LMFDB - Newform orbit 1728.2.r.d

Newspace parameters

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1728.r (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$13.7981494693$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.12960000.1
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{4} q^{5} + (\beta_{7} - \beta_{5}) q^{7}+O(q^{10}) $ q - b4 * q^5 + (b7 - b5) * q^7	$ q - \beta_{4} q^{5} + (\beta_{7} - \beta_{5}) q^{7} - 3 \beta_1 q^{11} - \beta_{4} q^{13} - 2 \beta_{3} q^{19} + \beta_{7} q^{23} + 10 \beta_{2} q^{25} - \beta_{6} q^{29} - \beta_{7} q^{31} + 15 \beta_{3} q^{35} + (2 \beta_{6} + 2 \beta_{4}) q^{37} + ( - 9 \beta_{2} + 9) q^{41} - 7 \beta_1 q^{43} + ( - \beta_{7} + \beta_{5}) q^{47} + (8 \beta_{2} - 8) q^{49} + (2 \beta_{6} + 2 \beta_{4}) q^{53} + 3 \beta_{5} q^{55} + ( - 3 \beta_{3} - 3 \beta_1) q^{59} - 3 \beta_{6} q^{61} + 15 \beta_{2} q^{65} + ( - 11 \beta_{3} - 11 \beta_1) q^{67} - 2 \beta_{5} q^{71} + 8 q^{73} + 3 \beta_{4} q^{77} + ( - \beta_{7} + \beta_{5}) q^{79} + 3 \beta_1 q^{83} + 12 q^{89} + 15 \beta_{3} q^{91} - 2 \beta_{7} q^{95} - \beta_{2} q^{97}+O(q^{100}) $ q - b4 * q^5 + (b7 - b5) * q^7 - 3b1 q^11 - b4 * q^13 - 2b3 q^19 + b7 * q^23 + 10b2 q^25 - b6 * q^29 - b7 * q^31 + 15b3 q^35 + (2b6 + 2b4) * q^37 + (-9b2 + 9) q^41 - 7b1 q^43 + (-b7 + b5) * q^47 + (8b2 - 8) q^49 + (2b6 + 2b4) * q^53 + 3b5 q^55 + (-3b3 - 3b1) * q^59 - 3b6 q^61 + 15b2 q^65 + (-11b3 - 11b1) * q^67 - 2b5 q^71 + 8 * q^73 + 3b4 q^77 + (-b7 + b5) * q^79 + 3b1 q^83 + 12 * q^89 + 15b3 q^91 - 2b7 q^95 - b2 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 40 q^{25} + 36 q^{41} - 32 q^{49} + 60 q^{65} + 64 q^{73} + 96 q^{89} - 4 q^{97}+O(q^{100}) $ 8 * q + 40 * q^25 + 36 * q^41 - 32 * q^49 + 60 * q^65 + 64 * q^73 + 96 * q^89 - 4 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3*x^6 + 8*x^4 - 3*x^2 + 1 :

$\beta_{1}$	$=$	$ ( \nu^{7} + 13\nu ) / 8 $ (v^7 + 13*v) / 8
$\beta_{2}$	$=$	$ ( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 9 ) / 8 $ (-3v^6 + 8v^4 - 24*v^2 + 9) / 8
$\beta_{3}$	$=$	$ ( -3\nu^{7} + 8\nu^{5} - 20\nu^{3} + \nu ) / 4 $ (-3v^7 + 8v^5 - 20*v^3 + v) / 4
$\beta_{4}$	$=$	$ ( -5\nu^{6} + 24\nu^{4} - 56\nu^{2} + 39 ) / 8 $ (-5v^6 + 24v^4 - 56*v^2 + 39) / 8
$\beta_{5}$	$=$	$ ( -5\nu^{7} + 16\nu^{5} - 44\nu^{3} + 31\nu ) / 4 $ (-5v^7 + 16v^5 - 44v^3 + 31v) / 4
$\beta_{6}$	$=$	$ ( -11\nu^{6} + 24\nu^{4} - 56\nu^{2} - 15 ) / 8 $ (-11v^6 + 24v^4 - 56*v^2 - 15) / 8
$\beta_{7}$	$=$	$ ( 13\nu^{7} - 32\nu^{5} + 88\nu^{3} + 25\nu ) / 8 $ (13v^7 - 32v^5 + 88v^3 + 25v) / 8

