LMFDB - Newform orbit 3204.1.cd.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3204,1,Mod(35,3204)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3204, base_ring=CyclotomicField(88))

chi = DirichletCharacter(H, H._module([44, 44, 63]))

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

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

chi := DirichletCharacter("3204.35");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3204 = 2^{2} \cdot 3^{2} \cdot 89 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3204.cd (of order $88$, degree $40$, 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:	$1.59900430048$
Analytic rank:	$0$
Dimension:	$40$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{88}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{88} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3204\mathbb{Z}\right)^\times$.

$n$	$181$	$713$	$1603$
$\chi(n)$	$-\zeta_{88}^{41}$	$-1$	$-1$

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} $

35.1

0.0713392	−	0.997452i
−0.0713392	+	0.997452i
−0.997452	−	0.0713392i
−0.479249	+	0.877679i
−0.977147	+	0.212565i
−0.212565	−	0.977147i
0.877679	−	0.479249i
0.599278	+	0.800541i
0.936950	−	0.349464i
−0.349464	−	0.936950i
−0.936950	−	0.349464i
0.599278	−	0.800541i
−0.997452	+	0.0713392i
0.0713392	+	0.997452i
0.800541	+	0.599278i
−0.479249	−	0.877679i
−0.212565	+	0.977147i
−0.877679	−	0.479249i
−0.800541	+	0.599278i
−0.349464	+	0.936950i

