LMFDB - Newform orbit 1000.2.v.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1000,2,Mod(43,1000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1000, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([10, 10, 19]))

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

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

chi := DirichletCharacter("1000.43");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1000 = 2^{3} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1000.v (of order $20$, degree $8$, not 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:	$7.98504020213$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

$q$-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 $\zeta_{20}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \zeta_{20}^{5} + 1) q^{2} + ( - \zeta_{20}^{7} + \zeta_{20}^{6} + \cdots - 1) q^{3}+ \cdots + (\zeta_{20}^{7} - 2 \zeta_{20}^{3} + 2 \zeta_{20}) q^{9}+O(q^{10}) $ q + (-z^5 + 1) * q^2 + (-z^7 + z^6 + z^5 - 2z^4 - 2z^3 + z^2 + z - 1) * q^3 - 2z^5 q^4 + (2z^6 - 4z^4 + 2z^2 - 2) q^6 + (z^5 - 2z^4 + 2z - 1) * q^7 + (-2z^5 - 2) q^8 + (z^7 - 2z^3 + 2z) * q^9	$ q + ( - \zeta_{20}^{5} + 1) q^{2} + ( - \zeta_{20}^{7} + \zeta_{20}^{6} + \cdots - 1) q^{3}+ \cdots + ( - 3 \zeta_{20}^{7} - \zeta_{20}^{6} + \cdots + 2) q^{99}+O(q^{100}) $ q + (-z^5 + 1) * q^2 + (-z^7 + z^6 + z^5 - 2z^4 - 2z^3 + z^2 + z - 1) * q^3 - 2z^5 q^4 + (2z^6 - 4z^4 + 2z^2 - 2) q^6 + (z^5 - 2z^4 + 2z - 1) * q^7 + (-2z^5 - 2) q^8 + (z^7 - 2z^3 + 2z) * q^9 + (-2z^7 + z^5 + z^4 - z^3 - z^2 + z) q^11 + (2z^7 + 2z^6 - 2z^5 - 4z^4 + 4z^3 + 2z^2 - 2z - 2) q^12 + (-z^7 - 3z^6 + 2z^5 + 2z^4 - 3z^3 - z^2) * q^13 + (2z^7 - 2z^6 - 2z^4 + 2z^3) * q^14 - 4 * q^16 + (-z^7 + z^6 + 2z^5 - 2z^4 - 2z^3 + 2z^2) * q^17 + (z^7 - 2z^4 - 2z^3 + 3z^2 + 2z - 2) * q^18 + (z^7 - 4z^6 - z^5 + z^4 + z^3 + 2z^2 - z + 2) * q^19 + (4z^7 - 4z^5 + 2z^2 - 2z + 2) * q^21 + (-2z^7 + 2z^5 - 2z^3 - 2z^2 + 2z) q^22 + (4z^7 - 4z^4) * q^23 + (4z^7 - 4z^5 + 8z^3 - 4z) * q^24 + (-2z^7 + 4z^5 - z^4 - 5z^3 + z^2 - z - 1) q^26 + (4z^7 - 2z^5 - 2z^4 + 2z^3 - 2z^2 - 2z) * q^27 + (4z^7 - 4z^6 - 2z^5 + 4z^3 - 4z + 2) q^28 + (2z^7 - 2z^6 + 2z^3 + z^2 - z - 1) q^29 + (2z^7 + 5z^6 - 3z^5 - z^4 + 3z^3 - 3z^2 - 2z - 2) * q^31 + (4z^5 - 4) q^32 + (-2z^6 + 2z^5 + 2z^4 - 4z^2 - 2z + 2) q^33 + (-z^7 + 3z^6 - 4z^4 + 3z^2 - z) q^34 + (-4z^4 + 6z^2 - 4) * q^36 + (4z^7 - z^5 + z^3 - 3z^2 - z - 1) * q^37 + (-2z^7 - 4z^6 - 2z^5 + 2z^4 + 2z^2 - 4z + 2) * q^38 + (2z^7 - z^6 - 2z^5 + 9z^4 + z^3 - 9z^2 + 1) * q^39 + (-4z^7 + 8z^5 + 2z^4 - 3z^3 + 2z^2 + 5z + 2) * q^41 + (2z^7 + 2z^6 - 6z^5 + 6z^2 - 2z - 2) q^42 + (-4z^7 - 4z^6 + 4z^5 + 2z^4 - 4z^2 + 2z) * q^43 + (2z^5 - 2z^4 - 2z^3 - 2z^2 + 2z) q^44 + (8z^7 - 4z^5 - 4z^4 + 4z^3 + 4z^2 - 4z) * q^46 + (-2z^7 - 2z^6 + 2z^4 + 2z^3 - 2z - 2) q^47 + (4z^7 - 4z^6 - 4z^5 + 8z^4 + 8z^3 - 4z^2 - 4z + 4) q^48 + (-4z^7 + 8z^6 + z^5 - 4z^3 + 8z^2 - 4) * q^49 + (z^7 - z^6 + 3z^5 + z^4 - 7z^3 + 6z - 2) q^51 + (-2z^7 + 6z^6 + 4z^5 - 6z^4 - 4z^3 + 4z^2 - 2z - 2) q^52 + (-6z^7 - 3z^6 + 3z^5 + 4z^4 - 4z^3 - 3z^2 + 3z + 6) q^53 + (8z^7 - 4z^5 + 4z^3 - 4z) * q^54 + (4z^7 - 4z^6 - 4z^5 + 4z^4 + 4z^3 - 8z + 4) * q^56 + (-8z^5 + 2z^3 + 2z^2 - 8) q^57 + (z^7 - 3z^6 + z^5 + 2z^4 + 2z^3 + z^2 - 3z + 1) * q^58 + (-3z^7 - 10z^6 + 6z^4 - 4z^3 - 5z^2 + 4z + 4) * q^59 + (z^6 + 6z^5 - 3z^3 + 6z + 1) q^61 + (6z^7 + 4z^6 - 2z^5 + 2z^4 + 4z^3 - 4z^2 + 2z - 2) q^62 + (3z^7 + 2z^6 - 4z^5 - 4z^4 + 2z^3 + 3z^2) * q^63 + 8z^5 q^64 + (2z^7 + 2z^5 + 2z^4 - 2z^3 - 4z^2 - 2z + 4) * q^66 + (-4z^7 - z^6 + 3z^5 - 3z^4 - z^3 + z^2 + 2z - 2) * q^67 + (4z^6 - 4z^5 - 4z^4 + 4z^3 + 2z^2 - 2z) * q^68 + (4z^7 + 8z^6 - 4z^5 - 4z^4 - 4z^3 + 8z^2 + 4z) q^69 + (7z^7 + z^6 - 7z^5 - 2z^4 - z^2 - z - 3) q^71 + (-2z^7 - 4z^4 + 4z^3 + 6z^2 - 4z - 4) q^72 + (-3z^7 + 6z^6 + z^5 - 8z^4 + 5z^3 + 5z^2 + z - 6) q^73 + (7z^7 + z^4 + z^3 - z - 1) q^74 + (-6z^7 - 2z^5 + 2z^4 - 2z^3 - 6z) q^76 + (2z^7 - 2z^6 + 4z^4 - 2z^3 - 2z^2 - 4z + 6) * q^77 + (2z^7 - 2z^6 + 6z^5 + 10z^4 - 8z^3 - 8z^2 + 8z) q^78 + (z^7 - 3z^5 + 4z^3 - 5z^2 - 2z + 5) * q^79 + (2z^6 + 5z^4 + 2z^2) q^81 + (-8z^7 - 2z^6 + 8z^5 - z^4 - 5z^3 + z^2 + 7z + 7) q^82 + (z^7 + 9z^6 - 10z^5 - 10z^4 + 4z^3 + 5z^2 - 5z - 6) * q^83 + (-4z^7 + 4z^6 - 4z^5 + 8z^2 - 8) * q^84 + (-2z^7 - 6z^6 + 6z^5 + 2z^4 - 2z^3 - 8z^2 + 4) * q^86 + (-2z^6 - 6z^5 + 5z^4 + 3z^3 + z^2 - z - 1) * q^87 + (4z^7 - 4z^4) * q^88 + (z^6 + 2z^5 - 4z^4 - 2z^3 - 4z^2 + 2z + 1) q^89 + (-2z^7 + 4z^5 + z^4 - z^3 - 7z^2 + 3z + 1) * q^91 + (8z^7 - 8z^5 + 8z^3 + 8z^2 - 8z) q^92 + (-4z^7 + 4z^6 + 10z^5 - 6z^4 + 2z^3 + 6z^2 - 2z + 6) q^93 + (-4z^7 - 2z^6 + 4z^5 + 4z^4 - 4z^2 - 2z) * q^94 + (-8z^6 + 16z^4 - 8z^2 + 8) q^96 + (-8z^7 - 6z^6 + 6z^5 + 5z^4 - 5z^3 - 6z^2 + 6z + 8) q^97 + (-12z^7 + 12z^6 + 5z^5 - 4z^4 - 4z^3 + 8z^2 + 8z - 7) q^98 + (-3z^7 - z^6 + 5z^5 + 3z^4 - 3z^3 - 4z^2 + 2) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 4 q^{7} - 16 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 - 4 * q^7 - 16 * q^8	$ 8 q + 8 q^{2} - 4 q^{7} - 16 q^{8} - 4 q^{11} - 12 q^{13} - 32 q^{16} + 10 q^{17} - 6 q^{18} + 10 q^{19} + 20 q^{21} - 4 q^{22} + 8 q^{23} - 4 q^{26} + 8 q^{28} - 10 q^{29} - 10 q^{31} - 32 q^{32} + 20 q^{34} - 12 q^{36} - 14 q^{37} + 8 q^{38} - 30 q^{39} + 16 q^{41} - 20 q^{43} + 16 q^{46} - 24 q^{47} - 20 q^{51} + 16 q^{52} + 28 q^{53} + 16 q^{56} - 60 q^{57} - 10 q^{59} + 10 q^{61} - 20 q^{62} + 18 q^{63} + 20 q^{66} - 10 q^{67} + 20 q^{68} + 40 q^{69} - 20 q^{71} - 12 q^{72} - 10 q^{73} - 10 q^{74} - 4 q^{76} + 32 q^{77} - 40 q^{78} + 30 q^{79} - 2 q^{81} + 56 q^{82} - 40 q^{84} - 20 q^{87} + 8 q^{88} + 10 q^{89} - 8 q^{91} + 16 q^{92} + 80 q^{93} - 20 q^{94} + 30 q^{97} - 8 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 - 4 * q^7 - 16 * q^8 - 4 * q^11 - 12 * q^13 - 32 * q^16 + 10 * q^17 - 6 * q^18 + 10 * q^19 + 20 * q^21 - 4 * q^22 + 8 * q^23 - 4 * q^26 + 8 * q^28 - 10 * q^29 - 10 * q^31 - 32 * q^32 + 20 * q^34 - 12 * q^36 - 14 * q^37 + 8 * q^38 - 30 * q^39 + 16 * q^41 - 20 * q^43 + 16 * q^46 - 24 * q^47 - 20 * q^51 + 16 * q^52 + 28 * q^53 + 16 * q^56 - 60 * q^57 - 10 * q^59 + 10 * q^61 - 20 * q^62 + 18 * q^63 + 20 * q^66 - 10 * q^67 + 20 * q^68 + 40 * q^69 - 20 * q^71 - 12 * q^72 - 10 * q^73 - 10 * q^74 - 4 * q^76 + 32 * q^77 - 40 * q^78 + 30 * q^79 - 2 * q^81 + 56 * q^82 - 40 * q^84 - 20 * q^87 + 8 * q^88 + 10 * q^89 - 8 * q^91 + 16 * q^92 + 80 * q^93 - 20 * q^94 + 30 * q^97 - 8 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times$.

