LMFDB - Newform orbit 1680.3.n.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,3,Mod(1471,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.1471"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1680.n (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-48,0,0,0,48,0,0,0,-96] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$45.7766844125$
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} - 7 x^{15} + 6 x^{14} + 115 x^{13} - 535 x^{12} + 605 x^{11} + 2453 x^{10} - 10997 x^{9} + \cdots + 92164 $ x^16 - 7x^15 + 6x^14 + 115x^13 - 535x^12 + 605x^11 + 2453x^10 - 10997x^9 + 23723x^8 - 49514x^7 + 98500x^6 - 121640x^5 + 87351x^4 - 81896x^3 + 155185x^2 - 191418*x + 92164
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{36} $
Twist minimal:	yes
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_{7} q^{3} + \beta_1 q^{5} + \beta_{8} q^{7} - 3 q^{9} + ( - \beta_{13} + \beta_{10} + \beta_{7}) q^{11} + ( - \beta_{3} - \beta_1 + 3) q^{13} - \beta_{10} q^{15} + ( - \beta_{5} - 2 \beta_{2} - 6) q^{17}+ \cdots + (3 \beta_{13} - 3 \beta_{10} - 3 \beta_{7}) q^{99}+O(q^{100}) $ q - b7 * q^3 + b1 * q^5 + b8 * q^7 - 3 * q^9 + (-b13 + b10 + b7) * q^11 + (-b3 - b1 + 3) * q^13 - b10 * q^15 + (-b5 - 2b2 - 6) q^17 + (-b15 - b11 - b8) * q^19 + b2 * q^21 + (b13 + b12 - b11 + 3b10 - b8 + b7) q^23 + 5 * q^25 + 3b7 q^27 + (b9 + b6 + b4 + b3 - 3b2 - 3b1 + 7) * q^29 + (2b13 - b12 - 3b11 + 2b10 + b8) q^31 + (b5 - b4 + b3 + 3b1 + 3) q^33 + b11 * q^35 + (2b6 + b5 + b4 - 2b2 - 2b1 - 14) q^37 + (b15 - b13 - b12 + b10 - 3b7) q^39 + (-2b9 + 2b6 + 2b5 - b4 + 2b3 + 2b1 + 10) q^41 + (b15 - b14 - 5b11 - 5b10 + 5b8 - b7) q^43 - 3b1 q^45 + (b15 + b14 + b11 - b10 - 5b8 - 9b7) * q^47 - 7 * q^49 + (b14 - b13 + 6b8 + 6b7) * q^51 + (-b9 + 3b6 - 2b5 - 2b3 + b2 + 4b1 - 6) * q^53 + (b13 + 2b12 + b10 + 5b7) * q^55 + (b9 + b6 - 2b3 - b2) q^57 + (-2b14 - 4b11 - 2b10 - 8b8 - 6b7) q^59 + (b9 + 3b6 - 2b5 - 2b4 + b3 - 3b2 - 7b1 + 5) q^61 - 3b8 q^63 + (2b5 + b3 + 3b1 - 5) * q^65 + (b15 + 3b14 - 2b13 - 5b11 + b10 + 5b8 - 11b7) q^67 + (-b9 + b6 - b3 - b2 + 9b1 + 3) q^69 + (b15 - 3b13 - 3b12 - 2b11 - 7b10 - 10b8 + 19b7) * q^71 + (4b6 - b4 + b3 + 10b2 - 5b1 - 3) q^73 - 5b7 q^75 + (b9 + b6 - 2b5 - b2) q^77 + (-2b15 - 2b14 - 2b12 + 2b10 - 12b8 + 6b7) * q^79 + 9 * q^81 + (-4b15 - 2b14 + 4b13 - 6b10 + 4b8 - 2b7) * q^83 + (2b6 - b5 + 2b3 - 6b1) q^85 + (-b15 - b14 + 2b13 + 4b12 + 3b11 + 3b10 + 9b8 - 7b7) * q^87 + (2b9 + 6b6 - 2b5 + 5b4 - 2b3 + 8b2 + 10b1 - 10) q^89 + (-b15 - b14 + b13 + 2b12 - b11 + 3b8) * q^91 + (b9 + 3b6 - 3b5 + 3b4 - 2b3 + b2 + 6b1) q^93 + (b15 + 2b14 - b11 - 5b8) * q^95 + (-2b9 + 6b6 + 2b5 - b4 - 3b3 - 4b2 + 11b1 + 37) * q^97 + (3b13 - 3b10 - 