LMFDB - Newform orbit 696.2.bh.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [696,2,Mod(13,696)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(696, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([0, 7, 0, 9]))

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

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

chi := DirichletCharacter("696.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 696 = 2^{3} \cdot 3 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	696.bh (of order $14$, degree $6$, 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:	$5.55758798068$
Analytic rank:	$0$
Dimension:	$180$
Relative dimension:	$30$ over $\Q(\zeta_{14})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

$\operatorname{Tr}(f)(q) = $ $ 180 q + 30 q^{3} - 2 q^{4} - 4 q^{7} - 3 q^{8} - 30 q^{9}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 180 q + 30 q^{3} - 2 q^{4} - 4 q^{7} - 3 q^{8} - 30 q^{9} - 8 q^{10} + 2 q^{12} + 3 q^{14} - 2 q^{16} + 4 q^{21} + 8 q^{23} + 10 q^{24} + 26 q^{25} - 10 q^{26} + 30 q^{27} - 2 q^{28} + 8 q^{30} + 25 q^{32} + 5 q^{34} - 2 q^{36} - 16 q^{37} - 24 q^{38} + 44 q^{40} - 3 q^{42} + 24 q^{46} + 30 q^{48} - 30 q^{49} - 49 q^{50} - 70 q^{52} - 7 q^{54} - 28 q^{56} + 54 q^{58} + 42 q^{60} - 16 q^{61} - 115 q^{62} - 4 q^{63} - 59 q^{64} + 30 q^{65} - 77 q^{66} + 66 q^{68} - 8 q^{69} + 52 q^{70} + 8 q^{71} + 4 q^{72} + 70 q^{73} - 31 q^{74} + 184 q^{75} + 57 q^{76} - 36 q^{77} - 25 q^{78} + 28 q^{79} - 48 q^{80} - 30 q^{81} + 109 q^{82} + 9 q^{84} + 56 q^{85} + 60 q^{86} + 12 q^{88} - 8 q^{90} - 17 q^{92} - 10 q^{94} - 32 q^{96} - 14 q^{97} + 19 q^{98}+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} $
13.1	−1.37238	−	0.341442i	−0.623490	+	0.781831i	1.76684	+	0.937173i	0.123565	−	0.256585i	1.12261	−	0.860082i	−1.65260	+	2.07229i	−2.10477	−	1.88943i	−0.222521	−	0.974928i	−0.257187	+	0.309941i
13.2	−1.35236	−	0.413655i	−0.623490	+	0.781831i	1.65778	+	1.11883i	0.0253992	−	0.0527420i	1.16659	−	0.799411i	2.50170	−	3.13703i	−1.77911	−	2.19881i	−0.222521	−	0.974928i	−0.0561660	+	0.0608199i
13.3	−1.35044	+	0.419910i	−0.623490	+	0.781831i	1.64735	−	1.13412i	0.131101	−	0.272235i	0.513684	−	1.31762i	1.27533	−	1.59922i	−1.74841	+	2.22330i	−0.222521	−	0.974928i	−0.0627300	+	0.422686i
13.4	−1.34211	+	0.445812i	−0.623490	+	0.781831i	1.60250	−	1.19666i	1.84657	−	3.83444i	0.488240	−	1.32726i	0.774228	−	0.970852i	−1.61725	+	2.32046i	−0.222521	−	0.974928i	−0.768852	+	5.96945i
13.5	−1.32715	−	0.488548i	−0.623490	+	0.781831i	1.52264	+	1.29675i	−1.82770	+	3.79526i	1.20943	−	0.733001i	−0.416445	+	0.522205i	−1.38724	−	2.46486i	−0.222521	−	0.974928i	4.27979	−	4.14395i
13.6	−1.19119	+	0.762278i	−0.623490	+	0.781831i	0.837866	−	1.81603i	−0.922939	+	1.91650i	0.146722	−	1.40658i	0.951846	−	1.19358i	0.386265	+	2.80193i	−0.222521	−	0.974928i	−0.361512	−	2.98646i
