LMFDB - Newform orbit 666.2.w.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [666,2,Mod(7,666)]

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([12, 16]))

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

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

chi := DirichletCharacter("666.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 666 = 2 \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	666.w (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 + 6 q^{3} - 3 q^{7} - 57 q^{8} + 6 q^{9}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 114 q + 6 q^{3} - 3 q^{7} - 57 q^{8} + 6 q^{9} - 3 q^{10} - 6 q^{11} + 6 q^{12} - 3 q^{13} - 6 q^{15} - 3 q^{18} + 24 q^{21} + 18 q^{23} - 3 q^{24} - 18 q^{26} + 3 q^{27} - 3 q^{28} + 3 q^{30} - 6 q^{31} - 21 q^{33} + 9 q^{35} - 3 q^{37} - 15 q^{38} - 42 q^{39} - 6 q^{41} - 39 q^{42} - 42 q^{43} + 45 q^{45} - 6 q^{46} - 27 q^{49} - 3 q^{52} + 3 q^{53} + 63 q^{54} - 3 q^{56} - 6 q^{57} + 9 q^{58} + 3 q^{59} - 18 q^{60} + 48 q^{61} + 15 q^{62} - 69 q^{63} - 57 q^{64} - 18 q^{65} + 9 q^{66} + 60 q^{67} - 69 q^{69} - 18 q^{70} + 27 q^{71} - 3 q^{72} - 72 q^{73} + 12 q^{74} + 3 q^{75} + 12 q^{77} + 21 q^{78} - 54 q^{79} + 6 q^{80} + 126 q^{81} - 24 q^{82} + 9 q^{83} - 3 q^{86} + 27 q^{87} + 12 q^{88} - 30 q^{89} - 18 q^{90} + 9 q^{91} - 6 q^{92} - 66 q^{93} - 6 q^{94} - 33 q^{95} - 3 q^{96} + 72 q^{97} - 27 q^{98} - 63 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} $
7.1	0.173648	−	0.984808i	−1.72428	−	0.163903i	−0.939693	−	0.342020i	2.37564	−	1.99340i	−0.460830	+	1.66962i	1.61113	+	0.586405i	−0.500000	+	0.866025i	2.94627	+	0.565228i	−1.55059	−	2.68570i
7.2	0.173648	−	0.984808i	−1.66660	+	0.471646i	−0.939693	−	0.342020i	−2.90385	+	2.43662i	0.175079	+	1.72318i	−4.13231	−	1.50404i	−0.500000	+	0.866025i	2.55510	−	1.57209i	1.89535	+	3.28285i
7.3	0.173648	−	0.984808i	−1.62635	+	0.595817i	−0.939693	−	0.342020i	2.32451	−	1.95050i	0.304353	+	1.70510i	−3.53023	−	1.28490i	−0.500000	+	0.866025i	2.29001	−	1.93801i	−1.51722	−	2.62790i
7.4	0.173648	−	0.984808i	−1.53891	+	0.794835i	−0.939693	−	0.342020i	−0.497918	+	0.417803i	0.515531	+	1.65355i	1.71269	+	0.623369i	−0.500000	+	0.866025i	1.73647	−	2.44636i	0.324993	+	0.562904i
7.5	0.173648	−	0.984808i	−1.43984	−	0.962739i	−0.939693	−	0.342020i	0.128566	−	0.107879i	−1.19814	+	1.25079i	0.874320	+	0.318227i	−0.500000	+	0.866025i	1.14627	+	2.77238i	−0.0839153	−	0.145346i
7.6	0.173648	−	0.984808i	−0.772157	−	1.55041i	−0.939693	−	0.342020i	−1.53679	+	1.28952i	−1.66094	+	0.491200i	−2.28087	−	0.830169i	−0.500000	+	0.866025i	−1.80755	+	2.39432i	1.00307	+	1.73736i
