LMFDB - Newform orbit 930.2.bt.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [930,2,Mod(13,930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, base_ring=CyclotomicField(60))

chi = DirichletCharacter(H, H._module([0, 45, 22]))

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

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

chi := DirichletCharacter("930.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	930.bt (of order $60$, degree $16$, 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:	$7.42608738798$
Analytic rank:	$0$
Dimension:	$256$
Relative dimension:	$16$ over $\Q(\zeta_{60})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

$q$-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) = $ $ 256 q + 8 q^{7}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 256 q + 8 q^{7} + 4 q^{10} - 24 q^{14} - 8 q^{15} + 64 q^{16} + 36 q^{17} - 40 q^{19} + 12 q^{20} - 16 q^{21} + 8 q^{22} - 32 q^{24} + 20 q^{25} + 4 q^{28} - 16 q^{29} - 8 q^{31} - 4 q^{33} - 24 q^{35} + 128 q^{36} + 4 q^{37} + 4 q^{38} + 48 q^{41} + 32 q^{42} - 24 q^{43} + 4 q^{44} - 12 q^{45} - 20 q^{46} - 168 q^{47} - 40 q^{49} + 48 q^{50} - 64 q^{53} + 64 q^{54} + 64 q^{55} + 28 q^{57} + 68 q^{58} - 24 q^{59} + 4 q^{62} + 4 q^{63} - 60 q^{65} + 12 q^{66} - 76 q^{67} - 4 q^{68} + 12 q^{69} - 16 q^{70} - 48 q^{71} + 52 q^{73} + 68 q^{74} + 48 q^{75} + 48 q^{76} + 84 q^{77} - 56 q^{79} - 32 q^{81} - 32 q^{82} + 136 q^{83} + 68 q^{85} + 24 q^{86} + 20 q^{87} + 56 q^{88} - 224 q^{89} + 40 q^{91} - 48 q^{93} - 4 q^{95} + 8 q^{97} + 64 q^{98} + 4 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} $
13.1	−0.156434	+	0.987688i	0.358368	−	0.933580i	−0.951057	−	0.309017i	−1.68012	−	1.47553i	0.866025	+	0.500000i	−1.74433	−	0.0914165i	0.453990	−	0.891007i	−0.743145	−	0.669131i	1.72019	−	1.42861i
13.2	−0.156434	+	0.987688i	0.358368	−	0.933580i	−0.951057	−	0.309017i	−1.65687	+	1.50160i	0.866025	+	0.500000i	−2.43455	−	0.127590i	0.453990	−	0.891007i	−0.743145	−	0.669131i	−1.22392	−	1.87137i
13.3	−0.156434	+	0.987688i	0.358368	−	0.933580i	−0.951057	−	0.309017i	−1.65366	+	1.50512i	0.866025	+	0.500000i	0.140670	+	0.00737222i	0.453990	−	0.891007i	−0.743145	−	0.669131i	−1.22790	−	1.86876i
13.4	−0.156434	+	0.987688i	0.358368	−	0.933580i	−0.951057	−	0.309017i	−1.12100	−	1.93477i	0.866025	+	0.500000i	0.00733325		0.000384319i	0.453990	−	0.891007i	−0.743145	−	0.669131i	2.08632	−	0.804537i
13.5	−0.156434	+	0.987688i	0.358368	−	0.933580i	−0.951057	−	0.309017i	1.57500	−	1.58726i	0.866025	+	0.500000i	4.42218	+	0.231757i	0.453990	−	0.891007i	−0.743145	−	0.669131i	1.32133	+	1.80391i
13.6	−0.156434	+	0.987688i	0.358368	−	0.933580i	−0.951057	−	0.309017i	1.66702	+	1.49031i	0.866025	+	0.500000i	2.09913	+	0.110011i	0.453990	−	0.891007i	−0.743145	−	0.669131i	−1.73274	+	1.41336i
