LMFDB - Newform orbit 686.2.c.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [686,2,Mod(361,686)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(686, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("686.361"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$686 = 2 \cdot 7^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	686.c (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [6,3,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.47773757866$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.64827.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1$ x^6 - x^5 + 3x^4 + 5x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/686\mathbb{Z}\right)^\times$ .

$n$	$3$
$\chi(n)$	$-\beta_{5}$

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}

361.1

−0.623490	+	1.07992i
0.222521	−	0.385418i
0.900969	−	1.56052i
−0.623490	−	1.07992i
0.222521	+	0.385418i
0.900969	+	1.56052i

0.500000

0.866025i

−0.623490

1.07992i

−0.500000

0.866025i

0.678448

1.17511i

−1.24698

−1.00000

0.722521

1.25144i

−0.678448

1.17511i

361.2

0.500000

0.866025i

0.222521

−

0.385418i

−0.500000

0.866025i

−1.52446

−

2.64044i

0.445042

−1.00000

1.40097

2.42655i

1.52446

−

2.64044i

361.3

0.500000

0.866025i

0.900969

−

1.56052i

−0.500000

0.866025i

0.846011

1.46533i

1.80194

−1.00000

−0.123490

−

0.213891i

−0.846011

1.46533i

667.1

0.500000

−

0.866025i

−0.623490

−

1.07992i

−0.500000

−

0.866025i

0.678448

−

1.17511i

−1.24698

−1.00000

0.722521

−

1.25144i

−0.678448

−

1.17511i

667.2

0.500000

−

0.866025i

0.222521

0.385418i

−0.500000

−

0.866025i

−1.52446

2.64044i

0.445042

−1.00000

1.40097

−

2.42655i

1.52446

2.64044i

667.3

0.500000

−

0.866025i

0.900969

1.56052i

−0.500000

−

0.866025i

0.846011

−

1.46533i

1.80194

−1.00000

−0.123490

0.213891i

−0.846011

−

1.46533i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	686.2.c.d		6
7.b	odd	2	1		686.2.c.c		6
7.c	even	3	1		686.2.a.a	✓	3
7.c	even	3	1	inner	686.2.c.d		6
7.d	odd	6	1		686.2.a.b	yes	3
7.d	odd	6	1		686.2.c.c		6
21.g	even	6	1		6174.2.a.l		3
21.h	odd	6	1		6174.2.a.k		3
28.f	even	6	1		5488.2.a.b		3
28.g	odd	6	1		5488.2.a.e		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
686.2.a.a	✓	3	7.c	even	3	1
686.2.a.b	yes	3	7.d	odd	6	1
686.2.c.c		6	7.b	odd	2	1
686.2.c.c		6	7.d	odd	6	1
686.2.c.d		6	1.a	even	1	1	trivial
686.2.c.d		6	7.c	even	3	1	inner
5488.2.a.b		3	28.f	even	6	1
5488.2.a.e		3	28.g	odd	6	1
6174.2.a.k		3	21.h	odd	6	1
6174.2.a.l		3	21.g	even	6	1

