LMFDB - Newform orbit 369.2.n.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [369,2,Mod(64,369)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(369, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 9]))

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

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

chi := DirichletCharacter("369.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 369 = 3^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	369.n (of order $10$, 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:	$2.94647983459$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 41)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/369\mathbb{Z}\right)^\times$.

$n$	$83$	$334$
$\chi(n)$	$1$	$\zeta_{15}^{2} + \zeta_{15}^{7}$

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} $

64.1

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

−0.413545

−

1.27276i

0.169131

−

0.122881i

2.78716

−

2.02499i

−4.07890

−

1.32531i

−2.39169

−

1.73767i

−3.72995

−

2.70997i

64.2

0.604528

1.86055i

−1.47815

1.07394i

1.13989

−

0.828176i

2.26988

0.737529i

0.273659

0.198825i

2.22995

1.62016i

127.1

−0.169131

0.122881i

−0.604528

1.86055i

−0.222562

0.684977i

−1.79833

2.47520i

−0.255585

−

0.786610i

−0.0465282

−

0.143199i

127.2

1.47815

−

1.07394i

0.413545

−

1.27276i

0.795511

−

2.44833i

1.10735

−

1.52414i

0.373619

1.14988i

−1.45347

−

4.47333i

154.1

−0.169131

−

0.122881i

−0.604528

−

1.86055i

−0.222562

−

0.684977i

−1.79833

−

2.47520i

−0.255585

0.786610i

−0.0465282

0.143199i

154.2

1.47815

1.07394i

0.413545

1.27276i

0.795511

2.44833i

1.10735

1.52414i

0.373619

−

1.14988i

−1.45347

4.47333i

271.1

−0.413545

1.27276i

0.169131

0.122881i

2.78716

2.02499i

−4.07890

1.32531i

−2.39169

1.73767i

−3.72995

2.70997i

271.2

0.604528

−

1.86055i

−1.47815

−

1.07394i

1.13989

0.828176i

2.26988

−

0.737529i

0.273659

−

0.198825i

2.22995

−

1.62016i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
41.f	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	369.2.n.c		8
3.b	odd	2	1		41.2.f.a	✓	8
12.b	even	2	1		656.2.be.a		8
41.f	even	10	1	inner	369.2.n.c		8
123.l	odd	10	1		41.2.f.a	✓	8
123.m	odd	20	2		1681.2.a.h		8
492.v	even	10	1		656.2.be.a		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
41.2.f.a	✓	8	3.b	odd	2	1
41.2.f.a	✓	8	123.l	odd	10	1
369.2.n.c		8	1.a	even	1	1	trivial
369.2.n.c		8	41.f	even	10	1	inner
656.2.be.a		8	12.b	even	2	1
656.2.be.a		8	492.v	even	10	1
1681.2.a.h		8	123.m	odd	20	2