7.7	0.173648	−	0.984808i	−0.684075	−	1.59124i	−0.939693	−	0.342020i	−0.301888	+	0.253314i	−1.68585	+	0.397366i	4.06439	+	1.47932i	−0.500000	+	0.866025i	−2.06408	+	2.17705i	0.197044	+	0.341289i
7.8	0.173648	−	0.984808i	−0.621754	+	1.61661i	−0.939693	−	0.342020i	1.90464	−	1.59818i	1.48408	+	0.893030i	2.88514	+	1.05010i	−0.500000	+	0.866025i	−2.22684	−	2.01027i	−1.24316	−	2.15322i
7.9	0.173648	−	0.984808i	−0.305113	+	1.70497i	−0.939693	−	0.342020i	−2.60790	+	2.18829i	1.62608	+	0.596542i	1.00158	+	0.364544i	−0.500000	+	0.866025i	−2.81381	−	1.04041i	1.70219	+	2.94828i
7.10	0.173648	−	0.984808i	−0.102438	+	1.72902i	−0.939693	−	0.342020i	0.922145	−	0.773771i	1.68496	+	0.401123i	−2.93878	−	1.06963i	−0.500000	+	0.866025i	−2.97901	−	0.354236i	−0.601887	−	1.04250i
7.11	0.173648	−	0.984808i	0.0637916	−	1.73088i	−0.939693	−	0.342020i	2.51042	−	2.10650i	−1.69350	−	0.363386i	−1.06083	−	0.386112i	−0.500000	+	0.866025i	−2.99186	−	0.220831i	−1.63856	−	2.83807i
7.12	0.173648	−	0.984808i	0.486532	−	1.66231i	−0.939693	−	0.342020i	−2.95041	+	2.47569i	−1.55257	−	0.767798i	3.61514	+	1.31580i	−0.500000	+	0.866025i	−2.52657	−	1.61754i	1.92574	+	3.33549i
7.13	0.173648	−	0.984808i	0.956963	+	1.44368i	−0.939693	−	0.342020i	0.0800741	−	0.0671901i	1.58793	−	0.691731i	−2.05069	−	0.746389i	−0.500000	+	0.866025i	−1.16844	+	2.76310i	−0.0522646	−	0.0905250i
7.14	0.173648	−	0.984808i	1.25876	−	1.18976i	−0.939693	−	0.342020i	−1.23626	+	1.03735i	−0.953101	−	1.44624i	−2.26337	−	0.823798i	−0.500000	+	0.866025i	0.168954	−	2.99524i	0.806912	+	1.39761i
7.15	0.173648	−	0.984808i	1.32896	−	1.11079i	−0.939693	−	0.342020i	2.03024	−	1.70358i	−0.863143	−	1.50166i	1.95890	+	0.712981i	−0.500000	+	0.866025i	0.532287	−	2.95240i	−1.32515	−	2.29522i
7.16	0.173648	−	0.984808i	1.34819	+	1.08737i	−0.939693	−	0.342020i	−2.20391	+	1.84930i	1.30496	−	1.13889i	−2.05935	−	0.749542i	−0.500000	+	0.866025i	0.635250	+	2.93197i	1.43850	+	2.49156i
7.17	0.173648	−	0.984808i	1.68178	+	0.414256i	−0.939693	−	0.342020i	1.31309	−	1.10181i	0.700001	−	1.58430i	3.04546	+	1.10846i	−0.500000	+	0.866025i	2.65678	+	1.39338i	−0.857058	−	1.48447i
7.18	0.173648	−	0.984808i	1.68540	+	0.399268i	−0.939693	−	0.342020i	2.86220	−	2.40167i	0.685869	−	1.59047i	−2.67541	−	0.973768i	−0.500000	+	0.866025i	2.68117	+	1.34585i	−1.86817	−	3.23576i
7.19	0.173648	−	0.984808i	1.73142	+	0.0465820i	−0.939693	−	0.342020i	−1.44655	+	1.21380i	0.346533	−	1.69703i	1.89673	+	0.690353i	−0.500000	+	0.866025i	2.99566	+	0.161306i	0.944168	+	1.63535i
49.1	−0.939693	−	0.342020i	−1.73073	+	0.0676331i	0.766044	+	0.642788i	0.421842	−	2.39239i	1.64949	+	0.528390i	−3.44654	−	2.89199i	−0.500000	−	0.866025i	2.99085	−	0.234109i	−1.21465	+	2.10383i

