LMFDB - Newform orbit 672.2.bs.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(155,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([4, 1, 4, 0]))

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

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

chi := DirichletCharacter("672.155");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.bs (of order $8$, degree $4$, 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:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$192$
Relative dimension:	$48$ over $\Q(\zeta_{8})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-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) = $ $ 192 q + 16 q^{4}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 192 q + 16 q^{4} + 8 q^{10} + 16 q^{16} + 16 q^{22} + 24 q^{27} + 44 q^{30} - 104 q^{36} - 32 q^{37} + 24 q^{39} + 32 q^{46} - 32 q^{48} - 80 q^{52} - 32 q^{55} - 72 q^{58} - 28 q^{60} + 80 q^{61} + 64 q^{64} + 76 q^{66} - 32 q^{67} - 56 q^{69} - 24 q^{70} + 60 q^{72} - 80 q^{75} - 56 q^{76} - 76 q^{78} - 32 q^{79} + 120 q^{82} - 56 q^{87} + 40 q^{88} + 60 q^{90} + 80 q^{93} - 120 q^{94} - 32 q^{96} - 64 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} $
155.1	−1.41421	−	0.00182353i	1.66131	+	0.489956i	1.99999	+	0.00515773i	−0.328767	+	0.793714i	−2.34855	−	0.695932i	0.707107	−	0.707107i	−2.82841	−	0.0109412i	2.51989	+	1.62794i	0.466394	−	1.12188i
155.2	−1.41292	−	0.0604352i	0.257532	+	1.71280i	1.99270	+	0.170780i	0.740432	−	1.78756i	−0.260359	−	2.43561i	0.707107	−	0.707107i	−2.80520	−	0.361728i	−2.86735	+	0.882200i	−1.15420	+	2.48093i
155.3	−1.41064	+	0.100442i	−1.73160	−	0.0395123i	1.97982	−	0.283377i	1.24365	−	3.00244i	2.44664	−	0.118189i	0.707107	−	0.707107i	−2.76436	+	0.598602i	2.99688	+	0.136839i	−1.45278	+	4.36029i
155.4	−1.38411	−	0.290248i	−0.626099	−	1.61493i	1.83151	+	0.803469i	1.22449	−	2.95618i	0.397859	+	2.41696i	0.707107	−	0.707107i	−2.30181	−	1.64368i	−2.21600	+	2.02221i	−2.55285	+	3.73627i
155.5	−1.36323	−	0.376307i	−0.709246	+	1.58018i	1.71679	+	1.02599i	−0.958822	+	2.31480i	1.56150	−	1.88725i	0.707107	−	0.707107i	−1.95428	−	2.04469i	−1.99394	−	2.24147i	2.17817	−	2.79479i
155.6	−1.34336	−	0.442023i	−1.14969	−	1.29546i	1.60923	+	1.18759i	−1.30330	+	3.14645i	0.971830	+	2.24845i	0.707107	−	0.707107i	−1.63683	−	2.30668i	−0.356409	+	2.97875i	3.14161	−	3.65073i
155.7	−1.31896	+	0.510228i	0.712074	−	1.57891i	1.47933	−	1.34595i	−0.745289	+	1.79929i	−0.133596	+	2.44584i	0.707107	−	0.707107i	−1.26445	+	2.53005i	−1.98590	−	2.24860i	0.0649625	−	2.75346i
