LMFDB - Newform orbit 1568.3.c.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,3,Mod(97,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1568.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.c (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:	no
Analytic conductor:	$42.7249054517$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 26 x^{14} - 16 x^{13} + 469 x^{12} + 144 x^{11} - 4526 x^{10} + 4440 x^{9} + 32608 x^{8} + \cdots + 208849 $ x^16 - 26x^14 - 16x^13 + 469x^12 + 144x^11 - 4526x^10 + 4440x^9 + 32608x^8 - 33728x^7 - 49760x^6 + 203528x^5 + 27401x^4 - 156928x^3 + 114964x^2 - 248608x + 208849
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{28}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{6} q^{3} + \beta_1 q^{5} + ( - \beta_{4} - 5) q^{9}+O(q^{10}) $ q + b6 * q^3 + b1 * q^5 + (-b4 - 5) * q^9	$ q + \beta_{6} q^{3} + \beta_1 q^{5} + ( - \beta_{4} - 5) q^{9} + ( - \beta_{12} + \beta_{5}) q^{11} + ( - \beta_{10} - \beta_{8} - \beta_1) q^{13} + ( - \beta_{13} + \beta_{12} + \beta_{5}) q^{15} + (\beta_{14} + 2 \beta_{8} + 2 \beta_1) q^{17} + (\beta_{15} - 2 \beta_{6}) q^{19} + (\beta_{12} + \beta_{11} - 2 \beta_{5}) q^{23} + ( - 2 \beta_{9} + 4 \beta_{7} + \cdots - 10) q^{25}+ \cdots + ( - 2 \beta_{13} - 3 \beta_{12} + \cdots - 12 \beta_{5}) q^{99}+O(q^{100}) $ q + b6 * q^3 + b1 * q^5 + (-b4 - 5) * q^9 + (-b12 + b5) * q^11 + (-b10 - b8 - b1) * q^13 + (-b13 + b12 + b5) * q^15 + (b14 + 2b8 + 2b1) * q^17 + (b15 - 2b6) q^19 + (b12 + b11 - 2b5) q^23 + (-2b9 + 4b7 - 2b4 - 10) q^25 + (-b15 - 6b6 - 2b2) * q^27 + (2b9 - b7 + b4 - 1) q^29 + (-b6 - b3) * q^31 + (-2b14 - 2b10 - 11b8 + 2b1) * q^33 + (-3b9 + 4b7 - b4 - 9) * q^37 + (b13 - b11 - 5b5) q^39 + (3b14 - 2b10 - 5b8) q^41 + (2b12 + 2b11) * q^43 + (2b14 - 7b10 - 13b8 - 13b1) * q^45 + (-2b15 + 9b6 + b3 - 2b2) q^47 + (-2b13 + b12 + 7b5) * q^51 + (3b9 + 6b7 + b4 - 5) * q^53 + (b15 + 7b6 + b3 - 3b2) * q^55 + (-4b9 + 6b7 + 23) * q^57 + (-5b6 + 2b3 - 2b2) q^59 + (-4b14 + b10 + 9b8 - 6b1) q^61 + (4b7 + 3b4 + 21) * q^65 + (2b12 + 9b5) * q^67 + (6b14 - 6b10 + 22b8 - 7b1) * q^69 + (-4b12 + 2b11 + 4b5) q^71 + (8b14 - 8b10 + 13b8 - 2b1) * q^73 + (-2b15 - 22b6 - 2b3 + 2b2) * q^75 + (-2b13 - 3b12 + b11 - 2b5) q^79 + (-4b9 + 20b7 + 5b4 + 48) q^81 + (4b15 + 6b6 - 2b3 + 2b2) * q^83 + (b9 - 3b4 - 67) q^85 + (b15 + 4b6 + 2b3 - b2) * q^87 + (14b10 + b8 + 4b1) * q^89 + (7b9 + 5b4 + 21) * q^93 + (3b12 + b11 - 12b5) * q^95 + (7b14 - 6b10 - 37b8 - 8b1) * q^97 + (-2b13 - 3b12 - 2b11 - 12b5) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 80 q^{9}+O(q^{10}) $ 16 * q - 80 * q^9	$ 16 q - 80 q^{9} - 160 q^{25} - 16 q^{29} - 144 q^{37} - 80 q^{53} + 368 q^{57} + 336 q^{65} + 768 q^{81} - 1072 q^{85} + 336 q^{93}+O(q^{100}) $ 16 * q - 80 * q^9 - 160 * q^25 - 16 * q^29 - 144 * q^37 - 80 * q^53 + 368 * q^57 + 336 * q^65 + 768 * q^81 - 1072 * q^85 + 336 * q^93

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 26 x^{14} - 16 x^{13} + 469 x^{12} + 144 x^{11} - 4526 x^{10} + 4440 x^{9} + 32608 x^{8} + \cdots + 208849 $ x^16 - 26*x^14 - 16*x^13 + 469*x^12 + 144*x^11 - 4526*x^10 + 4440*x^9 + 32608*x^8 - 33728*x^7 - 49760*x^6 + 203528*x^5 + 27401*x^4 - 156928*x^3 + 114964*x^2 - 248608*x + 208849 :

