LMFDB - Newform orbit 666.2.bj.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [666,2,Mod(289,666)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(666, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 7]))

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

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

chi := DirichletCharacter("666.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 666 = 2 \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	666.bj (of order $18$, degree $6$, 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:	$5.31803677462$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{18})$
Coefficient field:	$\Q(\zeta_{36})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{6} + 1 $ x^12 - x^6 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/666\mathbb{Z}\right)^\times$.

$n$	$371$	$631$
$\chi(n)$	$1$	$\zeta_{36}^{2}$

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} $

289.1

−0.342020	−	0.939693i
0.342020	+	0.939693i
−0.984808	+	0.173648i
0.984808	−	0.173648i
−0.642788	−	0.766044i
0.642788	+	0.766044i
−0.984808	−	0.173648i
0.984808	+	0.173648i
−0.642788	+	0.766044i
0.642788	−	0.766044i
−0.342020	+	0.939693i
0.342020	−	0.939693i

−0.342020

−

0.939693i

−0.766044

0.642788i

1.16679

−

0.205737i

0.120615

0.684040i

0.866025

0.500000i

−0.592396

−

1.02606i

289.2

0.342020

0.939693i

−0.766044

0.642788i

−1.16679

0.205737i

0.120615

0.684040i

−0.866025

−

0.500000i

−0.592396

−

1.02606i

361.1

−0.984808

0.173648i

0.939693

−

0.342020i

−2.19285

2.61334i

2.34730

1.96962i

−0.866025

0.500000i

1.70574

−

2.95442i

361.2

0.984808

−

0.173648i

0.939693

−

0.342020i

2.19285

−

2.61334i

2.34730

1.96962i

0.866025

−

0.500000i

1.70574

−

2.95442i

469.1

−0.642788

−

0.766044i

−0.173648

0.984808i

−0.761570

−

2.09240i

3.53209

−

1.28558i

0.866025

−

0.500000i

−1.11334

1.92836i

469.2

0.642788

0.766044i

−0.173648

0.984808i

0.761570

2.09240i

3.53209

−

1.28558i

−0.866025

0.500000i

−1.11334

1.92836i

559.1

−0.984808

−

0.173648i

0.939693

0.342020i

−2.19285

−

2.61334i

2.34730

−

1.96962i

−0.866025

−

0.500000i

1.70574

2.95442i

559.2

0.984808

0.173648i

0.939693

0.342020i

2.19285

2.61334i

2.34730

−

1.96962i

0.866025

0.500000i

1.70574

