LMFDB - Newform orbit 2940.1.be.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2940,1,Mod(1979,2940)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2940, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 3, 5])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2940.1979"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2940.be (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,8,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.46725113714$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{48})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{8}$
Projective field:	Galois closure of 8.2.2134623456000.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q - \zeta_{48}^{11} q^{2} - \zeta_{48}^{16} q^{3} + \zeta_{48}^{22} q^{4} + \zeta_{48}^{20} q^{5} - \zeta_{48}^{3} q^{6} + \zeta_{48}^{9} q^{8} - \zeta_{48}^{8} q^{9} + \zeta_{48}^{7} q^{10} + \zeta_{48}^{14} q^{12} + \cdots + \zeta_{48}^{23} q^{96} +O(q^{100}) $ q - z^11 * q^2 - z^16 * q^3 + z^22 * q^4 + z^20 * q^5 - z^3 * q^6 + z^9 * q^8 - z^8 * q^9 + z^7 * q^10 + z^14 * q^12 + z^12 * q^15 - z^20 * q^16 + (z^22 - z^10) * q^17 + z^19 * q^18 + (z^23 - z^17) * q^19 - z^18 * q^20 + (z^23 + z^17) * q^23 + z * q^24 - z^16 * q^25 - q^27 - z^23 * q^30 + (z^19 + z^13) * q^31 - z^7 * q^32 + (z^21 + z^9) * q^34 + z^6 * q^36 + (z^10 - z^4) * q^38 - z^5 * q^40 + z^4 * q^45 + (z^10 + z^4) * q^46 + (-z^14 - z^2) * q^47 - z^12 * q^48 - z^3 * q^50 + (z^14 - z^2) * q^51 + (z^7 - z) * q^53 + z^11 * q^54 + (z^15 - z^9) * q^57 - z^10 * q^60 + (z^11 - z^5) * q^61 + (z^6 + 1) * q^62 + z^18 * q^64 + (-z^20 + z^8) * q^68 + (z^15 + z^9) * q^69 - z^17 * q^72 - z^8 * q^75 + (-z^21 + z^15) * q^76 + (-z^14 + z^2) * q^79 + z^16 * q^80 + z^16 * q^81 + (z^18 - z^6) * q^83 + (-z^18 + z^6) * q^85 - z^15 * q^90 + (-z^21 - z^15) * q^92 + (z^11 + z^5) * q^93 + (z^13 - z) * q^94 + (-z^19 + z^13) * q^95 + z^23 * q^96
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{3} - 8 q^{9} + 8 q^{25} - 16 q^{27} + 16 q^{62} + 8 q^{68} - 8 q^{75} - 8 q^{80} - 8 q^{81}+O(q^{100}) $ 16 * q + 8 * q^3 - 8 * q^9 + 8 * q^25 - 16 * q^27 + 16 * q^62 + 8 * q^68 - 8 * q^75 - 8 * q^80 - 8 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times$.

$n$	$1081$	$1177$	$1471$	$1961$
$\chi(n)$	$\zeta_{48}^{8}$	$-1$	$-1$	$-1$

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} $

1979.1

−0.130526	−	0.991445i
−0.608761	+	0.793353i
0.793353	+	0.608761i
0.991445	−	0.130526i
−0.991445	+	0.130526i
−0.793353	−	0.608761i
0.608761	−	0.793353i
0.130526	+	0.991445i
−0.130526	+	0.991445i
−0.608761	−	0.793353i
0.793353	−	0.608761i
0.991445	+	0.130526i
−0.991445	−	0.130526i
−0.793353	+	0.608761i
0.608761	+	0.793353i
0.130526	−	0.991445i