$\beta_{1}$	$=$	$ ( - 81\!\cdots\!73 \nu^{15} + \cdots + 73\!\cdots\!63 ) / 38\!\cdots\!33 $ (-8161932957484190343665273v^15 - 44439187972352536048517534v^14 + 285301886851739906742316053v^13 + 1112658279033969221325555216v^12 - 4814066789420379222535386834v^11 - 19373728230501465036814065604v^10 + 62897025351342307275446286156v^9 + 109397774130481521364257407586v^8 - 727155270139808660498290086006v^7 - 358917417736567570954159885656v^6 + 2663256405889607525415415377260v^5 - 3729807910919652247800742026214v^4 - 5104731955019441877585892054673v^3 + 3653225388259983436664143334390v^2 - 8403237557696221261342105089459*v + 7385860188895574436656992849363) / 3898470889735654960653011126733
$\beta_{2}$	$=$	$ ( - 48\!\cdots\!54 \nu^{15} + \cdots - 53\!\cdots\!31 ) / 17\!\cdots\!38 $ (-48938175240500379766654v^15 + 1018957400875668923115623v^14 + 835513112149775321093486v^13 - 23340323391993423584168283v^12 - 31382921689458755274762972v^11 + 422624529274148148210626700v^10 + 197405447555300005513129088v^9 - 3858789009588295178665052992v^8 + 3862874530793152901565835124v^7 + 24367659029785269951151662472v^6 - 25736669438053538668038727184v^5 - 2151112781037815870259768302v^4 + 101958581123719058935672801610v^3 - 13157434736815645928245008145v^2 + 29704651808335598695839562814*v - 53023773050719004012849091831) / 17061141749390174882507707338
$\beta_{3}$	$=$	$ ( - 27\!\cdots\!43 \nu^{15} + \cdots - 81\!\cdots\!70 ) / 85\!\cdots\!69 $ (-27397357799034800546743v^15 + 1558767510707882111058699v^14 - 57160957226276830090407v^13 - 36625662950316359863762512v^12 - 24393580635427761438745686v^11 + 656801861543589915205908300v^10 + 82184390631076727554594640v^9 - 5718427399084193023996909296v^8 + 7363052759282056504592478762v^7 + 34759620118464264079239940438v^6 - 40592915588254877698413087792v^5 + 1806702359859996318952592874v^4 + 155317510002317643741095004221v^3 - 13272903298917380376827852085v^2 + 39835236593548766642892768057*v - 81435337105865920777942382070) / 8530570874695087441253853669
$\beta_{4}$	$=$	$ ( 17\!\cdots\!66 \nu^{15} + \cdots - 58\!\cdots\!09 ) / 44\!\cdots\!59 $ (175897262115818474166v^15 - 39700912768241219437v^14 - 4343208416441587651818v^13 - 2550483807844585241654v^12 + 77457246760524776820756v^11 + 16735111331723529813756v^10 - 685333887138161379193776v^9 + 746345850366636811491008v^8 + 4542541386163315725553572v^7 - 4695086562310980701080952v^6 - 2941504366413742418249552v^5 + 16539384619686044550131514v^4 - 22763625087679702581672874v^3 + 22641835053978957045423419v^2 - 19254675737460801888584922*v - 584502903330940031442410309) / 44662674736623494456826459
$\beta_{5}$	$=$	$ ( 17\!\cdots\!22 \nu^{15} + \cdots - 43\!\cdots\!47 ) / 40\!\cdots\!26 $ (172377580471175462040822v^15 + 212941530455158990989389v^14 - 4750870910547457913371838v^13 - 8575075886270639069659203v^12 + 83520657283426435935288420v^11 + 136031226314224464198067740v^10 - 850858232324077448090833992v^9 - 333463624084075292528090992v^8 + 7392719700656286585102150460v^7 + 828047657109777994066016808v^6 - 22830414462343664397603819704v^5 + 23325051206048404659865393490v^4 + 45252411702446011114980056022v^3 - 71624910281823322549240613167v^2 + 11496599637194480127236068690*v - 43231667292315975023081438847) / 40821684709273873933539383526
$\beta_{6}$	$=$	$ ( 87\!\cdots\!58 \nu^{15} + \cdots - 36\!\cdots\!77 ) / 17\!\cdots\!38 $ (87428286339522377943258v^15 - 19379552843836720073999v^14 - 2106362162310834996966530v^13 - 1203281739031657890480621v^12 + 37567134568520370754247220v^11 + 8063335103197233251269908v^10 - 327897110740258740307119480v^9 + 362422523161540239768290176v^8 + 2202446682825009641121408292v^7 - 1724782533273118586212671000v^6 - 1423302805080590158619506672v^5 + 8021116023302597726209967294v^4 + 9591582687049399777297302042v^3 + 10981892368336939613161395337v^2 - 9341327103233175018294227090*v - 3686481877316456037642549777) / 17061141749390174882507707338
$\beta_{7}$	$=$	$ ( - 351675045716698 \nu^{15} + 84134512548244 \nu^{14} + \cdots + 24\!\cdots\!04 ) / 45\!\cdots\!31 $ (-351675045716698v^15 + 84134512548244v^14 + 8997681016250754v^13 + 4445925380328116v^12 - 160496373722473788v^11 - 33874814941456464v^10 + 1467376262123572708v^9 - 1553113751898048416v^8 - 9401792267255930460v^7 + 10200336255773301928v^6 + 6044728226531394560v^5 - 34262171840112781800v^4 + 46812769807129599650v^3 - 46922766637244959676v^2 + 39937799358222111906*v + 240280225958040430804) / 45742705762379025631
$\beta_{8}$	$=$	$ ( 24\!\cdots\!00 \nu^{15} + \cdots - 35\!\cdots\!43 ) / 24\!\cdots\!53 $ (248197947118550681400v^15 + 60099576570060185087v^14 - 6110172991187791613532v^13 - 5610165377095918036704v^12 + 107123109897527970241392v^11 + 59136687103919879272608v^10 - 966301607286958942241280v^9 + 861897353924990552955686v^8 + 7059892149907340387084928v^7 - 4980438225173244013489056v^6 - 5677044395320575039843008v^5 + 40095443257835374287069696v^4 + 13407128082587646721614288v^3 - 5618854490528863822096597v^2 + 33962809459441010571298260*v - 35592347497900753218122943) / 24174914515820036838745953
$\beta_{9}$	$=$	$ ( - 63\!\cdots\!17 \nu^{15} + \cdots + 32\!\cdots\!77 ) / 44\!\cdots\!59 $ (-631161306625356779617v^15 + 150505234538692630725v^14 + 16500767577615813975981v^13 + 7419045073782499938700v^12 - 294271138216409917948542v^11 - 63717891799250964885804v^10 + 2750860404617240971368368v^9 - 2834324749199366123952240v^8 - 17259606953295549222511902v^7 + 19235282507483501622825296v^6 + 11183900242770749268101232v^5 - 62836954235436549118005834v^4 + 85357101295984988485619803v^3 - 86018275869553972168815051v^2 + 73144181174155227325195629*v + 327917369601661120754045677) / 44662674736623494456826459
$\beta_{10}$	$=$	$ ( - 10\!\cdots\!23 \nu^{15} + \cdots + 56\!\cdots\!52 ) / 38\!\cdots\!33 $ (-107274637135815474213267623v^15 + 40440310585103025551191222v^14 + 2509657568814871159359711123v^13 + 970451882572829151852741190v^12 - 44287196461473841197989452438v^11 + 1749022612776647015483582528v^10 + 369261641263869769858666117920v^9 - 588689505051165656418132450628v^8 - 2281678710917973731743246639250v^7 + 2977319895231402273537853555688v^6 - 1436373880362576535141563797140v^5 - 13591200467210556482109017274192v^4 + 1529669380689354048836851412717v^3 - 3020774732138613521871714908018v^2 - 3259759170842048260323290216941*v + 5648824684655197559037381780752) / 3898470889735654960653011126733
$\beta_{11}$	$=$	$ ( 13\!\cdots\!76 \nu^{15} + \cdots - 74\!\cdots\!23 ) / 40\!\cdots\!26 $ (1384364485533756987372876v^15 - 728768123650168136784649v^14 - 35908961898282986992714064v^13 - 472622045688405467009363v^12 + 658611626622471479423905864v^11 - 200247607529712853033798708v^10 - 6422914281900997227071722896v^9 + 10421619794726874885163185080v^8 + 42733803632554989723902755304v^7 - 75737920268645070599379844900v^6 - 46248813080430974509909364096v^5 + 348113674580553310579782267406v^4 - 10929072089602277258238994884v^3 - 188529834431551320226184211949v^2 - 272111133428858037728504337072*v - 74244525959635624072959944923) / 40821684709273873933539383526
$\beta_{12}$	$=$	$ ( 15\!\cdots\!48 \nu^{15} + \cdots - 25\!\cdots\!19 ) / 29\!\cdots\!09 $ (150357722430704850757448v^15 + 60075198633659217891900v^14 - 4028295388456629283530112v^13 - 4156719365271510392861673v^12 + 72661216673303491288979384v^11 + 56042699511608033717055244v^10 - 723159070367915974211775172v^9 + 303091942964013600088722988v^8 + 5658945163844004844545788056v^7 - 3056389735529669058506661454v^6 - 13723062855387011820116672024v^5 + 27731587236969783024247115660v^4 + 22370325291012881548177717004v^3 - 44749936787945170476365911956v^2 - 6602631145691352621161837336*v - 25283341868433551078267787519) / 2915834622090990995252813109
$\beta_{13}$	$=$	$ ( 12\!\cdots\!67 \nu^{15} + \cdots - 19\!\cdots\!50 ) / 20\!\cdots\!63 $ (1239986586879849068452267v^15 + 446611647240208516332506v^14 - 33172367614068978380700341v^13 - 31579135335561447776626614v^12 + 595924841732233384601337874v^11 + 409213280791450479744422696v^10 - 5938580497464899800282006892v^9 + 3285672381231400343517307320v^8 + 45850802484701684033452880102v^7 - 29875274922222088196649094936v^6 - 104669895774889538889453816484v^5 + 237551379027465268422197142796v^4 + 162608902664302276970871278667v^3 - 346649214519037359708573123958v^2 - 75641461576722423956309523053*v - 192223036278949580449972700050) / 20410842354636936966769691763
$\beta_{14}$	$=$	$ ( - 26\!\cdots\!62 \nu^{15} + \cdots + 54\!\cdots\!03 ) / 38\!\cdots\!33 $ (-267063145461629828893524862v^15 - 170988238389309380274226828v^14 + 6785108178874501898339214806v^13 + 8460206761577540248730427138v^12 - 118392865570954626911808600220v^11 - 109560588073441856744426807204v^10 + 1114274795933677419493107459288v^9 - 553827182119669552363089749314v^8 - 8791627040505418243644692523444v^7 + 3847948136217721768781922320392v^6 + 12900955768739878151945366278840v^5 - 48759551313770278706482388246430v^4 - 27203310203993899440149864599118v^3 + 15595962883292813976143238673104v^2 - 56018922452006827133376345549914*v + 54779321279479592533993047161903) / 3898470889735654960653011126733
$\beta_{15}$	$=$	$ ( - 17\!\cdots\!82 \nu^{15} + \cdots + 20\!\cdots\!29 ) / 17\!\cdots\!38 $ (-1724264633383462570433182v^15 - 2048070062977697491721035v^14 + 42252483719801659728501638v^13 + 81709401945744643690593333v^12 - 713861670067369782746122716v^11 - 1187832908714722814839439772v^10 + 6406234029560720562724506800v^9 + 1681944271704283897258555520v^8 - 55593224006113528318853699116v^7 - 23875599108767871968486476460v^6 + 91927619257102208322535794256v^5 - 163382527833887830227336488522v^4 - 453859953990828211998061613734v^3 - 199082612062867058370460165843v^2 + 124732580360973654215585228342*v + 207448752980846201883934977429) / 17061141749390174882507707338

