LMFDB - Newform orbit 136.2.s.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [136,2,Mod(3,136)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(136, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([8, 8, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("136.3"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 136 = 2^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	136.s (of order $16$, degree $8$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$1.08596546749$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{16}]$

$q$-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 $\zeta_{16}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\zeta_{16}^{5} - \zeta_{16}) q^{2} + (\zeta_{16}^{7} - \zeta_{16}^{5} + \cdots - 1) q^{3} - 2 \zeta_{16}^{6} q^{4} + ( - \zeta_{16}^{7} - 2 \zeta_{16}^{5} + \cdots + 1) q^{6} + (2 \zeta_{16}^{7} + 2 \zeta_{16}^{3}) q^{8} + \cdots + (3 \zeta_{16}^{7} + 4 \zeta_{16}^{6} + \cdots - 6) q^{99} +O(q^{100}) $ q + (z^5 - z) * q^2 + (z^7 - z^5 + z^4 - z^2 - z - 1) * q^3 - 2z^6 q^4 + (-z^7 - 2z^5 - z^4 + z^3 + 2z^2 + 1) * q^6 + (2z^7 + 2z^3) * q^8 + (-z^6 + z^4 + 2z^2 + 3z + 2) * q^9 + (z^6 - z^5 + 3z^4 + 3z^3 - z^2 + z) * q^11 + (2z^7 + 2z^6 + 2z^5 - 2z^3 + 2z^2 - 2) q^12 - 4z^4 q^16 + (-3z^7 + 2z^5 + 2z) q^17 + (3z^7 + 3z^6 + z^5 - z^3 - 3z^2 - 3z) * q^18 + (-6z^7 + z^5 - z) q^19 + (-2z^7 + 2z^6 - 3z^5 - 3z^4 - 3z - 3) q^22 + (-2z^6 - 2z^5 - 4z^3 - 2z^2 + 2z + 4) q^24 + 5z^5 q^25 + (z^7 + 2z^6 - 2z^5 - z^4 - 4z^3 - 3z^2 - 3z - 4) q^27 + (4z^5 + 4z) * q^32 + (4z^7 - 5z^6 - 5z^5 - 4z^4 - 5z^3 - 5z^2 + 4z) q^33 + (3z^4 - 4z^2 - 3) * q^34 + (-6z^7 - 4z^6 - 2z^4 + 2z^2 + 4) * q^36 + (-2z^6 + 6z^4 - 6) * q^38 + (4z^7 + 4z^4 + 4z^3 + 3z^2 - 3z - 4) q^41 + (3z^6 - 3z^4 + 8z^2 + 8) q^43 + (-2z^7 + 2z^4 - 2z^3 + 6z^2 + 6z - 2) q^44 + (4z^6 + 4z^5 + 4z^4 + 4z^3 - 4z + 4) q^48 - 7z^3 q^49 + (-5z^6 - 5z^2) * q^50 + (z^7 - z^6 - 5z^4 + z^3 - 7z - 5) * q^51 + (-5z^7 - z^6 - 3z^5 + 3z^4 + z^3 + 5z^2 + 5z + 5) q^54 + (5z^7 + 6z^6 - 2z^5 - 7z^4 + 7z^3 + 2z^2 - 6z - 5) q^57 + (2z^6 + 2z^4 - 5z^2 + 5) q^59 - 8z^2 q^64 + (9z^6 + 4z^5 + z^4 + 10z^3 + z^2 + 4z + 9) * q^66 + (10z^7 + 3z^5 + 3z^3 + 10z) * q^67 + (-4z^7 - 6z^5 + 4z^3) q^68 + (6z^7 + 6z^5 + 6z^4 + 2z^3 - 2z - 6) q^72 + (6z^5 - 6z^4 + z^3 + z^2 - 6z + 6) q^73 + (-5z^7 - 5z^6 - 5z^5 - 5z^4 + 5z^2 - 5z) * q^75 + (2z^7 - 12z^5 + 2z^3) q^76 + (-3z^7 + 3z^5 + 7z^4 + 6z^3 + z^2 + 6z + 7) q^81 + (3z^7 - 3z^6 - 8z^5 - 8z^4 - 3z^3 + 3z^2) * q^82 + (-8z^6 + z^4 + z^2 - 8) q^83 + (5z^7 + 11z^5 - 11z^3 - 5z) * q^86 + (6z^7 + 6z^6 - 4z^5 + 4z^4 - 6z^3 - 6z^2) * q^88 + (-11z^7 - 11z^5 + 2z^3 - 2z) * q^89 + (-4z^7 - 8z^6 - 4z^4 - 4z^3 - 8z - 4) q^96 + (-6z^7 - 6z^6 + 5z^4 - 6z^3 + 6z^2 - 5z) * q^97 + (7z^4 + 7) q^98 + (3z^7 + 4z^6 + 14z^5 + 14z^4 + 4z^3 + 3z^2 + 6z - 6) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 8 q^{6} + 16 q^{9} - 16 q^{12} - 24 q^{22} + 32 q^{24} - 32 q^{27} - 24 q^{34} + 32 q^{36} - 48 q^{38} - 32 q^{41} + 64 q^{43} - 16 q^{44} + 32 q^{48} - 40 q^{51} + 40 q^{54} - 40 q^{57}+ \cdots - 48 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 + 8 * q^6 + 16 * q^9 - 16 * q^12 - 24 * q^22 + 32 * q^24 - 32 * q^27 - 24 * q^34 + 32 * q^36 - 48 * q^38 - 32 * q^41 + 64 * q^43 - 16 * q^44 + 32 * q^48 - 40 * q^51 + 40 * q^54 - 40 * q^57 + 40 * q^59 + 72 * q^66 - 48 * q^72 + 48 * q^73 + 56 * q^81 - 64 * q^83 - 32 * q^96 + 56 * q^98 - 48 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$.