13.7	−1.16568	+	0.800740i	−0.623490	+	0.781831i	0.717631	−	1.86682i	−0.845442	+	1.75558i	0.100747	−	1.41062i	−2.60180	+	3.26255i	0.658306	+	2.75075i	−0.222521	−	0.974928i	−0.420245	−	2.72343i
13.8	−1.01124	−	0.988628i	−0.623490	+	0.781831i	0.0452292	+	1.99949i	1.34322	−	2.78922i	1.40344	−	0.174223i	−2.13782	+	2.68075i	1.93101	−	2.06669i	−0.222521	−	0.974928i	−4.11582	+	1.49264i
13.9	−0.920626	−	1.07352i	−0.623490	+	0.781831i	−0.304897	+	1.97662i	−0.496717	+	1.03144i	1.41331	−	0.0504443i	0.582611	−	0.730572i	2.40264	−	1.49242i	−0.222521	−	0.974928i	1.56457	−	0.416337i
13.10	−0.796276	+	1.16874i	−0.623490	+	0.781831i	−0.731888	−	1.86127i	0.779747	−	1.61916i	−0.417284	−	1.35125i	0.0498575	−	0.0625193i	2.75812	+	0.626705i	−0.222521	−	0.974928i	1.27148	+	2.20062i
13.11	−0.606898	+	1.27737i	−0.623490	+	0.781831i	−1.26335	−	1.55047i	0.179264	−	0.372246i	−0.620294	−	1.27092i	−0.847557	+	1.06280i	2.74724	−	0.672793i	−0.222521	−	0.974928i	0.366701	+	0.454902i
13.12	−0.564216	−	1.29679i	−0.623490	+	0.781831i	−1.36332	+	1.46334i	1.30230	−	2.70424i	1.36565	+	0.367413i	3.03113	−	3.80092i	2.66685	+	0.942302i	−0.222521	−	0.974928i	−4.24161	−	0.163025i
13.13	−0.496385	−	1.32424i	−0.623490	+	0.781831i	−1.50720	+	1.31466i	−1.25660	+	2.60937i	1.34482	+	0.437558i	−2.24331	+	2.81302i	2.48908	+	1.34331i	−0.222521	−	0.974928i	4.07918	+	0.368790i
13.14	−0.381888	+	1.36168i	−0.623490	+	0.781831i	−1.70832	−	1.04002i	−1.77519	+	3.68621i	−0.826498	−	1.14756i	2.66079	−	3.33653i	2.06855	−	1.92901i	−0.222521	−	0.974928i	−4.34150	−	3.82495i
13.15	0.0448919	−	1.41350i	−0.623490	+	0.781831i	−1.99597	−	0.126909i	0.636735	−	1.32219i	1.07713	+	0.916401i	−0.344476	+	0.431959i	−0.268989	+	2.81561i	−0.222521	−	0.974928i	−1.84034	−	0.959381i
13.16	0.0523923	+	1.41324i	−0.623490	+	0.781831i	−1.99451	+	0.148086i	1.62544	−	3.37527i	−1.13758	−	0.840181i	−2.34334	+	2.93845i	−0.313778	−	2.81097i	−0.222521	−	0.974928i	4.85524	+	2.12031i
13.17	0.162898	+	1.40480i	−0.623490	+	0.781831i	−1.94693	+	0.457679i	0.210199	−	0.436483i	−1.19988	−	0.748520i	2.56937	−	3.22189i	−0.960100	−	2.66049i	−0.222521	−	0.974928i	0.647413	+	0.224186i
13.18	0.371337	−	1.36459i	−0.623490	+	0.781831i	−1.72422	−	1.01345i	−0.700352	+	1.45430i	0.835355	+	1.14113i	0.975976	−	1.22384i	−2.02321	+	1.97652i	−0.222521	−	0.974928i	1.72445	+	1.49573i
13.19	0.413484	+	1.35242i	−0.623490	+	0.781831i	−1.65806	+	1.11840i	−0.358162	+	0.743730i	−1.31516	−	0.519943i	−0.962826	+	1.20735i	−2.19813	−	1.77995i	−0.222521	−	0.974928i	−1.15393	−	0.176863i
13.20	0.770167	−	1.18610i	−0.623490	+	0.781831i	−0.813684	−	1.82700i	0.0735173	−	0.152660i	0.447142	+	1.34166i	−3.03108	+	3.80085i	−2.79368	−	0.441980i	−0.222521	−	0.974928i	−0.124450	−	0.204773i