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( -\beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{2} + \beta_1 ) / 16 $ (-b14 + b13 - b12 - b11 + b10 + b7 - b6 - b5 - b2 + b1) / 16
$\nu^{2}$	$=$	$ ( -\beta_{14} - 2\beta_{10} - 13\beta_{8} + 2\beta_{6} - 2\beta_{5} + \beta_{4} - 2\beta _1 + 13 ) / 4 $ (-b14 - 2b10 - 13b8 + 2b6 - 2b5 + b4 - 2*b1 + 13) / 4
$\nu^{3}$	$=$	$ ( -2\beta_{15} - 6\beta_{9} + 19\beta_{7} - 37\beta_{6} + 6\beta_{4} + 2\beta_{3} - 11\beta_{2} + 24 ) / 8 $ (-2b15 - 6b9 + 19b7 - 37b6 + 6b4 + 2b3 - 11*b2 + 24) / 8
$\nu^{4}$	$=$	$ ( - 2 \beta_{15} - 5 \beta_{14} + 4 \beta_{13} + 2 \beta_{12} - 4 \beta_{11} - 34 \beta_{10} + 2 \beta_{9} + \cdots - 131 ) / 4 $ (-2b15 - 5b14 + 4b13 + 2b12 - 4b11 - 34b10 + 2b9 - 131b8 - 10b7 - 46b6 - 48b5 - 13b4 - 4b2 - 40b1 - 131) / 4
$\nu^{5}$	$=$	$ ( - 13 \beta_{15} + 22 \beta_{14} - 66 \beta_{13} + 27 \beta_{12} + 39 \beta_{11} - 277 \beta_{10} + \cdots + 680 ) / 8 $ (-13b15 + 22b14 - 66b13 + 27b12 + 39b11 - 277b10 - 65b9 - 680b8 + 172b7 - 290b6 + 303b5 + 85b4 + 9b3 - 57b2 - 472*b1 + 680) / 8
$\nu^{6}$	$=$	$ ( -43\beta_{15} + 48\beta_{9} - 163\beta_{7} - 921\beta_{6} - 135\beta_{4} + 6\beta_{3} - 122\beta_{2} - 1261 ) / 2 $ (-43b15 + 48b9 - 163b7 - 921b6 - 135b4 + 6b3 - 122*b2 - 1261) / 2
$\nu^{7}$	$=$	$ ( 268 \beta_{15} - \beta_{14} - 1033 \beta_{13} + 229 \beta_{12} + 733 \beta_{11} - 11339 \beta_{10} + \cdots - 34104 ) / 16 $ (268b15 - b14 - 1033b13 + 229b12 + 733b11 - 11339b10 + 2156b9 - 34104b8 - 5935b7 + 5873b6 + 6141b5 - 3780b4 - 100b3 + 933b2 - 17807b1 - 34104) / 16
$\nu^{8}$	$=$	$ ( - 788 \beta_{15} - 55 \beta_{14} - 2836 \beta_{13} + 472 \beta_{12} + 2164 \beta_{11} + 2392 \beta_{10} + \cdots - 6799 ) / 4 $ (-788b15 - 55b14 - 2836b13 + 472b12 + 2164b11 + 2392b10 + 478b9 + 6799b8 - 1312b7 - 16832b6 + 17620b5 - 779b4 + 224b3 - 2612b2 + 3826*b1 - 6799) / 4
$\nu^{9}$	$=$	$ ( 490 \beta_{15} + 33570 \beta_{9} - 97945 \beta_{7} + 9937 \beta_{6} - 70938 \beta_{4} + 526 \beta_{3} + \cdots - 660288 ) / 8 $ (490b15 + 33570b9 - 97945b7 + 9937b6 - 70938b4 + 526b3 - 369*b2 - 660288) / 8
$\nu^{10}$	$=$	$ ( 13059 \beta_{15} - 4297 \beta_{14} - 51234 \beta_{13} + 12057 \beta_{12} + 36840 \beta_{11} + \cdots - 125022 ) / 4 $ (13059b15 - 4297b14 - 51234b13 + 12057b12 + 36840b11 - 33095b10 + 2292b9 - 125022b8 - 10459b7 + 280281b6 + 293340b5 - 12464b4 - 4798b3 + 46436b2 - 39971*b1 - 125022) / 4
$\nu^{11}$	$=$	$ ( - 32613 \beta_{15} + 74343 \beta_{14} - 152793 \beta_{13} + 54954 \beta_{12} + 97494 \beta_{11} + \cdots - 5445440 ) / 8 $ (-32613b15 + 74343b14 - 152793b13 + 54954b12 + 97494b11 + 1662216b10 + 249249b9 + 5445440b8 - 753759b7 - 709441b6 + 742054b5 - 578853b4 + 18433b3 - 134360b2 + 2409963*b1 - 5445440) / 8
$\nu^{12}$	$=$	$ 97132 \beta_{15} + 112200 \beta_{9} - 363803 \beta_{7} + 2090869 \beta_{6} - 308517 \beta_{4} + \cdots - 2949878 $ 97132b15 + 112200b9 - 363803b7 + 2090869b6 - 308517b4 - 39428b3 + 355911*b2 - 2949878
$\nu^{13}$	$=$	$ ( 2229108 \beta_{15} + 2332695 \beta_{14} - 9498203 \beta_{13} + 2810879 \beta_{12} + 6449951 \beta_{11} + \cdots + 154362832 ) / 16 $ (2229108b15 + 2332695b14 - 9498203b13 + 2810879b12 + 6449951b11 + 46693893b10 - 6750900b9 + 154362832b8 + 20760401b7 + 48161131b6 + 50390239b5 + 16342196b4 - 1016084b3 + 8482119b2 + 66946593*b1 + 154362832) / 16
$\nu^{14}$	$=$	$ ( 2480988 \beta_{15} + 2503213 \beta_{14} + 10250770 \beta_{13} - 2807806 \beta_{12} - 7108918 \beta_{11} + \cdots - 153106199 ) / 4 $ (2480988b15 + 2503213b14 + 10250770b13 - 2807806b12 - 7108918b11 + 45953948b10 + 6433504b9 + 153106199b8 - 20082678b7 + 53474096b6 - 55955084b5 - 16152387b4 - 1047284b3 + 9203486b2 + 65254460*b1 - 153106199) / 4
$\nu^{15}$	$=$	$ ( 52228326 \beta_{15} - 75835062 \beta_{9} + 234972317 \beta_{7} + 1126170861 \beta_{6} + \cdots + 1769317544 ) / 8 $ (52228326b15 - 75835062b9 + 234972317b7 + 1126170861b6 + 186984966b4 - 22305174b3 + 194464767*b2 + 1769317544) / 8

