LMFDB - Newform orbit 798.2.cc.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [798,2,Mod(317,798)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(798, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 6, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("798.317"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 798 = 2 \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	798.cc (of order $18$, degree $6$, minimal)

Newform invariants

comment:select newform

sage:traces = [162,0,0,0,0,0,0,81] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)

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

Self dual:	no
Analytic conductor:	$6.37206208130$
Analytic rank:	$0$
Dimension:	$162$
Relative dimension:	$27$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

$\operatorname{Tr}(f)(q) = $ $ 162 q + 81 q^{8} + 15 q^{13} - 9 q^{14} + 3 q^{15} - 12 q^{17} + 18 q^{18} - 36 q^{19} - 6 q^{22} + 12 q^{25} + 9 q^{27} - 12 q^{28} - 6 q^{29} - 12 q^{33} + 6 q^{34} + 12 q^{35} - 18 q^{37} - 9 q^{39}+ \cdots + 39 q^{99}+O(q^{100}) $

162 * q + 81 * q^8 + 15 * q^13 - 9 * q^14 + 3 * q^15 - 12 * q^17 + 18 * q^18 - 36 * q^19 - 6 * q^22 + 12 * q^25 + 9 * q^27 - 12 * q^28 - 6 * q^29 - 12 * q^33 + 6 * q^34 + 12 * q^35 - 18 * q^37 - 9 * q^39 - 9 * q^43 - 6 * q^44 + 18 * q^45 - 18 * q^46 - 12 * q^47 + 24 * q^49 - 87 * q^50 + 24 * q^51 + 12 * q^52 + 36 * q^53 + 18 * q^54 - 18 * q^57 - 24 * q^59 - 27 * q^60 - 9 * q^61 - 36 * q^62 + 6 * q^63 - 81 * q^64 + 72 * q^65 + 12 * q^66 - 63 * q^67 - 9 * q^69 + 12 * q^70 - 12 * q^71 - 51 * q^73 + 42 * q^74 - 18 * q^78 + 21 * q^79 - 12 * q^81 + 3 * q^84 - 6 * q^85 - 36 * q^86 - 84 * q^87 + 54 * q^89 + 12 * q^90 + 75 * q^91 - 66 * q^93 + 72 * q^94 - 12 * q^97 - 12 * q^98 + 39 * 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} $
317.1	−0.173648	−	0.984808i	−1.66333	−	0.483049i	−0.939693	+	0.342020i	0.727990	−	2.00013i	−0.186877	+	1.72194i	−2.16840	−	1.51592i	0.500000	+	0.866025i	2.53333	+	1.60694i	−2.09616	−	0.369610i
317.2	−0.173648	−	0.984808i	−1.63247	−	0.578829i	−0.939693	+	0.342020i	0.104841	−	0.288049i	−0.286560	+	1.70818i	2.04725	−	1.67593i	0.500000	+	0.866025i	2.32991	+	1.88984i	−0.301878	−	0.0532293i
317.3	−0.173648	−	0.984808i	−1.59272	+	0.680625i	−0.939693	+	0.342020i	0.424869	−	1.16732i	0.946857	+	1.45033i	0.842300	+	2.50809i	0.500000	+	0.866025i	2.07350	−	2.16809i	−1.22336	−	0.215712i
317.4	−0.173648	−	0.984808i	−1.56298	+	0.746396i	−0.939693	+	0.342020i	0.834649	−	2.29318i	1.00646	+	1.40962i	1.57431	−	2.12639i	0.500000	+	0.866025i	1.88578	−	2.33320i	−2.40328	−	0.423763i
317.5	−0.173648	−	0.984808i	−1.55093	+	0.771120i	−0.939693	+	0.342020i	−1.48651	+	4.08415i	1.02872	+	1.39346i	1.98637	−	1.74766i	0.500000	+	0.866025i	1.81075	−	2.39190i	4.28024	+	0.754721i
317.6	−0.173648	−	0.984808i	−1.52684	−	0.817784i	−0.939693	+	0.342020i	−1.22401	+	3.36293i	−0.540228	+	1.64565i	−2.57164	+	0.621829i	0.500000	+	0.866025i	1.66246	+	2.49725i	3.52438	+	0.621444i
