LMFDB - Newform orbit 3584.1.cd.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3584,1,Mod(13,3584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3584, base_ring=CyclotomicField(128))

chi = DirichletCharacter(H, H._module([0, 111, 64]))

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

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

chi := DirichletCharacter("3584.13");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3584 = 2^{9} \cdot 7 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3584.cd (of order $128$, degree $64$, 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:	$1.78864900528$
Analytic rank:	$0$
Dimension:	$64$
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:	$D_{128}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{128} - \cdots)$

$q$-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}^{59} q^{2} - \zeta_{128}^{54} q^{4} + \zeta_{128}^{63} q^{7} - \zeta_{128}^{49} q^{8} - \zeta_{128}^{25} q^{9} +O(q^{10}) $ q - z^59 * q^2 - z^54 * q^4 + z^63 * q^7 - z^49 * q^8 - z^25 * q^9	\( q - \zeta_{128}^{59} q^{2} - \zeta_{128}^{54} q^{4} + \zeta_{128}^{63} q^{7} - \zeta_{128}^{49} q^{8} - \zeta_{128}^{25} q^{9} + ( - \zeta_{128}^{13} - \zeta_{128}^{2}) q^{11} + \zeta_{128}^{58} q^{14} - \zeta_{128}^{44} q^{16} - \zeta_{128}^{20} q^{18} + (\zeta_{128}^{61} - \zeta_{128}^{8}) q^{22} + ( - \zeta_{128}^{46} + \zeta_{128}^{28}) q^{23} - \zeta_{128}^{19} q^{25} + \zeta_{128}^{53} q^{28} + ( - \zeta_{128}^{55} - \zeta_{128}^{42}) q^{29} - \zeta_{128}^{39} q^{32} - \zeta_{128}^{15} q^{36} + ( - \zeta_{128}^{21} - \zeta_{128}^{6}) q^{37} + (\zeta_{128}^{36} + \zeta_{128}^{35}) q^{43} + (\zeta_{128}^{56} - \zeta_{128}^{3}) q^{44} + ( - \zeta_{128}^{41} + \zeta_{128}^{23}) q^{46} - \zeta_{128}^{62} q^{49} - \zeta_{128}^{14} q^{50} + (\zeta_{128}^{52} - \zeta_{128}^{43}) q^{53} + \zeta_{128}^{48} q^{56} + ( - \zeta_{128}^{50} - \zeta_{128}^{37}) q^{58} + \zeta_{128}^{24} q^{63} - \zeta_{128}^{34} q^{64} + ( - \zeta_{128}^{40} - \zeta_{128}^{33}) q^{67} + (\zeta_{128}^{37} + \zeta_{128}^{9}) q^{71} - \zeta_{128}^{10} q^{72} + ( - \zeta_{128}^{16} - \zeta_{128}) q^{74} + (\zeta_{128}^{12} + \zeta_{128}) q^{77} + ( - \zeta_{128}^{17} + \zeta_{128}^{11}) q^{79} + \zeta_{128}^{50} q^{81} + (\zeta_{128}^{31} + \zeta_{128}^{30}) q^{86} + (\zeta_{128}^{62} + \zeta_{128}^{51}) q^{88} + ( - \zeta_{128}^{36} + \zeta_{128}^{18}) q^{92} - \zeta_{128}^{57} q^{98} + (\zeta_{128}^{38} + \zeta_{128}^{27}) q^{99} +O(q^{100}) \) q - z^59 * q^2 - z^54 * q^4 + z^63 * q^7 - z^49 * q^8 - z^25 * q^9 + (-z^13 - z^2) * q^11 + z^58 * q^14 - z^44 * q^16 - z^20 * q^18 + (z^61 - z^8) * q^22 + (-z^46 + z^28) * q^23 - z^19 * q^25 + z^53 * q^28 + (-z^55 - z^42) * q^29 - z^39 * q^32 - z^15 * q^36 + (-z^21 - z^6) * q^37 + (z^36 + z^35) * q^43 + (z^56 - z^3) * q^44 + (-z^41 + z^23) * q^46 - z^62 * q^49 - z^14 * q^50 + (z^52 - z^43) * q^53 + z^48 * q^56 + (-z^50 - z^37) * q^58 + z^24 * q^63 - z^34 * q^64 + (-z^40 - z^33) * q^67 + (z^37 + z^9) * q^71 - z^10 * q^72 + (-z^16 - z) * q^74 + (z^12 + z) * q^77 + (-z^17 + z^11) * q^79 + z^50 * q^81 + (z^31 + z^30) * q^86 + (z^62 + z^51) * q^88 + (-z^36 + z^18) * q^92 - z^57 * q^98 + (z^38 + z^27) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q+O(q^{10}) $ 64 * q	$ 64 q+O(q^{100}) $ 64 * q

