LMFDB - Newform orbit 390.2.bn.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(67,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

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

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

chi := DirichletCharacter("390.67");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	390.bn (of order $12$, 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:	$3.11416567883$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/390\mathbb{Z}\right)^\times$.

$n$	$131$	$157$	$301$
$\chi(n)$	$1$	$\zeta_{24}^{6}$	$\zeta_{24}^{2}$

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.

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} $

67.1

−0.965926	−	0.258819i
0.965926	+	0.258819i
−0.258819	−	0.965926i
0.258819	+	0.965926i
−0.965926	+	0.258819i
0.965926	−	0.258819i
−0.258819	+	0.965926i
0.258819	−	0.965926i

0.500000

0.866025i

−0.965926

−

0.258819i

−0.500000

0.866025i

0.707107

−

2.12132i

−0.258819

−

0.965926i

−1.46945

−

0.848387i

−1.00000

0.866025

0.500000i

2.19067

−

0.448288i

67.2

0.500000

0.866025i

0.965926

0.258819i

−0.500000

0.866025i

−0.707107

2.12132i

0.258819

0.965926i

−3.26260

−

1.88366i

−1.00000

0.866025

0.500000i

−2.19067

0.448288i

97.1

0.500000

−

0.866025i

−0.258819

−

0.965926i

−0.500000

−

0.866025i

−0.707107

2.12132i

−0.965926

−

0.258819i

−3.98004

2.29788i

−1.00000

−0.866025

0.500000i

1.48356

1.67303i

97.2

0.500000

−

0.866025i

0.258819

0.965926i

−0.500000

−

0.866025i

0.707107

−

2.12132i

0.965926

0.258819i

2.71209

−

1.56583i

−1.00000

−0.866025

0.500000i

−1.48356

−

1.67303i

163.1

0.500000

−

0.866025i

−0.965926

0.258819i

−0.500000

−

0.866025i

0.707107

2.12132i

−0.258819

0.965926i

−1.46945

0.848387i

−1.00000

0.866025

−

0.500000i

2.19067

0.448288i

163.2

0.500000

−

0.866025i

0.965926

−

0.258819i

−0.500000

−

0.866025i

−0.707107

−

2.12132i

0.258819

−

0.965926i

−3.26260

1.88366i

−1.00000

0.866025

−

0.500000i

−2.19067

−

0.448288i

193.1

0.500000

0.866025i

−0.258819

0.965926i

−0.500000

0.866025i

−0.707107

−

2.12132i

−0.965926

0.258819i

−3.98004

−

2.29788i

−1.00000

−0.866025

−

0.500000i

1.48356

−

1.67303i

193.2

0.500000

0.866025i

0.258819

−

0.965926i

−0.500000

0.866025i

