LMFDB - Newform orbit 666.2.y.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [666,2,Mod(229,666)]

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([6, 16]))

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

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

chi := DirichletCharacter("666.229");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 666 = 2 \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	666.y (of order $9$, 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:	$114$
Relative dimension:	$19$ over $\Q(\zeta_{9})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $q$-expansion, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) = $ $ 114 q - 3 q^{3} - 3 q^{7} - 57 q^{8} - 3 q^{9}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 114 q - 3 q^{3} - 3 q^{7} - 57 q^{8} - 3 q^{9} - 3 q^{10} + 12 q^{11} - 3 q^{12} + 6 q^{13} + 3 q^{15} - 3 q^{18} - 12 q^{21} - 36 q^{23} + 6 q^{24} - 18 q^{26} + 3 q^{27} - 3 q^{28} - 6 q^{30} - 6 q^{31} - 21 q^{33} + 9 q^{35} - 3 q^{37} - 15 q^{38} + 21 q^{39} + 30 q^{41} + 15 q^{42} - 42 q^{43} + 36 q^{45} - 6 q^{46} + 9 q^{49} - 3 q^{52} + 3 q^{53} - 18 q^{54} + 6 q^{56} + 39 q^{57} + 9 q^{58} + 3 q^{59} + 9 q^{60} - 24 q^{61} + 15 q^{62} + 75 q^{63} - 57 q^{64} + 36 q^{65} + 9 q^{66} - 30 q^{67} - 42 q^{69} + 9 q^{70} + 27 q^{71} - 3 q^{72} - 72 q^{73} + 12 q^{74} + 3 q^{75} + 12 q^{77} - 42 q^{78} - 27 q^{79} + 6 q^{80} + 45 q^{81} - 24 q^{82} - 18 q^{83} - 3 q^{86} + 72 q^{87} - 6 q^{88} - 30 q^{89} - 9 q^{90} + 9 q^{91} - 6 q^{92} + 24 q^{93} - 33 q^{94} - 60 q^{95} - 3 q^{96} - 36 q^{97} - 27 q^{98} + 36 q^{99}+O(q^{100}) $

