LMFDB - Newform orbit 891.2.n.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [891,2,Mod(136,891)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(891, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([20, 6]))

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

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

chi := DirichletCharacter("891.136");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 891 = 3^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	891.n (of order $15$, degree $8$, not 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:	$7.11467082010$
Analytic rank:	$0$
Dimension:	$8$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/891\mathbb{Z}\right)^\times$.

$n$	$244$	$650$
$\chi(n)$	$-\zeta_{15}^{2} - \zeta_{15}^{7}$	$-1 - \zeta_{15}^{5}$

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} $

136.1

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

1.47815

−

0.658114i

0.413545

−

0.459289i

−0.348943

−

0.155360i

−2.93444

−

0.623735i

−0.690983

2.12663i

−0.618034

190.1

1.47815

0.658114i

0.413545

0.459289i

−0.348943

0.155360i

−2.93444

0.623735i

−0.690983

−

2.12663i

−0.618034

379.1

0.604528

0.128496i

−1.47815

−

0.658114i

2.56082

−

0.544320i

−0.313585

−

2.98357i

−1.80902

−

1.31433i

1.61803

433.1

−0.169131

1.60917i

−0.604528

−

0.128496i

0.0399263

0.379874i

2.00739

−

2.22943i

−0.690983

2.12663i

−0.618034

460.1

−0.413545

−

0.459289i

0.169131

−

1.60917i

−1.75181

1.94558i

2.74064

−

1.22021i

−1.80902

1.31433i

1.61803

676.1

−0.413545

0.459289i

0.169131

1.60917i

−1.75181

−

1.94558i

2.74064

1.22021i

−1.80902

−

1.31433i

1.61803

757.1

0.604528

−

0.128496i

−1.47815

0.658114i

2.56082

0.544320i

−0.313585

2.98357i

−1.80902

1.31433i

1.61803

784.1

−0.169131

−

1.60917i

−0.604528

0.128496i

0.0399263

−

0.379874i

2.00739

2.22943i

−0.690983

−

2.12663i

−0.618034

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
11.c	even	5	1	inner
99.m	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	891.2.n.d		8
3.b	odd	2	1		891.2.n.a		8
9.c	even	3	1		33.2.e.a	✓	4
9.c	even	3	1	inner	891.2.n.d		8
9.d	odd	6	1		99.2.f.b		4
