LMFDB - Newform orbit 714.2.bl.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [714,2,Mod(31,714)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(714, base_ring=CyclotomicField(48)) chi = DirichletCharacter(H, H._module([0, 8, 27])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("714.31"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 714 = 2 \cdot 3 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	714.bl (of order $48$, degree $16$, minimal)

Newform invariants

comment:select newform

sage:traces = [192,0,0,0,0,0,0,0,0,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.70131870432$
Analytic rank:	$0$
Dimension:	$192$
Relative dimension:	$12$ over $\Q(\zeta_{48})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{48}]$

$q$-expansion

The algebraic $q$-expansion of this newform has not been computed, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) = $ $ 192 q - 48 q^{10} + 16 q^{11} - 16 q^{14} + 32 q^{15} - 96 q^{18} + 32 q^{22} + 16 q^{25} + 32 q^{28} - 32 q^{29} + 64 q^{35} - 48 q^{37} - 64 q^{43} - 16 q^{46} + 48 q^{47} + 64 q^{49} + 16 q^{53} + 16 q^{58}+ \cdots - 32 q^{99}+O(q^{100}) $

192 * q - 48 * q^10 + 16 * q^11 - 16 * q^14 + 32 * q^15 - 96 * q^18 + 32 * q^22 + 16 * q^25 + 32 * q^28 - 32 * q^29 + 64 * q^35 - 48 * q^37 - 64 * q^43 - 16 * q^46 + 48 * q^47 + 64 * q^49 + 16 * q^53 + 16 * q^58 + 48 * q^61 - 64 * q^65 + 16 * q^70 - 32 * q^71 + 240 * q^73 + 48 * q^74 + 16 * q^77 - 64 * q^78 - 64 * q^79 + 48 * q^80 - 64 * q^85 + 32 * q^86 + 144 * q^87 + 16 * q^88 + 144 * q^89 - 64 * q^91 + 48 * q^94 + 80 * q^95 - 80 * q^98 - 32 * q^99

