LMFDB - Newform orbit 666.2.bj.a

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

−0.787884

0.138925i

−0.625565

−

3.54776i

0.866025

0.500000i

0.400019

0.692853i

289.2

0.342020

0.939693i

−0.766044

0.642788i

4.19936

−

0.740460i

0.504951

2.86372i

−0.866025

−

0.500000i

2.13207

3.69285i

361.1

−0.984808

0.173648i

0.939693

−

0.342020i

−0.548083

0.653180i

−1.63445

−

1.37147i

−0.866025

0.500000i

0.426333

−

0.738430i

361.2

0.984808

−

0.173648i

0.939693

−

0.342020i

−1.67860

2.00048i

−0.712846

−

0.598149i

0.866025

−

0.500000i

−1.30572

2.26157i

469.1

−0.642788

−

0.766044i

−0.173648

0.984808i

−0.131594

−

0.361551i

−4.25967

1.55039i

0.866025

−

0.500000i

−0.192377

0.333207i

469.2

0.642788

0.766044i

−0.173648

0.984808i

−1.05320

−

2.89364i

0.727576

−

0.264816i

−0.866025

0.500000i

1.53967

−

2.66679i

559.1

−0.984808

−

0.173648i

0.939693

0.342020i

−0.548083

−

0.653180i

−1.63445

1.37147i

−0.866025

−

0.500000i

0.426333

0.738430i

559.2

0.984808

0.173648i

0.939693

0.342020i

−1.67860

−

2.00048i

−0.712846

0.598149i

0.866025

0.500000i

−1.30572

−

2.26157i

595.1

−0.642788

0.766044i

−0.173648

−

0.984808i

−0.131594

0.361551i

−4.25967

−

1.55039i

0.866025

0.500000i

−0.192377

−

0.333207i

595.2

0.642788

−

0.766044i

−0.173648

−

0.984808i

−1.05320

2.89364i

0.727576

0.264816i

−0.866025

−

0.500000i

1.53967

2.66679i

613.1

−0.342020

0.939693i

−0.766044

−

0.642788i

−0.787884

−

0.138925i

−0.625565

3.54776i

0.866025

−

0.500000i

0.400019

−

0.692853i

613.2

0.342020

−

0.939693i

−0.766044

−

0.642788i

4.19936

0.740460i

0.504951

−

2.86372i

−0.866025

0.500000i

2.13207

−

3.69285i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
37.h	even	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.a	✓	12
3.b	odd	2	1		666.2.bj.b	yes	12
37.h	even	18	1	inner	666.2.bj.a	✓	12
111.n	odd	18	1		666.2.bj.b	yes	12

