LMFDB - Newform orbit 3577.1.p.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3577,1,Mod(656,3577)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3577.656"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3577, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 3577 = 7^{2} \cdot 73 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3577.p (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,1,0,-2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.78515555019$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	6.0.64827.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 $ x^6 - x^5 + 3x^4 + 5x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 511)
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.133432831.1
Artin image:	$C_3\times D_7$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{21} - \cdots)$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{2} - \beta_1) q^{2} + ( - \beta_{4} - \beta_1) q^{4} + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots + \beta_1) q^{5} + (\beta_{3} - \beta_{2}) q^{8} - \beta_{5} q^{9} + (\beta_{4} + \beta_1) q^{10}+ \cdots + ( - \beta_{5} - \beta_1 + 1) q^{94}+O(q^{100}) $ q + (b2 - b1) * q^2 + (-b4 - b1) * q^4 + (b5 + b4 + b3 - b2 + b1) * q^5 + (b3 - b2) * q^8 - b5 * q^9 + (b4 + b1) * q^10 + b3 * q^13 + (-b2 + b1) * q^16 + b1 * q^17 + b1 * q^18 + (b2 + 1) * q^20 + (-b4 - b3) * q^23 + (b5 + b4 - 1) * q^25 + (b5 - b2 + b1) * q^26 + b1 * q^31 + (-b5 + 1) * q^32 + (-b3 + b2 + 1) * q^34 + (-b3 + b2) * q^36 + (b5 + b4 + b3 - b2 + b1) * q^37 + (b5 + b2 - b1) * q^40 + (-b5 - b4 - b1 + 1) * q^45 + (-b5 - b1 + 1) * q^46 + (-b4 - b3) * q^47 - q^50 + (-b5 + 1) * q^52 + b4 * q^59 + (-b3 + b2 + 1) * q^62 + (b5 + b4 + b3) * q^65 + b1 * q^67 + (-b4 - b3 + 2b2 - 2b1) * q^68 + (-b3 + b2 - 1) * q^71 + (-b4 - b3 + b2 - b1) * q^72 + (b5 - 1) * q^73 + (b4 + b1) * q^74 + (b2 - b1) * q^79 + (-b4 - b1) * q^80 + (b5 - 1) * q^81 - b2 * q^83 + (b3 - b2) * q^85 + (b3 - b2) * q^90 - q^92 + (-b5 - b1 + 1) * q^94
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} - 2 q^{4} + q^{5} - 4 q^{8} - 3 q^{9} + 2 q^{10} - 2 q^{13} - q^{16} + q^{17} + q^{18} + 8 q^{20} + q^{23} - 2 q^{25} + 2 q^{26} + q^{31} + 3 q^{32} + 10 q^{34} + 4 q^{36} + q^{37} + 4 q^{40}+ \cdots + 2 q^{94}+O(q^{100}) $ 6 * q + q^2 - 2 * q^4 + q^5 - 4 * q^8 - 3 * q^9 + 2 * q^10 - 2 * q^13 - q^16 + q^17 + q^18 + 8 * q^20 + q^23 - 2 * q^25 + 2 * q^26 + q^31 + 3 * q^32 + 10 * q^34 + 4 * q^36 + q^37 + 4 * q^40 + q^45 + 2 * q^46 + q^47 - 6 * q^50 + 3 * q^52 + q^59 + 10 * q^62 + 2 * q^65 + q^67 + 3 * q^68 - 2 * q^71 + 2 * q^72 - 3 * q^73 + 2 * q^74 + q^79 - 2 * q^80 - 3 * q^81 - 2 * q^83 - 4 * q^85 - 4 * q^90 - 6 * q^92 + 2 * q^94

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 $ x^6 - x^5 + 3*x^4 + 5*x^2 - 2*x + 1 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13 $ (-v^5 + 3v^4 - 9v^3 + 5v^2 - 2v + 6) / 13
$\beta_{3}$	$=$	$ ( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13 $ (-3v^5 + 9v^4 - 14v^3 + 15v^2 - 6*v + 18) / 13
$\beta_{4}$	$=$	$ ( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13 $ (-4v^5 - v^4 - 10v^3 - 6v^2 - 34v - 2) / 13
$\beta_{5}$	$=$	$ ( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13 $ (-6v^5 + 5v^4 - 15v^3 - 9v^2 - 25*v + 10) / 13