2.95442i

595.1

−0.642788

0.766044i

−0.173648

−

0.984808i

−0.761570

2.09240i

3.53209

1.28558i

0.866025

0.500000i

−1.11334

−

1.92836i

595.2

0.642788

−

0.766044i

−0.173648

−

0.984808i

0.761570

−

2.09240i

3.53209

1.28558i

−0.866025

−

0.500000i

−1.11334

−

1.92836i

613.1

−0.342020

0.939693i

−0.766044

−

0.642788i

1.16679

0.205737i

0.120615

−

0.684040i

0.866025

−

0.500000i

−0.592396

1.02606i

613.2

0.342020

−

0.939693i

−0.766044

−

0.642788i

−1.16679

−

0.205737i

0.120615

−

0.684040i

−0.866025

0.500000i

−0.592396

1.02606i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
37.h	even	18	1	inner
111.n	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	666.2.bj.d	✓	12
3.b	odd	2	1	inner	666.2.bj.d	✓	12
37.h	even	18	1	inner	666.2.bj.d	✓	12
111.n	odd	18	1	inner	666.2.bj.d	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
666.2.bj.d	✓	12	1.a	even	1	1	trivial
666.2.bj.d	✓	12	3.b	odd	2	1	inner
666.2.bj.d	✓	12	37.h	even	18	1	inner
666.2.bj.d	✓	12	111.n	odd	18	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{12} + 9T_{5}^{10} + 162T_{5}^{8} + 648T_{5}^{6} + 729T_{5}^{4} - 6561T_{5}^{2} + 6561 $ T5^12 + 9*T5^10 + 162*T5^8 + 648*T5^6 + 729*T5^4 - 6561*T5^2 + 6561 acting on $S_{2}^{\mathrm{new}}(666, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} - T^{6} + 1 $ T^12 - T^6 + 1
$3$	$ T^{12} $ T^12
$5$	$ T^{12} + 9 T^{10} + \cdots + 6561 $ T^12 + 9T^10 + 162T^8 + 648T^6 + 729T^4 - 6561*T^2 + 6561
$7$	$ (T^{6} - 12 T^{5} + \cdots + 64)^{2} $ (T^6 - 12T^5 + 60T^4 - 152T^3 + 192T^2 - 96*T + 64)^2
$11$	$ T^{12} $ T^12
$13$	$ (T^{6} - 12 T^{5} + \cdots + 867)^{2} $ (T^6 - 12T^5 + 78T^4 - 303T^3 + 774T^2 - 1224*T + 867)^2
$17$	$ T^{12} + 9 T^{10} + \cdots + 6561 $ T^12 + 9T^10 + 162T^8 + 648T^6 + 729T^4 - 6561*T^2 + 6561
$19$	$ (T^{6} + 18 T^{5} + \cdots + 1728)^{2} $ (T^6 + 18T^5 + 144T^4 + 720T^3 + 2160T^2 + 2592*T + 1728)^2
$23$	$ T^{12} - 72 T^{10} + \cdots + 26873856 $ T^12 - 72T^10 + 3888T^8 - 82944T^6 + 1306368T^4 - 6718464*T^2 + 26873856
$29$	$ T^{12} + \cdots + 547981281 $ T^12 - 126T^10 + 11583T^8 - 494100T^6 + 15480315T^4 - 100494837*T^2 + 547981281
$31$	$ (T^{6} + 24 T^{4} + \cdots + 192)^{2} $ (T^6 + 24T^4 + 144T^2 + 192)^2
$37$	$ (T^{6} + 12 T^{5} + \cdots + 50653)^{2} $ (T^6 + 12T^5 - 30T^4 - 803T^3 - 1110T^2 + 16428*T + 50653)^2
