LMFDB - Newform orbit 3840.1.dy.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,1,Mod(29,3840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3840, base_ring=CyclotomicField(64))

chi = DirichletCharacter(H, H._module([0, 59, 32, 32]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3840.29");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3840.dy (of order $64$, degree $32$, 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:	$1.91640964851$
Analytic rank:	$0$
Dimension:	$64$
Relative dimension:	$2$ over $\Q(\zeta_{64})$
Coefficient field:	$\Q(\zeta_{128})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{64} + 1 $ x^64 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{64}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{64} + \cdots)$

$q$-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_{128}^{29} q^{2} - \zeta_{128} q^{3} + \zeta_{128}^{58} q^{4} - \zeta_{128}^{11} q^{5} + \zeta_{128}^{30} q^{6} + \zeta_{128}^{23} q^{8} + \zeta_{128}^{2} q^{9} + \zeta_{128}^{40} q^{10} - \zeta_{128}^{59} q^{12} + \cdots + \zeta_{128}^{57} q^{98} +O(q^{100}) $ q - z^29 * q^2 - z * q^3 + z^58 * q^4 - z^11 * q^5 + z^30 * q^6 + z^23 * q^8 + z^2 * q^9 + z^40 * q^10 - z^59 * q^12 + z^12 * q^15 - z^52 * q^16 + (-z^25 + z^15) * q^17 - z^31 * q^18 + (z^38 - z^20) * q^19 + z^5 * q^20 + (-z^45 - z^7) * q^23 - z^24 * q^24 + z^22 * q^25 - z^3 * q^27 - z^41 * q^30 + (z^62 - z^50) * q^31 - z^17 * q^32 + (z^54 - z^44) * q^34 + z^60 * q^36 + (z^49 + z^3) * q^38 - z^34 * q^40 - z^13 * q^45 + (z^36 - z^10) * q^46 + (-z^53 - z^19) * q^47 + z^53 * q^48 - z^28 * q^49 - z^51 * q^50 + (z^26 - z^16) * q^51 + (z^41 - z^5) * q^53 + z^32 * q^54 + (-z^39 + z^21) * q^57 - z^6 * q^60 + (-z^56 + z^42) * q^61 + (z^27 - z^15) * q^62 + z^46 * q^64 + (z^19 - z^9) * q^68 + (z^46 + z^8) * q^69 + z^25 * q^72 - z^23 * q^75 + (-z^32 + z^14) * q^76 + (z^48 - z^8) * q^79 + z^63 * q^80 + z^4 * q^81 + (z^47 + z^43) * q^83 + (z^36 - z^26) * q^85 + z^42 * q^90 + (z^39 + z) * q^92 + (-z^63 + z^51) * q^93 + (z^48 - z^18) * q^94 + (-z^49 + z^31) * q^95 + z^18 * q^96 + z^57 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q+O(q^{100}) $ 64 * q

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$.

$n$	$511$	$1537$	$2561$	$2821$
$\chi(n)$	$1$	$-1$	$-1$	$\zeta_{128}^{22}$

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} $

29.1

0.671559	+	0.740951i
−0.671559	−	0.740951i
−0.941544	−	0.336890i
0.941544	+	0.336890i
−0.595699	+	0.803208i
0.595699	−	0.803208i
−0.989177	−	0.146730i
0.989177	+	0.146730i
0.903989	+	0.427555i
−0.903989	−	0.427555i
0.998795	−	0.0490677i
−0.998795	+	0.0490677i
0.242980	−	0.970031i
−0.242980	+	0.970031i
0.970031	−	0.242980i
−0.970031	+	0.242980i
0.998795	+	0.0490677i
−0.998795	−	0.0490677i
0.903989	−	0.427555i
−0.903989	+	0.427555i