3b7) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9} + 48 q^{13} - 96 q^{17} + 80 q^{25} + 112 q^{29} + 48 q^{33} - 224 q^{37} + 160 q^{41} - 112 q^{49} - 96 q^{53} + 80 q^{61} - 80 q^{65} + 48 q^{69} - 48 q^{73} + 144 q^{81} - 160 q^{89}+ \cdots + 592 q^{97}+O(q^{100}) $ 16 * q - 48 * q^9 + 48 * q^13 - 96 * q^17 + 80 * q^25 + 112 * q^29 + 48 * q^33 - 224 * q^37 + 160 * q^41 - 112 * q^49 - 96 * q^53 + 80 * q^61 - 80 * q^65 + 48 * q^69 - 48 * q^73 + 144 * q^81 - 160 * q^89 + 592 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 7 x^{15} + 6 x^{14} + 115 x^{13} - 535 x^{12} + 605 x^{11} + 2453 x^{10} - 10997 x^{9} + \cdots + 92164 $ x^16 - 7*x^15 + 6*x^14 + 115*x^13 - 535*x^12 + 605*x^11 + 2453*x^10 - 10997*x^9 + 23723*x^8 - 49514*x^7 + 98500*x^6 - 121640*x^5 + 87351*x^4 - 81896*x^3 + 155185*x^2 - 191418*x + 92164 :

$\beta_{1}$	$=$	$ ( 20\!\cdots\!92 \nu^{15} + \cdots - 37\!\cdots\!72 ) / 72\!\cdots\!60 $ (20164875834556192v^15 - 166629305861773791v^14 + 181175436945772383v^13 + 2742331780242735387v^12 - 13111368657365316102v^11 + 13032706252959764862v^10 + 69759297553716596239v^9 - 269345535648590571303v^8 + 468567568782019144854v^7 - 925421268323338367342v^6 + 2132809003228625893332v^5 - 2081777642093942931102v^4 + 227291479495220812064v^3 - 1574984849740851348531v^2 + 4061422187595623919846*v - 3733556287735812568972) / 728857387274279840960
$\beta_{2}$	$=$	$ ( - 13\!\cdots\!08 \nu^{15} + \cdots - 96\!\cdots\!44 ) / 33\!\cdots\!44 $ (-13315832128803808v^15 + 59073746351088553v^14 + 79143204792591959v^13 - 1347743712860600077v^12 + 3545289848794874698v^11 + 1846138178823774190v^10 - 28509207278162011385v^9 + 65612820364527807265v^8 - 123549104184020541754v^7 + 340451203286101107746v^6 - 545112784774331855884v^5 + 389791842556079290354v^4 - 219515096247192064736v^3 + 512035600332855575605v^2 - 905004298314197108714*v - 960792939111490724844) / 330001530540658574144
$\beta_{3}$	$=$	$ ( 58\!\cdots\!88 \nu^{15} + \cdots - 58\!\cdots\!68 ) / 10\!\cdots\!00 $ (582366636854799619277038288v^15 - 4837795869840362887322373939v^14 + 5062240346976378465338941107v^13 + 80363911153991564520095169583v^12 - 376636950071278654339502297518v^11 + 351500279045276145087660834998v^10 + 2047079366247964120042697573891v^9 - 7557931495378116458522850003147v^8 + 12859012329887969351220669033406v^7 - 27279004757497234835721254928918v^6 + 64987502795795310503445777258308v^5 - 59991302776270061945409663913478v^4 - 997146869759303676291781327824v^3 - 46203664582101451968299121982279v^2 + 131670842043932010093829929246094*v - 58763604128606214919009061815068) / 10472574836347715713591894692800
