LMFDB - Newform orbit 2664.2.r.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2664,2,Mod(433,2664)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2664, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2664.433"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2664 = 2^{3} \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2664.r (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,-3,0,-3,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$21.2721470985$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.50898483.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 8x^{4} - 10x^{3} + 64x^{2} - 40x + 25 $ x^6 + 8x^4 - 10x^3 + 64x^2 - 40x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 888)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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_{3} + \beta_1) q^{5} - \beta_{3} q^{7} + ( - \beta_{4} + \beta_{2}) q^{11} + (2 \beta_{3} - \beta_1) q^{13} + (\beta_{5} + 3 \beta_{3} - 3) q^{17} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{19}+ \cdots + ( - \beta_{4} + 5 \beta_{2} + 1) q^{97}+O(q^{100}) $ q + (-b3 + b1) * q^5 - b3 * q^7 + (-b4 + b2) * q^11 + (2b3 - b1) q^13 + (b5 + 3b3 - 3) q^17 + (-b5 + b4 + 2b3 - b1) q^19 + (2b4 + b2 + 1) q^23 + (b5 + b3 - 3b2 - 3b1 - 1) * q^25 + (-b4 + 2b2 - 5) q^29 + b4 * q^31 + (b3 - b2 - b1 - 1) * q^35 + (-b5 + b4 + 3b3 + 2b2 - 1) * q^37 + (3b5 - 3b4 - b1) * q^41 - b2 * q^43 + (b4 - 2b2 + 3) q^47 + (-6b3 + 6) q^49 + (b5 - 5b3 + 2b2 + 2b1 + 5) q^53 + (-b5 + b4 - 5b3 + 4b1) * q^55 + (b5 + 2b3 - 3b2 - 3b1 - 2) q^59 + (2b5 - 2b4 - 2b3 - 2b1) * q^61 + (-b5 - 7b3 + 4b2 + 4b1 + 7) q^65 + (b5 - b4 - b3 - 3b1) q^67 + (-6b3 - 4b1) * q^71 + (3b4 - 2b2) * q^73 + (-b5 + b4 + b1) * q^77 + (3b5 - 3b4 + b3 + b1) * q^79 + (2b5 - 4b3 + 4b2 + 4b1 + 4) * q^83 + (5b2 + 3) q^85 + (2b5 + 2b2 + 2b1) q^89 + (-2b3 + b2 + b1 + 2) q^91 + (-b5 - 7b3 + 2b2 + 2b1 + 7) q^95 + (-b4 + 5b2 + 1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{5} - 3 q^{7} + 2 q^{11} + 6 q^{13} - 10 q^{17} + 5 q^{19} + 2 q^{23} - 4 q^{25} - 28 q^{29} - 2 q^{31} - 3 q^{35} + 2 q^{37} + 3 q^{41} + 16 q^{47} + 18 q^{49} + 14 q^{53} - 16 q^{55} - 7 q^{59}+ \cdots + 8 q^{97}+O(q^{100}) $ 6 * q - 3 * q^5 - 3 * q^7 + 2 * q^11 + 6 * q^13 - 10 * q^17 + 5 * q^19 + 2 * q^23 - 4 * q^25 - 28 * q^29 - 2 * q^31 - 3 * q^35 + 2 * q^37 + 3 * q^41 + 16 * q^47 + 18 * q^49 + 14 * q^53 - 16 * q^55 - 7 * q^59 - 4 * q^61 + 22 * q^65 - 2 * q^67 - 18 * q^71 - 6 * q^73 - q^77 + 6 * q^79 + 10 * q^83 + 18 * q^85 - 2 * q^89 + 6 * q^91 + 22 * q^95 + 8 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} + 8x^{4} - 10x^{3} + 64x^{2} - 40x + 25 $ x^6 + 8*x^4 - 10*x^3 + 64*x^2 - 40*x + 25 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{3} - 5 ) / 8 $ (v^3 - 5) / 8
$\beta_{3}$	$=$	$ ( \nu^{5} + 8\nu^{3} - 5\nu^{2} + 64\nu ) / 40 $ (v^5 + 8v^3 - 5v^2 + 64*v) / 40
$\beta_{4}$	$=$	$ ( \nu^{4} + \nu^{3} + 8\nu^{2} - 5\nu + 35 ) / 8 $ (v^4 + v^3 + 8v^2 - 5v + 35) / 8
$\beta_{5}$	$=$	$ ( -\nu^{5} - 7\nu^{3} + 13\nu^{2} - 56\nu + 35 ) / 8 $ (-v^5 - 7v^3 + 13v^2 - 56*v + 35) / 8