0.989821

−

0.142315i

0.959493

−

0.281733i

−0.377869

1.01311i

0.909632

−

0.415415i

−0.229843

1.05657i

143.1

0.989821

−

0.142315i

0.959493

−

0.281733i

0.377869

−

1.01311i

0.909632

−

0.415415i

0.229843

−

1.05657i

323.1

−0.989821

0.142315i

0.959493

−

0.281733i

−1.01311

−

0.377869i

−0.909632

0.415415i

1.05657

0.229843i

359.1

0.540641

−

0.841254i

−0.415415

−

0.909632i

0.905808

1.21002i

−0.989821

−

0.142315i

1.50765

−

0.107829i

431.1

−0.909632

−

0.415415i

0.654861

0.755750i

0.948742

−

1.73749i

−0.281733

−

0.959493i

−1.58479

1.18636i

503.1

0.909632

0.415415i

0.654861

0.755750i

−1.73749

−

0.948742i

0.281733

0.959493i

−1.18636

−

1.58479i

575.1

−0.540641

−

0.841254i

−0.415415

0.909632i

−1.21002

−

0.905808i

0.989821

−

0.142315i

−0.107829

1.50765i

647.1

0.281733

0.959493i

−0.841254

0.540641i

−0.129785

−

1.81463i

−0.755750

−

0.654861i

1.70456

−

0.635768i

683.1

−0.755750

−

0.654861i

0.142315

0.989821i

−0.119773

−

0.550588i

0.540641

−

0.841254i

−0.270040

0.494541i

719.1

0.755750

0.654861i

0.142315

0.989821i

0.550588

−

0.119773i

−0.540641

0.841254i

0.494541

0.270040i

755.1

−0.755750

0.654861i

0.142315

−

0.989821i

0.119773

−

0.550588i

0.540641

0.841254i

0.270040

0.494541i

827.1

0.281733

−

0.959493i

−0.841254

−

0.540641i

−0.129785

1.81463i

−0.755750

0.654861i

1.70456

0.635768i

863.1

−0.989821

−

0.142315i

0.959493

0.281733i

−1.01311

0.377869i

−0.909632

−

0.415415i

1.05657

−

0.229843i

1007.1

0.989821

0.142315i

0.959493

0.281733i

−0.377869

−

1.01311i

0.909632

0.415415i

−0.229843

−

1.05657i

1151.1

−0.281733

0.959493i

−0.841254

−

0.540641i

1.81463

0.129785i

0.755750

−

0.654861i

−0.635768

1.70456i

1187.1

0.540641

0.841254i

−0.415415

0.909632i

0.905808

−

1.21002i

−0.989821

0.142315i

1.50765

0.107829i

1223.1

0.909632

−

0.415415i

0.654861

−

0.755750i

−1.73749

0.948742i

0.281733

−

0.959493i

−1.18636

1.58479i

1259.1

−0.540641

0.841254i

−0.415415

−

0.909632i

1.21002

−

0.905808i

0.989821

0.142315i

0.107829

1.50765i

1439.1

−0.281733

−

0.959493i

−0.841254

0.540641i

−1.81463

0.129785i

0.755750

0.654861i

0.635768

1.70456i

1475.1

0.755750

−

0.654861i

0.142315

−

0.989821i

0.550588

0.119773i

−0.540641

−

0.841254i

0.494541

−

0.270040i

See all 40 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
267.p	even	88	1	inner
1068.bf	odd	88	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3204.1.cd.b	yes	40
3.b	odd	2	1		3204.1.cd.a	✓	40
4.b	odd	2	1	CM	3204.1.cd.b	yes	40
12.b	even	2	1		3204.1.cd.a	✓	40
89.h	odd	88	1		3204.1.cd.a	✓	40
267.p	even	88	1	inner	3204.1.cd.b	yes	40
356.o	even	88	1		3204.1.cd.a	✓	40
1068.bf	odd	88	1	inner	3204.1.cd.b	yes	40

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3204.1.cd.a	✓	40	3.b	odd	2	1
3204.1.cd.a	✓	40	12.b	even	2	1
3204.1.cd.a	✓	40	89.h	odd	88	1
3204.1.cd.a	✓	40	356.o	even	88	1
3204.1.cd.b	yes	40	1.a	even	1	1	trivial
3204.1.cd.b	yes	40	4.b	odd	2	1	CM
3204.1.cd.b	yes	40	267.p	even	88	1	inner
3204.1.cd.b	yes	40	1068.bf	odd	88	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{17}^{20} - 2 T_{17}^{19} + 2 T_{17}^{18} - 4 T_{17}^{16} - 36 T_{17}^{15} + 80 T_{17}^{14} - 88 T_{17}^{13} + \cdots + 1 $ T17^20 - 2*T17^19 + 2*T17^18 - 4*T17^16 - 36*T17^15 + 80*T17^14 - 88*T17^13 + 16*T17^12 + 166*T17^11 + 142*T17^10 - 614*T17^9 + 944*T17^8 - 660*T17^7 + 37*T17^6 + 36*T17^5 - 69*T17^4 + 66*T17^3 + 61*T17^2 + 10*T17 + 1 acting on $S_{1}^{\mathrm{new}}(3204, [\chi])$.

