LMFDB - Newform orbit 405.2.m.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,2,Mod(53,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([10, 9]))

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

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

chi := DirichletCharacter("405.53");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	405.m (of order $12$, degree $4$, 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:	$3.23394128186$
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:	no (minimal twist has level 45)
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} q^{2} + \zeta_{24}^{2} q^{4} + (\zeta_{24}^{7} + \cdots - \zeta_{24}^{3}) q^{5}+ \cdots + (3 \zeta_{24}^{5} - 3 \zeta_{24}) q^{8}+O(q^{10}) $ q + z^7 * q^2 + z^2 * q^4 + (z^7 + 2z^5 - z^3) q^5 + (-2z^6 + 2z^4 + 2z^2) q^7 + (3z^5 - 3z) * q^8	$ q + \zeta_{24}^{7} q^{2} + \zeta_{24}^{2} q^{4} + (\zeta_{24}^{7} + \cdots - \zeta_{24}^{3}) q^{5}+ \cdots + (\zeta_{24}^{5} - \zeta_{24}) q^{98}+O(q^{100}) $ q + z^7 * q^2 + z^2 * q^4 + (z^7 + 2z^5 - z^3) q^5 + (-2z^6 + 2z^4 + 2z^2) q^7 + (3z^5 - 3z) * q^8 + (-z^6 - 2) * q^10 + (-2z^7 + 2z) * q^11 + (z^4 - z^2 - 1) * q^13 + (2z^7 + 2z^5 - 2z^3) q^14 - z^4 * q^16 - 4z^3 q^17 + (2z^7 - z) q^20 + (2z^4 + 2z^2 - 2) * q^22 + 4z^5 q^23 + (3z^6 - 4z^4 - 3z^2) q^25 + (-z^5 - z^3 + z) * q^26 + (2z^6 + 2) q^28 + (3z^7 + 3z) * q^29 + (-4z^4 + 4) q^31 + (5z^7 - 5z^3) * q^32 + (-4z^6 + 4z^2) * q^34 + (6z^5 + 2z^3 - 6z) q^35 + (-z^6 + 1) * q^37 + (-3z^4 - 6z^2 + 3) * q^40 + (-z^7 + z^5 + z^3) * q^41 + (8z^6 + 8z^4 - 8z^2) q^43 + (-2z^5 + 2z^3 + 2z) q^44 - 4 * q^46 - 8z^7 q^47 + z^2 * q^49 + (-4z^7 - 3z^5 + 4z^3) q^50 + (z^6 - z^4 - z^2) * q^52 + (-4z^5 + 4z) * q^53 + (6z^6 + 2) q^55 + (6z^7 - 6z) * q^56 + (3z^4 - 3z^2 - 3) * q^58 + (-6z^7 - 6z^5 + 6z^3) q^59 - 8z^4 q^61 + 4z^3 q^62 - 7z^6 q^64 + (-3z^7 - z) q^65 + (4z^4 + 4z^2 - 4) * q^67 - 4z^5 q^68 + (2z^6 - 6z^4 - 2z^2) q^70 + (-4z^5 - 4z^3 + 4z) q^71 + (z^6 + 1) * q^73 + (z^7 + z) * q^74 + (-8z^7 + 8z^3) * q^77 + (-12z^6 + 12z^2) * q^79 + (-2z^5 + z^3 + 2z) * q^80 + (z^6 - 1) * q^82 - 4z q^83 + (-8z^4 + 4z^2 + 8) * q^85 + (8z^7 - 8z^5 - 8z^3) q^86 + (6z^6 + 6z^4 - 6z^2) q^88 + (-9z^5 + 9z^3 + 9z) q^89 - 4 * q^91 + 4z^7 q^92 + 8z^2 q^94 + (-11z^6 + 11z^4 + 11z^2) q^97 + (z^5 - z) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7}+O(q^{10}) $ 8 * q + 8 * q^7	$ 8 q + 8 q^{7} - 16 q^{10} - 4 q^{13} - 4 q^{16} - 8 q^{22} - 16 q^{25} + 16 q^{28} + 16 q^{31} + 8 q^{37} + 12 q^{40} + 32 q^{43} - 32 q^{46} - 4 q^{52} + 16 q^{55} - 12 q^{58} - 32 q^{61} - 16 q^{67} - 24 q^{70} + 8 q^{73} - 8 q^{82} + 32 q^{85} + 24 q^{88} - 32 q^{91} + 44 q^{97}+O(q^{100}) $ 8 * q + 8 * q^7 - 16 * q^10 - 4 * q^13 - 4 * q^16 - 8 * q^22 - 16 * q^25 + 16 * q^28 + 16 * q^31 + 8 * q^37 + 12 * q^40 + 32 * q^43 - 32 * q^46 - 4 * q^52 + 16 * q^55 - 12 * q^58 - 32 * q^61 - 16 * q^67 - 24 * q^70 + 8 * q^73 - 8 * q^82 + 32 * q^85 + 24 * q^88 - 32 * q^91 + 44 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$-\zeta_{24}^{6}$	$1 - \zeta_{24}^{4}$

