LMFDB - Newform orbit 294.3.o.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [294,3,Mod(61,294)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(294, base_ring=CyclotomicField(42))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("294.61");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 294 = 2 \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	294.o (of order $42$, 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:	$8.01091977219$
Analytic rank:	$0$
Dimension:	$96$
Relative dimension:	$8$ over $\Q(\zeta_{42})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

$q$-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) = $ $ 96 q + 24 q^{3} + 16 q^{4} - 24 q^{9}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 96 q + 24 q^{3} + 16 q^{4} - 24 q^{9} + 12 q^{10} + 70 q^{11} - 48 q^{12} + 42 q^{15} + 32 q^{16} - 86 q^{17} - 24 q^{19} - 28 q^{20} - 42 q^{21} - 64 q^{22} + 4 q^{23} - 44 q^{25} + 88 q^{26} + 40 q^{29} - 12 q^{30} + 12 q^{31} - 84 q^{33} - 14 q^{35} + 96 q^{36} + 352 q^{37} + 304 q^{38} + 336 q^{39} + 88 q^{40} + 14 q^{41} + 168 q^{42} - 80 q^{43} + 112 q^{44} + 84 q^{45} - 12 q^{46} + 338 q^{47} - 56 q^{49} - 32 q^{50} - 180 q^{51} - 224 q^{52} - 204 q^{53} - 560 q^{55} - 56 q^{56} + 300 q^{57} - 340 q^{58} - 258 q^{59} - 168 q^{60} - 642 q^{61} - 392 q^{62} - 210 q^{63} - 128 q^{64} - 610 q^{65} + 180 q^{66} - 44 q^{67} - 24 q^{68} - 140 q^{70} + 316 q^{71} + 30 q^{73} - 288 q^{74} + 132 q^{75} + 840 q^{77} - 120 q^{78} + 36 q^{79} + 72 q^{81} - 240 q^{82} + 784 q^{83} - 248 q^{85} + 956 q^{86} - 186 q^{87} + 176 q^{88} + 286 q^{89} + 364 q^{91} - 352 q^{92} - 222 q^{93} - 32 q^{94} - 116 q^{95}+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	−1.16848	+	0.796655i	1.17809	+	1.26968i	0.730682	−	1.86175i	−1.47403	+	4.77870i	−2.38808	−	0.545063i	−6.51751	+	2.55383i	0.629384	+	2.75751i	−0.224190	+	2.99161i	−2.08460	−	6.75810i
61.2	−1.16848	+	0.796655i	1.17809	+	1.26968i	0.730682	−	1.86175i	−0.154599	+	0.501196i	−2.38808	−	0.545063i	5.44495	+	4.39915i	0.629384	+	2.75751i	−0.224190	+	2.99161i	−0.218635	−	0.708799i
61.3	−1.16848	+	0.796655i	1.17809	+	1.26968i	0.730682	−	1.86175i	0.281198	−	0.911622i	−2.38808	−	0.545063i	0.685618	−	6.96634i	0.629384	+	2.75751i	−0.224190	+	2.99161i	0.397674	+	1.28923i
61.4	−1.16848	+	0.796655i	1.17809	+	1.26968i	0.730682	−	1.86175i	2.80386	−	9.08989i	−2.38808	−	0.545063i	−3.32869	+	6.15791i	0.629384	+	2.75751i	−0.224190	+	2.99161i	3.96526	+	12.8550i
61.5	1.16848	−	0.796655i	1.17809	+	1.26968i	0.730682	−	1.86175i	−1.62177	+	5.25766i	2.38808	+	0.545063i	−5.04280	+	4.85491i	−0.629384	−	2.75751i	−0.224190	+	2.99161i	2.29353	+	7.43545i
61.6	1.16848	−	0.796655i	1.17809	+	1.26968i	0.730682	−	1.86175i	−1.30075	+	4.21692i	2.38808	+	0.545063i	6.78420	−	1.72471i	−0.629384	−	2.75751i	−0.224190	+	2.99161i	1.83953	+	5.96362i
61.7	1.16848	−	0.796655i	1.17809	+	1.26968i	0.730682	−	1.86175i	0.854208	−	2.76928i	2.38808	+	0.545063i	−5.34846	−	4.51597i	−0.629384	−	2.75751i	−0.224190	+	2.99161i	−1.20803	−	3.91635i
61.8	1.16848	−	0.796655i	1.17809	+	1.26968i	0.730682	−	1.86175i	1.86201	−	6.03650i	2.38808	+	0.545063i	6.33701	+	2.97360i	−0.629384	−	2.75751i	−0.224190	+	2.99161i	−2.63328	−	8.53689i
73.1	−1.35138	+	0.416847i	−1.61232	+	0.632789i	1.65248	−	1.12664i	−1.24657	+	8.27048i	1.91509	−	1.52723i	−1.03853	+	6.92253i	−1.76350	+	2.21135i	2.19916	−	2.04052i	−1.76292	−	11.6962i