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$1$	$-1$

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} $

97.1

3.86852	+	1.41699i
−3.16141	+	2.64174i
2.08703	−	2.02145i
−2.79414	−	0.796701i
−1.90990	−	0.286185i
1.20279	−	1.51093i
−0.162551	+	0.910345i
0.869658	−	0.314400i
0.869658	+	0.314400i
−0.162551	−	0.910345i
1.20279	+	1.51093i
−1.90990	+	0.286185i
−2.79414	+	0.796701i
2.08703	+	2.02145i
−3.16141	−	2.64174i
3.86852	−	1.41699i

−

5.73991i

−

6.30408i

−23.9466

97.2

−

5.73991i

6.30408i

−23.9466

97.3

−

3.98546i

−

9.01776i

−6.88390

97.4

−

3.98546i

9.01776i

−6.88390

97.5

−

2.54150i

−

4.11878i

2.54076

97.6

−

2.54150i

4.11878i

2.54076

97.7

−

0.842794i

−

1.40510i

8.28970

97.8

−

0.842794i

1.40510i

8.28970

97.9

0.842794i

−

1.40510i

8.28970

97.10

0.842794i

1.40510i

8.28970

97.11

2.54150i

−

4.11878i

2.54076

97.12

2.54150i

4.11878i

2.54076

97.13

3.98546i

−

9.01776i

−6.88390

97.14

3.98546i

9.01776i

−6.88390

97.15

5.73991i

−

6.30408i

−23.9466

97.16

5.73991i

6.30408i

−23.9466

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.b	odd	2	1	inner
28.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1568.3.c.g		16
4.b	odd	2	1	inner	1568.3.c.g		16
7.b	odd	2	1	inner	1568.3.c.g		16
7.c	even	3	1		224.3.s.b	✓	16
7.d	odd	6	1		224.3.s.b	✓	16
28.d	even	2	1	inner	1568.3.c.g		16
28.f	even	6	1		224.3.s.b	✓	16
28.g	odd	6	1		224.3.s.b	✓	16
56.j	odd	6	1		448.3.s.h		16
56.k	odd	6	1		448.3.s.h		16
56.m	even	6	1		448.3.s.h		16
56.p	even	6	1		448.3.s.h		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
224.3.s.b	✓	16	7.c	even	3	1
224.3.s.b	✓	16	7.d	odd	6	1
224.3.s.b	✓	16	28.f	even	6	1
224.3.s.b	✓	16	28.g	odd	6	1
448.3.s.h		16	56.j	odd	6	1
448.3.s.h		16	56.k	odd	6	1
448.3.s.h		16	56.m	even	6	1
448.3.s.h		16	56.p	even	6	1
1568.3.c.g		16	1.a	even	1	1	trivial
1568.3.c.g		16	4.b	odd	2	1	inner
1568.3.c.g		16	7.b	odd	2	1	inner
1568.3.c.g		16	28.d	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(1568, [\chi])$:

$ T_{3}^{8} + 56T_{3}^{6} + 878T_{3}^{4} + 3976T_{3}^{2} + 2401 $

$ T_{5}^{8} + 140T_{5}^{6} + 5558T_{5}^{4} + 65260T_{5}^{2} + 108241 $

$ T_{11}^{8} - 536T_{11}^{6} + 80958T_{11}^{4} - 2585576T_{11}^{2} + 19018321 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 56 T^{6} + \cdots + 2401)^{2} $ (T^8 + 56T^6 + 878T^4 + 3976*T^2 + 2401)^2
$5$	$ (T^{8} + 140 T^{6} + \cdots + 108241)^{2} $ (T^8 + 140T^6 + 5558T^4 + 65260*T^2 + 108241)^2
$7$	$ T^{16} $ T^16
$11$	$ (T^{8} - 536 T^{6} + \cdots + 19018321)^{2} $ (T^8 - 536T^6 + 80958T^4 - 2585576*T^2 + 19018321)^2
$13$	$ (T^{8} + 296 T^{6} + \cdots + 12544)^{2} $ (T^8 + 296T^6 + 10928T^4 + 65152*T^2 + 12544)^2
$17$	$ (T^{8} + 844 T^{6} + \cdots + 717007729)^{2} $ (T^8 + 844T^6 + 231638T^4 + 24172652*T^2 + 717007729)^2
$19$	$ (T^{8} + 1448 T^{6} + \cdots + 6387046561)^{2} $ (T^8 + 1448T^6 + 625070T^4 + 106934296*T^2 + 6387046561)^2
$23$	$ (T^{8} - 2584 T^{6} + \cdots + 10882871041)^{2} $ (T^8 - 2584T^6 + 2064270T^4 - 536705512*T^2 + 10882871041)^2
$29$	$ (T^{4} + 4 T^{3} + \cdots + 5008)^{4} $ (T^4 + 4T^3 - 756T^2 - 1520*T + 5008)^4
$31$	$ (T^{8} + 2744 T^{6} + \cdots + 36725506321)^{2} $ (T^8 + 2744T^6 + 2005310T^4 + 482048392*T^2 + 36725506321)^2
$37$	$ (T^{4} + 36 T^{3} + \cdots - 25079)^{4} $ (T^4 + 36T^3 - 1522T^2 + 13236*T - 25079)^4
$41$	$ (T^{8} + 2632 T^{6} + \cdots + 31899904)^{2} $ (T^8 + 2632T^6 + 1251632T^4 + 118404224*T^2 + 31899904)^2
$43$	$ (T^{8} + \cdots + 12876384903424)^{2} $ (T^8 - 8800T^6 + 27334368T^4 - 34105899520*T^2 + 12876384903424)^2
$47$	$ (T^{8} + \cdots + 2641538328961)^{2} $ (T^8 + 10472T^6 + 32918030T^4 + 27781550104*T^2 + 2641538328961)^2
$53$	$ (T^{4} + 20 T^{3} + \cdots + 779737)^{4} $ (T^4 + 20T^3 - 4738T^2 + 74116*T + 779737)^4
$59$	$ (T^{8} + \cdots + 18597147378481)^{2} $ (T^8 + 14792T^6 + 73219550T^4 + 125198557624*T^2 + 18597147378481)^2
$61$	$ (T^{8} + \cdots + 11484080659489)^{2} $ (T^8 + 10076T^6 + 29256038T^4 + 31679470108*T^2 + 11484080659489)^2
$67$	$ (T^{8} + \cdots + 112936720954561)^{2} $ (T^8 - 16664T^6 + 83257230T^4 - 164393487272*T^2 + 112936720954561)^2
$71$	$ (T^{8} + \cdots + 159740883210496)^{2} $ (T^8 - 16096T^6 + 89738208T^4 - 205483188736*T^2 + 159740883210496)^2
$73$	$ (T^{8} + \cdots + 46497574501569)^{2} $ (T^8 + 22428T^6 + 116029062T^4 + 178264268316*T^2 + 46497574501569)^2
$79$	$ (T^{8} + \cdots + 85293924591841)^{2} $ (T^8 - 15288T^6 + 78813230T^4 - 154037472456*T^2 + 85293924591841)^2
$83$	$ (T^{8} + \cdots + 5740355977216)^{2} $ (T^8 + 19744T^6 + 76252928T^4 + 53618991104*T^2 + 5740355977216)^2
$89$	$ (T^{8} + \cdots + 32\!\cdots\!25)^{2} $ (T^8 + 46156T^6 + 648827126T^4 + 2695955513900*T^2 + 3212489203380625)^2
$97$	$ (T^{8} + \cdots + 16556565688576)^{2} $ (T^8 + 35752T^6 + 328917680T^4 + 170189944448*T^2 + 16556565688576)^2
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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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{3} + \beta_1 q^{5} + ( - \beta_{4} - 5) q^{9}+O(q^{10}) \) q + b6 * q^3 + b1 * q^5 + (-b4 - 5) * q^9	\( q + \beta_{6} q^{3} + \beta_1 q^{5} + ( - \beta_{4} - 5) q^{9} + ( - \beta_{12} + \beta_{5}) q^{11} + ( - \beta_{10} - \beta_{8} - \beta_1) q^{13} + ( - \beta_{13} + \beta_{12} + \beta_{5}) q^{15} + (\beta_{14} + 2 \beta_{8} + 2 \beta_1) q^{17} + (\beta_{15} - 2 \beta_{6}) q^{19} + (\beta_{12} + \beta_{11} - 2 \beta_{5}) q^{23} + ( - 2 \beta_{9} + 4 \beta_{7} + \cdots - 10) q^{25}+ \cdots + ( - 2 \beta_{13} - 3 \beta_{12} + \cdots - 12 \beta_{5}) q^{99}+O(q^{100}) \) q + b6 * q^3 + b1 * q^5 + (-b4 - 5) * q^9 + (-b12 + b5) * q^11 + (-b10 - b8 - b1) * q^13 + (-b13 + b12 + b5) * q^15 + (b14 + 2b8 + 2b1) * q^17 + (b15 - 2b6) q^19 + (b12 + b11 - 2b5) q^23 + (-2b9 + 4b7 - 2b4 - 10) q^25 + (-b15 - 6b6 - 2b2) * q^27 + (2b9 - b7 + b4 - 1) q^29 + (-b6 - b3) * q^31 + (-2b14 - 2b10 - 11b8 + 2b1) * q^33 + (-3b9 + 4b7 - b4 - 9) * q^37 + (b13 - b11 - 5b5) q^39 + (3b14 - 2b10 - 5b8) q^41 + (2b12 + 2b11) * q^43 + (2b14 - 7b10 - 13b8 - 13b1) * q^45 + (-2b15 + 9b6 + b3 - 2b2) q^47 + (-2b13 + b12 + 7b5) * q^51 + (3b9 + 6b7 + b4 - 5) * q^53 + (b15 + 7b6 + b3 - 3b2) * q^55 + (-4b9 + 6b7 + 23) * q^57 + (-5b6 + 2b3 - 2b2) q^59 + (-4b14 + b10 + 9b8 - 6b1) q^61 + (4b7 + 3b4 + 21) * q^65 + (2b12 + 9b5) * q^67 + (6b14 - 6b10 + 22b8 - 7b1) * q^69 + (-4b12 + 2b11 + 4b5) q^71 + (8b14 - 8b10 + 13b8 - 2b1) * q^73 + (-2b15 - 22b6 - 2b3 + 2b2) * q^75 + (-2b13 - 3b12 + b11 - 2b5) q^79 + (-4b9 + 20b7 + 5b4 + 48) q^81 + (4b15 + 6b6 - 2b3 + 2b2) * q^83 + (b9 - 3b4 - 67) q^85 + (b15 + 4b6 + 2b3 - b2) * q^87 + (14b10 + b8 + 4b1) * q^89 + (7b9 + 5b4 + 21) * q^93 + (3b12 + b11 - 12b5) * q^95 + (7b14 - 6b10 - 37b8 - 8b1) * q^97 + (-2b13 - 3b12 - 2b11 - 12b5) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 80 q^{9}+O(q^{10}) \) 16 * q - 80 * q^9	\( 16 q - 80 q^{9} - 160 q^{25} - 16 q^{29} - 144 q^{37} - 80 q^{53} + 368 q^{57} + 336 q^{65} + 768 q^{81} - 1072 q^{85} + 336 q^{93}+O(q^{100}) \) 16 * q - 80 * q^9 - 160 * q^25 - 16 * q^29 - 144 * q^37 - 80 * q^53 + 368 * q^57 + 336 * q^65 + 768 * q^81 - 1072 * q^85 + 336 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1568.3.c.g

