LMFDB - Newform orbit 2576.1.cq.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2576,1,Mod(13,2576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2576, base_ring=CyclotomicField(44))

chi = DirichletCharacter(H, H._module([0, 33, 22, 28]))

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

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

chi := DirichletCharacter("2576.13");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2576 = 2^{4} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2576.cq (of order $44$, 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:	$1.28559147254$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{44})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
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 $\left(\mathbb{Z}/2576\mathbb{Z}\right)^\times$.

$n$	$645$	$1473$	$1569$	$2255$
$\chi(n)$	$\zeta_{44}^{11}$	$-1$	$\zeta_{44}^{16}$	$1$

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} $

13.1

0.989821	−	0.142315i
0.755750	−	0.654861i
0.281733	+	0.959493i
0.540641	+	0.841254i
−0.909632	−	0.415415i
−0.909632	+	0.415415i
−0.540641	+	0.841254i
−0.755750	−	0.654861i
0.989821	+	0.142315i
0.281733	−	0.959493i
−0.989821	+	0.142315i
−0.755750	+	0.654861i
−0.281733	−	0.959493i
−0.540641	−	0.841254i
0.909632	+	0.415415i
0.909632	−	0.415415i
0.540641	−	0.841254i
0.755750	+	0.654861i
−0.989821	−	0.142315i
−0.281733	+	0.959493i