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^{40} $ T^40
$5$	$ T^{40} + 44 T^{32} + \cdots + 14641 $ T^40 + 44T^32 - 2904T^28 + 2178T^24 + 149314T^20 + 692120T^16 - 102487T^12 + 1449459T^8 - 190333T^4 + 14641
$7$	$ T^{40} $ T^40
$11$	$ T^{40} $ T^40
$13$	$ T^{40} - 4 T^{39} + \cdots + 1 $ T^40 - 4T^39 + 10T^38 - 20T^37 + 34T^36 - 48T^35 + 48T^34 - 164T^32 - 56T^31 + 1112T^30 - 3452T^29 + 7688T^28 - 14376T^27 + 22960T^26 - 29432T^25 + 27101T^24 + 5100T^23 - 28434T^22 + 33596T^21 - 30187T^20 + 22708T^19 + 12780T^18 - 137200T^17 + 441662T^16 - 527428T^15 - 17020T^14 + 363508T^13 - 125524T^12 - 278684T^11 + 448526T^10 - 352640T^9 + 234546T^8 - 59076T^7 + 31268T^6 - 3520T^5 + 3753T^4 - 856T^3 + 12T^2 + 20T + 1
$17$	$ (T^{20} - 2 T^{19} + \cdots + 1)^{2} $ (T^20 - 2T^19 + 2T^18 - 4T^16 - 36T^15 + 80T^14 - 88T^13 + 16T^12 + 166T^11 + 142T^10 - 614T^9 + 944T^8 - 660T^7 + 37T^6 + 36T^5 - 69T^4 + 66T^3 + 61T^2 + 10T + 1)^2
$19$	$ T^{40} $ T^40
$23$	$ T^{40} $ T^40
$29$	$ T^{40} - 4 T^{39} + \cdots + 1 $ T^40 - 4T^39 + 10T^38 - 20T^37 + 34T^36 - 48T^35 + 48T^34 - 164T^32 - 56T^31 + 1112T^30 - 3452T^29 + 7688T^28 - 14376T^27 + 22960T^26 - 29432T^25 + 27101T^24 + 5100T^23 - 28434T^22 + 33596T^21 - 30187T^20 + 22708T^19 + 12780T^18 - 137200T^17 + 441662T^16 - 527428T^15 - 17020T^14 + 363508T^13 - 125524T^12 - 278684T^11 + 448526T^10 - 352640T^9 + 234546T^8 - 59076T^7 + 31268T^6 - 3520T^5 + 3753T^4 - 856T^3 + 12T^2 + 20T + 1
$31$	$ T^{40} $ T^40
$37$	$ T^{40} + 4 T^{39} + \cdots + 1 $ T^40 + 4T^39 + 10T^38 + 20T^37 + 34T^36 + 48T^35 + 48T^34 + 221T^32 + 980T^31 + 2498T^30 + 4992T^29 + 8458T^28 + 11956T^27 + 12048T^26 + 304T^25 - 1444T^24 + 16680T^23 + 52504T^22 + 110988T^21 + 198536T^20 + 279176T^19 + 264438T^18 - 45136T^17 - 205105T^16 - 50204T^15 + 320812T^14 + 537304T^13 + 433738T^12 + 378784T^11 + 426900T^10 + 95240T^9 - 105497T^8 - 43400T^7 + 11952T^6 + 9900T^5 + 5282T^4 + 328T^3 + 78T^2 - 20T + 1
$41$	$ T^{40} + 40 T^{39} + \cdots + 1 $ T^40 + 40T^39 + 780T^38 + 9880T^37 + 91389T^36 + 657972T^35 + 3837750T^34 + 18636420T^33 + 76845781T^32 + 273061920T^31 + 845713232T^30 + 2303458720T^29 + 5556629099T^28 + 11939280948T^27 + 22953648798T^26 + 39627902340T^25 + 61610921776T^24 + 86449539744T^23 + 109648100376T^22 + 125842348304T^21 + 130761352070T^20 + 123025420984T^19 + 104763573396T^18 + 80679475704T^17 + 56115618802T^16 + 35187015072T^15 + 19843874256T^14 + 10034783904T^13 + 4533126870T^12 + 1820878984T^11 + 646647244T^10 + 201588264T^9 + 54676417T^8 + 12756936T^7 + 2522988T^6 + 414744T^5 + 55145T^4 + 5700T^3 + 430T^2 + 20*T + 1
$43$	$ T^{40} $ T^40
$47$	$ T^{40} $ T^40
$53$	$ (T^{20} - 2 T^{19} + \cdots + 1)^{2} $ (T^20 - 2T^19 + 2T^18 - 4T^16 + 8T^15 - 8T^14 - 66T^13 + 148T^12 - 164T^11 + 494T^10 - 658T^9 + 328T^8 + 176T^7 - 403T^6 + 454T^5 + 63T^4 - 66T^3 + 61T^2 - 12T + 1)^2
$59$	$ T^{40} $ T^40