317.7	−0.173648	−	0.984808i	−1.28388	−	1.16261i	−0.939693	+	0.342020i	1.12250	−	3.08403i	−0.922003	+	1.46626i	−1.00637	+	2.44688i	0.500000	+	0.866025i	0.296682	+	2.98529i	−3.23210	−	0.569906i
317.8	−0.173648	−	0.984808i	−1.16044	+	1.28583i	−0.939693	+	0.342020i	−0.749562	+	2.05941i	1.46781	+	0.919531i	−1.89892	+	1.84231i	0.500000	+	0.866025i	−0.306742	−	2.98428i	2.15828	+	0.380563i
317.9	−0.173648	−	0.984808i	−0.850014	−	1.50913i	−0.939693	+	0.342020i	−0.726838	+	1.99697i	−1.33860	+	1.09916i	2.31032	−	1.28935i	0.500000	+	0.866025i	−1.55495	+	2.56557i	2.09285	+	0.369026i
317.10	−0.173648	−	0.984808i	−0.807441	+	1.53233i	−0.939693	+	0.342020i	−0.0331429	+	0.0910594i	1.64926	+	0.529087i	−1.96662	−	1.76986i	0.500000	+	0.866025i	−1.69608	−	2.47453i	0.0954312	+	0.0168271i
317.11	−0.173648	−	0.984808i	−0.593593	−	1.62716i	−0.939693	+	0.342020i	−0.662980	+	1.82152i	−1.49936	+	0.867129i	−1.38490	−	2.25434i	0.500000	+	0.866025i	−2.29529	+	1.93174i	1.90898	+	0.336604i
317.12	−0.173648	−	0.984808i	−0.560498	−	1.63885i	−0.939693	+	0.342020i	−0.0666847	+	0.183215i	−1.51663	+	0.836567i	1.64561	+	2.07170i	0.500000	+	0.866025i	−2.37168	+	1.83715i	0.192011	+	0.0338567i
317.13	−0.173648	−	0.984808i	−0.218213	+	1.71825i	−0.939693	+	0.342020i	0.282752	−	0.776854i	1.73004	−	0.0834729i	2.59722	+	0.504428i	0.500000	+	0.866025i	−2.90477	−	0.749890i	−0.814151	−	0.143557i
317.14	−0.173648	−	0.984808i	−0.0146433	+	1.73199i	−0.939693	+	0.342020i	1.33157	−	3.65844i	1.70822	−	0.286336i	−1.16329	+	2.37629i	0.500000	+	0.866025i	−2.99957	−	0.0507239i	−3.83409	−	0.676053i
317.15	−0.173648	−	0.984808i	0.0195594	−	1.73194i	−0.939693	+	0.342020i	1.17404	−	3.22564i	−1.70902	+	0.281486i	2.53058	+	0.772120i	0.500000	+	0.866025i	−2.99923	−	0.0677514i	−3.38050	−	0.596074i
317.16	−0.173648	−	0.984808i	0.291399	+	1.70736i	−0.939693	+	0.342020i	−0.687483	+	1.88884i	1.63082	−	0.583453i	−0.708845	−	2.54903i	0.500000	+	0.866025i	−2.83017	+	0.995048i	1.97953	+	0.349044i
317.17	−0.173648	−	0.984808i	0.687834	−	1.58962i	−0.939693	+	0.342020i	0.890265	−	2.44598i	−1.68491	−	0.401350i	−2.59902	−	0.495088i	0.500000	+	0.866025i	−2.05377	−	2.18679i	−2.56342	−	0.452000i
317.18	−0.173648	−	0.984808i	0.983012	−	1.42607i	−0.939693	+	0.342020i	−0.835832	+	2.29643i	−1.57511	−	0.720442i	1.05193	+	2.42764i	0.500000	+	0.866025i	−1.06738	−	2.80370i	2.40668	+	0.424363i
317.19	−0.173648	−	0.984808i	1.07604	+	1.35726i	−0.939693	+	0.342020i	0.303001	−	0.832489i	1.14978	−	1.29538i	−2.63559	+	0.231649i	0.500000	+	0.866025i	−0.684287	+	2.92092i	−0.872457	−	0.153838i
317.20	−0.173648	−	0.984808i	1.11980	+	1.32138i	−0.939693	+	0.342020i	−0.0590761	+	0.162310i	1.10686	−	1.33224i	2.55580	−	0.684024i	0.500000	+	0.866025i	−0.492096	+	2.95937i	0.170103	+	0.0299937i

See next 80 embeddings (of 162 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
399.bv	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	798.2.cc.b	yes	162
3.b	odd	2	1		798.2.cc.a	yes	162
7.c	even	3	1		798.2.bu.b	yes	162
19.f	odd	18	1		798.2.bu.a	✓	162
21.h	odd	6	1		798.2.bu.a	✓	162
57.j	even	18	1		798.2.bu.b	yes	162
133.bd	odd	18	1		798.2.cc.a	yes	162
399.bv	even	18	1	inner	798.2.cc.b	yes	162