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

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{12} - 9 T_{5}^{10} - 72 T_{5}^{9} - 72 T_{5}^{8} + 378 T_{5}^{7} + 2286 T_{5}^{6} + 4860 T_{5}^{5} + \cdots + 81 $ T5^12 - 9*T5^10 - 72*T5^9 - 72*T5^8 + 378*T5^7 + 2286*T5^6 + 4860*T5^5 + 5670*T5^4 + 4050*T5^3 + 1782*T5^2 + 486*T5 + 81 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 + 81 $ T^12 - 9T^10 - 72T^9 - 72T^8 + 378T^7 + 2286T^6 + 4860T^5 + 5670T^4 + 4050T^3 + 1782T^2 + 486T + 81
$7$	$ T^{12} + 12 T^{11} + \cdots + 5329 $ T^12 + 12T^11 + 75T^10 + 344T^9 + 1206T^8 + 3042T^7 + 6456T^6 + 8820T^5 + 4716T^4 - 8530T^3 - 7014T^2 + 1752*T + 5329
$11$	$ T^{12} + 6 T^{11} + \cdots + 2653641 $ T^12 + 6T^11 + 72T^10 + 300T^9 + 2889T^8 + 11232T^7 + 59778T^6 + 122472T^5 + 422901T^4 + 777384T^3 + 2044521T^2 + 2287116*T + 2653641
$13$	$ T^{12} + 12 T^{11} + \cdots + 12321 $ T^12 + 12T^11 + 30T^10 - 66T^9 - 288T^8 - 1098T^7 + 672T^6 + 2682T^5 + 8838T^4 + 10116T^3 + 20655T^2 + 17982*T + 12321
$17$	$ T^{12} - 18 T^{10} + \cdots + 17665209 $ T^12 - 18T^10 - 414T^9 - 1368T^8 + 15606T^7 + 43839T^6 - 40014T^5 + 566676T^4 + 2621322T^3 + 10262214T^2 + 15433416T + 17665209
$19$	$ T^{12} - 9 T^{10} + \cdots + 431649 $ T^12 - 9T^10 - 36T^9 - 18T^8 - 594T^7 - 2088T^6 + 15552T^5 + 34344T^4 + 47466T^3 + 460566T^2 + 567648T + 431649
$23$	$ T^{12} + \cdots + 212838921 $ T^12 - 90T^10 + 5787T^8 - 1512T^7 - 178560T^6 + 83268T^5 + 4016223T^4 - 3996864T^3 - 32749029T^2 + 25209792*T + 212838921
$29$	$ T^{12} + \cdots + 121154049 $ T^12 - 36T^11 + 549T^10 - 4212T^9 + 13347T^8 + 23490T^7 - 248328T^6 - 89748T^5 + 3902985T^4 - 4009500T^3 - 23707242T^2 + 19614474*T + 121154049
$31$	$ T^{12} + 210 T^{10} + \cdots + 25796241 $ T^12 + 210T^10 + 15741T^8 + 517002T^6 + 7260498T^4 + 32560920*T^2 + 25796241
$37$	$ T^{12} + \cdots + 2565726409 $ T^12 + 6T^11 + 30T^10 + 266T^9 - 954T^8 - 15066T^7 - 53697T^6 - 557442T^5 - 1306026T^4 + 13473698T^3 + 56224830T^2 + 416063742*T + 2565726409
$41$	$ T^{12} + \cdots + 1768034304 $ T^12 + 12T^11 + 72T^10 - 336T^9 - 4752T^8 - 38016T^7 + 143424T^6 + 3048192T^5 + 29279232T^4 + 159307776T^3 + 804639744T^2 + 1961791488*T + 1768034304