Display 100 coefficients 10 coefficients

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	$ a_{2} $			$ a_{3} $			$ a_{4} $			$ a_{5} $			$ a_{6} $			$ a_{7} $			$ a_{8} $			$ a_{9} $			$ a_{10} $
229.1	0.766044	+	0.642788i	−1.72182	−	0.188009i	0.173648	+	0.984808i	−0.157709	−	0.0574015i	−1.19814	−	1.25079i	−0.712752	+	0.598070i	−0.500000	+	0.866025i	2.92931	+	0.647433i	−0.0839153	−	0.145346i
229.2	0.766044	+	0.642788i	−1.58809	+	0.691350i	0.173648	+	0.984808i	1.88515	+	0.686137i	−1.66094	−	0.491200i	1.85938	−	1.56021i	−0.500000	+	0.866025i	2.04407	−	2.19586i	1.00307	+	1.73736i
229.3	0.766044	+	0.642788i	−1.54686	+	0.779245i	0.173648	+	0.984808i	0.370321	+	0.134786i	−1.68585	−	0.397366i	−3.31332	+	2.78021i	−0.500000	+	0.866025i	1.78555	−	2.41077i	0.197044	+	0.341289i
229.4	0.766044	+	0.642788i	−1.42623	−	0.982788i	0.173648	+	0.984808i	−2.91415	−	1.06067i	−0.460830	−	1.66962i	−1.31341	+	1.10208i	−0.500000	+	0.866025i	1.06826	+	2.80336i	−1.55059	−	2.68570i
229.5	0.766044	+	0.642788i	−1.06372	+	1.36693i	0.173648	+	0.984808i	−3.07949	−	1.12084i	−1.69350	+	0.363386i	0.864799	−	0.725653i	−0.500000	+	0.866025i	−0.737007	−	2.90806i	−1.63856	−	2.83807i
229.6	0.766044	+	0.642788i	−0.973520	−	1.43257i	0.173648	+	0.984808i	3.56210	+	1.29650i	0.175079	−	1.72318i	3.36869	−	2.82666i	−0.500000	+	0.866025i	−1.10452	+	2.78927i	1.89535	+	3.28285i
229.7	0.766044	+	0.642788i	−0.862870	−	1.50182i	0.173648	+	0.984808i	−2.85143	−	1.03784i	0.304353	−	1.70510i	2.87787	−	2.41482i	−0.500000	+	0.866025i	−1.51091	+	2.59175i	−1.51722	−	2.62790i
229.8	0.766044	+	0.642788i	−0.695809	+	1.58614i	0.173648	+	0.984808i	3.61922	+	1.31729i	−1.55257	+	0.767798i	−2.94709	+	2.47290i	−0.500000	+	0.866025i	−2.03170	−	2.20731i	1.92574	+	3.33549i
229.9	0.766044	+	0.642788i	−0.667962	−	1.59807i	0.173648	+	0.984808i	0.610787	+	0.222308i	0.515531	−	1.65355i	−1.39620	+	1.17155i	−0.500000	+	0.866025i	−2.10765	+	2.13490i	0.324993	+	0.562904i
229.10	0.766044	+	0.642788i	0.199505	+	1.72052i	0.173648	+	0.984808i	1.51650	+	0.551960i	−0.953101	+	1.44624i	1.84511	−	1.54823i	−0.500000	+	0.866025i	−2.92040	+	0.686505i	0.806912	+	1.39761i
229.11	0.766044	+	0.642788i	0.304042	+	1.70516i	0.173648	+	0.984808i	−2.49046	−	0.906454i	−0.863143	+	1.50166i	−1.59691	+	1.33997i	−0.500000	+	0.866025i	−2.81512	+	1.03688i	−1.32515	−	2.29522i
229.12	0.766044	+	0.642788i	0.562844	−	1.63805i	0.173648	+	0.984808i	−2.33638	−	0.850373i	1.48408	−	0.893030i	−2.35199	+	1.97355i	−0.500000	+	0.866025i	−2.36641	−	1.84393i	−1.24316	−	2.15322i
229.13	0.766044	+	0.642788i	0.862200	−	1.50220i	0.173648	+	0.984808i	3.19907	+	1.16437i	1.62608	−	0.596542i	−0.816492	+	0.685118i	−0.500000	+	0.866025i	−1.51322	−	2.59040i	1.70219	+	2.94828i
229.14	0.766044	+	0.642788i	1.03292	−	1.39035i	0.173648	+	0.984808i	−1.13118	−	0.411715i	1.68496	−	0.401123i	2.39571	−	2.01024i	−0.500000	+	0.866025i	−0.866155	−	2.87224i	−0.601887	−	1.04250i
229.15	0.766044	+	0.642788i	1.35629	+	1.07725i	0.173648	+	0.984808i	1.77445	+	0.645849i	0.346533	+	1.69703i	−1.54623	+	1.29744i	−0.500000	+	0.866025i	0.679047	+	2.92214i	0.944168	+	1.63535i
229.16	0.766044	+	0.642788i	1.54774	+	0.777500i	0.173648	+	0.984808i	−3.51100	−	1.27790i	0.685869	+	1.59047i	2.18101	−	1.83009i	−0.500000	+	0.866025i	1.79099	+	2.40673i	−1.86817	−	3.23576i
229.17	0.766044	+	0.642788i	1.55460	+	0.763690i	0.173648	+	0.984808i	−1.61074	−	0.586262i	0.700001	+	1.58430i	−2.48268	+	2.08321i	−0.500000	+	0.866025i	1.83355	+	2.37446i	−0.857058	−	1.48447i
229.18	0.766044	+	0.642788i	1.66106	−	0.490802i	0.173648	+	0.984808i	−0.0982254	−	0.0357511i	1.58793	+	0.691731i	1.67174	−	1.40275i	−0.500000	+	0.866025i	2.51823	−	1.63050i	−0.0522646	−	0.0905250i
229.19	0.766044	+	0.642788i	1.73172	+	0.0336277i	0.173648	+	0.984808i	2.70350	+	0.983993i	1.30496	+	1.13889i	1.67880	−	1.40868i	−0.500000	+	0.866025i	2.99774	+	0.116468i	1.43850	+	2.49156i
349.1	0.766044	−	0.642788i	−1.72182	+	0.188009i	0.173648	−	0.984808i	−0.157709	+	0.0574015i	−1.19814	+	1.25079i	−0.712752	−	0.598070i	−0.500000	−	0.866025i	2.92931	−	0.647433i	−0.0839153	+	0.145346i

See next 80 embeddings (of 114 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
333.y	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	666.2.y.a	yes	114
9.c	even	3	1		666.2.w.a	✓	114
37.f	even	9	1		666.2.w.a	✓	114
333.y	even	9	1	inner	666.2.y.a	yes	114

