i

−0.936950

0.349464i

0.989821

0.142315i

−0.281733

−

0.959493i

−

1.00000i

−1.18636

0.170572i

293.3

0.909632

0.415415i

−0.707107

−

0.707107i

0.654861

0.755750i

1.34692

−

0.865611i

−0.349464

−

0.936950i

−0.989821

−

0.142315i

0.281733

0.959493i

1.00000i

1.58479

−

0.227858i

293.4

0.909632

0.415415i

0.707107

0.707107i

0.654861

0.755750i

−1.34692

0.865611i

0.349464

0.936950i

−0.989821

−

0.142315i

0.281733

0.959493i

1.00000i

−1.58479

0.227858i

797.1

−0.989821

−

0.142315i

−0.707107

−

0.707107i

0.959493

0.281733i

0.278401

0.321292i

0.599278

0.800541i

−0.540641

0.841254i

−0.909632

−

0.415415i

1.00000i

−0.229843

−

0.357643i

797.2

−0.989821

−

0.142315i

0.707107

0.707107i

0.959493

0.281733i

−0.278401

−

0.321292i

−0.599278

−

0.800541i

−0.540641

0.841254i

−0.909632

−

0.415415i

1.00000i

0.229843

0.357643i

797.3

0.989821

0.142315i

−0.707107

0.707107i

0.959493

0.281733i

1.27979

1.47696i

−0.800541

0.599278i

0.540641

−

0.841254i

0.909632

0.415415i

−

1.00000i

1.05657

1.64406i

797.4

0.989821

0.142315i

0.707107

−

0.707107i

0.959493

0.281733i

−1.27979

−

1.47696i

0.800541

−

0.599278i

0.540641

−

0.841254i

0.909632

0.415415i

−

1.00000i

−1.05657

−

1.64406i

1469.1

−0.989821

0.142315i

−0.707107

0.707107i

0.959493

−

0.281733i

0.278401

−

0.321292i

0.599278

−

0.800541i

−0.540641

−

0.841254i

−0.909632

0.415415i

−

1.00000i

−0.229843

0.357643i

1469.2

−0.989821

0.142315i

0.707107

−

0.707107i

0.959493

−

0.281733i

−0.278401

0.321292i

−0.599278

0.800541i

−0.540641

−

0.841254i

−0.909632

0.415415i

−

1.00000i

0.229843

−

0.357643i

1469.3

0.989821

−

0.142315i

−0.707107

−

0.707107i

0.959493

−

0.281733i

1.27979

−

1.47696i

−0.800541

−

0.599278i

0.540641

0.841254i

0.909632

−

0.415415i

1.00000i

1.05657

−

1.64406i

1469.4

0.989821

−

0.142315i

0.707107

0.707107i

0.959493

−

0.281733i

−1.27979

1.47696i

0.800541

0.599278i

0.540641

0.841254i

0.909632

−

0.415415i

1.00000i

−1.05657

1.64406i

1805.1

−0.540641

−

0.841254i

−0.707107

0.707107i

−0.415415

0.909632i

−1.91410

0.562029i

0.977147

0.212565i

−0.755750

0.654861i

0.989821

−

0.142315i

−

1.00000i

1.50765

1.30638i

1805.2

−0.540641

−

0.841254i

0.707107

−

0.707107i

−0.415415

0.909632i

1.91410

−

0.562029i

−0.977147

−

0.212565i

−0.755750

0.654861i

0.989821

−

0.142315i

−

1.00000i

−1.50765

−

1.30638i

1805.3

0.540641

0.841254i

−0.707107

−

0.707107i

−0.415415

0.909632i

0.136899

−

0.0401971i

0.212565

−

0.977147i

0.755750

−

0.654861i

−0.989821

0.142315i

1.00000i

0.107829

0.0934345i

1805.4

0.540641

0.841254i

0.707107

0.707107i

−0.415415

0.909632i

−0.136899

0.0401971i

−0.212565

0.977147i

0.755750

−

0.654861i

−0.989821

0.142315i

1.00000i

−0.107829

−

0.0934345i

See all 40 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
56.h	odd	2	1	CM by $\Q(\sqrt{-14}) $
7.b	odd	2	1	inner
8.b	even	2	1	inner
69.g	even	22	1	inner
483.w	odd	22	1	inner
552.bf	even	22	1	inner
3864.cx	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3864.1.cx.b	yes	40
3.b	odd	2	1		3864.1.cx.a	✓	40
7.b	odd	2	1	inner	3864.1.cx.b	yes	40
8.b	even	2	1	inner	3864.1.cx.b	yes	40
21.c	even	2	1		3864.1.cx.a	✓	40
23.d	odd	22	1		3864.1.cx.a	✓	40
24.h	odd	2	1		3864.1.cx.a	✓	40
56.h	odd	2	1	CM	3864.1.cx.b	yes	40
69.g	even	22	1	inner	3864.1.cx.b	yes	40
161.k	even	22	1		3864.1.cx.a	✓	40
168.i	even	2	1		3864.1.cx.a	✓	40
184.m	odd	22	1		3864.1.cx.a	✓	40
483.w	odd	22	1	inner	3864.1.cx.b	yes	40
552.bf	even	22	1	inner	3864.1.cx.b	yes	40
1288.bk	even	22	1		3864.1.cx.a	✓	40
3864.cx	odd	22	1	inner	3864.1.cx.b	yes	40