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$-1$	$-\zeta_{128}^{19}$

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} $

13.1

0.857729	−	0.514103i
0.514103	−	0.857729i
0.671559	+	0.740951i
0.336890	−	0.941544i
0.595699	−	0.803208i
−0.146730	+	0.989177i
0.903989	+	0.427555i
0.0490677	+	0.998795i
0.242980	−	0.970031i
−0.242980	−	0.970031i
0.998795	+	0.0490677i
−0.427555	−	0.903989i
−0.146730	−	0.989177i
0.595699	+	0.803208i
0.941544	−	0.336890i
0.740951	+	0.671559i
−0.514103	−	0.857729i
−0.857729	−	0.514103i
0.740951	−	0.671559i
−0.941544	−	0.336890i

−0.903989

0.427555i

0.634393

−

0.773010i

−0.857729

−

0.514103i

−0.242980

0.970031i

−0.595699

0.803208i

69.1

0.427555

−

0.903989i

−0.634393

−

0.773010i

−0.514103

−

0.857729i

−0.970031

0.242980i

−0.803208

0.595699i

125.1

−0.514103

0.857729i

−0.471397

−

0.881921i

−0.671559

0.740951i

0.998795

0.0490677i

0.427555

−

0.903989i

181.1

0.989177

−

0.146730i

0.956940

−

0.290285i

−0.336890

−

0.941544i

0.903989

−

0.427555i

−0.740951

−

0.671559i

237.1

−0.0490677

−

0.998795i

−0.995185

0.0980171i

−0.595699

−

0.803208i

0.146730

0.989177i

0.242980

−

0.970031i

293.1

−0.671559

−

0.740951i

−0.0980171

0.995185i

0.146730

0.989177i

0.803208

−

0.595699i

−0.514103

0.857729i

349.1

−0.595699

−

0.803208i

−0.290285

0.956940i

−0.903989

0.427555i

0.941544

−

0.336890i

−0.0490677

0.998795i

405.1

0.242980

−

0.970031i

−0.881921

−

0.471397i

−0.0490677

0.998795i

−0.671559

0.740951i

−0.941544

−

0.336890i

461.1

0.941544

0.336890i

0.773010

0.634393i

−0.242980

−

0.970031i

0.514103

0.857729i

0.146730

0.989177i

517.1

−0.941544

0.336890i

0.773010

−

0.634393i

0.242980

−

0.970031i

−0.514103

0.857729i

−0.146730

0.989177i

573.1

0.970031

−

0.242980i

0.881921

−

0.471397i

−0.998795

0.0490677i

0.740951

−

0.671559i

−0.336890

−

0.941544i

629.1

−0.803208

−

0.595699i

0.290285

0.956940i

0.427555

−

0.903989i

0.336890

−

0.941544i

−0.998795

0.0490677i

685.1

−0.671559

0.740951i

−0.0980171

−

0.995185i

0.146730

−

0.989177i

0.803208

0.595699i

−0.514103

−

0.857729i

741.1

−0.0490677

0.998795i

−0.995185

−

0.0980171i

−0.595699

0.803208i

0.146730

−

0.989177i

0.242980

0.970031i

797.1

−0.146730

0.989177i

−0.956940

−

0.290285i

−0.941544

−

0.336890i

0.427555

−

0.903989i

0.671559

0.740951i

853.1

−0.857729

0.514103i

0.471397

−

0.881921i

−0.740951

0.671559i

0.0490677

0.998795i

−0.903989

0.427555i

909.1

−0.427555

−

0.903989i

−0.634393

0.773010i

0.514103

−

0.857729i

0.970031

0.242980i

0.803208

0.595699i

965.1

0.903989

0.427555i

0.634393

0.773010i

0.857729

−

0.514103i

0.242980

0.970031i

0.595699

0.803208i

1021.1

−0.857729

−

0.514103i

0.471397

0.881921i

−0.740951

−

0.671559i

0.0490677

−

0.998795i

−0.903989

−

0.427555i

1077.1

0.146730

0.989177i

−0.956940

0.290285i

0.941544

−

0.336890i

−0.427555

−

0.903989i

−0.671559

0.740951i

See all 64 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $
512.o	even	128	1	inner
3584.cd	odd	128	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3584.1.cd.a	✓	64
7.b	odd	2	1	CM	3584.1.cd.a	✓	64
512.o	even	128	1	inner	3584.1.cd.a	✓	64
3584.cd	odd	128	1	inner	3584.1.cd.a	✓	64