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{5} + 5\beta_{3} - \beta_{2} - \beta _1 - 5 $ b5 + 5*b3 - b2 - b1 - 5
$\nu^{3}$	$=$	$ 8\beta_{2} + 5 $ 8*b2 + 5
$\nu^{4}$	$=$	$ -8\beta_{5} + 8\beta_{4} - 40\beta_{3} + 13\beta_1 $ -8b5 + 8b4 - 40b3 + 13b1
$\nu^{5}$	$=$	$ 5\beta_{5} + 65\beta_{3} - 69\beta_{2} - 69\beta _1 - 65 $ 5b5 + 65b3 - 69b2 - 69b1 - 65

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2664\mathbb{Z}\right)^\times$.

$n$	$1297$	$1333$	$1999$	$2369$
$\chi(n)$	$-\beta_{3}$	$1$	$1$	$1$

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} $

433.1

−1.55022	+	2.68505i
0.330560	−	0.572547i
1.21966	−	2.11251i
−1.55022	−	2.68505i
0.330560	+	0.572547i
1.21966	+	2.11251i

−2.05022

3.55108i

−0.500000

0.866025i

433.2

−0.169440

0.293478i

−0.500000

0.866025i

433.3

0.719656

−

1.24648i

−0.500000

0.866025i

1009.1

−2.05022

−

3.55108i

−0.500000

−

0.866025i

1009.2

−0.169440

−

0.293478i

−0.500000

−

0.866025i

1009.3

0.719656

1.24648i

−0.500000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
37.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2664.2.r.h		6
3.b	odd	2	1		888.2.q.h	✓	6
12.b	even	2	1		1776.2.q.m		6
37.c	even	3	1	inner	2664.2.r.h		6
111.i	odd	6	1		888.2.q.h	✓	6
444.t	even	6	1		1776.2.q.m		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
888.2.q.h	✓	6	3.b	odd	2	1
888.2.q.h	✓	6	111.i	odd	6	1
1776.2.q.m		6	12.b	even	2	1
1776.2.q.m		6	444.t	even	6	1
2664.2.r.h		6	1.a	even	1	1	trivial
2664.2.r.h		6	37.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2664, [\chi])$:

$ T_{5}^{6} + 3T_{5}^{5} + 14T_{5}^{4} - 11T_{5}^{3} + 31T_{5}^{2} + 10T_{5} + 4 $

T5^6 + 3*T5^5 + 14*T5^4 - 11*T5^3 + 31*T5^2 + 10*T5 + 4

$ T_{11}^{3} - T_{11}^{2} - 21T_{11} + 20 $

T11^3 - T11^2 - 21*T11 + 20

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} + 3 T^{5} + \cdots + 4 $ T^6 + 3T^5 + 14T^4 - 11T^3 + 31T^2 + 10*T + 4
$7$	$ (T^{2} + T + 1)^{3} $ (T^2 + T + 1)^3
$11$	$ (T^{3} - T^{2} - 21 T + 20)^{2} $ (T^3 - T^2 - 21*T + 20)^2
$13$	$ T^{6} - 6 T^{5} + \cdots + 9 $ T^6 - 6T^5 + 32T^4 - 30T^3 + 34T^2 + 12*T + 9
$17$	$ T^{6} + 10 T^{5} + \cdots + 676 $ T^6 + 10T^5 + 81T^4 + 242T^3 + 621T^2 - 494*T + 676
$19$	$ T^{6} - 5 T^{5} + \cdots + 5184 $ T^6 - 5T^5 + 40T^4 - 69T^3 + 585T^2 - 1080*T + 5184
$23$	$ (T^{3} - T^{2} - 67 T + 72)^{2} $ (T^3 - T^2 - 67*T + 72)^2
$29$	$ (T^{3} + 14 T^{2} + \cdots - 180)^{2} $ (T^3 + 14T^2 + 21T - 180)^2
$31$	$ (T^{3} + T^{2} - 14 T - 20)^{2} $ (T^3 + T^2 - 14*T - 20)^2
$37$	$ T^{6} - 2 T^{5} + \cdots + 50653 $ T^6 - 2T^5 - 34T^4 + 178T^3 - 1258T^2 - 2738*T + 50653
$41$	$ T^{6} - 3 T^{5} + \cdots + 532900 $ T^6 - 3T^5 + 140T^4 - 1067T^3 + 19351T^2 - 95630*T + 532900
$43$	$ (T^{3} - 8 T + 5)^{2} $ (T^3 - 8*T + 5)^2
$47$	$ (T^{3} - 8 T^{2} + \cdots + 174)^{2} $ (T^3 - 8T^2 - 23T + 174)^2
$53$	$ T^{6} - 14 T^{5} + \cdots + 57600 $ T^6 - 14T^5 + 179T^4 - 718T^3 + 3649T^2 + 4080*T + 57600
$59$	$ T^{6} + 7 T^{5} + \cdots + 9216 $ T^6 + 7T^5 + 116T^4 - 661T^3 + 3817T^2 - 6432*T + 9216
$61$	$ T^{6} + 4 T^{5} + \cdots + 64 $ T^6 + 4T^5 + 96T^4 - 304T^3 + 6432T^2 + 640*T + 64
$67$	$ T^{6} + 2 T^{5} + \cdots + 110889 $ T^6 + 2T^5 + 86T^4 + 502T^3 + 7390T^2 + 27306*T + 110889
$71$	$ T^{6} + 18 T^{5} + \cdots + 760384 $ T^6 + 18T^5 + 344T^4 + 1384T^3 + 16096T^2 + 17440*T + 760384
$73$	$ (T^{3} + 3 T^{2} + \cdots - 740)^{2} $ (T^3 + 3T^2 - 152T - 740)^2
$79$	$ T^{6} - 6 T^{5} + \cdots + 110889 $ T^6 - 6T^5 + 164T^4 + 102T^3 + 18382T^2 - 42624*T + 110889
$83$	$ T^{6} - 10 T^{5} + \cdots + 2359296 $ T^6 - 10T^5 + 260T^4 - 1472T^3 + 40960T^2 - 245760*T + 2359296
$89$	$ T^{6} + 2 T^{5} + \cdots + 57600 $ T^6 + 2T^5 + 96T^4 - 664T^3 + 7984T^2 - 22080*T + 57600
$97$	$ (T^{3} - 4 T^{2} + \cdots - 873)^{2} $ (T^3 - 4T^2 - 204T - 873)^2
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Belyi maps