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3864.1.cx.a	✓	40	3.b	odd	2	1
3864.1.cx.a	✓	40	21.c	even	2	1
3864.1.cx.a	✓	40	23.d	odd	22	1
3864.1.cx.a	✓	40	24.h	odd	2	1
3864.1.cx.a	✓	40	161.k	even	22	1
3864.1.cx.a	✓	40	168.i	even	2	1
3864.1.cx.a	✓	40	184.m	odd	22	1
3864.1.cx.a	✓	40	1288.bk	even	22	1
3864.1.cx.b	yes	40	1.a	even	1	1	trivial
3864.1.cx.b	yes	40	7.b	odd	2	1	inner
3864.1.cx.b	yes	40	8.b	even	2	1	inner
3864.1.cx.b	yes	40	56.h	odd	2	1	CM
3864.1.cx.b	yes	40	69.g	even	22	1	inner
3864.1.cx.b	yes	40	483.w	odd	22	1	inner
3864.1.cx.b	yes	40	552.bf	even	22	1	inner
3864.1.cx.b	yes	40	3864.cx	odd	22	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{137}^{5} - T_{137}^{4} - 4T_{137}^{3} + 3T_{137}^{2} + 3T_{137} - 1 $ T137^5 - T137^4 - 4*T137^3 + 3*T137^2 + 3*T137 - 1 acting on $S_{1}^{\mathrm{new}}(3864, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{20} - T^{18} + T^{16} + \cdots + 1)^{2} $ (T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1)^2
$3$	$ (T^{4} + 1)^{10} $ (T^4 + 1)^10
$5$	$ T^{40} + 4 T^{38} + \cdots + 1 $ T^40 + 4T^38 + 12T^36 + 32T^34 + 80T^32 + 720T^30 + 4375T^28 + 14664T^26 + 41156T^24 + 105924T^22 + 313192T^20 + 376528T^18 + 575325T^16 + 763728T^14 + 762929T^12 + 410140T^10 + 102466T^8 + 12216T^6 + 1912T^4 - 76*T^2 + 1
$7$	$ (T^{20} - T^{18} + T^{16} + \cdots + 1)^{2} $ (T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1)^2
$11$	$ T^{40} $ T^40
$13$	$ T^{40} + 4 T^{38} + \cdots + 1 $ T^40 + 4T^38 + 12T^36 + 32T^34 + 80T^32 + 720T^30 + 4375T^28 + 14664T^26 + 41156T^24 + 105924T^22 + 313192T^20 + 376528T^18 + 575325T^16 + 763728T^14 + 762929T^12 + 410140T^10 + 102466T^8 + 12216T^6 + 1912T^4 - 76*T^2 + 1
$17$	$ T^{40} $ T^40
$19$	$ T^{40} - 4 T^{38} + \cdots + 1 $ T^40 - 4T^38 + 12T^36 - 32T^34 + 80T^32 - 720T^30 + 4375T^28 - 14664T^26 + 41156T^24 - 105924T^22 + 313192T^20 - 376528T^18 + 575325T^16 - 763728T^14 + 762929T^12 - 410140T^10 + 102466T^8 - 12216T^6 + 1912T^4 + 76*T^2 + 1
$23$	$ (T^{10} - T^{9} + T^{8} + \cdots + 1)^{4} $ (T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^4
$29$	$ T^{40} $ T^40