−0.595699

0.803208i

−0.671559

−

0.740951i

−0.290285

−

0.956940i

0.970031

−

0.242980i

0.995185

−

0.0980171i

0.941544

0.336890i

−0.0980171

0.995185i

−0.382683

0.923880i

29.2

0.595699

−

0.803208i

0.671559

0.740951i

−0.290285

−

0.956940i

−0.970031

0.242980i

0.995185

−

0.0980171i

−0.941544

−

0.336890i

−0.0980171

0.995185i

−0.382683

0.923880i

149.1

−0.857729

−

0.514103i

0.941544

0.336890i

0.471397

0.881921i

−0.803208

−

0.595699i

−0.634393

−

0.773010i

0.0490677

−

0.998795i

0.773010

0.634393i

0.382683

0.923880i

149.2

0.857729

0.514103i

−0.941544

−

0.336890i

0.471397

0.881921i

0.803208

0.595699i

−0.634393

−

0.773010i

−0.0490677

0.998795i

0.773010

0.634393i

0.382683

0.923880i

269.1

−0.336890

−

0.941544i

0.595699

−

0.803208i

−0.773010

0.634393i

−0.671559

0.740951i

−0.956940

−

0.290285i

0.857729

0.514103i

−0.290285

−

0.956940i

0.923880

0.382683i

269.2

0.336890

0.941544i

−0.595699

0.803208i

−0.773010

0.634393i

0.671559

−

0.740951i

−0.956940

−

0.290285i

−0.857729

−

0.514103i

−0.290285

−

0.956940i

0.923880

0.382683i

389.1

−0.427555

−

0.903989i

0.989177

0.146730i

−0.634393

0.773010i

−0.0490677

0.998795i

−0.290285

−

0.956940i

0.970031

0.242980i

0.956940

0.290285i

0.923880

−

0.382683i

389.2

0.427555

0.903989i

−0.989177

−

0.146730i

−0.634393

0.773010i

0.0490677

−

0.998795i

−0.290285

−

0.956940i

−0.970031

−

0.242980i

0.956940

0.290285i

0.923880

−

0.382683i

509.1

−0.970031

−

0.242980i

−0.903989

−

0.427555i

0.881921

0.471397i

−0.146730

0.989177i

0.773010

0.634393i

−0.740951

−

0.671559i

0.634393

0.773010i

0.382683

−

0.923880i

509.2

0.970031

0.242980i

0.903989

0.427555i

0.881921

0.471397i

0.146730

−

0.989177i

0.773010

0.634393i

0.740951

0.671559i

0.634393

0.773010i

0.382683

−

0.923880i

629.1

−0.146730

0.989177i

−0.998795

0.0490677i

−0.956940

−

0.290285i

−0.857729

0.514103i

0.0980171

−

0.995185i

0.427555

−

0.903989i

0.995185

−

0.0980171i

−0.382683

−

0.923880i

629.2

0.146730

−

0.989177i

0.998795

−

0.0490677i

−0.956940

−

0.290285i

0.857729

−

0.514103i

0.0980171

−

0.995185i

−0.427555

0.903989i

0.995185

−

0.0980171i

−0.382683

−

0.923880i

749.1

−0.740951

0.671559i

−0.242980

0.970031i

0.0980171

−

0.995185i

0.427555

0.903989i

−0.471397

−

0.881921i

0.595699

0.803208i

−0.881921

−

0.471397i

−0.923880

−

0.382683i

749.2

0.740951

−

0.671559i

0.242980

−

0.970031i

0.0980171

−

0.995185i

−0.427555

−

0.903989i

−0.471397

−

0.881921i

−0.595699

−

0.803208i

−0.881921

−

0.471397i

−0.923880

−

0.382683i

869.1

−0.671559

0.740951i

−0.970031

0.242980i

−0.0980171

−

0.995185i

0.903989

0.427555i

0.471397

−

0.881921i

0.803208

0.595699i

0.881921

−

0.471397i

−0.923880

0.382683i

869.2

0.671559

−

0.740951i

0.970031

−

0.242980i

−0.0980171

−

0.995185i

−0.903989

−

0.427555i

0.471397

−

