LMFDB - Newform orbit 810.2.q.d

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:	$180$
Relative dimension:	$10$ 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) = $ $ 180 q+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 180 q - 18 q^{13} + 18 q^{18} + 72 q^{21} + 45 q^{23} + 18 q^{26} + 18 q^{27} - 9 q^{28} + 45 q^{29} + 18 q^{30} + 9 q^{33} - 54 q^{34} - 9 q^{38} - 9 q^{41} + 72 q^{42} - 36 q^{43} - 36 q^{45} + 27 q^{47} - 54 q^{49} + 36 q^{52} + 90 q^{53} - 27 q^{54} - 27 q^{55} + 27 q^{57} + 9 q^{58} - 9 q^{59} + 72 q^{61} + 81 q^{63} - 18 q^{65} - 27 q^{66} - 45 q^{67} + 18 q^{68} - 126 q^{69} + 9 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} - 180 q^{80} + 72 q^{82} + 27 q^{83} + 27 q^{85} - 72 q^{86} - 90 q^{87} + 9 q^{88} - 36 q^{89} - 36 q^{90} - 162 q^{93} - 18 q^{94} - 36 q^{95} + 54 q^{97} - 36 q^{98} + 108 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	−1.68518	−	0.400215i	−0.686242	+	0.727374i	0.0581448	−	0.998308i	0.299982	+	1.70588i	0.405172	−	0.0960275i	0.939693	+	0.342020i	2.67966	+	1.34887i	−0.939693	+	0.342020i
31.2	−0.396080	−	0.918216i	−1.59706	+	0.670362i	−0.686242	+	0.727374i	0.0581448	−	0.998308i	1.24810	+	1.20093i	−0.677468	+	0.160563i	0.939693	+	0.342020i	2.10123	−	2.14122i	−0.939693	+	0.342020i
31.3	−0.396080	−	0.918216i	−1.34637	−	1.08963i	−0.686242	+	0.727374i	0.0581448	−	0.998308i	−0.467246	+	1.66784i	−0.175757	+	0.0416553i	0.939693	+	0.342020i	0.625414	+	2.93409i	−0.939693	+	0.342020i
31.4	−0.396080	−	0.918216i	−0.761820	+	1.55552i	−0.686242	+	0.727374i	0.0581448	−	0.998308i	1.73004	+	0.0834073i	−1.40875	+	0.333879i	0.939693	+	0.342020i	−1.83926	−	2.37005i	−0.939693	+	0.342020i
31.5	−0.396080	−	0.918216i	−0.284741	−	1.70849i	−0.686242	+	0.727374i	0.0581448	−	0.998308i	−1.45598	+	0.938150i	−4.03005	+	0.955139i	0.939693	+	0.342020i	−2.83785	+	0.972951i	−0.939693	+	0.342020i
31.6	−0.396080	−	0.918216i	−0.247139	+	1.71433i	−0.686242	+	0.727374i	0.0581448	−	0.998308i	1.67201	−	0.452084i	4.90991	−	1.16367i	0.939693	+	0.342020i	−2.87784	−	0.847356i	−0.939693	+	0.342020i
31.7	−0.396080	−	0.918216i	0.861686	−	1.50250i	−0.686242	+	0.727374i	0.0581448	−	0.998308i	−1.72091	−	0.196105i	1.23546	−	0.292810i	0.939693	+	0.342020i	−1.51499	−	2.58936i	−0.939693	+	0.342020i
31.8	−0.396080	−	0.918216i	1.13985	+	1.30413i	−0.686242	+	0.727374i	0.0581448	−	0.998308i	0.746000	−	1.56316i	−3.78984	+	0.898208i	0.939693	+	0.342020i	−0.401495	+	2.97301i	−0.939693	+	0.342020i
31.9	−0.396080	−	0.918216i	1.27326	−	1.17422i	−0.686242	+	0.727374i	0.0581448	−	0.998308i	−1.58250	−	0.704046i	−0.454915	+	0.107817i	0.939693	+	0.342020i	0.242403	−	2.99019i	−0.939693	+	0.342020i
31.10	−0.396080	−	0.918216i	1.70782	+	0.288696i	−0.686242	+	0.727374i	0.0581448	−	0.998308i	−0.411348	−	1.68250i	3.22018	−	0.763197i	0.939693	+	0.342020i	2.83331	+	0.986083i	−0.939693	+	0.342020i
61.1	0.993238	−	0.116093i	−1.69755	+	0.343988i	0.973045	−	0.230616i	−0.893633	−	0.448799i	−1.64614	+	0.538736i	1.03413	+	3.45423i	0.939693	−	0.342020i	2.76334	−	1.16787i	−0.939693	−	0.342020i
61.2	0.993238	−	0.116093i	−1.67210	−	0.451758i	0.973045	−	0.230616i	−0.893633	−	0.448799i	−1.71324	−	0.254584i	0.0104735	+	0.0349840i	0.939693	−	0.342020i	2.59183	+	1.51077i	−0.939693	−	0.342020i
61.3	0.993238	−	0.116093i	−1.44718	+	0.951666i	0.973045	−	0.230616i	−0.893633	−	0.448799i	−1.32691	+	1.11324i	−1.42675	−	4.76567i	0.939693	−	0.342020i	1.18866	−	2.75447i	−0.939693	−	0.342020i
61.4	0.993238	−	0.116093i	−1.26093	−	1.18746i	0.973045	−	0.230616i	−0.893633	−	0.448799i	−1.39026	−	1.03305i	−0.324284	−	1.08318i	0.939693	−	0.342020i	0.179873	+	2.99460i	−0.939693	−	0.342020i
61.5	0.993238	−	0.116093i	−0.458946	+	1.67014i	0.973045	−	0.230616i	−0.893633	−	0.448799i	−0.261952	+	1.71213i	−0.641586	−	2.14305i	0.939693	−	0.342020i	−2.57874	−	1.53301i	−0.939693	−	0.342020i
61.6	0.993238	−	0.116093i	0.571466	+	1.63506i	0.973045	−	0.230616i	−0.893633	−	0.448799i	0.757421	+	1.55766i	0.622109	+	2.07799i	0.939693	−	0.342020i	−2.34685	+	1.86876i	−0.939693	−	0.342020i
61.7	0.993238	−	0.116093i	0.681259	−	1.59245i	0.973045	−	0.230616i	−0.893633	−	0.448799i	0.491781	−	1.66077i	1.03725	+	3.46464i	0.939693	−	0.342020i	−2.07177	−	2.16974i	−0.939693	−	0.342020i
61.8	0.993238	−	0.116093i	0.937056	−	1.45668i	0.973045	−	0.230616i	−0.893633	−	0.448799i	0.761610	−	1.55562i	−0.759219	−	2.53597i	0.939693	−	0.342020i	−1.24385	−	2.72999i	−0.939693	−	0.342020i
61.9	0.993238	−	0.116093i	1.67521	+	0.440086i	0.973045	−	0.230616i	−0.893633	−	0.448799i	1.71497	+	0.242631i	−1.21128	−	4.04594i	0.939693	−	0.342020i	2.61265	+	1.47447i	−0.939693	−	0.342020i
61.10	0.993238	−	0.116093i	1.73202	−	0.0105741i	0.973045	−	0.230616i	−0.893633	−	0.448799i	1.71908	−	0.211578i	0.893111	+	2.98320i	0.939693	−	0.342020i	2.99978	−	0.0366290i	−0.939693	−	0.342020i

See next 80 embeddings (of 180 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.d	✓	180
81.g	even	27	1	inner	810.2.q.d	✓	180

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
810.2.q.d	✓	180	1.a	even	1	1	trivial
810.2.q.d	✓	180	81.g	even	27	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{180} + 27 T_{7}^{178} - 57 T_{7}^{177} + 558 T_{7}^{176} + 63 T_{7}^{175} + 7488 T_{7}^{174} + \cdots + 16\!\cdots\!84 $ T7^180 + 27*T7^178 - 57*T7^177 + 558*T7^176 + 63*T7^175 + 7488*T7^174 - 27531*T7^173 - 18477*T7^172 - 66820*T7^171 + 2527227*T7^170 + 5572314*T7^169 + 90553254*T7^168 + 464782716*T7^167 + 4499036982*T7^166 + 11716264575*T7^165 + 21515361192*T7^164 + 139625710866*T7^163 + 1297213550974*T7^162 + 7435287959283*T7^161 + 20453126456241*T7^160 - 12339096387012*T7^159 + 254274071633010*T7^158 + 1342716468946281*T7^157 + 7028912272758039*T7^156 - 608167185566094*T7^155 - 60805546203695607*T7^154 - 184018039129991865*T7^153 - 806352174534226554*T7^152 - 2655994636271537226*T7^151 - 21541622864129232786*T7^150 - 40573830918205744551*T7^149 + 381379970532792878199*T7^148 - 285697423331130676743*T7^147 - 11694777561442014164091*T7^146 - 11898543069726952462713*T7^145 + 72899531779667755165914*T7^144 + 112573318366692435172653*T7^143 - 28831855723034590474002*T7^142 + 897047665359095355076503*T7^141 + 63160176658092017672968218*T7^140 + 215514813589185322763656263*T7^139 + 1453383850173986600962038*T7^138 + 374442576543029052554635170*T7^137 + 6808628042429156118858389643*T7^136 - 18120745831355974880289270111*T7^135 - 158131521734929583814813737439*T7^134 + 525675877771782861951829196877*T7^133 + 3851531315032237545554962269084*T7^132 + 4522934122075170191907971568666*T7^131 - 5472858319241218005627669563667*T7^130 - 82514776939950377705506282334628*T7^129 - 597180032923975900470333486418707*T7^128 - 3019686484658229737783020019389158*T7^127 - 4704161114017755179315274568120968*T7^126 + 40909720563749444738172990211768986*T7^125 + 147512911454035826525197256315968551*T7^124 - 106802339276821561108175831623743312*T7^123 + 181077195490062793281807082768034550*T7^122 - 1879521200011386231983745183988502931*T7^121 - 31783491330362658860819186650204423557*T7^120 - 26805199674494059165844581356679382541*T7^119 + 452200377170011963139796457540656474138*T7^118 + 1211050751181287374311484440054392840289*T7^117 + 1050929385862216601185394818928013031149*T7^116 + 9566808615037403694774282251876970565683*T7^115 - 10796172667819668626238933160472392389417*T7^114 - 213634552485069626814679340186938879952625*T7^113 - 53334137051481530839516122355376180961519*T7^112 + 1799533262470620011277162200743938794929695*T7^111 + 2811092392207349523716199528387970491153066*T7^110 + 15282504101671359943863549072736459661684013*T7^109 + 48681417051299346372993609313904410545571296*T7^108 - 148932879304455368020806515158986478358741304*T7^107 - 510704001825841166245336227677487298128683217*T7^106 - 114596471196736459758075100326479696051343441*T7^105 + 1663394352023286687353693988517218709548302529*T7^104 + 13965896062716240947354289952586783203212777861*T7^103 + 69019939632404647778254210449275694644815090119*T7^102 + 97319032766302665242633577741123399163725742764*T7^101 + 23990765343540801275435651586750438837605790324*T7^100 + 28194786345605836585097461844522031679961206220*T7^99 - 501986984577488970692803685853399706480361331768*T7^98 + 4916251826933849124412206923363264859215775258624*T7^97 + 33938985308908857505699401929102891036559582527598*T7^96 + 84214982129428578575580199973162296623142078414248*T7^95 + 210878116847654330401007941092116965291418349094734*T7^94 + 537090314266501658408876362558466464958424483437352*T7^93 + 17520602289697972638674955367425806253896537990262*T7^92 - 1748395771031312812876788445476624090209003353994703*T7^91 - 8191996088123987893832732293251594943151066231706617*T7^90 - 48497108008701708843622127071918636503227001218731998*T7^89 - 127273092473641050603038328296315402312174209544423208*T7^88 - 242968479811270881682634020341035601913712720575472280*T7^87 - 345362510728142318943212447292567115588963829982070488*T7^86 + 154520828237831477200340007191125427774571682173345777*T7^85 + 1863258721271737080519696153911844138146423740274607222*T7^84 + 5153396957180914086090731595070454794449961077861414282*T7^83 + 23099737072815638714308766016739031473772160340679306340*T7^82 + 54320421303628532822215957413285230197464626463613005716*T7^81 + 127015864250197743375781526013076400322562862158155203553*T7^80 + 253207866061519810605977476814341284804241188524577620904*T7^79 + 339194142015608601451053975902268951902210791113619013160*T7^78 + 313087512604770733035901161860414781973025666921668680075*T7^77 + 224504755825557920719897700957135269077544995490144669959*T7^76 - 1909605644469296744732728476744646732112662002542406969606*T7^75 - 5594353921897601678781538166620924595843249453457056397483*T7^74 - 16941090569271901075338048798211155707118502866496067810823*T7^73 - 48568438720823379798537301743709765767885527344677681630462*T7^72 - 114481072159294898406621351708814262003285119136301654331848*T7^71 - 244946771550825485567560319514413155560674695740974051921268*T7^70 - 411320001773049044495604303822606228663227933483028045756588*T7^69 - 458781339467870080396736769765726765591629730318933629426310*T7^68 - 331925508067259500480949454740169539301576774937098132190953*T7^67 + 345833429181029942118300056153925820699450631105451079507353*T7^66 + 4544159711625358582627066169585504050923120231777658097706765*T7^65 + 17324872388121480149192626829498111178342519697199711137647198*T7^64 + 43258481421543765936730272254579634491791428379607835605105394*T7^63 + 90175107038670824068032564666555737193096359262753627200914683*T7^62 + 171320293449658072125169243538122031399817444098305651556129970*T7^61 + 304998607809854720527174923931582320652026952019671667627971140*T7^60 + 574084588760452362057636776240356113542431299714747763526547801*T7^59 + 1190679472607160078023249165607105162944105663587514600802795564*T7^58 + 2351212655457309036515524040506313457087139777383470160823768161*T7^57 + 4225911479723911412318049840390871824598274383817767435300682559*T7^56 + 7228874990455517784310606030656147326499143236779424812224451492*T7^55 + 11417828647945436375236283461671561917587088534798183331392915055*T7^54 + 14764308503864297429752138598667772310327597634692076413005299693*T7^53 + 12512300864535945637133253519449418127896794679167430125945577915*T7^52 - 1666391601992041282150585404149170797066441791540272924709580591*T7^51 - 33964460769394619746851653792206464537044675760846012415255099584*T7^50 - 87609129408300985193099897364543914209157630293962965152220477505*T7^49 - 158421480421909455803763144463289670264721627744777913201753998896*T7^48 - 222878403239067092938507130225148915895202157769245123904076045595*T7^47 - 219272107239630826148986225565534842900611597504639780662341508880*T7^46 - 57908498430191482414230126518535329228529132903656488017875302875*T7^45 + 310219801854097988567216711407672060398518568949117969445644398270*T7^44 + 819152862791348871146714414965157690679265165994446128068559078192*T7^43 + 1300967059451840917297491868387918078168363998902584384554472650069*T7^42 + 1561358204721925566927777578171606789590372464363420874607624184981*T7^41 + 1370170665115847579247070425652213656765478604807054265216622172997*T7^40 + 342019338086345516335329800802143320548022187659229702267648902254*T7^39 - 1813001143869717093345030782768452219661188230466480181512403257677*T7^38 - 4548953361647164313684909938252852676551292206703003418350061271104*T7^37 - 6216773725167693444634146580281497632723760657490908410303051559250*T7^36 - 4972117473639949000385503074775997092109034519701200293670580083767*T7^35 - 530290270050947099461133024972985021315603370583368669404442190427*T7^34 + 5008949912804945269350946361227356990965085420709194825193213303221*T7^33 + 8694314362283547385015476333632546953933890076291002278239325714748*T7^32 + 9047966094893991527986902398547504918054691104628350704569331010981*T7^31 + 6903375301250804456693455638409766032854878024079333559640042889153*T7^30 + 4235875638306699053924305692766786449688805928160459128569253717009*T7^29 + 2320877071846039416819116689500409116239496022224087849049745366127*T7^28 + 1202064336690337356291848017498630000203195441116908765991971248222*T7^27 + 533004570712335788075668973049092087415299461493840434943466271077*T7^26 + 167074749041502146956988863932016852928129696062342235337912672984*T7^25 + 32086968978714279431884746606778924889180151914874119218792462548*T7^24 + 4155138181416358979919047094948686258165401191096153106712732199*T7^23 - 4225558775446107439787446936679263361099565752340316033860382212*T7^22 - 8061381861113860386511897311197848995368891523811266766180256265*T7^21 - 4939248657539280686902612200814210879835743247484016695891130383*T7^20 - 789680785625550726563484335179083338348347293596376423410866302*T7^19 + 610266207758794234619589919206114701611038046880682231373897765*T7^18 + 438902316286852099422252312494898145877186365584756068793454788*T7^17 + 142739580628666093833401113678168610921306797676467822588242060*T7^16 + 28177409097001785883522545592534908964100892126465624108902712*T7^15 + 3514937409659385304141646093882690463728786258638709395442736*T7^14 + 262540104496304876824271635517455112982729107517907734254400*T7^13 + 9785803477256440218623246204775653214625960281931197956544*T7^12 + 115098565903385527944696299701714675758094725715750803072*T7^11 + 2660106873667762183103474975154486361282976397721884672*T7^10 - 12239860678460795976579801764841132607503735540087808*T7^9 + 46196533758040179127437646588211583432908074170964992*T7^8 - 1786320641854723005438784728113890759847532408539136*T7^7 + 353912298508670850329452614491322465953253468008448*T7^6 - 14487869505478704878475783272703457980885701320704*T7^5 + 751034340799515642011026589343914301168691101696*T7^4 - 18266228012390014908829111754564761725125885952*T7^3 + 349568811215366969636950224192507629691273216*T7^2 - 3522624061936580257297452258593682588303360*T7 + 16648054751873509601715667724677719261184 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 31.10
Significant digits:
Format:

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