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} - 3T_{2}^{7} + 8T_{2}^{6} - 11T_{2}^{5} + 15T_{2}^{4} - 11T_{2}^{3} + 18T_{2}^{2} + 7T_{2} + 1 $ T2^8 - 3*T2^7 + 8*T2^6 - 11*T2^5 + 15*T2^4 - 11*T2^3 + 18*T2^2 + 7*T2 + 1 acting on $S_{2}^{\mathrm{new}}(369, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} - 3 T^{7} + \cdots + 1 $ T^8 - 3T^7 + 8T^6 - 11T^5 + 15T^4 - 11T^3 + 18T^2 + 7*T + 1
$3$	$ T^{8} $ T^8
$5$	$ T^{8} - 9 T^{7} + \cdots + 81 $ T^8 - 9T^7 + 42T^6 - 117T^5 + 225T^4 - 243T^3 + 162T^2 - 81*T + 81
$7$	$ T^{8} + 5 T^{7} + \cdots + 3481 $ T^8 + 5T^7 - 3T^6 - 45T^5 - 6T^4 + 185T^3 + 383T^2 - 2065*T + 3481
$11$	$ T^{8} - 3 T^{6} + \cdots + 3481 $ T^8 - 3T^6 + 50T^5 + 139T^4 - 150T^3 - 797T^2 - 590T + 3481
$13$	$ T^{8} + 5 T^{7} + \cdots + 1 $ T^8 + 5T^7 + 18T^6 - 15T^5 - 51T^4 - 145T^3 + 332T^2 + 35*T + 1
$17$	$ T^{8} - 75 T^{6} + \cdots + 93025 $ T^8 - 75T^6 + 95T^5 + 2110T^4 - 7125T^3 + 13225T^2 + 28975T + 93025
$19$	$ T^{8} + 20 T^{7} + \cdots + 961 $ T^8 + 20T^7 + 147T^6 + 480T^5 + 894T^4 + 260T^3 - 532T^2 - 1240*T + 961
$23$	$ T^{8} + 15 T^{7} + \cdots + 50625 $ T^8 + 15T^7 + 120T^6 + 675T^5 + 4275T^4 + 16875T^3 + 40500T^2 + 50625*T + 50625
$29$	$ T^{8} - 25 T^{7} + \cdots + 2163841 $ T^8 - 25T^7 + 233T^6 - 1225T^5 + 9624T^4 - 90675T^3 + 498267T^2 - 1471000*T + 2163841
$31$	$ T^{8} - 6 T^{7} + \cdots + 20736 $ T^8 - 6T^7 + 72T^6 - 648T^5 + 3600T^4 - 11232T^3 + 24192T^2 - 31104*T + 20736
$37$	$ T^{8} + 4 T^{7} + \cdots + 516961 $ T^8 + 4T^7 + 37T^6 - 28T^5 + 465T^4 + 5548T^3 + 54157T^2 + 175436*T + 516961
$41$	$ T^{8} + T^{7} + \cdots + 2825761 $ T^8 + T^7 + 70T^6 - 151T^5 + 3039T^4 - 6191T^3 + 117670T^2 + 68921T + 2825761
$43$	$ T^{8} + 13 T^{7} + \cdots + 3721 $ T^8 + 13T^7 + 148T^6 + 881T^5 + 3915T^4 + 10861T^3 + 17878T^2 + 12383*T + 3721
$47$	$ T^{8} - 20 T^{7} + \cdots + 961 $ T^8 - 20T^7 + 217T^6 - 1320T^5 + 4849T^4 - 9640T^3 + 6153T^2 + 5580*T + 961
$53$	$ T^{8} + 20 T^{7} + \cdots + 961 $ T^8 + 20T^7 + 183T^6 + 960T^5 + 2934T^4 + 4580T^3 + 2132T^2 - 1240*T + 961
$59$	$ T^{8} + 7 T^{7} + \cdots + 961 $ T^8 + 7T^7 + 48T^6 + 119T^5 + 155T^4 - 121T^3 + 1248T^2 - 1643*T + 961