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3584.1.cd.a	✓	64	1.a	even	1	1	trivial
3584.1.cd.a	✓	64	7.b	odd	2	1	CM
3584.1.cd.a	✓	64	512.o	even	128	1	inner
3584.1.cd.a	✓	64	3584.cd	odd	128	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3584, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{64} + 1 $ T^64 + 1
$3$	$ T^{64} $ T^64
$5$	$ T^{64} $ T^64
$7$	$ T^{64} + 1 $ T^64 + 1
$11$	$ T^{64} + 1344 T^{54} + \cdots + 2 $ T^64 + 1344T^54 - 39168T^45 + 500064T^44 - 640T^43 + 46800T^36 + 23710848T^35 + 47366368T^34 + 659904T^33 + 2T^32 - 13440T^27 + 33782144T^26 - 695521152T^25 + 582936240T^24 - 27083776T^23 + 36928T^22 + 1440T^18 + 4396480T^17 + 154686612T^16 + 461122560T^15 + 227026688T^14 + 22373632T^13 + 378528T^12 + 384T^11 - 64T^9 + 51480T^8 - 496128T^7 + 861120T^6 - 523776T^5 + 146224T^4 - 21120T^3 + 1632T^2 - 64T + 2
$13$	$ T^{64} $ T^64
$17$	$ T^{64} $ T^64
$19$	$ T^{64} $ T^64
$23$	$ (T^{32} + 2 T^{16} + \cdots + 2)^{2} $ (T^32 + 2T^16 - 256T^15 + 5440T^14 - 45696T^13 + 201552T^12 - 537472T^11 + 940576T^10 - 1136960T^9 + 980628T^8 - 615296T^7 + 283360T^6 - 95680T^5 + 23400T^4 - 4032T^3 + 464T^2 - 32T + 2)^2
$29$	$ T^{64} - 192 T^{57} + \cdots + 2 $ T^64 - 192T^57 + 13856T^50 + 2560T^47 - 466432T^43 + 1469736T^40 - 64T^37 + 7396112T^36 + 84723008T^33 + 2T^32 + 757504T^30 - 49600256T^29 + 777461696T^26 + 832T^25 - 54995968T^23 + 109390592T^22 + 33616T^20 + 1022678528T^19 + 21280T^18 + 150983820T^16 - 48161280T^15 + 1176704T^13 + 111131072T^12 + 46592T^11 + 352T^10 - 10762816T^9 + 1523544T^8 + 170880T^6 + 181824T^5 + 2640T^4 - 768T^3 + 864T^2 - 64T + 2
$31$	$ T^{64} $ T^64
$37$	$ T^{64} + 2640 T^{52} + \cdots + 2 $ T^64 + 2640T^52 - 46592T^47 + 64T^45 + 21280T^42 + 1523544T^40 - 832T^37 + 48161280T^35 - 181824T^33 + 2T^32 + 109390592T^30 + 111131072T^28 + 864T^26 + 49600256T^25 - 1022678528T^23 + 10762816T^21 + 7396112T^20 + 777461696T^18 + 150983820T^16 + 466432T^15 + 170880T^14 - 84723008T^13 + 54995968T^11 + 13856T^10 - 1176704T^9 + 1469736T^8 + 768T^7 + 757504T^6 + 192T^5 + 33616T^4 - 2560T^3 + 352T^2 + 64T + 2
$41$	$ T^{64} $ T^64