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{6} - T_{3}^{5} + 3T_{3}^{4} + 5T_{3}^{2} - 2T_{3} + 1$ T3^6 - T3^5 + 3*T3^4 + 5*T3^2 - 2*T3 + 1 acting on $S_{2}^{\mathrm{new}}(686, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{2} - T + 1)^{3}$ (T^2 - T + 1)^3
$3$	$T^{6} - T^{5} + 3 T^{4} + \cdots + 1$ T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1
$5$	$T^{6} + 7 T^{4} + \cdots + 49$ T^6 + 7T^4 - 14T^3 + 49T^2 - 49T + 49
$7$	$T^{6}$ T^6
$11$	$T^{6} + 21 T^{4} + \cdots + 49$ T^6 + 21T^4 - 14T^3 + 441T^2 - 147T + 49
$13$	$(T^{3} + 9 T^{2} + \cdots - 113)^{2}$ (T^3 + 9*T^2 - T - 113)^2
$17$	$T^{6} - 4 T^{5} + \cdots + 169$ T^6 - 4T^5 + 41T^4 + 126T^3 + 573T^2 + 325*T + 169
$19$	$T^{6} - 13 T^{5} + \cdots + 1849$ T^6 - 13T^5 + 122T^4 - 525T^3 + 1650T^2 - 2021*T + 1849
$23$	$T^{6} + 7 T^{5} + \cdots + 8281$ T^6 + 7T^5 + 63T^4 + 84T^3 + 833T^2 + 1274*T + 8281
$29$	$(T^{3} - 3 T^{2} - 4 T - 1)^{2}$ (T^3 - 3T^2 - 4T - 1)^2
$31$	$T^{6} - 16 T^{5} + \cdots + 1681$ T^6 - 16T^5 + 187T^4 - 1022T^3 + 4105T^2 - 2829*T + 1681
$37$	$T^{6} + 6 T^{5} + \cdots + 729$ T^6 + 6T^5 + 45T^4 + 243T^2 + 243T + 729
$41$	$(T^{3} + 11 T^{2} + \cdots + 13)^{2}$ (T^3 + 11T^2 + 31T + 13)^2
$43$	$(T^{3} + 11 T^{2} + \cdots - 29)^{2}$ (T^3 + 11T^2 + 24T - 29)^2
$47$	$T^{6} + 8 T^{5} + \cdots + 1$ T^6 + 8T^5 + 59T^4 + 42T^3 + 33T^2 - 5*T + 1
$53$	$T^{6} - 3 T^{5} + \cdots + 613089$ T^6 - 3T^5 + 153T^4 - 1134T^3 + 23085T^2 - 112752*T + 613089
$59$	$T^{6} + 4 T^{5} + \cdots + 44521$ T^6 + 4T^5 + 111T^4 + 42T^3 + 9869T^2 + 20045*T + 44521
$61$	$T^{6} - 31 T^{5} + \cdots + 829921$ T^6 - 31T^5 + 657T^4 - 7602T^3 + 64175T^2 - 276944*T + 829921
$67$	$T^{6} + 23 T^{5} + \cdots + 94249$ T^6 + 23T^5 + 369T^4 + 3066T^3 + 18539T^2 + 49120*T + 94249
$71$	$(T^{3} - 7 T^{2} + \cdots + 637)^{2}$ (T^3 - 7T^2 - 98T + 637)^2
$73$	$T^{6} - 27 T^{5} + \cdots + 57121$ T^6 - 27T^5 + 535T^4 - 4760T^3 + 31183T^2 - 46366*T + 57121
$79$	$T^{6} - 32 T^{5} + \cdots + 1129969$ T^6 - 32T^5 + 699T^4 - 8274T^3 + 71609T^2 - 345475*T + 1129969
$83$	$(T^{3} - 11 T^{2} + \cdots - 181)^{2}$ (T^3 - 11T^2 - 102T - 181)^2
$89$	$T^{6} - 10 T^{5} + \cdots + 841$ T^6 - 10T^5 + 153T^4 + 588T^3 + 2519T^2 + 1537*T + 841
$97$	$(T^{3} + 5 T^{2} + 6 T + 1)^{2}$ (T^3 + 5T^2 + 6T + 1)^2
show more
show less