$43$	$ T^{12} + \cdots + 504496521 $ T^12 + 390T^10 + 55809T^8 + 3516270T^6 + 89302374T^4 + 556265124*T^2 + 504496521
$47$	$ T^{12} + \cdots + 13231670841 $ T^12 + 12T^11 + 279T^10 + 2148T^9 + 39987T^8 + 289008T^7 + 3234060T^6 + 14628924T^5 + 119811879T^4 + 479935584T^3 + 2924606250T^2 + 6323374188*T + 13231670841
$53$	$ T^{12} + \cdots + 1998358209 $ T^12 + 48T^11 + 927T^10 + 8952T^9 + 46764T^8 + 178254T^7 + 884250T^6 + 3227040T^5 + 8580978T^4 + 35976042T^3 + 300918726T^2 - 98972442*T + 1998358209
$59$	$ T^{12} + 36 T^{11} + \cdots + 927369 $ T^12 + 36T^11 + 585T^10 + 5778T^9 + 39132T^8 + 191970T^7 + 700155T^6 + 1905444T^5 + 3797037T^4 + 5304690T^3 + 4787424T^2 + 2496096*T + 927369
$61$	$ T^{12} - 36 T^{11} + \cdots + 46416969 $ T^12 - 36T^11 + 513T^10 - 5130T^9 + 58437T^8 - 495072T^7 + 1974852T^6 - 3775086T^5 + 7533000T^4 + 27046872T^3 + 140330637T^2 + 103748364*T + 46416969
$67$	$ T^{12} + \cdots + 254306809 $ T^12 - 42T^11 + 786T^10 - 8062T^9 + 47610T^8 - 168426T^7 + 474042T^6 - 1538154T^5 + 5453460T^4 - 22318144T^3 + 74848155T^2 - 125247738*T + 254306809
$71$	$ T^{12} - 12 T^{11} + \cdots + 331776 $ T^12 - 12T^11 - 36T^10 + 336T^9 + 3456T^8 + 53568T^7 + 200448T^6 - 1078272T^5 + 5204736T^4 - 6027264T^3 + 45702144T^2 + 5971968*T + 331776
$73$	$ (T^{6} - 96 T^{4} + \cdots + 11584)^{2} $ (T^6 - 96T^4 - 272T^3 + 1872T^2 + 9600T + 11584)^2
$79$	$ T^{12} + \cdots + 359276763609 $ T^12 - 42T^11 + 732T^10 - 7734T^9 + 122139T^8 - 2842560T^7 + 37810905T^6 - 236835594T^5 + 1087001244T^4 - 8691622956T^3 + 32853372741T^2 + 409987548*T + 359276763609
$83$	$ T^{12} - 6 T^{11} + \cdots + 110889 $ T^12 - 6T^11 + 9T^10 + 384T^9 + 3105T^8 + 20034T^7 + 55512T^6 - 18630T^5 - 23004T^4 + 244512T^3 + 341577T^2 + 161838*T + 110889
$89$	$ T^{12} + \cdots + 86755578849 $ T^12 - 36T^11 + 729T^10 - 10710T^9 + 125145T^8 - 1240920T^7 + 10297260T^6 - 66627198T^5 + 311854536T^4 - 1137874716T^3 + 5829944949T^2 - 33973767792*T + 86755578849
$97$	$ T^{12} + \cdots + 241771401 $ T^12 - 18T^11 + 39T^10 + 1242T^9 - 4005T^8 - 66330T^7 + 328524T^6 + 1424898T^5 - 6753285T^4 - 25696440T^3 + 93026286T^2 + 288278460*T + 241771401
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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)