$n$	$69$	$103$	$105$
$\chi(n)$	$-1$	$-1$	$\zeta_{16}$

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} $

3.1

0.923880	+	0.382683i
−0.923880	+	0.382683i
0.382683	+	0.923880i
−0.382683	−	0.923880i
0.923880	−	0.382683i
−0.923880	−	0.382683i
−0.382683	+	0.923880i
0.382683	−	0.923880i

−1.30656

0.541196i

−3.17218

−

0.630986i

1.41421

−

1.41421i

4.48614

−

0.892349i

−1.08239

2.61313i

6.89296

2.85516i

11.1

1.30656

0.541196i

−0.242031

−

1.21677i

1.41421

1.41421i

0.342284

−

1.72078i

1.08239

2.61313i

1.34968

−

0.559056i

27.1

0.541196

−

1.30656i

−1.98214

−

1.32442i

−1.41421

−

1.41421i

−2.80317

1.87302i

−2.61313

1.08239i

1.02673

2.47875i

75.1

−0.541196

1.30656i

1.39635

−

2.08979i

−1.41421

−

1.41421i

1.97474

2.95541i

2.61313

−

1.08239i

−1.26937

−

3.06453i

91.1

−1.30656

−

0.541196i

−3.17218

0.630986i

1.41421

1.41421i

4.48614

0.892349i

−1.08239

−

2.61313i

6.89296

−

2.85516i

99.1

1.30656

−

0.541196i

−0.242031

1.21677i

1.41421

−

1.41421i

0.342284

1.72078i

1.08239

−

2.61313i

1.34968

0.559056i

107.1

−0.541196

−

1.30656i

1.39635

2.08979i

−1.41421

1.41421i

1.97474

−

2.95541i

2.61313

1.08239i

−1.26937

3.06453i

131.1

0.541196

1.30656i

−1.98214

1.32442i

−1.41421

1.41421i

−2.80317

−

1.87302i

−2.61313

−

1.08239i

1.02673

−

2.47875i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
17.e	odd	16	1	inner
136.s	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	136.2.s.a	✓	8
4.b	odd	2	1		544.2.cc.b		8
8.b	even	2	1		544.2.cc.b		8
8.d	odd	2	1	CM	136.2.s.a	✓	8
17.e	odd	16	1	inner	136.2.s.a	✓	8
68.i	even	16	1		544.2.cc.b		8
136.q	odd	16	1		544.2.cc.b		8
136.s	even	16	1	inner	136.2.s.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
136.2.s.a	✓	8	1.a	even	1	1	trivial
136.2.s.a	✓	8	8.d	odd	2	1	CM
136.2.s.a	✓	8	17.e	odd	16	1	inner
136.2.s.a	✓	8	136.s	even	16	1	inner
544.2.cc.b		8	4.b	odd	2	1
544.2.cc.b		8	8.b	even	2	1
544.2.cc.b		8	68.i	even	16	1
544.2.cc.b		8	136.q	odd	16	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 8T_{3}^{7} + 24T_{3}^{6} + 48T_{3}^{5} + 146T_{3}^{4} + 416T_{3}^{3} + 692T_{3}^{2} + 680T_{3} + 578 $ T3^8 + 8*T3^7 + 24*T3^6 + 48*T3^5 + 146*T3^4 + 416*T3^3 + 692*T3^2 + 680*T3 + 578 acting on $S_{2}^{\mathrm{new}}(136, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + 16 $ T^8 + 16