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

$\nu$	$=$	$ ( \beta_{7} + \beta_{5} - 3\beta_1 ) / 6 $ (b7 + b5 - 3*b1) / 6
$\nu^{2}$	$=$	$ ( 2\beta_{6} + \beta_{4} - 9\beta_{2} + 9 ) / 6 $ (2b6 + b4 - 9b2 + 9) / 6
$\nu^{3}$	$=$	$ ( 2\beta_{7} - \beta_{5} + 6\beta_{3} ) / 3 $ (2b7 - b5 + 6b3) / 3
$\nu^{4}$	$=$	$ ( \beta_{6} + 2\beta_{4} - 7\beta_{2} ) / 2 $ (b6 + 2b4 - 7b2) / 2
$\nu^{5}$	$=$	$ ( 5\beta_{7} - 10\beta_{5} + 33\beta_{3} + 33\beta_1 ) / 6 $ (5b7 - 10b5 + 33b3 + 33b1) / 6
$\nu^{6}$	$=$	$ ( -4\beta_{6} + 4\beta_{4} - 27 ) / 3 $ (-4b6 + 4b4 - 27) / 3
$\nu^{7}$	$=$	$ ( -13\beta_{7} - 13\beta_{5} + 87\beta_1 ) / 6 $ (-13b7 - 13b5 + 87*b1) / 6

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$1$	$-1 + \beta_{2}$

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.

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

289.1

0.535233	+	0.309017i
−0.535233	−	0.309017i
1.40126	+	0.809017i
−1.40126	−	0.809017i
0.535233	−	0.309017i
−0.535233	+	0.309017i
1.40126	−	0.809017i
−1.40126	+	0.809017i