Level	$1568$
Weight	$3$
Character orbit	1568.c
Analytic conductor	$42.725$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(42.7249054517\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 26 x^{14} - 16 x^{13} + 469 x^{12} + 144 x^{11} - 4526 x^{10} + 4440 x^{9} + 32608 x^{8} + \cdots + 208849 \) x^16 - 26x^14 - 16x^13 + 469x^12 + 144x^11 - 4526x^10 + 4440x^9 + 32608x^8 - 33728x^7 - 49760x^6 + 203528x^5 + 27401x^4 - 156928x^3 + 114964x^2 - 248608x + 208849
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{28}\cdot 7^{2} \)
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 81\!\cdots\!73 \nu^{15} + \cdots + 73\!\cdots\!63 ) / 38\!\cdots\!33 \) (-8161932957484190343665273v^15 - 44439187972352536048517534v^14 + 285301886851739906742316053v^13 + 1112658279033969221325555216v^12 - 4814066789420379222535386834v^11 - 19373728230501465036814065604v^10 + 62897025351342307275446286156v^9 + 109397774130481521364257407586v^8 - 727155270139808660498290086006v^7 - 358917417736567570954159885656v^6 + 2663256405889607525415415377260v^5 - 3729807910919652247800742026214v^4 - 5104731955019441877585892054673v^3 + 3653225388259983436664143334390v^2 - 8403237557696221261342105089459*v + 7385860188895574436656992849363) / 3898470889735654960653011126733
\(\beta_{2}\)	\(=\)	\( ( - 48\!\cdots\!54 \nu^{15} + \cdots - 53\!\cdots\!31 ) / 17\!\cdots\!38 \) (-48938175240500379766654v^15 + 1018957400875668923115623v^14 + 835513112149775321093486v^13 - 23340323391993423584168283v^12 - 31382921689458755274762972v^11 + 422624529274148148210626700v^10 + 197405447555300005513129088v^9 - 3858789009588295178665052992v^8 + 3862874530793152901565835124v^7 + 24367659029785269951151662472v^6 - 25736669438053538668038727184v^5 - 2151112781037815870259768302v^4 + 101958581123719058935672801610v^3 - 13157434736815645928245008145v^2 + 29704651808335598695839562814*v - 53023773050719004012849091831) / 17061141749390174882507707338
\(\beta_{3}\)	\(=\)	\( ( - 27\!\cdots\!43 \nu^{15} + \cdots - 81\!\cdots\!70 ) / 85\!\cdots\!69 \) (-27397357799034800546743v^15 + 1558767510707882111058699v^14 - 57160957226276830090407v^13 - 36625662950316359863762512v^12 - 24393580635427761438745686v^11 + 656801861543589915205908300v^10 + 82184390631076727554594640v^9 - 5718427399084193023996909296v^8 + 7363052759282056504592478762v^7 + 34759620118464264079239940438v^6 - 40592915588254877698413087792v^5 + 1806702359859996318952592874v^4 + 155317510002317643741095004221v^3 - 13272903298917380376827852085v^2 + 39835236593548766642892768057*v - 81435337105865920777942382070) / 8530570874695087441253853669
\(\beta_{4}\)	\(=\)	\( ( 17\!\cdots\!66 \nu^{15} + \cdots - 58\!\cdots\!09 ) / 44\!\cdots\!59 \) (175897262115818474166v^15 - 39700912768241219437v^14 - 4343208416441587651818v^13 - 2550483807844585241654v^12 + 77457246760524776820756v^11 + 16735111331723529813756v^10 - 685333887138161379193776v^9 + 746345850366636811491008v^8 + 4542541386163315725553572v^7 - 4695086562310980701080952v^6 - 2941504366413742418249552v^5 + 16539384619686044550131514v^4 - 22763625087679702581672874v^3 + 22641835053978957045423419v^2 - 19254675737460801888584922*v - 584502903330940031442410309) / 44662674736623494456826459
\(\beta_{5}\)	\(=\)	\( ( 17\!\cdots\!22 \nu^{15} + \cdots - 43\!\cdots\!47 ) / 40\!\cdots\!26 \) (172377580471175462040822v^15 + 212941530455158990989389v^14 - 4750870910547457913371838v^13 - 8575075886270639069659203v^12 + 83520657283426435935288420v^11 + 136031226314224464198067740v^10 - 850858232324077448090833992v^9 - 333463624084075292528090992v^8 + 7392719700656286585102150460v^7 + 828047657109777994066016808v^6 - 22830414462343664397603819704v^5 + 23325051206048404659865393490v^4 + 45252411702446011114980056022v^3 - 71624910281823322549240613167v^2 + 11496599637194480127236068690*v - 43231667292315975023081438847) / 40821684709273873933539383526
\(\beta_{6}\)	\(=\)	\( ( 87\!\cdots\!58 \nu^{15} + \cdots - 36\!\cdots\!77 ) / 17\!\cdots\!38 \) (87428286339522377943258v^15 - 19379552843836720073999v^14 - 2106362162310834996966530v^13 - 1203281739031657890480621v^12 + 37567134568520370754247220v^11 + 8063335103197233251269908v^10 - 327897110740258740307119480v^9 + 362422523161540239768290176v^8 + 2202446682825009641121408292v^7 - 1724782533273118586212671000v^6 - 1423302805080590158619506672v^5 + 8021116023302597726209967294v^4 + 9591582687049399777297302042v^3 + 10981892368336939613161395337v^2 - 9341327103233175018294227090*v - 3686481877316456037642549777) / 17061141749390174882507707338
\(\beta_{7}\)	\(=\)	\( ( - 351675045716698 \nu^{15} + 84134512548244 \nu^{14} + \cdots + 24\!\cdots\!04 ) / 45\!\cdots\!31 \) (-351675045716698v^15 + 84134512548244v^14 + 8997681016250754v^13 + 4445925380328116v^12 - 160496373722473788v^11 - 33874814941456464v^10 + 1467376262123572708v^9 - 1553113751898048416v^8 - 9401792267255930460v^7 + 10200336255773301928v^6 + 6044728226531394560v^5 - 34262171840112781800v^4 + 46812769807129599650v^3 - 46922766637244959676v^2 + 39937799358222111906*v + 240280225958040430804) / 45742705762379025631
\(\beta_{8}\)	\(=\)	\( ( 24\!\cdots\!00 \nu^{15} + \cdots - 35\!\cdots\!43 ) / 24\!\cdots\!53 \) (248197947118550681400v^15 + 60099576570060185087v^14 - 6110172991187791613532v^13 - 5610165377095918036704v^12 + 107123109897527970241392v^11 + 59136687103919879272608v^10 - 966301607286958942241280v^9 + 861897353924990552955686v^8 + 7059892149907340387084928v^7 - 4980438225173244013489056v^6 - 5677044395320575039843008v^5 + 40095443257835374287069696v^4 + 13407128082587646721614288v^3 - 5618854490528863822096597v^2 + 33962809459441010571298260*v - 35592347497900753218122943) / 24174914515820036838745953
\(\beta_{9}\)	\(=\)	\( ( - 63\!\cdots\!17 \nu^{15} + \cdots + 32\!\cdots\!77 ) / 44\!\cdots\!59 \) (-631161306625356779617v^15 + 150505234538692630725v^14 + 16500767577615813975981v^13 + 7419045073782499938700v^12 - 294271138216409917948542v^11 - 63717891799250964885804v^10 + 2750860404617240971368368v^9 - 2834324749199366123952240v^8 - 17259606953295549222511902v^7 + 19235282507483501622825296v^6 + 11183900242770749268101232v^5 - 62836954235436549118005834v^4 + 85357101295984988485619803v^3 - 86018275869553972168815051v^2 + 73144181174155227325195629*v + 327917369601661120754045677) / 44662674736623494456826459
\(\beta_{10}\)	\(=\)	\( ( - 10\!\cdots\!23 \nu^{15} + \cdots + 56\!\cdots\!52 ) / 38\!\cdots\!33 \) (-107274637135815474213267623v^15 + 40440310585103025551191222v^14 + 2509657568814871159359711123v^13 + 970451882572829151852741190v^12 - 44287196461473841197989452438v^11 + 1749022612776647015483582528v^10 + 369261641263869769858666117920v^9 - 588689505051165656418132450628v^8 - 2281678710917973731743246639250v^7 + 2977319895231402273537853555688v^6 - 1436373880362576535141563797140v^5 - 13591200467210556482109017274192v^4 + 1529669380689354048836851412717v^3 - 3020774732138613521871714908018v^2 - 3259759170842048260323290216941*v + 5648824684655197559037381780752) / 3898470889735654960653011126733
\(\beta_{11}\)	\(=\)	\( ( 13\!\cdots\!76 \nu^{15} + \cdots - 74\!\cdots\!23 ) / 40\!\cdots\!26 \) (1384364485533756987372876v^15 - 728768123650168136784649v^14 - 35908961898282986992714064v^13 - 472622045688405467009363v^12 + 658611626622471479423905864v^11 - 200247607529712853033798708v^10 - 6422914281900997227071722896v^9 + 10421619794726874885163185080v^8 + 42733803632554989723902755304v^7 - 75737920268645070599379844900v^6 - 46248813080430974509909364096v^5 + 348113674580553310579782267406v^4 - 10929072089602277258238994884v^3 - 188529834431551320226184211949v^2 - 272111133428858037728504337072*v - 74244525959635624072959944923) / 40821684709273873933539383526
\(\beta_{12}\)	\(=\)	\( ( 15\!\cdots\!48 \nu^{15} + \cdots - 25\!\cdots\!19 ) / 29\!\cdots\!09 \) (150357722430704850757448v^15 + 60075198633659217891900v^14 - 4028295388456629283530112v^13 - 4156719365271510392861673v^12 + 72661216673303491288979384v^11 + 56042699511608033717055244v^10 - 723159070367915974211775172v^9 + 303091942964013600088722988v^8 + 5658945163844004844545788056v^7 - 3056389735529669058506661454v^6 - 13723062855387011820116672024v^5 + 27731587236969783024247115660v^4 + 22370325291012881548177717004v^3 - 44749936787945170476365911956v^2 - 6602631145691352621161837336*v - 25283341868433551078267787519) / 2915834622090990995252813109
\(\beta_{13}\)	\(=\)	\( ( 12\!\cdots\!67 \nu^{15} + \cdots - 19\!\cdots\!50 ) / 20\!\cdots\!63 \) (1239986586879849068452267v^15 + 446611647240208516332506v^14 - 33172367614068978380700341v^13 - 31579135335561447776626614v^12 + 595924841732233384601337874v^11 + 409213280791450479744422696v^10 - 5938580497464899800282006892v^9 + 3285672381231400343517307320v^8 + 45850802484701684033452880102v^7 - 29875274922222088196649094936v^6 - 104669895774889538889453816484v^5 + 237551379027465268422197142796v^4 + 162608902664302276970871278667v^3 - 346649214519037359708573123958v^2 - 75641461576722423956309523053*v - 192223036278949580449972700050) / 20410842354636936966769691763
\(\beta_{14}\)	\(=\)	\( ( - 26\!\cdots\!62 \nu^{15} + \cdots + 54\!\cdots\!03 ) / 38\!\cdots\!33 \) (-267063145461629828893524862v^15 - 170988238389309380274226828v^14 + 6785108178874501898339214806v^13 + 8460206761577540248730427138v^12 - 118392865570954626911808600220v^11 - 109560588073441856744426807204v^10 + 1114274795933677419493107459288v^9 - 553827182119669552363089749314v^8 - 8791627040505418243644692523444v^7 + 3847948136217721768781922320392v^6 + 12900955768739878151945366278840v^5 - 48759551313770278706482388246430v^4 - 27203310203993899440149864599118v^3 + 15595962883292813976143238673104v^2 - 56018922452006827133376345549914*v + 54779321279479592533993047161903) / 3898470889735654960653011126733
\(\beta_{15}\)	\(=\)	\( ( - 17\!\cdots\!82 \nu^{15} + \cdots + 20\!\cdots\!29 ) / 17\!\cdots\!38 \) (-1724264633383462570433182v^15 - 2048070062977697491721035v^14 + 42252483719801659728501638v^13 + 81709401945744643690593333v^12 - 713861670067369782746122716v^11 - 1187832908714722814839439772v^10 + 6406234029560720562724506800v^9 + 1681944271704283897258555520v^8 - 55593224006113528318853699116v^7 - 23875599108767871968486476460v^6 + 91927619257102208322535794256v^5 - 163382527833887830227336488522v^4 - 453859953990828211998061613734v^3 - 199082612062867058370460165843v^2 + 124732580360973654215585228342*v + 207448752980846201883934977429) / 17061141749390174882507707338