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} $

53.1

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
−0.258819	+	0.965926i

−0.258819

0.965926i

0.866025

0.500000i

−0.448288

2.19067i

2.73205

0.732051i

−2.12132

2.12132i

−2.00000

−

1.00000i

53.2

0.258819

−

0.965926i

0.866025

0.500000i

0.448288

−

2.19067i

2.73205

0.732051i

2.12132

−

2.12132i

−2.00000

−

1.00000i

107.1

−0.258819

−

0.965926i

0.866025

−

0.500000i

−0.448288

−

2.19067i

2.73205

−

0.732051i

−2.12132

−

2.12132i

−2.00000

1.00000i

107.2

0.258819

0.965926i

0.866025

−

0.500000i

0.448288

2.19067i

2.73205

−

0.732051i

2.12132

2.12132i

−2.00000

1.00000i

188.1

−0.965926

0.258819i

−0.866025

0.500000i

1.67303

1.48356i

−0.732051

−

2.73205i

2.12132

−

2.12132i

−2.00000

−

1.00000i

188.2

0.965926

−

0.258819i

−0.866025

0.500000i

−1.67303

−

1.48356i

−0.732051

−

2.73205i

−2.12132

2.12132i

−2.00000

−

1.00000i

377.1

−0.965926

−

0.258819i

−0.866025

−

0.500000i

1.67303

−

1.48356i

−0.732051

2.73205i

2.12132

2.12132i

−2.00000

1.00000i

377.2

0.965926

0.258819i

−0.866025

−

0.500000i

−1.67303

1.48356i

−0.732051

2.73205i

−2.12132

−

2.12132i

−2.00000

1.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
9.c	even	3	1	inner
9.d	odd	6	1	inner
15.e	even	4	1	inner
45.k	odd	12	1	inner
45.l	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	405.2.m.a		8
3.b	odd	2	1	inner	405.2.m.a		8
5.c	odd	4	1	inner	405.2.m.a		8
9.c	even	3	1		45.2.f.a	✓	4
9.c	even	3	1	inner	405.2.m.a		8
9.d	odd	6	1		45.2.f.a	✓	4
9.d	odd	6	1	inner	405.2.m.a		8
15.e	even	4	1	inner	405.2.m.a		8
36.f	odd	6	1		720.2.w.d		4
36.h	even	6	1		720.2.w.d		4
45.h	odd	6	1		225.2.f.a		4
45.j	even	6	1		225.2.f.a		4
45.k	odd	12	1		45.2.f.a	✓	4
45.k	odd	12	1		225.2.f.a		4
45.k	odd	12	1	inner	405.2.m.a		8
45.l	even	12	1		45.2.f.a	✓	4
45.l	even	12	1		225.2.f.a		4
45.l	even	12	1	inner	405.2.m.a		8
72.j	odd	6	1		2880.2.w.b		4
72.l	even	6	1		2880.2.w.k		4
72.n	even	6	1		2880.2.w.b		4
72.p	odd	6	1		2880.2.w.k		4
180.n	even	6	1		3600.2.w.b		4
180.p	odd	6	1		3600.2.w.b		4
180.v	odd	12	1		720.2.w.d		4
180.v	odd	12	1		3600.2.w.b		4
180.x	even	12	1		720.2.w.d		4
180.x	even	12	1		3600.2.w.b		4
360.bo	even	12	1		2880.2.w.k		4
360.br	even	12	1		2880.2.w.b		4
360.bt	odd	12	1		2880.2.w.k		4
360.bu	odd	12	1		2880.2.w.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
45.2.f.a	✓	4	9.c	even	3	1
45.2.f.a	✓	4	9.d	odd	6	1
45.2.f.a	✓	4	45.k	odd	12	1
45.2.f.a	✓	4	45.l	even	12	1
225.2.f.a		4	45.h	odd	6	1
225.2.f.a		4	45.j	even	6	1
225.2.f.a		4	45.k	odd	12	1
225.2.f.a		4	45.l	even	12	1
