LMFDB - Newform orbit 402.2.j.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [402,2,Mod(5,402)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(402, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 5]))

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

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

chi := DirichletCharacter("402.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 402 = 2 \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	402.j (of order $22$, degree $10$, 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:	$3.20998616126$
Analytic rank:	$0$
Dimension:	$120$
Relative dimension:	$12$ over $\Q(\zeta_{22})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

$q$-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) = $ $ 120 q + 12 q^{2} - q^{3} - 12 q^{4} + q^{6} + 12 q^{8} - 5 q^{9}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 120 q + 12 q^{2} - q^{3} - 12 q^{4} + q^{6} + 12 q^{8} - 5 q^{9} - 6 q^{11} - q^{12} + 33 q^{14} + 4 q^{15} - 12 q^{16} - 33 q^{17} - 6 q^{18} - 2 q^{19} - 55 q^{21} + 6 q^{22} + 22 q^{23} - 10 q^{24} - 24 q^{25} - 22 q^{26} + 5 q^{27} - 4 q^{30} - 22 q^{31} + 12 q^{32} + 11 q^{33} + 22 q^{35} - 5 q^{36} + 22 q^{37} - 20 q^{38} + 27 q^{39} + 8 q^{41} - 11 q^{42} + 22 q^{43} - 6 q^{44} - 9 q^{45} - 12 q^{48} + 2 q^{49} + 46 q^{50} - 6 q^{51} - 33 q^{52} - 22 q^{53} + 6 q^{54} - 36 q^{55} + 4 q^{57} - 22 q^{58} + 33 q^{59} + 4 q^{60} + 22 q^{61} - 55 q^{62} - 19 q^{63} - 12 q^{64} + 11 q^{66} - 14 q^{67} + 10 q^{69} + 22 q^{70} - 6 q^{72} - 79 q^{73} + 44 q^{74} + 46 q^{75} - 2 q^{76} + 44 q^{77} + 6 q^{78} - 77 q^{79} - 253 q^{81} - 19 q^{82} + 99 q^{83} + 33 q^{84} - 80 q^{87} + 6 q^{88} + 22 q^{89} - 13 q^{90} + 8 q^{91} + 67 q^{93} - 22 q^{94} + 62 q^{95} + q^{96} - 2 q^{98} - 19 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} $
5.1	0.654861	−	0.755750i	−1.71690	+	0.228591i	−0.142315	−	0.989821i	0.242870	−	0.0713131i	−0.951573	+	1.44724i	−2.47201	−	2.14201i	−0.841254	−	0.540641i	2.89549	−	0.784935i	0.105151	−	0.230249i
5.2	0.654861	−	0.755750i	−1.50108	−	0.864158i	−0.142315	−	0.989821i	2.06705	−	0.606940i	−1.63608	+	0.568535i	2.70814	+	2.34662i	−0.841254	−	0.540641i	1.50646	+	2.59433i	0.894933	−	1.95963i
5.3	0.654861	−	0.755750i	−1.35285	+	1.08158i	−0.142315	−	0.989821i	0.740480	−	0.217425i	−0.0685239	+	1.73069i	1.96535	+	1.70298i	−0.841254	−	0.540641i	0.660382	−	2.92641i	0.320593	−	0.702000i
5.4	0.654861	−	0.755750i	−1.14781	−	1.29712i	−0.142315	−	0.989821i	−2.61258	+	0.767123i	−1.73196	+	0.0180248i	1.12824	+	0.977627i	−0.841254	−	0.540641i	−0.365051	+	2.97771i	−1.13112	+	2.47681i
5.5	0.654861	−	0.755750i	−0.840609	+	1.51439i	−0.142315	−	0.989821i	−2.77179	+	0.813870i	0.594018	+	1.62700i	−0.377481	−	0.327089i	−0.841254	−	0.540641i	−1.58675	−	2.54602i	−1.20005	+	2.62775i
