LMFDB - Newform orbit 16.30.a.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,30,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 30, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 30);

N := Newforms(S);

Level:	$N$	$=$	$16 = 2^{4}$
Weight:	$k$	$=$	$30$
Character orbit:	$[\chi]$	$=$	16.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:	$85.2448678129$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 97534740x + 130260063600$ x^3 - x^2 - 97534740*x + 130260063600
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{22}\cdot 3^{3}\cdot 5$
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

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

9125.19
1361.37
−10485.6

−1.27749e7

−2.40170e10

−6.66574e11

9.45684e13

1.2

−3.83101e6

1.31219e10

3.69184e11

−5.39537e13

1.3

9.81666e6

−6.33613e9

−1.81947e11

2.77365e13

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	16.30.a.d		3
4.b	odd	2	1		8.30.a.a	✓	3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8.30.a.a	✓	3	4.b	odd	2	1
16.30.a.d		3	1.a	even	1	1	trivial

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{3} + 6789276T_{3}^{2} - 114073987833936T_{3} - 480436149154912885440$ T3^3 + 6789276*T3^2 - 114073987833936*T3 - 480436149154912885440 acting on $S_{30}^{\mathrm{new}}(\Gamma_0(16))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{3}$ T^3
$3$	$T^{3} + \cdots - 48\!\cdots\!40$ T^3 + 6789276T^2 - 114073987833936T - 480436149154912885440
$5$	$T^{3} + \cdots - 19\!\cdots\!00$ T^3 + 17231192190T^2 - 246117382355727772500T - 1996829770931266412096159375000
$7$	$T^{3} + \cdots - 44\!\cdots\!84$ T^3 + 479336718840T^2 - 191979674786451020035392T - 44775085662834674528915629027075584
$11$	$T^{3} + \cdots - 98\!\cdots\!16$ T^3 + 1095790254198036T^2 - 1203612919147278080513987099856T - 984740220809516550855524947849140874209418816
$13$	$T^{3} + \cdots + 60\!\cdots\!48$ T^3 + 9395719012271046T^2 - 89820145805830423873586107007220T + 60335544781994317281560560598074149666919837448
$17$	$T^{3} + \cdots + 55\!\cdots\!00$ T^3 - 230104394673067638T^2 - 932063329781931393868309213371116340T + 55410322744338882061859921652148406830946234935763000
$19$	$T^{3} + \cdots + 36\!\cdots\!04$ T^3 - 2792352047895578484T^2 - 10304606497681667671802367931179285456T + 3679207720984007390740673138952823079781187362885437504
$23$	$T^{3} + \cdots + 69\!\cdots\!24$ T^3 - 27100487402581130520T^2 - 2717552255605046368547951484270534733632T + 69368369373172296462355878833292821267086780436738562113024
$29$	$T^{3} + \cdots - 29\!\cdots\!52$ T^3 + 2298307146058432367958T^2 - 3064425334541646583621148062924414103173044T - 2934924721308322421671472107176350945679011815980869165942125752
$31$	$T^{3} + \cdots + 11\!\cdots\!00$ T^3 - 5018849875397488231968T^2 - 37399186593146851151054766097087730706600960T + 119161499227786370010525381660783144564730280650339046387565363200
$37$	$T^{3} + \cdots + 46\!\cdots\!76$ T^3 - 3703979203240558724706T^2 - 6991446771257974409972788654886744387235856596T + 46602919240401032708695440780183499936453933136276397460291695475176
$41$	$T^{3} + \cdots + 17\!\cdots\!44$ T^3 + 459998129854300609354722T^2 + 59233502254882946932033299932806809258058567980T + 1715084305004576113357345729114163572370582149671899698775868569349144
$43$	$T^{3} + \cdots - 10\!\cdots\!92$ T^3 - 570966042804973431629580T^2 - 185614434004169926234946675835201529840396240848T - 10162404725978352150119357179896302218367287453907403878580047424483392
$47$	$T^{3} + \cdots - 70\!\cdots\!84$ T^3 - 4501973542027319162506032T^2 + 3843681461539973380998785297633018843019452375808T - 707842001108137570349696852045945431793656491782460261031095515760758784
$53$	$T^{3} + \cdots - 68\!\cdots\!24$ T^3 + 19837676923566709761266478T^2 + 43076159283342091433466095437735030730182758386028T - 68045490493110891652268633540015825111772665159922055077206948773257636824
$59$	$T^{3} + \cdots + 36\!\cdots\!24$ T^3 - 49071572383510413313137948T^2 - 813182811578495544084385132112584447742429935651920T + 36419322893031009441564997099522821136241682957078030276145365744817245573824
$61$	$T^{3} + \cdots + 61\!\cdots\!00$ T^3 + 15568873338675101441145462T^2 - 4400511267788168214943023507600507149730936514550580T + 61390234525342686791155087722837532973607819668948730115449068107330767302600
$67$	$T^{3} + \cdots + 18\!\cdots\!92$ T^3 + 470996379403916295765578844T^2 + 55009042518059625545277477018557312183690470123312560T + 1817472213541593031643678703330848502434836423485536898210619349378968829264192
$71$	$T^{3} + \cdots + 57\!\cdots\!80$ T^3 + 1512226782440279957980190136T^2 + 638165078122127428433957083736670347555699131022894784T + 57581026140799226523973939243499837344894760561192587089773766508430303415585280
$73$	$T^{3} + \cdots + 96\!\cdots\!76$ T^3 - 1023856508018785695598856190T^2 - 657671210916441810913649146391603376070969037206173012T + 96602594478320620490508874122428860621043596096463444071575227861094706756890776
$79$	$T^{3} + \cdots + 22\!\cdots\!80$ T^3 + 2019098233239353642934908592T^2 - 19716268751866620880755065974638634181440425115143400704T + 22514688219355540273073704709089431962211165721847883638089883246698665025307054080
$83$	$T^{3} + \cdots - 50\!\cdots\!36$ T^3 + 11053375188446267413995969612T^2 - 42680682272028016590267688921316347834525857093593476560T - 505263754238586767489639372237373589623711421110213470441456768353001849100798950336
$89$	$T^{3} + \cdots - 17\!\cdots\!84$ T^3 + 47291945286900614421880043154T^2 + 401738844970786586801155412825688985489872861570409975404T - 1753090164047314153267608675255814869743894746937344494597066797755240542865434662184
$97$	$T^{3} + \cdots + 18\!\cdots\!36$ T^3 + 182408046998735020224949890138T^2 + 10494407127570630117297935157132753045318638074712345041548T + 189052221615176833291846081281197230331082607215157519843311940347007467619071929948536
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$f(q)$	$=$	$q + ( - \beta_1 - 2263092) q^{3} + ( - \beta_{2} - 600 \beta_1 - 5743730730) q^{5} + ( - 28 \beta_{2} - 16338 \beta_1 - 159778906280) q^{7} + (4050 \beta_{2} + \cdots + 22783704059133) q^{9} + ( - 71832 \beta_{2} + \cdots - 365263418066012) q^{11}+ \cdots + ( - 11\!\cdots\!12 \beta_{2} + \cdots - 70\!\cdots\!20) q^{99}+O(q^{100})$ q + (-b1 - 2263092) * q^3 + (-b2 - 600b1 - 5743730730) q^5 + (-28b2 - 16338b1 - 159778906280) * q^7 + (4050b2 + 2218608b1 + 22783704059133) * q^9 + (-71832b2 - 22190259b1 - 365263418066012) * q^11 + (-264121b2 - 871984152b1 - 3131906337423682) * q^13 + (7000668b2 + 15055643050b1 + 64781869128341640) * q^15 + (44591938b2 - 52463075856b1 + 76701464891022546) * q^17 + (54410776b2 + 304728934491b1 + 930784015965192828) * q^19 + (194147604b2 + 420533002848b1 + 1771659017175622176) * q^21 + (2919191612b2 - 1799103026742b1 + 9033495800860376840) * q^23 + (6375949940b2 + 22590866844000b1 + 76785068076961810575) * q^25 + (-27496567800b2 + 8124024848022b1 - 87725493251387060616) * q^27 + (-115598280245b2 - 80385612729912b1 - 766102382019477455986) * q^29 + (-6292289440b2 + 594734994358584b1 + 1672949958465829410656) * q^31 + (418190772726b2 + 1035086817167696b1 + 2742036441049021498032) * q^33 + (175292284048b2 + 625588561149300b1 + 7335459067977775537040) * q^35 + (305652603251b2 + 7337101164464904b1 + 1234659734413519574902) * q^37 + (4738745218428b2 + 5559638083415234b1 + 82335536126001752199528) * q^39 + (-23731897324b2 - 9343067341309344b1 - 153332709951433536451574) * q^41 + (-31384877239888b2 + 1471231204774149b1 + 190322014268324477209860) * q^43 + (-24342706193841b2 - 88310364710592600b1 - 1051656408651658221420930) * q^45 + (52738811048776b2 - 126832636806353580b1 + 1500657847342439720835344) * q^47 + (4818487601928b2 + 17322193372456896b1 - 3015331409279446643165479) * q^49 + (8660513142216b2 - 495461625854272210b1 + 4353240347804912438711640) * q^51 + (537852482667235b2 + 116449457340015240b1 - 6612558974522236587088826) * q^53 + (311619291291716b2 + 1307948760513232350b1 + 17540849379205911004198680) * q^55 + (-1482845777406918b2 - 1425349074794205904b1 - 