LMFDB - Newform orbit 495.2.n.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,2,Mod(91,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 0, 8]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("495.91");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	495.n (of order $5$, degree $4$, 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:	$3.95259490005$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{5})$
Coefficient field:	$\Q(\zeta_{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 $ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/495\mathbb{Z}\right)^\times$.

$n$	$46$	$56$	$397$
$\chi(n)$	$-1 + \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + \zeta_{15}^{7}$	$1$	$1$

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} $

91.1

0.669131	−	0.743145i
−0.978148	−	0.207912i
0.669131	+	0.743145i
−0.978148	+	0.207912i
−0.104528	+	0.994522i
0.913545	−	0.406737i
−0.104528	−	0.994522i
0.913545	+	0.406737i

−1.58268

1.14988i

0.564602

−

1.73767i

0.809017

0.587785i

0.478148

−

1.47159i

−0.104528

−

0.321706i

−1.95630

91.2

1.08268

−

0.786610i

−0.0646021

0.198825i

0.809017

0.587785i

−1.16913

3.59821i

0.913545

2.81160i

1.33826

136.1

−1.58268

−

1.14988i

0.564602

1.73767i

0.809017

−

0.587785i

0.478148

1.47159i

−0.104528

0.321706i

−1.95630

136.2

1.08268

0.786610i

−0.0646021

−

0.198825i

0.809017

−

0.587785i

−1.16913

−

3.59821i

0.913545

−

2.81160i

1.33826

181.1

−0.564602

−

1.73767i

−1.08268

0.786610i

−0.309017

0.951057i

−1.41355

1.02700i

−0.978148

−

0.710666i

1.82709

181.2

0.0646021

0.198825i

1.58268

−

1.14988i

−0.309017

0.951057i

−0.395472

0.287327i

0.669131

0.486152i

−0.209057

361.1

−0.564602

1.73767i

−1.08268

−

0.786610i

−0.309017

−

0.951057i

−1.41355

−

1.02700i

−0.978148

0.710666i

1.82709

361.2

0.0646021

−

0.198825i

1.58268

1.14988i

−0.309017

−

0.951057i

−0.395472

−

0.287327i

0.669131

−

0.486152i

−0.209057

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	495.2.n.c		8
3.b	odd	2	1		165.2.m.c	✓	8
11.c	even	5	1	inner	495.2.n.c		8
11.c	even	5	1		5445.2.a.bj		4
11.d	odd	10	1		5445.2.a.bq		4
15.d	odd	2	1		825.2.n.j		8
15.e	even	4	2		825.2.bx.e		16
33.f	even	10	1		1815.2.a.q		4