$43$	$ T^{64} + 896 T^{55} + \cdots + 2 $ T^64 + 896T^55 + 4T^48 + 245824T^46 + 15504T^44 - 28800T^39 + 21690112T^37 - 8210496T^35 + 9280T^33 + 6T^32 + 6143616T^30 + 434291296T^28 + 293356416T^26 + 5454392T^24 + 33920T^23 + 1312T^22 - 81624576T^21 + 865095808T^19 - 493084032T^17 + 4T^16 + 31155456T^15 + 912960T^14 - 233728T^13 + 24807120T^12 + 64T^11 + 33984128T^10 + 8121928T^8 - 1920T^7 + 601440T^6 + 23296T^5 + 17808T^4 - 4928T^3 + 224T^2 + 64*T + 2
$47$	$ T^{64} $ T^64
$53$	$ T^{64} + 288 T^{54} + \cdots + 2 $ T^64 + 288T^54 - 3520T^51 + 4T^48 + 25968T^44 + 2748544T^41 + 2350752T^38 + 38336T^35 + 782880T^34 + 6T^32 - 119459328T^31 + 425251232T^28 + 64T^27 - 124151552T^25 + 5627448T^24 + 4751328T^22 + 341240320T^21 - 23360T^19 + 949049408T^18 - 3648T^17 + 4T^16 + 396793344T^15 + 3691072T^14 + 45378352T^12 - 18333120T^11 + 1964160T^9 + 5732936T^8 + 4224T^7 + 37472T^6 - 204288T^5 + 7280T^4 + 320T^3 + 800T^2 + 64*T + 2
$59$	$ T^{64} $ T^64
$61$	$ T^{64} $ T^64
$67$	$ T^{64} + 8 T^{56} + \cdots + 2 $ T^64 + 8T^56 + 960T^53 + 28T^48 - 246912T^45 + 243744T^42 + 56T^40 - 64T^39 + 3252544T^37 + 67776672T^34 + 70T^32 + 13878528T^31 - 8012544T^29 + 43280T^28 + 537086784T^26 + 56T^24 - 738280320T^23 + 4688448T^21 + 70900368T^20 + 275294016T^18 - 707904T^17 + 28T^16 + 453097728T^15 + 480T^14 - 569472T^13 + 87415664T^12 + 7001760T^10 + 4122240T^9 + 8T^8 - 1982016T^7 + 68320T^6 + 6336T^5 + 43248T^4 + 448T^3 + 544T^2 - 64*T + 2
$71$	$ 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
$73$	$ T^{64} $ T^64
$79$	$ T^{64} + 768 T^{54} + \cdots + 4 $ T^64 + 768T^54 + 352T^52 + 170880T^44 - 1176704T^42 + 33616T^40 + 864T^36 + 10762816T^34 + 150983820T^32 + 54995968T^30 + 757504T^28 - 181760T^26 + 111131072T^24 - 1022678528T^22 + 777461696T^20 - 84722944T^18 + 2993280T^16 + 48158720T^14 + 109390592T^12 + 49600256T^10 + 7398752T^8 + 419840T^6 + 35136T^4 - 640*T^2 + 4
$83$	$ T^{64} $ T^64
$89$	$ T^{64} $ T^64
$97$	$ T^{64} $ T^64
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Level:	\( N \)	\(=\)	\( 3584 = 2^{9} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3584.cd (of order \(128\), degree \(64\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{128}^{59} q^{2} - \zeta_{128}^{54} q^{4} + \zeta_{128}^{63} q^{7} - \zeta_{128}^{49} q^{8} - \zeta_{128}^{25} q^{9} +O(q^{10}) \) q - z^59 * q^2 - z^54 * q^4 + z^63 * q^7 - z^49 * q^8 - z^25 * q^9	\( q - \zeta_{128}^{59} q^{2} - \zeta_{128}^{54} q^{4} + \zeta_{128}^{63} q^{7} - \zeta_{128}^{49} q^{8} - \zeta_{128}^{25} q^{9} + ( - \zeta_{128}^{13} - \zeta_{128}^{2}) q^{11} + \zeta_{128}^{58} q^{14} - \zeta_{128}^{44} q^{16} - \zeta_{128}^{20} q^{18} + (\zeta_{128}^{61} - \zeta_{128}^{8}) q^{22} + ( - \zeta_{128}^{46} + \zeta_{128}^{28}) q^{23} - \zeta_{128}^{19} q^{25} + \zeta_{128}^{53} q^{28} + ( - \zeta_{128}^{55} - \zeta_{128}^{42}) q^{29} - \zeta_{128}^{39} q^{32} - \zeta_{128}^{15} q^{36} + ( - \zeta_{128}^{21} - \zeta_{128}^{6}) q^{37} + (\zeta_{128}^{36} + \zeta_{128}^{35}) q^{43} + (\zeta_{128}^{56} - \zeta_{128}^{3}) q^{44} + ( - \zeta_{128}^{41} + \zeta_{128}^{23}) q^{46} - \zeta_{128}^{62} q^{49} - \zeta_{128}^{14} q^{50} + (\zeta_{128}^{52} - \zeta_{128}^{43}) q^{53} + \zeta_{128}^{48} q^{56} + ( - \zeta_{128}^{50} - \zeta_{128}^{37}) q^{58} + \zeta_{128}^{24} q^{63} - \zeta_{128}^{34} q^{64} + ( - \zeta_{128}^{40} - \zeta_{128}^{33}) q^{67} + (\zeta_{128}^{37} + \zeta_{128}^{9}) q^{71} - \zeta_{128}^{10} q^{72} + ( - \zeta_{128}^{16} - \zeta_{128}) q^{74} + (\zeta_{128}^{12} + \zeta_{128}) q^{77} + ( - \zeta_{128}^{17} + \zeta_{128}^{11}) q^{79} + \zeta_{128}^{50} q^{81} + (\zeta_{128}^{31} + \zeta_{128}^{30}) q^{86} + (\zeta_{128}^{62} + \zeta_{128}^{51}) q^{88} + ( - \zeta_{128}^{36} + \zeta_{128}^{18}) q^{92} - \zeta_{128}^{57} q^{98} + (\zeta_{128}^{38} + \zeta_{128}^{27}) q^{99} +O(q^{100}) \) q - z^59 * q^2 - z^54 * q^4 + z^63 * q^7 - z^49 * q^8 - z^25 * q^9 + (-z^13 - z^2) * q^11 + z^58 * q^14 - z^44 * q^16 - z^20 * q^18 + (z^61 - z^8) * q^22 + (-z^46 + z^28) * q^23 - z^19 * q^25 + z^53 * q^28 + (-z^55 - z^42) * q^29 - z^39 * q^32 - z^15 * q^36 + (-z^21 - z^6) * q^37 + (z^36 + z^35) * q^43 + (z^56 - z^3) * q^44 + (-z^41 + z^23) * q^46 - z^62 * q^49 - z^14 * q^50 + (z^52 - z^43) * q^53 + z^48 * q^56 + (-z^50 - z^37) * q^58 + z^24 * q^63 - z^34 * q^64 + (-z^40 - z^33) * q^67 + (z^37 + z^9) * q^71 - z^10 * q^72 + (-z^16 - z) * q^74 + (z^12 + z) * q^77 + (-z^17 + z^11) * q^79 + z^50 * q^81 + (z^31 + z^30) * q^86 + (z^62 + z^51) * q^88 + (-z^36 + z^18) * q^92 - z^57 * q^98 + (z^38 + z^27) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \) 64 * q	\( 64 q+O(q^{100}) \) 64 * q