0.881921i

−0.803208

−

0.595699i

0.881921

−

0.471397i

−0.923880

0.382683i

989.1

−0.146730

−

0.989177i

−0.998795

−

0.0490677i

−0.956940

0.290285i

−0.857729

−

0.514103i

0.0980171

0.995185i

0.427555

0.903989i

0.995185

0.0980171i

−0.382683

0.923880i

989.2

0.146730

0.989177i

0.998795

0.0490677i

−0.956940

0.290285i

0.857729

0.514103i

0.0980171

0.995185i

−0.427555

−

0.903989i

0.995185

0.0980171i

−0.382683

0.923880i

1109.1

−0.970031

0.242980i

−0.903989

0.427555i

0.881921

−

0.471397i

−0.146730

−

0.989177i

0.773010

−

0.634393i

−0.740951

0.671559i

0.634393

−

0.773010i

0.382683

0.923880i

1109.2

0.970031

−

0.242980i

0.903989

−

0.427555i

0.881921

−

0.471397i

0.146730

0.989177i

0.773010

−

0.634393i

0.740951

−

0.671559i

0.634393

−

0.773010i

0.382683

0.923880i

See all 64 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $
3.b	odd	2	1	inner
5.b	even	2	1	inner
256.m	even	64	1	inner
768.y	odd	64	1	inner
1280.bw	even	64	1	inner
3840.dy	odd	64	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3840.1.dy.a	✓	64
3.b	odd	2	1	inner	3840.1.dy.a	✓	64
5.b	even	2	1	inner	3840.1.dy.a	✓	64
15.d	odd	2	1	CM	3840.1.dy.a	✓	64
256.m	even	64	1	inner	3840.1.dy.a	✓	64
768.y	odd	64	1	inner	3840.1.dy.a	✓	64
1280.bw	even	64	1	inner	3840.1.dy.a	✓	64
3840.dy	odd	64	1	inner	3840.1.dy.a	✓	64

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3840.1.dy.a	✓	64	1.a	even	1	1	trivial
3840.1.dy.a	✓	64	3.b	odd	2	1	inner
3840.1.dy.a	✓	64	5.b	even	2	1	inner
3840.1.dy.a	✓	64	15.d	odd	2	1	CM
3840.1.dy.a	✓	64	256.m	even	64	1	inner
3840.1.dy.a	✓	64	768.y	odd	64	1	inner
3840.1.dy.a	✓	64	1280.bw	even	64	1	inner
3840.1.dy.a	✓	64	3840.dy	odd	64	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3840, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{64} + 1 $ T^64 + 1
$3$	$ T^{64} + 1 $ T^64 + 1
$5$	$ T^{64} + 1 $ T^64 + 1
$7$	$ T^{64} $ T^64
$11$	$ T^{64} $ T^64
$13$	$ T^{64} $ T^64
$17$	$ T^{64} - 1664 T^{50} + \cdots + 4 $ T^64 - 1664T^50 + 51480T^48 - 30464T^46 + 608T^44 + 64T^38 + 523328T^36 + 30875520T^34 + 173626516T^32 + 177494528T^30 + 43439744T^28 + 2733056T^26 + 78336T^24 - 11162880T^22 + 227793152T^20 - 938361600T^18 + 1264445104T^16 - 732034560T^14 + 214599872T^12 - 35108864T^10 + 3811328T^8 + 82048T^6 + 43648T^4 + 768*T^2 + 4
$19$	$ (T^{32} + 80 T^{26} + \cdots + 2)^{2} $ (T^32 + 80T^26 + 160T^21 + 1880T^20 + 2T^16 - 10496T^15 + 12736T^14 - 32T^13 + 2320T^10 + 28864T^9 + 13452T^8 + 384T^7 - 96T^5 + 1320T^4 - 1280T^3 + 336T^2 - 32T + 2)^2
$23$	$ T^{64} + 64 T^{42} + \cdots + 4 $ T^64 + 64T^42 + 28336T^40 + 1619200T^38 + 29995680T^36 + 255487680T^34 + 1202160780T^32 + 3493585920T^30 + 6748972800T^28 + 9119916288T^26 + 8938965568T^24 + 6522760192T^22 + 3609101440T^20 + 1532729088T^18 + 505093248T^16 + 116441088T^14 + 48829440T^12 - 16248960T^10 + 7524000T^8 - 1014272T^6 + 60352T^4 - 896*T^2 + 4
$29$	$ T^{64} $ T^64
$31$	$ (T^{32} + 280 T^{24} + \cdots + 4)^{2} $ (T^32 + 280T^24 + 32T^22 - 832T^18 + 13460T^16 - 6272T^14 + 512T^12 + 15104T^10 + 30384T^8 + 11328T^6 + 2048T^4 - 128*T^2 + 4)^2