33.h	odd	10	1		165.2.m.c	✓	8
33.h	odd	10	1		1815.2.a.u		4
165.o	odd	10	1		825.2.n.j		8
165.o	odd	10	1		9075.2.a.co		4
165.r	even	10	1		9075.2.a.df		4
165.v	even	20	2		825.2.bx.e		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
165.2.m.c	✓	8	3.b	odd	2	1
165.2.m.c	✓	8	33.h	odd	10	1
495.2.n.c		8	1.a	even	1	1	trivial
495.2.n.c		8	11.c	even	5	1	inner
825.2.n.j		8	15.d	odd	2	1
825.2.n.j		8	165.o	odd	10	1
825.2.bx.e		16	15.e	even	4	2
825.2.bx.e		16	165.v	even	20	2
1815.2.a.q		4	33.f	even	10	1
1815.2.a.u		4	33.h	odd	10	1
5445.2.a.bj		4	11.c	even	5	1
5445.2.a.bq		4	11.d	odd	10	1
9075.2.a.co		4	165.o	odd	10	1
9075.2.a.df		4	165.r	even	10	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} + 2T_{2}^{7} + 3T_{2}^{6} - T_{2}^{5} - T_{2}^{3} + 23T_{2}^{2} - 3T_{2} + 1 $ T2^8 + 2*T2^7 + 3*T2^6 - T2^5 - T2^3 + 23*T2^2 - 3*T2 + 1 acting on $S_{2}^{\mathrm{new}}(495, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + 2 T^{7} + \cdots + 1 $ T^8 + 2T^7 + 3T^6 - T^5 - T^3 + 23T^2 - 3T + 1
$3$	$ T^{8} $ T^8
$5$	$ (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} $ (T^4 - T^3 + T^2 - T + 1)^2
$7$	$ T^{8} + 5 T^{7} + \cdots + 25 $ T^8 + 5T^7 + 25T^6 + 55T^5 + 90T^4 + 125T^3 + 175T^2 + 100*T + 25
$11$	$ T^{8} - T^{7} + \cdots + 14641 $ T^8 - T^7 + 10T^6 + 31T^5 - 51T^4 + 341T^3 + 1210T^2 - 1331T + 14641
$13$	$ T^{8} + 16 T^{7} + \cdots + 128881 $ T^8 + 16T^7 + 147T^6 + 883T^5 + 3900T^4 + 12757T^3 + 36137T^2 + 82929*T + 128881
$17$	$ T^{8} - 4 T^{7} + \cdots + 3481 $ T^8 - 4T^7 + 57T^6 - 392T^5 + 1925T^4 - 5768T^3 + 11697T^2 - 9676*T + 3481
$19$	$ T^{8} - 2 T^{7} + \cdots + 57121 $ T^8 - 2T^7 + 33T^6 + 131T^5 + 290T^4 - 7039T^3 + 36963T^2 - 38957*T + 57121
$23$	$ (T^{4} + 5 T^{3} - 25 T^{2} + \cdots - 5)^{2} $ (T^4 + 5T^3 - 25T^2 + 25*T - 5)^2
$29$	$ T^{8} - 16 T^{7} + \cdots + 175561 $ T^8 - 16T^7 + 147T^6 - 1073T^5 + 7400T^4 - 35807T^3 + 104937T^2 - 142879*T + 175561
$31$	$ T^{8} + 5 T^{7} + \cdots + 25 $ T^8 + 5T^7 + 15T^6 - 95T^5 + 590T^4 + 3925T^3 + 54825T^2 + 725*T + 25
$37$	$ T^{8} + 5 T^{7} + \cdots + 24025 $ T^8 + 5T^7 + 25T^6 + 85T^5 + 1440T^4 - 4225T^3 + 23725T^2 - 36425*T + 24025
$41$	$ T^{8} + 5 T^{7} + \cdots + 25 $ T^8 + 5T^7 + 75T^6 - 115T^5 - 510T^4 + 2525T^3 + 20975T^2 + 450*T + 25