155.8	−1.24057	+	0.678963i	0.613308	−	1.61983i	1.07802	−	1.68460i	1.04531	−	2.52359i	0.338955	+	2.42592i	0.707107	−	0.707107i	−0.193574	+	2.82180i	−2.24771	−	1.98691i	0.416652	+	3.84041i
155.9	−1.19021	+	0.763800i	−1.51796	+	0.834142i	0.833219	−	1.81817i	−1.24406	+	3.00343i	1.16958	−	2.15223i	0.707107	−	0.707107i	0.397011	+	2.80043i	1.60842	−	2.53239i	−0.813320	−	4.52493i
155.10	−1.16411	−	0.803019i	0.712483	−	1.57872i	0.710322	+	1.86961i	−0.879803	+	2.12403i	−2.09716	+	1.26568i	0.707107	−	0.707107i	0.674435	−	2.74684i	−1.98474	−	2.24963i	2.72983	−	1.76612i
155.11	−1.14196	+	0.834221i	−1.41654	−	0.996701i	0.608152	−	1.90530i	0.0343602	−	0.0829528i	2.44910	−	0.0435137i	0.707107	−	0.707107i	0.894951	+	2.68311i	1.01317	+	2.82373i	0.0299630	+	0.123393i
155.12	−1.12590	−	0.855778i	1.58528	−	0.697781i	0.535287	+	1.92704i	0.618840	−	1.49401i	−2.38200	−	0.571016i	0.707107	−	0.707107i	1.04644	−	2.62773i	2.02620	−	2.21235i	−1.97529	+	1.15251i
155.13	−1.09166	+	0.899047i	0.650725	+	1.60517i	0.383428	−	1.96290i	−0.365005	+	0.881200i	−2.15349	−	1.16726i	0.707107	−	0.707107i	1.34617	+	2.48753i	−2.15311	+	2.08904i	−0.393780	−	1.29012i
155.14	−0.770079	−	1.18616i	−0.845496	+	1.51167i	−0.813957	+	1.82688i	−0.484034	+	1.16856i	2.44418	−	0.161208i	0.707107	−	0.707107i	2.79378	−	0.441355i	−1.57027	−	2.55622i	1.75885	−	0.325742i
155.15	−0.768409	−	1.18724i	−1.35115	−	1.08370i	−0.819096	+	1.82458i	0.0800577	−	0.193276i	−0.248380	+	2.43686i	0.707107	−	0.707107i	2.79562	−	0.429554i	0.651199	+	2.92847i	−0.290983	+	0.0534673i
155.16	−0.684676	+	1.23742i	1.72238	−	0.182818i	−1.06244	−	1.69447i	0.622207	−	1.50214i	−0.953046	+	2.25648i	0.707107	−	0.707107i	2.82420	−	0.154520i	2.93315	−	0.629763i	1.43277	+	1.79841i
155.17	−0.658473	+	1.25156i	−1.56926	+	0.733092i	−1.13283	−	1.64824i	0.544247	−	1.31393i	0.115802	−	2.44675i	0.707107	−	0.707107i	2.80882	−	0.332485i	1.92515	−	2.30082i	1.28610	+	1.54635i
155.18	−0.584571	−	1.28774i	1.72479	+	0.158475i	−1.31655	+	1.50555i	−1.55272	+	3.74859i	−0.804185	−	2.31372i	0.707107	−	0.707107i	2.70838	+	0.815281i	2.94977	+	0.546669i	5.73489	−	0.191818i
155.19	−0.540503	+	1.30685i	1.42862	−	0.979314i	−1.41571	−	1.41271i	−1.40353	+	3.38841i	0.507644	+	2.39631i	0.707107	−	0.707107i	2.61140	−	1.08655i	1.08189	−	2.79813i	−3.66954	−	3.66565i
155.20	−0.503558	−	1.32153i	0.984099	+	1.42532i	−1.49286	+	1.33093i	0.483125	−	1.16637i	1.38805	−	2.01825i	0.707107	−	0.707107i	2.51060	+	1.30265i	−1.06310	+	2.80532i	−1.78467	−	0.0511282i

