LMFDB - Newform orbit 693.2.by.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,2,Mod(37,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([0, 10, 6]))

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

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

chi := DirichletCharacter("693.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	693.by (of order $15$, degree $8$, 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:	$5.53363286007$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 $ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/693\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$442$
$\chi(n)$	$1$	$-1 - \zeta_{15}^{5}$	$-1 + \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + \zeta_{15}^{7}$

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} $

37.1

0.669131	+	0.743145i
−0.104528	−	0.994522i
−0.978148	+	0.207912i
−0.978148	−	0.207912i
0.913545	+	0.406737i
0.913545	−	0.406737i
0.669131	−	0.743145i
−0.104528	+	0.994522i

0.564602

−

0.251377i

−1.08268

1.20243i

−0.233733

−

2.22382i

1.59618

2.11002i

−0.690983

2.12663i

−0.690983

−

1.19682i

163.1

−1.08268

−

1.20243i

−0.0646021

0.614648i

2.18720

0.464905i

−2.53158

−

0.768834i

−1.80902

1.31433i

−1.80902

−

3.13331i

235.1

−0.0646021

0.614648i

1.58268

0.336408i

2.04275

0.909491i

−0.0510966

2.64526i

−0.690983

2.12663i

−0.690983

1.19682i

289.1

−0.0646021

−

0.614648i

1.58268

−

0.336408i

2.04275

−

0.909491i

−0.0510966

−

2.64526i

−0.690983

−

2.12663i

−0.690983

−

1.19682i

361.1

1.58268

−

0.336408i

0.564602

−

0.251377i

−1.49622

1.66172i

−1.51351

−

2.17009i

−1.80902

1.31433i

−1.80902

3.13331i

478.1

1.58268

0.336408i

0.564602

0.251377i

−1.49622

−

1.66172i

−1.51351

2.17009i

−1.80902

−

1.31433i

−1.80902

−

3.13331i

487.1

0.564602

0.251377i

−1.08268

−

1.20243i

−0.233733

2.22382i

1.59618

−

2.11002i

−0.690983

−

2.12663i

−0.690983

1.19682i

676.1

−1.08268

1.20243i

−0.0646021

−

0.614648i

2.18720

−

0.464905i

−2.53158

0.768834i

−1.80902

−

1.31433i

−1.80902

3.13331i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
11.c	even	5	1	inner
77.m	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	693.2.by.a		8
3.b	odd	2	1		77.2.m.a	✓	8
7.c	even	3	1	inner	693.2.by.a		8
