LMFDB - Newform orbit 490.2.l.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [490,2,Mod(117,490)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(490, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([3, 10]))

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

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

chi := DirichletCharacter("490.117");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 490 = 2 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	490.l (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:	$3.91266969904$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{12})$
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:	$\Z[a_1, \ldots, a_{5}]$
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 $\left(\mathbb{Z}/490\mathbb{Z}\right)^\times$.

$n$	$101$	$197$
$\chi(n)$	$\zeta_{48}^{8}$	$\zeta_{48}^{12}$

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} $

117.1

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

−0.965926

0.258819i

−0.676327

2.52409i

0.866025

−

0.500000i

0.977945

−

2.01088i

−

2.61313i

−0.707107

0.707107i

−3.31552

−

1.91421i

−0.424170

2.19547i

117.2

−0.965926

0.258819i

0.676327

−

2.52409i

0.866025

−

0.500000i

−0.977945

2.01088i

2.61313i

−0.707107

0.707107i

−3.31552

−

1.91421i

0.424170

−

2.19547i

117.3

0.965926

−

0.258819i

−0.280144

1.04551i

0.866025

−

0.500000i

−2.01088

−

0.977945i

1.08239i

0.707107

−

0.707107i

1.58346

0.914214i

−2.19547

−

0.424170i

117.4

0.965926

−

0.258819i

0.280144

−

1.04551i

0.866025

−

0.500000i

2.01088

0.977945i

−

1.08239i

0.707107

−

0.707107i

1.58346

0.914214i

2.19547

0.424170i

227.1

−0.258819

0.965926i

−1.04551

0.280144i

−0.866025

−

0.500000i

−1.85236

1.25250i

−

1.08239i

0.707107

−

0.707107i

−1.58346

0.914214i

−0.730392

−

2.11342i

227.2

−0.258819

0.965926i

1.04551

−

0.280144i

−0.866025

−

0.500000i

1.85236

−

1.25250i

1.08239i

0.707107

−

0.707107i

−1.58346

0.914214i

0.730392

2.11342i

227.3

0.258819

−

0.965926i

−2.52409

0.676327i

−0.866025

−

0.500000i

−1.25250

−

1.85236i

2.61313i

−0.707107

0.707107i

3.31552

−

1.91421i

−2.11342

0.730392i

227.4

0.258819

−

0.965926i

2.52409

−

0.676327i

−0.866025

−

0.500000i

1.25250

1.85236i

−

2.61313i

−0.707107

0.707107i

3.31552

−

1.91421i

2.11342

−

0.730392i

313.1

−0.258819

−

0.965926i

−1.04551

−

0.280144i

−0.866025

0.500000i

−1.85236

−

1.25250i

1.08239i

0.707107

0.707107i

−1.58346

−

0.914214i

−0.730392

2.11342i

313.2

−0.258819

−

0.965926i

1.04551

0.280144i

−0.866025

0.500000i

1.85236

1.25250i

−

1.08239i

0.707107

0.707107i

−1.58346

−

0.914214i

0.730392

−

2.11342i

313.3

0.258819

0.965926i

−2.52409

−

0.676327i

−0.866025

0.500000i

−1.25250

1.85236i

−

2.61313i

−0.707107

−

0.707107i

3.31552

1.91421i

−2.11342

−

0.730392i

313.4

0.258819

0.965926i

2.52409

0.676327i

−0.866025

0.500000i

1.25250

−

1.85236i

2.61313i

−0.707107

−

0.707107i

3.31552

1.91421i

2.11342

0.730392i

423.1

−0.965926

−

0.258819i

−0.676327

−

2.52409i

0.866025

0.500000i

0.977945

2.01088i

2.61313i

−0.707107

−

0.707107i

−3.31552

1.91421i

−0.424170

−

2.19547i

423.2

−0.965926

−

0.258819i

0.676327

2.52409i

0.866025

0.500000i

−0.977945

−

2.01088i

−

2.61313i

−0.707107

−

0.707107i

−3.31552

1.91421i

0.424170

2.19547i

423.3

0.965926

0.258819i

−0.280144

−

1.04551i

0.866025

0.500000i

−2.01088

0.977945i

−

1.08239i

0.707107

0.707107i

1.58346

−

0.914214i

−2.19547

0.424170i

423.4

0.965926

0.258819i

0.280144

1.04551i

0.866025

0.500000i

2.01088

−

0.977945i

1.08239i

0.707107

0.707107i

1.58346

−