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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	3584.1.cd.a

Level	$3584$
Weight	$1$
Character orbit	3584.cd
Analytic conductor	$1.789$
Analytic rank	$0$
Dimension	$64$
Projective image	$D_{128}$
CM discriminant	-7
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.78864900528\)
Analytic rank:	\(0\)
Dimension:	\(64\)
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:	\(D_{128}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{128} - \cdots)\)

\(n\)	\(1023\)	\(1025\)	\(1541\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-\zeta_{128}^{19}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 13.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
512.o	even	128	1	inner
3584.cd	odd	128	1	inner

$p$	$F_p(T)$
$2$	\( T^{64} + 1 \) T^64 + 1
$3$	\( T^{64} \) T^64
$5$	\( T^{64} \) T^64
$7$	\( T^{64} + 1 \) T^64 + 1
$11$	\( T^{64} + 1344 T^{54} + \cdots + 2 \) T^64 + 1344T^54 - 39168T^45 + 500064T^44 - 640T^43 + 46800T^36 + 23710848T^35 + 47366368T^34 + 659904T^33 + 2T^32 - 13440T^27 + 33782144T^26 - 695521152T^25 + 582936240T^24 - 27083776T^23 + 36928T^22 + 1440T^18 + 4396480T^17 + 154686612T^16 + 461122560T^15 + 227026688T^14 + 22373632T^13 + 378528T^12 + 384T^11 - 64T^9 + 51480T^8 - 496128T^7 + 861120T^6 - 523776T^5 + 146224T^4 - 21120T^3 + 1632T^2 - 64T + 2
$13$	\( T^{64} \) T^64
$17$	\( T^{64} \) T^64
$19$	\( T^{64} \) T^64
$23$	\( (T^{32} + 2 T^{16} + \cdots + 2)^{2} \) (T^32 + 2T^16 - 256T^15 + 5440T^14 - 45696T^13 + 201552T^12 - 537472T^11 + 940576T^10 - 1136960T^9 + 980628T^8 - 615296T^7 + 283360T^6 - 95680T^5 + 23400T^4 - 4032T^3 + 464T^2 - 32T + 2)^2
$29$	\( T^{64} - 192 T^{57} + \cdots + 2 \) T^64 - 192T^57 + 13856T^50 + 2560T^47 - 466432T^43 + 1469736T^40 - 64T^37 + 7396112T^36 + 84723008T^33 + 2T^32 + 757504T^30 - 49600256T^29 + 777461696T^26 + 832T^25 - 54995968T^23 + 109390592T^22 + 33616T^20 + 1022678528T^19 + 21280T^18 + 150983820T^16 - 48161280T^15 + 1176704T^13 + 111131072T^12 + 46592T^11 + 352T^10 - 10762816T^9 + 1523544T^8 + 170880T^6 + 181824T^5 + 2640T^4 - 768T^3 + 864T^2 - 64T + 2
$31$	\( T^{64} \) T^64
