LMFDB - Newform orbit 3150.2.bf.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(1151,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 1]))

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

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

chi := DirichletCharacter("3150.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3150.bf (of order $6$, degree $2$, 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:	$25.1528766367$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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 $\zeta_{24}$. We also show the integral $q$-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$1$	$1 - \zeta_{24}^{4}$	$-1$

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} $

1151.1

0.258819	+	0.965926i
−0.258819	−	0.965926i
−0.965926	+	0.258819i
0.965926	−	0.258819i
0.258819	−	0.965926i
−0.258819	+	0.965926i
−0.965926	−	0.258819i
0.965926	+	0.258819i

−0.866025

0.500000i

0.500000

−

0.866025i

−2.63896

0.189469i

1.00000i

1151.2

−0.866025

0.500000i

0.500000

−

0.866025i

2.63896

−

0.189469i

1.00000i

1151.3

0.866025

−

0.500000i

0.500000

−

0.866025i

−0.189469

−

2.63896i

−

1.00000i

1151.4

0.866025

−

0.500000i

0.500000

−

0.866025i

0.189469

2.63896i

−

1.00000i

1601.1

−0.866025

−

0.500000i

0.500000

0.866025i

−2.63896

−

0.189469i

−

1.00000i

1601.2

−0.866025

−

0.500000i

0.500000

0.866025i

2.63896

0.189469i

−

1.00000i

1601.3

0.866025

0.500000i

0.500000

0.866025i

−0.189469

2.63896i

1.00000i

1601.4

0.866025

0.500000i

0.500000

0.866025i

0.189469

−

2.63896i

1.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
21.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3150.2.bf.b		8
3.b	odd	2	1		3150.2.bf.c		8
5.b	even	2	1		630.2.be.a	✓	8
5.c	odd	4	1		3150.2.bp.a		8
5.c	odd	4	1		3150.2.bp.d		8
7.d	odd	6	1		3150.2.bf.c		8
15.d	odd	2	1		630.2.be.b	yes	8
15.e	even	4	1		3150.2.bp.c		8
15.e	even	4	1		3150.2.bp.f		8
21.g	even	6	1	inner	3150.2.bf.b		8
35.i	odd	6	1		630.2.be.b	yes	8
35.i	odd	6	1		4410.2.b.b		8
35.j	even	6	1		4410.2.b.e		8
35.k	even	12	1		3150.2.bp.c		8
35.k	even	12	1		3150.2.bp.f		8
105.o	odd	6	1		4410.2.b.b		8
105.p	even	6	1		630.2.be.a	✓	8
105.p	even	6	1		4410.2.b.e		8
105.w	odd	12	1		3150.2.bp.a		8
105.w	odd	12	1		3150.2.bp.d		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
630.2.be.a	✓	8	5.b	even	2	1
630.2.be.a	✓	8	105.p	even	6	1
630.2.be.b	yes	8	15.d	odd	2	1
630.2.be.b	yes	8	35.i	odd	6	1
3150.2.bf.b		8	1.a	even	1	1	trivial
3150.2.bf.b		8	21.g	even	6	1	inner
3150.2.bf.c		8	3.b	odd	2	1
3150.2.bf.c		8	7.d	odd	6	1
3150.2.bp.a		8	5.c	odd	4	1
3150.2.bp.a		8	105.w	odd	12	1
3150.2.bp.c		8	15.e	even	4	1
3150.2.bp.c		8	35.k	even	12	1
3150.2.bp.d		8	5.c	odd	4	1
3150.2.bp.d		8	105.w	odd	12	1
3150.2.bp.f		8	15.e	even	4	1
3150.2.bp.f		8	35.k	even	12	1
4410.2.b.b		8	35.i	odd	6	1
4410.2.b.b		8	105.o	odd	6	1
4410.2.b.e		8	35.j	even	6	1
4410.2.b.e		8	105.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3150, [\chi])$:

$ T_{11}^{8} + 24 T_{11}^{7} + 260 T_{11}^{6} + 1632 T_{11}^{5} + 6447 T_{11}^{4} + 16320 T_{11}^{3} + \cdots + 9409 $