0.755750

0.654861i

0.142315

0.989821i

−0.281733

0.959493i

−0.540641

0.841254i

−0.909632

0.415415i

349.1

−0.909632

−

0.415415i

0.654861

0.755750i

−0.989821

0.142315i

−0.281733

−

0.959493i

0.540641

0.841254i

629.1

0.989821

−

0.142315i

0.959493

−

0.281733i

−0.540641

0.841254i

0.909632

−

0.415415i

0.755750

0.654861i

685.1

0.281733

0.959493i

−0.841254

0.540641i

0.909632

−

0.415415i

−0.755750

−

0.654861i

0.989821

−

0.142315i

853.1

0.540641

0.841254i

−0.415415

0.909632i

−0.755750

−

0.654861i

−0.989821

0.142315i

0.281733

0.959493i

909.1

0.540641

−

0.841254i

−0.415415

−

0.909632i

−0.755750

0.654861i

−0.989821

−

0.142315i

0.281733

−

0.959493i

1021.1

−0.281733

0.959493i

−0.841254

−

0.540641i

−0.909632

−

0.415415i

0.755750

−

0.654861i

−0.989821

−

0.142315i

1133.1

0.909632

−

0.415415i

0.654861

−

0.755750i

0.989821

0.142315i

0.281733

−

0.959493i

−0.540641

0.841254i

1189.1

0.755750

−

0.654861i

0.142315

−

0.989821i

−0.281733

−

0.959493i

−0.540641

−

0.841254i

−0.909632

−

0.415415i

1245.1

0.989821

0.142315i

0.959493

0.281733i

−0.540641

−

0.841254i

0.909632

0.415415i

0.755750

−

0.654861i

1301.1

−0.755750

−

0.654861i

0.142315

0.989821i

0.281733

−

0.959493i

0.540641

−

0.841254i

0.909632

−

0.415415i

1637.1

0.909632

0.415415i

0.654861

0.755750i

0.989821

−

0.142315i

0.281733

0.959493i

−0.540641

−

0.841254i

1917.1

−0.989821

0.142315i

0.959493

−

0.281733i

0.540641

−

0.841254i

−0.909632

0.415415i

−0.755750

−

0.654861i

1973.1

−0.281733

−

0.959493i

−0.841254

0.540641i

−0.909632

0.415415i

0.755750

0.654861i

−0.989821

0.142315i

2141.1

−0.540641

−

0.841254i

−0.415415

0.909632i

0.755750

0.654861i

0.989821

−

0.142315i

−0.281733

−

0.959493i

2197.1

−0.540641

0.841254i

−0.415415

−

0.909632i

0.755750

−

0.654861i

0.989821

0.142315i

−0.281733

0.959493i

2309.1

0.281733

−

0.959493i

−0.841254

−

0.540641i

0.909632

0.415415i

−0.755750

0.654861i

0.989821

0.142315i

2421.1

−0.909632

0.415415i

0.654861

−

0.755750i

−0.989821

−

0.142315i

−0.281733

0.959493i

0.540641

−

0.841254i

2477.1

−0.755750

0.654861i

0.142315

−

0.989821i

0.281733

0.959493i

0.540641

0.841254i

0.909632

0.415415i

2533.1

−0.989821

−

0.142315i

0.959493

0.281733i

0.540641

0.841254i

−0.909632

−

0.415415i

−0.755750

0.654861i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $
368.w	even	44	1	inner
2576.cq	odd	44	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2576.1.cq.b	yes	20
7.b	odd	2	1	CM	2576.1.cq.b	yes	20
16.e	even	4	1		2576.1.cq.a	✓	20
23.c	even	11	1		2576.1.cq.a	✓	20
112.l	odd	4	1		2576.1.cq.a	✓	20
161.l	odd	22	1		2576.1.cq.a	✓	20
368.w	even	44	1	inner	2576.1.cq.b	yes	20
2576.cq	odd	44	1	inner	2576.1.cq.b	yes	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2576.1.cq.a	✓	20	16.e	even	4	1
2576.1.cq.a	✓	20	23.c	even	11	1
2576.1.cq.a	✓	20	112.l	odd	4	1
2576.1.cq.a	✓	20	161.l	odd	22	1
2576.1.cq.b	yes	20	1.a	even	1	1	trivial
2576.1.cq.b	yes	20	7.b	odd	2	1	CM
2576.1.cq.b	yes	20	368.w	even	44	1	inner
2576.1.cq.b	yes	20	2576.cq	odd	44	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{11}^{20} - 2 T_{11}^{19} + 2 T_{11}^{18} - 22 T_{11}^{17} + 40 T_{11}^{16} - 36 T_{11}^{15} + \cdots + 1 $ T11^20 - 2*T11^19 + 2*T11^18 - 22*T11^17 + 40*T11^16 - 36*T11^15 + 179*T11^14 - 286*T11^13 + 214*T11^12 - 626*T11^11 + 824*T11^10 - 394*T11^9 + 713*T11^8 - 638*T11^7 - 51*T11^6 - 74*T11^5 + 250*T11^4 - 44*T11^3 + 50*T11^2 - 12*T11 + 1 acting on $S_{1}^{\mathrm{new}}(2576, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} - T^{18} + \cdots + 1 $ T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$3$	$ T^{20} $ T^20
$5$	$ T^{20} $ T^20
$7$	$ T^{20} - T^{18} + \cdots + 1 $ T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$11$	$ T^{20} - 2 T^{19} + \cdots + 1 $ T^20 - 2T^19 + 2T^18 - 22T^17 + 40T^16 - 36T^15 + 179T^14 - 286T^13 + 214T^12 - 626T^11 + 824T^10 - 394T^9 + 713T^8 - 638T^7 - 51T^6 - 74T^5 + 250T^4 - 44T^3 + 50T^2 - 12*T + 1
$13$	$ T^{20} $ T^20
$17$	$ T^{20} $ T^20
$19$	$ T^{20} $ T^20
$23$	$ (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} $ (T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$29$	$ T^{20} - 2 T^{19} + \cdots + 1 $ 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