9.d	odd	6	1		891.2.n.a		8
11.c	even	5	1	inner	891.2.n.d		8
33.h	odd	10	1		891.2.n.a		8
36.f	odd	6	1		528.2.y.f		4
45.j	even	6	1		825.2.n.f		4
45.k	odd	12	2		825.2.bx.b		8
99.h	odd	6	1		363.2.e.j		4
99.m	even	15	1		33.2.e.a	✓	4
99.m	even	15	1		363.2.a.h		2
99.m	even	15	2		363.2.e.h		4
99.m	even	15	1	inner	891.2.n.d		8
99.n	odd	30	1		99.2.f.b		4
99.n	odd	30	1		891.2.n.a		8
99.n	odd	30	1		1089.2.a.m		2
99.o	odd	30	1		363.2.a.e		2
99.o	odd	30	2		363.2.e.c		4
99.o	odd	30	1		363.2.e.j		4
99.p	even	30	1		1089.2.a.s		2
396.be	odd	30	1		528.2.y.f		4
396.be	odd	30	1		5808.2.a.bl		2
396.bf	even	30	1		5808.2.a.bm		2
495.bl	even	30	1		825.2.n.f		4
495.bl	even	30	1		9075.2.a.x		2
495.br	odd	30	1		9075.2.a.bv		2
495.bt	odd	60	2		825.2.bx.b		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
33.2.e.a	✓	4	9.c	even	3	1
33.2.e.a	✓	4	99.m	even	15	1
99.2.f.b		4	9.d	odd	6	1
99.2.f.b		4	99.n	odd	30	1
363.2.a.e		2	99.o	odd	30	1
363.2.a.h		2	99.m	even	15	1
363.2.e.c		4	99.o	odd	30	2
363.2.e.h		4	99.m	even	15	2
363.2.e.j		4	99.h	odd	6	1
363.2.e.j		4	99.o	odd	30	1
528.2.y.f		4	36.f	odd	6	1
528.2.y.f		4	396.be	odd	30	1
825.2.n.f		4	45.j	even	6	1
825.2.n.f		4	495.bl	even	30	1
825.2.bx.b		8	45.k	odd	12	2
825.2.bx.b		8	495.bt	odd	60	2
891.2.n.a		8	3.b	odd	2	1
891.2.n.a		8	9.d	odd	6	1
891.2.n.a		8	33.h	odd	10	1
891.2.n.a		8	99.n	odd	30	1
891.2.n.d		8	1.a	even	1	1	trivial
891.2.n.d		8	9.c	even	3	1	inner
891.2.n.d		8	11.c	even	5	1	inner
891.2.n.d		8	99.m	even	15	1	inner
1089.2.a.m		2	99.n	odd	30	1
1089.2.a.s		2	99.p	even	30	1
5808.2.a.bl		2	396.be	odd	30	1
5808.2.a.bm		2	396.bf	even	30	1
9075.2.a.x		2	495.bl	even	30	1
9075.2.a.bv		2	495.br	odd	30	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} - 3T_{2}^{7} + 5T_{2}^{6} - 8T_{2}^{5} + 9T_{2}^{4} - 2T_{2}^{3} - 2T_{2} + 1 $ T2^8 - 3*T2^7 + 5*T2^6 - 8*T2^5 + 9*T2^4 - 2*T2^3 - 2*T2 + 1 acting on $S_{2}^{\mathrm{new}}(891, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} - 3 T^{7} + \cdots + 1 $ T^8 - 3T^7 + 5T^6 - 8T^5 + 9T^4 - 2T^3 - 2T + 1
$3$	$ T^{8} $ T^8
$5$	$ T^{8} - T^{7} - 5 T^{6} + \cdots + 1 $ T^8 - T^7 - 5T^6 - 14T^5 + 39T^4 + 26T^3 + 10T^2 + 4T + 1
$7$	$ T^{8} - 3 T^{7} + \cdots + 6561 $ T^8 - 3T^7 + 27T^5 - 81T^4 + 243T^3 - 2187*T + 6561
