LMFDB - Newform orbit 1050.2.bc.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,2,Mod(157,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("1050.157");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1050.bc (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:	$8.38429221223$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{24}$. We also show the integral $q$-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$\zeta_{24}^{6}$	$1 - \zeta_{24}^{4}$	$1$

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} $

157.1

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

−0.258819

0.965926i

0.965926

−

0.258819i

−0.866025

−

0.500000i

1.00000i

−2.19067

1.48356i

0.707107

−

0.707107i

0.866025

−

0.500000i

157.2

0.258819

−

0.965926i

−0.965926

0.258819i

−0.866025

−

0.500000i

1.00000i

2.19067

−

1.48356i

−0.707107

0.707107i

0.866025

−

0.500000i

493.1

−0.965926

−

0.258819i

0.258819

0.965926i

0.866025

0.500000i

−

1.00000i

−1.48356

−

2.19067i

−0.707107

−

0.707107i

−0.866025

0.500000i

493.2

0.965926

0.258819i

−0.258819

−

0.965926i

0.866025

0.500000i

−

1.00000i

1.48356

2.19067i

0.707107

0.707107i

−0.866025

0.500000i

607.1

−0.965926

0.258819i

0.258819

−

0.965926i

0.866025

−

0.500000i

1.00000i

−1.48356

2.19067i

−0.707107

0.707107i

−0.866025

−

0.500000i

607.2

0.965926

−

0.258819i

−0.258819

0.965926i

0.866025

−

0.500000i

1.00000i

1.48356

−

2.19067i

0.707107

−

0.707107i

−0.866025

−

0.500000i

943.1

−0.258819

−

0.965926i

0.965926

0.258819i

−0.866025

0.500000i

−

1.00000i

−2.19067

−

1.48356i

0.707107

0.707107i

0.866025

0.500000i

943.2

0.258819

0.965926i

−0.965926

−

0.258819i

−0.866025

0.500000i

−

1.00000i

2.19067

1.48356i

−0.707107

−

0.707107i

0.866025

0.500000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
35.k	even	12	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1050.2.bc.b	✓	8
5.b	even	2	1	inner	1050.2.bc.b	✓	8
5.c	odd	4	2		1050.2.bc.c	yes	8
7.d	odd	6	1		1050.2.bc.c	yes	8
35.i	odd	6	1		1050.2.bc.c	yes	8
35.k	even	12	2	inner	1050.2.bc.b	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1050.2.bc.b	✓	8	1.a	even	1	1	trivial
1050.2.bc.b	✓	8	5.b	even	2	1	inner
1050.2.bc.b	✓	8	35.k	even	12	2	inner
1050.2.bc.c	yes	8	5.c	odd	4	2
1050.2.bc.c	yes	8	7.d	odd	6	1
1050.2.bc.c	yes	8	35.i	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1050, [\chi])$:

$ T_{11}^{4} + 2T_{11}^{3} + 6T_{11}^{2} - 4T_{11} + 4 $

$ T_{13}^{8} + 896T_{13}^{4} + 4096 $

$ T_{17}^{8} + 48T_{17}^{6} + 599T_{17}^{4} - 8112T_{17}^{2} + 28561 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} - T^{4} + 1 $ T^8 - T^4 + 1
$3$	$ T^{8} - T^{4} + 1 $ T^8 - T^4 + 1
$5$	$ T^{8} $ T^8
$7$	$ T^{8} + 71T^{4} + 2401 $ T^8 + 71*T^4 + 2401
$11$	$ (T^{4} + 2 T^{3} + 6 T^{2} + \cdots + 4)^{2} $ (T^4 + 2T^3 + 6T^2 - 4*T + 4)^2
$13$	$ T^{8} + 896T^{4} + 4096 $ T^8 + 896*T^4 + 4096
$17$	$ T^{8} + 48 T^{6} + \cdots + 28561 $ T^8 + 48T^6 + 599T^4 - 8112*T^2 + 28561
$19$	$ (T^{4} + 2 T^{3} + \cdots + 676)^{2} $ (T^4 + 2T^3 + 30T^2 - 52*T + 676)^2
$23$	$ T^{8} + 24 T^{6} + \cdots + 1 $ T^8 + 24T^6 + 191T^4 - 24*T^2 + 1
$29$	$ (T^{4} + 72 T^{2} + 324)^{2} $ (T^4 + 72*T^2 + 324)^2
$31$	$ (T^{4} - 24 T^{3} + \cdots + 2209)^{2} $ (T^4 - 24T^3 + 239T^2 - 1128*T + 2209)^2
$37$	$ T^{8} - 36 T^{6} + \cdots + 1296 $ T^8 - 36T^6 + 396T^4 + 1296*T^2 + 1296
$41$	$ (T^{4} + 26 T^{2} + 121)^{2} $ (T^4 + 26*T^2 + 121)^2
$43$	$ T^{8} + 10808T^{4} + 16 $ T^8 + 10808*T^4 + 16
$47$	$ T^{8} - 14641 T^{4} + 214358881 $ T^8 - 14641*T^4 + 214358881
$53$	$ T^{8} + 84 T^{6} + \cdots + 4477456 $ T^8 + 84T^6 + 236T^4 - 177744*T^2 + 4477456
$59$	$ (T^{4} - 8 T^{3} + 60 T^{2} + \cdots + 16)^{2} $ (T^4 - 8T^3 + 60T^2 - 32*T + 16)^2
$61$	$ (T^{4} - 18 T^{3} + \cdots + 484)^{2} $ (T^4 - 18T^3 + 86T^2 + 396*T + 484)^2
$67$	$ T^{8} - 256 T^{4} + 65536 $ T^8 - 256*T^4 + 65536
$71$	$ (T^{2} - 2 T - 191)^{4} $ (T^2 - 2*T - 191)^4
$73$	$ T^{8} - 4096 T^{4} + 16777216 $ T^8 - 4096*T^4 + 16777216
$79$	$ (T^{4} + 42 T^{3} + \cdots + 17161)^{2} $ (T^4 + 42T^3 + 719T^2 + 5502*T + 17161)^2
$83$	$ T^{8} + 19512 T^{4} + 18974736 $ T^8 + 19512*T^4 + 18974736
$89$	$ (T^{4} - 6 T^{3} + 39 T^{2} + \cdots + 9)^{2} $ (T^4 - 6T^3 + 39T^2 + 18*T + 9)^2
$97$	$ (T^{4} + 2401)^{2} $ (T^4 + 2401)^2
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1050.bc (of order \(12\), degree \(4\), minimal)