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
798.2.bu.a	✓	162	19.f	odd	18	1
798.2.bu.a	✓	162	21.h	odd	6	1
798.2.bu.b	yes	162	7.c	even	3	1
798.2.bu.b	yes	162	57.j	even	18	1
798.2.cc.a	yes	162	3.b	odd	2	1
798.2.cc.a	yes	162	133.bd	odd	18	1
798.2.cc.b	yes	162	1.a	even	1	1	trivial
798.2.cc.b	yes	162	399.bv	even	18	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{162} - 6 T_{5}^{160} - 18 T_{5}^{159} - 36 T_{5}^{158} + 12 T_{5}^{157} - 13004 T_{5}^{156} + \cdots + 56\!\cdots\!68 $ T5^162 - 6*T5^160 - 18*T5^159 - 36*T5^158 + 12*T5^157 - 13004*T5^156 + 1314*T5^155 + 85980*T5^154 + 238932*T5^153 + 387831*T5^152 - 168534*T5^151 + 111984507*T5^150 - 10797594*T5^149 - 851175288*T5^148 - 1959037914*T5^147 - 4409545803*T5^146 + 3923021316*T5^145 - 529067999700*T5^144 + 113624531196*T5^143 + 4523842361991*T5^142 + 8828102213646*T5^141 + 23699032776498*T5^140 - 32664880329846*T5^139 + 1798871074279744*T5^138 - 380469508100028*T5^137 - 17301426773495481*T5^136 - 26899253017543818*T5^135 - 82450792849208877*T5^134 + 146614265313077394*T5^133 - 3715937608093786704*T5^132 + 959712770563097124*T5^131 + 41474118585208885554*T5^130 + 47227656544481479566*T5^129 + 116059452862294668777*T5^128 - 352581999879664379268*T5^127 + 5981137570174938751265*T5^126 - 556943555288305150728*T5^125 - 61380397398281028773667*T5^124 - 46761860822133120046248*T5^123 - 122729506448970308731611*T5^122 + 464725091095897236847374*T5^121 - 6928419376848978792578027*T5^120 - 236140484762166879346992*T5^119 + 62529383154409730692030047*T5^118 + 25620483065576954692242138*T5^117 + 79982061835671953747278401*T5^116 - 358082263087150911990206238*T5^115 + 6102519167919206533965549606*T5^114 + 305822865511197452332627236*T5^113 - 38906962794598670910662411436*T5^112 + 6361400153064692445127351620*T5^111 - 57142130744429306429338235148*T5^110 + 196236194229805910930016506412*T5^109 - 3315070200053412084959477709400*T5^108 + 150672205687682983650534671832*T5^107 + 13764227878350968860178140507539*T5^106 - 9339813529842702807397838157048*T5^105 + 36408975891652586151842788640232*T5^104 - 107536611890530456701423251862018*T5^103 + 1315909839143670642876080065023939*T5^102 - 92182805847284910066461477852184*T5^101 - 2156028579134071250017652813222571*T5^100 + 3960764405561015827000278752310744*T5^99 - 3259026157843386621688289459762352*T5^98 + 42911863997221041290131783570067478*T5^97 - 295703037763245762615525150000803277*T5^96 + 78244211787091694226006228121895190*T5^95 + 253213678589806130215340101169875833*T5^94 - 617391314053241494423186941318261024*T5^93 + 135001166752808959368561503650683153*T5^92 - 9832086518006293110790435115068955892*T5^91 + 48152945716438720776958496516013305707*T5^90 - 20528656822455727034910371934997225524*T5^89 - 2732048878083806824314403024115597232*T5^88 - 3994274423710740055048152264509056158*T5^87 + 201313000960780655755777763138421814464*T5^86 + 1095570119388838641151227200084678901732*T5^85 - 4330395006196097507015850882223071839964*T5^84 + 959487579247817291137697384074676512140*T5^83 + 1403471861672526669317458682512657892316*T5^82 + 4010824390735942275075264294074704249314*T5^81 - 19576487163245495584079366693504879789679*T5^80 - 93217339707674164939052716437977112927648*T5^79 + 267921868770004607591862964331872029386254*T5^78 + 54049528303682876946156747199411126603968*T5^77 - 226355191242347780820213891306598775428407*T5^76 - 370783751255731298232876658382684294008536*T5^75 + 1177229633498593881445921459275118146927927*T5^74 + 5144105739887758258955580828503200321576698*T5^73 - 9321687981013562839196475483171370561345933*T5^72 - 10195191104568406280269836305135288132865816*T5^71 + 12840489573169229788735898810309990919316761*T5^70 + 29433699904842651470667834858686928147814836*T5^69 - 38077577244270632150224414022303575339014024*T5^68 - 174968963167624487384674817804663129770523034*T5^67 + 