LMFDB - Newform orbit 960.2.ci.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,2,Mod(61,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))

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

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

chi := DirichletCharacter("960.61");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	960.ci (of order $16$, 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:	$7.66563859404$
Analytic rank:	$0$
Dimension:	$272$
Relative dimension:	$34$ over $\Q(\zeta_{16})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

$q$-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) = $ $ 272 q - 48 q^{8}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 272 q - 48 q^{8} + 16 q^{22} + 112 q^{26} + 80 q^{28} + 80 q^{32} + 80 q^{34} + 80 q^{38} + 48 q^{40} - 64 q^{44} - 16 q^{51} - 48 q^{52} - 16 q^{56} - 160 q^{58} + 64 q^{59} - 96 q^{62} + 32 q^{63} - 48 q^{64} - 48 q^{66} + 32 q^{67} - 96 q^{68} + 64 q^{71} - 16 q^{74} + 112 q^{76} - 48 q^{78} - 112 q^{79} - 160 q^{86} + 80 q^{88} - 64 q^{93}+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} $
61.1	−1.41394	−	0.0280623i	−0.195090	+	0.980785i	1.99843	+	0.0793566i	−0.831470	−	0.555570i	0.303368	−	1.38129i	−3.27450	+	1.35634i	−2.82342	−	0.168286i	−0.923880	−	0.382683i	1.16005	+	0.808873i
61.2	−1.40285	−	0.178912i	−0.195090	+	0.980785i	1.93598	+	0.501975i	−0.831470	−	0.555570i	0.449157	−	1.34099i	3.47847	−	1.44083i	−2.62608	−	1.05057i	−0.923880	−	0.382683i	1.06703	+	0.928142i
61.3	−1.37744	+	0.320394i	0.195090	−	0.980785i	1.79470	−	0.882649i	0.831470	+	0.555570i	0.0455121	+	1.41348i	3.48253	−	1.44251i	−2.18929	+	1.79081i	−0.923880	−	0.382683i	−1.32330	−	0.498868i
61.4	−1.36674	−	0.363364i	0.195090	−	0.980785i	1.73593	+	0.993244i	0.831470	+	0.555570i	−0.623019	+	1.26959i	−0.670965	+	0.277923i	−2.01165	−	1.98828i	−0.923880	−	0.382683i	−0.934525	−	1.06144i
61.5	−1.22171	+	0.712345i	−0.195090	+	0.980785i	0.985128	−	1.74055i	−0.831470	−	0.555570i	−0.460315	−	1.33720i	−2.53202	+	1.04880i	0.0363389	+	2.82819i	−0.923880	−	0.382683i	1.41157	+	0.0864495i
61.6	−1.21876	+	0.717377i	0.195090	−	0.980785i	0.970740	−	1.74862i	0.831470	+	0.555570i	0.465825	+	1.33529i	−2.96409	+	1.22777i	0.0713212	+	2.82753i	−0.923880	−	0.382683i	−1.41191	−	0.0806282i
61.7	−1.17892	+	0.781120i	−0.195090	+	0.980785i	0.779702	−	1.84176i	−0.831470	−	0.555570i	−0.536115	−	1.30866i	3.04818	−	1.26260i	0.519426	+	2.78032i	−0.923880	−	0.382683i	1.41420	+	0.00549483i