\(\nu\)	\(=\)	\( ( -\beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{2} + \beta_1 ) / 16 \) (-b14 + b13 - b12 - b11 + b10 + b7 - b6 - b5 - b2 + b1) / 16
\(\nu^{2}\)	\(=\)	\( ( -\beta_{14} - 2\beta_{10} - 13\beta_{8} + 2\beta_{6} - 2\beta_{5} + \beta_{4} - 2\beta _1 + 13 ) / 4 \) (-b14 - 2b10 - 13b8 + 2b6 - 2b5 + b4 - 2*b1 + 13) / 4
\(\nu^{3}\)	\(=\)	\( ( -2\beta_{15} - 6\beta_{9} + 19\beta_{7} - 37\beta_{6} + 6\beta_{4} + 2\beta_{3} - 11\beta_{2} + 24 ) / 8 \) (-2b15 - 6b9 + 19b7 - 37b6 + 6b4 + 2b3 - 11*b2 + 24) / 8
\(\nu^{4}\)	\(=\)	\( ( - 2 \beta_{15} - 5 \beta_{14} + 4 \beta_{13} + 2 \beta_{12} - 4 \beta_{11} - 34 \beta_{10} + 2 \beta_{9} + \cdots - 131 ) / 4 \) (-2b15 - 5b14 + 4b13 + 2b12 - 4b11 - 34b10 + 2b9 - 131b8 - 10b7 - 46b6 - 48b5 - 13b4 - 4b2 - 40b1 - 131) / 4
\(\nu^{5}\)	\(=\)	\( ( - 13 \beta_{15} + 22 \beta_{14} - 66 \beta_{13} + 27 \beta_{12} + 39 \beta_{11} - 277 \beta_{10} + \cdots + 680 ) / 8 \) (-13b15 + 22b14 - 66b13 + 27b12 + 39b11 - 277b10 - 65b9 - 680b8 + 172b7 - 290b6 + 303b5 + 85b4 + 9b3 - 57b2 - 472*b1 + 680) / 8
\(\nu^{6}\)	\(=\)	\( ( -43\beta_{15} + 48\beta_{9} - 163\beta_{7} - 921\beta_{6} - 135\beta_{4} + 6\beta_{3} - 122\beta_{2} - 1261 ) / 2 \) (-43b15 + 48b9 - 163b7 - 921b6 - 135b4 + 6b3 - 122*b2 - 1261) / 2
\(\nu^{7}\)	\(=\)	\( ( 268 \beta_{15} - \beta_{14} - 1033 \beta_{13} + 229 \beta_{12} + 733 \beta_{11} - 11339 \beta_{10} + \cdots - 34104 ) / 16 \) (268b15 - b14 - 1033b13 + 229b12 + 733b11 - 11339b10 + 2156b9 - 34104b8 - 5935b7 + 5873b6 + 6141b5 - 3780b4 - 100b3 + 933b2 - 17807b1 - 34104) / 16
\(\nu^{8}\)	\(=\)	\( ( - 788 \beta_{15} - 55 \beta_{14} - 2836 \beta_{13} + 472 \beta_{12} + 2164 \beta_{11} + 2392 \beta_{10} + \cdots - 6799 ) / 4 \) (-788b15 - 55b14 - 2836b13 + 472b12 + 2164b11 + 2392b10 + 478b9 + 6799b8 - 1312b7 - 16832b6 + 17620b5 - 779b4 + 224b3 - 2612b2 + 3826*b1 - 6799) / 4
\(\nu^{9}\)	\(=\)	\( ( 490 \beta_{15} + 33570 \beta_{9} - 97945 \beta_{7} + 9937 \beta_{6} - 70938 \beta_{4} + 526 \beta_{3} + \cdots - 660288 ) / 8 \) (490b15 + 33570b9 - 97945b7 + 9937b6 - 70938b4 + 526b3 - 369*b2 - 660288) / 8
\(\nu^{10}\)	\(=\)	\( ( 13059 \beta_{15} - 4297 \beta_{14} - 51234 \beta_{13} + 12057 \beta_{12} + 36840 \beta_{11} + \cdots - 125022 ) / 4 \) (13059b15 - 4297b14 - 51234b13 + 12057b12 + 36840b11 - 33095b10 + 2292b9 - 125022b8 - 10459b7 + 280281b6 + 293340b5 - 12464b4 - 4798b3 + 46436b2 - 39971*b1 - 125022) / 4
\(\nu^{11}\)	\(=\)	\( ( - 32613 \beta_{15} + 74343 \beta_{14} - 152793 \beta_{13} + 54954 \beta_{12} + 97494 \beta_{11} + \cdots - 5445440 ) / 8 \) (-32613b15 + 74343b14 - 152793b13 + 54954b12 + 97494b11 + 1662216b10 + 249249b9 + 5445440b8 - 753759b7 - 709441b6 + 742054b5 - 578853b4 + 18433b3 - 134360b2 + 2409963*b1 - 5445440) / 8
\(\nu^{12}\)	\(=\)	\( 97132 \beta_{15} + 112200 \beta_{9} - 363803 \beta_{7} + 2090869 \beta_{6} - 308517 \beta_{4} + \cdots - 2949878 \) 97132b15 + 112200b9 - 363803b7 + 2090869b6 - 308517b4 - 39428b3 + 355911*b2 - 2949878
\(\nu^{13}\)	\(=\)	\( ( 2229108 \beta_{15} + 2332695 \beta_{14} - 9498203 \beta_{13} + 2810879 \beta_{12} + 6449951 \beta_{11} + \cdots + 154362832 ) / 16 \) (2229108b15 + 2332695b14 - 9498203b13 + 2810879b12 + 6449951b11 + 46693893b10 - 6750900b9 + 154362832b8 + 20760401b7 + 48161131b6 + 50390239b5 + 16342196b4 - 1016084b3 + 8482119b2 + 66946593*b1 + 154362832) / 16
\(\nu^{14}\)	\(=\)	\( ( 2480988 \beta_{15} + 2503213 \beta_{14} + 10250770 \beta_{13} - 2807806 \beta_{12} - 7108918 \beta_{11} + \cdots - 153106199 ) / 4 \) (2480988b15 + 2503213b14 + 10250770b13 - 2807806b12 - 7108918b11 + 45953948b10 + 6433504b9 + 153106199b8 - 20082678b7 + 53474096b6 - 55955084b5 - 16152387b4 - 1047284b3 + 9203486b2 + 65254460*b1 - 153106199) / 4
\(\nu^{15}\)	\(=\)	\( ( 52228326 \beta_{15} - 75835062 \beta_{9} + 234972317 \beta_{7} + 1126170861 \beta_{6} + \cdots + 1769317544 ) / 8 \) (52228326b15 - 75835062b9 + 234972317b7 + 1126170861b6 + 186984966b4 - 22305174b3 + 194464767*b2 + 1769317544) / 8