0.914214i

2.19547

−

0.424170i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
7.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner
35.f	even	4	1	inner
35.k	even	12	1	inner
35.l	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	490.2.l.b		16
5.c	odd	4	1	inner	490.2.l.b		16
7.b	odd	2	1	inner	490.2.l.b		16
7.c	even	3	1		490.2.g.a	✓	8
7.c	even	3	1	inner	490.2.l.b		16
7.d	odd	6	1		490.2.g.a	✓	8
7.d	odd	6	1	inner	490.2.l.b		16
35.f	even	4	1	inner	490.2.l.b		16
35.k	even	12	1		490.2.g.a	✓	8
35.k	even	12	1	inner	490.2.l.b		16
35.l	odd	12	1		490.2.g.a	✓	8
35.l	odd	12	1	inner	490.2.l.b		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
490.2.g.a	✓	8	7.c	even	3	1
490.2.g.a	✓	8	7.d	odd	6	1
490.2.g.a	✓	8	35.k	even	12	1
490.2.g.a	✓	8	35.l	odd	12	1
490.2.l.b		16	1.a	even	1	1	trivial
490.2.l.b		16	5.c	odd	4	1	inner
490.2.l.b		16	7.b	odd	2	1	inner
490.2.l.b		16	7.c	even	3	1	inner
490.2.l.b		16	7.d	odd	6	1	inner
490.2.l.b		16	35.f	even	4	1	inner
490.2.l.b		16	35.k	even	12	1	inner
490.2.l.b		16	35.l	odd	12	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{16} - 48T_{3}^{12} + 2240T_{3}^{8} - 3072T_{3}^{4} + 4096 $ T3^16 - 48*T3^12 + 2240*T3^8 - 3072*T3^4 + 4096 acting on $S_{2}^{\mathrm{new}}(490, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} - T^{4} + 1)^{2} $ (T^8 - T^4 + 1)^2
$3$	$ T^{16} - 48 T^{12} + \cdots + 4096 $ T^16 - 48T^12 + 2240T^8 - 3072*T^4 + 4096
$5$	$ T^{16} + 48 T^{12} + \cdots + 390625 $ T^16 + 48T^12 + 1679T^8 + 30000*T^4 + 390625
$7$	$ T^{16} $ T^16
$11$	$ (T^{4} - 4 T^{3} + 20 T^{2} + \cdots + 16)^{4} $ (T^4 - 4T^3 + 20T^2 + 16*T + 16)^4
$13$	$ (T^{8} + 1548 T^{4} + 334084)^{2} $ (T^8 + 1548*T^4 + 334084)^2
$17$	$ T^{16} + \cdots + 111612119056 $ T^16 - 1548T^12 + 2062220T^8 - 517162032*T^4 + 111612119056
$19$	$ (T^{8} + 72 T^{6} + \cdots + 419904)^{2} $ (T^8 + 72T^6 + 4536T^4 + 46656*T^2 + 419904)^2
$23$	$ (T^{8} - 8 T^{7} + \cdots + 256)^{2} $ (T^8 - 8T^7 + 32T^6 - 192T^5 + 752T^4 - 768T^3 + 512T^2 - 512*T + 256)^2
$29$	$ (T^{4} + 44 T^{2} + 196)^{4} $ (T^4 + 44*T^2 + 196)^4
$31$	$ (T^{8} - 16 T^{6} + \cdots + 1024)^{2} $ (T^8 - 16T^6 + 224T^4 - 512*T^2 + 1024)^2
$37$	$ (T^{8} + 8 T^{7} + \cdots + 4096)^{2} $ (T^8 + 8T^7 + 32T^6 + 384T^5 + 1472T^4 - 3072T^3 + 2048T^2 - 4096*T + 4096)^2
$41$	$ (T^{4} + 116 T^{2} + 2)^{4} $ (T^4 + 116*T^2 + 2)^4
$43$	$ (T^{4} + 24 T^{3} + \cdots + 4624)^{4} $ (T^4 + 24T^3 + 288T^2 + 1632*T + 4624)^4
$47$	$ T^{16} + \cdots + 6044831973376 $ T^16 - 3264T^12 + 8195072T^8 - 8024948736*T^4 + 6044831973376
$53$	$ (T^{8} - 4 T^{7} + \cdots + 1336336)^{2} $ (T^8 - 4T^7 + 8T^6 - 304T^5 - 548T^4 + 10336T^3 + 9248T^2 + 157216*T + 1336336)^2
$59$	$ (T^{8} + 72 T^{6} + \cdots + 419904)^{2} $ (T^8 + 72T^6 + 4536T^4 + 46656*T^2 + 419904)^2
$61$	$ (T^{8} - 52 T^{6} + \cdots + 334084)^{2} $ (T^8 - 52T^6 + 2126T^4 - 30056*T^2 + 334084)^2
$67$	$ (T^{4} + 12 T^{3} + \cdots + 5184)^{4} $ (T^4 + 12T^3 + 72T^2 + 864*T + 5184)^4
$71$	$ (T^{2} + 4 T - 68)^{8} $ (T^2 + 4*T - 68)^8
$73$	$ T^{16} + \cdots + 531726889113616 $ T^16 - 49932T^12 + 2470145420T^8 - 1151392174128*T^4 + 531726889113616
$79$	$ (T^{8} - 264 T^{6} + \cdots + 21381376)^{2} $ (T^8 - 264T^6 + 65072T^4 - 1220736*T^2 + 21381376)^2
$83$	$ (T^{8} + 35376 T^{4} + 5345344)^{2} $ (T^8 + 35376*T^4 + 5345344)^2
$89$	$ (T^{8} + 340 T^{6} + \cdots + 354117124)^{2} $ (T^8 + 340T^6 + 96782T^4 + 6398120*T^2 + 354117124)^2
$97$	$ (T^{8} + 4428 T^{4} + 9604)^{2} $ (T^8 + 4428*T^4 + 9604)^2
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Overview	Random
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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:	\( N \)	\(=\)	\( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	490.l (of order \(12\), degree \(4\), not minimal)