0.707107

2.12132i

0.965926

−

0.258819i

2.71209

1.56583i

−1.00000

−0.866025

−

0.500000i

−1.48356

1.67303i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
65.o	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	390.2.bn.a	yes	8
5.c	odd	4	1		390.2.bd.a	✓	8
13.f	odd	12	1		390.2.bd.a	✓	8
65.o	even	12	1	inner	390.2.bn.a	yes	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
390.2.bd.a	✓	8	5.c	odd	4	1
390.2.bd.a	✓	8	13.f	odd	12	1
390.2.bn.a	yes	8	1.a	even	1	1	trivial
390.2.bn.a	yes	8	65.o	even	12	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{8} + 12T_{7}^{7} + 48T_{7}^{6} - 388T_{7}^{4} + 4800T_{7}^{2} + 11040T_{7} + 8464 $ T7^8 + 12*T7^7 + 48*T7^6 - 388*T7^4 + 4800*T7^2 + 11040*T7 + 8464 acting on $S_{2}^{\mathrm{new}}(390, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} - T + 1)^{4} $ (T^2 - T + 1)^4
$3$	$ T^{8} - T^{4} + 1 $ T^8 - T^4 + 1
$5$	$ (T^{4} + 8 T^{2} + 25)^{2} $ (T^4 + 8*T^2 + 25)^2
$7$	$ T^{8} + 12 T^{7} + \cdots + 8464 $ T^8 + 12T^7 + 48T^6 - 388T^4 + 4800T^2 + 11040*T + 8464
$11$	$ T^{8} + 16 T^{7} + \cdots + 36481 $ T^8 + 16T^7 + 128T^6 + 696T^5 + 2687T^4 + 7416T^3 + 16928T^2 + 33616*T + 36481
$13$	$ (T^{4} + 24 T^{2} + 169)^{2} $ (T^4 + 24*T^2 + 169)^2
$17$	$ T^{8} - 12 T^{7} + \cdots + 10000 $ T^8 - 12T^7 + 72T^6 - 288T^5 - 4T^4 + 2880T^3 + 7200T^2 + 12000*T + 10000
$19$	$ T^{8} - 24 T^{7} + \cdots + 150544 $ T^8 - 24T^7 + 300T^6 - 2496T^5 + 13868T^4 - 49344T^3 + 111696T^2 - 167616*T + 150544
$23$	$ T^{8} + 4 T^{7} + \cdots + 5041 $ T^8 + 4T^7 + 38T^6 + 484T^5 + 691T^4 - 4012T^3 + 7442T^2 - 9088*T + 5041
$29$	$ T^{8} - 24 T^{7} + \cdots + 2211169 $ T^8 - 24T^7 + 204T^6 - 288T^5 - 4705T^4 + 9504T^3 + 191244T^2 - 1177704*T + 2211169
$31$	$ T^{8} + 4 T^{7} + \cdots + 529 $ T^8 + 4T^7 + 8T^6 - 92T^5 + 2350T^4 + 4892T^3 + 5000T^2 + 2300*T + 529
$37$	$ T^{8} - 24 T^{7} + \cdots + 529 $ T^8 - 24T^7 + 228T^6 - 864T^5 + 1127T^4 + 864T^3 - 636T^2 - 552*T + 529
$41$	$ T^{8} - 28 T^{7} + \cdots + 8464 $ T^8 - 28T^7 + 344T^6 - 2640T^5 + 13436T^4 - 31776T^3 + 26336T^2 - 736*T + 8464
$43$	$ T^{8} + 16 T^{7} + \cdots + 625 $ T^8 + 16T^7 + 80T^6 + 168T^5 + 551T^4 - 840T^3 + 2000T^2 - 2000*T + 625
$47$	$ T^{8} + 108 T^{6} + \cdots + 96721 $ T^8 + 108T^6 + 3830T^4 + 47052*T^2 + 96721
$53$	$ (T^{4} + 8 T^{3} + \cdots + 7744)^{2} $ (T^4 + 8T^3 + 32T^2 - 704*T + 7744)^2