$n$	$377$	$501$	$751$
$\chi(n)$	$\zeta_{20}$	$-1$	$-1$

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} $

43.1

0.951057	−	0.309017i
−0.587785	−	0.809017i
−0.587785	+	0.809017i
0.587785	−	0.809017i
0.587785	+	0.809017i
−0.951057	+	0.309017i
−0.951057	−	0.309017i
0.951057	+	0.309017i

1.00000

1.00000i

−0.754763

1.48131i

2.00000i

−2.23607

0.726543i

0.284079

0.284079i

−2.00000

2.00000i

0.138757

0.190983i

107.1

1.00000

−

1.00000i

−0.420808

2.65688i

−

2.00000i

2.23607

3.07768i

−0.557537

0.557537i

−2.00000

−

2.00000i

−4.02874

−

1.30902i

243.1

1.00000

1.00000i

−0.420808

−

2.65688i

2.00000i

2.23607

−

3.07768i

−0.557537

−

0.557537i

−2.00000

2.00000i

−4.02874

1.30902i

507.1

1.00000

−

1.00000i

2.65688

−

0.420808i

−

2.00000i

2.23607

−

3.07768i

1.79360

−

1.79360i

−2.00000

−

2.00000i

4.02874

−

1.30902i

643.1

1.00000

1.00000i

2.65688

0.420808i

2.00000i

2.23607

3.07768i

1.79360

1.79360i

−2.00000

2.00000i

4.02874

1.30902i

707.1

1.00000

−

1.00000i

−1.48131

−

0.754763i

−

2.00000i

−2.23607

0.726543i

−3.52015

3.52015i

−2.00000

−

2.00000i

−0.138757

−

0.190983i

843.1

1.00000

1.00000i

−1.48131

0.754763i

2.00000i

−2.23607

−

0.726543i

−3.52015

−

3.52015i

−2.00000

2.00000i

−0.138757

0.190983i

907.1

1.00000

−

1.00000i

−0.754763

−

1.48131i

−

2.00000i

−2.23607

−

0.726543i

0.284079

−

0.284079i

−2.00000

−

2.00000i

0.138757

−

0.190983i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
200.v	even	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1000.2.v.e		8
5.b	even	2	1		1000.2.v.a		8
5.c	odd	4	1		200.2.v.a	✓	8
5.c	odd	4	1		1000.2.v.f		8
8.d	odd	2	1		1000.2.v.d		8
20.e	even	4	1		800.2.bp.b		8
25.d	even	5	1		200.2.v.b	yes	8
25.e	even	10	1		1000.2.v.c		8
25.f	odd	20	1		1000.2.v.b		8
25.f	odd	20	1		1000.2.v.d		8
40.e	odd	2	1		1000.2.v.b		8
40.i	odd	4	1		800.2.bp.a		8
40.k	even	4	1		200.2.v.b	yes	8
40.k	even	4	1		1000.2.v.c		8
100.j	odd	10	1		800.2.bp.a		8
200.n	odd	10	1		200.2.v.a	✓	8
200.s	odd	10	1		1000.2.v.f		8
200.t	even	10	1		800.2.bp.b		8
200.v	even	20	1		1000.2.v.a		8
200.v	even	20	1	inner	1000.2.v.e		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.2.v.a	✓	8	5.c	odd	4	1
200.2.v.a	✓	8	200.n	odd	10	1
200.2.v.b	yes	8	25.d	even	5	1
200.2.v.b	yes	8	40.k	even	4	1
800.2.bp.a		8	40.i	odd	4	1
800.2.bp.a		8	100.j	odd	10	1
800.2.bp.b		8	20.e	even	4	1
800.2.bp.b		8	200.t	even	10	1
1000.2.v.a		8	5.b	even	2	1
1000.2.v.a		8	200.v	even	20	1
1000.2.v.b		8	25.f	odd	20	1
1000.2.v.b		8	40.e	odd	2	1
1000.2.v.c		8	25.e	even	10	1
1000.2.v.c		8	40.k	even	4	1
1000.2.v.d		8	8.d	odd	2	1
1000.2.v.d		8	25.f	odd	20	1
1000.2.v.e		8	1.a	even	1	1	trivial
1000.2.v.e		8	200.v	even	20	1	inner
1000.2.v.f		8	5.c	odd	4	1
1000.2.v.f		8	200.s	odd	10	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1000, [\chi])$:

$ T_{3}^{8} - 20T_{3}^{5} - 40T_{3}^{4} + 200T_{3}^{2} + 400T_{3} + 400 $

$ T_{7}^{8} + 4T_{7}^{7} + 8T_{7}^{6} - 40T_{7}^{5} + 136T_{7}^{4} + 80T_{7}^{3} + 32T_{7}^{2} - 32T_{7} + 16 $

$ T_{13}^{8} + 12T_{13}^{7} + 62T_{13}^{6} + 140T_{13}^{5} + 116T_{13}^{4} - 1370T_{13}^{3} - 597T_{13}^{2} - 7784T_{13} + 19321 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} - 2 T + 2)^{4} $ (T^2 - 2*T + 2)^4
$3$	$ T^{8} - 20 T^{5} + \cdots + 400 $ T^8 - 20T^5 - 40T^4 + 200T^2 + 400T + 400
$5$	$ T^{8} $ T^8
$7$	$ T^{8} + 4 T^{7} + \cdots + 16 $ T^8 + 4T^7 + 8T^6 - 40T^5 + 136T^4 + 80T^3 + 32T^2 - 32*T + 16
$11$	$ T^{8} + 4 T^{7} + \cdots + 16 $ T^8 + 4T^7 + 12T^6 + 12T^5 - 32T^3 + 72T^2 + 16T + 16
$13$	$ T^{8} + 12 T^{7} + \cdots + 19321 $ T^8 + 12T^7 + 62T^6 + 140T^5 + 116T^4 - 1370T^3 - 597T^2 - 7784*T + 19321
$17$	$ T^{8} - 10 T^{7} + \cdots + 25 $ T^8 - 10T^7 + 70T^6 - 330T^5 + 860T^4 - 1150T^3 + 725T^2 - 150*T + 25
$19$	$ T^{8} - 10 T^{7} + \cdots + 309136 $ T^8 - 10T^7 + 14T^6 - 80T^5 + 2116T^4 - 6320T^3 - 3016T^2 - 55600*T + 309136
$23$	$ T^{8} - 8 T^{7} + \cdots + 65536 $ T^8 - 8T^7 + 32T^6 - 1024T^4 + 10240T^3 + 53248T^2 + 65536T + 65536
$29$	$ T^{8} + 10 T^{7} + \cdots + 9025 $ T^8 + 10T^7 + 60T^6 + 290T^5 + 1310T^4 + 4100T^3 + 8475T^2 + 10450*T + 9025
$31$	$ T^{8} + 10 T^{7} + \cdots + 1392400 $ T^8 + 10T^7 + 10T^6 + 680T^5 + 11760T^4 + 45400T^3 + 2800T^2 + 94400*T + 1392400
$37$	$ T^{8} + 14 T^{7} + \cdots + 128881 $ T^8 + 14T^7 + 93T^6 + 410T^5 + 776T^4 - 590T^3 - 5238T^2 - 20822*T + 128881
$41$	$ T^{8} - 16 T^{7} + \cdots + 436921 $ T^8 - 16T^7 + 152T^6 - 828T^5 + 9650T^4 - 13032T^3 + 173237T^2 + 446836*T + 436921
$43$	$ T^{8} + 20 T^{7} + \cdots + 160000 $ T^8 + 20T^7 + 200T^6 + 800T^5 + 1200T^4 + 80000T^2 + 160000T + 160000
$47$	$ T^{8} + 24 T^{7} + \cdots + 256 $ T^8 + 24T^7 + 188T^6 + 560T^5 + 1136T^4 + 1600T^3 + 1472T^2 + 768*T + 256
$53$	$ T^{8} - 28 T^{7} + \cdots + 177241 $ T^8 - 28T^7 + 377T^6 - 3230T^5 + 20776T^4 - 82280T^3 + 225718T^2 - 325854*T + 177241
$59$	$ T^{8} + 10 T^{7} + \cdots + 3671056 $ T^8 + 10T^7 - 6T^6 + 1880T^5 + 34116T^4 + 115520T^3 - 14936T^2 + 421520*T + 3671056
$61$	$ T^{8} - 10 T^{7} + \cdots + 3258025 $ T^8 - 10T^7 + 330T^5 + 210T^4 - 15100T^3 - 8125T^2 + 776150T + 3258025