$ T_{37}^{8} + 24 T_{37}^{7} + 364 T_{37}^{6} + 3456 T_{37}^{5} + 24207 T_{37}^{4} + 117648 T_{37}^{3} + \cdots + 1329409 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} - T^{2} + 1)^{2} $ (T^4 - T^2 + 1)^2
$3$	$ T^{8} $ T^8
$5$	$ T^{8} $ T^8
$7$	$ T^{8} - 94T^{4} + 2401 $ T^8 - 94*T^4 + 2401
$11$	$ T^{8} + 24 T^{7} + \cdots + 9409 $ T^8 + 24T^7 + 260T^6 + 1632T^5 + 6447T^4 + 16320T^3 + 25796T^2 + 23280*T + 9409
$13$	$ T^{8} + 24 T^{6} + \cdots + 1 $ T^8 + 24T^6 + 146T^4 + 216*T^2 + 1
$17$	$ T^{8} + 40 T^{6} + \cdots + 1024 $ T^8 + 40T^6 + 192T^5 + 1632T^4 + 3840T^3 + 7936T^2 + 3072T + 1024
$19$	$ T^{8} - 18 T^{6} + \cdots + 1 $ T^8 - 18T^6 + 323T^4 + 864T^3 + 750T^2 - 48*T + 1
$23$	$ (T^{4} - 6 T^{3} + 11 T^{2} + \cdots + 1)^{2} $ (T^4 - 6T^3 + 11T^2 + 6*T + 1)^2
$29$	$ (T^{4} + 80 T^{2} + 64)^{2} $ (T^4 + 80*T^2 + 64)^2
$31$	$ T^{8} - 24 T^{6} + \cdots + 256 $ T^8 - 24T^6 + 560T^4 - 384*T^2 + 256
$37$	$ T^{8} + 24 T^{7} + \cdots + 1329409 $ T^8 + 24T^7 + 364T^6 + 3456T^5 + 24207T^4 + 117648T^3 + 421420T^2 + 940848*T + 1329409
$41$	$ (T^{4} + 16 T^{3} + \cdots - 71)^{2} $ (T^4 + 16T^3 + 68T^2 + 32*T - 71)^2
$43$	$ (T^{4} - 8 T^{3} + 8 T^{2} + \cdots - 32)^{2} $ (T^4 - 8T^3 + 8T^2 + 32*T - 32)^2
$47$	$ T^{8} + 8 T^{7} + \cdots + 330625 $ T^8 + 8T^7 + 110T^6 + 512T^5 + 6211T^4 + 29440T^3 + 167150T^2 + 253000*T + 330625
$53$	$ T^{8} - 24 T^{7} + \cdots + 2745649 $ T^8 - 24T^7 + 150T^6 + 1008T^5 - 7381T^4 - 39312T^3 + 222438T^2 + 1550952*T + 2745649
$59$	$ T^{8} - 24 T^{7} + \cdots + 22090000 $ T^8 - 24T^7 + 460T^6 - 4224T^5 + 35436T^4 - 142080T^3 + 1063600T^2 - 3384000*T + 22090000
$61$	$ T^{8} - 120 T^{6} + \cdots + 2262016 $ T^8 - 120T^6 + 12896T^4 - 57600T^3 - 103680T^2 + 721920*T + 2262016
$67$	$ T^{8} - 24 T^{7} + \cdots + 442008576 $ T^8 - 24T^7 + 600T^6 - 6912T^5 + 111456T^4 - 1099008T^3 + 13512960T^2 - 78713856*T + 442008576
$71$	$ T^{8} + 288 T^{6} + \cdots + 1364224 $ T^8 + 288T^6 + 23072T^4 + 446976*T^2 + 1364224
$73$	$ T^{8} - 136 T^{6} + \cdots + 256 $ T^8 - 136T^6 + 18480T^4 - 2176*T^2 + 256
$79$	$ T^{8} + 24 T^{7} + \cdots + 171295744 $ T^8 + 24T^7 + 568T^6 + 5376T^5 + 75360T^4 + 607488T^3 + 6823168T^2 + 33924096*T + 171295744
$83$	$ (T^{4} + 8 T^{3} + \cdots + 7648)^{2} $ (T^4 + 8T^3 - 184T^2 - 800*T + 7648)^2
$89$	$ T^{8} - 16 T^{7} + \cdots + 2062096 $ T^8 - 16T^7 + 308T^6 - 1024T^5 + 18988T^4 - 94208T^3 + 786512T^2 - 1332608*T + 2062096
$97$	$ T^{8} + 800 T^{6} + \cdots + 610287616 $ T^8 + 800T^6 + 221184T^4 + 23367680*T^2 + 610287616
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Level:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.bf (of order \(6\), degree \(2\), minimal)

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

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Newspace parameters

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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	3150.2.bf.b