$3$	$ T^{8} + 8 T^{7} + \cdots + 578 $ T^8 + 8T^7 + 24T^6 + 48T^5 + 146T^4 + 416T^3 + 692T^2 + 680*T + 578
$5$	$ T^{8} $ T^8
$7$	$ T^{8} $ T^8
$11$	$ T^{8} + 4 T^{6} + \cdots + 167042 $ T^8 + 4T^6 + 88T^5 + 54T^4 + 1496T^3 + 16428T^2 + 76296T + 167042
$13$	$ T^{8} $ T^8
$17$	$ T^{8} - 48 T^{6} + \cdots + 83521 $ T^8 - 48T^6 + 1152T^4 - 13872*T^2 + 83521
$19$	$ T^{8} + 48 T^{6} + \cdots + 1336336 $ T^8 + 48T^6 + 1152T^4 - 55488*T^2 + 1336336
$23$	$ T^{8} $ T^8
$29$	$ T^{8} $ T^8
$31$	$ T^{8} $ T^8
$37$	$ T^{8} $ T^8
$41$	$ T^{8} + 32 T^{7} + \cdots + 555458 $ T^8 + 32T^7 + 416T^6 + 2336T^5 - 3806T^4 - 62736T^3 + 359540T^2 - 282472*T + 555458
$43$	$ (T^{4} - 32 T^{3} + \cdots + 21218)^{2} $ (T^4 - 32T^3 + 498T^2 - 4532*T + 21218)^2
$47$	$ T^{8} $ T^8
$53$	$ T^{8} $ T^8
$59$	$ (T^{4} - 20 T^{3} + 118 T^{2} + \cdots + 2)^{2} $ (T^4 - 20T^3 + 118T^2 - 12*T + 2)^2
$61$	$ T^{8} $ T^8
$67$	$ (T^{4} + 436 T^{2} + 45602)^{2} $ (T^4 + 436*T^2 + 45602)^2
$71$	$ T^{8} $ T^8
$73$	$ T^{8} - 48 T^{7} + \cdots + 7380482 $ T^8 - 48T^7 + 1200T^6 - 19048T^5 + 206114T^4 - 1209408T^3 + 2197740T^2 + 3304120*T + 7380482
$79$	$ T^{8} $ T^8
$83$	$ (T^{4} + 32 T^{3} + \cdots + 4418)^{2} $ (T^4 + 32T^3 + 354T^2 + 1316*T + 4418)^2
$89$	$ T^{8} + \cdots + 2687592964 $ T^8 + 146316*T^4 + 2687592964
$97$	$ T^{8} + 52 T^{6} + \cdots + 856069442 $ T^8 + 52T^6 + 4656T^5 + 11334T^4 + 404296T^3 + 1260196T^2 - 80273320T + 856069442
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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	136.s (of order \(16\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{16}^{5} - \zeta_{16}) q^{2} + (\zeta_{16}^{7} - \zeta_{16}^{5} + \cdots - 1) q^{3} - 2 \zeta_{16}^{6} q^{4} + ( - \zeta_{16}^{7} - 2 \zeta_{16}^{5} + \cdots + 1) q^{6} + (2 \zeta_{16}^{7} + 2 \zeta_{16}^{3}) q^{8} + \cdots + (3 \zeta_{16}^{7} + 4 \zeta_{16}^{6} + \cdots - 6) q^{99} +O(q^{100}) \) q + (z^5 - z) * q^2 + (z^7 - z^5 + z^4 - z^2 - z - 1) * q^3 - 2z^6 q^4 + (-z^7 - 2z^5 - z^4 + z^3 + 2z^2 + 1) * q^6 + (2z^7 + 2z^3) * q^8 + (-z^6 + z^4 + 2z^2 + 3z + 2) * q^9 + (z^6 - z^5 + 3z^4 + 