11.c	even	5	1	inner	693.2.by.a		8
21.c	even	2	1		539.2.q.a		8
21.g	even	6	1		539.2.f.b		4
21.g	even	6	1		539.2.q.a		8
21.h	odd	6	1		77.2.m.a	✓	8
21.h	odd	6	1		539.2.f.a		4
33.d	even	2	1		847.2.n.b		8
33.f	even	10	1		847.2.e.b		4
33.f	even	10	2		847.2.n.a		8
33.f	even	10	1		847.2.n.b		8
33.h	odd	10	1		77.2.m.a	✓	8
33.h	odd	10	1		847.2.e.a		4
33.h	odd	10	2		847.2.n.c		8
77.m	even	15	1	inner	693.2.by.a		8
231.l	even	6	1		847.2.n.b		8
231.u	even	10	1		539.2.q.a		8
231.z	odd	30	1		77.2.m.a	✓	8
231.z	odd	30	1		539.2.f.a		4
231.z	odd	30	1		847.2.e.a		4
231.z	odd	30	2		847.2.n.c		8
231.z	odd	30	1		5929.2.a.q		2
231.bc	even	30	1		539.2.f.b		4
231.bc	even	30	1		539.2.q.a		8
231.bc	even	30	1		5929.2.a.o		2
231.be	even	30	1		847.2.e.b		4
231.be	even	30	2		847.2.n.a		8
231.be	even	30	1		847.2.n.b		8
231.be	even	30	1		5929.2.a.l		2
231.bf	odd	30	1		5929.2.a.j		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.2.m.a	✓	8	3.b	odd	2	1
77.2.m.a	✓	8	21.h	odd	6	1
77.2.m.a	✓	8	33.h	odd	10	1
77.2.m.a	✓	8	231.z	odd	30	1
539.2.f.a		4	21.h	odd	6	1
539.2.f.a		4	231.z	odd	30	1
539.2.f.b		4	21.g	even	6	1
539.2.f.b		4	231.bc	even	30	1
539.2.q.a		8	21.c	even	2	1
539.2.q.a		8	21.g	even	6	1
539.2.q.a		8	231.u	even	10	1
539.2.q.a		8	231.bc	even	30	1
693.2.by.a		8	1.a	even	1	1	trivial
693.2.by.a		8	7.c	even	3	1	inner
693.2.by.a		8	11.c	even	5	1	inner
693.2.by.a		8	77.m	even	15	1	inner
847.2.e.a		4	33.h	odd	10	1
847.2.e.a		4	231.z	odd	30	1
847.2.e.b		4	33.f	even	10	1
847.2.e.b		4	231.be	even	30	1
847.2.n.a		8	33.f	even	10	2
847.2.n.a		8	231.be	even	30	2
847.2.n.b		8	33.d	even	2	1
847.2.n.b		8	33.f	even	10	1
847.2.n.b		8	231.l	even	6	1
847.2.n.b		8	231.be	even	30	1
847.2.n.c		8	33.h	odd	10	2
847.2.n.c		8	231.z	odd	30	2
5929.2.a.j		2	231.bf	odd	30	1
5929.2.a.l		2	231.be	even	30	1
5929.2.a.o		2	231.bc	even	30	1
5929.2.a.q		2	231.z	odd	30	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} - 2T_{2}^{7} - 2T_{2}^{5} + 9T_{2}^{4} - 8T_{2}^{3} + 5T_{2}^{2} - 3T_{2} + 1 $ T2^8 - 2*T2^7 - 2*T2^5 + 9*T2^4 - 8*T2^3 + 5*T2^2 - 3*T2 + 1 acting on $S_{2}^{\mathrm{new}}(693, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} - 2 T^{7} + \cdots + 1 $ T^8 - 2T^7 - 2T^5 + 9T^4 - 8T^3 + 5T^2 - 3T + 1
$3$	$ T^{8} $ T^8
$5$	$ T^{8} - 5 T^{7} + \cdots + 625 $ T^8 - 5T^7 + 10T^6 - 25T^5 + 75T^4 - 125T^3 + 250T^2 - 625*T + 625