61.8	−1.15662	−	0.813770i	0.195090	−	0.980785i	0.675556	+	1.88245i	0.831470	+	0.555570i	−1.02378	+	0.975641i	4.16771	−	1.72632i	0.750520	−	2.72703i	−0.923880	−	0.382683i	−0.509591	−	1.31921i
61.9	−1.05653	−	0.940074i	−0.195090	+	0.980785i	0.232522	+	1.98644i	−0.831470	−	0.555570i	1.12813	−	0.852832i	0.409756	−	0.169726i	1.62173	−	2.31732i	−0.923880	−	0.382683i	0.356198	+	1.36862i
61.10	−1.01035	−	0.989537i	0.195090	−	0.980785i	0.0416344	+	1.99957i	0.831470	+	0.555570i	−1.16763	+	0.797892i	−3.97809	+	1.64778i	1.93658	−	2.06147i	−0.923880	−	0.382683i	−0.290322	−	1.38409i
61.11	−0.931319	+	1.06426i	0.195090	−	0.980785i	−0.265292	−	1.98233i	0.831470	+	0.555570i	0.862118	+	1.12105i	0.933346	−	0.386604i	2.35678	+	1.56384i	−0.923880	−	0.382683i	−1.36563	+	0.367486i
61.12	−0.755415	+	1.19555i	−0.195090	+	0.980785i	−0.858697	−	1.80628i	−0.831470	−	0.555570i	−1.02521	−	0.974141i	−0.0327702	+	0.0135738i	2.80817	+	0.337871i	−0.923880	−	0.382683i	1.29232	−	0.574381i
61.13	−0.444331	−	1.34260i	0.195090	−	0.980785i	−1.60514	+	1.19312i	0.831470	+	0.555570i	−1.40349	+	0.173866i	−1.12942	+	0.467822i	2.31509	+	1.62492i	−0.923880	−	0.382683i	0.376460	−	1.36319i
61.14	−0.428552	+	1.34772i	0.195090	−	0.980785i	−1.63269	−	1.15513i	0.831470	+	0.555570i	1.23822	+	0.683244i	−2.89139	+	1.19765i	2.25649	−	1.70537i	−0.923880	−	0.382683i	−1.10508	+	0.882496i
61.15	−0.289298	−	1.38431i	−0.195090	+	0.980785i	−1.83261	+	0.800956i	−0.831470	−	0.555570i	1.41415	−	0.0136747i	1.35817	−	0.562573i	1.63894	+	2.30518i	−0.923880	−	0.382683i	−0.528537	+	1.31173i
61.16	0.0905265	−	1.41131i	−0.195090	+	0.980785i	−1.98361	−	0.255522i	−0.831470	−	0.555570i	1.36653	+	0.364121i	−2.65455	+	1.09955i	−0.540191	+	2.77636i	−0.923880	−	0.382683i	−0.859354	+	1.12317i
61.17	0.146617	+	1.40659i	−0.195090	+	0.980785i	−1.95701	+	0.412460i	−0.831470	−	0.555570i	−1.40817	−	0.130613i	−4.71557	+	1.95325i	−0.867094	−	2.69224i	−0.923880	−	0.382683i	0.659554	−	1.25100i
61.18	0.230620	+	1.39528i	−0.195090	+	0.980785i	−1.89363	+	0.643560i	−0.831470	−	0.555570i	−1.41346	−	0.0460176i	3.62626	−	1.50205i	−1.33466	−	2.49373i	−0.923880	−	0.382683i	0.583424	−	1.28826i
61.19	0.312058	+	1.37935i	0.195090	−	0.980785i	−1.80524	+	0.860878i	0.831470	+	0.555570i	1.41373	−	0.0369635i	3.73405	−	1.54669i	−1.75080	−	2.22142i	−0.923880	−	0.382683i	−0.506861	+	1.32026i
61.20	0.454066	−	1.33934i	0.195090	−	0.980785i	−1.58765	−	1.21629i	0.831470	+	0.555570i	−1.22502	−	0.706633i	0.461200	−	0.191035i	−2.34992	+	1.57412i	−0.923880	−	0.382683i	1.12164	−	0.861353i