$61$	$ T^{40} - 2 T^{38} + \cdots + 1 $ T^40 - 2T^38 + 4T^37 + 2T^36 + 72T^35 + 16T^34 - 144T^33 + 252T^32 + 208T^31 + 1836T^30 + 880T^29 - 3344T^28 + 5168T^27 + 6536T^26 + 21476T^25 + 14288T^24 - 27144T^23 + 44630T^22 + 68488T^21 + 136302T^20 + 95832T^19 - 87910T^18 + 185108T^17 + 286544T^16 + 329392T^15 + 343164T^14 + 116780T^13 + 380133T^12 + 480312T^11 + 534148T^10 + 326392T^9 + 97691T^8 + 70640T^7 + 41890T^6 + 10036T^5 + 5213T^4 + 400T^3 + 58T^2 - 20T + 1
$67$	$ T^{40} $ T^40
$71$	$ T^{40} $ T^40
$73$	$ (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
$79$	$ T^{40} $ T^40
$83$	$ T^{40} $ T^40
$89$	$ T^{40} - T^{36} + \cdots + 1 $ T^40 - T^36 + T^32 - T^28 + T^24 - T^20 + T^16 - T^12 + T^8 - T^4 + 1
$97$	$ 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
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Overview	Random
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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:	\( N \)	\(=\)	\( 3204 = 2^{2} \cdot 3^{2} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3204.cd (of order \(88\), degree \(40\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{88}^{42} q^{2} - \zeta_{88}^{40} q^{4} + ( - \zeta_{88}^{35} + \zeta_{88}^{19}) q^{5} + \zeta_{88}^{38} q^{8} +O(q^{10}) \) q + z^42 * q^2 - z^40 * q^4 + (-z^35 + z^19) * q^5 + z^38 * q^8	\( q + \zeta_{88}^{42} q^{2} - \zeta_{88}^{40} q^{4} + ( - \zeta_{88}^{35} + \zeta_{88}^{19}) q^{5} + \zeta_{88}^{38} q^{8} + (\zeta_{88}^{33} - \zeta_{88}^{17}) q^{10} + (\zeta_{88}^{12} + \zeta_{88}^{7}) q^{13} - \zeta_{88}^{36} q^{16} + (\zeta_{88}^{36} - \zeta_{88}^{34}) q^{17} + ( - \zeta_{88}^{31} + \zeta_{88}^{15}) q^{20} + (\zeta_{88}^{38} + \cdots + \zeta_{88}^{10}) q^{25} + \cdots - \zeta_{88}^{19} q^{98} +O(q^{100}) \) q + z^42 * q^2 - z^40 * q^4 + (-z^35 + z^19) * q^5 + z^38 * q^8 + (z^33 - z^17) * q^10 + (z^12 + z^7) * q^13 - z^36 * q^16 + (z^36 - z^34) * q^17 + (-z^31 + z^15) * q^20 + (z^38 - z^26 + z^10) * q^25 + (-z^10 - z^5) * q^26 + (z^39 + z^4) * q^29 + z^34 * q^32 + (-z^34 + z^32) * q^34 + (-z^35 - z^20) * q^37 + (z^29 - z^13) * q^40 + (z^25 - 1) * q^41 + z^21 * q^49 + (-z^36 + z^24 - z^8) * q^50 + (z^8 + z^3) * q^52 + (-z^24 + z^6) * q^53 + (-z^37 - z^2) * q^58 + (z^39 + z^18) * q^61 - z^32 * q^64 + (-z^42 + z^31 + z^26 + z^3) * q^65 + (z^32 - z^30) * q^68 + (-z^18 - z^10) * q^73 + (z^33 + z^18) * q^74 + (-z^27 + z^11) * q^80 + (-z^42 - z^23) * q^82 + (z^27 - z^25 - z^11 + z^9) * q^85 + z^17 * q^89 + (z^31 + z) * q^97 - z^19 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 4 q^{4}+O(q^{10}) \) 40 * q + 4 * q^4	\( 40 q + 4 q^{4} + 4 q^{13} - 4 q^{16} + 4 q^{17} + 4 q^{29} - 4 q^{34} - 4 q^{37} - 40 q^{41} - 4 q^{50} - 4 q^{52} + 4 q^{53} + 4 q^{64} - 4 q^{68}+O(q^{100}) \) 40 * q + 4 * q^4 + 4 * q^13 - 4 * q^16 + 4 * q^17 + 4 * q^29 - 4 * q^34 - 4 * q^37 - 40 * q^41 - 4 * q^50 - 4 * q^52 + 4 * q^53 + 4 * q^64 - 4 * q^68