$31$	$ T^{40} $ T^40
$37$	$ T^{40} $ T^40
$41$	$ T^{40} $ T^40
$43$	$ T^{40} $ T^40
$47$	$ T^{40} $ T^40
$53$	$ T^{40} $ T^40
$59$	$ T^{40} - 4 T^{38} + \cdots + 1 $ T^40 - 4T^38 + 12T^36 - 76T^34 + 256T^32 - 676T^30 + 2406T^28 - 6920T^26 + 38142T^24 - 130432T^22 + 369710T^20 - 483272T^18 + 473729T^16 - 130260T^14 + 97759T^12 + 104836T^10 + 32231T^8 - 20708T^6 + 2066T^4 + 32*T^2 + 1
$61$	$ (T^{20} - 2 T^{18} + \cdots + 1024)^{2} $ (T^20 - 2T^18 + 4T^16 - 8T^14 + 16T^12 - 32T^10 + 64T^8 - 128T^6 + 256T^4 - 512*T^2 + 1024)^2
$67$	$ T^{40} $ T^40
$71$	$ (T^{20} + 7 T^{18} + \cdots + 1)^{2} $ (T^20 + 7T^18 + 27T^16 + 57T^14 + 102T^12 + 54T^10 + 224T^8 - 236T^6 + 185T^4 - 25*T^2 + 1)^2
$73$	$ T^{40} $ T^40
$79$	$ (T^{20} - 4 T^{18} + \cdots + 1)^{2} $ (T^20 - 4T^18 + 16T^16 - 64T^14 + 234T^12 - 474T^10 + 444T^8 - 203T^6 + 119T^4 + 19*T^2 + 1)^2
$83$	$ T^{40} + 4 T^{38} + \cdots + 1 $ T^40 + 4T^38 + 12T^36 + 76T^34 + 256T^32 + 676T^30 + 2406T^28 + 6920T^26 + 38142T^24 + 130432T^22 + 369710T^20 + 483272T^18 + 473729T^16 + 130260T^14 + 97759T^12 - 104836T^10 + 32231T^8 + 20708T^6 + 2066T^4 - 32*T^2 + 1
$89$	$ T^{40} $ T^40
$97$	$ T^{40} $ T^40
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:	\( N \)	\(=\)	\( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3864.cx (of order \(22\), degree \(10\), minimal)

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

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	3864.1.cx.b

Level	$3864$
Weight	$1$
Character orbit	3864.cx
Analytic conductor	$1.928$
Analytic rank	$0$
Dimension	$40$
Projective image	$D_{44}$
CM discriminant	-56
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.92838720881\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(4\) over \(\Q(\zeta_{22})\)
Coefficient field:	\(\Q(\zeta_{88})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{40} - x^{36} + x^{32} - x^{28} + x^{24} - x^{20} + x^{16} - x^{12} + x^{8} - x^{4} + 1 \) x^40 - x^36 + x^32 - x^28 + x^24 - x^20 + x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{44}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{44} - \cdots)\)