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ -\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 $ -b5 + b4 + b3 - b2 + b1
$\nu^{3}$	$=$	$ \beta_{3} - 3\beta_{2} $ b3 - 3*b2
$\nu^{4}$	$=$	$ 2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2 $ 2b5 - 3b4 - 4*b1 - 2
$\nu^{5}$	$=$	$ \beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1 $ b5 - 4b4 - 4b3 + 9b2 - 9b1

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3577\mathbb{Z}\right)^\times$.

$n$	$589$	$3286$
$\chi(n)$	$-1$	$\beta_{5}$

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} $

656.1

−0.623490	−	1.07992i
0.222521	+	0.385418i
0.900969	+	1.56052i
−0.623490	+	1.07992i
0.222521	−	0.385418i
0.900969	−	1.56052i

−0.623490

1.07992i

−0.277479

−

0.480608i

0.222521

−

0.385418i

−0.554958

−0.500000

0.866025i

0.277479

0.480608i

656.2

0.222521

−

0.385418i

0.400969

0.694498i

0.900969

−

1.56052i

0.801938

−0.500000

0.866025i

−0.400969

−

0.694498i

656.3

0.900969

−

1.56052i

−1.12349

−

1.94594i

−0.623490

1.07992i

−2.24698

−0.500000

0.866025i

1.12349

1.94594i

2481.1

−0.623490

−

1.07992i

−0.277479

0.480608i

0.222521

0.385418i

−0.554958

−0.500000

−

0.866025i

0.277479

−

0.480608i

2481.2

0.222521

0.385418i

0.400969

−

0.694498i

0.900969

1.56052i

0.801938

−0.500000

−

0.866025i

−0.400969

0.694498i

2481.3

0.900969

1.56052i

−1.12349

1.94594i

−0.623490

−

1.07992i

−2.24698

−0.500000

−

0.866025i

1.12349

−

1.94594i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
511.c	odd	2	1	CM by $\Q(\sqrt{-511}) $
7.c	even	3	1	inner
511.p	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3577.1.p.b		6
7.b	odd	2	1		3577.1.p.a		6
7.c	even	3	1		511.1.c.a	✓	3
7.c	even	3	1	inner	3577.1.p.b		6
7.d	odd	6	1		511.1.c.b	yes	3
7.d	odd	6	1		3577.1.p.a		6
73.b	even	2	1		3577.1.p.a		6
511.c	odd	2	1	CM	3577.1.p.b		6
511.n	even	6	1		511.1.c.b	yes	3
511.n	even	6	1		3577.1.p.a		6
511.p	odd	6	1		511.1.c.a	✓	3
511.p	odd	6	1	inner	3577.1.p.b		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
511.1.c.a	✓	3	7.c	even	3	1
511.1.c.a	✓	3	511.p	odd	6	1
511.1.c.b	yes	3	7.d	odd	6	1
511.1.c.b	yes	3	511.n	even	6	1
3577.1.p.a		6	7.b	odd	2	1
3577.1.p.a		6	7.d	odd	6	1
3577.1.p.a		6	73.b	even	2	1
3577.1.p.a		6	511.n	even	6	1
3577.1.p.b		6	1.a	even	1	1	trivial
3577.1.p.b		6	7.c	even	3	1	inner
3577.1.p.b		6	511.c	odd	2	1	CM
3577.1.p.b		6	511.p	odd	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{6} - T_{5}^{5} + 3T_{5}^{4} + 5T_{5}^{2} - 2T_{5} + 1 $ T5^6 - T5^5 + 3*T5^4 + 5*T5^2 - 2*T5 + 1 acting on $S_{1}^{\mathrm{new}}(3577, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} - T^{5} + 3 T^{4} + \cdots + 1 $ T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$3$	$ T^{6} $ T^6
$5$	$ T^{6} - T^{5} + 3 T^{4} + \cdots + 1 $ T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$7$	$ T^{6} $ T^6
$11$	$ T^{6} $ T^6
$13$	$ (T^{3} + T^{2} - 2 T - 1)^{2} $ (T^3 + T^2 - 2*T - 1)^2
$17$	$ T^{6} - T^{5} + 3 T^{4} + \cdots + 1 $ T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$19$	$ T^{6} $ T^6
