LMFDB - Newform orbit 4.22.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4,22,Mod(1,4)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("4.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$N$	$=$	$4 = 2^{2}$
Weight:	$k$	$=$	$22$
Character orbit:	$[\chi]$	$=$	4.a (trivial)

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:	$11.1790937715$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2161})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 540$ x^2 - x - 540
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2^{8}\cdot 3\cdot 7$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of $\beta = 2688\sqrt{2161}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta + 32820) q^{3} + ( - 204 \beta + 6844662) q^{5} + (5886 \beta - 130254040) q^{7} + ( - 65640 \beta + 6230767581) q^{9} + (314445 \beta + 72717981660) q^{11} + (324756 \beta + 714450208670) q^{13}+ \cdots + ( - 28\!\cdots\!55 \beta + 13\!\cdots\!60) q^{99}+O(q^{100})$ q + (-b + 32820) * q^3 + (-204b + 6844662) q^5 + (5886b - 130254040) q^7 + (-65640b + 6230767581) q^9 + (314445b + 72717981660) q^11 + (324756b + 714450208670) q^13 + (-13539942b + 3409891357176) q^15 + (84858072b + 920310288210) q^17 + (-265449285b - 8390371964284) q^19 + (323432560b - 96178755501024) q^21 + (627682938b - 159845962713480) q^23 + (-2792622096b + 219803147959663) q^25 + (2075280822b + 886085884611720) q^27 + (4723712580b + 1871055887883246) q^29 + (4595859000b + 56021176731296) q^31 + (-62397896760b - 2523130130425680) q^33 + (66859504692b - 19639923731212176) q^35 + (245612021796b - 16681352818773610) q^37 + (-703791716750b + 18377525932035096) q^39 + (265794737520b + 87564872161566666) q^41 + (1278047140965b + 7673306708264060) q^43 + (-1720360200204b + 251727278576557662) q^45 + (553734430452b - 342424409644227600) q^47 + (-1533350558880b - 633876939155943) q^49 + (1864731634830b - 1294766669676143448) q^51 + (7229502808932b + 337652697122210790) q^53 + (-12682198516050b - 503855789070504600) q^55 + (-321673569416b + 3869344735677604560) q^57 + (-7341605116095b + 521222625217717452) q^59 + (32044341559860b + 4532998914868234382) q^61 + (45224173167366b - 6844149257222100600) q^63 + (-143524997516208b + 3855741291206701524) q^65 + (-19726418252529b - 15232150523401387660) q^67 + (180446516738640b - 15046766045364645792) q^69 + (115464309246510b - 4099546751415259032) q^71 + (-226051490503944b + 12707543329687396970) q^73 + (-311457005150383b + 50817852431439952524) q^75 + (387060308442960b + 19426885130290589280) q^77 + (99610802523180b + 60602176612530082448) q^79 + (-131357583788760b - 68498060032734793191) q^81 + (218924689501323b + 54371468378516904900) q^83 + (393081522016824b - 263994922822459877172) q^85 + (-1716023641007646b - 12347844638894937000) q^87 + (-1052618322986760b - 92103999637482684486) q^89 + (4162953147217380b - 63213709769507333456) q^91 + (94814915648704b - 69920982103000721280) q^93 + (-105274753252734b + 788092955533462657752) q^95 + (-5190622720371144b + 369748892594238233570) q^97 + (-2813974604154855b + 130813883985288961260) q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 65640 q^{3} + 13689324 q^{5} - 260508080 q^{7} + 12461535162 q^{9} + 145435963320 q^{11} + 1428900417340 q^{13} + 6819782714352 q^{15} + 1840620576420 q^{17} - 16780743928568 q^{19} - 192357511002048 q^{21}+ \cdots + 26\!\cdots\!20 q^{99}+O(q^{100})$ 2 * q + 65640 * q^3 + 13689324 * q^5 - 260508080 * q^7 + 12461535162 * q^9 + 145435963320 * q^11 + 1428900417340 * q^13 + 6819782714352 * q^15 + 1840620576420 * q^17 - 16780743928568 * q^19 - 192357511002048 * q^21 - 319691925426960 * q^23 + 439606295919326 * q^25 + 1772171769223440 * q^27 + 3742111775766492 * q^29 + 112042353462592 * q^31 - 5046260260851360 * q^33 - 39279847462424352 * q^35 - 33362705637547220 * q^37 + 36755051864070192 * q^39 + 175129744323133332 * q^41 + 15346613416528120 * q^43 + 503454557153115324 * q^45 - 684848819288455200 * q^47 - 1267753878311886 * q^49 - 2589533339352286896 * q^51 + 675305394244421580 * q^53 - 1007711578141009200 * q^55 + 7738689471355209120 * q^57 + 1042445250435434904 * q^59 + 9065997829736468764 * q^61 - 13688298514444201200 * q^63 + 7711482582413403048 * q^65 - 30464301046802775320 * q^67 - 30093532090729291584 * q^69 - 8199093502830518064 * q^71 + 25415086659374793940 * q^73 + 101635704862879905048 * q^75 + 38853770260581178560 * q^77 + 121204353225060164896 * q^79 - 136996120065469586382 * q^81 + 108742936757033809800 * q^83 - 527989845644919754344 * q^85 - 24695689277789874000 * q^87 - 184207999274965368972 * q^89 - 126427419539014666912 * q^91 - 139841964206001442560 * q^93 + 1576185911066925315504 * q^95 + 739497785188476467140 * q^97 + 261627767970577922520 * q^99