$7$	$ T^{8} + 5 T^{7} + \cdots + 2401 $ T^8 + 5T^7 + 18T^6 + 55T^5 + 149T^4 + 385T^3 + 882T^2 + 1715*T + 2401
$11$	$ T^{8} + 4 T^{7} + \cdots + 14641 $ T^8 + 4T^7 + 10T^6 - 64T^5 - 261T^4 - 704T^3 + 1210T^2 + 5324*T + 14641
$13$	$ (T^{4} + 5 T^{3} + 40 T^{2} + \cdots + 25)^{2} $ (T^4 + 5T^3 + 40T^2 + 50*T + 25)^2
$17$	$ T^{8} - 4 T^{7} + \cdots + 256 $ T^8 - 4T^7 - 16T^5 + 144T^4 - 256T^3 + 320T^2 - 384T + 256
$19$	$ T^{8} + 3 T^{7} + \cdots + 6561 $ T^8 + 3T^7 - 45T^6 + 378T^5 + 3159T^4 - 6318T^3 + 7290T^2 - 8748*T + 6561
$23$	$ (T^{4} + 8 T^{3} + \cdots + 121)^{2} $ (T^4 + 8T^3 + 53T^2 + 88*T + 121)^2
$29$	$ (T^{4} + 12 T^{3} + \cdots + 81)^{2} $ (T^4 + 12T^3 + 54T^2 - 27*T + 81)^2
$31$	$ T^{8} - 8 T^{7} + \cdots + 707281 $ T^8 - 8T^7 + 30T^6 - 98T^5 - 381T^4 + 10498T^3 - 21025T^2 - 73167*T + 707281
$37$	$ T^{8} + 13 T^{7} + \cdots + 923521 $ T^8 + 13T^7 + 75T^6 + 478T^5 + 3039T^4 + 9982T^3 + 48050T^2 + 357492*T + 923521
$41$	$ (T^{4} + T^{3} + 16 T^{2} + \cdots + 121)^{2} $ (T^4 + T^3 + 16T^2 + 66T + 121)^2
$43$	$ (T^{2} - 7 T + 1)^{4} $ (T^2 - 7*T + 1)^4
$47$	$ T^{8} + 6 T^{7} + \cdots + 256 $ T^8 + 6T^7 - 40T^6 + 344T^5 + 5424T^4 + 4064T^3 + 1920T^2 + 896*T + 256
$53$	$ T^{8} - 12 T^{7} + \cdots + 923521 $ T^8 - 12T^7 + 50T^6 - 322T^5 + 3039T^4 - 14818T^3 + 72075T^2 - 387283*T + 923521
$59$	$ T^{8} - 18 T^{7} + \cdots + 1679616 $ T^8 - 18T^7 + 180T^6 - 1728T^5 + 11664T^4 - 15552T^3 - 559872T + 1679616
$61$	$ T^{8} - 18 T^{7} + \cdots + 33362176 $ T^8 - 18T^7 + 80T^6 - 1048T^5 + 23664T^4 - 200032T^3 + 1386240T^2 - 9657472*T + 33362176
$67$	$ (T^{4} + 19 T^{3} + \cdots + 7921)^{2} $ (T^4 + 19T^3 + 272T^2 + 1691*T + 7921)^2
$71$	$ (T^{4} - 8 T^{3} + 24 T^{2} + \cdots + 16)^{2} $ (T^4 - 8T^3 + 24T^2 + 8*T + 16)^2
$73$	$ T^{8} - 15 T^{7} + \cdots + 4100625 $ T^8 - 15T^7 + 135T^6 - 1350T^5 + 6075T^4 + 60750T^3 - 182250T^2 + 4100625
$79$	$ T^{8} - 9 T^{7} + \cdots + 6561 $ T^8 - 9T^7 - 90T^6 - 1161T^5 + 27459T^4 - 30861T^3 + 21870T^2 - 15309*T + 6561
$83$	$ (T^{4} + 9 T^{3} + \cdots + 6561)^{2} $ (T^4 + 9T^3 + 81T^2 + 729*T + 6561)^2
$89$	$ (T^{4} - 12 T^{3} + \cdots + 256)^{2} $ (T^4 - 12T^3 + 128T^2 - 192*T + 256)^2
$97$	$ (T^{4} + 7 T^{3} + \cdots + 121)^{2} $ (T^4 + 7T^3 + 34T^2 + 88*T + 121)^2
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Overview	Random
Universe	Knowledge

