LMFDB - Newform orbit 930.2.bj.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [930,2,Mod(277,930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([0, 5, 6]))

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

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

chi := DirichletCharacter("930.277");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	930.bj (of order $20$, degree $8$, 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:	$128$
Relative dimension:	$16$ over $\Q(\zeta_{20})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

$q$-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) = $ $ 128 q - 16 q^{7}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 128 q - 16 q^{7} - 4 q^{10} + 4 q^{15} + 32 q^{16} - 12 q^{17} - 40 q^{19} + 40 q^{21} + 16 q^{22} - 32 q^{24} - 8 q^{25} - 4 q^{28} + 8 q^{29} + 20 q^{31} + 4 q^{33} + 24 q^{35} - 128 q^{36} + 64 q^{37} - 16 q^{38} - 24 q^{41} + 4 q^{42} + 24 q^{43} - 8 q^{44} + 20 q^{46} + 108 q^{47} - 100 q^{49} - 24 q^{50} + 16 q^{53} - 32 q^{54} + 12 q^{55} + 16 q^{57} - 40 q^{58} - 16 q^{62} - 4 q^{63} + 36 q^{65} - 12 q^{66} - 32 q^{67} + 8 q^{68} - 32 q^{70} + 24 q^{71} + 60 q^{73} - 16 q^{74} + 24 q^{75} - 24 q^{76} - 20 q^{77} - 56 q^{79} + 32 q^{81} - 16 q^{82} - 8 q^{83} - 132 q^{85} - 20 q^{87} - 4 q^{88} + 136 q^{89} - 40 q^{91} - 64 q^{93} + 108 q^{95} + 64 q^{97} - 16 q^{98} - 8 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_{7} $			$ a_{8} $			$ a_{9} $			$ a_{10} $
277.1	−0.156434	−	0.987688i	0.987688	+	0.156434i	−0.951057	+	0.309017i	−2.22990	+	0.166033i	−	1.00000i	0.154498	−	0.303220i	0.453990	+	0.891007i	0.951057	+	0.309017i	0.512821	+	2.17647i
277.2	−0.156434	−	0.987688i	0.987688	+	0.156434i	−0.951057	+	0.309017i	−1.27713	+	1.83547i	−	1.00000i	2.18789	−	4.29397i	0.453990	+	0.891007i	0.951057	+	0.309017i	2.01266	+	0.974271i
277.3	−0.156434	−	0.987688i	0.987688	+	0.156434i	−0.951057	+	0.309017i	−0.620092	−	2.14837i	−	1.00000i	−0.961148	+	1.88636i	0.453990	+	0.891007i	0.951057	+	0.309017i	−2.02491	+	0.948537i
277.4	−0.156434	−	0.987688i	0.987688	+	0.156434i	−0.951057	+	0.309017i	−0.426558	+	2.19501i	−	1.00000i	−0.653972	+	1.28349i	0.453990	+	0.891007i	0.951057	+	0.309017i	2.23471	+	0.0779318i
277.5	−0.156434	−	0.987688i	0.987688	+	0.156434i	−0.951057	+	0.309017i	0.591821	−	2.15633i	−	1.00000i	−0.754948	+	1.48167i	0.453990	+	0.891007i	0.951057	+	0.309017i	−2.22236	−	0.247211i
277.6	−0.156434	−	0.987688i	0.987688	+	0.156434i	−0.951057	+	0.309017i	1.90676	−	1.16802i	−	1.00000i	0.193795	−	0.380343i	0.453990	+	0.891007i	0.951057	+	0.309017i	−1.45193	−	1.70056i
