LMFDB - Newform orbit 3.127.b.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3,127,Mod(2,3)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 127, names="a")

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

chi := DirichletCharacter("3.2");

S:= CuspForms(chi, 127);

N := Newforms(S);

Level:	$ N $	$=$	$ 3 $
Weight:	$ k $	$=$	$ 127 $
Character orbit:	$[\chi]$	$=$	3.b (of order $2$, degree $1$, 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:	yes
Analytic conductor:	$301.614165769$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$ q - 11\!\cdots\!27 q^{3}+ \cdots + 13\!\cdots\!29 q^{9}+O(q^{10}) $

$ q - 11\!\cdots\!27 q^{3}+ \cdots - 17\!\cdots\!46 q^{97}+O(q^{100}) $

q - 1144561273430837494885949696427 * q^3 + 85070591730234615865843651857942052864 * q^4 - 133746816060358027855113634867182735459029690869320286 * q^7 + 1310020508637620352391208095712502073964245732475093456566329 * q^9 - 97368504802272205153595678239454304952368475046930119079817985916928 * q^12 + 19841152603074149275090890324610684502792057287637401479180276695474906 * q^13 + 7237005577332262213973186563042994240829374041602535252466099000494570602496 * q^16 + 248413807005157169623594265860731517953056726650496983345647595696300433347959818 * q^19 + 153081426107363372378034878033871214039187629661588061819279514625257245109532818122 * q^21 + 11754943508222875079687365372222456778186655567720875215087517062784172594547271728515625 * q^25 - 1499398741586788200414239710724876101933611366003344657118522818557991334322919287339806483 * q^27 - 11377920784289503950344172165320471889062733033492861495797252862532508668556148927159599104 * q^28 + 12207955576895410177078746105884359289551992249631070124755742603043947057559870128900839756882 * q^31 + 111444219848545291112918149658401217019177846881717006276548100629318214534968256903948922840416256 * q^36 + 734533449758416293284480586742435705708429425991594634597411399495065583918408604592199076978772394 * q^37 - 22709414889710124490848362915957187470839219376156744786542092312916064197151234540087419695296360862 * q^39 - 9006624125890382043581238932901517512827276643126554359599073463266735063482239535734631733454950460614 * q^43 - 8283196319417487337008426752360107305013961134870888947274555954431444947226090508990539733167049688481792 * q^48 - 12475373830402708882796875810939096792271349298565654815342353711612893527056667668119541803181872714123853 * q^49 + 1687898592553402745095638857895772589299402508131341244822543042419990777358811439785874533977085033037430784 * q^52 - 284324823283624989934641680710315944127226167193949841281358852711463614243867567236537152611429640016894170286 * q^57 + 12928544661978676515567214119262538014281953071012853122568865302526649720057594215620260586912402382611085354362 * q^61 - 175211072004052474300241690532019553423155743543627180098308146797219661493240082651435227148013196846979704250094 * q^63 + 615656346818663737691860001564743965704370926101022604186692084441339402679643915803347910232576806887603562348544 * q^64 - 3550883955185995535260131886000200461572020298001289300655721288015382221977117243078218356783070576406952803859926 * q^67 + 4776169238538897334756708673990993217518179156023578435929862176833550696907021884892214232859514314588593678434767666 * q^73 - 13454253110879130282535547877881199324765662402628312634839322480709209088156445843775799176000873558223247528076171875 * q^75 + 21132709555889021401949178344837918689097770460853957333738975686251386618283843986153808266769637495056968681103818752 * q^76 + 673769460124715522094826134804328616181084020643334565469667280636453962637209424904604978026996045808986984800412236978 * q^79 + 1716153733051169540307898602341497569311343837142335036933392894197976205305758468407429248816168018787798340006776536241 * q^81 + 13022727501861587929658524398951489107207418925289270426266985952563427546617126726242615644038195710433253937567821201408 * q^84 - 2653690987628852118946024598839052224571196459844107129430507828188853723190781685201514401518089092763801686428520189743116 * q^91 - 13972753181078505058562982422334288573673041285212128636827159171282048604124319327288923105991994042213854475502980384060614 * q^93 - 177521256401008613760602548539776127707825557097806625459006632438577147736100184623290276729892494738927513605078509673355646 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

2.1

−1.14456e30

8.50706e37

−1.33747e53

1.31002e60

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3.127.b.a	✓	1
3.b	odd	2	1	CM	3.127.b.a	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3.127.b.a	✓	1	1.a	even	1	1	trivial
3.127.b.a	✓	1	3.b	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} $ T2 acting on $S_{127}^{\mathrm{new}}(3, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 11\!\cdots\!27 $ T + 1144561273430837494885949696427
$5$	$ T $ T
$7$	$ T + 13\!\cdots\!86 $ T + 133746816060358027855113634867182735459029690869320286
$11$	$ T $ T
$13$	$ T - 19\!\cdots\!06 $ T - 19841152603074149275090890324610684502792057287637401479180276695474906
$17$	$ T $ T
$19$	$ T - 24\!\cdots\!18 $ T - 248413807005157169623594265860731517953056726650496983345647595696300433347959818
$23$	$ T $ T
$29$	$ T $ T
$31$	$ T - 12\!\cdots\!82 $ T - 12207955576895410177078746105884359289551992249631070124755742603043947057559870128900839756882
$37$	$ T - 73\!\cdots\!94 $ T - 734533449758416293284480586742435705708429425991594634597411399495065583918408604592199076978772394
$41$	$ T $ T
$43$	$ T + 90\!\cdots\!14 $ T + 9006624125890382043581238932901517512827276643126554359599073463266735063482239535734631733454950460614
$47$	$ T $ T
$53$	$ T $ T
$59$	$ T $ T
$61$	$ T - 12\!\cdots\!62 $ T - 12928544661978676515567214119262538014281953071012853122568865302526649720057594215620260586912402382611085354362
$67$	$ T + 35\!\cdots\!26 $ T + 3550883955185995535260131886000200461572020298001289300655721288015382221977117243078218356783070576406952803859926
$71$	$ T $ T
$73$	$ T - 47\!\cdots\!66 $ T - 4776169238538897334756708673990993217518179156023578435929862176833550696907021884892214232859514314588593678434767666
$79$	$ T - 67\!\cdots\!78 $ T - 673769460124715522094826134804328616181084020643334565469667280636453962637209424904604978026996045808986984800412236978
$83$	$ T $ T
$89$	$ T $ T
$97$	$ T + 17\!\cdots\!46 $ T + 177521256401008613760602548539776127707825557097806625459006632438577147736100184623290276729892494738927513605078509673355646
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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:	\( N \)	\(=\)	\( 3 \)
Weight:	\( k \)	\(=\)	\( 127 \)
Character orbit:	\([\chi]\)	\(=\)	3.b (of order \(2\), degree \(1\), minimal)

\(n\)	\(2\)
\(\chi(n)\)	\(-1\)

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