5.6	0.654861	−	0.755750i	−0.482917	−	1.66337i	−0.142315	−	0.989821i	2.50946	−	0.736845i	−1.57333	−	0.724309i	−2.21975	−	1.92343i	−0.841254	−	0.540641i	−2.53358	+	1.60654i	1.08648	−	2.37906i
5.7	0.654861	−	0.755750i	0.812042	+	1.52990i	−0.142315	−	0.989821i	4.19490	−	1.23173i	1.68799	+	0.388170i	2.39898	+	2.07873i	−0.841254	−	0.540641i	−1.68118	+	2.48468i	1.81619	−	3.97690i
5.8	0.654861	−	0.755750i	1.12510	−	1.31687i	−0.142315	−	0.989821i	−0.167968	+	0.0493200i	−0.258445	−	1.71266i	3.46402	+	3.00159i	−0.841254	−	0.540641i	−0.468312	−	2.96322i	−0.0727223	+	0.159240i
5.9	0.654861	−	0.755750i	1.24596	−	1.20316i	−0.142315	−	0.989821i	2.22460	−	0.653202i	−0.0933572	−	1.72953i	−0.639153	−	0.553829i	−0.841254	−	0.540641i	0.104822	−	2.99817i	0.963147	−	2.10900i
5.10	0.654861	−	0.755750i	1.30634	+	1.13731i	−0.142315	−	0.989821i	−3.41108	+	1.00158i	1.71499	−	0.242485i	3.24167	+	2.80892i	−0.841254	−	0.540641i	0.413047	+	2.97143i	−1.47684	+	3.23382i
5.11	0.654861	−	0.755750i	1.32337	−	1.11744i	−0.142315	−	0.989821i	−3.81422	+	1.11996i	0.0221185	−	1.73191i	−2.93792	−	2.54572i	−0.841254	−	0.540641i	0.502640	−	2.95759i	−1.65138	+	3.61601i
5.12	0.654861	−	0.755750i	1.64476	+	0.542907i	−0.142315	−	0.989821i	0.798282	−	0.234397i	1.48739	−	0.887502i	−1.77174	−	1.53522i	−0.841254	−	0.540641i	2.41050	+	1.78591i	0.345618	−	0.756798i
53.1	−0.415415	−	0.909632i	−1.70193	−	0.321604i	−0.654861	+	0.755750i	−0.487223	−	3.38871i	0.414466	+	1.68173i	2.37486	−	1.08456i	0.959493	+	0.281733i	2.79314	+	1.09470i	−2.88008	+	1.85091i
53.2	−0.415415	−	0.909632i	−1.57028	+	0.730902i	−0.654861	+	0.755750i	−0.0640359	−	0.445379i	1.31717	+	1.12475i	−1.48804	+	0.679564i	0.959493	+	0.281733i	1.93156	−	2.29544i	−0.378530	+	0.243266i
53.3	−0.415415	−	0.909632i	−1.09017	+	1.34593i	−0.654861	+	0.755750i	0.621342	+	4.32153i	1.67717	+	0.432535i	0.522788	−	0.238749i	0.959493	+	0.281733i	−0.623054	−	2.93459i	3.67289	−	2.36042i
53.4	−0.415415	−	0.909632i	−1.06328	−	1.36728i	−0.654861	+	0.755750i	−0.0682677	−	0.474812i	−0.802016	+	1.53518i	−2.74677	+	1.25441i	0.959493	+	0.281733i	−0.738882	+	2.90759i	−0.403545	+	0.259343i
53.5	−0.415415	−	0.909632i	−0.891192	−	1.48519i	−0.654861	+	0.755750i	0.275987	+	1.91953i	−0.980758	+	1.42763i	3.39983	−	1.55265i	0.959493	+	0.281733i	−1.41155	+	2.64717i	1.63142	−	1.04845i
53.6	−0.415415	−	0.909632i	0.158062	+	1.72482i	−0.654861	+	0.755750i	−0.574881	−	3.99839i	1.50329	−	0.860296i	−3.24959	+	1.48404i	0.959493	+	0.281733i	−2.95003	+	0.545259i	−3.39825	+	2.18392i
53.7	−0.415415	−	0.909632i	0.227861	−	1.71700i	−0.654861	+	0.755750i	0.280377	+	1.95006i	−1.65649	+	0.505997i	1.38208	−	0.631177i	0.959493	+	0.281733i	−2.89616	−	0.782473i	1.65737	−	1.06512i
53.8	−0.415415	−	0.909632i	1.06320	+	1.36734i	−0.654861	+	0.755750i	0.294281	+	2.04677i	0.802103	−	1.53513i	−0.870333	+	0.397468i	0.959493	+	0.281733i	−0.739213	+	2.90750i	1.73956	−	1.11795i