See next 80 embeddings (of 180 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
232.o	even	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	696.2.bh.b	yes	180
8.b	even	2	1		696.2.bh.a	✓	180
29.e	even	14	1		696.2.bh.a	✓	180
232.o	even	14	1	inner	696.2.bh.b	yes	180

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
696.2.bh.a	✓	180	8.b	even	2	1
696.2.bh.a	✓	180	29.e	even	14	1
696.2.bh.b	yes	180	1.a	even	1	1	trivial
696.2.bh.b	yes	180	232.o	even	14	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{180} - 88 T_{5}^{178} + 4339 T_{5}^{176} - 160095 T_{5}^{174} + 4955922 T_{5}^{172} + \cdots + 12\!\cdots\!29 $ T5^180 - 88*T5^178 + 4339*T5^176 - 160095*T5^174 + 4955922*T5^172 - 17444*T5^171 - 136199798*T5^170 + 2557940*T5^169 + 3428675315*T5^168 - 151103484*T5^167 - 79901480891*T5^166 + 6176019276*T5^165 + 1731440052539*T5^164 - 203853243776*T5^163 - 35089872508224*T5^162 + 5794329766540*T5^161 + 669401537098454*T5^160 - 147226163902244*T5^159 - 12083213575784000*T5^158 + 3426626125628776*T5^157 + 207065774019647968*T5^156 - 73679616290805560*T5^155 - 3373244175772569122*T5^154 + 1470486900726259248*T5^153 + 52294816679551390479*T5^152 - 27317965849579405028*T5^151 - 772707244957422071450*T5^150 + 473901295561535638916*T5^149 + 10900471623246751088431*T5^148 - 7709500970452074802176*T5^147 - 146990326173826487052309*T5^146 + 118113047962849709712572*T5^145 + 1895930800514716734599986*T5^144 - 1708721645323252178480652*T5^143 - 23395750194432095491141310*T5^142 + 23377373408953839087465716*T5^141 + 276327217782416220043259480*T5^140 - 302634300940889567143383596*T5^139 - 3126523768240323089462522121*T5^138 + 3709646751076244848261525476*T5^137 + 33924745151779539567600303720*T5^136 - 43119215637365581063732428256*T5^135 - 353352614898748181268549365685*T5^134 + 476213624559662235621521010760*T5^133 + 3535444840784564666260336540395*T5^132 - 5006309900108163210243275598436*T5^131 - 33995879027941053398557891690428*T5^130 + 50161843805820949175073374192912*T5^129 + 314318053124224975946062465841680*T5^128 - 479291200933008715549949120966244*T5^127 - 2795633170964387882042186249067082*T5^126 + 4368427561607509772672219662380568*T5^125 + 23927059155853955610411962087112444*T5^124 - 37997151176870412783343932282992036*T5^123 - 197066123405313979138679232700502174*T5^122 + 315559119782769588018031215665423116*T5^121 + 1561543958890953695862069520188146950*T5^120 - 2502182753915681262285627780950300636*T5^119 - 11899912411342107756931840570441279452*T5^118 + 18931523285274495441983762843746543756*T5^117 + 87176714229988625560183554449378617984*T5^116 - 136504717713812486114570816224639746912*T5^115 - 613575289521448986846741499801323841296*T5^114 + 936910427438788269576944872654914392004*T5^113 + 4145361979488072053194940667343334620953*T5^112 - 6115573010051095816353177416289422120096*T5^111 - 26851675708627418907084569049370434924472*T5^110 + 37922359839327492248484840578749312872148*T5^109 + 166536729765871356452370102771880253827149*T5^108 - 222979050548621889544049166259331962553908*T5^107 - 987618385160197970676584494155507563421835*T5^106 + 1239746565844544348591189634824965268067756*T5^105 + 5593279902300937033437666277452107462719783*T5^104 - 6497992402387357533231823770714937038305720*T5^103 - 30205597076996271097520079496946458699612020*T5^102 + 32059298509570298372012120033926507474000068*T5^101 + 155241694146693925858605675294332268194698501*T5^100 - 149023727587628738575247756083249400177274148*T5^99 - 757618922559684389736901537584496143412414376*T5^98 + 654800048437995142389204788605207017235994324*T5^97 + 3504469787338735747290490284763425141911593265*T5^96 - 2732535192762519623161966140834923054615944472*T5^95 - 15355559299806619859373599305785679007148376451*T5^94 + 10884603907413052035353610961683082789435242208*T5^93 + 63804012242637353885876092413734298827129642330*T5^92 - 41648576106716036584246964067736098366020991428*T5^91 - 251822989577566170826143223298797168107299363976*T5^90 + 154367093009661121161151831735713617871158547480*T5^89 + 944337734198007526170583651179479671885011390553*T5^88 - 556927470611313333187602174779099694723712931664*T5^87 - 3361271867481330668200987359206647500443749008749*T5^86 + 1950922636848440718662638357317723488532763719324*T5^85 + 11355720236972157721949795419122993212306545019152*T5^84 - 6619506324709584704661216537203682771292982598200*T5^83 - 36424487803378902902726432759272361545613056264874*T5^82 + 21737312846170100351619248428787718262573849962184*T5^81 + 110935606741874826392802056223263814617967189391299*T5^80 - 69091272044675832339783657071802483452640724085824*T5^79 - 320476835108937289257501484423893700362247135156491*T5^78 + 212442854439798926552769398404801329523969170353848*T5^77 + 874008702908838920683355194955949538346237362687824*T5^76 - 