−0.991445

−

0.130526i

0.500000

0.866025i

0.965926

0.258819i

−0.866025

−

0.500000i

−0.382683

−

0.923880i

−0.923880

−

0.382683i

−0.500000

0.866025i

0.793353

0.608761i

1979.2

−0.793353

0.608761i

0.500000

0.866025i

0.258819

−

0.965926i

0.866025

0.500000i

−0.923880

−

0.382683i

0.382683

0.923880i

−0.500000

0.866025i

−0.991445

0.130526i

1979.3

−0.608761

−

0.793353i

0.500000

0.866025i

−0.258819

0.965926i

0.866025

0.500000i

0.382683

−

0.923880i

0.923880

−

0.382683i

−0.500000

0.866025i

−0.130526

−

0.991445i

1979.4

−0.130526

0.991445i

0.500000

0.866025i

−0.965926

−

0.258819i

−0.866025

−

0.500000i

−0.923880

0.382683i

0.382683

−

0.923880i

−0.500000

0.866025i

0.608761

−

0.793353i

1979.5

0.130526

−

0.991445i

0.500000

0.866025i

−0.965926

−

0.258819i

−0.866025

−

0.500000i

0.923880

−

0.382683i

−0.382683

0.923880i

−0.500000

0.866025i

−0.608761

0.793353i

1979.6

0.608761

0.793353i

0.500000

0.866025i

−0.258819

0.965926i

0.866025

0.500000i

−0.382683

0.923880i

−0.923880

0.382683i

−0.500000

0.866025i

0.130526

0.991445i

1979.7

0.793353

−

0.608761i

0.500000

0.866025i

0.258819

−

0.965926i

0.866025

0.500000i

0.923880

0.382683i

−0.382683

−

0.923880i

−0.500000

0.866025i

0.991445

−

0.130526i

1979.8

0.991445

0.130526i

0.500000

0.866025i

0.965926

0.258819i

−0.866025

−

0.500000i

0.382683

0.923880i

0.923880

0.382683i

−0.500000

0.866025i

−0.793353

−

0.608761i

2579.1

−0.991445

0.130526i

0.500000

−

0.866025i

0.965926

−

0.258819i

−0.866025

0.500000i

−0.382683

0.923880i

−0.923880

0.382683i

−0.500000

−

0.866025i

0.793353

−

0.608761i

2579.2

−0.793353

−

0.608761i

0.500000

−

0.866025i

0.258819

0.965926i

0.866025

−

0.500000i

−0.923880

0.382683i

0.382683

−

0.923880i

−0.500000

−

0.866025i

−0.991445

−

0.130526i

2579.3

−0.608761

0.793353i

0.500000

−

0.866025i

−0.258819

−

0.965926i

0.866025

−

0.500000i

0.382683

0.923880i

0.923880

0.382683i

−0.500000

−

0.866025i

−0.130526

0.991445i

2579.4

−0.130526

−

0.991445i

0.500000

−

0.866025i

−0.965926

0.258819i

−0.866025

0.500000i

−0.923880

−

0.382683i

0.382683

0.923880i

−0.500000

−

0.866025i

0.608761

0.793353i

2579.5

0.130526

0.991445i

0.500000

−

0.866025i

−0.965926

0.258819i

−0.866025

0.500000i

0.923880

0.382683i

−0.382683

−

0.923880i

−0.500000

−

0.866025i

−0.608761

−

0.793353i

2579.6

0.608761

−

0.793353i

0.500000

−

0.866025i

−0.258819

−

0.965926i

0.866025

−

0.500000i

−0.382683

−

0.923880i

−0.923880

−

0.382683i

−0.500000

−

0.866025i

0.130526

−

0.991445i

2579.7

0.793353

0.608761i

0.500000

−

0.866025i

0.258819

0.965926i

0.866025

−

0.500000i

0.923880

−

0.382683i

−0.382683

0.923880i

−0.500000

−

0.866025i

0.991445

0.130526i

2579.8

0.991445

−

0.130526i

0.500000

−

0.866025i

0.965926

−

0.258819i

−0.866025

0.500000i

0.382683

−

0.923880i

0.923880

−

0.382683i

−0.500000

−

0.866025i

−0.793353

0.608761i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $
7.c	even	3	1	inner
12.b	even	2	1	inner
20.d	odd	2	1	inner
21.c	even	2	1	inner
21.g	even	6	1	inner
28.d	even	2	1	inner
28.f	even	6	1	inner
35.c	odd	2	1	inner
35.i	odd	6	1	inner