$67$	$ T^{8} + 10 T^{7} + \cdots + 144400 $ T^8 + 10T^7 + 70T^6 + 520T^5 + 2760T^4 + 7400T^3 + 33600T^2 + 121600*T + 144400
$71$	$ T^{8} + 20 T^{7} + \cdots + 144400 $ T^8 + 20T^7 + 180T^6 + 1180T^5 + 15360T^4 + 199600T^3 + 1066600T^2 - 767600*T + 144400
$73$	$ T^{8} + 10 T^{7} + \cdots + 17598025 $ T^8 + 10T^7 - 130T^6 - 1030T^5 + 8660T^4 + 156950T^3 + 1489725T^2 + 3901350*T + 17598025
$79$	$ T^{8} - 30 T^{7} + \cdots + 144400 $ T^8 - 30T^7 + 410T^6 - 2800T^5 + 13360T^4 - 43400T^3 + 101200T^2 - 152000*T + 144400
$83$	$ T^{8} - 40 T^{6} + \cdots + 537312400 $ T^8 - 40T^6 - 1500T^5 - 20040T^4 - 222000T^3 + 2102200T^2 + 34770000T + 537312400
$89$	$ T^{8} - 10 T^{7} + \cdots + 1745041 $ T^8 - 10T^7 - 11T^6 - 320T^5 + 7086T^4 - 3130T^3 + 15124T^2 - 660500*T + 1745041
$97$	$ T^{8} - 30 T^{7} + \cdots + 9025 $ T^8 - 30T^7 + 410T^6 - 3370T^5 + 23360T^4 + 21250T^3 + 94025T^2 - 59850*T + 9025
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1000 = 2^{3} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1000.v (of order \(20\), degree \(8\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{20}^{5} + 1) q^{2} + ( - \zeta_{20}^{7} + \zeta_{20}^{6} + \cdots - 1) q^{3}+ \cdots + (\zeta_{20}^{7} - 2 \zeta_{20}^{3} + 2 \zeta_{20}) q^{9}+O(q^{10}) \) q + (-z^5 + 1) * q^2 + (-z^7 + z^6 + z^5 - 2z^4 - 2z^3 + z^2 + z - 1) * q^3 - 2z^5 q^4 + (2z^6 - 4z^4 + 2z^2 - 2) q^6 + (z^5 - 2z^4 + 2z - 1) * q^7 + (-2z^5 - 2) q^8 + (z^7 - 2z^3 + 2z) * q^9	\( q + ( - \zeta_{20}^{5} + 1) q^{2} + ( - \zeta_{20}^{7} + \zeta_{20}^{6} + \cdots - 1) q^{3}+ \cdots + ( - 3 \zeta_{20}^{7} - \zeta_{20}^{6} + \cdots + 2) q^{99}+O(q^{100}) \) q + (-z^5 + 1) * q^2 + (-z^7 + z^6 + z^5 - 2z^4 - 2z^3 + z^2 + z - 1) * q^3 - 2z^5 q^4 + (2z^6 - 4z^4 + 2z^2 - 2) q^6 + (z^5 - 2z^4 + 2z - 1) * q^7 + (-2z^5 - 2) q^8 + (z^7 - 2z^3 + 2z) * q^9 + (-2z^7 + z^5 + z^4 - z^3 - z^2 + z) q^11 + (2z^7 + 2z^6 - 2z^5 - 4z^4 + 4z^3 + 2z^2 - 2z - 2) q^12 + (-z^7 - 3z^6 + 2z^5 + 2z^4 - 3z^3 - z^2) * q^13 + (2z^7 - 2z^6 - 2z^4 + 2z^3) * q^14 - 4 * q^16 + (-z^7 + z^6 + 2z^5 - 2z^4 - 2z^3 + 2z^2) * q^17 + (z^7 - 2z^4 - 2z^3 + 3z^2 + 2z - 2) * q^18 + (z^7 - 4z^6 - z^5 + z^4 + z^3 + 2z^2 - z + 2) * q^19 + (4z^7 - 4z^5 + 2z^2 - 2z + 2) * q^21 + (-2z^7 + 2z^5 - 2z^3 - 2z^2 + 2z) q^22 + (4z^7 - 4z^4) * q^23 + (4z^7 - 4z^5 + 8z^3 - 4z) * q^24 + (-2z^7 + 4z^5 - z^4 - 5z^3 + z^2 - z - 1) q^26 + (4z^7 - 2z^5 - 2z^4 + 2z^3 - 2z^2 - 2z) * q^27 + (4z^7 - 4z^6 - 2z^5 + 4z^3 - 4z + 2) q^28 + (2z^7 - 2z^6 + 2z^3 + z^2 - z - 1) q^29 + (2z^7 + 5z^6 - 3z^5 - z^4 + 3z^3 - 3z^2 - 2z - 2) * q^31 + (4z^5 - 4) q^32 + (-2z^6 + 2z^5 + 2z^4 - 4z^2 - 2z + 2) q^33 + (-z^7 + 3z^6 - 4z^4 + 3z^2 - z) q^34 + (-4z^4 + 6z^2 - 4) * q^36 + (4z^7 - z^5 + z^3 - 3z^2 - z - 1) * q^37 + (-2z^7 - 4z^6 - 2z^5 + 2z^4 + 2z^2 - 4z + 2) * q^38 + (2z^7 - z^6 - 2z^5 + 9z^4 + z^3 - 9z^2 + 1) * q^39 + (-4z^7 + 8z^5 + 2z^4 - 3z^3 + 2z^2 + 5z + 2) * q^41 + (2z^7 + 2z^6 - 6z^5 + 6z^2 - 2z - 2) q^42 + (-4z^7 - 4z^6 + 4z^5 + 2z^4 - 4z^2 + 2z) * q^43 + (2z^5 - 2z^4 - 2z^3 - 2z^2 + 2z) q^44 + (8z^7 - 4z^5 - 4z^4 + 4z^3 + 4z^2 - 4z) * q^46 + (-2z^7 - 2z^6 + 2z^4 + 2z^3 - 2z - 2) q^47 + (4z^7 - 4z^6 - 4z^5 + 8z^4 + 8z^3 - 4z^2 - 4z + 4) q^48 + (-4z^7 + 8z^6 + z^5 - 4z^3 + 8z^2 - 4) * q^49 + (z^7 - z^6 + 3z^5 + z^4 - 7z^3 + 6z - 2) q^51 + (-2z^7 + 6z^6 + 4z^5 - 6z^4 - 4z^3 + 4z^2 - 2z - 2) q^52 + (-6z^7 - 3z^6 + 3z^5 + 4z^4 - 4z^3 - 3z^2 + 3z + 6) q^53 + (8z^7 - 4z^5 + 4z^3 - 4z) * q^54 + (4z^7 - 4z^6 - 4z^5 + 4z^4 + 4z^3 - 8z + 4) * q^56 + (-8z^5 + 2z^3 + 2z^2 - 8) q^57 + (z^7 - 3z^6 + z^5 + 2z^4 + 2z^3 + z^2 - 3z + 1) * q^58 + (-3z^7 - 10z^6 + 6z^4 - 4z^3 - 5z^2 + 4z + 4) * q^59 + (z^6 + 6z^5 - 3z^3 + 6z + 1) q^61 + (6z^7 + 4z^6 - 2z^5 + 2z^4 + 4z^3 - 4z^2 + 2z - 2) q^62 + (3z^7 + 2z^6 - 4z^5 - 4z^4 + 2z^3 + 3z^2) * q^63 + 8z^5 q^64 + (2z^7 + 2z^5 + 2z^4 - 2z^3 - 4z^2 - 2z + 4) * q^66 + (-4z^7 - z^6 + 3z^5 - 3z^4 - z^3 + z^2 + 2z - 2) * q^67 + (4z^6 - 4z^5 - 4z^4 + 4z^3 + 2z^2 - 2z) * q^68 + (4z^7 + 8z^6 - 4z^5 - 4z^4 - 4z^3 + 8z^2 + 4z) q^69 + (7z^7 + z^6 - 7z^5 - 2z^4 - z^2 - z - 3) q^71 + (-2z^7 - 4z^4 + 4z^3 + 6z^2 - 4z - 4) q^72 + (-3z^7 + 6z^6 + z^5 - 8z^4 + 5z^3 + 5z^2 + z - 6) q^73 + (7z^7 + z^4 + z^3 - z - 1) q^74 + (-6z^7 - 2z^5 + 2z^4 - 2z^3 - 6z) q^76 + (2z^7 - 2z^6 + 4z^4 - 2z^3 - 2z^2 - 4z + 6) * q^77 + (2z^7 - 2z^6 + 6z^5 + 10z^4 - 8z^3 - 8z^2 + 8z) q^78 + (z^7 - 3z^5 + 4z^3 - 5z^2 - 2z + 5) * q^79 + (2z^6 + 5z^4 + 2z^2) q^81 + (-8z^7 - 2z^6 + 8z^5 - z^4 - 5z^3 + z^2 + 7z + 7) q^82 + (z^7 + 9z^6 - 10z^5 - 10z^4 + 4z^3 + 5z^2 - 5z - 6) * q^83 + (-4z^7 + 4z^6 - 4z^5 + 8z^2 - 8) * q^84 + (-2z^7 - 6z^6 + 6z^5 + 2z^4 - 2z^3 - 8z^2 + 4) * q^86 + (-2z^6 - 6z^5 + 5z^4 + 3z^3 + z^2 - z - 1) * q^87 + (4z^7 - 4z^4) * q^88 + (z^6 + 2z^5 - 4z^4 - 2z^3 - 4z^2 + 2z + 1) q^89 + (-2z^7 + 4z^5 + z^4 - z^3 - 7z^2 + 3z + 1) * q^91 + (8z^7 - 8z^5 + 8z^3 + 8z^2 - 8z) q^92 + (-4z^7 + 4z^6 + 10z^5 - 6z^4 + 2z^3 + 6z^2 - 2z + 6) q^93 + (-4z^7 - 2z^6 + 4z^5 + 4z^4 - 4z^2 - 2z) * q^94 + (-8z^6 + 16z^4 - 8z^2 + 8) q^96 + (-8z^7 - 6z^6 + 6z^5 + 5z^4 - 5z^3 - 6z^2 + 6z + 8) q^97 + (-12z^7 + 12z^6 + 5z^5 - 4z^4 - 4z^3 + 8z^2 + 8z - 7) q^98 + (-3z^7 - z^6 + 5z^5 + 3z^4 - 3z^3 - 4z^2 + 2) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 4 q^{7} - 16 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 - 4 * q^7 - 16 * q^8	\( 8 q + 8 q^{2} - 4 q^{7} - 16 q^{8} - 4 q^{11} - 12 q^{13} - 32 q^{16} + 10 q^{17} - 6 q^{18} + 10 q^{19} + 20 q^{21} - 4 q^{22} + 8 q^{23} - 4 q^{26} + 8 q^{28} - 10 q^{29} - 10 q^{31} - 32 q^{32} + 20 q^{34} - 12 q^{36} - 14 q^{37} + 8 q^{38} - 30 q^{39} + 16 q^{41} - 20 q^{43} + 16 q^{46} - 24 q^{47} - 20 q^{51} + 16 q^{52} + 28 q^{53} + 16 q^{56} - 60 q^{57} - 10 q^{59} + 10 q^{61} - 20 q^{62} + 18 q^{63} + 20 q^{66} - 10 q^{67} + 20 q^{68} + 40 q^{69} - 20 q^{71} - 12 q^{72} - 10 q^{73} - 10 q^{74} - 4 q^{76} + 32 q^{77} - 40 q^{78} + 30 q^{79} - 2 q^{81} + 56 q^{82} - 40 q^{84} - 20 q^{87} + 8 q^{88} + 10 q^{89} - 8 q^{91} + 16 q^{92} + 80 q^{93} - 20 q^{94} + 30 q^{97} - 8 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 - 4 * q^7 - 16 * q^8 - 4 * q^11 - 12 * q^13 - 32 * q^16 + 10 * q^17 - 6 * q^18 + 10 * q^19 + 20 * q^21 - 4 * q^22 + 8 * q^23 - 4 * q^26 + 8 * q^28 - 10 * q^29 - 10 * q^31 - 32 * q^32 + 20 * q^34 - 12 * q^36 - 14 * q^37 + 8 * q^38 - 30 * q^39 + 16 * q^41 - 20 * q^43 + 16 * q^46 - 24 * q^47 - 20 * q^51 + 16 * q^52 + 28 * q^53 + 16 * q^56 - 60 * q^57 - 10 * q^59 + 10 * q^61 - 20 * q^62 + 18 * q^63 + 20 * q^66 - 10 * q^67 + 20 * q^68 + 40 * q^69 - 20 * q^71 - 12 * q^72 - 10 * q^73 - 10 * q^74 - 4 * q^76 + 32 * q^77 - 40 * q^78 + 30 * q^79 - 2 * q^81 + 56 * q^82 - 40 * q^84 - 20 * q^87 + 8 * q^88 + 10 * q^89 - 8 * q^91 + 16 * q^92 + 80 * q^93 - 20 * q^94 + 30 * q^97 - 8 * q^98