−3.35410

1.93649i

−1.93649

3.35410i

289.2

−3.35410

1.93649i

1.93649

−

3.35410i

289.3

3.35410

−

1.93649i

−1.93649

3.35410i

289.4

3.35410

−

1.93649i

1.93649

−

3.35410i

1441.1

−3.35410

−

1.93649i

−1.93649

−

3.35410i

1441.2

−3.35410

−

1.93649i

1.93649

3.35410i

1441.3

3.35410

1.93649i

−1.93649

−

3.35410i

1441.4

3.35410

1.93649i

1.93649

3.35410i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
9.c	even	3	1	inner
36.f	odd	6	1	inner
72.n	even	6	1	inner
72.p	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1728.2.r.d		8
3.b	odd	2	1		576.2.r.d	✓	8
4.b	odd	2	1	inner	1728.2.r.d		8
8.b	even	2	1	inner	1728.2.r.d		8
8.d	odd	2	1	inner	1728.2.r.d		8
9.c	even	3	1	inner	1728.2.r.d		8
9.c	even	3	1		5184.2.d.g		4
9.d	odd	6	1		576.2.r.d	✓	8
9.d	odd	6	1		5184.2.d.h		4
12.b	even	2	1		576.2.r.d	✓	8
24.f	even	2	1		576.2.r.d	✓	8
24.h	odd	2	1		576.2.r.d	✓	8
36.f	odd	6	1	inner	1728.2.r.d		8
36.f	odd	6	1		5184.2.d.g		4
36.h	even	6	1		576.2.r.d	✓	8
36.h	even	6	1		5184.2.d.h		4
72.j	odd	6	1		576.2.r.d	✓	8
72.j	odd	6	1		5184.2.d.h		4
72.l	even	6	1		576.2.r.d	✓	8
72.l	even	6	1		5184.2.d.h		4
72.n	even	6	1	inner	1728.2.r.d		8
72.n	even	6	1		5184.2.d.g		4
72.p	odd	6	1	inner	1728.2.r.d		8
72.p	odd	6	1		5184.2.d.g		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
576.2.r.d	✓	8	3.b	odd	2	1
576.2.r.d	✓	8	9.d	odd	6	1
576.2.r.d	✓	8	12.b	even	2	1
576.2.r.d	✓	8	24.f	even	2	1
576.2.r.d	✓	8	24.h	odd	2	1
576.2.r.d	✓	8	36.h	even	6	1
576.2.r.d	✓	8	72.j	odd	6	1
576.2.r.d	✓	8	72.l	even	6	1
1728.2.r.d		8	1.a	even	1	1	trivial
1728.2.r.d		8	4.b	odd	2	1	inner
1728.2.r.d		8	8.b	even	2	1	inner
1728.2.r.d		8	8.d	odd	2	1	inner
1728.2.r.d		8	9.c	even	3	1	inner
1728.2.r.d		8	36.f	odd	6	1	inner
1728.2.r.d		8	72.n	even	6	1	inner
1728.2.r.d		8	72.p	odd	6	1	inner
5184.2.d.g		4	9.c	even	3	1
5184.2.d.g		4	36.f	odd	6	1
5184.2.d.g		4	72.n	even	6	1
5184.2.d.g		4	72.p	odd	6	1
5184.2.d.h		4	9.d	odd	6	1
5184.2.d.h		4	36.h	even	6	1
5184.2.d.h		4	72.j	odd	6	1
5184.2.d.h		4	72.l	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{4} - 15T_{5}^{2} + 225 $ T5^4 - 15*T5^2 + 225 acting on $S_{2}^{\mathrm{new}}(1728, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} $ T^8
$5$	$ (T^{4} - 15 T^{2} + 225)^{2} $ (T^4 - 15*T^2 + 225)^2
$7$	$ (T^{4} + 15 T^{2} + 225)^{2} $ (T^4 + 15*T^2 + 225)^2
$11$	$ (T^{4} - 9 T^{2} + 81)^{2} $ (T^4 - 9*T^2 + 81)^2
$13$	$ (T^{4} - 15 T^{2} + 225)^{2} $ (T^4 - 15*T^2 + 225)^2
$17$	$ T^{8} $ T^8
$19$	$ (T^{2} + 4)^{4} $ (T^2 + 4)^4
$23$	$ (T^{4} + 15 T^{2} + 225)^{2} $ (T^4 + 15*T^2 + 225)^2
$29$	$ (T^{4} - 15 T^{2} + 225)^{2} $ (T^4 - 15*T^2 + 225)^2
$31$	$ (T^{4} + 15 T^{2} + 225)^{2} $ (T^4 + 15*T^2 + 225)^2
$37$	$ (T^{2} + 60)^{4} $ (T^2 + 60)^4
$41$	$ (T^{2} - 9 T + 81)^{4} $ (T^2 - 9*T + 81)^4
$43$	$ (T^{4} - 49 T^{2} + 2401)^{2} $ (T^4 - 49*T^2 + 2401)^2
$47$	$ (T^{4} + 15 T^{2} + 225)^{2} $ (T^4 + 15*T^2 + 225)^2
$53$	$ (T^{2} + 60)^{4} $ (T^2 + 60)^4
$59$	$ (T^{4} - 9 T^{2} + 81)^{2} $ (T^4 - 9*T^2 + 81)^2
$61$	$ (T^{4} - 135 T^{2} + 18225)^{2} $ (T^4 - 135*T^2 + 18225)^2
$67$	$ (T^{4} - 121 T^{2} + 14641)^{2} $ (T^4 - 121*T^2 + 14641)^2
$71$	$ (T^{2} - 60)^{4} $ (T^2 - 60)^4
$73$	$ (T - 8)^{8} $ (T - 8)^8
$79$	$ (T^{4} + 15 T^{2} + 225)^{2} $ (T^4 + 15*T^2 + 225)^2
$83$	$ (T^{4} - 9 T^{2} + 81)^{2} $ (T^4 - 9*T^2 + 81)^2
$89$	$ (T - 12)^{8} $ (T - 12)^8
$97$	$ (T^{2} + T + 1)^{4} $ (T^2 + T + 1)^4
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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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.r (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{5} + (\beta_{7} - \beta_{5}) q^{7}+O(q^{10}) \) q - b4 * q^5 + (b7 - b5) * q^7	\( q - \beta_{4} q^{5} + (\beta_{7} - \beta_{5}) q^{7} - 3 \beta_1 q^{11} - \beta_{4} q^{13} - 2 \beta_{3} q^{19} + \beta_{7} q^{23} + 10 \beta_{2} q^{25} - \beta_{6} q^{29} - \beta_{7} q^{31} + 15 \beta_{3} q^{35} + (2 \beta_{6} + 2 \beta_{4}) q^{37} + ( - 9 \beta_{2} + 9) q^{41} - 7 \beta_1 q^{43} + ( - \beta_{7} + \beta_{5}) q^{47} + (8 \beta_{2} - 8) q^{49} + (2 \beta_{6} + 2 \beta_{4}) q^{53} + 3 \beta_{5} q^{55} + ( - 3 \beta_{3} - 3 \beta_1) q^{59} - 3 \beta_{6} q^{61} + 15 \beta_{2} q^{65} + ( - 11 \beta_{3} - 11 \beta_1) q^{67} - 2 \beta_{5} q^{71} + 8 q^{73} + 3 \beta_{4} q^{77} + ( - \beta_{7} + \beta_{5}) q^{79} + 3 \beta_1 q^{83} + 12 q^{89} + 15 \beta_{3} q^{91} - 2 \beta_{7} q^{95} - \beta_{2} q^{97}+O(q^{100}) \) q - b4 * q^5 + (b7 - b5) * q^7 - 3b1 q^11 - b4 * q^13 - 2b3 q^19 + b7 * q^23 + 10b2 q^25 - b6 * q^29 - b7 * q^31 + 15b3 q^35 + (2b6 + 2b4) * q^37 + (-9b2 + 9) q^41 - 7b1 q^43 + (-b7 + b5) * q^47 + (8b2 - 8) q^49 + (2b6 + 2b4) * q^53 + 3b5 q^55 + (-3b3 - 3b1) * q^59 - 3b6 q^61 + 15b2 q^65 + (-11b3 - 11b1) * q^67 - 2b5 q^71 + 8 * q^73 + 3b4 q^77 + (-b7 + b5) * q^79 + 3b1 q^83 + 12 * q^89 + 15b3 q^91 - 2b7 q^95 - b2 * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 40 q^{25} + 36 q^{41} - 32 q^{49} + 60 q^{65} + 64 q^{73} + 96 q^{89} - 4 q^{97}+O(q^{100}) \) 8 * q + 40 * q^25 + 36 * q^41 - 32 * q^49 + 60 * q^65 + 64 * q^73 + 96 * q^89 - 4 * 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	1728.2.r.d