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	3204.1.cd.b

Level	$3204$
Weight	$1$
Character orbit	3204.cd
Analytic conductor	$1.599$
Analytic rank	$0$
Dimension	$40$
Projective image	$D_{88}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.59900430048\)
Analytic rank:	\(0\)
Dimension:	\(40\)
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:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{88}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{88} - \cdots)\)

\(n\)	\(181\)	\(713\)	\(1603\)
\(\chi(n)\)	\(-\zeta_{88}^{41}\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 35.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
267.p	even	88	1	inner
1068.bf	odd	88	1	inner

$p$	$F_p(T)$
$2$	\( (T^{20} - T^{18} + T^{16} + \cdots + 1)^{2} \) (T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1)^2
$3$	\( T^{40} \) T^40
$5$	\( T^{40} + 44 T^{32} + \cdots + 14641 \) T^40 + 44T^32 - 2904T^28 + 2178T^24 + 149314T^20 + 692120T^16 - 102487T^12 + 1449459T^8 - 190333T^4 + 14641
$7$	\( T^{40} \) T^40
$11$	\( T^{40} \) T^40
$13$	\( T^{40} - 4 T^{39} + \cdots + 1 \) T^40 - 4T^39 + 10T^38 - 20T^37 + 34T^36 - 48T^35 + 48T^34 - 164T^32 - 56T^31 + 1112T^30 - 3452T^29 + 7688T^28 - 14376T^27 + 22960T^26 - 29432T^25 + 27101T^24 + 5100T^23 - 28434T^22 + 33596T^21 - 30187T^20 + 22708T^19 + 12780T^18 - 137200T^17 + 441662T^16 - 527428T^15 - 17020T^14 + 363508T^13 - 125524T^12 - 278684T^11 + 448526T^10 - 352640T^9 + 234546T^8 - 59076T^7 + 31268T^6 - 3520T^5 + 3753T^4 - 856T^3 + 12T^2 + 20T + 1
$17$	\( (T^{20} - 2 T^{19} + \cdots + 1)^{2} \) (T^20 - 2T^19 + 2T^18 - 4T^16 - 36T^15 + 80T^14 - 88T^13 + 16T^12 + 166T^11 + 142T^10 - 614T^9 + 944T^8 - 660T^7 + 37T^6 + 36T^5 - 69T^4 + 66T^3 + 61T^2 + 10T + 1)^2
$19$	\( T^{40} \) T^40
$23$	\( T^{40} \) T^40
$29$	\( T^{40} - 4 T^{39} + \cdots + 1 \) T^40 - 4T^39 + 10T^38 - 20T^37 + 34T^36 - 48T^35 + 48T^34 - 164T^32 - 56T^31 + 1112T^30 - 3452T^29 + 7688T^28 - 14376T^27 + 22960T^26 - 29432T^25 + 27101T^24 + 5100T^23 - 28434T^22 + 33596T^21 - 30187T^20 + 22708T^19 + 12780T^18 - 137200T^17 + 441662T^16 - 527428T^15 - 17020T^14 + 363508T^13 - 125524T^12 - 278684T^11 + 448526T^10 - 352640T^9 + 234546T^8 - 59076T^7 + 31268T^6 - 3520T^5 + 3753T^4 - 856T^3 + 12T^2 + 20T + 1
$31$	\( T^{40} \) T^40