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 $\F_{q}$
Belyi maps

$f(q)$	$=$	$q + \beta_{5} q^{2} + \beta_1 q^{3} + (\beta_{5} - 1) q^{4} + ( - \beta_{5} - 2 \beta_{4} + \cdots - \beta_1) q^{5} + \beta_{2} q^{6} - q^{8} + (2 \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{9} + ( - \beta_{5} - 2 \beta_{4} - \beta_1 + 1) q^{10}+ \cdots + (5 \beta_{3} - 2 \beta_{2} + 7) q^{99}+O(q^{100})$ q + b5 * q^2 + b1 * q^3 + (b5 - 1) * q^4 + (-b5 - 2b4 - 2b3 + b2 - b1) * q^5 + b2 * q^6 - q^8 + (2b5 + b4 + b3 - b2 + b1) q^9 + (-b5 - 2b4 - b1 + 1) q^10 + (-2b5 - 3b4 - 3b1 + 2) q^11 + (b2 - b1) * q^12 + (-2b3 + 4b2 - 5) * q^13 + (-b3 + 2b2 - 1) q^15 - b5 * q^16 + (-3b5 - 4b4 - b1 + 3) * q^17 + (2b5 + b4 + b1 - 2) q^18 + (3b5 - 2b4 - 2b3 + 2b2 - 2b1) q^19 + (2b3 - b2 + 1) q^20 + (3b3 - 3b2 + 2) * q^22 + (-3b5 - 3b4 - 3b3 - b2 + b1) q^23 - b1 * q^24 + (b5 + 3b4 + b1 - 1) q^25 + (-5b5 - 2b4 - 2b3 + 4b2 - 4b1) q^26 + (b3 + 3b2) q^27 + (-b3 - b2 + 1) * q^29 + (-b5 - b4 - b3 + 2b2 - 2b1) * q^30 + (-5b5 - b4 + 2b1 + 5) * q^31 + (-b5 + 1) * q^32 + (-3b4 - 3b3 + 4b2 - 4b1) * q^33 + (4b3 - b2 + 3) q^34 + (-b3 + b2 - 2) * q^36 + (-3b5 + 3b2 - 3b1) q^37 + (3b5 - 2b4 - 2b1 - 3) q^38 + (-2b5 + 4b4 + b1 + 2) * q^39 + (b5 + 2b4 + 2b3 - b2 + b1) * q^40 + (2b3 - 3) q^41 + (3b3 - 2b2 - 2) * q^43 + (2b5 + 3b4 + 3b3 - 3b2 + 3b1) q^44 + (-4b5 - 4b4 - b1 + 4) * q^45 + (-3b5 - 3b4 + b1 + 3) * q^46 + (-b5 + 2b4 + 2b3 - 3b2 + 3b1) * q^47 - b2 * q^48 + (-3b3 + b2 - 1) q^50 + (-3b5 - b4 - b3 + 2b2 - 2b1) q^51 + (-5b5 - 2b4 - 4b1 + 5) q^52 + (-3b4 + 6b1) * q^53 + (b4 + b3 + 3b2 - 3b1) * q^54 + (-4b3 - b2 - 8) q^55 + (-2b3 + 7b2) * q^57 + (b5 - b4 - b3 - b2 + b1) * q^58 + (3b5 + 6b4 - b1 - 3) * q^59 + (-b5 - b4 - 2b1 + 1) q^60 + (9b5 - 3b4 - 3b3 + b2 - b1) q^61 + (b3 + 2b2 + 5) q^62 + q^64 + (7b5 + 8b4 + 8b3 + b2 - b1) q^65 + (-3b4 - 4b1) * q^66 + (8b5 + 2b4 - b1 - 8) * q^67 + (3b5 + 4b4 + 4b3 - b2 + b1) q^68 + (b3 - b2 - 4) * q^69 + (3b3 - 8b2 + 6) * q^71 + (-2b5 - b4 - b3 + b2 - b1) q^72 + (-6b5 + 5b4 + 4b1 + 6) q^73 + (-3b5 - 3b1 + 3) * q^74 + (2b5 + b4 + b3 - 3b2 + 3b1) q^75 + (2b3 - 2b2 - 3) * q^76 + (-4b3 + b2 + 2) q^78 + (9b5 - 2b4 - 2b3 + 3b2 - 3b1) q^79 + (b5 + 2b4 + b1 - 1) q^80 + (2b5 + 6b4 + 5b1 - 2) q^81 + (-3b5 + 2b4 + 2b3) q^82 + (-4b3 - 5b2 + 4) * q^83 + (-9b3 + 5b2 - 14) * q^85 + (-2b5 + 3b4 + 3b3 - 2b2 + 2b1) q^86 + (2b5 - b4 + b1 - 2) q^87 + (2b5 + 3b4 + 3b1 - 