\(f(q)\)	\(=\)	\( q + \zeta_{36} q^{2} + \zeta_{36}^{2} q^{4} + ( - \zeta_{36}^{8} + \cdots - \zeta_{36}^{2}) q^{5}+ \cdots + \zeta_{36}^{3} q^{8}+O(q^{10}) \) q + z * q^2 + z^2 * q^4 + (-z^8 - z^7 + z^5 - z^3 - z^2) * q^5 + (z^10 - z^9 + z^7 - z^3 + z - 1) * q^7 + z^3 * q^8	\( q + \zeta_{36} q^{2} + \zeta_{36}^{2} q^{4} + ( - \zeta_{36}^{8} + \cdots - \zeta_{36}^{2}) q^{5}+ \cdots + ( - \zeta_{36}^{11} + 2 \zeta_{36}^{10} + \cdots - 4) q^{98}+O(q^{100}) \) q + z * q^2 + z^2 * q^4 + (-z^8 - z^7 + z^5 - z^3 - z^2) * q^5 + (z^10 - z^9 + z^7 - z^3 + z - 1) * q^7 + z^3 * q^8 + (-z^9 - z^8 + z^6 - z^4 - z^3) * q^10 + (-3z^10 - 2z^9 - 3z^8 + z^6 + z^4 + z^3 + 2z^2 - 1) * q^11 + (z^11 - 2z^9 - 2z^7 + z^4 + 2z^3 - 1) q^13 + (z^11 - z^10 + z^8 - z^4 + z^2 - z) * q^14 + z^4 * q^16 + (2z^11 - z^10 + z^9 - z^8 - 2z^7 - 2z^6 + z^5 - z^4 + 2z^3 - z^2 + 1) * q^17 + (2z^11 + z^10 + z^9 - 2z^6 - 2z^5 + z^4 - z^3 - z + 1) q^19 + (-z^10 - z^9 + z^7 - z^5 - z^4) * q^20 + (-3z^11 - 2z^10 - 3z^9 + z^7 + z^5 + z^4 + 2z^3 - z) * q^22 + (-z^11 - 2z^10 - 2z^9 + 2z^8 - z^7 + z^5 - 2z^4 + 2z^3 + 2z^2 + z) * q^23 + (2z^11 + z^10 + 4z^9 - z^8 - 4z^7 - z^6 + 2z^5 - 2z^3 - z^2 + 2z + 2) * q^25 + (-2z^10 - 2z^8 + z^6 + z^5 + 2z^4 - z - 1) q^26 + (-z^11 + z^9 + z^6 - z^5 + z^3 - z^2 - 1) * q^28 + (z^11 - z^10 + z^8 - z^7 + 2z^6 + 2z^5 - z^4 - z^3 - 2z^2 + 3z + 2) * q^29 + (-z^11 - z^8 + z^7 - 6z^6 + 2z^5 - z^4 + z^2 - 2z + 3) q^31 + z^5 * q^32 + (-z^11 + z^10 - z^9 - 2z^8 - 2z^7 + 3z^6 - z^5 + 2z^4 - z^3 + z - 2) * q^34 + (2z^11 + z^10 - 2z^9 - 3z^8 + 3z^6 - 2z^4 + 3z^2 + z) * q^35 + (-2z^11 - 3z^9 + z^8 - z^7 - z^6 + 2z^5 - 3z^4 - z^3 + z^2 + 3z) q^37 + (z^11 + z^10 - 2z^7 + z^5 - z^4 - z^2 + z - 2) q^38 + (-z^11 - z^10 + z^8 - z^6 - z^5) * q^40 + (-2z^10 - 2z^9 + 2z^7 + 2z^6 - 2z^3 + 4z^2 - 4z - 2) q^41 + (-z^11 - z^10 + 3z^9 + 2z^8 + 