See next 80 embeddings (of 272 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
64.i	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	960.2.ci.b	✓	272
64.i	even	16	1	inner	960.2.ci.b	✓	272

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
960.2.ci.b	✓	272	1.a	even	1	1	trivial
960.2.ci.b	✓	272	64.i	even	16	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{272} + 480 T_{7}^{267} - 6912 T_{7}^{266} + 9344 T_{7}^{265} + 1834432 T_{7}^{264} + \cdots + 22\!\cdots\!76 $ T7^272 + 480*T7^267 - 6912*T7^266 + 9344*T7^265 + 1834432*T7^264 + 48896*T7^263 - 3376256*T7^262 + 3466240*T7^261 + 28372992*T7^260 + 646108480*T7^259 - 10817840640*T7^258 + 17816212608*T7^257 + 1438617766816*T7^256 + 100100957440*T7^255 - 5301042277888*T7^254 + 5533979919360*T7^253 + 44901747142656*T7^252 + 312680930060672*T7^251 - 6975584556135936*T7^250 + 13960839855461632*T7^249 + 637975481576363392*T7^248 + 76265127746374144*T7^247 - 3448817603291358208*T7^246 + 3555170927031626752*T7^245 + 29242547861368258560*T7^244 + 58001523413589320960*T7^243 - 2441062012606625393664*T7^242 + 5962567534863137512960*T7^241 + 178118496918703228997568*T7^240 + 29066010267776612484096*T7^239 - 1225297628270579746666496*T7^238 + 1216286180080660895707136*T7^237 + 10326665714176776120156160*T7^236 - 2262397295315089337119232*T7^235 - 515101275219027529850976256*T7^234 + 1562125929242886254956135424*T7^233 + 32923605602938622356948449792*T7^232 + 6314594634266719636625311744*T7^231 - 264100433425274770135430639616*T7^230 + 247674560089175765884327677952*T7^229 + 2202075155749254123356594110464*T7^228 - 2837728741589714136431885270016*T7^227 - 68669652137950651184452684525568*T7^226 + 266587253235451841170516756305920*T7^225 + 4120797845045991361957487191482880*T7^224 + 833695499543854968867217190195200*T7^223 - 36223175464850873679481995771207680*T7^222 + 31777901237484930789433928635056128*T7^221 + 298420479475610845120848256285081600*T7^220 - 533321035382685366821127319282681856*T7^219 - 5856115847361791129577937504422387712*T7^218 + 30523026444616132510813930396190068736*T7^217 + 351413820420164109650559690493109966848*T7^216 + 68662813194411001278017863679594471424*T7^215 - 3217702968183351267379882190281382510592*T7^214 + 2660064713580747352052724643881786785792*T7^213 + 26265380189811265970156236939631924084736*T7^212 - 51667079838366065552372573622644460908544*T7^211 - 313366731790967889588420458820709019205632*T7^210 + 2371147208814828496832217996260399248203776*T7^209 + 20335086388193364512193525681778737702454784*T7^208 + 3542252162468846530356135527506746906460160*T7^207 - 184447839866439662652765631889632505766182912*T7^206 + 149629829794421710378590790109789327308685312*T7^205 + 1504729778722907468807642007928756171033018368*T7^204 - 2966359873073335465112385672358632023150223360*T7^203 - 9781559517924272333281071969262910303877693440*T7^202 + 124972222563745114099481014316384946797438976000*T7^201 + 791605311177574082443175936684348946029479403520*T7^200 + 115855498834824661520714502018658800206481489920*T7^199 - 6670898959815650306435113524387296149887439863808*T7^198 + 5844120448589908105820275634392353004740228218880*T7^197 + 55427036362896921429070405189339854241777181523968*T7^196 - 105175317259282411672811445653073125140055587962880*T7^195 - 134255373250855399572791233007612258423298941714432*T7^194 + 4438899353741550218251522804785690411006431720669184*T7^193 + 20650733091188534349678833233627279965600171674542080*T7^192 + 2724288378699496924565812213772308307991076629053440*T7^191 - 145804316821546932480357935117153553290583634909724672*T7^190 + 162624583906264809525023378595010707282851920036757504*T7^189 + 1285313683321398588913839488569177907707034086958694400*T7^188 - 2323567324428160235653716873140662838579178295821828096*T7^187 + 1247360905773630976999404654891846462293584711884079104*T7^186 + 105131342520020717912540803392952495927107122483455262720*T7^185 + 365373530716710007258776632843292131450078485578416685056*T7^184 + 64327737819699736963310456712126580991566035241832153088*T7^183 - 1774260145315134016794281493649946478944237232800364494848*T7^182 + 3199262132287570589081290629120724781089660506688028147712*T7^181 + 18370419900761640830752499599049119757173857128521033318400*T7^180 - 31078795392723649323662002165447507290151804183656952561664*T7^179 + 81246091992436655793987063420356162407733788566068654768128*T7^178 + 1634071737268813245633496327443928151264308038565824148668416*T7^177 + 4468694313203083244589007792473164215050303689340060787654656*T7^176 + 1524313190619838487952484329708857210556838021216960946896896*T7^175 - 8692166167136324100886206131144861959312419447343435045404672*T7^174 + 43631482868407000554831729179245484267177163165298885697470464*T7^173 + 157260235834971127817959171651391777760627825222582523393998848*T7^172 - 233958799607083263642871474620546316628055872098895930461650944*T7^171 + 1409701489430043403936055512775143049797693595957485809153081344*T7^170 + 16328914175059124935025618753857106438478621422689955619176972288*T7^169 + 38116747596315193901312469317312270531451768610696924140880658432*T7^168 + 