28402693568351908352221488) * q^57 + (-1540377801974112b2 + 2648181191916508641b1 + 16357190794503471104379316) * q^59 + (-594567654618113b2 + 5814337683725775144b1 - 5189624446225033813715154) * q^61 + (-668892336197148b2 - 2444736264485356170b1 - 29334093747116850336540168) * q^63 + (6679203911866044b2 + 16709172069495578400b1 + 115700243967732263411217620) * q^65 + (7964166949093784b2 + 196808256112191087b1 - 156998793134638765255192948) * q^67 + (-6056288428531716b2 - 36374697827926352608b1 + 134782746080694913753870944) * q^69 + (17192755212880884b2 + 16648730720763368094b1 - 504075594146759985993396712) * q^71 + (3932966254774746b2 - 88005097640409140688b1 + 341285502672928565199618730) * q^73 + (-120635361078559920b2 - 135322599408541151375b1 - 2123243568342182254404089100) * q^75 + (8532135612886844b2 + 36144355059330925344b1 + 489876959716070967604108384) * q^77 + (265411751002526056b2 - 19573797134978895732b1 - 673032744413117880978302864) * q^79 + (-185177646408974850b2 + 192602500955722544400b1 - 2065951791624764236434271911) * q^81 + (-527400934801870720b2 - 54458601206695580469b1 - 3684458396148755804665323204) * q^83 + (270409081513186330b2 + 185397665112552774000b1 - 6597155401585347301847530100) * q^85 + (853923091926997260b2 + 1842052922746079316978b1 + 8671334745299102675750909736) * q^87 + (971811855729881562b2 + 691354259604314758704b1 - 15763981762300204807293347718) * q^89 + (184828407747467120b2 + 465288260219395993380b1 + 3201567795692228321529566032) * q^91 + (-2379916761162119280b2 - 1587732575697370883200b1 - 55107157846322642911042715520) * q^93 + (-2254194928422148004b2 - 5327072433954361527150b1 - 31953191247863372885224056920) * q^95 + (-1103580828188668342b2 + 1341710403747186721200b1 - 60802682332911673408316630046) * q^97 + (-1173655525448894112b2 - 5078383277737567849191b1 - 70460793280440244058450435820) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 6789276 q^{3} - 17231192190 q^{5} - 479336718840 q^{7} + 68351112177399 q^{9} - 10\!\cdots\!36 q^{11} - 93\!\cdots\!46 q^{13} + 19\!\cdots\!20 q^{15} + 23\!\cdots\!38 q^{17} + 27\!\cdots\!84 q^{19}+ \cdots - 21\!\cdots\!60 q^{99}+O(q^{100})$ 3 * q - 6789276 * q^3 - 17231192190 * q^5 - 479336718840 * q^7 + 68351112177399 * q^9 - 1095790254198036 * q^11 - 9395719012271046 * q^13 + 194345607385024920 * q^15 + 230104394673067638 * q^17 + 2792352047895578484 * q^19 + 5314977051526866528 * q^21 + 27100487402581130520 * q^23 + 230355204230885431725 * q^25 - 263176479754161181848 * q^27 - 2298307146058432367958 * q^29 + 5018849875397488231968 * q^31 + 8226109323147064494096 * q^33 + 22006377203933326611120 * q^35 + 3703979203240558724706 * q^37 + 247006608378005256598584 * q^39 - 459998129854300609354722 * q^41 + 570966042804973431629580 * q^43 - 3154969225954974664262790 * q^45 + 4501973542027319162506032 * q^47 - 9045994227838339929496437 * q^49 + 13059721043414737316134920 * q^51 - 19837676923566709761266478 * q^53 + 52622548137617733012596040 * q^55 - 85208080705055725056664464 * q^57 + 49071572383510413313137948 * q^59 - 15568873338675101441145462 * q^61 - 88002281241350551009620504 * q^63 + 347100731903196790233652860 * q^65 - 470996379403916295765578844 * q^67 + 404348238242084741261612832 * q^69 - 1512226782440279957980190136 * q^71 + 1023856508018785695598856190 * q^73 - 6369730705026546763212267300 * q^75 + 1469630879148212902812325152 * q^77 - 2019098233239353642934908592 * q^79 - 6197855374874292709302815733 * q^81 - 11053375188446267413995969612 * q^83 - 19791466204756041905542590300 * q^85 + 26014004235897308027252729208 * q^87 - 47291945286900614421880043154 * q^89 + 9604703387076684964588698096 * q^91 - 165321473538967928733128146560 * q^93 - 95859573743590118655672170760 * q^95 - 182408046998735020224949890138 * q^97 - 211382379841320732175351307460 * q^99

$\beta_{1}$	$=$	$1152\nu - 384$ 1152*v - 384
$\beta_{2}$	$=$	$( 8192\nu^{2} + 16403968\nu - 532675197440 ) / 25$ (8192v^2 + 16403968v - 532675197440) / 25

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