$31$	$ T^{20} $ T^20
$37$	$ T^{20} - 2 T^{19} + \cdots + 1 $ T^20 - 2T^19 + 2T^18 - 4T^16 + 8T^15 + 47T^14 - 132T^13 + 170T^12 - 76T^11 - 188T^10 + 530T^9 - 79T^8 + 66T^7 + 125T^6 - 382T^5 + 514T^4 - 330T^3 + 94T^2 - 12T + 1
$41$	$ T^{20} $ T^20
$43$	$ T^{20} + 2 T^{19} + \cdots + 1 $ T^20 + 2T^19 + 2T^18 - 4T^16 - 8T^15 - 8T^14 + 93T^12 - 122T^11 - 331T^10 - 420T^9 - 178T^8 + 484T^7 + 1324T^6 + 1702T^5 + 1218T^4 + 484T^3 + 105T^2 + 12*T + 1
$47$	$ T^{20} $ T^20
$53$	$ T^{20} + 2 T^{19} + \cdots + 1 $ T^20 + 2T^19 + 2T^18 - 4T^16 - 8T^15 + 47T^14 + 132T^13 + 170T^12 + 76T^11 - 188T^10 - 530T^9 - 79T^8 - 66T^7 + 125T^6 + 382T^5 + 514T^4 + 330T^3 + 94T^2 + 12T + 1
$59$	$ T^{20} $ T^20
$61$	$ T^{20} $ T^20
$67$	$ T^{20} + 2 T^{19} + \cdots + 1 $ T^20 + 2T^19 + 2T^18 - 4T^16 - 30T^15 - 52T^14 - 44T^13 + 93T^12 + 274T^11 + 505T^10 + 460T^9 - 90T^8 + 110T^7 + 400T^6 + 316T^5 + 338T^4 + 44T^3 + 17T^2 - 10T + 1
$71$	$ T^{20} - 4 T^{18} + \cdots + 1 $ T^20 - 4T^18 + 16T^16 - 9T^14 + 36T^12 + 120T^10 + 125T^8 - 335T^6 + 130T^4 + 8*T^2 + 1
$73$	$ T^{20} $ T^20
$79$	$ T^{20} + 55 T^{14} + \cdots + 121 $ T^20 + 55T^14 - 264T^10 + 605T^8 + 121T^6 + 1210T^4 + 484T^2 + 121
$83$	$ T^{20} $ T^20
$89$	$ T^{20} $ T^20
$97$	$ T^{20} $ T^20
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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:	\( N \)	\(=\)	\( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2576.cq (of order \(44\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{44}^{17} q^{2} - \zeta_{44}^{12} q^{4} - \zeta_{44}^{9} q^{7} - \zeta_{44}^{7} q^{8} - \zeta_{44}^{3} q^{9} +O(q^{10}) \) q - z^17 * q^2 - z^12 * q^4 - z^9 * q^7 - z^7 * q^8 - z^3 * q^9	\( q - \zeta_{44}^{17} q^{2} - \zeta_{44}^{12} q^{4} - \zeta_{44}^{9} q^{7} - \zeta_{44}^{7} q^{8} - \zeta_{44}^{3} q^{9} + (\zeta_{44}^{21} + \zeta_{44}^{2}) q^{11} - \zeta_{44}^{4} q^{14} - \zeta_{44}^{2} q^{16} + \zeta_{44}^{20} q^{18} + ( - \zeta_{44}^{19} + \zeta_{44}^{16}) q^{22} - \zeta_{44}^{8} q^{23} - \zeta_{44}^{5} q^{25} + \zeta_{44}^{21} q^{28} + (\zeta_{44}^{18} - \zeta_{44}^{17}) q^{29} + \zeta_{44}^{19} q^{32} + \zeta_{44}^{15} q^{36} + ( - \zeta_{44}^{16} + \zeta_{44}) q^{37} + (\zeta_{44}^{15} - \zeta_{44}^{10}) q^{43} + ( - \zeta_{44}^{14} + \zeta_{44}^{11}) q^{44} - \zeta_{44}^{3} q^{46} + \zeta_{44}^{18} q^{49} - q^{50} + (\zeta_{44}^{4} + \zeta_{44}^{3}) q^{53} + \zeta_{44}^{16} q^{56} + (\zeta_{44}^{13} - \zeta_{44}^{12}) q^{58} + \zeta_{44}^{12} q^{63} + \zeta_{44}^{14} q^{64} + ( - \zeta_{44}^{15} - \zeta_{44}^{6}) q^{67} + (\zeta_{44}^{9} + \zeta_{44}) q^{71} + \zeta_{44}^{10} q^{72} + ( - \zeta_{44}^{18} - \zeta_{44}^{11}) q^{74} + ( - \zeta_{44}^{11} + \zeta_{44}^{8}) q^{77} + (\zeta_{44}^{19} - \zeta_{44}^{7}) q^{79} + \zeta_{44}^{6} q^{81} + (\zeta_{44}^{10} - \zeta_{44}^{5}) q^{86} + ( - \zeta_{44}^{9} + \zeta_{44}^{6}) q^{88} + \zeta_{44}^{20} q^{92} + \zeta_{44}^{13} q^{98} + ( - \zeta_{44}^{5} + \zeta_{44}^{2}) q^{99} +O(q^{100}) \) q - z^17 * q^2 - z^12 * q^4 - z^9 * q^7 - z^7 * q^8 - z^3 * q^9 + (z^21 + z^2) * q^11 - z^4 * q^14 - z^2 * q^16 + z^20 * q^18 + (-z^19 + z^16) * q^22 - z^8 * q^23 - z^5 * q^25 + z^21 * q^28 + (z^18 - z^17) * q^29 + z^19 * q^32 + z^15 * q^36 + (-z^16 + z) * q^37 + (z^15 - z^10) * q^43 + (-z^14 + z^11) * q^44 - z^3 * q^46 + z^18 * q^49 - q^50 + (z^4 + z^3) * q^53 + z^16 * q^56 + (z^13 - z^12) * q^58 + z^12 * q^63 + z^14 * q^64 + (-z^15 - z^6) * q^67 + (z^9 + z) * q^71 + z^10 * q^72 + (-z^18 - z^11) * q^74 + (-z^11 + z^8) * q^77 + (z^19 - z^7) * q^79 + z^6 * q^81 + (z^10 - z^5) * q^86 + (-z^9 + z^6) * q^88 + z^20 * q^92 + z^13 * q^98 + (-z^5 + z^2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{4}+O(q^{10}) \) 20 * q + 2 * q^4	\( 20 q + 2 q^{4} + 2 q^{11} + 2 q^{14} - 2 q^{16} - 2 q^{18} - 2 q^{22} + 2 q^{23} + 2 q^{29} + 2 q^{37} - 2 q^{43} - 2 q^{44} + 2 q^{49} - 20 q^{50} - 2 q^{53} - 2 q^{56} + 2 q^{58} - 2 q^{63} + 2 q^{64} - 2 q^{67} + 2 q^{72} - 2 q^{74} - 2 q^{77} + 2 q^{81} + 2 q^{86} + 2 q^{88} - 2 q^{92} + 2 q^{99}+O(q^{100}) \) 20 * q + 2 * q^4 + 2 * q^11 + 2 * q^14 - 2 * q^16 - 2 * q^18 - 2 * q^22 + 2 * q^23 + 2 * q^29 + 2 * q^37 - 2 * q^43 - 2 * q^44 + 2 * q^49 - 20 * q^50 - 2 * q^53 - 2 * q^56 + 2 * q^58 - 2 * q^63 + 2 * q^64 - 2 * q^67 + 2 * q^72 - 2 * q^74 - 2 * q^77 + 2 * q^81 + 2 * q^86 + 2 * q^88 - 2 * q^92 + 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	2576.1.cq.b