13.7	−0.156434	+	0.987688i	0.358368	−	0.933580i	−0.951057	−	0.309017i	2.05562	+	0.880016i	0.866025	+	0.500000i	−0.216795	−	0.0113618i	0.453990	−	0.891007i	−0.743145	−	0.669131i	−1.19075	+	1.89265i
13.8	−0.156434	+	0.987688i	0.358368	−	0.933580i	−0.951057	−	0.309017i	2.21938	+	0.272664i	0.866025	+	0.500000i	−4.57122	−	0.239568i	0.453990	−	0.891007i	−0.743145	−	0.669131i	−0.616494	+	2.14940i
13.9	0.156434	−	0.987688i	−0.358368	+	0.933580i	−0.951057	−	0.309017i	−2.21293	−	0.320863i	0.866025	+	0.500000i	−2.17593	−	0.114036i	−0.453990	+	0.891007i	−0.743145	−	0.669131i	−0.663091	+	2.13549i
13.10	0.156434	−	0.987688i	−0.358368	+	0.933580i	−0.951057	−	0.309017i	−1.90371	+	1.17298i	0.866025	+	0.500000i	1.91384	+	0.100300i	−0.453990	+	0.891007i	−0.743145	−	0.669131i	0.860728	+	2.06377i
13.11	0.156434	−	0.987688i	−0.358368	+	0.933580i	−0.951057	−	0.309017i	−1.16716	−	1.90728i	0.866025	+	0.500000i	2.78313	+	0.145858i	−0.453990	+	0.891007i	−0.743145	−	0.669131i	−2.06639	+	0.854429i
13.12	0.156434	−	0.987688i	−0.358368	+	0.933580i	−0.951057	−	0.309017i	−0.443582	+	2.19163i	0.866025	+	0.500000i	−2.50424	−	0.131242i	−0.453990	+	0.891007i	−0.743145	−	0.669131i	2.09525	+	0.780967i
13.13	0.156434	−	0.987688i	−0.358368	+	0.933580i	−0.951057	−	0.309017i	0.904456	−	2.04498i	0.866025	+	0.500000i	−2.34346	−	0.122815i	−0.453990	+	0.891007i	−0.743145	−	0.669131i	−1.87832	−	1.21323i
13.14	0.156434	−	0.987688i	−0.358368	+	0.933580i	−0.951057	−	0.309017i	1.00881	+	1.99557i	0.866025	+	0.500000i	4.83852	+	0.253576i	−0.453990	+	0.891007i	−0.743145	−	0.669131i	2.12882	−	0.684211i
13.15	0.156434	−	0.987688i	−0.358368	+	0.933580i	−0.951057	−	0.309017i	1.45393	−	1.69884i	0.866025	+	0.500000i	3.84003	+	0.201247i	−0.453990	+	0.891007i	−0.743145	−	0.669131i	−1.45048	−	1.70179i
13.16	0.156434	−	0.987688i	−0.358368	+	0.933580i	−0.951057	−	0.309017i	1.76830	+	1.36862i	0.866025	+	0.500000i	−0.704223	−	0.0369068i	−0.453990	+	0.891007i	−0.743145	−	0.669131i	1.62840	−	1.53243i
43.1	−0.987688	+	0.156434i	0.933580	−	0.358368i	0.951057	−	0.309017i	−1.95765	+	1.08055i	−0.866025	+	0.500000i	−0.179844	−	3.43163i	−0.891007	+	0.453990i	0.743145	−	0.669131i	1.76452	−	1.37349i
43.2	−0.987688	+	0.156434i	0.933580	−	0.358368i	0.951057	−	0.309017i	−1.92766	+	1.13319i	−0.866025	+	0.500000i	0.0854612	+	1.63070i	−0.891007	+	0.453990i	0.743145	−	0.669131i	1.72666	−	1.42079i
43.3	−0.987688	+	0.156434i	0.933580	−	0.358368i	0.951057	−	0.309017i	−1.71760	−	1.43173i	−0.866025	+	0.500000i	0.0366609	+	0.699531i	−0.891007	+	0.453990i	0.743145	−	0.669131i	1.92043	+	1.14541i
43.4	−0.987688	+	0.156434i	0.933580	−	0.358368i	0.951057	−	0.309017i	−0.387188	+	2.20229i	−0.866025	+	0.500000i	−0.0918879	−	1.75333i	−0.891007	+	0.453990i	0.743145	−	0.669131i	0.0379066	−	2.23575i