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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.by (of order \(15\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{15}^{7} + \zeta_{15}^{5} + \cdots + 1) q^{2}+ \cdots + (2 \zeta_{15}^{7} - \zeta_{15}^{6} + \cdots - 2) q^{8}+O(q^{10}) \) q + (-z^7 + z^5 - z^4 + 1) * q^2 + (-z^6 + z^2 - z) * q^4 + (z^7 + z^5 + z^4 - z + 1) * q^5 + (2z^7 - 2z^6 - z^4 - 2z^3 + 2z^2 - 2) * q^7 + (2z^7 - z^6 + 2z^2 - 2) * q^8	\( q + ( - \zeta_{15}^{7} + \zeta_{15}^{5} + \cdots + 1) q^{2}+ \cdots + (5 \zeta_{15}^{7} + 8 \zeta_{15}^{4} + \cdots - 5) q^{98}+O(q^{100}) \) q + (-z^7 + z^5 - z^4 + 1) * q^2 + (-z^6 + z^2 - z) * q^4 + (z^7 + z^5 + z^4 - z + 1) * q^5 + (2z^7 - 2z^6 - z^4 - 2z^3 + 2z^2 - 2) * q^7 + (2z^7 - z^6 + 2z^2 - 2) * q^8 + (z^7 + 2z^5 + z^4 - z^3 + z^2 + z - 1) q^10 + (2z^7 - 2z^6 + z^5 + 2z^4 - 2z^3 + 4z^2 - 2) q^11 + (z^7 + 3z^6 + 3z^3 + z^2) * q^13 + (2z^7 - 3z^5 - z^2 - 2) * q^14 + (3z^7 + 3z^4) * q^16 + (-2z^7 + 2z^6 - 2z^4 + 2z^3 + 2) * q^17 + (3z^7 - 3z^5 + 6z - 3) q^19 + (3z^7 - z^6 - z^3 + 3z^2) * q^20 + (2z^7 + z^6 + 3z^3 + 2z^2) q^22 + (-5z^5 + 2z^4 + 2z - 5) q^23 + (4z^7 + 7z^4 + 4z) q^26 + (-3z^6 + 2z^5 - 2z - 1) q^28 + (3z^7 - 3z^6 - 6z^3 + 3z^2 - 6) * q^29 + (z^7 - z^6 - z^5 + z^4 - z^3 + 6z^2 - 1) q^31 + (-z^7 + 5z^5 - z^4 + z^3 - z^2 - z + 1) q^32 + (-2z^7 + 2z^3 - 2z^2 + 4) q^34 + (-4z^7 + 2z^6 + 6z^5 - 6z^4 + 2z^3 - z^2 + 6) q^35 + (-5z^7 - 3z^4 - 5z) q^37 + (3z^6 - 3z^5 + 3z) q^38 - 5z^4 q^40 + (-z^7 + 4z^6 - z^2 + 1) q^41 + (3z^7 - 3z^3 + 3z^2 + 2) q^43 + (-z^7 + 4z^5 - 3z + 4) * q^44 + (3z^7 - 3z^6 - 3z^5 + 3z^4 - 3z^3 + 2z^2 - 3) * q^46 + (4z^7 - 4z^5 + 6z - 4) q^47 + (5z^7 - 5z^5 + 5z^4 + 3z^3 + 5z - 5) q^49 + (2z^6 + z^5 + z^2 + 2z) * q^52 + (5z^7 - 5z^6 - 5z^5 + 5z^4 - 5z^3 + 2z^2 - 5) * q^53 + (6z^7 - 6z^6 - 9z^3 + 6z^2 - 4) * q^55 + (-4z^7 - z^5 + 2z^4 + 4z^3 - 4z^2 + 2z + 1) q^56 + (6z^7 - 6z^5 - 3z - 6) q^58 + (-6z^7 - 6z^4 + 6z^3 - 6z + 6) * q^59 + (6z^7 + 2z^5 + 6z^4 - 2z + 2) * q^61 + (z^7 - z^6 + 4z^3 + z^2 + 4) q^62 + (-z^7 - 2z^6 - 2z^3 - z^2) * q^64 + (-5z^5 - 5z^4 - 5z - 5) q^65 + (-z^7 + 10z^5 - z^4 + z^3 - z^2 - z + 1) q^67 - 2z^7 q^68 + (-8z^7 + 6z^6 - 6z^4 + 9z^3 - 6z^2 - 2z + 9) * q^70 + (2z^6 - 2z^3 + 2) * q^71 + (-9z^7 + 6z^6 - 9z^4 + 9z^3 - 6z^2 - 3z + 