73.2	−1.35138	+	0.416847i	−1.61232	+	0.632789i	1.65248	−	1.12664i	0.0118638	−	0.0787113i	1.91509	−	1.52723i	6.86278	+	1.37920i	−1.76350	+	2.21135i	2.19916	−	2.04052i	0.0167780	+	0.111315i
73.3	−1.35138	+	0.416847i	−1.61232	+	0.632789i	1.65248	−	1.12664i	0.199351	−	1.32261i	1.91509	−	1.52723i	−6.98745	+	0.418976i	−1.76350	+	2.21135i	2.19916	−	2.04052i	0.281925	+	1.87045i
73.4	−1.35138	+	0.416847i	−1.61232	+	0.632789i	1.65248	−	1.12664i	0.775701	−	5.14644i	1.91509	−	1.52723i	0.0677500	−	6.99967i	−1.76350	+	2.21135i	2.19916	−	2.04052i	1.09701	+	7.27816i
73.5	1.35138	−	0.416847i	−1.61232	+	0.632789i	1.65248	−	1.12664i	−1.26889	+	8.41853i	−1.91509	+	1.52723i	−5.87802	−	3.80116i	1.76350	−	2.21135i	2.19916	−	2.04052i	1.79448	+	11.9056i
73.6	1.35138	−	0.416847i	−1.61232	+	0.632789i	1.65248	−	1.12664i	−0.359413	+	2.38455i	−1.91509	+	1.52723i	3.31540	−	6.16507i	1.76350	−	2.21135i	2.19916	−	2.04052i	0.508286	+	3.37226i
73.7	1.35138	−	0.416847i	−1.61232	+	0.632789i	1.65248	−	1.12664i	0.204702	−	1.35811i	−1.91509	+	1.52723i	−0.513265	+	6.98116i	1.76350	−	2.21135i	2.19916	−	2.04052i	−0.289492	−	1.92065i
73.8	1.35138	−	0.416847i	−1.61232	+	0.632789i	1.65248	−	1.12664i	0.590481	−	3.91758i	−1.91509	+	1.52723i	−6.98353	+	0.479892i	1.76350	−	2.21135i	2.19916	−	2.04052i	−0.835066	−	5.54030i
103.1	−1.39842	+	0.210778i	0.975699	−	1.43109i	1.91115	−	0.589510i	−6.49804	−	0.486961i	−1.06279	+	2.20691i	6.99090	−	0.356834i	−2.54832	+	1.22721i	−1.09602	−	2.79262i	9.18961	−	0.688666i
103.2	−1.39842	+	0.210778i	0.975699	−	1.43109i	1.91115	−	0.589510i	−1.69652	−	0.127137i	−1.06279	+	2.20691i	−4.94257	−	4.95691i	−2.54832	+	1.22721i	−1.09602	−	2.79262i	2.39925	−	0.179799i
103.3	−1.39842	+	0.210778i	0.975699	−	1.43109i	1.91115	−	0.589510i	0.784539	+	0.0587931i	−1.06279	+	2.20691i	−6.16101	+	3.32295i	−2.54832	+	1.22721i	−1.09602	−	2.79262i	−1.10951	+	0.0831460i
103.4	−1.39842	+	0.210778i	0.975699	−	1.43109i	1.91115	−	0.589510i	2.93458	+	0.219916i	−1.06279	+	2.20691i	5.28548	−	4.58952i	−2.54832	+	1.22721i	−1.09602	−	2.79262i	−4.15012	+	0.311008i

See all 96 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
49.h	odd	42	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	294.3.o.a	✓	96
49.h	odd	42	1	inner	294.3.o.a	✓	96

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
294.3.o.a	✓	96	1.a	even	1	1	trivial
294.3.o.a	✓	96	49.h	odd	42	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{96} + 122 T_{5}^{94} + 812 T_{5}^{93} + 3584 T_{5}^{92} + 94462 T_{5}^{91} + \cdots + 35\!\cdots\!49 $ T5^96 + 122*T5^94 + 812*T5^93 + 3584*T5^92 + 94462*T5^91 + 40680*T5^90 + 2277268*T5^89 + 24914276*T5^88 + 40592390*T5^87 + 2353676487*T5^86 + 24945874306*T5^85 + 269751611510*T5^84 + 2393392945630*T5^83 + 15531586503600*T5^82 + 79575140494454*T5^81 + 147002170017198*T5^80 - 1066161242006814*T5^79 - 32308264720278665*T5^78 - 350068626976275952*T5^77 - 1967778424094714579*T5^76 - 9027773483041233622*T5^75 - 6286382881421632179*T5^74 + 204370782048830744730*T5^73 + 2177039451145748316046*T5^72 + 15671758155114798957180*T5^71 + 44674207468161559197015*T5^70 + 49774903774075061019610*T5^69 - 981628191267015426729382*T5^68 - 6708268602432023939908178*T5^67 - 33296273793049749793153000*T5^66 - 212762150452297338919492110*T5^65 - 313478037101162523172456898*T5^64 + 2482613561556169907881095166*T5^63 + 45017532218544476566486100955*T5^62 + 294399271942023239891007746598*T5^61 + 1396102947595195922103098811450*T5^60 + 3479563786129412252223390637536*T5^59 - 3461201735791474095499906682759*T5^58 - 116689270321342199460856802402840*T5^57 - 751246664824555126277086827578842*T5^56 - 4688813602869046792163481920209738*T5^55 - 24843583615990272948447582999914904*T5^54 - 116603231771896876417961907483991440*T5^53 - 234082449513970183888994321452862629*T5^52 + 726757956761806918921048244392521548*T5^51 + 9661088400559888538971060797682136360*T5^50 + 48376936291383982670977045794067053472*T5^49 + 189797324120478523114221768656730227768*T5^48 + 732898000729074439745388638764131234220*T5^47 + 2813197241644729118554424325125924958838*T5^46 + 8197911718925761352453892387144858270684*T5^45 + 5292069052903911830811531193540129296049*T5^44 - 108849537379663468646573028497396768606240*T5^43 - 739821377320161899010989232956830560307449*T5^42 - 2234129182545298960652882909482888325172242*T5^41 + 1032804563315025876365434230128171270338991*T5^40 + 41718363159145096325419669475070350444187274*T5^39 + 194019050721755553909680235264274957695051732*T5^38 + 428235200930241195902236603463876987459609718*T5^37 - 38166311712894205987189034224670059361497802*T5^36 - 4219950033716636209618416361338724870416777510*T5^35 - 19265519794104596920856445017067735659827345303*T5^34 - 50275179762250889660790026404396601929147008444*T5^33 - 60687810577363725383577228332294103573796098213*T5^32 + 117037389891969055507922587902298076439623381306*T5^31 + 801126312856003390862864336742551028637118747675*T5^30 + 2073142538367776997134679384369922389057609292638*T5^29 + 3030288015578390388061011994870668271910603874451*T5^28 + 1209447578932820525876717783635219333726773463012*T5^27 - 3834964474948719318340237700265062377365301293515*T5^26 - 5403186072781969549439526548747929140314426168448*T5^25 + 4920616223117172997318618133893886883844519233700*T5^24 + 32299906827174108139088982251086292428496906707440*T5^23 + 86222247176220086996226808793205673218704880538222*T5^22 + 169302296749319106617374976479688684264828357798408*T5^21 + 284337128289173783641172458920085934424970159666514*T5^20 + 340862338087008260746180629664212753295995166828214*T5^19 + 512644094333675846690899809203435717028351364781740*T5^18 + 730016988274774174037164553796069982616128626457912*T5^17 + 569726761765685108923241531919426377786172001016609*T5^16 + 794266827361902747508400295656042002681415178163624*T5^15 + 263805862195719662678821240953712959805158330091535*T5^14 + 701135393172100816745465173856086642999027091298498*T5^13 - 335073537738147235551398105186385304615120077362000*T5^12 - 647454608281172363636868707057962063154514672646034*T5^11 + 2018466920317154541897971188283617546634583486179878*T5^10 - 3805464134230585944728537186748326526056258432524108*T5^9 + 2879785187492812792524775353867803753007270135531955*T5^8 - 2329764604990885077446159752748195257000637463984198*T5^7 + 857656993084973340605932182215461785729984157206658*T5^6 - 474962152030089620002491954973365179574081529508232*T5^5 + 353363022397957986975940674647450855139564943052512*T5^4 + 50164375558021682669800578065009034307199610733164*T5^3 + 57131970147632811305227880563837896855277861847100*T5^2 - 952740403067309867840536723694325462248176038024*T5 + 357603899118066578280647240403268553785257538849 acting on $S_{3}^{\mathrm{new}}(294, [\chi])$.

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:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	294.o (of order \(42\), degree \(12\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 61.8
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more