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} $
31.1	−0.793353	−	0.608761i	−0.659346	+	0.751840i	0.258819	+	0.965926i	−1.16153	+	2.35536i	0.980785	−	0.195090i	1.07493	−	2.41754i	0.382683	−	0.923880i	−0.130526	−	0.991445i	2.35536	−	1.16153i
31.2	−0.793353	−	0.608761i	−0.659346	+	0.751840i	0.258819	+	0.965926i	−0.360559	+	0.731140i	0.980785	−	0.195090i	2.34297	−	1.22902i	0.382683	−	0.923880i	−0.130526	−	0.991445i	0.731140	−	0.360559i
31.3	−0.793353	−	0.608761i	−0.659346	+	0.751840i	0.258819	+	0.965926i	−0.0355436	+	0.0720753i	0.980785	−	0.195090i	−2.59947	−	0.492676i	0.382683	−	0.923880i	−0.130526	−	0.991445i	0.0720753	−	0.0355436i
31.4	−0.793353	−	0.608761i	−0.659346	+	0.751840i	0.258819	+	0.965926i	−0.00979683	+	0.0198660i	0.980785	−	0.195090i	0.596978	+	2.57752i	0.382683	−	0.923880i	−0.130526	−	0.991445i	0.0198660	−	0.00979683i
31.5	−0.793353	−	0.608761i	−0.659346	+	0.751840i	0.258819	+	0.965926i	1.19451	−	2.42223i	0.980785	−	0.195090i	−2.62745	−	0.310617i	0.382683	−	0.923880i	−0.130526	−	0.991445i	−2.42223	+	1.19451i
31.6	−0.793353	−	0.608761i	−0.659346	+	0.751840i	0.258819	+	0.965926i	1.56264	−	3.16873i	0.980785	−	0.195090i	2.34138	−	1.23204i	0.382683	−	0.923880i	−0.130526	−	0.991445i	−3.16873	+	1.56264i
31.7	−0.793353	−	0.608761i	0.659346	−	0.751840i	0.258819	+	0.965926i	−1.58464	+	3.21333i	−0.980785	+	0.195090i	−0.808165	−	2.51930i	0.382683	−	0.923880i	−0.130526	−	0.991445i	3.21333	−	1.58464i
31.8	−0.793353	−	0.608761i	0.659346	−	0.751840i	0.258819	+	0.965926i	−0.928428	+	1.88267i	−0.980785	+	0.195090i	2.18840	+	1.48691i	0.382683	−	0.923880i	−0.130526	−	0.991445i	1.88267	−	0.928428i
31.9	−0.793353	−	0.608761i	0.659346	−	0.751840i	0.258819	+	0.965926i	0.0475892	−	0.0965013i	−0.980785	+	0.195090i	−2.05854	−	1.66205i	0.382683	−	0.923880i	−0.130526	−	0.991445i	−0.0965013	+	0.0475892i
31.10	−0.793353	−	0.608761i	0.659346	−	0.751840i	0.258819	+	0.965926i	0.121926	−	0.247241i	−0.980785	+	0.195090i	−1.88236	+	1.85922i	0.382683	−	0.923880i	−0.130526	−	0.991445i	−0.247241	+	0.121926i
31.11	−0.793353	−	0.608761i	0.659346	−	0.751840i	0.258819	+	0.965926i	1.35865	−	2.75506i	−0.980785	+	0.195090i	−0.276034	+	2.63131i	0.382683	−	0.923880i	−0.130526	−	0.991445i	−2.75506	+	1.35865i
31.12	−0.793353	−	0.608761i	0.659346	−	0.751840i	0.258819	+	0.965926i	1.49760	−	3.03683i	−0.980785	+	0.195090i	1.91292	−	1.82777i	0.382683	−	0.923880i	−0.130526	−	0.991445i	−3.03683	+	1.49760i
61.1	−0.608761	−	0.793353i	−0.997859	−	0.0654031i	−0.258819	+	0.965926i	−3.98166	−	1.35159i	0.555570	+	0.831470i	1.67414	+	2.04872i	0.923880	−	0.382683i	0.991445	+	0.130526i	1.35159	+	3.98166i
61.2	−0.608761	−	0.793353i	−0.997859	−	0.0654031i	−0.258819	+	0.965926i	−2.03737	−	0.691594i	0.555570	+	0.831470i	−2.38560	−	1.14407i	0.923880	−	0.382683i	0.991445	+	0.130526i	0.691594	+	2.03737i
61.3	−0.608761	−	0.793353i	−0.997859	−	0.0654031i	−0.258819	+	0.965926i	−1.46767	−	0.498208i	0.555570	+	0.831470i	−2.46893	+	0.950993i	0.923880	−	0.382683i	0.991445	+	0.130526i	0.498208	+	1.46767i
61.4	−0.608761	−	0.793353i	−0.997859	−	0.0654031i	−0.258819	+	0.965926i	−0.619157	−	0.210176i	0.555570	+	0.831470i	0.818611	−	2.51592i	0.923880	−	0.382683i	0.991445	+	0.130526i	0.210176	+	0.619157i
61.5	−0.608761	−	0.793353i	−0.997859	−	0.0654031i	−0.258819	+	0.965926i	2.71295	+	0.920921i	0.555570	+	0.831470i	−0.695280	+	2.55276i	0.923880	−	0.382683i	0.991445	+	0.130526i	−0.920921	−	2.71295i
61.6	−0.608761	−	0.793353i	−0.997859	−	0.0654031i	−0.258819	+	0.965926i	3.00327	+	1.01947i	0.555570	+	0.831470i	2.54392	−	0.726955i	0.923880	−	0.382683i	0.991445	+	0.130526i	−1.01947	−	3.00327i
61.7	−0.608761	−	0.793353i	0.997859	+	0.0654031i	−0.258819	+	0.965926i	−3.70938	−	1.25917i	−0.555570	−	0.831470i	−1.85824	−	1.88333i	0.923880	−	0.382683i	0.991445	+	0.130526i	1.25917	+	3.70938i
61.8	−0.608761	−	0.793353i	0.997859	+	0.0654031i	−0.258819	+	0.965926i	−1.25442	−	0.425817i	−0.555570	−	0.831470i	−1.07390	+	2.41800i	0.923880	−	0.382683i	0.991445	+	0.130526i	0.425817	+	1.25442i