See next 80 embeddings (of 114 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
333.w	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.w.a	✓	114
9.c	even	3	1		666.2.y.a	yes	114
37.f	even	9	1		666.2.y.a	yes	114
333.w	even	9	1	inner	666.2.w.a	✓	114

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{114} + 13 T_{5}^{111} - 66 T_{5}^{110} - 57 T_{5}^{109} + 8698 T_{5}^{108} + \cdots + 86\!\cdots\!41 $ T5^114 + 13*T5^111 - 66*T5^110 - 57*T5^109 + 8698*T5^108 - 10971*T5^107 + 15756*T5^106 + 102445*T5^105 - 444306*T5^104 - 1828617*T5^103 + 51120763*T5^102 - 91340019*T5^101 + 69735774*T5^100 + 578507326*T5^99 - 1877562831*T5^98 - 10062612900*T5^97 + 177732724300*T5^96 - 430960523238*T5^95 + 314200309491*T5^94 + 1796656555285*T5^93 - 4484470105848*T5^92 - 41274610228938*T5^91 + 454397818340950*T5^90 - 1155078254453679*T5^89 + 481431344086356*T5^88 + 4984655834630735*T5^87 - 9695187872709690*T5^86 - 74182578740997864*T5^85 + 705873017556213896*T5^84 - 1842518666343547335*T5^83 + 421906586991244596*T5^82 + 8800548011938861076*T5^81 - 13053137034092620773*T5^80 - 106170790232860977072*T5^79 + 797592357201055378190*T5^78 - 1903299859036993782090*T5^77 - 94157810179573254270*T5^76 + 8935951269147526859837*T5^75 - 6824596606079006568090*T5^74 - 84615844555438946134803*T5^73 + 478713634996978180865993*T5^72 - 1110428158107723661777746*T5^71 + 340368211455989843961627*T5^70 + 3528982971925741370930186*T5^69 - 1474076055377917000439472*T5^68 - 41368292337934419786226491*T5^67 + 200082577629257810110508276*T5^66 - 447478617125376186434360076*T5^65 + 258928160447758162011205959*T5^64 + 805542632266838503899155222*T5^63 - 9929304247512819941775804*T5^62 - 10413695014933102729756884804*T5^61 + 43592166793114107389866272430*T5^60 - 103182746408703819356332898856*T5^59 + 140445781607630883687798031716*T5^58 - 141039639359743258371860202584*T5^57 + 469323464139979227884939511384*T5^56 - 1800037682828928065749941347217*T5^55 + 5080034859084624829046051285242*T5^54 - 10847734331475334357312332855015*T5^53 + 14302484873012769780381153347358*T5^52 - 10936564078053038850382040200721*T5^51 + 28618831341848313076117781949957*T5^50 - 107609516351937118884684262228137*T5^49 + 264657606482643246765161656553662*T5^48 - 506175536044603469634867494213619*T5^47 + 674071723534861306519999350745101*T5^46 - 604917379306685299959144271310267*T5^45 + 1281765668137800400949097525966612*T5^44 - 4096238528449968496868595960482007*T5^43 + 9448336676431128458346195742955230*T5^42 - 15673177345655324787881635559208369*T5^41 + 16658296282226727978737930018655960*T5^40 - 8692009638390382949871900700018973*T5^39 + 13993263255192251108491730016903213*T5^38 - 51094217321961162428136149291478210*T5^37 + 98077691539894208787133333683171025*T5^36 - 106987042240125885582788119452588993*T5^35 + 20566552995622175768728842620750121*T5^34 + 127826342158155546965229209771469417*T5^33 - 116415836587809922182244352473925106*T5^32 - 84540932655529266229263470044829640*T5^31 + 296762314460917795575040775858083605*T5^30 - 393929865384644063917084713015112794*T5^29 + 271688090916836511713148666384020535*T5^28 + 26586062951209391297652813147678021*T5^27 - 108959782810809575570295958862430078*T5^26 + 66805391335664076613685784216746883*T5^25 - 7515769770350186895630714406850307*T5^24 - 148797553049897320881170508755980503*T5^23 + 262880215075835497800821898232955874*T5^22 - 109783945066150039360249787065574301*T5^21 + 185092055469372768085564386197552838*T5^20 - 102177290856161717041647525311977434*T5^19 + 19081417412597679844409968616723553*T5^18 - 75435276636981306785636138415364110*T5^17 + 45656800482553557956961751338243111*T5^16 + 42302280207134845002909981512863215*T5^15 + 83397805996906989410954800158145755*T5^14 + 58200486002908270662639556892108943*T5^13 + 38762149969538106938481879053565354*T5^12 + 17715626778038039040013775522287632*T5^11 + 6689190811605983539600753026155529*T5^10 + 1841495259422101471885081304493075*T5^9 + 422552634893227916538385394602971*T5^8 + 44229065883991798841332500481317*T5^7 - 3032490697004221230619728291060*T5^6 + 841699096370211920456661209667*T5^5 + 574057084373745620992402940181*T5^4 + 35777652605412384836755443366*T5^3 - 5188697866780301740214307276*T5^2 + 88818921807256361918990943*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.w (of order \(9\), degree \(6\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 7.19
Significant digits:
Format:

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