$59$	$ T^{8} + 32 T^{7} + \cdots + 3411409 $ T^8 + 32T^7 + 464T^6 + 5384T^5 + 58999T^4 + 428824T^3 + 1529552T^2 + 2186848*T + 3411409
$61$	$ T^{8} + 8 T^{7} + \cdots + 234256 $ T^8 + 8T^7 + 188T^6 - 640T^5 + 16300T^4 + 14080T^3 + 90992T^2 - 85184*T + 234256
$67$	$ (T^{4} + 8 T^{3} + 66 T^{2} + \cdots + 4)^{2} $ (T^4 + 8T^3 + 66T^2 - 16*T + 4)^2
$71$	$ T^{8} - 8 T^{7} + \cdots + 37552384 $ T^8 - 8T^7 + 128T^6 + 384T^5 - 10000T^4 + 75264T^3 - 185344T^2 - 5686784*T + 37552384
$73$	$ (T^{4} + 4 T^{3} + \cdots + 4036)^{2} $ (T^4 + 4T^3 - 160T^2 - 568*T + 4036)^2
$79$	$ T^{8} + 324 T^{6} + \cdots + 7834401 $ T^8 + 324T^6 + 34470T^4 + 1270404*T^2 + 7834401
$83$	$ T^{8} + 552 T^{6} + \cdots + 238764304 $ T^8 + 552T^6 + 106712T^4 + 8501664*T^2 + 238764304
$89$	$ T^{8} - 4 T^{7} + \cdots + 10000 $ T^8 - 4T^7 + 200T^6 - 2592T^5 + 12476T^4 - 25920T^3 + 20000T^2 - 4000*T + 10000
$97$	$ T^{8} + 4 T^{7} + \cdots + 150544 $ T^8 + 4T^7 + 64T^6 + 16T^5 + 2332T^4 + 1888T^3 + 29440T^2 - 40352*T + 150544
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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:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	390.bn (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{24}^{4} q^{2} + \zeta_{24} q^{3} + (\zeta_{24}^{4} - 1) q^{4} + (2 \zeta_{24}^{5} + \cdots - 2 \zeta_{24}) q^{5}+ \cdots + \zeta_{24}^{2} q^{9}+O(q^{10}) \) q + z^4 * q^2 + z * q^3 + (z^4 - 1) * q^4 + (2z^5 + z^3 - 2z) * q^5 + z^5 * q^6 + (-2z^7 + 2z^5 - z^4 - z^2 - 2z - 1) q^7 - q^8 + z^2 * q^9	\( q + \zeta_{24}^{4} q^{2} + \zeta_{24} q^{3} + (\zeta_{24}^{4} - 1) q^{4} + (2 \zeta_{24}^{5} + \cdots - 2 \zeta_{24}) q^{5}+ \cdots + (2 \zeta_{24}^{7} + \cdots - 2 \zeta_{24}) q^{99}+O(q^{100}) \) q + z^4 * q^2 + z * q^3 + (z^4 - 1) * q^4 + (2z^5 + z^3 - 2z) * q^5 + z^5 * q^6 + (-2z^7 + 2z^5 - z^4 - z^2 - 2z - 1) q^7 - q^8 + z^2 * q^9 + (z^7 - 2z) q^10 + (2z^7 + 2z^5 + 2z^2 - z - 2) q^11 + (z^5 - z) * q^12 + (3z^5 + 2z^3 - 3z) q^13 + (-2z^7 - z^6 - 2z^4 + 2z^3 - 2z + 1) * q^14 + (2z^6 + z^4 - 2z^2) * q^15 - z^4 * q^16 + (2z^6 + 4z^5 + z^4 - z^2 + 1) * q^17 + z^6 * q^18 + (z^7 - 4z^6 - z^5 + 2z^4 - z^3 + 2z^2 + 2z + 2) * q^19 + (z^7 - 2z^5 - z^3) q^20 + (2z^6 - z^5 - 2z^4 - z^3 - 2z^2 - z + 2) q^21 + (2z^7 + 2z^6 + z^5 - 2z^4 - 2z^3 - 2z) q^22 + (-3z^6 - 4z^5 - 3z^4 + 2z^2 + 2z + 1) q^23 - z * q^24 + (-3z^6 - 4) q^25 + (2z^7 - 3z) * q^26 + z^3 * q^27 + (-z^6 - 2z^5 - z^4 + 2z^3 + z^2 + 2) * q^28 + (-z^7 - 2z^6 + 3z^5 - 2z^4 - 3z^3 + 2z^2 + z + 4) q^29 + (z^4 - 2z^2 - 1) q^30 + (-4z^7 + 2z^6 + 3z^4 + 2z^3 - 3z^2 - 2) q^31 + (-z^4 + 1) * q^32 + (2z^6 + 2z^4 + 2z^3 - z^2 - 2z - 2) * q^33 + (z^6 + 4z^5 + 2z^4 - 2z^2 - 4z - 1) * q^34 + (-3z^7 - 6z^6 - 3z^5 + 4z^4 + z^3 + 2z^2 + 4z - 2) * q^35 + (z^6 - z^2) * q^36 + (2z^7 + 2z^6 - z^5 - 2z^4 + z^3 - 2z^2 - 2z + 4) q^37 + (-2z^6 + z^5 + 4z^4 - z^3 + 4z^2 + z - 2) q^38 + (3z^6 + 2z^4 - 3z^2) q^39 + (-2z^5 - z^3 + 2z) * q^40 + (z^6 - z^4 + 3z^2 - 4z + 4) * q^41 + (-z^7 - 2z^5 - 2z^2 + z + 2) * q^42 + (-z^7 + 2z^6 - 2z^5 + z^3 - 2z^2 + 4z - 2) * q^43 + (2z^6 - z^5 - 2z^4 - 2z^3 - 2z^2 - z + 2) * q^44 + (2z^7 + z^5 - 2z^3) * q^45 + (-z^6 - 2z^5 - 2z^4 + 3z^2 + 4z + 3) * q^46 + (-2z^7 + 4z^6 - 2z^5 - 2z^4 + 1) * q^47 - z^5 * q^48 + (4z^7 - 2z^6 + 5z^4 - 2z^2 + 4z) q^49 + (-3z^6 - 4z^4 + 3z^2) q^50 + (2z^7 + 4z^6 + z^5 - z^3 + z) * q^51 + (2z^7 - 3z^5 - 2z^3) q^52 + (-2z^6 - 8z^4 + 8z^2 + 2) q^53 + z^7 * q^54 + (4z^7 - 2z^5 - 3z^4 - 6z^3 - 4z^2 + 4z - 2) * q^55 + (2z^7 - 2z^5 + z^4 + z^2 + 2z + 1) q^56 + (-4z^7 - z^6 + 2z^5 + 2z^3 + 2z^2 + 2z - 1) q^57 + (-4z^7 + 4z^5 + 2z^4 + z^3 + 2z^2 - 3z + 2) q^58 + (2z^7 - 2z^6 + 3z^5 + 4z^4 + 2z^3 + 6z^2 - 6) * q^59 + (-2z^6 - 1) q^60 + (-3z^7 + 8z^6 - 3z^5 + 2z^4 - z^3 - 4z^2 + 4z - 2) * q^61 + (-2z^7 - z^6 + z^4 + 4z^3 - 2z^2 - 3) q^62 + (2z^7 - z^6 - 2z^5 - z^4 - 2z^3 - z^2 + 2z) * q^63 + q^64 + (-4z^6 - 7) q^65 + (2z^7 + z^6 - 2z^5 - 2z^2 - 2) q^66 + (3z^7 - 4z^4 + 3z) q^67 + (-z^6 + z^4 - z^2 - 4z - 2) q^68 + (-3z^7 - 4z^6 - 3z^5 + 2z^3 + 2z^2 + z) q^69 + (-2z^7 - 4z^6 + z^5 + 2z^4 + 3z^3 + 6z^2 + 3z - 4) * q^70 + (-2z^7 + 4z^6 + 4z^5 + 6z^4 - 2z^3 + 2z^2 - 2) * q^71 - z^2 * q^72 + (-2z^7 + 5z^6 + 2z^5 - 10z^2 - 1) * q^73 + (3z^7 - 3z^5 + 2z^4 - 2z^3 - 2z^2 + z + 2) q^74 + (-3z^7 - 4z) * q^75 + (-z^7 + 2z^6 + 2z^5 + 2z^4 + 2z^2 - z - 4) * q^76 + (2z^7 - 4z^6 - 13z^5 - 2z^4 - z^3 + 2z^2 + 13z + 4) * q^77 + (2z^4 - 3z^2 - 2) * q^78 + (2z^7 - 3z^6 - 2z^5 - 8z^4 - 4z^3 + 4z + 4) * q^79 + (-z^7 + 2z) q^80 + z^4 * q^81 + (4z^6 - 4z^5 + 3z^4 - z^2 + 1) q^82 + (-4z^7 + 4z^6 - 7z^5 - 8z^4 - 3z^3 + 3z + 4) * q^83 + (-z^7 - 2z^6 - z^5 + 2z^4 + z^3 + 2z) q^84 + (-z^7 + 3z^5 + 4z^4 - z^3 - 8z^2 - 6z - 4) * q^85 + (2z^5 - 2z^4 + z^3 - 2z^2 + 2z) * q^86 + (-2z^7 + 3z^6 - 2z^5 - 4z^4 + 2z^3 + z^2 + 4z + 1) * q^87 + (-2z^7 - 2z^5 - 2z^2 + z + 2) q^88 + (-2z^7 - 5z^6 + 5z^4 + 4z^3 + 3z^2 - 4z - 2) * q^89 + (z^5 - 2z^3 - z) q^90 + (-5z^7 - 10z^6 - 5z^5 + 6z^4 + z^3 + 4z^2 + 6z - 4) * q^91 + (2z^6 + 2z^5 + z^4 + z^2 + 2z + 2) q^92 + (2z^7 + 3z^5 - 2z^4 - 3z^3 - 2z + 4) q^93 + (-2z^7 + 4z^6 - 2z^5 - z^4 + 2z^3 - 4z^2 + 2z + 2) * q^94 + (6z^7 + 4z^6 + 2z^5 - z^4 + 6z^3 - 3z^2 - 4z + 3) * q^95 + (-z^5 + z) * q^96 + (2z^7 + 2z^6 + 2z^5 + z^4 + 2z^3 - z^2 - 4z - 1) q^97 + (4z^7 - 4z^6 + 4z^5 + 5z^4 - 4z^3 + 2z^2 - 5) * q^98 + (2z^7 + 2z^5 + 2z^4 - z^3 - 2z^2 - 2z) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} - 4 q^{4} - 12 q^{7} - 8 q^{8}+O(q^{10}) \) 8 * q + 4 * q^2 - 4 * q^4 - 12 * q^7 - 8 * q^8	\( 8 q + 4 q^{2} - 4 q^{4} - 12 q^{7} - 8 q^{8} - 16 q^{11} + 4 q^{15} - 4 q^{16} + 12 q^{17} + 24 q^{19} + 8 q^{21} - 8 q^{22} - 4 q^{23} - 32 q^{25} + 12 q^{28} + 24 q^{29} - 4 q^{30} - 4 q^{31} + 4 q^{32} - 8 q^{33} + 24 q^{37} + 8 q^{39} + 28 q^{41} + 16 q^{42} - 16 q^{43} + 8 q^{44} + 16 q^{46} + 20 q^{49} - 16 q^{50} - 16 q^{53} - 28 q^{55} + 12 q^{56} - 8 q^{57} + 24 q^{58} - 32 q^{59} - 8 q^{60} - 8 q^{61} - 20 q^{62} - 4 q^{63} + 8 q^{64} - 56 q^{65} - 16 q^{66} - 16 q^{67} - 12 q^{68} - 24 q^{70} + 8 q^{71} - 8 q^{73} + 24 q^{74} - 24 q^{76} + 24 q^{77} - 8 q^{78} + 4 q^{81} + 20 q^{82} + 8 q^{84} - 16 q^{85} - 8 q^{86} - 8 q^{87} + 16 q^{88} + 4 q^{89} - 8 q^{91} + 20 q^{92} + 24 q^{93} + 12 q^{94} + 20 q^{95} - 4 q^{97} - 20 q^{98} + 8 q^{99}+O(q^{100}) \) 8 * q + 4 * q^2 - 4 * q^4 - 12 * q^7 - 8 * q^8 - 16 * q^11 + 4 * q^15 - 4 * q^16 + 12 * q^17 + 24 * q^19 + 8 * q^21 - 8 * q^22 - 4 * q^23 - 32 * q^25 + 12 * q^28 + 24 * q^29 - 4 * q^30 - 4 * q^31 + 4 * q^32 - 8 * q^33 + 24 * q^37 + 8 * q^39 + 28 * q^41 + 16 * q^42 - 16 * q^43 + 8 * q^44 + 16 * q^46 + 20 * q^49 - 16 * q^50 - 16 * q^53 - 28 * q^55 + 12 * q^56 - 8 * q^57 + 24 * q^58 - 32 * q^59 - 8 * q^60 - 8 * q^61 - 20 * q^62 - 4 * q^63 + 8 * q^64 - 56 * q^65 - 16 * q^66 - 16 * q^67 - 12 * q^68 - 24 * q^70 + 8 * q^71 - 8 * q^73 + 24 * q^74 - 24 * q^76 + 24 * q^77 - 8 * q^78 + 4 * q^81 + 20 * q^82 + 8 * q^84 - 16 * q^85 - 8 * q^86 - 8 * q^87 + 16 * q^88 + 4 * q^89 - 8 * q^91 + 20 * q^92 + 24 * q^93 + 12 * q^94 + 20 * q^95 - 4 * q^97 - 20 * q^98 + 8 * q^99