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 97.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 56 T^{6} + \cdots + 2401)^{2} \) (T^8 + 56T^6 + 878T^4 + 3976*T^2 + 2401)^2
$5$	\( (T^{8} + 140 T^{6} + \cdots + 108241)^{2} \) (T^8 + 140T^6 + 5558T^4 + 65260*T^2 + 108241)^2
$7$	\( T^{16} \) T^16
$11$	\( (T^{8} - 536 T^{6} + \cdots + 19018321)^{2} \) (T^8 - 536T^6 + 80958T^4 - 2585576*T^2 + 19018321)^2
$13$	\( (T^{8} + 296 T^{6} + \cdots + 12544)^{2} \) (T^8 + 296T^6 + 10928T^4 + 65152*T^2 + 12544)^2
$17$	\( (T^{8} + 844 T^{6} + \cdots + 717007729)^{2} \) (T^8 + 844T^6 + 231638T^4 + 24172652*T^2 + 717007729)^2
$19$	\( (T^{8} + 1448 T^{6} + \cdots + 6387046561)^{2} \) (T^8 + 1448T^6 + 625070T^4 + 106934296*T^2 + 6387046561)^2
$23$	\( (T^{8} - 2584 T^{6} + \cdots + 10882871041)^{2} \) (T^8 - 2584T^6 + 2064270T^4 - 536705512*T^2 + 10882871041)^2
$29$	\( (T^{4} + 4 T^{3} + \cdots + 5008)^{4} \) (T^4 + 4T^3 - 756T^2 - 1520*T + 5008)^4
$31$	\( (T^{8} + 2744 T^{6} + \cdots + 36725506321)^{2} \) (T^8 + 2744T^6 + 2005310T^4 + 482048392*T^2 + 36725506321)^2
$37$	\( (T^{4} + 36 T^{3} + \cdots - 25079)^{4} \) (T^4 + 36T^3 - 1522T^2 + 13236*T - 25079)^4
$41$	\( (T^{8} + 2632 T^{6} + \cdots + 31899904)^{2} \) (T^8 + 2632T^6 + 1251632T^4 + 118404224*T^2 + 31899904)^2
$43$	\( (T^{8} + \cdots + 12876384903424)^{2} \) (T^8 - 8800T^6 + 27334368T^4 - 34105899520*T^2 + 12876384903424)^2
$47$	\( (T^{8} + \cdots + 2641538328961)^{2} \) (T^8 + 10472T^6 + 32918030T^4 + 27781550104*T^2 + 2641538328961)^2
$53$	\( (T^{4} + 20 T^{3} + \cdots + 779737)^{4} \) (T^4 + 20T^3 - 4738T^2 + 74116*T + 779737)^4
$59$	\( (T^{8} + \cdots + 18597147378481)^{2} \) (T^8 + 14792T^6 + 73219550T^4 + 125198557624*T^2 + 18597147378481)^2
$61$	\( (T^{8} + \cdots + 11484080659489)^{2} \) (T^8 + 10076T^6 + 29256038T^4 + 31679470108*T^2 + 11484080659489)^2
$67$	\( (T^{8} + \cdots + 112936720954561)^{2} \) (T^8 - 16664T^6 + 83257230T^4 - 164393487272*T^2 + 112936720954561)^2
$71$	\( (T^{8} + \cdots + 159740883210496)^{2} \) (T^8 - 16096T^6 + 89738208T^4 - 205483188736*T^2 + 159740883210496)^2
$73$	\( (T^{8} + \cdots + 46497574501569)^{2} \) (T^8 + 22428T^6 + 116029062T^4 + 178264268316*T^2 + 46497574501569)^2
$79$	\( (T^{8} + \cdots + 85293924591841)^{2} \) (T^8 - 15288T^6 + 78813230T^4 - 154037472456*T^2 + 85293924591841)^2
$83$	\( (T^{8} + \cdots + 5740355977216)^{2} \) (T^8 + 19744T^6 + 76252928T^4 + 53618991104*T^2 + 5740355977216)^2
$89$	\( (T^{8} + \cdots + 32\!\cdots\!25)^{2} \) (T^8 + 46156T^6 + 648827126T^4 + 2695955513900*T^2 + 3212489203380625)^2
$97$	\( (T^{8} + \cdots + 16556565688576)^{2} \) (T^8 + 35752T^6 + 328917680T^4 + 170189944448*T^2 + 16556565688576)^2
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\(n\)	\(197\)	\(1471\)	\(1473\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)