$37$	$ T^{64} $ T^64
$41$	$ T^{64} $ T^64
$43$	$ T^{64} $ T^64
$47$	$ T^{64} - 1664 T^{50} + \cdots + 4 $ T^64 - 1664T^50 + 51480T^48 - 30464T^46 + 608T^44 + 64T^38 + 523328T^36 + 30875520T^34 + 173626516T^32 + 177494528T^30 + 43439744T^28 + 2733056T^26 + 78336T^24 - 11162880T^22 + 227793152T^20 - 938361600T^18 + 1264445104T^16 - 732034560T^14 + 214599872T^12 - 35108864T^10 + 3811328T^8 + 82048T^6 + 43648T^4 + 768*T^2 + 4
$53$	$ T^{64} + 64 T^{60} + \cdots + 16 $ T^64 + 64T^60 + 1824T^56 + 30528T^52 + 333200T^48 + 2490624T^44 + 13032832T^40 + 47783936T^36 + 131049600T^32 + 106153984T^28 + 777809408T^24 - 791304448T^20 + 701793728T^16 - 117660672T^12 + 7672576T^8 - 21504*T^4 + 16
$59$	$ T^{64} $ T^64
$61$	$ (T^{32} + 4 T^{24} + \cdots + 2)^{2} $ (T^32 + 4T^24 - 448T^23 + 144T^22 + 6T^16 + 2688T^15 + 23136T^14 + 20352T^13 + 2616T^12 + 32T^11 + 4T^8 - 960T^7 + 6864T^6 - 11648T^5 + 7752T^4 - 2464T^3 + 400T^2 - 32*T + 2)^2
$67$	$ T^{64} $ T^64
$71$	$ T^{64} $ T^64
$73$	$ T^{64} $ T^64
$79$	$ (T^{8} + 8 T^{5} + 2 T^{4} + \cdots + 2)^{8} $ (T^8 + 8T^5 + 2T^4 + 12T^2 - 8T + 2)^8
$83$	$ T^{64} + 7280 T^{48} + \cdots + 16 $ T^64 + 7280T^48 - 537472T^44 + 5230016T^40 - 15779712T^36 + 26499200T^32 + 180433920T^28 + 1110315648T^24 + 1521254912T^20 + 644145728T^16 + 42616832T^12 + 7228928T^8 - 17920T^4 + 16
$89$	$ T^{64} $ T^64
$97$	$ T^{64} $ T^64
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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:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3840.dy (of order \(64\), degree \(32\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{128}^{29} q^{2} - \zeta_{128} q^{3} + \zeta_{128}^{58} q^{4} - \zeta_{128}^{11} q^{5} + \zeta_{128}^{30} q^{6} + \zeta_{128}^{23} q^{8} + \zeta_{128}^{2} q^{9} + \zeta_{128}^{40} q^{10} - \zeta_{128}^{59} q^{12} + \cdots + \zeta_{128}^{57} q^{98} +O(q^{100}) \) q - z^29 * q^2 - z * q^3 + z^58 * q^4 - z^11 * q^5 + z^30 * q^6 + z^23 * q^8 + z^2 * q^9 + z^40 * q^10 - z^59 * q^12 + z^12 * q^15 - z^52 * q^16 + (-z^25 + z^15) * q^17 - z^31 * q^18 + (z^38 - z^20) * q^19 + z^5 * q^20 + (-z^45 - z^7) * q^23 - z^24 * q^24 + z^22 * q^25 - z^3 * q^27 - z^41 * q^30 + (z^62 - z^50) * q^31 - z^17 * q^32 + (z^54 - z^44) * q^34 + z^60 * q^36 + (z^49 + z^3) * q^38 - z^34 * q^40 - z^13 * q^45 + (z^36 - z^10) * q^46 + (-z^53 - z^19) * q^47 + z^53 * q^48 - z^28 * q^49 - z^51 * q^50 + (z^26 - z^16) * q^51 + (z^41 - z^5) * q^53 + z^32 * q^54 + (-z^39 + z^21) * q^57 - z^6 * q^60 + (-z^56 + z^42) * q^61 + (z^27 - z^15) * q^62 + z^46 * q^64 + (z^19 - z^9) * q^68 + (z^46 + z^8) * q^69 + z^25 * q^72 - z^23 * q^75 + (-z^32 + z^14) * q^76 + (z^48 - z^8) * q^79 + z^63 * q^80 + z^4 * q^81 + (z^47 + z^43) * q^83 + (z^36 - z^26) * q^85 + z^42 * q^90 + (z^39 + z) * q^92 + (-z^63 + z^51) * q^93 + (z^48 - z^18) * q^94 + (-z^49 + z^31) * q^95 + z^18 * q^96 + z^57 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{100}) \) 64 * q