See next 80 embeddings (of 256 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
155.x	even	60	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	930.2.bt.a	✓	256
5.c	odd	4	1		930.2.bt.b	yes	256
31.h	odd	30	1		930.2.bt.b	yes	256
155.x	even	60	1	inner	930.2.bt.a	✓	256

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
930.2.bt.a	✓	256	1.a	even	1	1	trivial
930.2.bt.a	✓	256	155.x	even	60	1	inner
930.2.bt.b	yes	256	5.c	odd	4	1
930.2.bt.b	yes	256	31.h	odd	30	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{256} - 8 T_{7}^{255} + 52 T_{7}^{254} - 348 T_{7}^{253} + 3362 T_{7}^{252} + \cdots + 18\!\cdots\!36 $ T7^256 - 8*T7^255 + 52*T7^254 - 348*T7^253 + 3362*T7^252 - 25332*T7^251 + 132364*T7^250 - 723220*T7^249 + 4676325*T7^248 - 30014104*T7^247 + 146092184*T7^246 - 661014544*T7^245 + 3550228426*T7^244 - 19952058636*T7^243 + 91452481898*T7^242 - 355234383396*T7^241 + 1604678397770*T7^240 - 8025282025596*T7^239 + 33360872435360*T7^238 - 104289727662040*T7^237 + 338360526072214*T7^236 - 1335167650464436*T7^235 + 3600315808400840*T7^234 + 5584594459422012*T7^233 - 105280650344675417*T7^232 + 717702484360934536*T7^231 - 4276726688293882214*T7^230 + 25888544561991727108*T7^229 - 143036163989511787689*T7^228 + 731084290928878186796*T7^227 - 3522187526107791277070*T7^226 + 16166109425695731046672*T7^225 - 74611129563919186090454*T7^224 + 334251890219908657801324*T7^223 - 1432747860886717894417642*T7^222 + 5584144072242765183477932*T7^221 - 21544071347322621233795487*T7^220 + 80439778952099880888793568*T7^219 - 274282214811521837159514194*T7^218 + 718260409824613876632605516*T7^217 - 967236046742927209513557609*T7^216 - 4623511220788617657699687904*T7^215 + 60688368027552595295582612838*T7^214 - 442572213711149319980075631700*T7^213 + 2729591716227322416683010976945*T7^212 - 15004147395359944683286246875840*T7^211 + 77995881892728435120462356309926*T7^210 - 383076535141466682954626339192040*T7^209 + 1839793892242585329116926343142452*T7^208 - 8615632660421986888390838236520384*T7^207 + 39426060677239443750819259553126186*T7^206 - 175103851070622992242732016946674560*T7^205 + 764429335670088687103410946204811495*T7^204 - 3290843789003457476438128782926436808*T7^203 + 13923403490782957460321054443408870376*T7^202 - 57824118298134147744931237952936389156*T7^201 + 236274516041942331271145744419807857989*T7^200 - 952819602024088766951685032455764339484*T7^199 + 3780667779177146975136728352937425214458*T7^198 - 14814392791779612811510924363895354569120*T7^197 + 57198387163393357326975453149315197170737*T7^196 - 217718230338697049090226213362895623582304*T7^195 + 815501362386315823511750473937636839111100*T7^194 - 3021790138124365378458433921218967082293952*T7^193 + 11024606235548026719923836342555404242511490*T7^192 - 39554255837024801888438869453363924433606208*T7^191 + 139522860347531305936195827350999376428365216*T7^190 - 485116888037163974366002605284295325492188720*T7^189 + 1652592860548222785589537475284230816243401916*T7^188 - 5508403196502972014516869766211975177379888904*T7^187 + 17984133402657456258613809434186966364392951280*T7^186 - 57227996862481936018365890698044263466643614256*T7^185 + 176499825417168812818144272477916619005033688233*T7^184 - 526451861808624651654780754422026448755776162524*T7^183 + 1515221374778116145795638364811247843606947540558*T7^182 - 4104045289527127046303249836496472710419010536532*T7^181 + 10389551067778985612314544931513566049840761749647*T7^180 - 23828909901599538023603319731993132418371861721776*T7^179 + 45638830524324206406797570017577522025741918455340*T7^178 - 43313892518701537281369792807513685643697109729016*T7^177 - 138023856293443669869862876513018817253475625467093*T7^176 + 1191082985369618983882989115276816781962251555844304*T7^175 - 5625086220940473026092684437821638613510839325863298*T7^174 + 22007213220899367319595601186025341454691012714251012*T7^173 - 72909492884610479505154270765386068190084823465057489*T7^172 + 230257415630103662253370377700595365907581448275953368*T7^171 - 663877108913184210637664732283447574129306959399431792*T7^170 + 1730231674644718250571870727236219786099161925869481740*T7^169 - 3931030506433567337935444155384032627025084875575083457*T7^168 + 8594544326392909717126048612860328966835820200652976268*T7^167 - 12718556794819711495153012232494488786783519369346777432*T7^166 - 1776533995837205665054728343353364604663916385633890036*T7^165 + 95611197456805073762551969554265211071914426080048279992*T7^164 - 447085231170120593470450719731214261963267783257590102432*T7^163 + 1859639859493223342138670133226167740956481061883750848532*T7^162 - 5877387387899023466871386565505150320897026666513903662180*T7^161 + 15144507522310420680377109152660559541309379214423531774375*T7^160 - 39078316621105687144548398192880507387470184510583402177556*T7^159 + 91973206337492547664499849237445076176116452702962764857922*T7^158 - 132955232200228570764042722496788294000434892749331400636568*T7^157 + 85987574695930222099494991566842761939470568760562399188085*T7^156 - 57143869371066031713321422836723950479093558484911201646520*T7^155 - 1853723757567630442191509105427851285650814490465283428083634*T7^154 + 10908063857904654052434747502466840936446905808088160716022600*T7^153 - 30272818030660267436120159085341215176044040270648235790567463*T7^152 + 53765414013464532850867598188175941591534383036129540643433784*T7^151 - 138991883524175253548232661372233831413075788724455936204775098*T7^150 + 188462945074297559762360140903686820391063896602041044815742828*T7^149 - 30649890002335785691926285666639868122328113411481102883923719*T7^148 - 347460541605561262640608112795666290637714212567411478450453212*T7^147 + 1030723484851571612613493834329859710121267134075939693693655930*T7^146 - 14971666422963622728654295082257954988950550816637125883835447112*T7^145 + 40555224017937005779316968937225822802434501663326857742312820458*T7^144 + 27766687415920824760830410663296948030779923266511410869198253452*T7^143 - 357601873260256363728274630050884776271178031956905580644933066730*T7^142 + 628264017420164899517199141997953352531267462793403262360544420932*T7^141 - 355217251108711856754215921014935772640012854915987491789704820087*T7^140 + 3211576190374902382009155350700624867430954754667298187857771118040*T7^139 - 4799335113722205102960618972955232000609514877027469379948223916584*T7^138 - 11348554525512046133212000578446719117317142253517163654632710857324*T7^137 + 79853050499741603361987989181079707634202859329441104425719744700000*T7^136 - 148325343169811469194151383053896276149286730446163838343657266322348*T7^135 + 823363679919425274490001480231414101993462880174844083901162358070910*T7^134 - 1504097532879794292836167185461684593076492332552202991131206781207304*T7^133 + 1227502486993566175427348555137374882699745420017727442254421208351981*T7^132 + 242864290009465852970297131119306742000142974796717453837441335333084*T7^131 + 7631519112759352044634610144478149973813354136486724875642527518194748*T7^130 + 46677493444859556891216545820633398348004068622140962530516640559058404*T7^129 - 207084730529974353232989934135591925511343177651339200585592687448595885*T7^128 + 252441339735009097363457811760728236290825192290256974138931391389076496*T7^127 - 188942545345246474802827407078766762169296280489464442865493977291214160*T7^126 + 1272386166772468431514413289736364544475155034133980832909287720722217100*T7^125 - 3487915912859101982003393989947601490022565674625056049056434661833831400*T7^124 - 3690619686844025942126437567462329689810233768726423176072868639231270624*T7^123 + 25448849682434733576342348465699868560001912757213719880447288464823362378*T7^122 - 87744776609966457365444043627459372079382867126851912785813338522130004988*T7^121 + 