3z^3 - z^2 + z) * q^11 + (2z^7 + 2z^6 + 2z^5 - 2z^3 + 2z^2 - 2) q^12 - 4z^4 q^16 + (-3z^7 + 2z^5 + 2z) q^17 + (3z^7 + 3z^6 + z^5 - z^3 - 3z^2 - 3z) * q^18 + (-6z^7 + z^5 - z) q^19 + (-2z^7 + 2z^6 - 3z^5 - 3z^4 - 3z - 3) q^22 + (-2z^6 - 2z^5 - 4z^3 - 2z^2 + 2z + 4) q^24 + 5z^5 q^25 + (z^7 + 2z^6 - 2z^5 - z^4 - 4z^3 - 3z^2 - 3z - 4) q^27 + (4z^5 + 4z) * q^32 + (4z^7 - 5z^6 - 5z^5 - 4z^4 - 5z^3 - 5z^2 + 4z) q^33 + (3z^4 - 4z^2 - 3) * q^34 + (-6z^7 - 4z^6 - 2z^4 + 2z^2 + 4) * q^36 + (-2z^6 + 6z^4 - 6) * q^38 + (4z^7 + 4z^4 + 4z^3 + 3z^2 - 3z - 4) q^41 + (3z^6 - 3z^4 + 8z^2 + 8) q^43 + (-2z^7 + 2z^4 - 2z^3 + 6z^2 + 6z - 2) q^44 + (4z^6 + 4z^5 + 4z^4 + 4z^3 - 4z + 4) q^48 - 7z^3 q^49 + (-5z^6 - 5z^2) * q^50 + (z^7 - z^6 - 5z^4 + z^3 - 7z - 5) * q^51 + (-5z^7 - z^6 - 3z^5 + 3z^4 + z^3 + 5z^2 + 5z + 5) q^54 + (5z^7 + 6z^6 - 2z^5 - 7z^4 + 7z^3 + 2z^2 - 6z - 5) q^57 + (2z^6 + 2z^4 - 5z^2 + 5) q^59 - 8z^2 q^64 + (9z^6 + 4z^5 + z^4 + 10z^3 + z^2 + 4z + 9) * q^66 + (10z^7 + 3z^5 + 3z^3 + 10z) * q^67 + (-4z^7 - 6z^5 + 4z^3) q^68 + (6z^7 + 6z^5 + 6z^4 + 2z^3 - 2z - 6) q^72 + (6z^5 - 6z^4 + z^3 + z^2 - 6z + 6) q^73 + (-5z^7 - 5z^6 - 5z^5 - 5z^4 + 5z^2 - 5z) * q^75 + (2z^7 - 12z^5 + 2z^3) q^76 + (-3z^7 + 3z^5 + 7z^4 + 6z^3 + z^2 + 6z + 7) q^81 + (3z^7 - 3z^6 - 8z^5 - 8z^4 - 3z^3 + 3z^2) * q^82 + (-8z^6 + z^4 + z^2 - 8) q^83 + (5z^7 + 11z^5 - 11z^3 - 5z) * q^86 + (6z^7 + 6z^6 - 4z^5 + 4z^4 - 6z^3 - 6z^2) * q^88 + (-11z^7 - 11z^5 + 2z^3 - 2z) * q^89 + (-4z^7 - 8z^6 - 4z^4 - 4z^3 - 8z - 4) q^96 + (-6z^7 - 6z^6 + 5z^4 - 6z^3 + 6z^2 - 5z) * q^97 + (7z^4 + 7) q^98 + (3z^7 + 4z^6 + 14z^5 + 14z^4 + 4z^3 + 3z^2 + 6z - 6) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 8 q^{6} + 16 q^{9} - 16 q^{12} - 24 q^{22} + 32 q^{24} - 32 q^{27} - 24 q^{34} + 32 q^{36} - 48 q^{38} - 32 q^{41} + 64 q^{43} - 16 q^{44} + 32 q^{48} - 40 q^{51} + 40 q^{54} - 40 q^{57}+ \cdots - 48 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 + 8 * q^6 + 16 * q^9 - 16 * q^12 - 24 * q^22 + 32 * q^24 - 32 * q^27 - 24 * q^34 + 32 * q^36 - 48 * q^38 - 32 * q^41 + 64 * q^43 - 16 * q^44 + 32 * q^48 - 40 * q^51 + 40 * q^54 - 40 * q^57 + 40 * q^59 + 72 * q^66 - 48 * q^72 + 48 * q^73 + 56 * q^81 - 64 * q^83 - 32 * q^96 + 56 * q^98 - 48 * q^99