$11$	$ T^{8} + 9 T^{7} + \cdots + 14641 $ T^8 + 9T^7 + 40T^6 + 171T^5 + 669T^4 + 1881T^3 + 4840T^2 + 11979*T + 14641
$13$	$ T^{8} - 9 T^{7} + \cdots + 14641 $ T^8 - 9T^7 + 50T^6 - 301T^5 + 939T^4 + 2519T^3 - 3630T^2 + 1331*T + 14641
$17$	$ (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 - 2T^3 + 4T^2 - 3*T + 1)^2
$19$	$ (T^{4} + 10 T^{3} + \cdots + 25)^{2} $ (T^4 + 10T^3 + 40T^2 + 25*T + 25)^2
$23$	$ (T^{4} - 2 T^{3} + \cdots + 361)^{2} $ (T^4 - 2T^3 + 23T^2 + 38*T + 361)^2
$29$	$ T^{8} - 10 T^{7} + \cdots + 160000 $ T^8 - 10T^7 + 40T^6 - 200T^5 + 1200T^4 - 4000T^3 + 16000T^2 - 80000*T + 160000
$31$	$ T^{8} + 8 T^{7} + \cdots + 14641 $ T^8 + 8T^7 + 30T^6 + 118T^5 + 419T^4 + 682T^3 + 1815T^2 + 9317*T + 14641
$37$	$ (T^{4} + 3 T^{3} + 19 T^{2} + \cdots + 1)^{2} $ (T^4 + 3T^3 + 19T^2 + 7*T + 1)^2
$41$	$ T^{8} + 23 T^{7} + \cdots + 25411681 $ T^8 + 23T^7 + 280T^6 + 3313T^5 + 29199T^4 + 68657T^3 + 201640T^2 + 6084487*T + 25411681
$43$	$ (T^{4} + 8 T^{3} + \cdots + 121)^{2} $ (T^4 + 8T^3 + 53T^2 + 88*T + 121)^2
$47$	$ T^{8} - 3 T^{7} + \cdots + 1 $ T^8 - 3T^7 + 5T^6 - 8T^5 + 9T^4 - 2T^3 - 2T + 1
$53$	$ (T^{4} - 6 T^{3} + \cdots + 5041)^{2} $ (T^4 - 6T^3 + 76T^2 - 781*T + 5041)^2
$59$	$ T^{8} - 20 T^{7} + \cdots + 9150625 $ T^8 - 20T^7 + 210T^6 - 2150T^5 + 16575T^4 - 35750T^3 + 105875T^2 - 2495625*T + 9150625
$61$	$ T^{8} + 3 T^{7} + \cdots + 6561 $ T^8 + 3T^7 - 45T^6 + 378T^5 + 3159T^4 - 6318T^3 + 7290T^2 - 8748*T + 6561
$67$	$ (T^{4} + T^{3} + \cdots + 10201)^{2} $ (T^4 + T^3 + 102T^2 - 101T + 10201)^2
$71$	$ (T^{4} + 27 T^{3} + \cdots + 6561)^{2} $ (T^4 + 27T^3 + 324T^2 + 1458*T + 6561)^2
$73$	$ (T^{4} - 6 T^{3} + 16 T^{2} + \cdots + 16)^{2} $ (T^4 - 6T^3 + 16T^2 - 16*T + 16)^2
$79$	$ T^{8} + 5 T^{7} + \cdots + 625 $ T^8 + 5T^7 - 60T^6 + 575T^5 + 7575T^4 - 6625T^3 + 3500T^2 - 1875*T + 625
$83$	$ T^{8} + 21 T^{7} + \cdots + 6561 $ T^8 + 21T^7 + 270T^6 + 3429T^5 + 27459T^4 + 10449T^3 - 7290T^2 + 6561*T + 6561
$89$	$ (T^{2} - 10 T + 5)^{4} $ (T^2 - 10*T + 5)^4
$97$	$ T^{8} + \cdots + 1982119441 $ T^8 - 33T^7 + 455T^6 - 7418T^5 + 134619T^4 - 1342382T^3 + 17363190T^2 - 300605792*T + 1982119441
show more
show less