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

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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Label	490.2.l.b

Level	$490$
Weight	$2$
Character orbit	490.l
Analytic conductor	$3.913$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(3.91266969904\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{12})\)
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:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(\zeta_{48}^{8}\)	\(\zeta_{48}^{12}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 117.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} - T^{4} + 1)^{2} \) (T^8 - T^4 + 1)^2
$3$	\( T^{16} - 48 T^{12} + \cdots + 4096 \) T^16 - 48T^12 + 2240T^8 - 3072*T^4 + 4096
$5$	\( T^{16} + 48 T^{12} + \cdots + 390625 \) T^16 + 48T^12 + 1679T^8 + 30000*T^4 + 390625
$7$	\( T^{16} \) T^16
$11$	\( (T^{4} - 4 T^{3} + 20 T^{2} + \cdots + 16)^{4} \) (T^4 - 4T^3 + 20T^2 + 16*T + 16)^4
$13$	\( (T^{8} + 1548 T^{4} + 334084)^{2} \) (T^8 + 1548*T^4 + 334084)^2
$17$	\( T^{16} + \cdots + 111612119056 \) T^16 - 1548T^12 + 2062220T^8 - 517162032*T^4 + 111612119056
$19$	\( (T^{8} + 72 T^{6} + \cdots + 419904)^{2} \) (T^8 + 72T^6 + 4536T^4 + 46656*T^2 + 419904)^2
$23$	\( (T^{8} - 8 T^{7} + \cdots + 256)^{2} \) (T^8 - 8T^7 + 32T^6 - 192T^5 + 752T^4 - 768T^3 + 512T^2 - 512*T + 256)^2
$29$	\( (T^{4} + 44 T^{2} + 196)^{4} \) (T^4 + 44*T^2 + 196)^4
$31$	\( (T^{8} - 16 T^{6} + \cdots + 1024)^{2} \) (T^8 - 16T^6 + 224T^4 - 512*T^2 + 1024)^2
$37$	\( (T^{8} + 8 T^{7} + \cdots + 4096)^{2} \) (T^8 + 8T^7 + 32T^6 + 384T^5 + 1472T^4 - 3072T^3 + 2048T^2 - 4096*T + 4096)^2
$41$	\( (T^{4} + 116 T^{2} + 2)^{4} \) (T^4 + 116*T^2 + 2)^4
$43$	\( (T^{4} + 24 T^{3} + \cdots + 4624)^{4} \) (T^4 + 24T^3 + 288T^2 + 1632*T + 4624)^4
$47$	\( T^{16} + \cdots + 6044831973376 \) T^16 - 3264T^12 + 8195072T^8 - 8024948736*T^4 + 6044831973376
$53$	\( (T^{8} - 4 T^{7} + \cdots + 1336336)^{2} \) (T^8 - 4T^7 + 8T^6 - 304T^5 - 548T^4 + 10336T^3 + 9248T^2 + 157216*T + 1336336)^2
$59$	\( (T^{8} + 72 T^{6} + \cdots + 419904)^{2} \) (T^8 + 72T^6 + 4536T^4 + 46656*T^2 + 419904)^2
$61$	\( (T^{8} - 52 T^{6} + \cdots + 334084)^{2} \) (T^8 - 52T^6 + 2126T^4 - 30056*T^2 + 334084)^2
$67$	\( (T^{4} + 12 T^{3} + \cdots + 5184)^{4} \) (T^4 + 12T^3 + 72T^2 + 864*T + 5184)^4
$71$	\( (T^{2} + 4 T - 68)^{8} \) (T^2 + 4*T - 68)^8
$73$	\( T^{16} + \cdots + 531726889113616 \) T^16 - 49932T^12 + 2470145420T^8 - 1151392174128*T^4 + 531726889113616
$79$	\( (T^{8} - 264 T^{6} + \cdots + 21381376)^{2} \) (T^8 - 264T^6 + 65072T^4 - 1220736*T^2 + 21381376)^2
$83$	\( (T^{8} + 35376 T^{4} + 5345344)^{2} \) (T^8 + 35376*T^4 + 5345344)^2
$89$	\( (T^{8} + 340 T^{6} + \cdots + 354117124)^{2} \) (T^8 + 340T^6 + 96782T^4 + 6398120*T^2 + 354117124)^2
$97$	\( (T^{8} + 4428 T^{4} + 9604)^{2} \) (T^8 + 4428*T^4 + 9604)^2
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