Level	$1728$
Weight	$2$
Character orbit	1728.r
Analytic conductor	$13.798$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(13.7981494693\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	8.0.12960000.1
Defining polynomial:	\( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index:	\( 2^{4}\cdot 3^{2} \)
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{7} + 13\nu ) / 8 \) (v^7 + 13*v) / 8
\(\beta_{2}\)	\(=\)	\( ( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 9 ) / 8 \) (-3v^6 + 8v^4 - 24*v^2 + 9) / 8
\(\beta_{3}\)	\(=\)	\( ( -3\nu^{7} + 8\nu^{5} - 20\nu^{3} + \nu ) / 4 \) (-3v^7 + 8v^5 - 20*v^3 + v) / 4
\(\beta_{4}\)	\(=\)	\( ( -5\nu^{6} + 24\nu^{4} - 56\nu^{2} + 39 ) / 8 \) (-5v^6 + 24v^4 - 56*v^2 + 39) / 8
\(\beta_{5}\)	\(=\)	\( ( -5\nu^{7} + 16\nu^{5} - 44\nu^{3} + 31\nu ) / 4 \) (-5v^7 + 16v^5 - 44v^3 + 31v) / 4
\(\beta_{6}\)	\(=\)	\( ( -11\nu^{6} + 24\nu^{4} - 56\nu^{2} - 15 ) / 8 \) (-11v^6 + 24v^4 - 56*v^2 - 15) / 8
\(\beta_{7}\)	\(=\)	\( ( 13\nu^{7} - 32\nu^{5} + 88\nu^{3} + 25\nu ) / 8 \) (13v^7 - 32v^5 + 88v^3 + 25v) / 8