Introduction

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Modular forms

Varieties

Fields

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Groups

Database

Properties

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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	3840.1.dy.a

Level	$3840$
Weight	$1$
Character orbit	3840.dy
Analytic conductor	$1.916$
Analytic rank	$0$
Dimension	$64$
Projective image	$D_{64}$
CM discriminant	-15
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.91640964851\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(2\) over \(\Q(\zeta_{64})\)
Coefficient field:	\(\Q(\zeta_{128})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{64} + 1 \) x^64 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{64}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{64} + \cdots)\)

\(n\)	\(511\)	\(1537\)	\(2561\)	\(2821\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(\zeta_{128}^{22}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 29.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{64} + 1 \) T^64 + 1
$3$	\( T^{64} + 1 \) T^64 + 1
$5$	\( T^{64} + 1 \) T^64 + 1
$7$	\( T^{64} \) T^64
$11$	\( T^{64} \) T^64
$13$	\( T^{64} \) T^64
$17$	\( T^{64} - 1664 T^{50} + \cdots + 4 \) T^64 - 1664T^50 + 51480T^48 - 30464T^46 + 608T^44 + 64T^38 + 523328T^36 + 30875520T^34 + 173626516T^32 + 177494528T^30 + 43439744T^28 + 2733056T^26 + 78336T^24 - 11162880T^22 + 227793152T^20 - 938361600T^18 + 1264445104T^16 - 732034560T^14 + 214599872T^12 - 35108864T^10 + 3811328T^8 + 82048T^6 + 43648T^4 + 768*T^2 + 4
$19$	\( (T^{32} + 80 T^{26} + \cdots + 2)^{2} \) (T^32 + 80T^26 + 160T^21 + 1880T^20 + 2T^16 - 10496T^15 + 12736T^14 - 32T^13 + 2320T^10 + 28864T^9 + 13452T^8 + 384T^7 - 96T^5 + 1320T^4 - 1280T^3 + 336T^2 - 32T + 2)^2
$23$	\( T^{64} + 64 T^{42} + \cdots + 4 \) T^64 + 64T^42 + 28336T^40 + 1619200T^38 + 29995680T^36 + 255487680T^34 + 1202160780T^32 + 3493585920T^30 + 6748972800T^28 + 9119916288T^26 + 8938965568T^24 + 6522760192T^22 + 3609101440T^20 + 1532729088T^18 + 505093248T^16 + 116441088T^14 + 48829440T^12 - 16248960T^10 + 7524000T^8 - 1014272T^6 + 60352T^4 - 896*T^2 + 4
$29$	\( T^{64} \) T^64
$31$	\( (T^{32} + 280 T^{24} + \cdots + 4)^{2} \) (T^32 + 280T^24 + 32T^22 - 832T^18 + 13460T^16 - 6272T^14 + 512T^12 + 15104T^10 + 30384T^8 + 11328T^6 + 2048T^4 - 128*T^2 + 4)^2
$37$	\( T^{64} \) T^64
$41$	\( T^{64} \) T^64
$43$	\( T^{64} \) T^64
$47$	\( T^{64} - 1664 T^{50} + \cdots + 4 \) T^64 - 1664T^50 + 51480T^48 - 30464T^46 + 608T^44 + 64T^38 + 523328T^36 + 30875520T^34 + 173626516T^32 + 177494528T^30 + 43439744T^28 + 2733056T^26 + 78336T^24 - 11162880T^22 + 227793152T^20 - 938361600T^18 + 1264445104T^16 - 732034560T^14 + 214599872T^12 - 35108864T^10 + 3811328T^8 + 82048T^6 + 43648T^4 + 768*T^2 + 4
$53$	\( T^{64} + 64 T^{60} + \cdots + 16 \) T^64 + 64T^60 + 1824T^56 + 30528T^52 + 333200T^48 + 2490624T^44 + 13032832T^40 + 47783936T^36 + 131049600T^32 + 106153984T^28 + 777809408T^24 - 791304448T^20 + 701793728T^16 - 117660672T^12 + 7672576T^8 - 21504*T^4 + 16
$59$	\( T^{64} \) T^64
$61$	\( (T^{32} + 4 T^{24} + \cdots + 2)^{2} \) (T^32 + 4T^24 - 448T^23 + 144T^22 + 6T^16 + 2688T^15 + 23136T^14 + 20352T^13 + 2616T^12 + 32T^11 + 4T^8 - 960T^7 + 6864T^6 - 11648T^5 + 7752T^4 - 2464T^3 + 400T^2 - 32*T + 2)^2
$67$	\( T^{64} \) T^64
$71$	\( T^{64} \) T^64
$73$	\( T^{64} \) T^64
$79$	\( (T^{8} + 8 T^{5} + 2 T^{4} + \cdots + 2)^{8} \) (T^8 + 8T^5 + 2T^4 + 12T^2 - 8T + 2)^8
$83$	\( T^{64} + 7280 T^{48} + \cdots + 16 \) T^64 + 7280T^48 - 537472T^44 + 5230016T^40 - 15779712T^36 + 26499200T^32 + 180433920T^28 + 1110315648T^24 + 1521254912T^20 + 644145728T^16 + 42616832T^12 + 7228928T^8 - 17920T^4 + 16
$89$	\( T^{64} \) T^64
$97$	\( T^{64} \) T^64
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