2) q^88 + (7b5 + 7b4 + 7b3 - 4b2 + 4b1) q^89 + (4b3 - b2 + 4) q^90 + (3b3 + b2 + 3) q^92 + (-3b5 + 2b4 + 2b3 - 6b2 + 6b1) q^93 + (-b5 + 2b4 + 3b1 + 1) * q^94 + (b5 - 6b4 - 5b1 - 1) * q^95 + (-b2 + b1) * q^96 + (b2 - 2) * q^97 + (5b3 - 2b2 + 7) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 3 q^{2} + q^{3} - 3 q^{4} + 2 q^{6} - 6 q^{8} + 4 q^{9} + q^{12} - 18 q^{13} - 3 q^{16} + 4 q^{17} - 4 q^{18} + 13 q^{19} - 7 q^{23} - q^{24} + q^{25} - 9 q^{26} + 4 q^{27} + 6 q^{29} + 16 q^{31}+ \cdots + 28 q^{99}+O(q^{100})$ 6 * q + 3 * q^2 + q^3 - 3 * q^4 + 2 * q^6 - 6 * q^8 + 4 * q^9 + q^12 - 18 * q^13 - 3 * q^16 + 4 * q^17 - 4 * q^18 + 13 * q^19 - 7 * q^23 - q^24 + q^25 - 9 * q^26 + 4 * q^27 + 6 * q^29 + 16 * q^31 + 3 * q^32 + 7 * q^33 + 8 * q^34 - 8 * q^36 - 6 * q^37 - 13 * q^38 + 11 * q^39 - 22 * q^41 - 22 * q^43 + 7 * q^45 + 7 * q^46 - 8 * q^47 - 2 * q^48 + 2 * q^50 - 6 * q^51 + 9 * q^52 + 3 * q^53 + 2 * q^54 - 42 * q^55 + 18 * q^57 + 3 * q^58 - 4 * q^59 + 31 * q^61 + 32 * q^62 + 6 * q^64 + 14 * q^65 - 7 * q^66 - 23 * q^67 + 4 * q^68 - 28 * q^69 + 14 * q^71 - 4 * q^72 + 27 * q^73 + 6 * q^74 + 2 * q^75 - 26 * q^76 + 22 * q^78 + 32 * q^79 + 5 * q^81 - 11 * q^82 + 22 * q^83 - 56 * q^85 - 11 * q^86 - 6 * q^87 + 10 * q^89 + 14 * q^90 + 14 * q^92 - 17 * q^93 + 8 * q^94 - 14 * q^95 - q^96 - 10 * q^97 + 28 * q^99

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13$ (-v^5 + 3v^4 - 9v^3 + 5v^2 - 2v + 6) / 13
$\beta_{3}$	$=$	$( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13$ (-3v^5 + 9v^4 - 14v^3 + 15v^2 - 6*v + 18) / 13
$\beta_{4}$	$=$	$( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13$ (-4v^5 - v^4 - 10v^3 - 6v^2 - 34v - 2) / 13
$\beta_{5}$	$=$	$( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13$ (-6v^5 + 5v^4 - 15v^3 - 9v^2 - 25*v + 10) / 13

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	686.2.c.d

Level	$686$
Weight	$2$
Character orbit	686.c
Analytic conductor	$5.478$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$-\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1$ -b5 + b4 + b3 - b2 + b1
$\nu^{3}$	$=$	$\beta_{3} - 3\beta_{2}$ b3 - 3*b2
$\nu^{4}$	$=$	$2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2$ 2b5 - 3b4 - 4*b1 - 2
$\nu^{5}$	$=$	$\beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1$ b5 - 4b4 - 4b3 + 9b2 - 9b1

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 361.3
Significant digits:
Format: