LMFDB - Newform orbit 570.2.bh.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [570,2,Mod(13,570)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(570, base_ring=CyclotomicField(36))

chi = DirichletCharacter(H, H._module([0, 27, 10]))

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

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

chi := DirichletCharacter("570.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 570 = 2 \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	570.bh (of order $36$, degree $12$, 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:	$4.55147291521$
Analytic rank:	$0$
Dimension:	$120$
Relative dimension:	$10$ over $\Q(\zeta_{36})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

$q$-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^{5} + 12 q^{7}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 120 q - 12 q^{5} + 12 q^{7} - 12 q^{10} + 48 q^{13} + 12 q^{17} + 36 q^{21} - 36 q^{22} - 96 q^{23} + 12 q^{25} + 12 q^{26} + 12 q^{30} + 24 q^{38} + 60 q^{41} + 96 q^{43} - 48 q^{47} - 24 q^{52} - 72 q^{53} + 108 q^{55} + 12 q^{57} + 24 q^{58} - 12 q^{60} - 24 q^{61} + 24 q^{62} + 24 q^{66} + 72 q^{67} + 12 q^{68} - 48 q^{70} + 36 q^{73} - 12 q^{76} - 24 q^{78} + 12 q^{80} - 72 q^{82} + 12 q^{83} - 108 q^{85} - 24 q^{86} + 12 q^{87} - 48 q^{91} - 84 q^{92} - 48 q^{93} - 204 q^{95} + 120 q^{96} + 24 q^{97} - 96 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	−0.996195	−	0.0871557i	−0.906308	+	0.422618i	0.984808	+	0.173648i	−1.79770	−	1.32977i	0.939693	−	0.342020i	−1.12737	−	4.20740i	−0.965926	−	0.258819i	0.642788	−	0.766044i	1.67496	+	1.48138i
13.2	−0.996195	−	0.0871557i	−0.906308	+	0.422618i	0.984808	+	0.173648i	−0.538816	−	2.17018i	0.939693	−	0.342020i	0.793037	+	2.95965i	−0.965926	−	0.258819i	0.642788	−	0.766044i	0.347622	+	2.20888i
13.3	−0.996195	−	0.0871557i	−0.906308	+	0.422618i	0.984808	+	0.173648i	−0.393795	+	2.20112i	0.939693	−	0.342020i	0.524562	+	1.95769i	−0.965926	−	0.258819i	0.642788	−	0.766044i	0.584136	−	2.15842i
13.4	−0.996195	−	0.0871557i	−0.906308	+	0.422618i	0.984808	+	0.173648i	1.51913	−	1.64081i	0.939693	−	0.342020i	0.148294	+	0.553442i	−0.965926	−	0.258819i	0.642788	−	0.766044i	−1.65636	+	1.50216i
13.5	−0.996195	−	0.0871557i	−0.906308	+	0.422618i	0.984808	+	0.173648i	2.18820	+	0.460190i	0.939693	−	0.342020i	−0.860238	−	3.21045i	−0.965926	−	0.258819i	0.642788	−	0.766044i	−2.13977	−	0.649154i
13.6	0.996195	+	0.0871557i	0.906308	−	0.422618i	0.984808	+	0.173648i	−2.19145	+	0.444477i	0.939693	−	0.342020i	−1.33986	−	5.00042i	0.965926	+	0.258819i	0.642788	−	0.766044i	−2.22185	+	0.251788i