9) * q^73 + (-5z^6 - 3z^5 - 3z^2 - 5z) * q^74 + (-6z^7 + 6z^3 - 6z^2 - 3) q^76 + (-7z^7 - 3z^6 + 4z^5 - 6z^4 + 9z^3 - 3z^2 - 8z + 7) q^77 + (-9z^7 + 6z^5 - 6z^4 + 6) q^79 + (-6z^6 - 3z^5 - 3z^2 - 6z) * q^80 + (2z^7 + z^5 + 2z^4 - z + 1) * q^82 + 9z^3 q^83 + (2z^7 + 4z^6 + 2z^2 - 2) q^85 + (z^7 + 2z^5 - 2z^4 + 2) * q^86 + (-2z^7 - 3z^6 + 2z^5 - 2z^4 + 2z^3 - 6z^2 - 5z + 2) q^88 + (8z^5 - 4z^4 - 4z + 8) q^89 + (-9z^7 - 2z^6 + 3z^5 - 6z^2 + z + 9) * q^91 + (-7z^7 + 7z^6 + 2z^3 - 7z^2 + 2) * q^92 + (2z^7 + 2z^6 - 4z^5 + 2z^4 - 2z^3 + 4z - 2) * q^94 + (-3z^7 + 12z^6 - 3z^4 + 3z^3 - 12z^2 + 9z + 3) * q^95 + (-z^7 + 3z^6 + 3z^3 - z^2) * q^97 + (5z^7 + 8z^4 - 5z^3 + 5z^2 + 8z - 5) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} + 2 q^{4} + 5 q^{5} - 5 q^{7} - 10 q^{8}+O(q^{10}) \) 8 * q + 2 * q^2 + 2 * q^4 + 5 * q^5 - 5 * q^7 - 10 * q^8	\( 8 q + 2 q^{2} + 2 q^{4} + 5 q^{5} - 5 q^{7} - 10 q^{8} - 10 q^{10} - 4 q^{11} - 10 q^{13} - 3 q^{14} + 6 q^{16} + 4 q^{17} - 3 q^{19} + 10 q^{20} - 4 q^{22} - 16 q^{23} + 15 q^{26} - 12 q^{28} - 24 q^{29} + 8 q^{31} - 18 q^{32} + 24 q^{34} + 5 q^{35} - 13 q^{37} + 9 q^{38} - 5 q^{40} - 2 q^{41} + 28 q^{43} + 12 q^{44} + 8 q^{46} - 6 q^{47} - 11 q^{49} - 5 q^{52} + 12 q^{53} + 10 q^{55} - 21 q^{58} + 18 q^{59} + 18 q^{61} + 28 q^{62} + 6 q^{64} - 30 q^{65} - 38 q^{67} - 2 q^{68} + 20 q^{70} + 16 q^{71} + 15 q^{73} + 14 q^{74} - 48 q^{76} + 4 q^{77} + 9 q^{79} + 15 q^{80} + 7 q^{82} - 18 q^{83} - 20 q^{85} + 7 q^{86} - 5 q^{88} + 24 q^{89} + 50 q^{91} - 16 q^{92} + 8 q^{94} - 15 q^{95} - 14 q^{97} - 4 q^{98}+O(q^{100}) \) 8 * q + 2 * q^2 + 2 * q^4 + 5 * q^5 - 5 * q^7 - 10 * q^8 - 10 * q^10 - 4 * q^11 - 10 * q^13 - 3 * q^14 + 6 * q^16 + 4 * q^17 - 3 * q^19 + 10 * q^20 - 4 * q^22 - 16 * q^23 + 15 * q^26 - 12 * q^28 - 24 * q^29 + 8 * q^31 - 18 * q^32 + 24 * q^34 + 5 * q^35 - 13 * q^37 + 9 * q^38 - 5 * q^40 - 2 * q^41 + 28 * q^43 + 12 * q^44 + 8 * q^46 - 6 * q^47 - 11 * q^49 - 5 * q^52 + 12 * q^53 + 10 * q^55 - 21 * q^58 + 18 * q^59 + 18 * q^61 + 28 * q^62 + 6 * q^64 - 30 * q^65 - 38 * q^67 - 2 * q^68 + 20 * q^70 + 16 * q^71 + 15 * q^73 + 14 * q^74 - 48 * q^76 + 4 * q^77 + 9 * q^79 + 15 * q^80 + 7 * q^82 - 18 * q^83 - 20 * q^85 + 7 * q^86 - 5 * q^88 + 24 * q^89 + 50 * q^91 - 16 * q^92 + 8 * q^94 - 15 * q^95 - 14 * q^97 - 4 * q^98