4z^7 + 5z^5 + 3z^4 - 3z^2 - 5z) q^43 + (-2z^11 - 3z^10 + z^8 - 2z^6 + z^5 + 2z^4 - z^2 + 3) * q^44 + (-2z^11 - 2z^10 + 2z^9 - z^8 - 2z^5 + 2z^4 + 2z^3 + z^2 + 1) * q^46 + (z^11 - 6z^10 - z^9 + 2z^8 + z^7 - 2z^6 + z^5 + 2z^4 - z^3 - 6z^2 + z) q^47 + (4z^11 - z^10 + 2z^9 + 3z^8 - 4z^7 + 3z^6 - 2z^5 + 2z^3 - z^2 + 2z - 2) * q^49 + (z^11 + 4z^10 - z^9 - 4z^8 - z^7 + 4z^6 - 2z^4 - z^3 + 2z^2 + 2z - 2) * q^50 + (-2z^11 - 2z^9 + z^7 + z^6 + 2z^5 - z^2 - z) q^52 + (-z^11 + z^9 - 4z^8 + z^7 + 2z^5 + 5z^4 + z^3 + z - 4) q^53 + (9z^11 + 5z^10 - 2z^9 - 2z^8 + z^7 + 2z^6 - 9z^5 - 7z^4 + z^3 + 4z^2 + z - 7) * q^55 + (z^10 + z^7 - 2z^6 + z^4 - z^3 - z + 1) q^56 + (-z^11 + z^9 - z^8 + 2z^7 + 3z^6 - z^5 - z^4 - 2z^3 + 3z^2 + 2z - 1) q^58 + (z^11 + z^9 + 3z^8 - z^7 - z^3 + z - 3) q^59 + (2z^11 + 3z^9 + 6z^8 + 3z^7 - 3z^4 - z^3 - 3z^2 - 2z + 3) q^61 + (-z^9 + z^8 - 6z^7 + z^6 - z^5 + z^3 - 2z^2 + 3z + 1) q^62 + z^6 * q^64 + (z^11 + 3z^10 + 5z^9 + 2z^8 - z^7 - 4z^6 - 5z^5 - 3z^4 - z^3 + 2z^2 + 2z + 2) * q^65 + (2z^11 - 3z^10 + 3z^9 + 3z^7 + z^6 - z^5 - z^4 - 3z^3 - 3z + 3) * q^67 + (z^11 - z^10 - 2z^9 - 2z^8 + 3z^7 - 2z^6 + 2z^5 - z^4 + z^2 - 2z + 1) * q^68 + (z^11 - 2z^10 - 3z^9 + 3z^7 + 2z^6 - 2z^5 + 3z^3 + z^2 - 2) * q^70 + (-2z^11 - 2z^9 - 2z^8 + 2z^7 - 2z^6 - 2z^5 - 2z^4 + 4z^3 - 4z + 2) q^71 + (-4z^11 - 2z^10 + 2z^8 + 2z^7 + 2z^5 + 2z) * q^73 + (-3z^10 + z^9 - z^8 - z^7 - 3z^5 - z^4 + z^3 + 3z^2 + 2) q^74 + (z^11 - 2z^8 + 2z^6 - z^5 - z^3 + z^2 - 2z - 1) q^76 + (-z^11 - 3z^9 + 4z^8 + 3z^7 + z^6 - z^5 + z^4 + 6z^3 - 6z - 1) q^77 + (3z^10 + z^9 - 3z^8 + 5z^7 + z^6 + 5z^5 + z^4 + 5z^3 - 3z^2 + z + 3) * q^79 + (-z^11 + z^9 - z^7 - 2z^6 + 1) q^80 + (-2z^11 - 2z^10 + 2z^8 + 2z^7 - 2z^4 + 4z^3 - 4z^2 - 2z) * q^82 + (2z^11 + 2z^10 + z^9 - 2z^8 - z^7 + 3z^6 - z^5 - 2z^4 - 2z^3 - z^2 + 2z - 1) q^83 + (-3z^11 + 5z^10 + 6z^9 + 5z^8 - 3z^7 - 3z^6 - z^4 - 3z^3 - 4z^2 + 3z + 3) q^85 + (-z^11 + 3z^10 + 2z^9 + 4z^8 + 4z^6 + 3z^5 - 3z^3 - 5z^2 + 1) q^86 + (-3z^11 + z^9 - 2z^7 - z^6 + 2z^5 - z^3 + 3z + 2) * q^88 + (2z^10 - 3z^8 - 6z^7 - 3z^5 - z^4 + 3z^3 + 6z + 3) * q^89 + (-2z^11 + z^10 + 3z^9 - 4z^8 + z^7 - z^6 + 3z^5 + z^4 - 2z^3 + 3z^2 - 3) * q^91 + (-2z^11 + 2z^10 - z^9 - 4z^6 + 2z^5 + 2z^4 + z^3 + z + 2) q^92 + (-6z^11 - z^10 + 2z^9 + z^8 - 2z^7 + 2z^6 + 2z^5 - z^4 - 6z^3 + z^2 - 1) * q^94 + (-5z^11 + 5z^9 + 4z^8 + z^7 - 2z^6 + 4z^5 - z^3 - 2z^2 + z + 4) * q^95 + (-2z^11 - 3z^10 + 2z^9 - 4z^8 - 2z^7 - z^6 + 2z^5 + 4z^4 - 2z^3 + 3z^2 + 2z + 2) * q^97 + (-z^11 + 2z^10 + 3z^9 - 4z^8 + 3z^7 + 2z^6 + 2z^4 - z^3 + 2z^2 - 2z - 4) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7}+O(q^{10}) \) 12 * q - 12 * q^7	\( 12 q - 12 q^{7} + 6 q^{10} - 6 q^{11} - 12 q^{13} + 18 q^{25} - 6 q^{26} - 6 q^{28} + 36 q^{29} - 6 q^{34} + 18 q^{35} - 6 q^{37} - 24 q^{38} - 6 q^{40} - 12 q^{41} + 24 q^{44} + 12 q^{46} - 12 q^{47} - 6 q^{49} + 6 q^{52} - 48 q^{53} - 72 q^{55} + 6 q^{58} - 36 q^{59} + 36 q^{61} + 18 q^{62} + 6 q^{64} + 42 q^{67} - 12 q^{70} + 12 q^{71} + 24 q^{74} - 6 q^{77} + 42 q^{79} + 6 q^{83} + 18 q^{85} + 36 q^{86} + 18 q^{88} + 36 q^{89} - 42 q^{91} + 36 q^{95} + 18 q^{97} - 36 q^{98}+O(q^{100}) \) 12 * q - 12 * q^7 + 6 * q^10 - 6 * q^11 - 12 * q^13 + 18 * q^25 - 6 * q^26 - 6 * q^28 + 36 * q^29 - 6 * q^34 + 18 * q^35 - 6 * q^37 - 24 * q^38 - 6 * q^40 - 12 * q^41 + 24 * q^44 + 12 * q^46 - 12 * q^47 - 6 * q^49 + 6 * q^52 - 48 * q^53 - 72 * q^55 + 6 * q^58 - 36 * q^59 + 36 * q^61 + 18 * q^62 + 6 * q^64 + 42 * q^67 - 12 * q^70 + 12 * q^71 + 24 * q^74 - 6 * q^77 + 42 * q^79 + 6 * q^83 + 18 * q^85 + 36 * q^86 + 18 * q^88 + 36 * q^89 - 42 * q^91 + 36 * q^95 + 18 * q^97 - 36 * 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	666.2.bj.a