277.7	−0.156434	−	0.987688i	0.987688	+	0.156434i	−0.951057	+	0.309017i	2.03470	+	0.927363i	−	1.00000i	−1.56419	+	3.06990i	0.453990	+	0.891007i	0.951057	+	0.309017i	0.597648	−	2.15472i
277.8	−0.156434	−	0.987688i	0.987688	+	0.156434i	−0.951057	+	0.309017i	2.08697	+	0.802838i	−	1.00000i	2.05326	−	4.02976i	0.453990	+	0.891007i	0.951057	+	0.309017i	0.466480	−	2.18687i
277.9	0.156434	+	0.987688i	−0.987688	−	0.156434i	−0.951057	+	0.309017i	−1.90323	−	1.17377i	−	1.00000i	0.771267	−	1.51370i	−0.453990	−	0.891007i	0.951057	+	0.309017i	0.861584	−	2.06341i
277.10	0.156434	+	0.987688i	−0.987688	−	0.156434i	−0.951057	+	0.309017i	−1.17179	+	1.90444i	−	1.00000i	1.50670	−	2.95707i	−0.453990	−	0.891007i	0.951057	+	0.309017i	−2.06431	−	0.859444i
277.11	0.156434	+	0.987688i	−0.987688	−	0.156434i	−0.951057	+	0.309017i	−0.980551	−	2.00961i	−	1.00000i	1.28197	−	2.51601i	−0.453990	−	0.891007i	0.951057	+	0.309017i	1.83147	−	1.28285i
277.12	0.156434	+	0.987688i	−0.987688	−	0.156434i	−0.951057	+	0.309017i	−0.683297	−	2.12911i	−	1.00000i	−2.06274	+	4.04835i	−0.453990	−	0.891007i	0.951057	+	0.309017i	1.99600	−	1.00795i
277.13	0.156434	+	0.987688i	−0.987688	−	0.156434i	−0.951057	+	0.309017i	−0.284153	+	2.21794i	−	1.00000i	−0.874752	+	1.71680i	−0.453990	−	0.891007i	0.951057	+	0.309017i	−2.23508	+	0.0663075i
277.14	0.156434	+	0.987688i	−0.987688	−	0.156434i	−0.951057	+	0.309017i	1.11441	+	1.93858i	−	1.00000i	−1.22295	+	2.40017i	−0.453990	−	0.891007i	0.951057	+	0.309017i	−1.74038	+	1.40395i
277.15	0.156434	+	0.987688i	−0.987688	−	0.156434i	−0.951057	+	0.309017i	1.96074	−	1.07493i	−	1.00000i	−0.462930	+	0.908551i	−0.453990	−	0.891007i	0.951057	+	0.309017i	1.36843	+	1.76845i
277.16	0.156434	+	0.987688i	−0.987688	−	0.156434i	−0.951057	+	0.309017i	2.23243	−	0.127537i	−	1.00000i	−0.416181	+	0.816801i	−0.453990	−	0.891007i	0.951057	+	0.309017i	0.475195	+	2.18499i
337.1	−0.453990	+	0.891007i	−0.891007	+	0.453990i	−0.587785	−	0.809017i	−2.23208	−	0.133471i	−	1.00000i	0.703148	+	0.111368i	0.987688	−	0.156434i	0.587785	−	0.809017i	1.13227	−	1.92820i
337.2	−0.453990	+	0.891007i	−0.891007	+	0.453990i	−0.587785	−	0.809017i	−1.80883	−	1.31459i	−	1.00000i	2.31722	+	0.367011i	0.987688	−	0.156434i	0.587785	−	0.809017i	1.99250	−	1.01486i
337.3	−0.453990	+	0.891007i	−0.891007	+	0.453990i	−0.587785	−	0.809017i	−1.30319	+	1.81705i	−	1.00000i	−1.71397	−	0.271467i	0.987688	−	0.156434i	0.587785	−	0.809017i	−1.02737	−	1.98608i
337.4	−0.453990	+	0.891007i	−0.891007	+	0.453990i	−0.587785	−	0.809017i	−0.789819	−	2.09193i	−	1.00000i	−4.01115	−	0.635304i	0.987688	−	0.156434i	0.587785	−	0.809017i	2.22250	+	0.245984i