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Newspace parameters

Newform invariants

$q$-expansion

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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	693.2.by.a

Level	$693$
Weight	$2$
Character orbit	693.by
Analytic conductor	$5.534$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(5.53363286007\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{15})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 77)
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

\(n\)	\(155\)	\(199\)	\(442\)
\(\chi(n)\)	\(1\)	\(-1 - \zeta_{15}^{5}\)	\(-1 + \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + \zeta_{15}^{7}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 37.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} - 2 T^{7} + \cdots + 1 \) T^8 - 2T^7 - 2T^5 + 9T^4 - 8T^3 + 5T^2 - 3T + 1
$3$	\( T^{8} \) T^8
$5$	\( T^{8} - 5 T^{7} + \cdots + 625 \) T^8 - 5T^7 + 10T^6 - 25T^5 + 75T^4 - 125T^3 + 250T^2 - 625*T + 625
$7$	\( T^{8} + 5 T^{7} + \cdots + 2401 \) T^8 + 5T^7 + 18T^6 + 55T^5 + 149T^4 + 385T^3 + 882T^2 + 1715*T + 2401
$11$	\( T^{8} + 4 T^{7} + \cdots + 14641 \) T^8 + 4T^7 + 10T^6 - 64T^5 - 261T^4 - 704T^3 + 1210T^2 + 5324*T + 14641
$13$	\( (T^{4} + 5 T^{3} + 40 T^{2} + \cdots + 25)^{2} \) (T^4 + 5T^3 + 40T^2 + 50*T + 25)^2
$17$	\( T^{8} - 4 T^{7} + \cdots + 256 \) T^8 - 4T^7 - 16T^5 + 144T^4 - 256T^3 + 320T^2 - 384T + 256
$19$	\( T^{8} + 3 T^{7} + \cdots + 6561 \) T^8 + 3T^7 - 45T^6 + 378T^5 + 3159T^4 - 6318T^3 + 7290T^2 - 8748*T + 6561
$23$	\( (T^{4} + 8 T^{3} + \cdots + 121)^{2} \) (T^4 + 8T^3 + 53T^2 + 88*T + 121)^2
$29$	\( (T^{4} + 12 T^{3} + \cdots + 81)^{2} \) (T^4 + 12T^3 + 54T^2 - 27*T + 81)^2
$31$	\( T^{8} - 8 T^{7} + \cdots + 707281 \) T^8 - 8T^7 + 30T^6 - 98T^5 - 381T^4 + 10498T^3 - 21025T^2 - 73167*T + 707281
$37$	\( T^{8} + 13 T^{7} + \cdots + 923521 \) T^8 + 13T^7 + 75T^6 + 478T^5 + 3039T^4 + 9982T^3 + 48050T^2 + 357492*T + 923521
$41$	\( (T^{4} + T^{3} + 16 T^{2} + \cdots + 121)^{2} \) (T^4 + T^3 + 16T^2 + 66T + 121)^2
$43$	\( (T^{2} - 7 T + 1)^{4} \) (T^2 - 7*T + 1)^4
$47$	\( T^{8} + 6 T^{7} + \cdots + 256 \) T^8 + 6T^7 - 40T^6 + 344T^5 + 5424T^4 + 4064T^3 + 1920T^2 + 896*T + 256
$53$	\( T^{8} - 12 T^{7} + \cdots + 923521 \) T^8 - 12T^7 + 50T^6 - 322T^5 + 3039T^4 - 14818T^3 + 72075T^2 - 387283*T + 923521
$59$	\( T^{8} - 18 T^{7} + \cdots + 1679616 \) T^8 - 18T^7 + 180T^6 - 1728T^5 + 11664T^4 - 15552T^3 - 559872T + 1679616
$61$	\( T^{8} - 18 T^{7} + \cdots + 33362176 \) T^8 - 18T^7 + 80T^6 - 1048T^5 + 23664T^4 - 200032T^3 + 1386240T^2 - 9657472*T + 33362176
$67$	\( (T^{4} + 19 T^{3} + \cdots + 7921)^{2} \) (T^4 + 19T^3 + 272T^2 + 1691*T + 7921)^2
$71$	\( (T^{4} - 8 T^{3} + 24 T^{2} + \cdots + 16)^{2} \) (T^4 - 8T^3 + 24T^2 + 8*T + 16)^2
$73$	\( T^{8} - 15 T^{7} + \cdots + 4100625 \) T^8 - 15T^7 + 135T^6 - 1350T^5 + 6075T^4 + 60750T^3 - 182250T^2 + 4100625
$79$	\( T^{8} - 9 T^{7} + \cdots + 6561 \) T^8 - 9T^7 - 90T^6 - 1161T^5 + 27459T^4 - 30861T^3 + 21870T^2 - 15309*T + 6561
$83$	\( (T^{4} + 9 T^{3} + \cdots + 6561)^{2} \) (T^4 + 9T^3 + 81T^2 + 729*T + 6561)^2
$89$	\( (T^{4} - 12 T^{3} + \cdots + 256)^{2} \) (T^4 - 12T^3 + 128T^2 - 192*T + 256)^2
$97$	\( (T^{4} + 7 T^{3} + \cdots + 121)^{2} \) (T^4 + 7T^3 + 34T^2 + 88*T + 121)^2
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