$37$	\( T^{64} + 2640 T^{52} + \cdots + 2 \) T^64 + 2640T^52 - 46592T^47 + 64T^45 + 21280T^42 + 1523544T^40 - 832T^37 + 48161280T^35 - 181824T^33 + 2T^32 + 109390592T^30 + 111131072T^28 + 864T^26 + 49600256T^25 - 1022678528T^23 + 10762816T^21 + 7396112T^20 + 777461696T^18 + 150983820T^16 + 466432T^15 + 170880T^14 - 84723008T^13 + 54995968T^11 + 13856T^10 - 1176704T^9 + 1469736T^8 + 768T^7 + 757504T^6 + 192T^5 + 33616T^4 - 2560T^3 + 352T^2 + 64T + 2
$41$	\( T^{64} \) T^64
$43$	\( T^{64} + 896 T^{55} + \cdots + 2 \) T^64 + 896T^55 + 4T^48 + 245824T^46 + 15504T^44 - 28800T^39 + 21690112T^37 - 8210496T^35 + 9280T^33 + 6T^32 + 6143616T^30 + 434291296T^28 + 293356416T^26 + 5454392T^24 + 33920T^23 + 1312T^22 - 81624576T^21 + 865095808T^19 - 493084032T^17 + 4T^16 + 31155456T^15 + 912960T^14 - 233728T^13 + 24807120T^12 + 64T^11 + 33984128T^10 + 8121928T^8 - 1920T^7 + 601440T^6 + 23296T^5 + 17808T^4 - 4928T^3 + 224T^2 + 64*T + 2
$47$	\( T^{64} \) T^64
$53$	\( T^{64} + 288 T^{54} + \cdots + 2 \) T^64 + 288T^54 - 3520T^51 + 4T^48 + 25968T^44 + 2748544T^41 + 2350752T^38 + 38336T^35 + 782880T^34 + 6T^32 - 119459328T^31 + 425251232T^28 + 64T^27 - 124151552T^25 + 5627448T^24 + 4751328T^22 + 341240320T^21 - 23360T^19 + 949049408T^18 - 3648T^17 + 4T^16 + 396793344T^15 + 3691072T^14 + 45378352T^12 - 18333120T^11 + 1964160T^9 + 5732936T^8 + 4224T^7 + 37472T^6 - 204288T^5 + 7280T^4 + 320T^3 + 800T^2 + 64*T + 2
$59$	\( T^{64} \) T^64
$61$	\( T^{64} \) T^64
$67$	\( T^{64} + 8 T^{56} + \cdots + 2 \) T^64 + 8T^56 + 960T^53 + 28T^48 - 246912T^45 + 243744T^42 + 56T^40 - 64T^39 + 3252544T^37 + 67776672T^34 + 70T^32 + 13878528T^31 - 8012544T^29 + 43280T^28 + 537086784T^26 + 56T^24 - 738280320T^23 + 4688448T^21 + 70900368T^20 + 275294016T^18 - 707904T^17 + 28T^16 + 453097728T^15 + 480T^14 - 569472T^13 + 87415664T^12 + 7001760T^10 + 4122240T^9 + 8T^8 - 1982016T^7 + 68320T^6 + 6336T^5 + 43248T^4 + 448T^3 + 544T^2 - 64*T + 2
$71$	\( T^{64} + 64 T^{60} + \cdots + 16 \) 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
$73$	\( T^{64} \) T^64
$79$	\( T^{64} + 768 T^{54} + \cdots + 4 \) T^64 + 768T^54 + 352T^52 + 170880T^44 - 1176704T^42 + 33616T^40 + 864T^36 + 10762816T^34 + 150983820T^32 + 54995968T^30 + 757504T^28 - 181760T^26 + 111131072T^24 - 1022678528T^22 + 777461696T^20 - 84722944T^18 + 2993280T^16 + 48158720T^14 + 109390592T^12 + 49600256T^10 + 7398752T^8 + 419840T^6 + 35136T^4 - 640*T^2 + 4
$83$	\( T^{64} \) T^64
$89$	\( T^{64} \) T^64
$97$	\( T^{64} \) T^64
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