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

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
$5$	\( T^{12} - 9 T^{10} + \cdots + 81 \) T^12 - 9T^10 - 72T^9 - 72T^8 + 378T^7 + 2286T^6 + 4860T^5 + 5670T^4 + 4050T^3 + 1782T^2 + 486T + 81
$7$	\( T^{12} + 12 T^{11} + \cdots + 5329 \) T^12 + 12T^11 + 75T^10 + 344T^9 + 1206T^8 + 3042T^7 + 6456T^6 + 8820T^5 + 4716T^4 - 8530T^3 - 7014T^2 + 1752*T + 5329
$11$	\( T^{12} + 6 T^{11} + \cdots + 2653641 \) T^12 + 6T^11 + 72T^10 + 300T^9 + 2889T^8 + 11232T^7 + 59778T^6 + 122472T^5 + 422901T^4 + 777384T^3 + 2044521T^2 + 2287116*T + 2653641
$13$	\( T^{12} + 12 T^{11} + \cdots + 12321 \) T^12 + 12T^11 + 30T^10 - 66T^9 - 288T^8 - 1098T^7 + 672T^6 + 2682T^5 + 8838T^4 + 10116T^3 + 20655T^2 + 17982*T + 12321
$17$	\( T^{12} - 18 T^{10} + \cdots + 17665209 \) T^12 - 18T^10 - 414T^9 - 1368T^8 + 15606T^7 + 43839T^6 - 40014T^5 + 566676T^4 + 2621322T^3 + 10262214T^2 + 15433416T + 17665209
$19$	\( T^{12} - 9 T^{10} + \cdots + 431649 \) T^12 - 9T^10 - 36T^9 - 18T^8 - 594T^7 - 2088T^6 + 15552T^5 + 34344T^4 + 47466T^3 + 460566T^2 + 567648T + 431649
$23$	\( T^{12} + \cdots + 212838921 \) T^12 - 90T^10 + 5787T^8 - 1512T^7 - 178560T^6 + 83268T^5 + 4016223T^4 - 3996864T^3 - 32749029T^2 + 25209792*T + 212838921
$29$	\( T^{12} + \cdots + 121154049 \) T^12 - 36T^11 + 549T^10 - 4212T^9 + 13347T^8 + 23490T^7 - 248328T^6 - 89748T^5 + 3902985T^4 - 4009500T^3 - 23707242T^2 + 19614474*T + 121154049
$31$	\( T^{12} + 210 T^{10} + \cdots + 25796241 \) T^12 + 210T^10 + 15741T^8 + 517002T^6 + 7260498T^4 + 32560920*T^2 + 25796241
$37$	\( T^{12} + \cdots + 2565726409 \) T^12 + 6T^11 + 30T^10 + 266T^9 - 954T^8 - 15066T^7 - 53697T^6 - 557442T^5 - 1306026T^4 + 13473698T^3 + 56224830T^2 + 416063742*T + 2565726409
$41$	\( T^{12} + \cdots + 1768034304 \) T^12 + 12T^11 + 72T^10 - 336T^9 - 4752T^8 - 38016T^7 + 143424T^6 + 3048192T^5 + 29279232T^4 + 159307776T^3 + 804639744T^2 + 1961791488*T + 1768034304
$43$	\( T^{12} + \cdots + 504496521 \) T^12 + 390T^10 + 55809T^8 + 3516270T^6 + 89302374T^4 + 556265124*T^2 + 504496521
$47$	\( T^{12} + \cdots + 13231670841 \) T^12 + 12T^11 + 279T^10 + 2148T^9 + 39987T^8 + 289008T^7 + 3234060T^6 + 14628924T^5 + 119811879T^4 + 479935584T^3 + 2924606250T^2 + 6323374188*T + 13231670841
$53$	\( T^{12} + \cdots + 1998358209 \) T^12 + 48T^11 + 927T^10 + 8952T^9 + 46764T^8 + 178254T^7 + 884250T^6 + 3227040T^5 + 8580978T^4 + 35976042T^3 + 300918726T^2 - 98972442*T + 1998358209
$59$	\( T^{12} + 36 T^{11} + \cdots + 927369 \) T^12 + 36T^11 + 585T^10 + 5778T^9 + 39132T^8 + 191970T^7 + 700155T^6 + 1905444T^5 + 3797037T^4 + 5304690T^3 + 4787424T^2 + 2496096*T + 927369
$61$	\( T^{12} - 36 T^{11} + \cdots + 46416969 \) T^12 - 36T^11 + 513T^10 - 5130T^9 + 58437T^8 - 495072T^7 + 1974852T^6 - 3775086T^5 + 7533000T^4 + 27046872T^3 + 140330637T^2 + 103748364*T + 46416969
$67$	\( T^{12} + \cdots + 254306809 \) T^12 - 42T^11 + 786T^10 - 8062T^9 + 47610T^8 - 168426T^7 + 474042T^6 - 1538154T^5 + 5453460T^4 - 22318144T^3 + 74848155T^2 - 125247738*T + 254306809
$71$	\( T^{12} - 12 T^{11} + \cdots + 331776 \) T^12 - 12T^11 - 36T^10 + 336T^9 + 3456T^8 + 53568T^7 + 200448T^6 - 1078272T^5 + 5204736T^4 - 6027264T^3 + 45702144T^2 + 5971968*T + 331776
$73$	\( (T^{6} - 96 T^{4} + \cdots + 11584)^{2} \) (T^6 - 96T^4 - 272T^3 + 1872T^2 + 9600T + 11584)^2
$79$	\( T^{12} + \cdots + 359276763609 \) T^12 - 42T^11 + 732T^10 - 7734T^9 + 122139T^8 - 2842560T^7 + 37810905T^6 - 236835594T^5 + 1087001244T^4 - 8691622956T^3 + 32853372741T^2 + 409987548*T + 359276763609
$83$	\( T^{12} - 6 T^{11} + \cdots + 110889 \) T^12 - 6T^11 + 9T^10 + 384T^9 + 3105T^8 + 20034T^7 + 55512T^6 - 18630T^5 - 23004T^4 + 244512T^3 + 341577T^2 + 161838*T + 110889
$89$	\( T^{12} + \cdots + 86755578849 \) T^12 - 36T^11 + 729T^10 - 10710T^9 + 125145T^8 - 1240920T^7 + 10297260T^6 - 66627198T^5 + 311854536T^4 - 1137874716T^3 + 5829944949T^2 - 33973767792*T + 86755578849
$97$	\( T^{12} + \cdots + 241771401 \) T^12 - 18T^11 + 39T^10 + 1242T^9 - 4005T^8 - 66330T^7 + 328524T^6 + 1424898T^5 - 6753285T^4 - 25696440T^3 + 93026286T^2 + 288278460*T + 241771401
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\(n\)	\(371\)	\(631\)
\(\chi(n)\)	\(1\)	\(\zeta_{36}^{2}\)