$43$	$ (T^{4} + 4 T^{3} + \cdots - 569)^{2} $ (T^4 + 4T^3 - 124T^2 - 541*T - 569)^2
$47$	$ T^{8} - T^{7} + \cdots + 14641 $ T^8 - T^7 + 27T^6 + 137T^5 + 300T^4 - 4807T^3 + 20207T^2 + 7986T + 14641
$53$	$ T^{8} - 15 T^{7} + \cdots + 625 $ T^8 - 15T^7 + 80T^6 + 175T^5 + 525T^4 + 625T^3 + 750T^2 + 625*T + 625
$59$	$ T^{8} + 3 T^{7} + \cdots + 1798281 $ T^8 + 3T^7 + 123T^6 - 549T^5 + 1980T^4 + 8721T^3 + 43983T^2 - 36207*T + 1798281
$61$	$ T^{8} + 11 T^{7} + \cdots + 43178041 $ T^8 + 11T^7 + 222T^6 + 3313T^5 + 35825T^4 + 273007T^3 + 2338152T^2 + 13792529*T + 43178041
$67$	$ (T^{4} - 3 T^{3} - 46 T^{2} + \cdots + 31)^{2} $ (T^4 - 3T^3 - 46T^2 - 72*T + 31)^2
$71$	$ T^{8} + T^{7} + \cdots + 151321 $ T^8 + T^7 + 27T^6 - 67T^5 + 590T^4 + 3377T^3 + 27117T^2 + 85969T + 151321
$73$	$ T^{8} + 35 T^{7} + \cdots + 17682025 $ T^8 + 35T^7 + 640T^6 + 7315T^5 + 58515T^4 + 320075T^3 + 1321900T^2 + 4562425*T + 17682025
$79$	$ T^{8} + 30 T^{7} + \cdots + 93025 $ T^8 + 30T^7 + 395T^6 + 2380T^5 + 7740T^4 + 15250T^3 + 58050T^2 + 51850*T + 93025
$83$	$ T^{8} + 31 T^{7} + \cdots + 215179561 $ T^8 + 31T^7 + 867T^6 + 15503T^5 + 250190T^4 + 3033527T^3 + 31113477T^2 + 124261099*T + 215179561
$89$	$ (T^{4} - 24 T^{3} + \cdots - 31869)^{2} $ (T^4 - 24T^3 - 129T^2 + 6066*T - 31869)^2
$97$	$ T^{8} - 11 T^{7} + \cdots + 3721 $ T^8 - 11T^7 + 222T^6 - 3833T^5 + 46515T^4 + 32923T^3 + 169412T^2 - 40809*T + 3721
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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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	495.n (of order \(5\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{15}^{7} + \zeta_{15}^{5} + \cdots + 1) q^{2}+ \cdots + ( - \zeta_{15}^{7} + \cdots - \zeta_{15}^{5}) q^{8}+O(q^{10}) \) q + (-z^7 + z^5 - z^4 + z^3 - 2z + 1) q^2 + (z^2 + z) * q^4 - z^3 * q^5 + (z^6 - 2z^5 + z^4 - z^3 + 2z - 2) * q^7 + (-z^7 + z^6 - z^5) * q^8	\( q + ( - \zeta_{15}^{7} + \zeta_{15}^{5} + \cdots + 1) q^{2}+ \cdots + ( - 2 \zeta_{15}^{7} - \zeta_{15}^{6} + \cdots - 3) q^{98}+O(q^{100}) \) q + (-z^7 + z^5 - z^4 + z^3 - 2z + 1) q^2 + (z^2 + z) * q^4 - z^3 * q^5 + (z^6 - 2z^5 + z^4 - z^3 + 2z - 2) * q^7 + (-z^7 + z^6 - z^5) * q^8 + (-z^6 + z^4 - z) * q^10 + (-3z^7 - z^6 + 2z^5 - 3z^4 + 3z^3 - z^2 - 4z + 3) q^11 + (-5z^7 + 3z^6 - z^5 - 3z^4 + 4z^3 - 4z^2 - 2z + 1) * q^13 + (-z^7 + 2z^6 - z^5 + 2z^4 - z^3 - 2z^2 + z + 1) q^14 + (2z^6 + 2z^5 + z^4 - 2z^3 + z^2 + 2z + 2) * q^16 + (2z^5 + 2z^4 - 3z^3 + 2z^2 + 2z) q^17 + (-6z^7 + 5z^6 + 3z^5 - 6z^4 + 3z^3 - 3z^2 - 3z + 6) q^19 + (-z^5 - z^4) * q^20 + (3z^7 - 2z^6 - 2z^4 - 2z^3 + 4z^2 + 2z - 3) * q^22 + (-z^7 + z^6 + 2z^5 - 3z^4 - z^3 + 3z^2 - z) q^23 + z^6 * q^25 + (2z^7 - 4z^5 + 2z^4 - 5z^3 + 4z^2 + 6z - 7) * q^26 + (-z^6 - z^5 + z^3 - z - 1) * q^28 + (-5z^7 + 3z^6 + 4z^5 - 2z^4 + 6z^3 - 4z^2 - 3z + 8) q^29 + (-z^6 + 5z^5 + z^4 + z^3 + 2z^2 - 4z + 2) q^31 + (-3z^7 - 2z^6 + 2z^4 + 3z^3 - 3z^2 - 2z + 1) * q^32 + (2z^7 - 3z^6 - 2z^5 + 5z^4 - 2z^2 - z + 2) q^34 + (2z^6 - 2z^5 + z^3 - z^2 + 2z - 1) q^35 + (6z^7 - 5z^6 - 2z^5 + z^4 - 4z^3 + 3z^2 - z - 5) q^37 + (-8z^6 + 5z^4 - 5z^3 + 5z^2 - 8) * q^38 + (z^6 - 2z^5 + z^4 - z^2 + z - 1) q^40 + (-2z^7 - 2z^6 + 4z^5 - 6z^4 + 3z^3 - 3z + 3) * q^41 + (-z^7 - 5z^6 + 3z^5 + 2z^4 - 2z^3 + 5z^2 - 8z - 1) * q^43 + (-6z^7 + 2z^6 + 4z^5 - 2z^4 + 2z^3 - 4z^2 - z + 7) * q^44 + (z^7 - 5z^6 + 3z^5 + 5z^4 - 6z^3 + 2z - 1) q^46 + (2z^7 + 2z^6 + 2z^5 + 2z^4 - z^3 - 2z^2 + z + 1) q^47 + (-z^6 + z^5 - 3z^4 - 2z^3 - 3z^2 + z - 1) q^49 + (-z^6 + z^4 - z^3 + z^2 - 1) * q^50 + (-5z^7 + z^6 + 3z^5 - 2z^4 + z^3 - 6z^2 - z + 7) * q^52 + (z^7 - 4z^6 + 4z^5 - 2z^4 - z^3 + 4z^2 - 6z + 3) q^53 + (2z^7 - 4z^6 + 2z^4 - 2z^3 + z^2 - 2z - 2) q^55 + (-z^7 + 2z^6 - 2z^5 + 3z^3 - 5z^2 + 4z - 2) q^56 + (2z^7 - 4z^6 - 3z^5 - 2z^4 + z^3 + 7z^2 - z - 6) q^58 + (2z^7 + 3z^6 - 10z^5 + 5z^4 - z^3 - 3z^2 + 5z - 6) * q^59 + (2z^6 + z^5 + 10z^3 + z + 2) * q^61 + (8z^7 - 7z^6 - 2z^5 + z^4 - 3z^3 + 8z^2 + 2z - 4) * q^62 + (4z^7 + 5z^6 - 5z^5 + 3z^4 + z^3 + 8z - 4) q^64 + (z^7 - z^6 - 2z^5 + 3z^4 + z^3 - 3z^2 + z - 3) q^65 + (-7z^7 + z^6 + 2z^5 - 3z^4 + 5z^3 - 3z^2 - z + 5) q^67 + (2z^7 + 4z^6 - z^5 - z^4 + 4z^3 + 2z^2) * q^68 + (-3z^7 + 3z^6 - 2z^2 + z) q^70 + (z^5 - 2z^4 - 3z^3 - 2z^2 + z) q^71 + (3z^7 - 6z^5 + 3z^4 - 8z^3 + 2z^2 + 5z - 11) * q^73 + (-z^7 + 2z^6 + 5z^5 - 4z^4 + 2z^3 - 2z^2 - 2z + 4) * q^74 + (z^7 + 3z^6 - 2z^5 - z^4 + z^3 - 3z^2 + 5z + 7) * q^76 + (z^5 + 4z^4 - 5z^3 + z^2 - 2z + 2) q^77 + (-5z^7 + 5z^6 - 2z^5 + 2z^4 + 3z^3 - 7z^2 + 4z - 2) q^79 + (-5z^7 + 4z^6 + z^5 - 4z^4 + 2z^3 - 2z^2 - 2z + 