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Newspace parameters

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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	390.2.bn.a

Level	$390$
Weight	$2$
Character orbit	390.bn
Analytic conductor	$3.114$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(3.11416567883\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\Q(\zeta_{24})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{4} + 1 \) x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

\(n\)	\(131\)	\(157\)	\(301\)
\(\chi(n)\)	\(1\)	\(\zeta_{24}^{6}\)	\(\zeta_{24}^{2}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 67.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} - T + 1)^{4} \) (T^2 - T + 1)^4
$3$	\( T^{8} - T^{4} + 1 \) T^8 - T^4 + 1
$5$	\( (T^{4} + 8 T^{2} + 25)^{2} \) (T^4 + 8*T^2 + 25)^2
$7$	\( T^{8} + 12 T^{7} + \cdots + 8464 \) T^8 + 12T^7 + 48T^6 - 388T^4 + 4800T^2 + 11040*T + 8464
$11$	\( T^{8} + 16 T^{7} + \cdots + 36481 \) T^8 + 16T^7 + 128T^6 + 696T^5 + 2687T^4 + 7416T^3 + 16928T^2 + 33616*T + 36481
$13$	\( (T^{4} + 24 T^{2} + 169)^{2} \) (T^4 + 24*T^2 + 169)^2
$17$	\( T^{8} - 12 T^{7} + \cdots + 10000 \) T^8 - 12T^7 + 72T^6 - 288T^5 - 4T^4 + 2880T^3 + 7200T^2 + 12000*T + 10000
$19$	\( T^{8} - 24 T^{7} + \cdots + 150544 \) T^8 - 24T^7 + 300T^6 - 2496T^5 + 13868T^4 - 49344T^3 + 111696T^2 - 167616*T + 150544
$23$	\( T^{8} + 4 T^{7} + \cdots + 5041 \) T^8 + 4T^7 + 38T^6 + 484T^5 + 691T^4 - 4012T^3 + 7442T^2 - 9088*T + 5041
$29$	\( T^{8} - 24 T^{7} + \cdots + 2211169 \) T^8 - 24T^7 + 204T^6 - 288T^5 - 4705T^4 + 9504T^3 + 191244T^2 - 1177704*T + 2211169
$31$	\( T^{8} + 4 T^{7} + \cdots + 529 \) T^8 + 4T^7 + 8T^6 - 92T^5 + 2350T^4 + 4892T^3 + 5000T^2 + 2300*T + 529
$37$	\( T^{8} - 24 T^{7} + \cdots + 529 \) T^8 - 24T^7 + 228T^6 - 864T^5 + 1127T^4 + 864T^3 - 636T^2 - 552*T + 529
$41$	\( T^{8} - 28 T^{7} + \cdots + 8464 \) T^8 - 28T^7 + 344T^6 - 2640T^5 + 13436T^4 - 31776T^3 + 26336T^2 - 736*T + 8464
$43$	\( T^{8} + 16 T^{7} + \cdots + 625 \) T^8 + 16T^7 + 80T^6 + 168T^5 + 551T^4 - 840T^3 + 2000T^2 - 2000*T + 625
$47$	\( T^{8} + 108 T^{6} + \cdots + 96721 \) T^8 + 108T^6 + 3830T^4 + 47052*T^2 + 96721
$53$	\( (T^{4} + 8 T^{3} + \cdots + 7744)^{2} \) (T^4 + 8T^3 + 32T^2 - 704*T + 7744)^2
$59$	\( T^{8} + 32 T^{7} + \cdots + 3411409 \) T^8 + 32T^7 + 464T^6 + 5384T^5 + 58999T^4 + 428824T^3 + 1529552T^2 + 2186848*T + 3411409
$61$	\( T^{8} + 8 T^{7} + \cdots + 234256 \) T^8 + 8T^7 + 188T^6 - 640T^5 + 16300T^4 + 14080T^3 + 90992T^2 - 85184*T + 234256
$67$	\( (T^{4} + 8 T^{3} + 66 T^{2} + \cdots + 4)^{2} \) (T^4 + 8T^3 + 66T^2 - 16*T + 4)^2
$71$	\( T^{8} - 8 T^{7} + \cdots + 37552384 \) T^8 - 8T^7 + 128T^6 + 384T^5 - 10000T^4 + 75264T^3 - 185344T^2 - 5686784*T + 37552384
$73$	\( (T^{4} + 4 T^{3} + \cdots + 4036)^{2} \) (T^4 + 4T^3 - 160T^2 - 568*T + 4036)^2
$79$	\( T^{8} + 324 T^{6} + \cdots + 7834401 \) T^8 + 324T^6 + 34470T^4 + 1270404*T^2 + 7834401
$83$	\( T^{8} + 552 T^{6} + \cdots + 238764304 \) T^8 + 552T^6 + 106712T^4 + 8501664*T^2 + 238764304
$89$	\( T^{8} - 4 T^{7} + \cdots + 10000 \) T^8 - 4T^7 + 200T^6 - 2592T^5 + 12476T^4 - 25920T^3 + 20000T^2 - 4000*T + 10000
$97$	\( T^{8} + 4 T^{7} + \cdots + 150544 \) T^8 + 4T^7 + 64T^6 + 16T^5 + 2332T^4 + 1888T^3 + 29440T^2 - 40352*T + 150544
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