25372719499343454380947376496478942411498592024047897149496098816*T7^167 + 43988958851803682476084695424819653073792373009908184579927179264*T7^166 + 401915806559210036189196813279773928753928847499357655867189100544*T7^165 + 765799831469684340356842993765719821169317977815201187380819132416*T7^164 - 774108715994035254007660285908151345130796883059595990495044829184*T7^163 + 12786163768167410246320896691812743089403450338663940762383787491328*T7^162 + 101818511210946368757975994035964628223886010882901120829538636922880*T7^161 + 223432957049533551943406598058524374936626613665712261293257050161152*T7^160 + 252177901918566125069689430111508525665678319719411079724237518274560*T7^159 + 826880225702938164075649564099210382167041549525796885092060788424704*T7^158 + 2411009427762906443522362525920916957978113697885107934620990444666880*T7^157 + 2009352445578558406372163898431307043118605928930877360901880402673664*T7^156 + 828668734738560161865049483975593006068262022209531024229674995154944*T7^155 + 65378433888720010625694823504523061670516096463959750042561675228348416*T7^154 + 380484625959273531054219263809794089900671005511632690841936794791968768*T7^153 + 865592702905133897251029842175439618918442857758368262264018914628861952*T7^152 + 1433527747957169555745343855962387671126330007893038748216810906719354880*T7^151 + 4456677720433866180413551916819854685533844863308784876141994393825443840*T7^150 + 9005967244771489739926153092039503487693582378976073238260655881688449024*T7^149 + 3902273158859861629985821526848543880454513315064671357934953297182982144*T7^148 + 13199538441930303971602654303727061829767983261944253081822913736257568768*T7^147 + 184932251032340335674935669403642217004121263269203107536091631959171661824*T7^146 + 827417553536381328953822873155385509113093462660073989942309240838939475968*T7^145 + 2108828193289381107960927901414578788650991172862909642837633186747190935552*T7^144 + 4491691249006282401693738491826500344702657976744366488477373140993210056704*T7^143 + 11571281314024114410117907697671165414211933304291870819982792917038464499712*T7^142 + 20284108031090247011860936412810730031813554561080064678918217598364012249088*T7^141 + 12492751689902250361213185611144647619701350402060084112412305247742380212224*T7^140 + 32365792298951499820671040209981156102947325245144739736709238089490995609600*T7^139 + 275271639951437141789208348924092290745859813556172132147196663450686621483008*T7^138 + 1105056812666376446133725230112315721533298653191327413304325385493680398270464*T7^137 + 3118631161996466072065879037937654711539111991628650613070638647147386768457728*T7^136 + 7415206430989098163662629357638930402075557669557251198805736583944119775133696*T7^135 + 17066723270616747053112065455914451710472089412117528516685843131884061077798912*T7^134 + 28411165213837409961882041463888532525724671094069170611305811112835761892753408*T7^133 + 24212926450266514283818010341648387546305480146740603998205746197666182004211712*T7^132 + 37581751028599802095101004167084754432686647246061008143228925962983384406818816*T7^131 + 223926814159184915842105937849498065627370314521269928772826721861095852001460224*T7^130 + 892563968892022810462131696822426621121025140934887189711647711565413440913670144*T7^129 + 2722574194225418141388675642293826119754174297122770785971278728993509442154659840*T7^128 + 6755285062953860329468247491269519178124960709820563951446281317998414255496364032*T7^127 + 14799141852759879764731618280629642156245396925921759458208259503888753411856793600*T7^126 + 24198272123884055264199027166259508498909215352684616770395875155639258705729421312*T7^125 + 23967890951167391922157754730771944243635461329952944579577969810847363441411227648*T7^124 + 24499900384057412403683514071801898072752063169597195192226041671884782151692976128*T7^123 + 99477560202022400543285891537931819716361572482455642805922492612089319238450282496*T7^122 + 416449392542161503575780335731548694780957732683835011316009624668622885429281030144*T7^121 + 1372258178131725237015182199038322037473314243819799034301539816333548222025666920448*T7^120 + 3477031347088264477042893914511642860978539958840176128746028796560261198712322129920*T7^119 + 7491527321980937981919335016354138143592201217009231313669677909357037229813327200256*T7^118 + 12105080174594792011028113380222977440392880494690798873524392676722061406988654346240*T7^117 + 12800096292337329561772415943400290744382355355071313334559587185474375335754999005184*T7^116 + 9425193453594592727749301240164311752623709005078478776274777835179990601334622519296*T7^115 + 22465784502249810124954422043449342047200892278734719043185366716871031351710463295488*T7^114 + 105651380218090523278597835965256195322653005233403555797254839805994048422326086664192*T7^113 + 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$S_{2}^{\mathrm{new}}(960, [\chi])$.

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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	960.ci (of order \(16\), degree \(8\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 61.34
Significant digits:
Format:

Introduction

L-functions

Modular forms

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