LMFDB - Newform orbit 810.2.q.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,2,Mod(31,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(54))

chi = DirichletCharacter(H, H._module([20, 0]))

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

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

chi := DirichletCharacter("810.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	810.q (of order $27$, degree $18$, 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:	$6.46788256372$
Analytic rank:	$0$
Dimension:	$162$
Relative dimension:	$9$ over $\Q(\zeta_{27})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{27}]$

$q$-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) = $ $ 162 q+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 162 q - 18 q^{13} - 18 q^{18} - 72 q^{21} + 63 q^{23} - 18 q^{26} + 36 q^{27} + 9 q^{28} + 9 q^{29} + 18 q^{30} + 9 q^{33} + 54 q^{34} - 9 q^{38} + 18 q^{41} - 81 q^{42} - 36 q^{43} + 36 q^{45} + 27 q^{47} - 9 q^{48} + 54 q^{49} + 36 q^{52} - 36 q^{53} + 27 q^{54} - 27 q^{55} + 27 q^{57} - 18 q^{58} - 9 q^{59} - 72 q^{61} - 99 q^{63} - 18 q^{65} - 45 q^{66} + 63 q^{67} + 18 q^{68} + 63 q^{69} + 18 q^{70} - 72 q^{71} - 9 q^{72} - 18 q^{73} - 36 q^{74} + 9 q^{76} - 72 q^{77} - 72 q^{78} - 18 q^{79} + 162 q^{80} - 144 q^{81} - 72 q^{82} - 198 q^{83} - 18 q^{84} + 27 q^{85} - 72 q^{86} - 36 q^{87} + 9 q^{88} - 36 q^{89} - 36 q^{90} - 18 q^{93} + 45 q^{94} - 36 q^{95} + 90 q^{97} - 36 q^{98} - 90 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} $
31.1	−0.396080	−	0.918216i	−1.67127	−	0.454831i	−0.686242	+	0.727374i	−0.0581448	+	0.998308i	0.244321	+	1.71473i	1.90982	−	0.452636i	0.939693	+	0.342020i	2.58626	+	1.52029i	0.939693	−	0.342020i
31.2	−0.396080	−	0.918216i	−1.53152	+	0.808981i	−0.686242	+	0.727374i	−0.0581448	+	0.998308i	1.34942	+	1.08584i	−4.46797	+	1.05893i	0.939693	+	0.342020i	1.69110	−	2.47794i	0.939693	−	0.342020i
31.3	−0.396080	−	0.918216i	−0.812412	−	1.52970i	−0.686242	+	0.727374i	−0.0581448	+	0.998308i	−1.08282	+	1.35185i	3.62367	−	0.858826i	0.939693	+	0.342020i	−1.67997	+	2.48550i	0.939693	−	0.342020i
31.4	−0.396080	−	0.918216i	−0.721884	+	1.57445i	−0.686242	+	0.727374i	−0.0581448	+	0.998308i	1.73161	+	0.0392394i	−0.596907	+	0.141470i	0.939693	+	0.342020i	−1.95777	−	2.27314i	0.939693	−	0.342020i
31.5	−0.396080	−	0.918216i	0.0129628	+	1.73200i	−0.686242	+	0.727374i	−0.0581448	+	0.998308i	1.58522	−	0.697914i	1.23412	−	0.292491i	0.939693	+	0.342020i	−2.99966	+	0.0449033i	0.939693	−	0.342020i
31.6	−0.396080	−	0.918216i	0.457047	−	1.67066i	−0.686242	+	0.727374i	−0.0581448	+	0.998308i	−1.71505	+	0.242047i	−1.11093	+	0.263296i	0.939693	+	0.342020i	−2.58222	−	1.52714i	0.939693	−	0.342020i
31.7	−0.396080	−	0.918216i	1.37358	+	1.05513i	−0.686242	+	0.727374i	−0.0581448	+	0.998308i	0.424789	−	1.67915i	2.01281	−	0.477046i	0.939693	+	0.342020i	0.773417	+	2.89859i	0.939693	−	0.342020i
31.8	−0.396080	−	0.918216i	1.44566	−	0.953971i	−0.686242	+	0.727374i	−0.0581448	+	0.998308i	−1.44855	−	0.949582i	3.83233	−	0.908278i	0.939693	+	0.342020i	1.17988	−	2.75824i	0.939693	−	0.342020i
31.9	−0.396080	−	0.918216i	1.55204	−	0.768881i	−0.686242	+	0.727374i	−0.0581448	+	0.998308i	−1.32073	−	1.12057i	−3.93181	+	0.931856i	0.939693	+	0.342020i	1.81764	−	2.38666i	0.939693	−	0.342020i
61.1	0.993238	−	0.116093i	−1.41520	+	0.998609i	0.973045	−	0.230616i	0.893633	+	0.448799i	−1.28970	+	1.15615i	−0.536847	−	1.79320i	0.939693	−	0.342020i	1.00556	−	2.82645i	0.939693	+	0.342020i
61.2	0.993238	−	0.116093i	−1.40036	−	1.01931i	0.973045	−	0.230616i	0.893633	+	0.448799i	−1.50922	−	0.849848i	0.446772	+	1.49232i	0.939693	−	0.342020i	0.922008	+	2.85480i	0.939693	+	0.342020i
61.3	0.993238	−	0.116093i	−0.906790	−	1.47571i	0.973045	−	0.230616i	0.893633	+	0.448799i	−1.07198	−	1.36046i	0.893266	+	2.98372i	0.939693	−	0.342020i	−1.35546	+	2.67633i	0.939693	+	0.342020i
61.4	0.993238	−	0.116093i	−0.659316	+	1.60166i	0.973045	−	0.230616i	0.893633	+	0.448799i	−0.468917	+	1.66737i	1.32137	+	4.41367i	0.939693	−	0.342020i	−2.13060	−	2.11199i	0.939693	+	0.342020i
61.5	0.993238	−	0.116093i	−0.269948	−	1.71089i	0.973045	−	0.230616i	0.893633	+	0.448799i	−0.466745	−	1.66798i	−1.17767	−	3.93368i	0.939693	−	0.342020i	−2.85426	+	0.923701i	0.939693	+	0.342020i
61.6	0.993238	−	0.116093i	1.33184	+	1.10734i	0.973045	−	0.230616i	0.893633	+	0.448799i	1.45139	+	0.945230i	0.508204	+	1.69752i	0.939693	−	0.342020i	0.547617	+	2.94960i	0.939693	+	0.342020i
61.7	0.993238	−	0.116093i	1.34364	−	1.09300i	0.973045	−	0.230616i	0.893633	+	0.448799i	1.20766	−	1.24159i	1.10743	+	3.69908i	0.939693	−	0.342020i	0.610713	−	2.93718i	0.939693	+	0.342020i
61.8	0.993238	−	0.116093i	1.34389	+	1.09268i	0.973045	−	0.230616i	0.893633	+	0.448799i	1.46166	+	0.929275i	−0.495456	−	1.65494i	0.939693	−	0.342020i	0.612101	+	2.93689i	0.939693	+	0.342020i
61.9	0.993238	−	0.116093i	1.51378	−	0.841700i	0.973045	−	0.230616i	0.893633	+	0.448799i	1.40583	−	1.01175i	−0.821784	−	2.74495i	0.939693	−	0.342020i	1.58308	−	2.54830i	0.939693	+	0.342020i
121.1	0.0581448	+	0.998308i	−1.72109	−	0.194558i	−0.993238	+	0.116093i	0.973045	+	0.230616i	0.0941566	−	1.72949i	2.76758	−	3.71751i	−0.173648	−	0.984808i	2.92429	+	0.669704i	−0.173648	+	0.984808i
121.2	0.0581448	+	0.998308i	−1.45014	+	0.947148i	−0.993238	+	0.116093i	0.973045	+	0.230616i	−1.02986	−	1.39262i	0.128210	−	0.172215i	−0.173648	−	0.984808i	1.20582	−	2.74700i	−0.173648	+	0.984808i

See next 80 embeddings (of 162 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
81.g	even	27	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	810.2.q.b	✓	162
81.g	even	27	1	inner	810.2.q.b	✓	162

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
810.2.q.b	✓	162	1.a	even	1	1	trivial
810.2.q.b	✓	162	81.g	even	27	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{162} - 27 T_{7}^{160} + 150 T_{7}^{159} + 756 T_{7}^{158} - 3852 T_{7}^{157} + \cdots + 15\!\cdots\!96 $ T7^162 - 27*T7^160 + 150*T7^159 + 756*T7^158 - 3852*T7^157 + 8346*T7^156 + 112122*T7^155 - 323442*T7^154 - 346239*T7^153 + 10634409*T7^152 + 1346841*T7^151 - 149984295*T7^150 + 277408881*T7^149 + 2220243795*T7^148 - 9119536230*T7^147 - 13180388922*T7^146 + 222383708991*T7^145 + 246142534410*T7^144 - 3383509282821*T7^143 - 4917963646251*T7^142 + 16409524625223*T7^141 + 307067185218060*T7^140 - 1555178597429877*T7^139 - 5405514939999843*T7^138 + 40773022531947837*T7^137 + 33672699179640873*T7^136 + 67955449374228471*T7^135 - 477805948495482033*T7^134 + 7718416380349300368*T7^133 - 6324450190571632506*T7^132 - 290812807194846890562*T7^131 + 398363229909323457984*T7^130 - 1944203647596720336588*T7^129 + 10983119231080747693575*T7^128 - 8035318673115983566668*T7^127 + 45944689018828619317176*T7^126 + 568161944246056422432168*T7^125 - 2292077664622264127428656*T7^124 + 4696152979505993206398942*T7^123 - 37174177987010623057561614*T7^122 + 80541511453433445903417114*T7^121 - 146262048015437604855115206*T7^120 + 145920701640748968700876584*T7^119 + 2801058714134869900161335445*T7^118 - 17222320600756820516690293878*T7^117 + 109114068223989895962114290043*T7^116 - 536039703484221986174548946547*T7^115 + 2142785305702933424003258285151*T7^114 - 8304277423105036996442758766718*T7^113 + 24225940856444771985424033898331*T7^112 - 43731829097784864405449261536161*T7^111 - 36241416339741458418810446380053*T7^110 + 915550876201612646828360838740598*T7^109 - 5271344267653556219228047545427068*T7^108 + 22756139978251317011947710775464732*T7^107 - 75455006694451794076250760837244974*T7^106 + 175793882758654223968459409572307646*T7^105 - 169223518010150532016785865768195926*T7^104 - 1451463907535244654237945029270678139*T7^103 + 12669200273269670574985462199365397949*T7^102 - 67958996944560323578685929921091956047*T7^101 + 307966750302974634927382547301163816752*T7^100 - 1252445129709247872836754090969322954350*T7^99 + 4692763550523161139366144038740702757172*T7^98 - 16648997241990245758188588854861142079872*T7^97 + 56061918862692321233973345775490387394192*T7^96 - 179767535574685978993978028196268836722226*T7^95 + 553931253436580389245708745064157165260688*T7^94 - 1639101924963431330955692088199483941734217*T7^93 + 4644799366577866462766895996231752432363037*T7^92 - 12672682079800485805632199397412937088905122*T7^91 + 33423982851177362464228709646015008716374690*T7^90 - 83752670482747112585975193262022091492608730*T7^89 + 203307769049088649831027668242800468266990690*T7^88 - 478708685984517903653453428523431363391529066*T7^87 + 1054627311989566081129692212582300823050020395*T7^86 - 2298639508756971139798627694551526049471428202*T7^85 + 4871095904781990959113156323445104383503991364*T7^84 - 9487013042061910951507272762347274111709703385*T7^83 + 19179997732129066770939722207513005797218201436*T7^82 - 37930874516458919918741480324459639586743121712*T7^81 + 65595485982699066717874444884888106014424860429*T7^80 - 131303226970901199674948715818518756039729241985*T7^79 + 248967045562640711204278711658950485271779085359*T7^78 - 357203162795868186787794235760412170324458058883*T7^77 + 783535529703337470031881477891652904114593603929*T7^76 - 1276013730361539625052392485393714884961459965517*T7^75 + 1346543066496946885319803209018580269287017194735*T7^74 - 3756331098915146856803333756205247504646485721439*T7^73 + 3859793204082472396310929716320334072761457844260*T7^72 - 2444553837260321130968708598738200942692786026381*T7^71 + 13270751260584311903296585200223377232564427087532*T7^70 - 1308539123435310265858239435719986182101151394404*T7^69 + 3360608681642987487438194106882384540831477349806*T7^68 - 28199097146907317274851007676634059242377041288536*T7^67 - 33572384785864847041190552513978642056568025380538*T7^66 + 21782802302284756882934437744630439735053739478198*T7^65 - 149496857898205528720546729842019550320056957138*T7^64 + 262442974082032894174099990112453929384488321007089*T7^63 - 353395199971041772201572749797224128086821857371833*T7^62 + 147555167692586191938938579180928336342784696176540*T7^61 - 1109217191014114850864529491388545722848689933485284*T7^60 + 1706546847527446599801538084642388096573022009883770*T7^59 + 2701725616978274207881542077782492321597359542453532*T7^58 + 2005226973179298537243528989531744231132446170752514*T7^57 + 2367296604807561542085532698734338310086580041873257*T7^56 - 10235735890926755156266010790474074324584769405127957*T7^55 + 10834254687748176590485847135641810749257555874207336*T7^54 + 17495846919681104919035293676356732585011448208751807*T7^53 + 49460244398506254351979612841058878363789346632595825*T7^52 + 31589147837831435651709040607412853050057622803654954*T7^51 + 69818321564793612213989632998080386247783859484917927*T7^50 + 137314102010313533907893184550240658590844370094086321*T7^49 - 110134991323133989964280393405339336896487131396810049*T7^48 + 53707121281540656256116125808774723582522507578556771*T7^47 + 548033359151386505711875274473553039486038185852834699*T7^46 + 693910655576130738365865513000103944987152553053941479*T7^45 + 667187613468726861309640214945211800235100559973601476*T7^44 - 872586339014096478320748129136110036035038434177712923*T7^43 - 965407664949227401784936614942423332589759147244678373*T7^42 + 7124087299744309791679030517037530441527700108042031423*T7^41 + 1331910504883660326964175021920366611652245240573883086*T7^40 - 14105845654502079644354609019067097537043219543425302788*T7^39 + 12356273637163353556955357360177702232774269524294719459*T7^38 + 22216061616995107258871120711530965627077387250043875306*T7^37 - 23085134353872546890458758215828486300147252339292999159*T7^36 - 11526019338123015558808983878461081781183424068692070946*T7^35 + 38117335216353810488831896775528454132012416218199190066*T7^34 + 11508482960150950640351531637816191770494501189178428684*T7^33 - 24326686063749010857589845633154610063683283506235524314*T7^32 + 11463911417641499068168925755723320159204836269822001628*T7^31 + 25320390353372965346474872894484178921069345727987398840*T7^30 + 3177651565839453257866430537604055377376804650840395364*T7^29 + 21303125338394301413283682699593726206533710772818218973*T7^28 + 26053543479574224590219127592591742500165132720125025140*T7^27 + 14668038451731405543852609084420265215918057504269055489*T7^26 + 29505719797409831234527059536955408904777550599217662434*T7^25 + 33845736632963064667691110632131132773149049771619572707*T7^24 + 22696057375892194803002442649245638510249995929229693533*T7^23 + 16467988026102423090590171054514979087550270080281046107*T7^22 + 6794796268193439183536690946603646684302598043017624086*T7^21 + 2226591531561805166166551587866695165661245934719886273*T7^20 + 1883376309670052659806702725727029981893103433764741289*T7^19 - 145884016322021453673584520767168759555222522139169983*T7^18 + 260111409548649913527914967971869420900491351644001072*T7^17 + 309658105743173560767163958225085973895380478435696388*T7^16 - 192644338745557644097467545601993906760143906819696312*T7^15 + 72296514697785442186953292335028584104056244449772400*T7^14 + 25913332566556922031233921333041252015825465657856256*T7^13 - 30796278961406454657234736226226359840886279564021312*T7^12 + 16531523620755242185387891852049112454153943579889792*T7^11 - 1854656789807318897472598984667750788603728458617344*T7^10 - 1973642092788589718249651038169501214341971742324736*T7^9 + 1326658694626547961018069806855930776771895036451840*T7^8 - 424468833006446394257295585598835037661483104995328*T7^7 + 86285293712270752292183743248821962539105533792256*T7^6 - 11018134815714374056054156204419327211743929966592*T7^5 + 782646420702181034727980745579075170753625636864*T7^4 + 2285986434803217502745081744113702399289819136*T7^3 - 4414258768586747008946271777942065409045233664*T7^2 + 263577217434479948637104880458110113530118144*T7 + 15585893134592289342038436984106138051280896 acting on $S_{2}^{\mathrm{new}}(810, [\chi])$.

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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	810.q (of order \(27\), degree \(18\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 31.9
Significant digits:
Format:

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