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

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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	1050.2.bc.b

Level	$1050$
Weight	$2$
Character orbit	1050.bc
Analytic conductor	$8.384$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(8.38429221223\)
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:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

\(n\)	\(127\)	\(451\)	\(701\)
\(\chi(n)\)	\(\zeta_{24}^{6}\)	\(1 - \zeta_{24}^{4}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 157.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} - T^{4} + 1 \) T^8 - T^4 + 1
$3$	\( T^{8} - T^{4} + 1 \) T^8 - T^4 + 1
$5$	\( T^{8} \) T^8
$7$	\( T^{8} + 71T^{4} + 2401 \) T^8 + 71*T^4 + 2401
$11$	\( (T^{4} + 2 T^{3} + 6 T^{2} + \cdots + 4)^{2} \) (T^4 + 2T^3 + 6T^2 - 4*T + 4)^2
$13$	\( T^{8} + 896T^{4} + 4096 \) T^8 + 896*T^4 + 4096
$17$	\( T^{8} + 48 T^{6} + \cdots + 28561 \) T^8 + 48T^6 + 599T^4 - 8112*T^2 + 28561
$19$	\( (T^{4} + 2 T^{3} + \cdots + 676)^{2} \) (T^4 + 2T^3 + 30T^2 - 52*T + 676)^2
$23$	\( T^{8} + 24 T^{6} + \cdots + 1 \) T^8 + 24T^6 + 191T^4 - 24*T^2 + 1
$29$	\( (T^{4} + 72 T^{2} + 324)^{2} \) (T^4 + 72*T^2 + 324)^2
$31$	\( (T^{4} - 24 T^{3} + \cdots + 2209)^{2} \) (T^4 - 24T^3 + 239T^2 - 1128*T + 2209)^2
$37$	\( T^{8} - 36 T^{6} + \cdots + 1296 \) T^8 - 36T^6 + 396T^4 + 1296*T^2 + 1296
$41$	\( (T^{4} + 26 T^{2} + 121)^{2} \) (T^4 + 26*T^2 + 121)^2
$43$	\( T^{8} + 10808T^{4} + 16 \) T^8 + 10808*T^4 + 16
$47$	\( T^{8} - 14641 T^{4} + 214358881 \) T^8 - 14641*T^4 + 214358881
$53$	\( T^{8} + 84 T^{6} + \cdots + 4477456 \) T^8 + 84T^6 + 236T^4 - 177744*T^2 + 4477456
$59$	\( (T^{4} - 8 T^{3} + 60 T^{2} + \cdots + 16)^{2} \) (T^4 - 8T^3 + 60T^2 - 32*T + 16)^2
$61$	\( (T^{4} - 18 T^{3} + \cdots + 484)^{2} \) (T^4 - 18T^3 + 86T^2 + 396*T + 484)^2
$67$	\( T^{8} - 256 T^{4} + 65536 \) T^8 - 256*T^4 + 65536
$71$	\( (T^{2} - 2 T - 191)^{4} \) (T^2 - 2*T - 191)^4
$73$	\( T^{8} - 4096 T^{4} + 16777216 \) T^8 - 4096*T^4 + 16777216
$79$	\( (T^{4} + 42 T^{3} + \cdots + 17161)^{2} \) (T^4 + 42T^3 + 719T^2 + 5502*T + 17161)^2
$83$	\( T^{8} + 19512 T^{4} + 18974736 \) T^8 + 19512*T^4 + 18974736
$89$	\( (T^{4} - 6 T^{3} + 39 T^{2} + \cdots + 9)^{2} \) (T^4 - 6T^3 + 39T^2 + 18*T + 9)^2
$97$	\( (T^{4} + 2401)^{2} \) (T^4 + 2401)^2
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