$61$	$ T^{8} - 29 T^{7} + \cdots + 6355441 $ T^8 - 29T^7 + 672T^6 - 9037T^5 + 91175T^4 - 608293T^3 + 2555322T^2 - 5926871*T + 6355441
$67$	$ T^{8} - 40 T^{7} + \cdots + 292681 $ T^8 - 40T^7 + 673T^6 - 6240T^5 + 35284T^4 - 101720T^3 + 91782T^2 + 470670*T + 292681
$71$	$ T^{8} - 30 T^{7} + \cdots + 68121 $ T^8 - 30T^7 + 363T^6 - 2205T^5 + 6894T^4 - 10665T^3 + 9747T^2 - 11745*T + 68121
$73$	$ (T^{4} + 24 T^{3} + \cdots - 1889)^{2} $ (T^4 + 24T^3 + 146T^2 - 111*T - 1889)^2
$79$	$ T^{8} + 187 T^{6} + \cdots + 2343961 $ T^8 + 187T^6 + 11654T^4 + 285428*T^2 + 2343961
$83$	$ (T^{4} - 28 T^{3} + \cdots - 13919)^{2} $ (T^4 - 28T^3 + 129T^2 + 1928*T - 13919)^2
$89$	$ T^{8} + 35 T^{7} + \cdots + 326041 $ T^8 + 35T^7 + 592T^6 + 5655T^5 + 29749T^4 + 72985T^3 + 36408T^2 - 25695*T + 326041
$97$	$ T^{8} - 20 T^{7} + \cdots + 20205025 $ T^8 - 20T^7 + 220T^6 + 3390T^5 - 33290T^4 - 96400T^3 + 2105325T^2 - 9844050*T + 20205025
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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:	\( N \)	\(=\)	\( 369 = 3^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	369.n (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{15}^{5} - \zeta_{15}^{2} + 1) q^{2} + (\zeta_{15}^{5} + \zeta_{15}^{4}) q^{4} + (2 \zeta_{15}^{7} + \cdots + 2 \zeta_{15}^{2}) q^{5}+ \cdots + ( - \zeta_{15}^{4} + \cdots - \zeta_{15}^{2}) q^{8}+O(q^{10}) \) q + (z^5 - z^2 + 1) * q^2 + (z^5 + z^4) * q^4 + (2z^7 - 2z^6 + z^5 + z^4 - 2z^3 + 2z^2) * q^5 + (2z^7 + z^6 + z^4 - z^2 + 3z - 1) * q^7 + (-z^4 + z^3 - z^2) * q^8	\( q + (\zeta_{15}^{5} - \zeta_{15}^{2} + 1) q^{2} + (\zeta_{15}^{5} + \zeta_{15}^{4}) q^{4} + (2 \zeta_{15}^{7} + \cdots + 2 \zeta_{15}^{2}) q^{5}+ \cdots + (5 \zeta_{15}^{7} - 7 \zeta_{15}^{6} + \cdots + 3) q^{98}+O(q^{100}) \) q + (z^5 - z^2 + 1) * q^2 + (z^5 + z^4) * q^4 + (2z^7 - 2z^6 + z^5 + z^4 - 2z^3 + 2z^2) * q^5 + (2z^7 + z^6 + z^4 - z^2 + 3z - 1) * q^7 + (-z^4 + z^3 - z^2) * q^8 + (2z^5 - z^4 - z^3 - z^2 + 2z) * q^10 + (z^6 - 2z^5 - z^4 + z^2 + 2z - 1) * q^11 + (z^7 - 2z^6 - 3z^3 + z^2 - z - 2) * q^13 + (-3z^7 + 3z^6 + z^4 - z^3 - 3z^2 + z + 1) q^14 + (-z^7 + 2z^6 - 2z^5 + z^4 + z^3 - 4z^2 - z) q^16 + (6z^7 - 3z^6 + 5z^4 - 3z^3 + z^2 - 3) * q^17 + (-2z^6 + 2z^4 + z^3 + 2z^2 - 2z - 3) * q^19 + (-z^7 + 2z^5 - z^4 - z^3 - z) q^20 + (z^7 + 4z^6 - 3z^4 + z^2 + 3) * q^22 + (-z^7 + 4z^6 + 3z^5 - 4z^4 + 2z^3 - 2z + 2) q^23 + (-2z^7 - z^6 + 6z^5 - 3z^4 + 2z^3 + z^2 - 3z + 5) q^25 + (-z^7 + 2z^5 - 2z^4 + z^2) * q^26 + (-z^7 - 2z^6 - 2z^3 - z^2 - 2z - 3) q^28 + (-3z^7 - z^6 - 2z^5 + 2z^4 + 4z^3 + 3) * q^29 + (2z^5 + 2z^4 - 4z^3 + 2z^2 + 2z) q^31 + (-3z^7 - 2z^6 + 2z^4 + 3z^3 - 3z^2 - 2z + 1) * q^32 + (-4z^6 + 4z^4 - 2z^3 - 3z^2 + 3z - 2) q^34 + (-z^7 + 4z^6 - 4z^5 - 5z^2 + 8z - 5) * q^35 + (3z^7 + 4z^6 - 4z^5 + 2z^3 + z^2 + 4z - 2) q^37 + (7z^7 - 6z^6 - 7z^5 + 3z^4 - 2z^3 + 7z^2 + 3z - 8) q^38 + (-z^7 + 2z^6 - 2z^5 + 3z^3 - 5z^2 + 4z - 1) q^40 + (-z^7 + 3z^5 - 5z^4 - 7z + 3) q^41 + (4z^7 - 4z^6 + z^5 + 3z^2 - 3) q^43 + (-3z^7 + 5z^6 + 3z^5 - 2z^4 + 4z^3 - 3z^2 - 2z + 5) q^44 + (-8z^7 + 5z^6 + 2z^5 - 6z^4 + 9z^3 - 4z^2 - 8z + 4) q^46 + (z^6 + 4z^3 - 2z + 4) * q^47 + (-3z^5 + 2z^4 - 2z^3 + 2z^2 - 3z) q^49 + (-3z^7 + z^6 - z^4 + 3z^3 - 3z^2 + z - 3) q^50 + (-6z^7 + z^6 + 2z^5 - 5z^4 + 3z^3 - z^2 - 2z + 5) q^52 + (3z^7 - 3z^6 - 2z^5 + 2z^4 - 3z^3 + 3z^2 - 6) * q^53 + (z^7 + 3z^6 - 7z^5 + 2z^4 + 2z^3 + z^2 + 2z - 3) q^55 + (-2z^7 + 3z^6 - 3z^5 + 3z^4 - z^3 + 2) * q^56 + (12z^7 - 7z^6 - 6z^5 + 7z^4 - 6z^3 + 5z^2 + 6z - 5) q^58 + (4z^7 - 2z^6 - 3z^5 + 4z^4 - 2z^3 + 3z^2 + 2z - 5) q^59 + (7z^7 - 8z^6 + 2z^5 - z^4 + 8z^2 - z + 1) * q^61 + (-2z^7 - 2z^6 + 8z^5 - 4z^4 - 4z^3 + 4z^2 - 2z) q^62 + (-3z^7 - z^6 - 5z^5 + 2z^4 - z^3 + z - 1) q^64 + (-2z^7 - z^6 + 4z^5 - 5z^4 + z^3 + 3z^2 - 4z + 3) q^65 + (-9z^7 + 7z^6 + 4z^5 - 4z^4 + 5z^3 - 3z^2 + 12) * q^67 + (-z^7 - 7z^6 - 2z^5 - z^4 - 7z^3 - z^2 - z - 4) q^68 + (-4z^7 + 7z^6 - z^5 + z^4 - 5z^3 + 2z^2 + 2) * q^70 + (-2z^7 - z^6 - 2z^5 + z^4 + z^3 - 3z^2 + 2z + 3) * q^71 + (-2z^7 - 3z^6 + 3z^5 - z^3 + 4z^2 - 6z - 5) q^73 + (7z^6 - 2z^5 - 3z^4 + 3z^3 - 3z^2 - 2z + 7) * q^74 + (8z^7 - 6z^5 - 5z^3 + 2z^2 + 5z - 3) q^76 + (9z^7 - 2z^6 - 2z^3 + 9z^2) * q^77 + (-5z^7 + 6z^6 - 3z^5 - 3z^4 + 