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:	\( N \)	\(=\)	\( 891 = 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	891.n (of order \(15\), degree \(8\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{15}^{5} - \zeta_{15}^{2}) q^{2} + ( - \zeta_{15}^{5} + \zeta_{15}^{4} - 1) q^{4} + (\zeta_{15}^{7} - \zeta_{15}^{4} + \zeta_{15}) q^{5} + 3 \zeta_{15}^{2} q^{7} + ( - 2 \zeta_{15}^{6} - \zeta_{15}^{3} - 2) q^{8}+O(q^{10}) \) q + (-z^5 - z^2) * q^2 + (-z^5 + z^4 - 1) * q^4 + (z^7 - z^4 + z) * q^5 + 3z^2 q^7 + (-2z^6 - z^3 - 2) q^8	\( q + ( - \zeta_{15}^{5} - \zeta_{15}^{2}) q^{2} + ( - \zeta_{15}^{5} + \zeta_{15}^{4} - 1) q^{4} + (\zeta_{15}^{7} - \zeta_{15}^{4} + \zeta_{15}) q^{5} + 3 \zeta_{15}^{2} q^{7} + ( - 2 \zeta_{15}^{6} - \zeta_{15}^{3} - 2) q^{8} + (\zeta_{15}^{7} + \cdots + \zeta_{15}^{2}) q^{10}+ \cdots + ( - 2 \zeta_{15}^{7} + 2 \zeta_{15}^{3} + \cdots + 2) q^{98}+O(q^{100}) \) q + (-z^5 - z^2) * q^2 + (-z^5 + z^4 - 1) * q^4 + (z^7 - z^4 + z) * q^5 + 3z^2 q^7 + (-2z^6 - z^3 - 2) q^8 + (z^7 - z^3 + z^2) * q^10 + (-z^7 + 3z^6 + 2z^5 - z^4 + z^3 - z^2 + 2z + 1) q^11 + (-2z^7 + 2z^5 + 3z + 2) q^13 + (-3z^7 - 3z^4) * q^14 + (3z^7 + 3z^4 - 3z^3 + 3z - 3) * q^16 + (z^7 - z^6 + z^2) * q^17 + (-3z^6 + z^3 - 3) q^19 + (-3z^6 + 2z^5 + 2z^2 - 3z) * q^20 + (-4z^7 + 4z^5 - 3z^4 - z + 4) q^22 + (-z^5 + 4z^4 + 4z - 1) * q^23 + (-3z^7 + 3z^6 - 3z^4 + 3z^3 + 3) * q^25 + (-2z^7 - 5z^6 - 5z^3 - 2z^2) * q^26 + (-3z^7 + 3z^6 - 3z^2) q^28 + (4z^7 - 4z^6 - 4z^5 + 4z^4 - 4z^3 + 2z^2 - 4) * q^29 + (3z^7 - 3z^5 + z - 3) * q^31 + (5z^5 - z^4 - z + 5) q^32 - z^5 * q^34 + (3z^7 - 6z^6 + 3z^2 - 3) q^35 + (z^7 + 2z^6 + 2z^3 + z^2) * q^37 + (2z^7 - 3z^6 + 2z^4 - 2z^3 + 3z^2 - z - 2) q^38 + (-z^7 - 3z^5 + 3z^4 - 3) * q^40 + (-z^7 - 7z^5 - z^4 + 7z - 7) * q^41 + (-2z^7 + 5z^5 - 2z^4 + 2z^3 - 2z^2 - 2z + 2) * q^43 + (3z^7 - 5z^6 - 2z^3 + 3z^2 - 3) * q^44 + (-3z^7 - 4z^6 - 3z^2 + 3) q^46 + (z^6 - z^5 + z) * q^47 + 2z^4 q^49 + (-3z^7 - 6z^4 - 3z) q^50 + (2z^7 - 3z^6 + z^5 + 2z^4 - 2z^3 - z - 2) * q^52 + (z^7 - 9z^6 + z^2 - 1) q^53 + (-2z^7 + 5z^6 - z^3 - 2z^2 + 5) q^55 + (-6z^7 + 3z^5 - 6z^4 + 6z^3 - 6z^2 - 6z + 6) * q^56 + (6z^7 - 4z^5 + 6z^4 + 4z - 4) * q^58 + (-z^7 + 7z^5 - 7z^4 + 7) * q^59 + (-3z^7 + 6z^6 - 3z^4 + 3z^3 - 6z^2 + 3z + 3) * q^61 + (3z^7 + 2z^6 + 2z^3 + 3z^2) * q^62 + (2z^7 + z^6 + 2z^2 - 2) * q^64 + (z^7 - 4z^5 + z^4 - z^3 + z^2 + z - 1) q^65 + (4z^5 - 9z^4 - 9z + 4) q^67 + (-z^7 + z^5 - 2z + 1) q^68 + (3z^7 - 3z^6 - 3z^5 + 3z^4 - 3z^3 + 3z^2 - 3) * q^70 + (-9z^3 - 9) q^71 + (2z^7 - 2z^6 - 2z^3 + 2z^2) * q^73 + (-5z^7 + 3z^6 + 2z^5 - 5z^4 + 5z^3 - 2z + 5) * q^74 + (7z^5 - 4z^4 - 4z + 7) q^76 + (12z^7 - 6z^5 + 6z^4 + 9z - 6) * q^77 + (-7z^6 + 4z^5 + 4z^2 - 7z) * q^79 + (3z^6 + 3) q^80 + (8z^7 - 8z^6 - 9z^3 + 8z^2 - 9) * q^82 + (9z^7 - 3z^6 + 9z^4 - 9z^3 + 3z^2 + 6z - 9) * q^83 + (z^7 - 3z^5 + z^4 + 3z - 3) * q^85 + (-3z^7 + 3z^5 - 2z + 3) q^86 + (-2z^7 - 4z^5 - 2z^4 + 2z^3 + 3z^2 - 2z + 2) * q^88 + (-4z^7 + 4z^3 - 4z^2 + 7) q^89 + (6z^6 + 15z^3 + 6) * q^91 + (-z^7 + z^6 + z^5 - z^4 + z^3 - 5z^2 + 1) q^92 - z^4 * q^94 + (-4z^7 - 7z^5 + 7z^4 - 7) q^95 + (6z^6 - 13z^5 - 13z^2 + 6z) * q^97 + (-2z^7 + 2z^3 - 2z^2 + 2) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{2} - 3 q^{4} + q^{5} + 3 q^{7} - 10 q^{8}+O(q^{10}) \) 8 * q + 3 * q^2 - 3 * q^4 + q^5 + 3 * q^7 - 10 * q^8	\( 8 q + 3 q^{2} - 3 q^{4} + q^{5} + 3 q^{7} - 10 q^{8} + 4 q^{10} - 9 q^{11} + 9 q^{13} - 6 q^{14} - 9 q^{16} + 4 q^{17} - 20 q^{19} - 3 q^{20} + 8 q^{22} + 4 q^{23} + 6 q^{25} + 16 q^{26} - 12 q^{28} + 10 q^{29} - 8 q^{31} + 18 q^{32} + 4 q^{34} - 6 q^{35} - 6 q^{37} - 10 q^{40} - 23 q^{41} - 16 q^{43} - 4 q^{44} + 26 q^{46} + 3 q^{47} + 2 q^{49} - 12 q^{50} - 7 q^{52} + 12 q^{53} + 28 q^{55} + 20 q^{59} - 3 q^{61} - 2 q^{62} - 14 q^{64} + 14 q^{65} - 2 q^{67} + q^{68} + 9 q^{70} - 54 q^{71} + 12 q^{73} + 4 q^{74} + 20 q^{76} + 3 q^{77} - 5 q^{79} + 18 q^{80} - 22 q^{82} - 21 q^{83} - 7 q^{85} + 7 q^{86} + 25 q^{88} + 40 q^{89} + 6 q^{91} - 7 q^{92} - q^{94} - 25 q^{95} + 33 q^{97} + 8 q^{98}+O(q^{100}) \) 8 * q + 3 * q^2 - 3 * q^4 + q^5 + 3 * q^7 - 10 * q^8 + 4 * q^10 - 9 * q^11 + 9 * q^13 - 6 * q^14 - 9 * q^16 + 4 * q^17 - 20 * q^19 - 3 * q^20 + 8 * q^22 + 4 * q^23 + 6 * q^25 + 16 * q^26 - 12 * q^28 + 10 * q^29 - 8 * q^31 + 18 * q^32 + 4 * q^34 - 6 * q^35 - 6 * q^37 - 10 * q^40 - 23 * q^41 - 16 * q^43 - 4 * q^44 + 26 * q^46 + 3 * q^47 + 2 * q^49 - 12 * q^50 - 7 * q^52 + 12 * q^53 + 28 * q^55 + 20 * q^59 - 3 * q^61 - 2 * q^62 - 14 * q^64 + 14 * q^65 - 2 * q^67 + q^68 + 9 * q^70 - 54 * q^71 + 12 * q^73 + 4 * q^74 + 20 * q^76 + 3 * q^77 - 5 * q^79 + 18 * q^80 - 22 * q^82 - 21 * q^83 - 7 * q^85 + 7 * q^86 + 25 * q^88 + 40 * q^89 + 6 * q^91 - 7 * q^92 - q^94 - 25 * q^95 + 33 * q^97 + 8 * q^98