Level:	\( N \)	\(=\)	\( 2664 = 2^{3} \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2664.r (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{3} + \beta_1) q^{5} - \beta_{3} q^{7} + ( - \beta_{4} + \beta_{2}) q^{11} + (2 \beta_{3} - \beta_1) q^{13} + (\beta_{5} + 3 \beta_{3} - 3) q^{17} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{19}+ \cdots + ( - \beta_{4} + 5 \beta_{2} + 1) q^{97}+O(q^{100}) \) q + (-b3 + b1) * q^5 - b3 * q^7 + (-b4 + b2) * q^11 + (2b3 - b1) q^13 + (b5 + 3b3 - 3) q^17 + (-b5 + b4 + 2b3 - b1) q^19 + (2b4 + b2 + 1) q^23 + (b5 + b3 - 3b2 - 3b1 - 1) * q^25 + (-b4 + 2b2 - 5) q^29 + b4 * q^31 + (b3 - b2 - b1 - 1) * q^35 + (-b5 + b4 + 3b3 + 2b2 - 1) * q^37 + (3b5 - 3b4 - b1) * q^41 - b2 * q^43 + (b4 - 2b2 + 3) q^47 + (-6b3 + 6) q^49 + (b5 - 5b3 + 2b2 + 2b1 + 5) q^53 + (-b5 + b4 - 5b3 + 4b1) * q^55 + (b5 + 2b3 - 3b2 - 3b1 - 2) q^59 + (2b5 - 2b4 - 2b3 - 2b1) * q^61 + (-b5 - 7b3 + 4b2 + 4b1 + 7) q^65 + (b5 - b4 - b3 - 3b1) q^67 + (-6b3 - 4b1) * q^71 + (3b4 - 2b2) * q^73 + (-b5 + b4 + b1) * q^77 + (3b5 - 3b4 + b3 + b1) * q^79 + (2b5 - 4b3 + 4b2 + 4b1 + 4) * q^83 + (5b2 + 3) q^85 + (2b5 + 2b2 + 2b1) q^89 + (-2b3 + b2 + b1 + 2) q^91 + (-b5 - 7b3 + 2b2 + 2b1 + 7) q^95 + (-b4 + 5b2 + 1) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{5} - 3 q^{7} + 2 q^{11} + 6 q^{13} - 10 q^{17} + 5 q^{19} + 2 q^{23} - 4 q^{25} - 28 q^{29} - 2 q^{31} - 3 q^{35} + 2 q^{37} + 3 q^{41} + 16 q^{47} + 18 q^{49} + 14 q^{53} - 16 q^{55} - 7 q^{59}+ \cdots + 8 q^{97}+O(q^{100}) \) 6 * q - 3 * q^5 - 3 * q^7 + 2 * q^11 + 6 * q^13 - 10 * q^17 + 5 * q^19 + 2 * q^23 - 4 * q^25 - 28 * q^29 - 2 * q^31 - 3 * q^35 + 2 * q^37 + 3 * q^41 + 16 * q^47 + 18 * q^49 + 14 * q^53 - 16 * q^55 - 7 * q^59 - 4 * q^61 + 22 * q^65 - 2 * q^67 - 18 * q^71 - 6 * q^73 - q^77 + 6 * q^79 + 10 * q^83 + 18 * q^85 - 2 * q^89 + 6 * q^91 + 22 * q^95 + 8 * q^97

Introduction

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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	2664.2.r.h

Level	$2664$
Weight	$2$
Character orbit	2664.r
Analytic conductor	$21.272$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(21.2721470985\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	6.0.50898483.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} + 8x^{4} - 10x^{3} + 64x^{2} - 40x + 25 \) x^6 + 8x^4 - 10x^3 + 64x^2 - 40x + 25
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 888)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} - 5 ) / 8 \) (v^3 - 5) / 8
\(\beta_{3}\)	\(=\)	\( ( \nu^{5} + 8\nu^{3} - 5\nu^{2} + 64\nu ) / 40 \) (v^5 + 8v^3 - 5v^2 + 64*v) / 40
\(\beta_{4}\)	\(=\)	\( ( \nu^{4} + \nu^{3} + 8\nu^{2} - 5\nu + 35 ) / 8 \) (v^4 + v^3 + 8v^2 - 5v + 35) / 8
\(\beta_{5}\)	\(=\)	\( ( -\nu^{5} - 7\nu^{3} + 13\nu^{2} - 56\nu + 35 ) / 8 \) (-v^5 - 7v^3 + 13v^2 - 56*v + 35) / 8

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{5} + 5\beta_{3} - \beta_{2} - \beta _1 - 5 \) b5 + 5*b3 - b2 - b1 - 5
\(\nu^{3}\)	\(=\)	\( 8\beta_{2} + 5 \) 8*b2 + 5
\(\nu^{4}\)	\(=\)	\( -8\beta_{5} + 8\beta_{4} - 40\beta_{3} + 13\beta_1 \) -8b5 + 8b4 - 40b3 + 13b1
\(\nu^{5}\)	\(=\)	\( 5\beta_{5} + 65\beta_{3} - 69\beta_{2} - 69\beta _1 - 65 \) 5b5 + 65b3 - 69b2 - 69b1 - 65