See next 80 embeddings (of 128 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
155.r	even	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	930.2.bj.a	✓	128
5.c	odd	4	1		930.2.bj.b	yes	128
31.f	odd	10	1		930.2.bj.b	yes	128
155.r	even	20	1	inner	930.2.bj.a	✓	128

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
930.2.bj.a	✓	128	1.a	even	1	1	trivial
930.2.bj.a	✓	128	155.r	even	20	1	inner
930.2.bj.b	yes	128	5.c	odd	4	1
930.2.bj.b	yes	128	31.f	odd	10	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{128} + 16 T_{7}^{127} + 178 T_{7}^{126} + 1624 T_{7}^{125} + 11835 T_{7}^{124} + \cdots + 34\!\cdots\!76 $ T7^128 + 16*T7^127 + 178*T7^126 + 1624*T7^125 + 11835*T7^124 + 75196*T7^123 + 421604*T7^122 + 2103280*T7^121 + 9959183*T7^120 + 46457988*T7^119 + 227962632*T7^118 + 1215614740*T7^117 + 6594911285*T7^116 + 34452834472*T7^115 + 163658432428*T7^114 + 673612516932*T7^113 + 2335570840722*T7^112 + 6298308305748*T7^111 + 10885486496408*T7^110 + 5047570866196*T7^109 + 8649565195249*T7^108 + 590716613755808*T7^107 + 5623839292769532*T7^106 + 29807590344549632*T7^105 + 96589884937671335*T7^104 + 70016214036612784*T7^103 - 1391895452655281998*T7^102 - 10173782164635584280*T7^101 - 37620106558826512508*T7^100 - 30876227654970268780*T7^99 + 679375123978378366460*T7^98 + 6165646343775655822536*T7^97 + 34714486125516038710083*T7^96 + 151177917784409847862912*T7^95 + 535573442930393670781832*T7^94 + 1540990422163787017706476*T7^93 + 3390589961155392643873372*T7^92 + 4438084221407186719720824*T7^91 - 3407419103170207303854904*T7^90 - 37766601375186355600130660*T7^89 - 91179438273316219502570543*T7^88 + 62692075226524934705848568*T7^87 + 1425608218117355940614891150*T7^86 + 6435891700738477641387527132*T7^85 + 16307082518468778237398291963*T7^84 + 11708596147054296207629786104*T7^83 - 113880532311549388807150430280*T7^82 - 689953615337398179939547811540*T7^81 - 2293909503209534502541858616991*T7^80 - 4726559081550795216021178722936*T7^79 - 2206097736962505918035720442384*T7^78 + 29319942178044304149233831614628*T7^77 + 149595024902845738429188958226165*T7^76 + 433474645813747479614720786110512*T7^75 + 765887254297576028946190883107668*T7^74 + 172025549056979845375208686662468*T7^73 - 4382968227539553605674259669016166*T7^72 - 18190800845078701263859170323448584*T7^71 - 41736268219435164435070320145946392*T7^70 - 43584281698249195467231744085992932*T7^69 + 94052636262898991307118781251774599*T7^68 + 604698425367348494222214641014691488*T7^67 + 1613831084908183374712308352839690732*T7^66 + 2182660366629188303162627259881017756*T7^65 - 1705129235633363596052533331903162219*T7^64 - 18644491734844209846711896651696085732*T7^63 - 55783972520155377949140411278459503482*T7^62 - 90758130786883415250120230682450846696*T7^61 - 10674513013754989614185017291209552904*T7^60 + 445434505894310428627530082337421742800*T7^59 + 1577249590809221018030975512448458224180*T7^58 + 3124151128284887034350892195683180450432*T7^57 + 2938782389693763913580590072935072398581*T7^56 - 4412502886415437666892755418808904280356*T7^55 - 26358403307984058984719659835008639127136*T7^54 - 63407698522630323449448073984101263649376*T7^53 - 89647679639468195624372867819367081775652*T7^52 - 32170541343530074063533093948534812075516*T7^51 + 217205590499578565590335876014772823473904*T7^50 + 708368641886584597736140228124189411244848*T7^49 + 1243459677894736233848775029323984901833579*T7^48 + 1205814790639668840260810804027520223483776*T7^47 - 294482104826379477384358444701994207656774*T7^46 - 3620792401933006343895910141170171969386196*T7^45 - 7217416056737966040332898512420823392327827*T7^44 - 6639629643715009528466698177074123055953792*T7^43 + 3974067227129216472129069172680295111002304*T7^42 + 26275230596023235893231120341866407024208104*T7^41 + 49271247757228224473154373353576019684383351*T7^40 + 45229433843128111551662134630731033475068712*T7^39 - 18174315003410609890388242916824838871607704*T7^38 - 142609641389950069913473353907825459497793460*T7^37 - 256427931579320902178260376337872801739736933*T7^36 - 214050037750900350368307286124920274717558492*T7^35 + 104754177830120460545146785132063133712859260*T7^34 + 610651935143799114013578226527663128311054808*T7^33 + 899477826969658858089676374493675106524704954*T7^32 + 468574202077335243039717983662101886546428776*T7^31 - 719005861664021779057021959663218630331092264*T7^30 - 1810766598393390197692406454341858910279264488*T7^29 - 1514197734585520485665445484316218651093897845*T7^28 + 537376408501042565963676471181643960865235208*T7^27 + 2804920302793642728745930186694517089907087648*T7^26 + 2878714411101063111408825057329990617731550648*T7^25 + 133073401045954754166218739271195588035384273*T7^24 - 2982672909812082374854292792731646369618425824*T7^23 - 3250066162927750989144576343387753376425379174*T7^22 - 438619211387299014499696769258696933097431932*T7^21 + 2292522181384042311532252726615543103770427878*T7^20 + 2224443765107187873464193574513344095314848940*T7^19 + 135002587633582782789753744789045873790472804*T7^18 - 1262978656112871007837390933410997803526390064*T7^17 - 874404467471893584214333183814169087950548119*T7^16 + 102006744153620937360733497318935762701046620*T7^15 + 422672444021052597311148383874270154055907160*T7^14 + 164093046374082703801165320905995627118431192*T7^13 - 54700105422076902414714487987149463206814788*T7^12 - 53389944773346589628998392343102504700616752*T7^11 - 5826497076005935107654406033689346461000768*T7^10 - 867356393228547895787885013827571894127936*T7^9 - 3801006341616126732289981473920533217726576*T7^8 + 56431004477785847027213957161969513784448*T7^7 + 1548091386536710263121051641800218310172928*T7^6 + 680798047468443225918447952651213990928000*T7^5 + 179974079810345180373620268185833832269760*T7^4 + 75962502169011607643665407508796967995136*T7^3 + 27970984721971465382058092049400890111488*T7^2 + 5063431556106291603058315448459828473344*T7 + 344148528744680629897567854960054866176 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.bj (of order \(20\), degree \(8\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 277.16
Significant digits:
Format:

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