84.n	even	6	1	inner
105.o	odd	6	1	inner
140.p	odd	6	1	inner
420.o	odd	2	1	inner
420.be	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2940.1.be.f		16
3.b	odd	2	1		2940.1.be.e		16
4.b	odd	2	1		2940.1.be.e		16
5.b	even	2	1		2940.1.be.e		16
7.b	odd	2	1		2940.1.be.e		16
7.c	even	3	1		2940.1.o.c	✓	8
7.c	even	3	1	inner	2940.1.be.f		16
7.d	odd	6	1		2940.1.o.d	yes	8
7.d	odd	6	1		2940.1.be.e		16
12.b	even	2	1	inner	2940.1.be.f		16
15.d	odd	2	1	CM	2940.1.be.f		16
20.d	odd	2	1	inner	2940.1.be.f		16
21.c	even	2	1	inner	2940.1.be.f		16
21.g	even	6	1		2940.1.o.c	✓	8
21.g	even	6	1	inner	2940.1.be.f		16
21.h	odd	6	1		2940.1.o.d	yes	8
21.h	odd	6	1		2940.1.be.e		16
28.d	even	2	1	inner	2940.1.be.f		16
28.f	even	6	1		2940.1.o.c	✓	8
28.f	even	6	1	inner	2940.1.be.f		16
28.g	odd	6	1		2940.1.o.d	yes	8
28.g	odd	6	1		2940.1.be.e		16
35.c	odd	2	1	inner	2940.1.be.f		16
35.i	odd	6	1		2940.1.o.c	✓	8
35.i	odd	6	1	inner	2940.1.be.f		16
35.j	even	6	1		2940.1.o.d	yes	8
35.j	even	6	1		2940.1.be.e		16
60.h	even	2	1		2940.1.be.e		16
84.h	odd	2	1		2940.1.be.e		16
84.j	odd	6	1		2940.1.o.d	yes	8
84.j	odd	6	1		2940.1.be.e		16
84.n	even	6	1		2940.1.o.c	✓	8
84.n	even	6	1	inner	2940.1.be.f		16
105.g	even	2	1		2940.1.be.e		16
105.o	odd	6	1		2940.1.o.c	✓	8
105.o	odd	6	1	inner	2940.1.be.f		16
105.p	even	6	1		2940.1.o.d	yes	8
105.p	even	6	1		2940.1.be.e		16
140.c	even	2	1		2940.1.be.e		16
140.p	odd	6	1		2940.1.o.c	✓	8
140.p	odd	6	1	inner	2940.1.be.f		16
140.s	even	6	1		2940.1.o.d	yes	8
140.s	even	6	1		2940.1.be.e		16
420.o	odd	2	1	inner	2940.1.be.f		16
420.ba	even	6	1		2940.1.o.d	yes	8
420.ba	even	6	1		2940.1.be.e		16
420.be	odd	6	1		2940.1.o.c	✓	8
420.be	odd	6	1	inner	2940.1.be.f		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2940.1.o.c	✓	8	7.c	even	3	1
2940.1.o.c	✓	8	21.g	even	6	1
2940.1.o.c	✓	8	28.f	even	6	1
2940.1.o.c	✓	8	35.i	odd	6	1
2940.1.o.c	✓	8	84.n	even	6	1
2940.1.o.c	✓	8	105.o	odd	6	1
2940.1.o.c	✓	8	140.p	odd	6	1
2940.1.o.c	✓	8	420.be	odd	6	1
2940.1.o.d	yes	8	7.d	odd	6	1
2940.1.o.d	yes	8	21.h	odd	6	1
2940.1.o.d	yes	8	28.g	odd	6	1
2940.1.o.d	yes	8	35.j	even	6	1
2940.1.o.d	yes	8	84.j	odd	6	1
2940.1.o.d	yes	8	105.p	even	6	1
2940.1.o.d	yes	8	140.s	even	6	1
2940.1.o.d	yes	8	420.ba	even	6	1
2940.1.be.e		16	3.b	odd	2	1
2940.1.be.e		16	4.b	odd	2	1
2940.1.be.e		16	5.b	even	2	1
2940.1.be.e		16	7.b	odd	2	1
2940.1.be.e		16	7.d	odd	6	1
2940.1.be.e		16	21.h	odd	6	1
2940.1.be.e		16	28.g	odd	6	1
2940.1.be.e		16	35.j	even	6	1
2940.1.be.e		16	60.h	even	2	1
2940.1.be.e		16	84.h	odd	2	1
2940.1.be.e		16	84.j	odd	6	1
2940.1.be.e		16	105.g	even	2	1
2940.1.be.e		16	105.p	even	6	1
2940.1.be.e		16	140.c	even	2	1
2940.1.be.e		16	140.s	even	6	1
2940.1.be.e		16	420.ba	even	6	1
2940.1.be.f		16	1.a	even	1	1	trivial
2940.1.be.f		16	7.c	even	3	1	inner
2940.1.be.f		16	12.b	even	2	1	inner
2940.1.be.f		16	15.d	odd	2	1	CM
2940.1.be.f		16	20.d	odd	2	1	inner
2940.1.be.f		16	21.c	even	2	1	inner
2940.1.be.f		16	21.g	even	6	1	inner
2940.1.be.f		16	28.d	even	2	1	inner
2940.1.be.f		16	28.f	even	6	1	inner
2940.1.be.f		16	35.c	odd	2	1	inner
2940.1.be.f		16	35.i	odd	6	1	inner
2940.1.be.f		16	84.n	even	6	1	inner
2940.1.be.f		16	105.o	odd	6	1	inner
2940.1.be.f		16	140.p	odd	6	1	inner
2940.1.be.f		16	420.o	odd	2	1	inner
2940.1.be.f		16	420.be	odd	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(2940, [\chi])$:

$ T_{17}^{4} - 2T_{17}^{2} + 4 $

T17^4 - 2*T17^2 + 4

$ T_{23}^{8} - 4T_{23}^{6} + 14T_{23}^{4} - 8T_{23}^{2} + 4 $

T23^8 - 4*T23^6 + 14*T23^4 - 8*T23^2 + 4

$ T_{41} $

T41

$ T_{983}^{2} + 2T_{983} + 4 $

T983^2 + 2*T983 + 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} - T^{8} + 1 $ T^16 - T^8 + 1
$3$	$ (T^{2} - T + 1)^{8} $ (T^2 - T + 1)^8
$5$	$ (T^{4} - T^{2} + 1)^{4} $ (T^4 - T^2 + 1)^4
$7$	$ T^{16} $ T^16
$11$	$ T^{16} $ T^16
$13$	$ T^{16} $ T^16
$17$	$ (T^{4} - 2 T^{2} + 4)^{4} $ (T^4 - 2*T^2 + 4)^4
$19$	$ (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} $ (T^8 + 4T^6 + 14T^4 + 8*T^2 + 4)^2
$23$	$ (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} $ (T^8 - 4T^6 + 14T^4 - 8*T^2 + 4)^2
$29$	$ T^{16} $ T^16
$31$	$ (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} $ (T^8 + 4T^6 + 14T^4 + 8*T^2 + 4)^2
$37$	$ T^{16} $ T^16
$41$	$ T^{16} $ T^16
$43$	$ T^{16} $ T^16
$47$	$ (T^{4} + 2 T^{2} + 4)^{4} $ (T^4 + 2*T^2 + 4)^4
$53$	$ (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} $ (T^8 + 4T^6 + 14T^4 + 8*T^2 + 4)^2
$59$	$ T^{16} $ T^16
$61$	$ (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} $ (T^8 - 4T^6 + 14T^4 - 8*T^2 + 4)^2
$67$	$ T^{16} $ T^16
$71$	$ T^{16} $ T^16
$73$	$ T^{16} $ T^16
$79$	$ (T^{4} - 2 T^{2} + 4)^{4} $ (T^4 - 2*T^2 + 4)^4
$83$	$ (T^{2} - 2)^{8} $ (T^2 - 2)^8
$89$	$ T^{16} $ T^16
$97$	$ T^{16} $ T^16
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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2940.be (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{48}^{11} q^{2} - \zeta_{48}^{16} q^{3} + \zeta_{48}^{22} q^{4} + \zeta_{48}^{20} q^{5} - \zeta_{48}^{3} q^{6} + \zeta_{48}^{9} q^{8} - \zeta_{48}^{8} q^{9} + \zeta_{48}^{7} q^{10} + \zeta_{48}^{14} q^{12} + \cdots + \zeta_{48}^{23} q^{96} +O(q^{100}) \) q - z^11 * q^2 - z^16 * q^3 + z^22 * q^4 + z^20 * q^5 - z^3 * q^6 + z^9 * q^8 - z^8 * q^9 + z^7 * q^10 + z^14 * q^12 + z^12 * q^15 - z^20 * q^16 + (z^22 - z^10) * q^17 + z^19 * q^18 + (z^23 - z^17) * q^19 - z^18 * q^20 + (z^23 + z^17) * q^23 + z * q^24 - z^16 * q^25 - q^27 - z^23 * q^30 + (z^19 + z^13) * q^31 - z^7 * q^32 + (z^21 + z^9) * q^34 + z^6 * q^36 + (z^10 - z^4) * q^38 - z^5 * q^40 + z^4 * q^45 + (z^10 + z^4) * q^46 + (-z^14 - z^2) * q^47 - z^12 * q^48 - z^3 * q^50 + (z^14 - z^2) * q^51 + (z^7 - z) * q^53 + z^11 * q^54 + (z^15 - z^9) * q^57 - z^10 * q^60 + (z^11 - z^5) * q^61 + (z^6 + 1) * q^62 + z^18 * q^64 + (-z^20 + z^8) * q^68 + (z^15 + z^9) * q^69 - z^17 * q^72 - z^8 * q^75 + (-z^21 + z^15) * q^76 + (-z^14 + z^2) * q^79 + z^16 * q^80 + z^16 * q^81 + (z^18 - z^6) * q^83 + (-z^18 + z^6) * q^85 - z^15 * q^90 + (-z^21 - z^15) * q^92 + (z^11 + z^5) * q^93 + (z^13 - z) * q^94 + (-z^19 + z^13) * q^95 + z^23 * q^96
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{3} - 8 q^{9} + 8 q^{25} - 16 q^{27} + 16 q^{62} + 8 q^{68} - 8 q^{75} - 8 q^{80} - 8 q^{81}+O(q^{100}) \) 16 * q + 8 * q^3 - 8 * q^9 + 8 * q^25 - 16 * q^27 + 16 * q^62 + 8 * q^68 - 8 * q^75 - 8 * q^80 - 8 * q^81