\(\nu\)	\(=\)	\( ( \beta_{7} + \beta_{5} - 3\beta_1 ) / 6 \) (b7 + b5 - 3*b1) / 6
\(\nu^{2}\)	\(=\)	\( ( 2\beta_{6} + \beta_{4} - 9\beta_{2} + 9 ) / 6 \) (2b6 + b4 - 9b2 + 9) / 6
\(\nu^{3}\)	\(=\)	\( ( 2\beta_{7} - \beta_{5} + 6\beta_{3} ) / 3 \) (2b7 - b5 + 6b3) / 3
\(\nu^{4}\)	\(=\)	\( ( \beta_{6} + 2\beta_{4} - 7\beta_{2} ) / 2 \) (b6 + 2b4 - 7b2) / 2
\(\nu^{5}\)	\(=\)	\( ( 5\beta_{7} - 10\beta_{5} + 33\beta_{3} + 33\beta_1 ) / 6 \) (5b7 - 10b5 + 33b3 + 33b1) / 6
\(\nu^{6}\)	\(=\)	\( ( -4\beta_{6} + 4\beta_{4} - 27 ) / 3 \) (-4b6 + 4b4 - 27) / 3
\(\nu^{7}\)	\(=\)	\( ( -13\beta_{7} - 13\beta_{5} + 87\beta_1 ) / 6 \) (-13b7 - 13b5 + 87*b1) / 6

\(n\)	\(325\)	\(703\)	\(1217\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1 + \beta_{2}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1441.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( T^{8} \) T^8
$5$	\( (T^{4} - 15 T^{2} + 225)^{2} \) (T^4 - 15*T^2 + 225)^2
$7$	\( (T^{4} + 15 T^{2} + 225)^{2} \) (T^4 + 15*T^2 + 225)^2
$11$	\( (T^{4} - 9 T^{2} + 81)^{2} \) (T^4 - 9*T^2 + 81)^2
$13$	\( (T^{4} - 15 T^{2} + 225)^{2} \) (T^4 - 15*T^2 + 225)^2
$17$	\( T^{8} \) T^8
$19$	\( (T^{2} + 4)^{4} \) (T^2 + 4)^4
$23$	\( (T^{4} + 15 T^{2} + 225)^{2} \) (T^4 + 15*T^2 + 225)^2
$29$	\( (T^{4} - 15 T^{2} + 225)^{2} \) (T^4 - 15*T^2 + 225)^2
$31$	\( (T^{4} + 15 T^{2} + 225)^{2} \) (T^4 + 15*T^2 + 225)^2
$37$	\( (T^{2} + 60)^{4} \) (T^2 + 60)^4
$41$	\( (T^{2} - 9 T + 81)^{4} \) (T^2 - 9*T + 81)^4
$43$	\( (T^{4} - 49 T^{2} + 2401)^{2} \) (T^4 - 49*T^2 + 2401)^2
$47$	\( (T^{4} + 15 T^{2} + 225)^{2} \) (T^4 + 15*T^2 + 225)^2
$53$	\( (T^{2} + 60)^{4} \) (T^2 + 60)^4
$59$	\( (T^{4} - 9 T^{2} + 81)^{2} \) (T^4 - 9*T^2 + 81)^2
$61$	\( (T^{4} - 135 T^{2} + 18225)^{2} \) (T^4 - 135*T^2 + 18225)^2
$67$	\( (T^{4} - 121 T^{2} + 14641)^{2} \) (T^4 - 121*T^2 + 14641)^2
$71$	\( (T^{2} - 60)^{4} \) (T^2 - 60)^4
$73$	\( (T - 8)^{8} \) (T - 8)^8
$79$	\( (T^{4} + 15 T^{2} + 225)^{2} \) (T^4 + 15*T^2 + 225)^2
$83$	\( (T^{4} - 9 T^{2} + 81)^{2} \) (T^4 - 9*T^2 + 81)^2
$89$	\( (T - 12)^{8} \) (T - 12)^8
$97$	\( (T^{2} + T + 1)^{4} \) (T^2 + T + 1)^4
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