122355487004267694100274447785248235456190311583966381243504734341942681923*T7^120 + 33707390068599119703120145260746146223107355980351072218779885897030703576*T7^119 - 595390974429934762692709077205894072566380849508719391419941923788186541526*T7^118 + 561123460686117630163649853668626215276628540046795223013997397962926694640*T7^117 - 765340573469259806526635961120032989147860885793743532845272919839319287589*T7^116 + 8070662137546124995094622679895479402914222600977097485439116748315171971416*T7^115 - 17151412717593359913320293262281742971900449748544331089909812169500027319292*T7^114 - 20514402863996721087142717094363787149990919355258674839396828833900686410752*T7^113 + 170763882127164641455115782590099188809310728335212824828694866896891730491185*T7^112 - 261543647673352476390037280636888099431142503519397226788258700392313716487952*T7^111 + 30841307250319014801789998807337034254690772551689121754305695034891344401656*T7^110 + 703150913473604785768846746622092217699365635908638580575525019628212977900172*T7^109 - 1961494081546566167994339580841918809679532710327917353792831887801833545424664*T7^108 + 4722581232158530343986993868397569452201468437185747448473382699111265561765260*T7^107 - 4788976233669932671149605689718666593894564710651132044362440165777201914613870*T7^106 - 13468327196009129782945107637218142088164295028128289131350832335244557289046832*T7^105 + 53596247279126765185274885890276715019523591212111582376619989338316059653375387*T7^104 - 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45881031146601161941833790805204983429710560829975282500545216081322908340945274432*T7^23 + 21670314696657173783741258093141188576034604666218038146030159541886656821881045632*T7^22 - 6287461952767061116686779869895458387900848207824226648268377524835125752999802880*T7^21 + 638724696752629124461993650212568034603981978918225659980609916353219941080760576*T7^20 + 303324746302481158954345491030439560905185485362607580682293365755747726834786816*T7^19 - 147775296523380816327873931662337176030332718255273859383200714143899149517256704*T7^18 + 24476095515222385815267898170861090909958394202480763993135757448369386865036800*T7^17 + 1298550383024854169991154033311251435332297046825414976232282103463617667790080*T7^16 - 1165685312056836983406433009132789879146648215446129734997033871939541472181248*T7^15 + 141741284039319183255060728946216444199597076034300714779086740254124555290624*T7^14 + 20945516853850764668317231224936940025449103450695802037630735424014873563136*T7^13 - 7289485517951280762091510507505827909869409510715695026494605600368837369856*T7^12 + 399285428885058261877317544117278205296031004773082154017282867722306469888*T7^11 + 147655032614029714542803543112026424117865754554047027634846194975168094208*T7^10 - 37405239962566883264108389666872242514742747434913035544776343797224087552*T7^9 + 3704113154817737095276849815788167708655270617088424939786280928933306368*T7^8 - 212546668531630876393223132356002326713256711502349914451385192427487232*T7^7 + 16467059278653076780392748590079851357399603124024594778478954630316032*T7^6 - 1326188476119064969405376026146670476483668703119209520348064937902080*T7^5 + 66933071865505009151572806260986208548552187533955751571822730166272*T7^4 - 1834510805513328702172357599125426303645511272082426057832684781568*T7^3 + 53014653253756582366982666328195733237471148393900555059484164096*T7^2 - 569948796561198427474421981528235815162228391822144380297150464*T7 + 1888675791296810408729165650912540149049306384976807460929536 acting on $S_{2}^{\mathrm{new}}(930, [\chi])$.

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:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.bt (of order \(60\), degree \(16\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 13.16
Significant digits:
Format:

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