Introduction

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Modular forms

Varieties

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Database

Properties

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Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

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	136.2.s.a

Level	$136$
Weight	$2$
Character orbit	136.s
Analytic conductor	$1.086$
Analytic rank	$0$
Dimension	$8$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.08596546749\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{16})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} + 1 \) x^8 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{16}]$

\(n\)	\(69\)	\(103\)	\(105\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(\zeta_{16}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 3.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
17.e	odd	16	1	inner
136.s	even	16	1	inner

$p$	$F_p(T)$
$2$	\( T^{8} + 16 \) T^8 + 16
$3$	\( T^{8} + 8 T^{7} + \cdots + 578 \) T^8 + 8T^7 + 24T^6 + 48T^5 + 146T^4 + 416T^3 + 692T^2 + 680*T + 578
$5$	\( T^{8} \) T^8
$7$	\( T^{8} \) T^8
$11$	\( T^{8} + 4 T^{6} + \cdots + 167042 \) T^8 + 4T^6 + 88T^5 + 54T^4 + 1496T^3 + 16428T^2 + 76296T + 167042
$13$	\( T^{8} \) T^8
$17$	\( T^{8} - 48 T^{6} + \cdots + 83521 \) T^8 - 48T^6 + 1152T^4 - 13872*T^2 + 83521
$19$	\( T^{8} + 48 T^{6} + \cdots + 1336336 \) T^8 + 48T^6 + 1152T^4 - 55488*T^2 + 1336336
$23$	\( T^{8} \) T^8
$29$	\( T^{8} \) T^8
$31$	\( T^{8} \) T^8
$37$	\( T^{8} \) T^8
$41$	\( T^{8} + 32 T^{7} + \cdots + 555458 \) T^8 + 32T^7 + 416T^6 + 2336T^5 - 3806T^4 - 62736T^3 + 359540T^2 - 282472*T + 555458
$43$	\( (T^{4} - 32 T^{3} + \cdots + 21218)^{2} \) (T^4 - 32T^3 + 498T^2 - 4532*T + 21218)^2
$47$	\( T^{8} \) T^8
$53$	\( T^{8} \) T^8
$59$	\( (T^{4} - 20 T^{3} + 118 T^{2} + \cdots + 2)^{2} \) (T^4 - 20T^3 + 118T^2 - 12*T + 2)^2
$61$	\( T^{8} \) T^8
$67$	\( (T^{4} + 436 T^{2} + 45602)^{2} \) (T^4 + 436*T^2 + 45602)^2
$71$	\( T^{8} \) T^8
$73$	\( T^{8} - 48 T^{7} + \cdots + 7380482 \) T^8 - 48T^7 + 1200T^6 - 19048T^5 + 206114T^4 - 1209408T^3 + 2197740T^2 + 3304120*T + 7380482
$79$	\( T^{8} \) T^8
$83$	\( (T^{4} + 32 T^{3} + \cdots + 4418)^{2} \) (T^4 + 32T^3 + 354T^2 + 1316*T + 4418)^2
$89$	\( T^{8} + \cdots + 2687592964 \) T^8 + 146316*T^4 + 2687592964
$97$	\( T^{8} + 52 T^{6} + \cdots + 856069442 \) T^8 + 52T^6 + 4656T^5 + 11334T^4 + 404296T^3 + 1260196T^2 - 80273320T + 856069442
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