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}

1.1

23.7433
−22.7433

−92135.9

−1.86463e7

6.05236e8

−1.97134e9

1.2

157776.

3.23357e7

−8.65744e8

1.44329e10

Atkin-Lehner signs

$p$	Sign
$2$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4.22.a.a	✓	2
3.b	odd	2	1		36.22.a.c		2
4.b	odd	2	1		16.22.a.d		2
8.b	even	2	1		64.22.a.i		2
8.d	odd	2	1		64.22.a.j		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4.22.a.a	✓	2	1.a	even	1	1	trivial
16.22.a.d		2	4.b	odd	2	1
36.22.a.c		2	3.b	odd	2	1
64.22.a.i		2	8.b	even	2	1
64.22.a.j		2	8.d	odd	2	1

Hecke kernels

This newform subspace is the entire newspace $S_{22}^{\mathrm{new}}(\Gamma_0(4))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + \cdots - 14536815984$ T^2 - 65640*T - 14536815984
$5$	$T^{2} + \cdots - 602941510374300$ T^2 - 13689324*T - 602941510374300
$7$	$T^{2} + \cdots - 52\!\cdots\!64$ T^2 + 260508080*T - 523979757271484864
$11$	$T^{2} + \cdots + 37\!\cdots\!00$ T^2 - 145435963320*T + 3744063458354550474000
$13$	$T^{2} + \cdots + 50\!\cdots\!76$ T^2 - 1428900417340*T + 508792350703839023859076
$17$	$T^{2} + \cdots - 11\!\cdots\!56$ T^2 - 1840620576420*T - 111587534986863099167066556
$19$	$T^{2} + \cdots - 10\!\cdots\!44$ T^2 + 16780743928568*T - 1029813754402613290511477744
$23$	$T^{2} + \cdots + 19\!\cdots\!04$ T^2 + 319691925426960*T + 19399048867628871982312970304
$29$	$T^{2} + \cdots + 31\!\cdots\!16$ T^2 - 3742111775766492*T + 3152448468197568105932233838916
$31$	$T^{2} + \cdots - 32\!\cdots\!84$ T^2 - 112042353462592*T - 326658618033233761411318160384
$37$	$T^{2} + \cdots - 66\!\cdots\!44$ T^2 + 33362705637547220*T - 663649252516730585773944853052444
$41$	$T^{2} + \cdots + 65\!\cdots\!56$ T^2 - 175129744323133332*T + 6564529271551297737155742807001956
$43$	$T^{2} + \cdots - 25\!\cdots\!00$ T^2 - 15346613416528120*T - 25445046500017259166904915417506800
$47$	$T^{2} + \cdots + 11\!\cdots\!64$ T^2 + 684848819288455200*T + 112466892925179871217951998752555264
$53$	$T^{2} + \cdots - 70\!\cdots\!16$ T^2 - 675305394244421580*T - 702065813129431455661427948572167516
$59$	$T^{2} + \cdots - 56\!\cdots\!96$ T^2 - 1042445250435434904*T - 569906843823145646513898806246653296
$61$	$T^{2} + \cdots + 45\!\cdots\!24$ T^2 - 9065997829736468764*T + 4515034583522915039640423128837795524
$67$	$T^{2} + \cdots + 22\!\cdots\!56$ T^2 + 30464301046802775320*T + 225942521425912212010106807399173184656
$71$	$T^{2} + \cdots - 19\!\cdots\!76$ T^2 + 8199093502830518064*T - 191359247694642928957929692279696381376
$73$	$T^{2} + \cdots - 63\!\cdots\!24$ T^2 - 25415086659374793940*T - 636380828039675818656768818811923639324
$79$	$T^{2} + \cdots + 35\!\cdots\!04$ T^2 - 121204353225060164896*T + 3517697144635126645604555333030146511104
$83$	$T^{2} + \cdots + 22\!\cdots\!64$ T^2 - 108742936757033809800*T + 2207909989750197501974459605125848411664
$89$	$T^{2} + \cdots - 88\!\cdots\!04$ T^2 + 184207999274965368972*T - 8817213503400738099413107050859805994204
$97$	$T^{2} + \cdots - 28\!\cdots\!24$ T^2 - 739497785188476467140*T - 283966102422014109258441421435503651097724
show more
show less

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 $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more