See next 80 embeddings (of 120 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
201.j	even	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	402.2.j.b	yes	120
3.b	odd	2	1		402.2.j.a	✓	120
67.f	odd	22	1		402.2.j.a	✓	120
201.j	even	22	1	inner	402.2.j.b	yes	120

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
402.2.j.a	✓	120	3.b	odd	2	1
402.2.j.a	✓	120	67.f	odd	22	1
402.2.j.b	yes	120	1.a	even	1	1	trivial
402.2.j.b	yes	120	201.j	even	22	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{120} + 42 T_{5}^{118} + 12 T_{5}^{117} + 747 T_{5}^{116} + 536 T_{5}^{115} + \cdots + 15\!\cdots\!24 $ T5^120 + 42*T5^118 + 12*T5^117 + 747*T5^116 + 536*T5^115 + 9813*T5^114 + 12870*T5^113 + 192835*T5^112 + 258364*T5^111 + 4467082*T5^110 + 6480706*T5^109 + 87549468*T5^108 + 161400838*T5^107 + 1617793286*T5^106 + 3585857830*T5^105 + 30216249489*T5^104 + 65967744450*T5^103 + 485519143164*T5^102 + 1065756025666*T5^101 + 7122192074934*T5^100 + 15855942378518*T5^99 + 94163502187691*T5^98 + 191270503624830*T5^97 + 1073476482904403*T5^96 + 1968752111057494*T5^95 + 11870869129377601*T5^94 + 17855399588583398*T5^93 + 126437763680276484*T5^92 + 119063618038860012*T5^91 + 1204288382677861440*T5^90 + 503558000288415182*T5^89 + 11119192953068537670*T5^88 - 442764075216199434*T5^87 + 97546324121344262162*T5^86 - 54133950183782260934*T5^85 + 843688060840796210065*T5^84 - 786127997930174693626*T5^83 + 7231827182542711880965*T5^82 - 9397788789050180140330*T5^81 + 60431693041738077015641*T5^80 - 95333288879987655652994*T5^79 + 515769279727484463533268*T5^78 - 940293454971504548621370*T5^77 + 4200921236831415407670164*T5^76 - 8408177060549398877863328*T5^75 + 34987222256840315067381857*T5^74 - 77457061181045395982471306*T5^73 + 275040128272987943975153919*T5^72 - 636668061318483349415427686*T5^71 + 2038618301141466530781315937*T5^70 - 5019335776340375048361636476*T5^69 + 14981978494035668031736027866*T5^68 - 37321271836597298622941131598*T5^67 + 106596499417979599620004011118*T5^66 - 264696545477356429746570670006*T5^65 + 740627706417514891824032740135*T5^64 - 1827750605380992841981633081528*T5^63 + 4992240454003217822202333330333*T5^62 - 12322825268040442887465165569626*T5^61 + 32721673449263075270122289996704*T5^60 - 81035820881220991993082943777940*T5^59 + 208551161115317709473383987514915*T5^58 - 513402302140474786735164972486558*T5^57 + 1281597730866437700222459528852393*T5^56 - 3082512154914253944635262225524530*T5^55 + 7389129861951380754267995530557685*T5^54 - 16924218415483626542188218847110712*T5^53 + 38300849809466025321967190982202011*T5^52 - 82861488912454316195349813013981174*T5^51 + 177881643405322353560386630845563326*T5^50 - 367584999146444124444953287299722622*T5^49 + 751561899394511292371824773602252017*T5^48 - 1476445989253917781720767855027370458*T5^47 + 2846585888179071727508991916558947406*T5^46 - 5277332448977184029375689992671945726*T5^45 + 9576825185175303742356665819776448347*T5^44 - 16735827840714647410356744135016624154*T5^43 + 28489235132112416669853519075444933456*T5^42 - 46672791506194468145308596437455946996*T5^41 + 74354394668682673666859078673401099429*T5^40 - 114330578466059099885572686439770984754*T5^39 + 170902685903873329443425384694281518223*T5^38 - 246661356234668184202081003751463409028*T5^37 + 345523998626757701331207701336714183741*T5^36 - 468327955259713103064452158773477728556*T5^35 + 616354932144310812024484607208201966634*T5^34 - 783404073875734928076395260943282908564*T5^33 + 961036428435631597048894727946880572199*T5^32 - 1135329345852351812939358384714995062804*T5^31 + 1291914382135375223015106915474144660856*T5^30 - 1405060216391610512353743150253279429704*T5^29 + 1444708999528562155928136571793446271144*T5^28 - 1395208120922279865691822985912824587942*T5^27 + 1260752594132805239503203382975535959786*T5^26 - 1048047954103980649736837714428379418612*T5^25 + 770809621551521438471826675365319521088*T5^24 - 486705485149609899809288505372352282682*T5^23 + 266007230105265728767653115643560345922*T5^22 - 126365104057079318981740440349270323020*T5^21 + 50454102479492632331180981942228650033*T5^20 - 16862959040502601903015127375753387820*T5^19 + 4798925312982370105350616595839678248*T5^18 - 1013639143437323009885656995732485280*T5^17 + 152968518088257975371121114006602688*T5^16 - 14106370374959235800153361961096512*T5^15 - 812277054073798909868077773311232*T5^14 + 394243333607468938133767280126976*T5^13 - 16772483936934865854391996668672*T5^12 + 6203379397566381565830487769088*T5^11 + 1168361663340038447581975701504*T5^10 - 1624478061531327491507115995136*T5^9 + 292313997038210232597450817536*T5^8 + 8924321543173460831282946048*T5^7 - 2503190437252742270456119296*T5^6 - 229457991967007032855461888*T5^5 + 145581859403783247655600128*T5^4 - 6610753902723793625677824*T5^3 + 1235822724948077042466816*T5^2 - 116006686364264155840512*T5 + 15070267630748433383424 acting on $S_{2}^{\mathrm{new}}(402, [\chi])$.

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:	\( N \)	\(=\)	\( 402 = 2 \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	402.j (of order \(22\), degree \(10\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 5.12
Significant digits:
Format:

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