$23$	$ T^{6} - T^{5} + 3 T^{4} + \cdots + 1 $ T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$29$	$ T^{6} $ T^6
$31$	$ T^{6} - T^{5} + 3 T^{4} + \cdots + 1 $ T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$37$	$ T^{6} - T^{5} + 3 T^{4} + \cdots + 1 $ T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$41$	$ T^{6} $ T^6
$43$	$ T^{6} $ T^6
$47$	$ T^{6} - T^{5} + 3 T^{4} + \cdots + 1 $ T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$53$	$ T^{6} $ T^6
$59$	$ T^{6} - T^{5} + 3 T^{4} + \cdots + 1 $ T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$61$	$ T^{6} $ T^6
$67$	$ T^{6} - T^{5} + 3 T^{4} + \cdots + 1 $ T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$71$	$ (T^{3} + T^{2} - 2 T - 1)^{2} $ (T^3 + T^2 - 2*T - 1)^2
$73$	$ (T^{2} + T + 1)^{3} $ (T^2 + T + 1)^3
$79$	$ T^{6} - T^{5} + 3 T^{4} + \cdots + 1 $ T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$83$	$ (T^{3} + T^{2} - 2 T - 1)^{2} $ (T^3 + T^2 - 2*T - 1)^2
$89$	$ T^{6} $ T^6
$97$	$ T^{6} $ T^6
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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3577 = 7^{2} \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3577.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{2} - \beta_1) q^{2} + ( - \beta_{4} - \beta_1) q^{4} + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots + \beta_1) q^{5} + (\beta_{3} - \beta_{2}) q^{8} - \beta_{5} q^{9} + (\beta_{4} + \beta_1) q^{10}+ \cdots + ( - \beta_{5} - \beta_1 + 1) q^{94}+O(q^{100}) \) q + (b2 - b1) * q^2 + (-b4 - b1) * q^4 + (b5 + b4 + b3 - b2 + b1) * q^5 + (b3 - b2) * q^8 - b5 * q^9 + (b4 + b1) * q^10 + b3 * q^13 + (-b2 + b1) * q^16 + b1 * q^17 + b1 * q^18 + (b2 + 1) * q^20 + (-b4 - b3) * q^23 + (b5 + b4 - 1) * q^25 + (b5 - b2 + b1) * q^26 + b1 * q^31 + (-b5 + 1) * q^32 + (-b3 + b2 + 1) * q^34 + (-b3 + b2) * q^36 + (b5 + b4 + b3 - b2 + b1) * q^37 + (b5 + b2 - b1) * q^40 + (-b5 - b4 - b1 + 1) * q^45 + (-b5 - b1 + 1) * q^46 + (-b4 - b3) * q^47 - q^50 + (-b5 + 1) * q^52 + b4 * q^59 + (-b3 + b2 + 1) * q^62 + (b5 + b4 + b3) * q^65 + b1 * q^67 + (-b4 - b3 + 2b2 - 2b1) * q^68 + (-b3 + b2 - 1) * q^71 + (-b4 - b3 + b2 - b1) * q^72 + (b5 - 1) * q^73 + (b4 + b1) * q^74 + (b2 - b1) * q^79 + (-b4 - b1) * q^80 + (b5 - 1) * q^81 - b2 * q^83 + (b3 - b2) * q^85 + (b3 - b2) * q^90 - q^92 + (-b5 - b1 + 1) * q^94
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} - 2 q^{4} + q^{5} - 4 q^{8} - 3 q^{9} + 2 q^{10} - 2 q^{13} - q^{16} + q^{17} + q^{18} + 8 q^{20} + q^{23} - 2 q^{25} + 2 q^{26} + q^{31} + 3 q^{32} + 10 q^{34} + 4 q^{36} + q^{37} + 4 q^{40}+ \cdots + 2 q^{94}+O(q^{100}) \) 6 * q + q^2 - 2 * q^4 + q^5 - 4 * q^8 - 3 * q^9 + 2 * q^10 - 2 * q^13 - q^16 + q^17 + q^18 + 8 * q^20 + q^23 - 2 * q^25 + 2 * q^26 + q^31 + 3 * q^32 + 10 * q^34 + 4 * q^36 + q^37 + 4 * q^40 + q^45 + 2 * q^46 + q^47 - 6 * q^50 + 3 * q^52 + q^59 + 10 * q^62 + 2 * q^65 + q^67 + 3 * q^68 - 2 * q^71 + 2 * q^72 - 3 * q^73 + 2 * q^74 + q^79 - 2 * q^80 - 3 * q^81 - 2 * q^83 - 4 * q^85 - 4 * q^90 - 6 * q^92 + 2 * q^94