$41$	$ T^{12} + \cdots + 387420489 $ T^12 + 81T^10 - 4374T^8 - 157464T^6 + 36669429T^4 - 43046721*T^2 + 387420489
$43$	$ (T^{6} + 96 T^{4} + \cdots + 12288)^{2} $ (T^6 + 96T^4 + 2304T^2 + 12288)^2
$47$	$ T^{12} + \cdots + 241864704 $ T^12 + 108T^10 + 9072T^8 + 248832T^6 + 5038848T^4 + 40310784*T^2 + 241864704
$53$	$ T^{12} + 108 T^{10} + \cdots + 59049 $ T^12 + 108T^10 + 2268T^8 - 69741T^6 + 761076T^4 - 78732*T^2 + 59049
$59$	$ T^{12} + 36 T^{10} + \cdots + 26873856 $ T^12 + 36T^10 + 25920T^8 - 565056T^6 + 2426112T^4 + 13436928*T^2 + 26873856
$61$	$ (T^{6} + 45 T^{4} + \cdots + 75843)^{2} $ (T^6 + 45T^4 + 801T^3 - 378T^2 - 4293T + 75843)^2
$67$	$ (T^{6} + 6 T^{5} + \cdots + 23104)^{2} $ (T^6 + 6T^5 - 12T^4 - 512T^3 + 1920T^2 - 1824*T + 23104)^2
$71$	$ T^{12} + \cdots + 19591041024 $ T^12 - 108T^10 + 23328T^8 - 1119744T^6 + 15116544T^4 + 1632586752*T^2 + 19591041024
$73$	$ (T^{3} - 9 T^{2} - 21 T + 53)^{4} $ (T^3 - 9T^2 - 21T + 53)^4
$79$	$ (T^{6} + 6 T^{5} + \cdots + 192)^{2} $ (T^6 + 6T^5 + 24T^4 - 168T^3 + 144T^2 + 192)^2
$83$	$ T^{12} + \cdots + 241864704 $ T^12 + 432T^10 + 36288T^8 - 4463424T^6 + 194835456T^4 - 80621568*T^2 + 241864704
$89$	$ T^{12} + \cdots + 44386483761 $ T^12 + 54T^10 - 24543T^8 - 6089337T^6 + 1099649844T^4 + 2201405769*T^2 + 44386483761
$97$	$ (T^{6} + 36 T^{5} + \cdots + 1083)^{2} $ (T^6 + 36T^5 + 537T^4 + 3780T^3 + 11709T^2 + 5985*T + 1083)^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:	\( N \)	\(=\)	\( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	666.bj (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{36} q^{2} + \zeta_{36}^{2} q^{4} + (\zeta_{36}^{9} + \zeta_{36}^{7} + \cdots + \zeta_{36}) q^{5}+ \cdots + \zeta_{36}^{3} q^{8}+O(q^{10}) \) q + z * q^2 + z^2 * q^4 + (z^9 + z^7 + z^3 + z) * q^5 + (-2z^10 + 2) q^7 + z^3 * q^8	\( q + \zeta_{36} q^{2} + \zeta_{36}^{2} q^{4} + (\zeta_{36}^{9} + \zeta_{36}^{7} + \cdots + \zeta_{36}) q^{5}+ \cdots + ( - \zeta_{36}^{11} + \cdots + 4 \zeta_{36}) q^{98}+O(q^{100}) \) q + z * q^2 + z^2 * q^4 + (z^9 + z^7 + z^3 + z) * q^5 + (-2z^10 + 2) q^7 + z^3 * q^8 + (z^10 + z^8 + z^4 + z^2) * q^10 + (-2z^8 - 2z^4 + z^2 + 2) * q^13 + (-2z^11 + 2z) * q^14 + z^4 * q^16 + (z^11 - 2z^9 - 2z^5 + z^3) * q^17 + (2z^10 + 2z^6 - 4z^4 - 4) q^19 + (z^11 + z^9 + z^5 + z^3) * q^20 + (2z^11 + 4z^7 - 4z^5 - 2z) * q^23 + (z^10 + 3z^8 + 3z^6 - 3) * q^25 + (-2z^9 - 2z^5 + z^3 + 2z) q^26 + (-2z^6 + 2z^2 + 2) * q^28 + (-2z^11 + 2z^7 + z^5 + 6z^3 - z) q^29 + (2z^8 + 2z^4 - 2z^2) q^31 + z^5 * q^32 + (-2z^10 - z^6 + z^4 - 1) q^34 + (-4z^11 + 2z^9 + 2z^5 + 2z^3 + 6z) q^35 + (4z^10 - 4z^8 - 2z^4 - z^2 - 2) q^37 + (2z^11 + 2z^7 - 4z^5 - 4z) * q^38 + (z^10 + 2z^6 + z^4 - 1) q^40 + (3z^9 - 3z^7 + 3z^3 + 6z) * q^41 + (-4z^10 - 4z^8) * q^43 + (4z^8 - 2z^6 - 2z^2 - 2) q^46 + (2z^11 - 2z^9 - 4z^7 - 4z^5 - 2z^3 + 2z) * q^47 + (-z^10 - 4z^2 + 4) q^49 + (z^11 + 3z^9 + 3z^7 - 3z) q^50 + (-2z^10 - 2z^6 + z^4 + 2z^2) q^52 + (4z^11 - 4z^9 + z^7 - 2z^5 + 2z^3 + z) * q^53 + (-2z^7 + 2z^3 + 2z) q^56 + (2z^8 - z^6 + 6z^4 - z^2 + 2) * q^58 + (4z^11 + 4z^9 + 6z^7 - 2z^5 - 2z^3 - 6z) * q^59 + (-2z^10 - 8z^8 - 2z^6 + z^4 + 4z^2 + 1) * q^61 + (2z^9 + 2z^5 - 2z^3) q^62 + z^6 * q^64 + (-5z^11 - z^9 - 2z^7 + z^5 + 5z^3 + 4z) * q^65 + (2z^10 + 6z^8 + 2z^6 - 2z^4 - 6z^2 - 2) q^67 + (-2z^11 - z^7 + z^5 - z) q^68 + (2z^10 - 2z^6 + 2z^4 + 6z^2 + 4) * q^70 + (6z^9 - 6z^3 + 6z) q^71 + (4z^8 - 4z^4 - 4z^2 + 3) q^73 + (4z^11 - 4z^9 - 2z^5 - z^3 - 2z) * q^74 + (2z^8 - 2z^6 - 4z^2 - 2) q^76 + (-2z^8 - 2z^6 - 2z^4 - 2z^2) * q^79 + (z^11 + 2z^7 + z^5 - z) q^80 + (3z^10 - 3z^8 + 3z^4 + 6z^2) * q^82 + (-4z^11 - 8z^9 + 2z^7 + 8z^5 + 4z^3 - 4z) * q^83 + (-3z^10 - 3z^8 - 3z^6 + 3z^4 + 3) * q^85 + (-4z^11 - 4z^9) * q^86 + (-3z^11 - 3z^9 + 9z^5 - 6z^3) * q^89 + (-4z^10 - 2z^6 - 4z^4 - 2z^2 + 2) * q^91 + (4z^9 - 2z^7 - 2z^3 - 2z) * q^92 + (-2z^10 - 4z^8 - 2z^6 - 2z^4 + 2z^2 - 2) q^94 + (-6z^7 - 6z^5 - 6z^3 - 6z) * q^95 + (4z^10 + z^8 + 4z^6 - z^4 - 4z^2 - 8) q^97 + (-z^11 - 4z^3 + 4z) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 24 q^{7}+O(q^{10}) \) 12 * q + 24 * q^7	\( 12 q + 24 q^{7} + 24 q^{13} - 36 q^{19} - 18 q^{25} + 12 q^{28} - 18 q^{34} - 24 q^{37} - 36 q^{46} + 48 q^{49} - 12 q^{52} + 18 q^{58} + 6 q^{64} - 12 q^{67} + 36 q^{70} + 36 q^{73} - 36 q^{76} - 12 q^{79} + 18 q^{85} + 12 q^{91} - 36 q^{94} - 72 q^{97}+O(q^{100}) \) 12 * q + 24 * q^7 + 24 * q^13 - 36 * q^19 - 18 * q^25 + 12 * q^28 - 18 * q^34 - 24 * q^37 - 36 * q^46 + 48 * q^49 - 12 * q^52 + 18 * q^58 + 6 * q^64 - 12 * q^67 + 36 * q^70 + 36 * q^73 - 36 * q^76 - 12 * q^79 + 18 * q^85 + 12 * q^91 - 36 * q^94 - 72 * q^97