405.2.m.a		8	1.a	even	1	1	trivial
405.2.m.a		8	3.b	odd	2	1	inner
405.2.m.a		8	5.c	odd	4	1	inner
405.2.m.a		8	9.c	even	3	1	inner
405.2.m.a		8	9.d	odd	6	1	inner
405.2.m.a		8	15.e	even	4	1	inner
405.2.m.a		8	45.k	odd	12	1	inner
405.2.m.a		8	45.l	even	12	1	inner
720.2.w.d		4	36.f	odd	6	1
720.2.w.d		4	36.h	even	6	1
720.2.w.d		4	180.v	odd	12	1
720.2.w.d		4	180.x	even	12	1
2880.2.w.b		4	72.j	odd	6	1
2880.2.w.b		4	72.n	even	6	1
2880.2.w.b		4	360.br	even	12	1
2880.2.w.b		4	360.bu	odd	12	1
2880.2.w.k		4	72.l	even	6	1
2880.2.w.k		4	72.p	odd	6	1
2880.2.w.k		4	360.bo	even	12	1
2880.2.w.k		4	360.bt	odd	12	1
3600.2.w.b		4	180.n	even	6	1
3600.2.w.b		4	180.p	odd	6	1
3600.2.w.b		4	180.v	odd	12	1
3600.2.w.b		4	180.x	even	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} - T_{2}^{4} + 1 $ T2^8 - T2^4 + 1 acting on $S_{2}^{\mathrm{new}}(405, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} - T^{4} + 1 $ T^8 - T^4 + 1
$3$	$ T^{8} $ T^8
$5$	$ T^{8} + 8 T^{6} + \cdots + 625 $ T^8 + 8T^6 + 39T^4 + 200*T^2 + 625
$7$	$ (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 64)^{2} $ (T^4 - 4T^3 + 8T^2 - 32*T + 64)^2
$11$	$ (T^{4} - 8 T^{2} + 64)^{2} $ (T^4 - 8*T^2 + 64)^2
$13$	$ (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} $ (T^4 + 2T^3 + 2T^2 + 4*T + 4)^2
$17$	$ (T^{4} + 256)^{2} $ (T^4 + 256)^2
$19$	$ T^{8} $ T^8
$23$	$ T^{8} - 256 T^{4} + 65536 $ T^8 - 256*T^4 + 65536
$29$	$ (T^{4} + 18 T^{2} + 324)^{2} $ (T^4 + 18*T^2 + 324)^2
$31$	$ (T^{2} - 4 T + 16)^{4} $ (T^2 - 4*T + 16)^4
$37$	$ (T^{2} - 2 T + 2)^{4} $ (T^2 - 2*T + 2)^4
$41$	$ (T^{4} - 2 T^{2} + 4)^{2} $ (T^4 - 2*T^2 + 4)^2
$43$	$ (T^{4} - 16 T^{3} + \cdots + 16384)^{2} $ (T^4 - 16T^3 + 128T^2 - 2048*T + 16384)^2
$47$	$ T^{8} - 4096 T^{4} + 16777216 $ T^8 - 4096*T^4 + 16777216
$53$	$ (T^{4} + 256)^{2} $ (T^4 + 256)^2
$59$	$ (T^{4} + 72 T^{2} + 5184)^{2} $ (T^4 + 72*T^2 + 5184)^2
$61$	$ (T^{2} + 8 T + 64)^{4} $ (T^2 + 8*T + 64)^4
$67$	$ (T^{4} + 8 T^{3} + \cdots + 1024)^{2} $ (T^4 + 8T^3 + 32T^2 + 256*T + 1024)^2
$71$	$ (T^{2} + 32)^{4} $ (T^2 + 32)^4
$73$	$ (T^{2} - 2 T + 2)^{4} $ (T^2 - 2*T + 2)^4
$79$	$ (T^{4} - 144 T^{2} + 20736)^{2} $ (T^4 - 144*T^2 + 20736)^2
$83$	$ T^{8} - 256 T^{4} + 65536 $ T^8 - 256*T^4 + 65536
$89$	$ (T^{2} - 162)^{4} $ (T^2 - 162)^4
$97$	$ (T^{4} - 22 T^{3} + \cdots + 58564)^{2} $ (T^4 - 22T^3 + 242T^2 - 5324*T + 58564)^2
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Classical	Maass
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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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.m (of order \(12\), degree \(4\), not minimal)

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

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Newspace parameters

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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	405.2.m.a

Level	$405$
Weight	$2$
Character orbit	405.m
Analytic conductor	$3.234$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(3.23394128186\)
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:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

\(n\)	\(82\)	\(326\)
\(\chi(n)\)	\(-\zeta_{24}^{6}\)	\(1 - \zeta_{24}^{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 53.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} - T^{4} + 1 \) T^8 - T^4 + 1
$3$	\( T^{8} \) T^8
$5$	\( T^{8} + 8 T^{6} + \cdots + 625 \) T^8 + 8T^6 + 39T^4 + 200*T^2 + 625
$7$	\( (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 64)^{2} \) (T^4 - 4T^3 + 8T^2 - 32*T + 64)^2
$11$	\( (T^{4} - 8 T^{2} + 64)^{2} \) (T^4 - 8*T^2 + 64)^2
$13$	\( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) (T^4 + 2T^3 + 2T^2 + 4*T + 4)^2
$17$	\( (T^{4} + 256)^{2} \) (T^4 + 256)^2
$19$	\( T^{8} \) T^8
$23$	\( T^{8} - 256 T^{4} + 65536 \) T^8 - 256*T^4 + 65536
$29$	\( (T^{4} + 18 T^{2} + 324)^{2} \) (T^4 + 18*T^2 + 324)^2
$31$	\( (T^{2} - 4 T + 16)^{4} \) (T^2 - 4*T + 16)^4
$37$	\( (T^{2} - 2 T + 2)^{4} \) (T^2 - 2*T + 2)^4
$41$	\( (T^{4} - 2 T^{2} + 4)^{2} \) (T^4 - 2*T^2 + 4)^2
$43$	\( (T^{4} - 16 T^{3} + \cdots + 16384)^{2} \) (T^4 - 16T^3 + 128T^2 - 2048*T + 16384)^2
$47$	\( T^{8} - 4096 T^{4} + 16777216 \) T^8 - 4096*T^4 + 16777216
$53$	\( (T^{4} + 256)^{2} \) (T^4 + 256)^2
$59$	\( (T^{4} + 72 T^{2} + 5184)^{2} \) (T^4 + 72*T^2 + 5184)^2
$61$	\( (T^{2} + 8 T + 64)^{4} \) (T^2 + 8*T + 64)^4
$67$	\( (T^{4} + 8 T^{3} + \cdots + 1024)^{2} \) (T^4 + 8T^3 + 32T^2 + 256*T + 1024)^2
$71$	\( (T^{2} + 32)^{4} \) (T^2 + 32)^4
$73$	\( (T^{2} - 2 T + 2)^{4} \) (T^2 - 2*T + 2)^4
$79$	\( (T^{4} - 144 T^{2} + 20736)^{2} \) (T^4 - 144*T^2 + 20736)^2
$83$	\( T^{8} - 256 T^{4} + 65536 \) T^8 - 256*T^4 + 65536
$89$	\( (T^{2} - 162)^{4} \) (T^2 - 162)^4
$97$	\( (T^{4} - 22 T^{3} + \cdots + 58564)^{2} \) (T^4 - 22T^3 + 242T^2 - 5324*T + 58564)^2
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