See next 80 embeddings (of 192 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner
17.e	odd	16	1	inner
119.s	even	48	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	714.2.bl.a	✓	192
7.d	odd	6	1	inner	714.2.bl.a	✓	192
17.e	odd	16	1	inner	714.2.bl.a	✓	192
119.s	even	48	1	inner	714.2.bl.a	✓	192

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
714.2.bl.a	✓	192	1.a	even	1	1	trivial
714.2.bl.a	✓	192	7.d	odd	6	1	inner
714.2.bl.a	✓	192	17.e	odd	16	1	inner
714.2.bl.a	✓	192	119.s	even	48	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{192} - 8 T_{5}^{190} + 4 T_{5}^{188} + 48 T_{5}^{187} + 7680 T_{5}^{186} - 21360 T_{5}^{185} + \cdots + 62\!\cdots\!01 $ T5^192 - 8*T5^190 + 4*T5^188 + 48*T5^187 + 7680*T5^186 - 21360*T5^185 - 60664*T5^184 + 168000*T5^183 + 856368*T5^182 + 1274928*T5^181 + 19949432*T5^180 - 268606896*T5^179 - 78334968*T5^178 + 2090940096*T5^177 - 2191756378*T5^176 - 10530862080*T5^175 + 93501341728*T5^174 - 716314991520*T5^173 + 2930082389028*T5^172 + 7301583172896*T5^171 - 47013919732672*T5^170 + 448019610960*T5^169 + 498415167584000*T5^168 - 1745053387468656*T5^167 + 2679477362419088*T5^166 - 11135439414352224*T5^165 - 85455913429105076*T5^164 + 572929631035550160*T5^163 + 548875634398393088*T5^162 - 7497189844241388336*T5^161 + 8509972201935659285*T5^160 + 33338122006981050048*T5^159 - 92744777732797514608*T5^158 + 903747187900143127296*T5^157 - 4007306996254812656588*T5^156 - 7116582134977897277376*T5^155 + 69415088613234237384080*T5^154 - 17259891499944592688592*T5^153 - 861083052157031320386992*T5^152 + 1018627175532222195654192*T5^151 + 3372668736184515985356624*T5^150 + 15597286918087350579465216*T5^149 - 28116731123478530213809196*T5^148 - 322906758004372919988352848*T5^147 + 300720417597898922682707248*T5^146 + 4196021660832853946823827088*T5^145 - 3883013876067761070228097614*T5^144 - 43701842068615444329462970176*T5^143 + 99448069143380878513178737432*T5^142 - 407934366791455143084376571808*T5^141 + 2527292896428196241860865385856*T5^140 + 1574196743346526448893673299056*T5^139 - 58809956655454433144895936510144*T5^138 + 101653165438586844660649981781088*T5^137 + 640819033989773421051410496732936*T5^136 - 2643259467949518234016017976922640*T5^135 + 1866490573849015033215457027621344*T5^134 + 6116749582703378327023079990680944*T5^133 - 20697528427867148679747025170645828*T5^132 + 98429174122651331930569411965506016*T5^131 - 333116844798022571416998484006640808*T5^130 + 77969345738347327040467654310060784*T5^129 + 2186620176051294815225355188998662538*T5^128 - 4664661758318605660283210710599479424*T5^127 + 4293929516040313086282480695041759208*T5^126 - 8090536139803081891449835122260486784*T5^125 - 52984457388567586332607773921699520284*T5^124 + 424686578729833379251225526802075927696*T5^123 - 320045136362744903277201535067796437184*T5^122 - 1839731742036996171352883296573436511024*T5^121 + 1243673224598529115946588237823615225944*T5^120 - 2821186785111899966834971628205223669728*T5^119 + 20487671937948948123637748209208110671984*T5^118 + 5335271528625576171921920262427267731408*T5^117 - 116301265968167824458936347997633912359824*T5^116 + 66306195127809701125125779514130015326480*T5^115 + 476789467981851665062533978285820008619128*T5^114 + 159723743088583175037982646138867409769440*T5^113 - 420165900940470245296977619422396484477314*T5^112 - 3205872958121241702947705835688876852309632*T5^111 + 7915336470359755290322164167273294249190992*T5^110 + 64272191997278212728743044731895408013250912*T5^109 - 55637553953337910333310740475545112401029092*T5^108 - 784166913322166187000201638500025952400001664*T5^107 - 1577138084302482251986491514862602365803488000*T5^106 + 440082142720168420486822647607291887959545200*T5^105 + 13416162445913373256248272597965109228522178128*T5^104 + 35352436079383318617318845255229618209533329744*T5^103 + 22212756601818961888999722549284629180839347376*T5^102 - 65684789816630641842898350917135938666933413696*T5^101 - 191016169789710186700154576002296182647164338516*T5^100 - 61436710175775018573084575167704809672289712848*T5^99 + 427039006288536795973531592105346009091546823632*T5^98 - 796511901297663235783357080106307294529974158320*T5^97 - 8084358456563415891729541209453546532742563700515*T5^96 - 21119497992658815023831546874174604967012636157120*T5^95 - 23776815465947895097046528810057899081940907790896*T5^94 + 12754747490627766904524007494256125160439668374144*T5^93 + 45500299663096381555429056932308382573475392660492*T5^92 - 184739350265403620083215239160156504561666120984960*T5^91 - 1058482708288666759403046893889003763254667760391152*T5^90 - 2407322588205859417654725702712292612653982492229168*T5^89 - 2124794503288176873641562265430122369592841697007664*T5^88 + 4785946383206801469808051299409392549762944532975504*T5^87 + 26142526942411445364906995411988436045009097669927600*T5^86 + 75593504882814945781627548424745028160169786238566976*T5^85 + 195691190430959804495707783974783713712619821347747036*T5^84 + 512841465411154365119419846945318023198864443810245776*T5^83 + 1335383515961668717833427754850719136871384762032018800*T5^82 + 3294908109851297867169179972710169155642999206194675760*T5^81 + 7661683874702929775990018152385883648281896249602225790*T5^80 + 17230051056901832982128702774917396643708452835457428544*T5^79 + 38689364919012971611806274325485413588197336343121953928*T5^78 + 87260457400049109240412820001455982161071445865780790944*T5^77 + 194240849407181075703405950005626804499251929760429077232*T5^76 + 417417274443701070152662363181962582987162443761850154128*T5^75 + 856555325167671728582495425007941589596988036047109243360*T5^74 + 1681437993748214729277995809309020179163514234148111815392*T5^73 + 3194241374282933373216819266159984969762450837070974259928*T5^72 + 5958153618821046748667094576679777723650814564479073857168*T5^71 + 11031311966727323087819683797013337169652965400438738541920*T5^70 + 20321447873763714148388856307134666624842060222872591839312*T5^69 + 37039795362477315660989904640400201757044437291501999491332*T5^68 + 66158924691091264658912821751248860222258834956808914299104*T5^67 + 114757089734699169540249160410289169677625037113112838207592*T5^66 + 192060230439599596304692422665785983151542862246317630170256*T5^65 + 308981670579347169045548631979998289528626809872837635086986*T5^64 + 476886938603437139373505394953825384771748247760172167780992*T5^63 + 705589566685692577269888005742494880260332378282044229037368*T5^62 + 1001040959818340736608286556389749343394293395589742642968192*T5^61 + 1364010712787183364794044977119187007661526908090006905179484*T5^60 + 1791068278215026529377372403729081636008024101850485049564784*T5^59 + 2278517450889529932987941002764995209443796834528551533538688*T5^58 + 2827552259380054132236132806194859907448806106756791860958000*T5^57 + 3446051002090014950055974725869109192819571363681810365189256*T5^56 + 4140821590924650602986773394928254432120509131439928826022816*T5^55 + 4899511725144749283428478061282838156039288604682581428909712*T5^54 + 5669820435342703401210158462729606446205443796912893870284912*T5^53 + 6350731332153473076999051964991905357719689349939160903536640*T5^52 + 6808267955477205377295533123371669866058084118954627074701360*T5^51 + 6917569217082772282361974757680327971022544572150836769748040*T5^50 + 6613110180618756507248846079716642559700664172344979698839392*T5^49 + 5919978442727241119750724748982429661141953478859644124182706*T5^48 + 4949376506984440338551929061678518174750791462473762639209344*T5^47 + 3861790322577620567203819676844478104454357841990427451206800*T5^46 + 2815609053661667787451096299059239445987575792040603129282976*T5^45 + 1923630865982682779664727192256496989313934763188930097179284*T5^44 + 1235473034235189186641332304683965420221454750013734845578560*T5^43 + 747382048694973105107590743680590634537638956351038813666784*T5^42 + 425672736852587438562185222855897505067915432985102237084112*T5^41 + 227746107343781544527688597980395811335558291573905328358864*T5^40 + 114120485985526905680302817819887434324353078855316025980720*T5^39 + 53365743726038424913665492876980323295249380756384342683920*T5^38 + 23156175537427776590153945590414272468290783970857396087424*T5^37 + 9228752405856322221786640605330017325932651746213859101716*T5^36 + 3319522403426756830509277013845347857185346115136862886096*T5^35 + 1043818547005171628794206584169631920091391537071544498192*T5^34 + 265659604067949479460746632364595466160451285464932163952*T5^33 + 39747177034927394691067743510860699065718304015861222933*T5^32 - 8303657982069684827625783847155863315456342857001510976*T5^31 - 9811953878980797771333641658807000946142978543991598368*T5^30 - 4582347528599569705732023577868472333132760325889470976*T5^29 - 1377951316383075712572266271737567686193222095755197908*T5^28 - 263956771778405029707283775690718768862103262657107424*T5^27 - 20407300108585857553653098850932346053432073131725872*T5^26 + 5162893009633672676665877473040627932232125939250064*T5^25 + 3641538396101648925673502141692702878621347609972608*T5^24 + 1509123936214898998741491850994650199732127994079664*T5^23 + 371464475069562129417654015579328749450542297193328*T5^22 + 38384143573877585073724534665733972817956241880480*T5^21 - 1789592701229969973592975289257449561282723164572*T5^20 - 627627551338699633532802409487128227057805711920*T5^19 - 25246274022492142456008036903444235001398289152*T5^18 - 5837327250229059562200347199474877374097788912*T5^17 - 423474170224121465391771999317766934726570074*T5^16 + 30230193423201512781995667232597301804931264*T5^15 + 4412424981684967590268196891929095444118872*T5^14 + 541739553716129290046561647191117171505056*T5^13 + 22472749624208451188607671510873963916664*T5^12 - 2574704707501469472598001516128488668112*T5^11 - 554055527144582422805109913018862366432*T5^10 - 44430362196574149254900672916396763200*T5^9 + 1154014659034411093139979338558247560*T5^8 + 470556954881865564717708953593130448*T5^7 + 39599343895565941128110221222119488*T5^6 + 1878341086513212801031396641908016*T5^5 + 58204283391684919566811865207204*T5^4 + 1227648197092838727178913404800*T5^3 + 17401118708866818329818789784*T5^2 + 152243688897554206399717200*T5 + 625031758776774379641601 acting on $S_{2}^{\mathrm{new}}(714, [\chi])$.

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	714.bl (of order \(48\), degree \(16\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 31.12
Significant digits:
Format:

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