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	1000.2.v.e

Level	$1000$
Weight	$2$
Character orbit	1000.v
Analytic conductor	$7.985$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(7.98504020213\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{20})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

\(n\)	\(377\)	\(501\)	\(751\)
\(\chi(n)\)	\(\zeta_{20}\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 43.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} - 2 T + 2)^{4} \) (T^2 - 2*T + 2)^4
$3$	\( T^{8} - 20 T^{5} + \cdots + 400 \) T^8 - 20T^5 - 40T^4 + 200T^2 + 400T + 400
$5$	\( T^{8} \) T^8
$7$	\( T^{8} + 4 T^{7} + \cdots + 16 \) T^8 + 4T^7 + 8T^6 - 40T^5 + 136T^4 + 80T^3 + 32T^2 - 32*T + 16
$11$	\( T^{8} + 4 T^{7} + \cdots + 16 \) T^8 + 4T^7 + 12T^6 + 12T^5 - 32T^3 + 72T^2 + 16T + 16
$13$	\( T^{8} + 12 T^{7} + \cdots + 19321 \) T^8 + 12T^7 + 62T^6 + 140T^5 + 116T^4 - 1370T^3 - 597T^2 - 7784*T + 19321
$17$	\( T^{8} - 10 T^{7} + \cdots + 25 \) T^8 - 10T^7 + 70T^6 - 330T^5 + 860T^4 - 1150T^3 + 725T^2 - 150*T + 25
$19$	\( T^{8} - 10 T^{7} + \cdots + 309136 \) T^8 - 10T^7 + 14T^6 - 80T^5 + 2116T^4 - 6320T^3 - 3016T^2 - 55600*T + 309136
$23$	\( T^{8} - 8 T^{7} + \cdots + 65536 \) T^8 - 8T^7 + 32T^6 - 1024T^4 + 10240T^3 + 53248T^2 + 65536T + 65536
$29$	\( T^{8} + 10 T^{7} + \cdots + 9025 \) T^8 + 10T^7 + 60T^6 + 290T^5 + 1310T^4 + 4100T^3 + 8475T^2 + 10450*T + 9025
$31$	\( T^{8} + 10 T^{7} + \cdots + 1392400 \) T^8 + 10T^7 + 10T^6 + 680T^5 + 11760T^4 + 45400T^3 + 2800T^2 + 94400*T + 1392400
$37$	\( T^{8} + 14 T^{7} + \cdots + 128881 \) T^8 + 14T^7 + 93T^6 + 410T^5 + 776T^4 - 590T^3 - 5238T^2 - 20822*T + 128881
$41$	\( T^{8} - 16 T^{7} + \cdots + 436921 \) T^8 - 16T^7 + 152T^6 - 828T^5 + 9650T^4 - 13032T^3 + 173237T^2 + 446836*T + 436921
$43$	\( T^{8} + 20 T^{7} + \cdots + 160000 \) T^8 + 20T^7 + 200T^6 + 800T^5 + 1200T^4 + 80000T^2 + 160000T + 160000
$47$	\( T^{8} + 24 T^{7} + \cdots + 256 \) T^8 + 24T^7 + 188T^6 + 560T^5 + 1136T^4 + 1600T^3 + 1472T^2 + 768*T + 256
$53$	\( T^{8} - 28 T^{7} + \cdots + 177241 \) T^8 - 28T^7 + 377T^6 - 3230T^5 + 20776T^4 - 82280T^3 + 225718T^2 - 325854*T + 177241
$59$	\( T^{8} + 10 T^{7} + \cdots + 3671056 \) T^8 + 10T^7 - 6T^6 + 1880T^5 + 34116T^4 + 115520T^3 - 14936T^2 + 421520*T + 3671056
$61$	\( T^{8} - 10 T^{7} + \cdots + 3258025 \) T^8 - 10T^7 + 330T^5 + 210T^4 - 15100T^3 - 8125T^2 + 776150T + 3258025
$67$	\( T^{8} + 10 T^{7} + \cdots + 144400 \) T^8 + 10T^7 + 70T^6 + 520T^5 + 2760T^4 + 7400T^3 + 33600T^2 + 121600*T + 144400
$71$	\( T^{8} + 20 T^{7} + \cdots + 144400 \) T^8 + 20T^7 + 180T^6 + 1180T^5 + 15360T^4 + 199600T^3 + 1066600T^2 - 767600*T + 144400
$73$	\( T^{8} + 10 T^{7} + \cdots + 17598025 \) T^8 + 10T^7 - 130T^6 - 1030T^5 + 8660T^4 + 156950T^3 + 1489725T^2 + 3901350*T + 17598025
$79$	\( T^{8} - 30 T^{7} + \cdots + 144400 \) T^8 - 30T^7 + 410T^6 - 2800T^5 + 13360T^4 - 43400T^3 + 101200T^2 - 152000*T + 144400
$83$	\( T^{8} - 40 T^{6} + \cdots + 537312400 \) T^8 - 40T^6 - 1500T^5 - 20040T^4 - 222000T^3 + 2102200T^2 + 34770000T + 537312400
$89$	\( T^{8} - 10 T^{7} + \cdots + 1745041 \) T^8 - 10T^7 - 11T^6 - 320T^5 + 7086T^4 - 3130T^3 + 15124T^2 - 660500*T + 1745041
$97$	\( T^{8} - 30 T^{7} + \cdots + 9025 \) T^8 - 30T^7 + 410T^6 - 3370T^5 + 23360T^4 + 21250T^3 + 94025T^2 - 59850*T + 9025
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