Introduction

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Varieties

Fields

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Properties

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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	666.2.bj.d

Level	$666$
Weight	$2$
Character orbit	666.bj
Analytic conductor	$5.318$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(5.31803677462\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{18})\)
Coefficient field:	\(\Q(\zeta_{36})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{6} + 1 \) x^12 - x^6 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 289.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} - T^{6} + 1 \) T^12 - T^6 + 1
$3$	\( T^{12} \) T^12
$5$	\( T^{12} + 9 T^{10} + \cdots + 6561 \) T^12 + 9T^10 + 162T^8 + 648T^6 + 729T^4 - 6561*T^2 + 6561
$7$	\( (T^{6} - 12 T^{5} + \cdots + 64)^{2} \) (T^6 - 12T^5 + 60T^4 - 152T^3 + 192T^2 - 96*T + 64)^2
$11$	\( T^{12} \) T^12
$13$	\( (T^{6} - 12 T^{5} + \cdots + 867)^{2} \) (T^6 - 12T^5 + 78T^4 - 303T^3 + 774T^2 - 1224*T + 867)^2
$17$	\( T^{12} + 9 T^{10} + \cdots + 6561 \) T^12 + 9T^10 + 162T^8 + 648T^6 + 729T^4 - 6561*T^2 + 6561
$19$	\( (T^{6} + 18 T^{5} + \cdots + 1728)^{2} \) (T^6 + 18T^5 + 144T^4 + 720T^3 + 2160T^2 + 2592*T + 1728)^2
$23$	\( T^{12} - 72 T^{10} + \cdots + 26873856 \) T^12 - 72T^10 + 3888T^8 - 82944T^6 + 1306368T^4 - 6718464*T^2 + 26873856
$29$	\( T^{12} + \cdots + 547981281 \) T^12 - 126T^10 + 11583T^8 - 494100T^6 + 15480315T^4 - 100494837*T^2 + 547981281
$31$	\( (T^{6} + 24 T^{4} + \cdots + 192)^{2} \) (T^6 + 24T^4 + 144T^2 + 192)^2
$37$	\( (T^{6} + 12 T^{5} + \cdots + 50653)^{2} \) (T^6 + 12T^5 - 30T^4 - 803T^3 - 1110T^2 + 16428*T + 50653)^2
$41$	\( T^{12} + \cdots + 387420489 \) T^12 + 81T^10 - 4374T^8 - 157464T^6 + 36669429T^4 - 43046721*T^2 + 387420489
$43$	\( (T^{6} + 96 T^{4} + \cdots + 12288)^{2} \) (T^6 + 96T^4 + 2304T^2 + 12288)^2
$47$	\( T^{12} + \cdots + 241864704 \) T^12 + 108T^10 + 9072T^8 + 248832T^6 + 5038848T^4 + 40310784*T^2 + 241864704
$53$	\( T^{12} + 108 T^{10} + \cdots + 59049 \) T^12 + 108T^10 + 2268T^8 - 69741T^6 + 761076T^4 - 78732*T^2 + 59049
$59$	\( T^{12} + 36 T^{10} + \cdots + 26873856 \) T^12 + 36T^10 + 25920T^8 - 565056T^6 + 2426112T^4 + 13436928*T^2 + 26873856
$61$	\( (T^{6} + 45 T^{4} + \cdots + 75843)^{2} \) (T^6 + 45T^4 + 801T^3 - 378T^2 - 4293T + 75843)^2
$67$	\( (T^{6} + 6 T^{5} + \cdots + 23104)^{2} \) (T^6 + 6T^5 - 12T^4 - 512T^3 + 1920T^2 - 1824*T + 23104)^2
$71$	\( T^{12} + \cdots + 19591041024 \) T^12 - 108T^10 + 23328T^8 - 1119744T^6 + 15116544T^4 + 1632586752*T^2 + 19591041024
$73$	\( (T^{3} - 9 T^{2} - 21 T + 53)^{4} \) (T^3 - 9T^2 - 21T + 53)^4
$79$	\( (T^{6} + 6 T^{5} + \cdots + 192)^{2} \) (T^6 + 6T^5 + 24T^4 - 168T^3 + 144T^2 + 192)^2
$83$	\( T^{12} + \cdots + 241864704 \) T^12 + 432T^10 + 36288T^8 - 4463424T^6 + 194835456T^4 - 80621568*T^2 + 241864704
$89$	\( T^{12} + \cdots + 44386483761 \) T^12 + 54T^10 - 24543T^8 - 6089337T^6 + 1099649844T^4 + 2201405769*T^2 + 44386483761
$97$	\( (T^{6} + 36 T^{5} + \cdots + 1083)^{2} \) (T^6 + 36T^5 + 537T^4 + 3780T^3 + 11709T^2 + 5985*T + 1083)^2
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\(n\)	\(371\)	\(631\)
\(\chi(n)\)	\(1\)	\(\zeta_{36}^{2}\)