177165966323054141164392326680040268455274303*T5^66 + 385568590804219378205159262874753712942597460*T5^65 - 164699893105740447383477610252334828647705087*T5^64 - 809586319883420632368999572794279310025724596*T5^63 + 427747800638371891653310936692509069613873003*T5^62 + 3040516611138657737104830871015741349083362234*T5^61 - 2173415427827358988992497089895066518457080994*T5^60 - 5795781567045887589706175723902485189852317640*T5^59 + 3909028051398582706327884416319744203590348926*T5^58 + 6877977816887399021796585665636912196369142848*T5^57 - 12131928099870035266048634346049450030842808863*T5^56 - 15550132362095633747312525824027452215547690984*T5^55 + 36693585772677534596636960380251344898619289038*T5^54 + 20097854363036086283217703882600942379433212502*T5^53 - 50736730769006149286430049623211958972237061273*T5^52 + 11856863635897951110997836476628905450298339246*T5^51 + 58897749125170188247064459866691307486521206533*T5^50 - 47611641821514813831409152122577352653146874984*T5^49 - 92641732447849123973635664098958966564203961421*T5^48 + 55729159150602772460310180659264777000672383920*T5^47 + 68506419028872907100888144772424723653712338045*T5^46 - 93635881628100161667059586283972183681822970844*T5^45 + 1937431177446572117371592156275660470156315465*T5^44 + 122299366208906241533429113151130121116476651922*T5^43 + 7132117262764088744275363826284569981169227979*T5^42 - 67962934742111113931381470049299789639877877636*T5^41 + 16225574997029872150386431654673869673189089943*T5^40 + 44450855006014992853670878685123702522673193362*T5^39 - 70947537897138180201056846490142187690814168243*T5^38 - 54162103522555986470557430742125837746553378196*T5^37 + 20081356750398307395610029189714583638297719418*T5^36 + 3581188747210668250972852589869571850857036010*T5^35 - 1317121067481283522979779480479256702030606287*T5^34 + 9695100592068849997680567175603441474800144402*T5^33 + 31014095362917855269730651908750295926565273837*T5^32 + 14295315037616780839975237398984062769660288426*T5^31 - 3197396912291618173115967320921931990831330798*T5^30 + 1305497268421809005855100322948402272008583900*T5^29 - 1635913092234695715945425504247134373268509222*T5^28 + 319980542411430270522713895636685116424026726*T5^27 - 3865600725618988949528514618303649364638934154*T5^26 - 609235623558495529772745351253582952805473982*T5^25 + 2361388072340660009361221289289905644418949383*T5^24 - 826734258821904999817311610870863615101018946*T5^23 + 167543572696273170451398807307249589669033610*T5^22 - 92935761448643219805975062796865943537567370*T5^21 + 66530067987883522221089109302728753253472741*T5^20 - 1334870830763006867261334890881600014045432*T5^19 - 7480719957197131164998823590801313434530011*T5^18 + 79206940090063337946632217652425110433486*T5^17 - 298055174779766279236924417994806932542817*T5^16 + 328793308939120459469322771282716538575400*T5^15 - 139512192370725345897189463729672668003036*T5^14 + 4904103244404629754335542075162617039804*T5^13 + 15687585747349525225050770807626657925187*T5^12 - 2476078858146988593091700922622721634702*T5^11 + 1539782059125087835426925093469891469644*T5^10 - 186328424637590385548123814793001645952*T5^9 + 158115001132771971047809308292015714635*T5^8 - 23194943579555112824982172448841580308*T5^7 + 1319165205573643700110767863822404773*T5^6 - 1897053578598586319926562885589002106*T5^5 + 78473105793304825640597157517222392*T5^4 - 5843034911806398782243982171080712*T5^3 + 7599293146376039621041622769875472*T5^2 + 388508741195462549864476927300800*T5 + 56673867154103092985137287576768 acting on $S_{2}^{\mathrm{new}}(798, [\chi])$.

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:	\( N \)	\(=\)	\( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	798.cc (of order \(18\), degree \(6\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 317.27
Significant digits:
Format:

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