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	2940.1.be.f

Level	$2940$
Weight	$1$
Character orbit	2940.be
Analytic conductor	$1.467$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{8}$
CM discriminant	-15
Inner twists	$16$

Self dual:	no
Analytic conductor:	\(1.46725113714\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{48})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - x^{8} + 1 \) x^16 - x^8 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{8}\)
Projective field:	Galois closure of 8.2.2134623456000.5

\(n\)	\(1081\)	\(1177\)	\(1471\)	\(1961\)
\(\chi(n)\)	\(\zeta_{48}^{8}\)	\(-1\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1979.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} - T^{8} + 1 \) T^16 - T^8 + 1
$3$	\( (T^{2} - T + 1)^{8} \) (T^2 - T + 1)^8
$5$	\( (T^{4} - T^{2} + 1)^{4} \) (T^4 - T^2 + 1)^4
$7$	\( T^{16} \) T^16
$11$	\( T^{16} \) T^16
$13$	\( T^{16} \) T^16
$17$	\( (T^{4} - 2 T^{2} + 4)^{4} \) (T^4 - 2*T^2 + 4)^4
$19$	\( (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) (T^8 + 4T^6 + 14T^4 + 8*T^2 + 4)^2
$23$	\( (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) (T^8 - 4T^6 + 14T^4 - 8*T^2 + 4)^2
$29$	\( T^{16} \) T^16
$31$	\( (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) (T^8 + 4T^6 + 14T^4 + 8*T^2 + 4)^2
$37$	\( T^{16} \) T^16
$41$	\( T^{16} \) T^16
$43$	\( T^{16} \) T^16
$47$	\( (T^{4} + 2 T^{2} + 4)^{4} \) (T^4 + 2*T^2 + 4)^4
$53$	\( (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) (T^8 + 4T^6 + 14T^4 + 8*T^2 + 4)^2
$59$	\( T^{16} \) T^16
$61$	\( (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) (T^8 - 4T^6 + 14T^4 - 8*T^2 + 4)^2
$67$	\( T^{16} \) T^16
$71$	\( T^{16} \) T^16
$73$	\( T^{16} \) T^16
$79$	\( (T^{4} - 2 T^{2} + 4)^{4} \) (T^4 - 2*T^2 + 4)^4
$83$	\( (T^{2} - 2)^{8} \) (T^2 - 2)^8
$89$	\( T^{16} \) T^16
$97$	\( T^{16} \) T^16
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