623697222634488662987832605310156425598452427944304*T5^75 - 2241233720523271013330362547006229680799488843092328*T5^74 + 1710569268566195091167988852138828983331517063969876*T5^73 + 5447956355058992732429507925287002473851488770038756*T5^72 - 4440983636345607043496158273572344752834678647579852*T5^71 - 12510540246903596504674330074704151017397772229220875*T5^70 + 10916294791464866557191760758106899811541419385429540*T5^69 + 27018737692370791491174832302780769959866230148079811*T5^68 - 25207187793724347292115857576899317534509537117030964*T5^67 - 54916339305822000668400496309830498178157850218408109*T5^66 + 54773032975586594607536200798227795710651271955871204*T5^65 + 104247167175936913889669477749131536835174597058568023*T5^64 - 111172885386694781619556762010233735602279066702202812*T5^63 - 180457650300172360508916485530633433016000828486746059*T5^62 + 193718649091178259981914690283943289485791265860606064*T5^61 + 321972194839659491276368727130011669837024166584244922*T5^60 - 360487023121931840767343049038746781285342702539015352*T5^59 - 459939675670153194051386870989576409775739713627269423*T5^58 + 478671484108860707646814765882984322628659393747567300*T5^57 + 826972563185692427471549315890902725231819344100347534*T5^56 - 882423641630272104357772054845113688783150160175006244*T5^55 - 928150551495568126198334028149393635893053825956006794*T5^54 + 874388468253869998449987666591124002784998290749332296*T5^53 + 1593195875882329184711331665454253593173236105298415098*T5^52 - 1454706067937605012063019763361056329185854125826177172*T5^51 - 1486852406442061227092779441947470539762129544068237771*T5^50 + 1079793032244479053243504385570472563613695283737271664*T5^49 + 2142239565522092550299872581369769726674793694996954761*T5^48 - 1414843772061008439675050869234894552187946148223304712*T5^47 - 1809611628217373068271758682825587630057832444019569033*T5^46 + 980239201956946791844135248121442150688296504929591224*T5^45 + 1748465528607698724563739048324684445591994872031622886*T5^44 - 889871056521781197192454057391052388385945314032737364*T5^43 - 1203531345184198286554344853213453120766427509239072129*T5^42 + 581056510447992696686010835845972812654095949308680412*T5^41 + 752917769699224548722841873666213885442118794921296386*T5^40 - 343574503968005972601949349313645367174772535458625920*T5^39 - 423933805888599297486361382935195611476214007848613034*T5^38 + 200622782704061945046172416595373295333022667158473944*T5^37 + 212633168812401830581487609045249363784591942111168970*T5^36 - 124257934205606617757166044589279615829821433865844064*T5^35 - 83500357493335362255537650425222481103311933984552581*T5^34 + 69266517917474555528390167738323598701728378345449868*T5^33 + 24231808721901547101727341562227447460827613142299282*T5^32 - 36890969658274052944841037544334246009585680265215712*T5^31 + 2774054426934942811132783246381535432206116644180857*T5^30 + 11509846328195418711044982990063928659001741224909884*T5^29 - 4584054803077470666846688883542514171873849344413631*T5^28 - 1748877143368316992917457228187856351403918854540660*T5^27 + 1757436826761020649927777831224854761643343306082860*T5^26 - 149931125841978749767926444964488461902540202903972*T5^25 - 349040491402142731258707138178980354959529318193561*T5^24 + 151437743936209420562751084321004232101610675339488*T5^23 + 16563566478736006964188079064332123808263451971822*T5^22 - 35667665672548325298526999757590005100294363165576*T5^21 + 12404345766675728244259888632080347975485910504227*T5^20 - 423364277872091149119491363472286416614563507200*T5^19 - 1093261808506785676065678727626007870353670381639*T5^18 + 376084709517713552734303437575954770562932852260*T5^17 - 26341306018046570802051595131246728202744989711*T5^16 - 17255323463714278288915312237902020946694197088*T5^15 + 5686885773868094586372806712049818958202816116*T5^14 - 472758446192573373743328265604153789981350356*T5^13 - 125501896779263137332340924508179464669568482*T5^12 + 37269262476555897138548127027530673327390884*T5^11 - 2386890394537567411761727659281127210480302*T5^10 - 483170753692693099457149992767880655821496*T5^9 + 101220817096048849391484008994340205931761*T5^8 - 4301500655176945973237048165348329386300*T5^7 - 580982521031057110121512873430255480404*T5^6 + 13644620270747688155832455368749334992*T5^5 + 10028331832441428321359457573559704175*T5^4 - 565180209875316300231169333586188488*T5^3 + 33345596026388708572964300139770043*T5^2 + 326199794692210512024070796853204*T5 + 1271212676327770108497341537929 acting on $S_{2}^{\mathrm{new}}(696, [\chi])$.

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:	\( N \)	\(=\)	\( 696 = 2^{3} \cdot 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	696.bh (of order \(14\), degree \(6\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 13.30
Significant digits:
Format:

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