Level	$3150$
Weight	$2$
Character orbit	3150.bf
Analytic conductor	$25.153$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(25.1528766367\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{24})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{4} + 1 \) x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(n\)	\(127\)	\(451\)	\(2801\)
\(\chi(n)\)	\(1\)	\(1 - \zeta_{24}^{4}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1151.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} - T^{2} + 1)^{2} \) (T^4 - T^2 + 1)^2
$3$	\( T^{8} \) T^8
$5$	\( T^{8} \) T^8
$7$	\( T^{8} - 94T^{4} + 2401 \) T^8 - 94*T^4 + 2401
$11$	\( T^{8} + 24 T^{7} + \cdots + 9409 \) T^8 + 24T^7 + 260T^6 + 1632T^5 + 6447T^4 + 16320T^3 + 25796T^2 + 23280*T + 9409
$13$	\( T^{8} + 24 T^{6} + \cdots + 1 \) T^8 + 24T^6 + 146T^4 + 216*T^2 + 1
$17$	\( T^{8} + 40 T^{6} + \cdots + 1024 \) T^8 + 40T^6 + 192T^5 + 1632T^4 + 3840T^3 + 7936T^2 + 3072T + 1024
$19$	\( T^{8} - 18 T^{6} + \cdots + 1 \) T^8 - 18T^6 + 323T^4 + 864T^3 + 750T^2 - 48*T + 1
$23$	\( (T^{4} - 6 T^{3} + 11 T^{2} + \cdots + 1)^{2} \) (T^4 - 6T^3 + 11T^2 + 6*T + 1)^2
$29$	\( (T^{4} + 80 T^{2} + 64)^{2} \) (T^4 + 80*T^2 + 64)^2
$31$	\( T^{8} - 24 T^{6} + \cdots + 256 \) T^8 - 24T^6 + 560T^4 - 384*T^2 + 256
$37$	\( T^{8} + 24 T^{7} + \cdots + 1329409 \) T^8 + 24T^7 + 364T^6 + 3456T^5 + 24207T^4 + 117648T^3 + 421420T^2 + 940848*T + 1329409
$41$	\( (T^{4} + 16 T^{3} + \cdots - 71)^{2} \) (T^4 + 16T^3 + 68T^2 + 32*T - 71)^2
$43$	\( (T^{4} - 8 T^{3} + 8 T^{2} + \cdots - 32)^{2} \) (T^4 - 8T^3 + 8T^2 + 32*T - 32)^2
$47$	\( T^{8} + 8 T^{7} + \cdots + 330625 \) T^8 + 8T^7 + 110T^6 + 512T^5 + 6211T^4 + 29440T^3 + 167150T^2 + 253000*T + 330625
$53$	\( T^{8} - 24 T^{7} + \cdots + 2745649 \) T^8 - 24T^7 + 150T^6 + 1008T^5 - 7381T^4 - 39312T^3 + 222438T^2 + 1550952*T + 2745649
$59$	\( T^{8} - 24 T^{7} + \cdots + 22090000 \) T^8 - 24T^7 + 460T^6 - 4224T^5 + 35436T^4 - 142080T^3 + 1063600T^2 - 3384000*T + 22090000
$61$	\( T^{8} - 120 T^{6} + \cdots + 2262016 \) T^8 - 120T^6 + 12896T^4 - 57600T^3 - 103680T^2 + 721920*T + 2262016
$67$	\( T^{8} - 24 T^{7} + \cdots + 442008576 \) T^8 - 24T^7 + 600T^6 - 6912T^5 + 111456T^4 - 1099008T^3 + 13512960T^2 - 78713856*T + 442008576
$71$	\( T^{8} + 288 T^{6} + \cdots + 1364224 \) T^8 + 288T^6 + 23072T^4 + 446976*T^2 + 1364224
$73$	\( T^{8} - 136 T^{6} + \cdots + 256 \) T^8 - 136T^6 + 18480T^4 - 2176*T^2 + 256
$79$	\( T^{8} + 24 T^{7} + \cdots + 171295744 \) T^8 + 24T^7 + 568T^6 + 5376T^5 + 75360T^4 + 607488T^3 + 6823168T^2 + 33924096*T + 171295744
$83$	\( (T^{4} + 8 T^{3} + \cdots + 7648)^{2} \) (T^4 + 8T^3 - 184T^2 - 800*T + 7648)^2
$89$	\( T^{8} - 16 T^{7} + \cdots + 2062096 \) T^8 - 16T^7 + 308T^6 - 1024T^5 + 18988T^4 - 94208T^3 + 786512T^2 - 1332608*T + 2062096
$97$	\( T^{8} + 800 T^{6} + \cdots + 610287616 \) T^8 + 800T^6 + 221184T^4 + 23367680*T^2 + 610287616
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