LMFDB - Newform orbit 966.2.s.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(97,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 11, 1]))

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

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

chi := DirichletCharacter("966.97");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 966 = 2 \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	966.s (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:	$7.71354883526$
Analytic rank:	$0$
Dimension:	$160$
Relative dimension:	$16$ 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) = $ $ 160 q - 16 q^{2} - 16 q^{4} - 16 q^{8} + 16 q^{9}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 160 q - 16 q^{2} - 16 q^{4} - 16 q^{8} + 16 q^{9} - 22 q^{14} - 16 q^{16} + 16 q^{18} - 36 q^{23} + 96 q^{25} + 22 q^{28} - 20 q^{29} - 16 q^{32} - 50 q^{35} + 16 q^{36} + 22 q^{37} + 4 q^{39} - 110 q^{43} + 8 q^{46} - 36 q^{50} + 22 q^{51} - 88 q^{53} + 22 q^{57} + 24 q^{58} - 16 q^{64} - 72 q^{70} + 48 q^{71} + 16 q^{72} - 22 q^{74} + 24 q^{77} + 4 q^{78} + 88 q^{79} - 16 q^{81} + 22 q^{84} + 76 q^{85} + 44 q^{86} - 44 q^{88} + 8 q^{92} - 14 q^{95} + 22 q^{98} - 22 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} $
97.1	−0.959493	−	0.281733i	−0.755750	−	0.654861i	0.841254	+	0.540641i	−0.480377	+	3.34109i	0.540641	+	0.841254i	−0.131654	+	2.64247i	−0.654861	−	0.755750i	0.142315	+	0.989821i	1.40221	−	3.07042i
97.2	−0.959493	−	0.281733i	−0.755750	−	0.654861i	0.841254	+	0.540641i	−0.341716	+	2.37669i	0.540641	+	0.841254i	−0.0945662	−	2.64406i	−0.654861	−	0.755750i	0.142315	+	0.989821i	0.997464	−	2.18414i
97.3	−0.959493	−	0.281733i	−0.755750	−	0.654861i	0.841254	+	0.540641i	−0.288108	+	2.00384i	0.540641	+	0.841254i	2.63618	+	0.224848i	−0.654861	−	0.755750i	0.142315	+	0.989821i	0.840984	−	1.84150i
97.4	−0.959493	−	0.281733i	−0.755750	−	0.654861i	0.841254	+	0.540641i	0.0153446	−	0.106724i	0.540641	+	0.841254i	−1.90691	−	1.83403i	−0.654861	−	0.755750i	0.142315	+	0.989821i	−0.0447907	+	0.0980779i
97.5	−0.959493	−	0.281733i	−0.755750	−	0.654861i	0.841254	+	0.540641i	0.0416256	−	0.289512i	0.540641	+	0.841254i	−0.445694	+	2.60794i	−0.654861	−	0.755750i	0.142315	+	0.989821i	−0.121504	+	0.266058i
97.6	−0.959493	−	0.281733i	−0.755750	−	0.654861i	0.841254	+	0.540641i	0.126102	−	0.877058i	0.540641	+	0.841254i	2.30089	−	1.30610i	−0.654861	−	0.755750i	0.142315	+	0.989821i	−0.368090	+	0.806004i
97.7	−0.959493	−	0.281733i	−0.755750	−	0.654861i	0.841254	+	0.540641i	0.286709	−	1.99411i	0.540641	+	0.841254i	−2.61261	−	0.417427i	−0.654861	−	0.755750i	0.142315	+	0.989821i	−0.836900	+	1.83256i