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
666.2.w.a	✓	114	9.c	even	3	1
666.2.w.a	✓	114	37.f	even	9	1
666.2.y.a	yes	114	1.a	even	1	1	trivial
666.2.y.a	yes	114	333.y	even	9	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{114} + 13 T_{5}^{111} + 42 T_{5}^{110} - 732 T_{5}^{109} + 8698 T_{5}^{108} + \cdots + 86\!\cdots\!41 $ T5^114 + 13*T5^111 + 42*T5^110 - 732*T5^109 + 8698*T5^108 + 11475*T5^107 - 30090*T5^106 + 67939*T5^105 + 438801*T5^104 - 3925518*T5^103 + 42199882*T5^102 + 79108254*T5^101 - 152903454*T5^100 + 254314654*T5^99 + 1886369703*T5^98 - 16200882171*T5^97 + 126907019989*T5^96 + 324284928093*T5^95 - 574170168636*T5^94 + 284823630610*T5^93 + 6224256534336*T5^92 - 28303683936816*T5^91 + 247538875067395*T5^90 + 670373435533293*T5^89 - 862981443363459*T5^88 + 578919255010163*T5^87 + 8862824678146494*T5^86 - 39402557050749369*T5^85 + 335314874112260447*T5^84 + 862961928834062850*T5^83 - 902588283412787151*T5^82 + 1512919979426290196*T5^81 + 7247139076827984453*T5^80 - 31764891752660258292*T5^79 + 323149963971997038128*T5^78 + 656926919879125035534*T5^77 - 232231420210957783962*T5^76 + 1788563986602271678508*T5^75 + 533430328986009510747*T5^74 - 19183369995187127617041*T5^73 + 173419953409977684417563*T5^72 + 269089553763545678479851*T5^71 + 77910354255377919782679*T5^70 + 976163276153214762336920*T5^69 - 119880133661052498052776*T5^68 - 5274965042589906409520589*T5^67 + 63014230909038009964071773*T5^66 + 64007040940325942566714539*T5^65 + 58249617605706404632487742*T5^64 + 222465941662543111817482349*T5^63 - 517537330020306503913786105*T5^62 - 1261789921493959348883717139*T5^61 + 12636399002708241525733516378*T5^60 + 2283071388972836963254389420*T5^59 + 14433784541295664653554856708*T5^58 - 9821342382556708920702293273*T5^57 - 94867960993111128740626954089*T5^56 - 90937988552184606671234738874*T5^55 + 1418697835675751625429863073283*T5^54 - 429858240875645508165984502944*T5^53 + 2290727109868581766071684684519*T5^52 - 3085634219952475197001707945977*T5^51 - 1039759085058275698453157008599*T5^50 - 19217492282444198077427870771037*T5^49 + 105070692489479364988749318621442*T5^48 - 123676190517905554044935183373228*T5^47 + 257778005720181707534373794642973*T5^46 - 300019049941166459463831335163875*T5^45 + 194924767150504713392074620151263*T5^44 - 723850685847286271264394658508907*T5^43 + 3579594849368542580451871069278607*T5^42 - 4483446153642088244242619167163442*T5^41 + 6245383512714234572477415088629669*T5^40 - 6039330530741068742776138869303530*T5^39 + 3940034704412710222154888838554805*T5^38 - 7202094153353189217852345471543621*T5^37 + 54584182150119433170559716584724499*T5^36 - 108237865982040020302050645445669851*T5^35 + 186850946743260315694136932590239076*T5^34 - 215186664161763599251643808556214580*T5^33 + 190915010303486378840090104018429395*T5^32 - 26339422558092947111677218935560623*T5^31 + 18609597971905335529176516112649955*T5^30 + 32714285285891290410663705347154381*T5^29 + 255134308257793564634480905936325457*T5^28 - 483922599039405530196119002549614753*T5^27 + 801628541229918392003187076733574804*T5^26 - 716192753594179510222326820156833423*T5^25 + 753356035689315459889517602140726567*T5^24 - 493423031840465363115163598026880193*T5^23 + 467869896317068053337537456277703150*T5^22 - 449244554008088393501162908880359341*T5^21 + 443567763995796531560068544902446999*T5^20 - 302430347466311697711805744915945428*T5^19 + 197440452974176099642949843862374823*T5^18 - 75123148301683916271215052124635993*T5^17 + 41542298547831367246434942238414689*T5^16 - 77533706589965649678221398554897186*T5^15 + 52510383390836858737086552015400392*T5^14 - 1515568779227480696142284952596673*T5^13 - 935835977115152021243380963451382*T5^12 - 5910183606472236500221512894727458*T5^11 + 3506793009368672701674258673157505*T5^10 - 451766318578877941406147992079412*T5^9 - 77376785242880883787907976074088*T5^8 + 17443458590454744527451327020019*T5^7 - 1888048279443092185410962326713*T5^6 + 592053818040003434478110692128*T5^5 + 428340601491087446224394293017*T5^4 + 53003734508060617578348881673*T5^3 + 9739840572008830101739175793*T5^2 + 1532167511039183913519854898*T5 + 86249130006858312188597841 acting on $S_{2}^{\mathrm{new}}(666, [\chi])$.

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.y (of order \(9\), degree \(6\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 229.19
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more