4) * q^80 + (-2z^6 + 3z^5 - 2z^4 - 2z^2 + 3z - 2) q^82 + (-8z^6 - 9z^5 - 2z^4 + 3z^3 - 2z^2 - 9z - 8) * q^83 + (-4z^7 + 3z^6 - 4z^4 + 2z^3 - 2z + 2) q^85 + (15z^7 - 8z^6 - 2z^5 + 2z^4 - 10z^3 + 13z^2 + 4z - 2) q^86 + (-3z^6 + 3z^5 - z^4 - 4z^3 + 4z^2 - 4z + 2) q^88 + (19z^7 - 2z^6 - 8z^5 + 10z^4 - 11z^3 + 3z^2 + 6z - 6) q^89 + (3z^7 - 5z^6 + 8z^5 - 2z^4 + z^3 - 3z^2 - z + 4) q^91 + (2z^7 - 4z^5 + 2z^4 + 3z^3 - 2z^2 + 1) q^92 + (-4z^6 - 3z^5 + 2z^4 + 4z^3 + 2z^2 - 3z - 4) * q^94 + (-2z^7 + 2z^6 + 2z^3 - 5z^2 - 3z + 2) q^95 + (9z^7 - 2z^6 + 6z^5 + 4z^4 - z^3 + 10z^2 - 2z + 1) * q^97 + (-2z^7 - z^6 - 2z^5 + 3z^4 + 4z^3 - 6z^2 + z - 3) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 5 q^{7} + q^{8}+O(q^{10}) \) 8 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 5 * q^7 + q^8	\( 8 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 5 q^{7} + q^{8} + 2 q^{10} + q^{11} - 16 q^{13} + 10 q^{14} + 12 q^{16} + 4 q^{17} + 2 q^{19} + 3 q^{20} - 9 q^{22} - 10 q^{23} - 2 q^{25} - 16 q^{26} - 5 q^{28} + 16 q^{29} - 5 q^{31} + 34 q^{34} - 5 q^{35} - 5 q^{37} - 28 q^{38} - q^{40} - 5 q^{41} - 8 q^{43} + 19 q^{44} + 10 q^{46} + q^{47} - 11 q^{49} - 2 q^{50} + 26 q^{52} + 15 q^{53} - q^{55} - 20 q^{56} - 24 q^{58} - 3 q^{59} - 11 q^{61} + 15 q^{62} - 9 q^{64} - 14 q^{65} + 6 q^{67} - 9 q^{68} - 10 q^{70} - q^{71} - 35 q^{73} - 5 q^{74} + 58 q^{76} + 25 q^{77} - 30 q^{79} + 3 q^{80} - 25 q^{82} - 31 q^{83} - 4 q^{85} + 62 q^{86} + 17 q^{88} + 48 q^{89} + 5 q^{91} + 20 q^{92} - 19 q^{94} - 2 q^{95} + 11 q^{97} - 26 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 5 * q^7 + q^8 + 2 * q^10 + q^11 - 16 * q^13 + 10 * q^14 + 12 * q^16 + 4 * q^17 + 2 * q^19 + 3 * q^20 - 9 * q^22 - 10 * q^23 - 2 * q^25 - 16 * q^26 - 5 * q^28 + 16 * q^29 - 5 * q^31 + 34 * q^34 - 5 * q^35 - 5 * q^37 - 28 * q^38 - q^40 - 5 * q^41 - 8 * q^43 + 19 * q^44 + 10 * q^46 + q^47 - 11 * q^49 - 2 * q^50 + 26 * q^52 + 15 * q^53 - q^55 - 20 * q^56 - 24 * q^58 - 3 * q^59 - 11 * q^61 + 15 * q^62 - 9 * q^64 - 14 * q^65 + 6 * q^67 - 9 * q^68 - 10 * q^70 - q^71 - 35 * q^73 - 5 * q^74 + 58 * q^76 + 25 * q^77 - 30 * q^79 + 3 * q^80 - 25 * q^82 - 31 * q^83 - 4 * q^85 + 62 * q^86 + 17 * q^88 + 48 * q^89 + 5 * q^91 + 20 * q^92 - 19 * q^94 - 2 * q^95 + 11 * q^97 - 26 * q^98

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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	495.2.n.c

Level	$495$
Weight	$2$
Character orbit	495.n
Analytic conductor	$3.953$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(3.95259490005\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\Q(\zeta_{15})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

\(n\)	\(46\)	\(56\)	\(397\)
\(\chi(n)\)	\(-1 + \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + \zeta_{15}^{7}\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 91.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} + 2 T^{7} + \cdots + 1 \) T^8 + 2T^7 + 3T^6 - T^5 - T^3 + 23T^2 - 3T + 1
$3$	\( T^{8} \) T^8
$5$	\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) (T^4 - T^3 + T^2 - T + 1)^2
$7$	\( T^{8} + 5 T^{7} + \cdots + 25 \) T^8 + 5T^7 + 25T^6 + 55T^5 + 90T^4 + 125T^3 + 175T^2 + 100*T + 25
$11$	\( T^{8} - T^{7} + \cdots + 14641 \) T^8 - T^7 + 10T^6 + 31T^5 - 51T^4 + 341T^3 + 1210T^2 - 1331T + 14641
$13$	\( T^{8} + 16 T^{7} + \cdots + 128881 \) T^8 + 16T^7 + 147T^6 + 883T^5 + 3900T^4 + 12757T^3 + 36137T^2 + 82929*T + 128881
$17$	\( T^{8} - 4 T^{7} + \cdots + 3481 \) T^8 - 4T^7 + 57T^6 - 392T^5 + 1925T^4 - 5768T^3 + 11697T^2 - 9676*T + 3481
$19$	\( T^{8} - 2 T^{7} + \cdots + 57121 \) T^8 - 2T^7 + 33T^6 + 131T^5 + 290T^4 - 7039T^3 + 36963T^2 - 38957*T + 57121
$23$	\( (T^{4} + 5 T^{3} - 25 T^{2} + \cdots - 5)^{2} \) (T^4 + 5T^3 - 25T^2 + 25*T - 5)^2
$29$	\( T^{8} - 16 T^{7} + \cdots + 175561 \) T^8 - 16T^7 + 147T^6 - 1073T^5 + 7400T^4 - 35807T^3 + 104937T^2 - 142879*T + 175561
$31$	\( T^{8} + 5 T^{7} + \cdots + 25 \) T^8 + 5T^7 + 15T^6 - 95T^5 + 590T^4 + 3925T^3 + 54825T^2 + 725*T + 25
$37$	\( T^{8} + 5 T^{7} + \cdots + 24025 \) T^8 + 5T^7 + 25T^6 + 85T^5 + 1440T^4 - 4225T^3 + 23725T^2 - 36425*T + 24025
$41$	\( T^{8} + 5 T^{7} + \cdots + 25 \) T^8 + 5T^7 + 75T^6 - 115T^5 - 510T^4 + 2525T^3 + 20975T^2 + 450*T + 25
$43$	\( (T^{4} + 4 T^{3} + \cdots - 569)^{2} \) (T^4 + 4T^3 - 124T^2 - 541*T - 569)^2
$47$	\( T^{8} - T^{7} + \cdots + 14641 \) T^8 - T^7 + 27T^6 + 137T^5 + 300T^4 - 4807T^3 + 20207T^2 + 7986T + 14641
$53$	\( T^{8} - 15 T^{7} + \cdots + 625 \) T^8 - 15T^7 + 80T^6 + 175T^5 + 525T^4 + 625T^3 + 750T^2 + 625*T + 625
$59$	\( T^{8} + 3 T^{7} + \cdots + 1798281 \) T^8 + 3T^7 + 123T^6 - 549T^5 + 1980T^4 + 8721T^3 + 43983T^2 - 36207*T + 1798281
$61$	\( T^{8} + 11 T^{7} + \cdots + 43178041 \) T^8 + 11T^7 + 222T^6 + 3313T^5 + 35825T^4 + 273007T^3 + 2338152T^2 + 13792529*T + 43178041
$67$	\( (T^{4} - 3 T^{3} - 46 T^{2} + \cdots + 31)^{2} \) (T^4 - 3T^3 - 46T^2 - 72*T + 31)^2
$71$	\( T^{8} + T^{7} + \cdots + 151321 \) T^8 + T^7 + 27T^6 - 67T^5 + 590T^4 + 3377T^3 + 27117T^2 + 85969T + 151321
$73$	\( T^{8} + 35 T^{7} + \cdots + 17682025 \) T^8 + 35T^7 + 640T^6 + 7315T^5 + 58515T^4 + 320075T^3 + 1321900T^2 + 4562425*T + 17682025
$79$	\( T^{8} + 30 T^{7} + \cdots + 93025 \) T^8 + 30T^7 + 395T^6 + 2380T^5 + 7740T^4 + 15250T^3 + 58050T^2 + 51850*T + 93025
$83$	\( T^{8} + 31 T^{7} + \cdots + 215179561 \) T^8 + 31T^7 + 867T^6 + 15503T^5 + 250190T^4 + 3033527T^3 + 31113477T^2 + 124261099*T + 215179561
$89$	\( (T^{4} - 24 T^{3} + \cdots - 31869)^{2} \) (T^4 - 24T^3 - 129T^2 + 6066*T - 31869)^2
$97$	\( T^{8} - 11 T^{7} + \cdots + 3721 \) T^8 - 11T^7 + 222T^6 - 3833T^5 + 46515T^4 + 32923T^3 + 169412T^2 - 40809*T + 3721
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