4z^3 - 5z^2 - 3z + 3) q^79 + (-3z^7 + 3z^6 - 3z^5 + 6z^4 - 3z^3 - 6z^2 + 3z + 3) q^80 + (-7z^7 + 2z^6 + 3z^5 - 5z^4 + 11z^3 - 6z^2 - 7z + 4) q^82 + (-z^7 + 2z^6 + 4z^5 - 6z^4 - 3z^3 + 7z^2 - 2z + 9) * q^83 + (z^7 - 9z^6 + 8z^5 - z^4 - 7z^3 + z^2 - z) q^85 + (2z^7 + 4z^6 - 7z^5 + z^4 - 2z^2 + 8z - 4) q^86 + (2z^7 - 3z^6 + 2z^5 - 4z^3 + 4z^2 - 2z) * q^88 + (5z^7 + 5z^4 + z^3 + z^2 + 4z - 6) q^89 + (-10z^7 + 4z^6 + 3z^5 - 7z^4 + 7z^3 - 4z^2 + z + 7) * q^91 + (-4z^6 - 5z^5 + 3z^3 - 5z - 4) * q^92 + (3z^7 - 2z^6 - 3z^5 + 3z^4 + 3z^3 - 4z^2 + 1) * q^94 + (-4z^7 - 2z^6 + 2z^5 + 3z^3 - 2z^2 - 7z + 1) * q^95 + (4z^7 + 3z^6 - 8z^5 + 8z^4 - 7z^3 - 4) q^97 + (5z^7 - 7z^6 + 4z^5 - 2z^4 + z^3 + 4z^2 - 5z + 3) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 5 q^{7} - 4 q^{8}+O(q^{10}) \) 8 * q + 3 * q^2 - 3 * q^4 + 9 * q^5 - 5 * q^7 - 4 * q^8	\( 8 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 5 q^{7} - 4 q^{8} - 6 q^{10} - 5 q^{13} - 3 q^{16} - 20 q^{19} - 9 q^{20} + 15 q^{22} - 15 q^{23} + 7 q^{25} - 10 q^{26} - 20 q^{28} + 25 q^{29} + 6 q^{31} - 30 q^{35} - 4 q^{37} - 12 q^{40} - q^{41} - 13 q^{43} - 30 q^{46} + 20 q^{47} + 17 q^{49} - 38 q^{50} + 10 q^{52} - 20 q^{53} + 25 q^{56} + 40 q^{58} - 7 q^{59} + 29 q^{61} - 24 q^{62} + 16 q^{64} + 40 q^{67} + 15 q^{70} + 30 q^{71} - 48 q^{73} + 36 q^{74} + 25 q^{76} + 26 q^{77} + 36 q^{80} - 31 q^{82} + 56 q^{83} - 3 q^{86} + 10 q^{88} - 35 q^{89} + 2 q^{91} - 15 q^{92} + 20 q^{94} - 15 q^{95} + 20 q^{97} + 22 q^{98}+O(q^{100}) \) 8 * q + 3 * q^2 - 3 * q^4 + 9 * q^5 - 5 * q^7 - 4 * q^8 - 6 * q^10 - 5 * q^13 - 3 * q^16 - 20 * q^19 - 9 * q^20 + 15 * q^22 - 15 * q^23 + 7 * q^25 - 10 * q^26 - 20 * q^28 + 25 * q^29 + 6 * q^31 - 30 * q^35 - 4 * q^37 - 12 * q^40 - q^41 - 13 * q^43 - 30 * q^46 + 20 * q^47 + 17 * q^49 - 38 * q^50 + 10 * q^52 - 20 * q^53 + 25 * q^56 + 40 * q^58 - 7 * q^59 + 29 * q^61 - 24 * q^62 + 16 * q^64 + 40 * q^67 + 15 * q^70 + 30 * q^71 - 48 * q^73 + 36 * q^74 + 25 * q^76 + 26 * q^77 + 36 * q^80 - 31 * q^82 + 56 * q^83 - 3 * q^86 + 10 * q^88 - 35 * q^89 + 2 * q^91 - 15 * q^92 + 20 * q^94 - 15 * q^95 + 20 * q^97 + 22 * q^98