13.7	0.996195	+	0.0871557i	0.906308	−	0.422618i	0.984808	+	0.173648i	−2.02193	+	0.954879i	0.939693	−	0.342020i	0.864180	+	3.22516i	0.965926	+	0.258819i	0.642788	−	0.766044i	−2.09746	+	0.775022i
13.8	0.996195	+	0.0871557i	0.906308	−	0.422618i	0.984808	+	0.173648i	−0.261022	−	2.22078i	0.939693	−	0.342020i	0.151755	+	0.566356i	0.965926	+	0.258819i	0.642788	−	0.766044i	−0.0664750	−	2.23508i
13.9	0.996195	+	0.0871557i	0.906308	−	0.422618i	0.984808	+	0.173648i	0.998061	+	2.00097i	0.939693	−	0.342020i	0.394140	+	1.47095i	0.965926	+	0.258819i	0.642788	−	0.766044i	0.819868	+	2.08034i
13.10	0.996195	+	0.0871557i	0.906308	−	0.422618i	0.984808	+	0.173648i	2.19391	−	0.432154i	0.939693	−	0.342020i	−0.280552	−	1.04704i	0.965926	+	0.258819i	0.642788	−	0.766044i	2.22323	−	0.239298i
67.1	−0.819152	−	0.573576i	0.996195	+	0.0871557i	0.342020	+	0.939693i	−2.23514	−	0.0643515i	−0.766044	−	0.642788i	2.48134	−	0.664873i	0.258819	−	0.965926i	0.984808	+	0.173648i	1.79401	+	1.33474i
67.2	−0.819152	−	0.573576i	0.996195	+	0.0871557i	0.342020	+	0.939693i	−1.90079	−	1.17771i	−0.766044	−	0.642788i	−2.42358	+	0.649396i	0.258819	−	0.965926i	0.984808	+	0.173648i	0.881533	+	2.05497i
67.3	−0.819152	−	0.573576i	0.996195	+	0.0871557i	0.342020	+	0.939693i	−0.0386474	+	2.23573i	−0.766044	−	0.642788i	2.25738	−	0.604862i	0.258819	−	0.965926i	0.984808	+	0.173648i	1.31402	−	1.80924i
67.4	−0.819152	−	0.573576i	0.996195	+	0.0871557i	0.342020	+	0.939693i	0.180509	−	2.22877i	−0.766044	−	0.642788i	−2.19282	+	0.587565i	0.258819	−	0.965926i	0.984808	+	0.173648i	−1.42623	+	1.72217i
67.5	−0.819152	−	0.573576i	0.996195	+	0.0871557i	0.342020	+	0.939693i	2.23464	+	0.0799806i	−0.766044	−	0.642788i	4.38798	−	1.17576i	0.258819	−	0.965926i	0.984808	+	0.173648i	−1.78463	−	1.34725i
67.6	0.819152	+	0.573576i	−0.996195	−	0.0871557i	0.342020	+	0.939693i	−2.19503	+	0.426412i	−0.766044	−	0.642788i	2.31403	−	0.620043i	−0.258819	+	0.965926i	0.984808	+	0.173648i	−2.04265	−	0.909723i
67.7	0.819152	+	0.573576i	−0.996195	−	0.0871557i	0.342020	+	0.939693i	−1.42376	+	1.72421i	−0.766044	−	0.642788i	−2.79653	+	0.749327i	−0.258819	+	0.965926i	0.984808	+	0.173648i	−2.15524	+	0.595756i
67.8	0.819152	+	0.573576i	−0.996195	−	0.0871557i	0.342020	+	0.939693i	−0.598083	−	2.15460i	−0.766044	−	0.642788i	−0.0495570	+	0.0132788i	−0.258819	+	0.965926i	0.984808	+	0.173648i	0.745906	−	2.10799i
67.9	0.819152	+	0.573576i	−0.996195	−	0.0871557i	0.342020	+	0.939693i	−0.299221	−	2.21596i	−0.766044	−	0.642788i	−3.99996	+	1.07179i	−0.258819	+	0.965926i	0.984808	+	0.173648i	1.02591	−	1.98683i
67.10	0.819152	+	0.573576i	−0.996195	−	0.0871557i	0.342020	+	0.939693i	1.51676	+	1.64300i	−0.766044	−	0.642788i	2.75377	−	0.737870i	−0.258819	+	0.965926i	0.984808	+	0.173648i	0.300075	+	2.21584i