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	891.2.n.d

Level	$891$
Weight	$2$
Character orbit	891.n
Analytic conductor	$7.115$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(7.11467082010\)
Analytic rank:	\(0\)
Dimension:	\(8\)
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:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

\(n\)	\(244\)	\(650\)
\(\chi(n)\)	\(-\zeta_{15}^{2} - \zeta_{15}^{7}\)	\(-1 - \zeta_{15}^{5}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 136.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} - 3 T^{7} + \cdots + 1 \) T^8 - 3T^7 + 5T^6 - 8T^5 + 9T^4 - 2T^3 - 2T + 1
$3$	\( T^{8} \) T^8
$5$	\( T^{8} - T^{7} - 5 T^{6} + \cdots + 1 \) T^8 - T^7 - 5T^6 - 14T^5 + 39T^4 + 26T^3 + 10T^2 + 4T + 1
$7$	\( T^{8} - 3 T^{7} + \cdots + 6561 \) T^8 - 3T^7 + 27T^5 - 81T^4 + 243T^3 - 2187*T + 6561
$11$	\( T^{8} + 9 T^{7} + \cdots + 14641 \) T^8 + 9T^7 + 40T^6 + 171T^5 + 669T^4 + 1881T^3 + 4840T^2 + 11979*T + 14641
$13$	\( T^{8} - 9 T^{7} + \cdots + 14641 \) T^8 - 9T^7 + 50T^6 - 301T^5 + 939T^4 + 2519T^3 - 3630T^2 + 1331*T + 14641
$17$	\( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) (T^4 - 2T^3 + 4T^2 - 3*T + 1)^2
$19$	\( (T^{4} + 10 T^{3} + \cdots + 25)^{2} \) (T^4 + 10T^3 + 40T^2 + 25*T + 25)^2
$23$	\( (T^{4} - 2 T^{3} + \cdots + 361)^{2} \) (T^4 - 2T^3 + 23T^2 + 38*T + 361)^2
$29$	\( T^{8} - 10 T^{7} + \cdots + 160000 \) T^8 - 10T^7 + 40T^6 - 200T^5 + 1200T^4 - 4000T^3 + 16000T^2 - 80000*T + 160000
$31$	\( T^{8} + 8 T^{7} + \cdots + 14641 \) T^8 + 8T^7 + 30T^6 + 118T^5 + 419T^4 + 682T^3 + 1815T^2 + 9317*T + 14641
$37$	\( (T^{4} + 3 T^{3} + 19 T^{2} + \cdots + 1)^{2} \) (T^4 + 3T^3 + 19T^2 + 7*T + 1)^2
$41$	\( T^{8} + 23 T^{7} + \cdots + 25411681 \) T^8 + 23T^7 + 280T^6 + 3313T^5 + 29199T^4 + 68657T^3 + 201640T^2 + 6084487*T + 25411681
$43$	\( (T^{4} + 8 T^{3} + \cdots + 121)^{2} \) (T^4 + 8T^3 + 53T^2 + 88*T + 121)^2
$47$	\( T^{8} - 3 T^{7} + \cdots + 1 \) T^8 - 3T^7 + 5T^6 - 8T^5 + 9T^4 - 2T^3 - 2T + 1
$53$	\( (T^{4} - 6 T^{3} + \cdots + 5041)^{2} \) (T^4 - 6T^3 + 76T^2 - 781*T + 5041)^2
$59$	\( T^{8} - 20 T^{7} + \cdots + 9150625 \) T^8 - 20T^7 + 210T^6 - 2150T^5 + 16575T^4 - 35750T^3 + 105875T^2 - 2495625*T + 9150625
$61$	\( T^{8} + 3 T^{7} + \cdots + 6561 \) T^8 + 3T^7 - 45T^6 + 378T^5 + 3159T^4 - 6318T^3 + 7290T^2 - 8748*T + 6561
$67$	\( (T^{4} + T^{3} + \cdots + 10201)^{2} \) (T^4 + T^3 + 102T^2 - 101T + 10201)^2
$71$	\( (T^{4} + 27 T^{3} + \cdots + 6561)^{2} \) (T^4 + 27T^3 + 324T^2 + 1458*T + 6561)^2
$73$	\( (T^{4} - 6 T^{3} + 16 T^{2} + \cdots + 16)^{2} \) (T^4 - 6T^3 + 16T^2 - 16*T + 16)^2
$79$	\( T^{8} + 5 T^{7} + \cdots + 625 \) T^8 + 5T^7 - 60T^6 + 575T^5 + 7575T^4 - 6625T^3 + 3500T^2 - 1875*T + 625
$83$	\( T^{8} + 21 T^{7} + \cdots + 6561 \) T^8 + 21T^7 + 270T^6 + 3429T^5 + 27459T^4 + 10449T^3 - 7290T^2 + 6561*T + 6561
$89$	\( (T^{2} - 10 T + 5)^{4} \) (T^2 - 10*T + 5)^4
$97$	\( T^{8} + \cdots + 1982119441 \) T^8 - 33T^7 + 455T^6 - 7418T^5 + 134619T^4 - 1342382T^3 + 17363190T^2 - 300605792*T + 1982119441
show more
show less