97.8	−0.959493	−	0.281733i	−0.755750	−	0.654861i	0.841254	+	0.540641i	0.486537	−	3.38394i	0.540641	+	0.841254i	2.17047	+	1.51296i	−0.654861	−	0.755750i	0.142315	+	0.989821i	−1.42020	+	3.10979i
97.9	−0.959493	−	0.281733i	0.755750	+	0.654861i	0.841254	+	0.540641i	−0.486537	+	3.38394i	−0.540641	−	0.841254i	0.277936	−	2.63111i	−0.654861	−	0.755750i	0.142315	+	0.989821i	1.42020	−	3.10979i
97.10	−0.959493	−	0.281733i	0.755750	+	0.654861i	0.841254	+	0.540641i	−0.286709	+	1.99411i	−0.540641	−	0.841254i	−1.39543	+	2.24784i	−0.654861	−	0.755750i	0.142315	+	0.989821i	0.836900	−	1.83256i
97.11	−0.959493	−	0.281733i	0.755750	+	0.654861i	0.841254	+	0.540641i	−0.126102	+	0.877058i	−0.540641	−	0.841254i	2.49385	−	0.883586i	−0.654861	−	0.755750i	0.142315	+	0.989821i	0.368090	−	0.806004i
97.12	−0.959493	−	0.281733i	0.755750	+	0.654861i	0.841254	+	0.540641i	−0.0416256	+	0.289512i	−0.540641	−	0.841254i	−2.26282	−	1.37100i	−0.654861	−	0.755750i	0.142315	+	0.989821i	0.121504	−	0.266058i
97.13	−0.959493	−	0.281733i	0.755750	+	0.654861i	0.841254	+	0.540641i	−0.0153446	+	0.106724i	−0.540641	−	0.841254i	0.137306	+	2.64219i	−0.654861	−	0.755750i	0.142315	+	0.989821i	0.0447907	−	0.0980779i
97.14	−0.959493	−	0.281733i	0.755750	+	0.654861i	0.841254	+	0.540641i	0.288108	−	2.00384i	−0.540641	−	0.841254i	1.55640	−	2.13954i	−0.654861	−	0.755750i	0.142315	+	0.989821i	−0.840984	+	1.84150i
97.15	−0.959493	−	0.281733i	0.755750	+	0.654861i	0.841254	+	0.540641i	0.341716	−	2.37669i	−0.540641	−	0.841254i	1.93632	+	1.80296i	−0.654861	−	0.755750i	0.142315	+	0.989821i	−0.997464	+	2.18414i
97.16	−0.959493	−	0.281733i	0.755750	+	0.654861i	0.841254	+	0.540641i	0.480377	−	3.34109i	−0.540641	−	0.841254i	−2.08326	−	1.63096i	−0.654861	−	0.755750i	0.142315	+	0.989821i	−1.40221	+	3.07042i
181.1	−0.142315	−	0.989821i	−0.909632	−	0.415415i	−0.959493	+	0.281733i	−2.74900	+	3.17251i	−0.281733	+	0.959493i	1.98395	+	1.75041i	0.415415	+	0.909632i	0.654861	+	0.755750i	3.53144	+	2.26952i
181.2	−0.142315	−	0.989821i	−0.909632	−	0.415415i	−0.959493	+	0.281733i	−1.32721	+	1.53168i	−0.281733	+	0.959493i	−2.59428	+	0.519336i	0.415415	+	0.909632i	0.654861	+	0.755750i	1.70498	+	1.09572i
181.3	−0.142315	−	0.989821i	−0.909632	−	0.415415i	−0.959493	+	0.281733i	−1.30557	+	1.50670i	−0.281733	+	0.959493i	2.10508	−	1.60270i	0.415415	+	0.909632i	0.654861	+	0.755750i	1.67717	+	1.07785i
181.4	−0.142315	−	0.989821i	−0.909632	−	0.415415i	−0.959493	+	0.281733i	−0.296662	+	0.342366i	−0.281733	+	0.959493i	2.35940	−	1.19718i	0.415415	+	0.909632i	0.654861	+	0.755750i	0.381101	+	0.244919i