Level	$2576$
Weight	$1$
Character orbit	2576.cq
Analytic conductor	$1.286$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{44}$
CM discriminant	-7
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.28559147254\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\Q(\zeta_{44})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{44}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{44} - \cdots)\)

\(n\)	\(645\)	\(1473\)	\(1569\)	\(2255\)
\(\chi(n)\)	\(\zeta_{44}^{11}\)	\(-1\)	\(\zeta_{44}^{16}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 13.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
368.w	even	44	1	inner
2576.cq	odd	44	1	inner

$p$	$F_p(T)$
$2$	\( T^{20} - T^{18} + \cdots + 1 \) T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$3$	\( T^{20} \) T^20
$5$	\( T^{20} \) T^20
$7$	\( T^{20} - T^{18} + \cdots + 1 \) T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$11$	\( T^{20} - 2 T^{19} + \cdots + 1 \) T^20 - 2T^19 + 2T^18 - 22T^17 + 40T^16 - 36T^15 + 179T^14 - 286T^13 + 214T^12 - 626T^11 + 824T^10 - 394T^9 + 713T^8 - 638T^7 - 51T^6 - 74T^5 + 250T^4 - 44T^3 + 50T^2 - 12*T + 1
$13$	\( T^{20} \) T^20
$17$	\( T^{20} \) T^20
$19$	\( T^{20} \) T^20
$23$	\( (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} \) (T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$29$	\( T^{20} - 2 T^{19} + \cdots + 1 \) 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
$31$	\( T^{20} \) T^20
$37$	\( T^{20} - 2 T^{19} + \cdots + 1 \) T^20 - 2T^19 + 2T^18 - 4T^16 + 8T^15 + 47T^14 - 132T^13 + 170T^12 - 76T^11 - 188T^10 + 530T^9 - 79T^8 + 66T^7 + 125T^6 - 382T^5 + 514T^4 - 330T^3 + 94T^2 - 12T + 1
$41$	\( T^{20} \) T^20
$43$	\( T^{20} + 2 T^{19} + \cdots + 1 \) T^20 + 2T^19 + 2T^18 - 4T^16 - 8T^15 - 8T^14 + 93T^12 - 122T^11 - 331T^10 - 420T^9 - 178T^8 + 484T^7 + 1324T^6 + 1702T^5 + 1218T^4 + 484T^3 + 105T^2 + 12*T + 1
$47$	\( T^{20} \) T^20
$53$	\( T^{20} + 2 T^{19} + \cdots + 1 \) T^20 + 2T^19 + 2T^18 - 4T^16 - 8T^15 + 47T^14 + 132T^13 + 170T^12 + 76T^11 - 188T^10 - 530T^9 - 79T^8 - 66T^7 + 125T^6 + 382T^5 + 514T^4 + 330T^3 + 94T^2 + 12T + 1
$59$	\( T^{20} \) T^20
$61$	\( T^{20} \) T^20
$67$	\( T^{20} + 2 T^{19} + \cdots + 1 \) T^20 + 2T^19 + 2T^18 - 4T^16 - 30T^15 - 52T^14 - 44T^13 + 93T^12 + 274T^11 + 505T^10 + 460T^9 - 90T^8 + 110T^7 + 400T^6 + 316T^5 + 338T^4 + 44T^3 + 17T^2 - 10T + 1
$71$	\( T^{20} - 4 T^{18} + \cdots + 1 \) T^20 - 4T^18 + 16T^16 - 9T^14 + 36T^12 + 120T^10 + 125T^8 - 335T^6 + 130T^4 + 8*T^2 + 1
$73$	\( T^{20} \) T^20
$79$	\( T^{20} + 55 T^{14} + \cdots + 121 \) T^20 + 55T^14 - 264T^10 + 605T^8 + 121T^6 + 1210T^4 + 484T^2 + 121
$83$	\( T^{20} \) T^20
$89$	\( T^{20} \) T^20
$97$	\( T^{20} \) T^20
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