See next 80 embeddings (of 192 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
32.h	odd	8	1	inner
96.o	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	672.2.bs.b	✓	192
3.b	odd	2	1	inner	672.2.bs.b	✓	192
32.h	odd	8	1	inner	672.2.bs.b	✓	192
96.o	even	8	1	inner	672.2.bs.b	✓	192

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
672.2.bs.b	✓	192	1.a	even	1	1	trivial
672.2.bs.b	✓	192	3.b	odd	2	1	inner
672.2.bs.b	✓	192	32.h	odd	8	1	inner
672.2.bs.b	✓	192	96.o	even	8	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{192} + 2832 T_{5}^{186} + 294224 T_{5}^{184} + 832576 T_{5}^{182} + 4010112 T_{5}^{180} + \cdots + 55\!\cdots\!76 $ T5^192 + 2832*T5^186 + 294224*T5^184 + 832576*T5^182 + 4010112*T5^180 + 600321952*T5^178 + 35449259008*T5^176 + 183369345856*T5^174 + 866831972864*T5^172 + 47118982873024*T5^170 + 2186512737667008*T5^168 + 14843633289267712*T5^166 + 69104829695327744*T5^164 + 1755043127804703872*T5^162 + 73267647970119709632*T5^160 + 564474359453460708096*T5^158 + 2598441739584289466368*T5^156 + 35584387309369322823424*T5^154 + 1416184072845997994669824*T5^152 + 11613870866540375413773312*T5^150 + 52910751976132398503077888*T5^148 + 419938607446795581514936832*T5^146 + 16349016161562821001865728000*T5^144 + 138051969409519682680454669312*T5^142 + 621816598133892774908658262016*T5^140 + 2976726759974852558021802699776*T5^138 + 113747259142705549562226639088640*T5^136 + 968777385659675028499378043461632*T5^134 + 4304199769802873300489439061991424*T5^132 + 12966883738642369837387697838499840*T5^130 + 471830263823653560512423380710295040*T5^128 + 3996555467570145897251694200027082752*T5^126 + 17469996055618163470864530731115249664*T5^124 + 35904846600443479497103719636322308096*T5^122 + 1139976969196328010575863998494896246784*T5^120 + 9526936491469354775920988326671073722368*T5^118 + 40957372548581854140333415791747400433664*T5^116 + 64551805650844202354464800987865157132288*T5^114 + 1557065921473115710020622153854588426321920*T5^112 + 12792190226001085308603208874508007067205632*T5^110 + 54231674856511075293510840202806923205935104*T5^108 + 72984769668794554748719206313847552474300416*T5^106 + 1136929735414846157591854102330318693638815744*T5^104 + 9144266018955611278444215554871136660037173248*T5^102 + 38414837251312415448460153365331825311938641920*T5^100 + 47640509739464372174444865701913978786663989248*T5^98 + 400242566685784312875572587077913760199741784064*T5^96 + 3113014932065540771343171909491885658569949904896*T5^94 + 13070115426457964322878450808703818475793949392896*T5^92 + 15620414197738305359909526915705521492717378797568*T5^90 + 60517281366505299464233197604590661314461977870336*T5^88 + 439219566257300387969930803612050843481047249453056*T5^86 + 1882287529192464265269172635091848815028529360535552*T5^84 + 2145740732403565177388557334024328604814057266348032*T5^82 + 4590068996601555840417210841459823841138397649305600*T5^80 + 29519876263780035126355500077754536772573490634817536*T5^78 + 129214123303091693628339697523388207456809142938763264*T5^76 + 142405346533711977796612637192773609920693313984528384*T5^74 + 188824963949033748617429283851706957275182529497006080*T5^72 + 991602067635310327950161440744826847166361329341562880*T5^70 + 4447601449573122734075045415887933426706604570328432640*T5^68 + 4807956718147554440472820672694117250219018988169986048*T5^66 + 4328334708632492466329292293833994447532042737703518208*T5^64 + 16232392727451853859352845611141845730975547692300632064*T5^62 + 75890032415427774666599633811964831804714179435553619968*T5^60 + 81497023982985433666466755505235165130069212556152537088*T5^58 + 54831365750208049530673755424922317001245200832748584960*T5^56 + 118264278468543859645229749018083321844529328203860279296*T5^54 + 615882647328823993610870347296535252269553885830841368576*T5^52 + 657583312797624541918027212075996916672729069923539615744*T5^50 + 361306123949909029769135227706594601226189153409462960128*T5^48 + 272990153001189752641782075555278491770856735849677062144*T5^46 + 2117748069046407608819723389442412114416259288995354116096*T5^44 + 2183885959132741382969091615387821712451027756391520534528*T5^42 + 1072880859462544038115280786313395201507837094301397942272*T5^40 - 70645917875139326149354892335101887285129807445870247936*T5^38 + 2728795100332810464479370852946208770620915471127496622080*T5^36 + 2605946320753566922829326543985615487244054531102795104256*T5^34 + 1200955649692477299928512662027868114497022298589860200448*T5^32 - 573379045138954278812566428283255105374044881043740164096*T5^30 + 531230609156090156402685544273833353312762041262885830656*T5^28 + 413525276169175920809897345003595296140201139335206010880*T5^26 + 209153323689832926290689016495150611819916748804444913664*T5^24 - 128339048804078713436460154672331029560061217873739120640*T5^22 + 32666900614517297086345042183638614826263558676824457216*T5^20 - 3184529925674223037151589132064699915882589110646341632*T5^18 + 164797236873444960582564217071221734657654994033967104*T5^16 - 2627625477270248311234155794865194304833670027935744*T5^14 + 93559039692904049502436681145000694559626928062464*T5^12 - 10850814519535715885430748509318310992590720204800*T5^10 + 703970920660013874688103410805575366052369399808*T5^8 - 3183071612861605962660976470463808414941184000*T5^6 + 3453180028519712958449614416754126715092992*T5^4 + 195248393611629357639533666649935154511872*T5^2 + 5519830256904331477510506425685404286976 acting on $S_{2}^{\mathrm{new}}(672, [\chi])$.

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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.bs (of order \(8\), degree \(4\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 155.48
Significant digits:
Format:

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