\(n\)	\(1297\)	\(1333\)	\(1999\)	\(2369\)
\(\chi(n)\)	\(-\beta_{3}\)	\(1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 433.3
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} \) T^6
$3$	\( T^{6} \) T^6
$5$	\( T^{6} + 3 T^{5} + \cdots + 4 \) T^6 + 3T^5 + 14T^4 - 11T^3 + 31T^2 + 10*T + 4
$7$	\( (T^{2} + T + 1)^{3} \) (T^2 + T + 1)^3
$11$	\( (T^{3} - T^{2} - 21 T + 20)^{2} \) (T^3 - T^2 - 21*T + 20)^2
$13$	\( T^{6} - 6 T^{5} + \cdots + 9 \) T^6 - 6T^5 + 32T^4 - 30T^3 + 34T^2 + 12*T + 9
$17$	\( T^{6} + 10 T^{5} + \cdots + 676 \) T^6 + 10T^5 + 81T^4 + 242T^3 + 621T^2 - 494*T + 676
$19$	\( T^{6} - 5 T^{5} + \cdots + 5184 \) T^6 - 5T^5 + 40T^4 - 69T^3 + 585T^2 - 1080*T + 5184
$23$	\( (T^{3} - T^{2} - 67 T + 72)^{2} \) (T^3 - T^2 - 67*T + 72)^2
$29$	\( (T^{3} + 14 T^{2} + \cdots - 180)^{2} \) (T^3 + 14T^2 + 21T - 180)^2
$31$	\( (T^{3} + T^{2} - 14 T - 20)^{2} \) (T^3 + T^2 - 14*T - 20)^2
$37$	\( T^{6} - 2 T^{5} + \cdots + 50653 \) T^6 - 2T^5 - 34T^4 + 178T^3 - 1258T^2 - 2738*T + 50653
$41$	\( T^{6} - 3 T^{5} + \cdots + 532900 \) T^6 - 3T^5 + 140T^4 - 1067T^3 + 19351T^2 - 95630*T + 532900
$43$	\( (T^{3} - 8 T + 5)^{2} \) (T^3 - 8*T + 5)^2
$47$	\( (T^{3} - 8 T^{2} + \cdots + 174)^{2} \) (T^3 - 8T^2 - 23T + 174)^2
$53$	\( T^{6} - 14 T^{5} + \cdots + 57600 \) T^6 - 14T^5 + 179T^4 - 718T^3 + 3649T^2 + 4080*T + 57600
$59$	\( T^{6} + 7 T^{5} + \cdots + 9216 \) T^6 + 7T^5 + 116T^4 - 661T^3 + 3817T^2 - 6432*T + 9216
$61$	\( T^{6} + 4 T^{5} + \cdots + 64 \) T^6 + 4T^5 + 96T^4 - 304T^3 + 6432T^2 + 640*T + 64
$67$	\( T^{6} + 2 T^{5} + \cdots + 110889 \) T^6 + 2T^5 + 86T^4 + 502T^3 + 7390T^2 + 27306*T + 110889
$71$	\( T^{6} + 18 T^{5} + \cdots + 760384 \) T^6 + 18T^5 + 344T^4 + 1384T^3 + 16096T^2 + 17440*T + 760384
$73$	\( (T^{3} + 3 T^{2} + \cdots - 740)^{2} \) (T^3 + 3T^2 - 152T - 740)^2
$79$	\( T^{6} - 6 T^{5} + \cdots + 110889 \) T^6 - 6T^5 + 164T^4 + 102T^3 + 18382T^2 - 42624*T + 110889
$83$	\( T^{6} - 10 T^{5} + \cdots + 2359296 \) T^6 - 10T^5 + 260T^4 - 1472T^3 + 40960T^2 - 245760*T + 2359296
$89$	\( T^{6} + 2 T^{5} + \cdots + 57600 \) T^6 + 2T^5 + 96T^4 - 664T^3 + 7984T^2 - 22080*T + 57600
$97$	\( (T^{3} - 4 T^{2} + \cdots - 873)^{2} \) (T^3 - 4T^2 - 204T - 873)^2
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