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	369.2.n.c

Level	$369$
Weight	$2$
Character orbit	369.n
Analytic conductor	$2.946$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(2.94647983459\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{10})\)
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:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 41)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

\(n\)	\(83\)	\(334\)
\(\chi(n)\)	\(1\)	\(\zeta_{15}^{2} + \zeta_{15}^{7}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 64.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} - 3 T^{7} + \cdots + 1 \) T^8 - 3T^7 + 8T^6 - 11T^5 + 15T^4 - 11T^3 + 18T^2 + 7*T + 1
$3$	\( T^{8} \) T^8
$5$	\( T^{8} - 9 T^{7} + \cdots + 81 \) T^8 - 9T^7 + 42T^6 - 117T^5 + 225T^4 - 243T^3 + 162T^2 - 81*T + 81
$7$	\( T^{8} + 5 T^{7} + \cdots + 3481 \) T^8 + 5T^7 - 3T^6 - 45T^5 - 6T^4 + 185T^3 + 383T^2 - 2065*T + 3481
$11$	\( T^{8} - 3 T^{6} + \cdots + 3481 \) T^8 - 3T^6 + 50T^5 + 139T^4 - 150T^3 - 797T^2 - 590T + 3481
$13$	\( T^{8} + 5 T^{7} + \cdots + 1 \) T^8 + 5T^7 + 18T^6 - 15T^5 - 51T^4 - 145T^3 + 332T^2 + 35*T + 1
$17$	\( T^{8} - 75 T^{6} + \cdots + 93025 \) T^8 - 75T^6 + 95T^5 + 2110T^4 - 7125T^3 + 13225T^2 + 28975T + 93025
$19$	\( T^{8} + 20 T^{7} + \cdots + 961 \) T^8 + 20T^7 + 147T^6 + 480T^5 + 894T^4 + 260T^3 - 532T^2 - 1240*T + 961
$23$	\( T^{8} + 15 T^{7} + \cdots + 50625 \) T^8 + 15T^7 + 120T^6 + 675T^5 + 4275T^4 + 16875T^3 + 40500T^2 + 50625*T + 50625
$29$	\( T^{8} - 25 T^{7} + \cdots + 2163841 \) T^8 - 25T^7 + 233T^6 - 1225T^5 + 9624T^4 - 90675T^3 + 498267T^2 - 1471000*T + 2163841
$31$	\( T^{8} - 6 T^{7} + \cdots + 20736 \) T^8 - 6T^7 + 72T^6 - 648T^5 + 3600T^4 - 11232T^3 + 24192T^2 - 31104*T + 20736
$37$	\( T^{8} + 4 T^{7} + \cdots + 516961 \) T^8 + 4T^7 + 37T^6 - 28T^5 + 465T^4 + 5548T^3 + 54157T^2 + 175436*T + 516961
$41$	\( T^{8} + T^{7} + \cdots + 2825761 \) T^8 + T^7 + 70T^6 - 151T^5 + 3039T^4 - 6191T^3 + 117670T^2 + 68921T + 2825761
$43$	\( T^{8} + 13 T^{7} + \cdots + 3721 \) T^8 + 13T^7 + 148T^6 + 881T^5 + 3915T^4 + 10861T^3 + 17878T^2 + 12383*T + 3721
$47$	\( T^{8} - 20 T^{7} + \cdots + 961 \) T^8 - 20T^7 + 217T^6 - 1320T^5 + 4849T^4 - 9640T^3 + 6153T^2 + 5580*T + 961
$53$	\( T^{8} + 20 T^{7} + \cdots + 961 \) T^8 + 20T^7 + 183T^6 + 960T^5 + 2934T^4 + 4580T^3 + 2132T^2 - 1240*T + 961
$59$	\( T^{8} + 7 T^{7} + \cdots + 961 \) T^8 + 7T^7 + 48T^6 + 119T^5 + 155T^4 - 121T^3 + 1248T^2 - 1643*T + 961
$61$	\( T^{8} - 29 T^{7} + \cdots + 6355441 \) T^8 - 29T^7 + 672T^6 - 9037T^5 + 91175T^4 - 608293T^3 + 2555322T^2 - 5926871*T + 6355441
$67$	\( T^{8} - 40 T^{7} + \cdots + 292681 \) T^8 - 40T^7 + 673T^6 - 6240T^5 + 35284T^4 - 101720T^3 + 91782T^2 + 470670*T + 292681
$71$	\( T^{8} - 30 T^{7} + \cdots + 68121 \) T^8 - 30T^7 + 363T^6 - 2205T^5 + 6894T^4 - 10665T^3 + 9747T^2 - 11745*T + 68121
$73$	\( (T^{4} + 24 T^{3} + \cdots - 1889)^{2} \) (T^4 + 24T^3 + 146T^2 - 111*T - 1889)^2
$79$	\( T^{8} + 187 T^{6} + \cdots + 2343961 \) T^8 + 187T^6 + 11654T^4 + 285428*T^2 + 2343961
$83$	\( (T^{4} - 28 T^{3} + \cdots - 13919)^{2} \) (T^4 - 28T^3 + 129T^2 + 1928*T - 13919)^2
$89$	\( T^{8} + 35 T^{7} + \cdots + 326041 \) T^8 + 35T^7 + 592T^6 + 5655T^5 + 29749T^4 + 72985T^3 + 36408T^2 - 25695*T + 326041
$97$	\( T^{8} - 20 T^{7} + \cdots + 20205025 \) T^8 - 20T^7 + 220T^6 + 3390T^5 - 33290T^4 - 96400T^3 + 2105325T^2 - 9844050*T + 20205025
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