See next 80 embeddings (of 120 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
19.f	odd	18	1	inner
95.r	even	36	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	570.2.bh.a	✓	120
5.c	odd	4	1	inner	570.2.bh.a	✓	120
19.f	odd	18	1	inner	570.2.bh.a	✓	120
95.r	even	36	1	inner	570.2.bh.a	✓	120

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
570.2.bh.a	✓	120	1.a	even	1	1	trivial
570.2.bh.a	✓	120	5.c	odd	4	1	inner
570.2.bh.a	✓	120	19.f	odd	18	1	inner
570.2.bh.a	✓	120	95.r	even	36	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{120} - 12 T_{7}^{119} + 72 T_{7}^{118} - 640 T_{7}^{117} + 3162 T_{7}^{116} + \cdots + 20\!\cdots\!41 $ T7^120 - 12*T7^119 + 72*T7^118 - 640*T7^117 + 3162*T7^116 - 1500*T7^115 - 4864*T7^114 + 330876*T7^113 - 3582399*T7^112 + 6729484*T7^111 - 18076752*T7^110 - 17621928*T7^109 + 2098928528*T7^108 - 5817635424*T7^107 + 14345943768*T7^106 - 73226793228*T7^105 - 787522965132*T7^104 + 2874166674240*T7^103 - 5295265517528*T7^102 + 47755050609108*T7^101 + 214685392075581*T7^100 - 1043913176701712*T7^99 + 1219543662201216*T7^98 - 17604193030729344*T7^97 - 45933635146914726*T7^96 + 294358285043131884*T7^95 - 165716549311563384*T7^94 + 4657446730279821112*T7^93 + 8437520062086062598*T7^92 - 66799054554988396428*T7^91 + 3098729160205661656*T7^90 - 955549970362292063688*T7^89 - 1443524507750494436700*T7^88 + 12253181427493570302172*T7^87 + 4862218284702789649200*T7^86 + 158895680625180869323692*T7^85 + 238504330782301755222131*T7^84 - 1817825812477256165518536*T7^83 - 1423399925047032523981248*T7^82 - 21797662444356064364766308*T7^81 - 36585226858574328944396124*T7^80 + 216760120064376330183921768*T7^79 + 249163393487453042711030768*T7^78 + 2497658900491387467779949180*T7^77 + 4914872496471654404338718454*T7^76 - 20515575443917496426933740016*T7^75 - 31217671074102446953311501768*T7^74 - 238643867753127903622667567676*T7^73 - 552030742280772025626816673387*T7^72 + 1511555394197798699003588081796*T7^71 + 2971461173553393528182214329472*T7^70 + 18996219223139322435283201742084*T7^69 + 50820929742413392409526859026777*T7^68 - 82691126363129198352652328298000*T7^67 - 215576703683328001597203038649656*T7^66 - 1244912082299424306536175436294164*T7^65 - 3783002771404781549482594701100257*T7^64 + 3008241103259644731439389078749628*T7^63 + 12006854476732670646437008929269616*T7^62 + 66755929346415109487942560523438412*T7^61 + 227455987735814554341560173123880218*T7^60 - 34117119550847480972061050623605288*T7^59 - 494802286955250103737148032604598952*T7^58 - 2874388674208226999898876673624550852*T7^57 - 10960183341559085302065231298565848641*T7^56 - 3853643122729128780427165214999248836*T7^55 + 14300156457598516329991517137992529552*T7^54 + 98063732475514361661477565530075172536*T7^53 + 422857947287250167473067438556194221278*T7^52 + 325164238287141937292436054130992615360*T7^51 - 210104503796148818834171131338623042976*T7^50 - 2538832759340217850046302040133265829736*T7^49 - 12857569961005635103733072424765950919841*T7^48 - 14451445890653679414342478596198443774220*T7^47 - 2947963347221957408766760631455303835280*T7^46 + 46912702128858797141462828650279885255064*T7^45 + 304574117145282031825278998185486570367814*T7^44 + 445676228718025762066382525460173707273908*T7^43 + 293162870607612315165387111408201848535632*T7^42 - 466790710708219613776207048268139286407564*T7^41 - 5369343991363960351329012690678352765644243*T7^40 - 9750639718558174024150983321916889725359716*T7^39 - 9194405083770475574965181378932302211466016*T7^38 - 2087188866860212943401375926424074589664520*T7^37 + 67005420577917821754469559286654838809274553*T7^36 + 153578719221442703936327584848071165367531540*T7^35 + 181971439826180573597468383184620617151218240*T7^34 + 178107349469296635579791893325719637367298432*T7^33 - 469723603662046208529858613407959345418890199*T7^32 - 1548371228221702678987702145594651153118271368*T7^31 - 2157235550795011815055916760161318489182858704*T7^30 - 2902669847057199102672325469443610530412736672*T7^29 + 698325028684955193000563469543225126757393464*T7^28 + 9374292658910279533754013083858493577967259684*T7^27 + 16103726097421663087808744576813052585824072160*T7^26 + 26595076166913346327122988109559855026204686932*T7^25 + 25327713341669925743688509228123213780161542243*T7^24 - 1394309273885981305714391894006216038405925376*T7^23 - 22603284671180307106844138679726599060724555672*T7^22 - 48730240804017613213924022312527696585153871452*T7^21 - 57253522528824393101299172621249520485220748032*T7^20 - 10087547092125390079194264137672039034203337264*T7^19 + 32991446458333613997940150727902253579542741584*T7^18 + 77190135436806543338021465069578816593903954288*T7^17 + 87672900422136569063015612922605941295009055465*T7^16 + 23770081658551748476177504784493128962176322740*T7^15 - 17185305334362432957527523586895525849510954184*T7^14 - 38660130008928585135964605661904356118444045988*T7^13 - 34277692873709293506760539591542507749245037500*T7^12 - 7622128992410166286162701217178170501289531604*T7^11 + 4224736436422084717273746606644688576187357368*T7^10 + 9507760611818551251349550765572076985070359420*T7^9 + 11699620079265835538667180297722323836961034427*T7^8 + 6977848868215209449077025519306792422449440268*T7^7 + 3117188872836824708169392414148532683183160088*T7^6 + 1470290762548030203570163771071368815768330020*T7^5 + 400519396012094631221776791057071482138824843*T7^4 + 25694516414360603787647333154756174735162304*T7^3 + 1050962695903605957105455948329475336152200*T7^2 + 66194377502129070155278157535474902605980*T7 + 2084610438587945491283539481158111206241 acting on $S_{2}^{\mathrm{new}}(570, [\chi])$.

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:	\( N \)	\(=\)	\( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	570.bh (of order \(36\), degree \(12\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 13.10
Significant digits:
Format:

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