\(n\)	\(967\)	\(1289\)	\(1933\)	\(2761\)	\(2857\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(-1\)	\(\zeta_{88}^{28}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 125.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{20} - T^{18} + T^{16} + \cdots + 1)^{2} \) (T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1)^2
$3$	\( (T^{4} + 1)^{10} \) (T^4 + 1)^10
$5$	\( T^{40} + 4 T^{38} + \cdots + 1 \) T^40 + 4T^38 + 12T^36 + 32T^34 + 80T^32 + 720T^30 + 4375T^28 + 14664T^26 + 41156T^24 + 105924T^22 + 313192T^20 + 376528T^18 + 575325T^16 + 763728T^14 + 762929T^12 + 410140T^10 + 102466T^8 + 12216T^6 + 1912T^4 - 76*T^2 + 1
$7$	\( (T^{20} - T^{18} + T^{16} + \cdots + 1)^{2} \) (T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1)^2
$11$	\( T^{40} \) T^40
$13$	\( T^{40} + 4 T^{38} + \cdots + 1 \) T^40 + 4T^38 + 12T^36 + 32T^34 + 80T^32 + 720T^30 + 4375T^28 + 14664T^26 + 41156T^24 + 105924T^22 + 313192T^20 + 376528T^18 + 575325T^16 + 763728T^14 + 762929T^12 + 410140T^10 + 102466T^8 + 12216T^6 + 1912T^4 - 76*T^2 + 1
$17$	\( T^{40} \) T^40
$19$	\( T^{40} - 4 T^{38} + \cdots + 1 \) T^40 - 4T^38 + 12T^36 - 32T^34 + 80T^32 - 720T^30 + 4375T^28 - 14664T^26 + 41156T^24 - 105924T^22 + 313192T^20 - 376528T^18 + 575325T^16 - 763728T^14 + 762929T^12 - 410140T^10 + 102466T^8 - 12216T^6 + 1912T^4 + 76*T^2 + 1
$23$	\( (T^{10} - T^{9} + T^{8} + \cdots + 1)^{4} \) (T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^4
$29$	\( T^{40} \) T^40
$31$	\( T^{40} \) T^40
$37$	\( T^{40} \) T^40
$41$	\( T^{40} \) T^40
$43$	\( T^{40} \) T^40
$47$	\( T^{40} \) T^40
$53$	\( T^{40} \) T^40
$59$	\( T^{40} - 4 T^{38} + \cdots + 1 \) T^40 - 4T^38 + 12T^36 - 76T^34 + 256T^32 - 676T^30 + 2406T^28 - 6920T^26 + 38142T^24 - 130432T^22 + 369710T^20 - 483272T^18 + 473729T^16 - 130260T^14 + 97759T^12 + 104836T^10 + 32231T^8 - 20708T^6 + 2066T^4 + 32*T^2 + 1
$61$	\( (T^{20} - 2 T^{18} + \cdots + 1024)^{2} \) (T^20 - 2T^18 + 4T^16 - 8T^14 + 16T^12 - 32T^10 + 64T^8 - 128T^6 + 256T^4 - 512*T^2 + 1024)^2
$67$	\( T^{40} \) T^40
$71$	\( (T^{20} + 7 T^{18} + \cdots + 1)^{2} \) (T^20 + 7T^18 + 27T^16 + 57T^14 + 102T^12 + 54T^10 + 224T^8 - 236T^6 + 185T^4 - 25*T^2 + 1)^2
$73$	\( T^{40} \) T^40
$79$	\( (T^{20} - 4 T^{18} + \cdots + 1)^{2} \) (T^20 - 4T^18 + 16T^16 - 64T^14 + 234T^12 - 474T^10 + 444T^8 - 203T^6 + 119T^4 + 19*T^2 + 1)^2
$83$	\( T^{40} + 4 T^{38} + \cdots + 1 \) T^40 + 4T^38 + 12T^36 + 76T^34 + 256T^32 + 676T^30 + 2406T^28 + 6920T^26 + 38142T^24 + 130432T^22 + 369710T^20 + 483272T^18 + 473729T^16 + 130260T^14 + 97759T^12 - 104836T^10 + 32231T^8 + 20708T^6 + 2066T^4 - 32*T^2 + 1
$89$	\( T^{40} \) T^40
$97$	\( T^{40} \) T^40
show more
show less