$37$	\( T^{40} + 4 T^{39} + \cdots + 1 \) T^40 + 4T^39 + 10T^38 + 20T^37 + 34T^36 + 48T^35 + 48T^34 + 221T^32 + 980T^31 + 2498T^30 + 4992T^29 + 8458T^28 + 11956T^27 + 12048T^26 + 304T^25 - 1444T^24 + 16680T^23 + 52504T^22 + 110988T^21 + 198536T^20 + 279176T^19 + 264438T^18 - 45136T^17 - 205105T^16 - 50204T^15 + 320812T^14 + 537304T^13 + 433738T^12 + 378784T^11 + 426900T^10 + 95240T^9 - 105497T^8 - 43400T^7 + 11952T^6 + 9900T^5 + 5282T^4 + 328T^3 + 78T^2 - 20T + 1
$41$	\( T^{40} + 40 T^{39} + \cdots + 1 \) T^40 + 40T^39 + 780T^38 + 9880T^37 + 91389T^36 + 657972T^35 + 3837750T^34 + 18636420T^33 + 76845781T^32 + 273061920T^31 + 845713232T^30 + 2303458720T^29 + 5556629099T^28 + 11939280948T^27 + 22953648798T^26 + 39627902340T^25 + 61610921776T^24 + 86449539744T^23 + 109648100376T^22 + 125842348304T^21 + 130761352070T^20 + 123025420984T^19 + 104763573396T^18 + 80679475704T^17 + 56115618802T^16 + 35187015072T^15 + 19843874256T^14 + 10034783904T^13 + 4533126870T^12 + 1820878984T^11 + 646647244T^10 + 201588264T^9 + 54676417T^8 + 12756936T^7 + 2522988T^6 + 414744T^5 + 55145T^4 + 5700T^3 + 430T^2 + 20*T + 1
$43$	\( T^{40} \) T^40
$47$	\( T^{40} \) T^40
$53$	\( (T^{20} - 2 T^{19} + \cdots + 1)^{2} \) (T^20 - 2T^19 + 2T^18 - 4T^16 + 8T^15 - 8T^14 - 66T^13 + 148T^12 - 164T^11 + 494T^10 - 658T^9 + 328T^8 + 176T^7 - 403T^6 + 454T^5 + 63T^4 - 66T^3 + 61T^2 - 12T + 1)^2
$59$	\( T^{40} \) T^40
$61$	\( T^{40} - 2 T^{38} + \cdots + 1 \) T^40 - 2T^38 + 4T^37 + 2T^36 + 72T^35 + 16T^34 - 144T^33 + 252T^32 + 208T^31 + 1836T^30 + 880T^29 - 3344T^28 + 5168T^27 + 6536T^26 + 21476T^25 + 14288T^24 - 27144T^23 + 44630T^22 + 68488T^21 + 136302T^20 + 95832T^19 - 87910T^18 + 185108T^17 + 286544T^16 + 329392T^15 + 343164T^14 + 116780T^13 + 380133T^12 + 480312T^11 + 534148T^10 + 326392T^9 + 97691T^8 + 70640T^7 + 41890T^6 + 10036T^5 + 5213T^4 + 400T^3 + 58T^2 - 20T + 1
$67$	\( T^{40} \) T^40
$71$	\( T^{40} \) T^40
$73$	\( (T^{20} - 4 T^{18} + \cdots + 1)^{2} \) (T^20 - 4T^18 + 16T^16 - 64T^14 + 234T^12 - 474T^10 + 444T^8 - 203T^6 + 119T^4 + 19*T^2 + 1)^2
$79$	\( T^{40} \) T^40
$83$	\( T^{40} \) T^40
$89$	\( T^{40} - T^{36} + \cdots + 1 \) T^40 - T^36 + T^32 - T^28 + T^24 - T^20 + T^16 - T^12 + T^8 - T^4 + 1
$97$	\( T^{40} + 4 T^{38} + \cdots + 1 \) 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
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