Introduction

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Varieties

Fields

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Groups

Database

Properties

Related objects

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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	3577.1.p.b

Level	$3577$
Weight	$1$
Character orbit	3577.p
Analytic conductor	$1.785$
Analytic rank	$0$
Dimension	$6$
Projective image	$D_{7}$
CM discriminant	-511
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.78515555019\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{6})\)
Coefficient field:	6.0.64827.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) x^6 - x^5 + 3x^4 + 5x^2 - 2*x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 511)
Projective image:	\(D_{7}\)
Projective field:	Galois closure of 7.1.133432831.1
Artin image:	$C_3\times D_7$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{21} - \cdots)\)

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13 \) (-v^5 + 3v^4 - 9v^3 + 5v^2 - 2v + 6) / 13
\(\beta_{3}\)	\(=\)	\( ( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13 \) (-3v^5 + 9v^4 - 14v^3 + 15v^2 - 6*v + 18) / 13
\(\beta_{4}\)	\(=\)	\( ( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13 \) (-4v^5 - v^4 - 10v^3 - 6v^2 - 34v - 2) / 13
\(\beta_{5}\)	\(=\)	\( ( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13 \) (-6v^5 + 5v^4 - 15v^3 - 9v^2 - 25*v + 10) / 13

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( -\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) -b5 + b4 + b3 - b2 + b1
\(\nu^{3}\)	\(=\)	\( \beta_{3} - 3\beta_{2} \) b3 - 3*b2
\(\nu^{4}\)	\(=\)	\( 2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2 \) 2b5 - 3b4 - 4*b1 - 2
\(\nu^{5}\)	\(=\)	\( \beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1 \) b5 - 4b4 - 4b3 + 9b2 - 9b1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 656.3
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$3$	\( T^{6} \) T^6
$5$	\( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$7$	\( T^{6} \) T^6
$11$	\( T^{6} \) T^6
$13$	\( (T^{3} + T^{2} - 2 T - 1)^{2} \) (T^3 + T^2 - 2*T - 1)^2
$17$	\( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$19$	\( T^{6} \) T^6
$23$	\( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$29$	\( T^{6} \) T^6
$31$	\( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$37$	\( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$41$	\( T^{6} \) T^6
$43$	\( T^{6} \) T^6
$47$	\( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$53$	\( T^{6} \) T^6
$59$	\( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$61$	\( T^{6} \) T^6
$67$	\( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$71$	\( (T^{3} + T^{2} - 2 T - 1)^{2} \) (T^3 + T^2 - 2*T - 1)^2
$73$	\( (T^{2} + T + 1)^{3} \) (T^2 + T + 1)^3
$79$	\( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$83$	\( (T^{3} + T^{2} - 2 T - 1)^{2} \) (T^3 + T^2 - 2*T - 1)^2
$89$	\( T^{6} \) T^6
$97$	\( T^{6} \) T^6
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\(n\)	\(589\)	\(3286\)
\(\chi(n)\)	\(-1\)	\(\beta_{5}\)