See next 80 embeddings (of 160 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
23.d	odd	22	1	inner
161.k	even	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	966.2.s.a	✓	160
7.b	odd	2	1	inner	966.2.s.a	✓	160
23.d	odd	22	1	inner	966.2.s.a	✓	160
161.k	even	22	1	inner	966.2.s.a	✓	160

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
966.2.s.a	✓	160	1.a	even	1	1	trivial
966.2.s.a	✓	160	7.b	odd	2	1	inner
966.2.s.a	✓	160	23.d	odd	22	1	inner
966.2.s.a	✓	160	161.k	even	22	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{160} - 8 T_{5}^{158} + 498 T_{5}^{156} + 2224 T_{5}^{154} + 73541 T_{5}^{152} + \cdots + 43\!\cdots\!36 $ T5^160 - 8*T5^158 + 498*T5^156 + 2224*T5^154 + 73541*T5^152 + 2421622*T5^150 + 15420420*T5^148 + 564993581*T5^146 + 11701982046*T5^144 + 131705820584*T5^142 + 2602961972147*T5^140 + 37975621173911*T5^138 + 484287032711002*T5^136 + 6710567457498103*T5^134 + 81170858726003749*T5^132 + 998125512970812625*T5^130 + 15469866572672035273*T5^128 + 215360836274905932643*T5^126 + 2054550825696217757396*T5^124 + 14462321513122230182488*T5^122 + 132763569394043025581341*T5^120 + 1643590889871844879743158*T5^118 + 18326162003091813764621308*T5^116 + 162119403502502058930228216*T5^114 + 1252636005829262013612202495*T5^112 + 9457312253495092585231272888*T5^110 + 74624839080546942791588761105*T5^108 + 600071598616950878321279255052*T5^106 + 4669881607547547219737001222829*T5^104 + 34490269975519995334711220945632*T5^102 + 242820155307850146485995366370628*T5^100 + 1629709569512942958144917165420711*T5^98 + 10435686899231532081741264706811368*T5^96 + 64456286725601597801241986495448111*T5^94 + 391567021064230216291339588575135021*T5^92 + 2351237994754715004486905652288943683*T5^90 + 13791200271203091596431945365285874504*T5^88 + 77650952568312072081313992034201605205*T5^86 + 415071147896749717920183388317408754004*T5^84 + 2096723874499400057016917199720241758404*T5^82 + 10005360268002749521380892884973214951453*T5^80 + 45242102645039916140367834971519007755328*T5^78 + 195589794276828894034933995249455400646115*T5^76 + 820586887708816184539400129899432491657563*T5^74 + 3386556904215110838623700101590507407967619*T5^72 + 13723258628568037127946131349768866151720886*T5^70 + 53440954789235981491767849358669793535680365*T5^68 + 193669664985396596295263362159781208955175960*T5^66 + 634171596652008580789583492587949736770877576*T5^64 + 1836370639758507214179150174007895873599570224*T5^62 + 4644974958897488502446716126837446231300227342*T5^60 + 10226989121002076862651561210395335536430416686*T5^58 + 19739928583761870934354814614753461983841391966*T5^56 + 34052760762565606487355792292327677159105377156*T5^54 + 54288004631538776818519716283218692059916365264*T5^52 + 82049624087299125400441522181162608990051759036*T5^50 + 120161379636113979282506956330283626367759529858*T5^48 + 164472993005736807733615577288712945362230216792*T5^46 + 243699935147199152254574691306848487502194802554*T5^44 + 319710248997086220751036188197620739056902182692*T5^42 + 526343946225145536678399860885491623802511224105*T5^40 + 363116062018561186542564645234004777962956106912*T5^38 + 966182384414736453301309271812965732859176916968*T5^36 - 83896031428253301491737103740461698479356252542*T5^34 + 1353782199686265874270535935890235163382339953066*T5^32 + 577730730561725342937900484379096686360337993266*T5^30 + 52292884459124213420066399280937799786437984957*T5^28 + 1270581177110749639568857648993221673824257285459*T5^26 + 579032047201957888903727493921636667267823418563*T5^24 + 146745589213470659711743811835403339796090908126*T5^22 + 550053059294495755528122224914018018049698138657*T5^20 + 206042005064988540981906664689915626446274610636*T5^18 + 62092339913966423190541946674457371240598839536*T5^16 + 11639009819881816131859292361691952553519574336*T5^14 + 1607968473951809760965779059955529189694615040*T5^12 + 141134966113666644366271857697126655695931392*T5^10 + 10508259391275068742848040207532125357277184*T5^8 + 900191879791909244203059774018028806275072*T5^6 + 49115235521898106267233726132504996413440*T5^4 + 814719745354515649892311091637102837760*T5^2 + 4354787894506867937804059060376436736 acting on $S_{2}^{\mathrm{new}}(966, [\chi])$.

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:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	966.s (of order \(22\), degree \(10\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 97.16
Significant digits:
Format:

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