$\beta_{4}$	$=$	$ ( - 56\!\cdots\!91 \nu^{15} + \cdots - 13\!\cdots\!64 ) / 65\!\cdots\!00 $ (-56047855803089679899249191v^15 + 48410555193735353235158113v^14 + 1337623935499866783763930141v^13 - 5064124786041075996290780736v^12 - 5668881192238634124444656554v^11 + 73834859004659402732742386279v^10 - 134319870208491950394222572437v^9 - 151111300778750901285336057416v^8 + 800798629550237246394499874218v^7 - 1373523274579063075077336490604v^6 + 4173261952334018417225284446534v^5 - 8316080536691764063780097332504v^4 + 7277399878649258614896011736553v^3 - 2332235659508133320431484132252v^2 - 2123284095330733218023153318248*v - 13094510897902093659282518643064) / 654535927271732232099493418300
$\beta_{5}$	$=$	$ ( - 65\!\cdots\!12 \nu^{15} + \cdots + 66\!\cdots\!52 ) / 52\!\cdots\!00 $ (-653440116454697544763613412v^15 + 5244670371690778643109741691v^14 - 5385508743974265099941034963v^13 - 87105072614453014787413678427v^12 + 411611358134911033669080881422v^11 - 402896536170218454400510024922v^10 - 2204197160777032400557374664059v^9 + 8503938559849503377979882470663v^8 - 14891326630481649206094730660574v^7 + 29023689290482498109339464485822v^6 - 65525031293538861698177571836812v^5 + 64667881361699864845135100833822v^4 - 9990891041239390842983902051204v^3 + 49284056043705051458968734753411v^2 - 121930406334169990493494597303686*v + 66329118670334543223990843406252) / 5236287418173857856795947346400
$\beta_{6}$	$=$	$ ( 40\!\cdots\!28 \nu^{15} + \cdots - 49\!\cdots\!53 ) / 18\!\cdots\!15 $ (4045270610244428v^15 - 27614563320597354v^14 + 13877382408238632v^13 + 498297775623043408v^12 - 2057313093604515718v^11 + 1445319702582612078v^10 + 12145458709178910626v^9 - 42065974347296857402v^8 + 73528198040302600146v^7 - 149532072502703865648v^6 + 310385991217264058438v^5 - 289805693373966048238v^4 + 61493275908093068816v^3 - 247894433283840714764v^2 + 573474421636053123784*v - 492657714220523891453) / 18239310734541970015
$\beta_{7}$	$=$	$ ( - 44\!\cdots\!59 \nu^{15} + \cdots + 28\!\cdots\!04 ) / 14\!\cdots\!00 $ (-4451688264821559v^15 + 25127507241894272v^14 + 7533833085618394v^13 - 503095680267227459v^12 + 1700012156067937384v^11 - 365530518185361819v^10 - 11503770502154506088v^9 + 33404637660227258311v^8 - 59867045232295275528v^7 + 137863064513700905734v^6 - 248404394072046332874v^5 + 194104553896890261954v^4 - 104048739539033487963v^3 + 206618222870149974907v^2 - 388400377630040507062*v + 289965463667049565204) / 14621984778467900800
$\beta_{8}$	$=$	$ ( 159676076414873 \nu^{15} - 914285067944359 \nu^{14} - 234643800713243 \nu^{13} + \cdots - 10\!\cdots\!88 ) / 36\!\cdots\!00 $ (159676076414873v^15 - 914285067944359v^14 - 234643800713243v^13 + 18191737755581248v^12 - 62099095508720498v^11 + 14598765205215543v^10 + 418108081976384011v^9 - 1220058645333238992v^8 + 2171271284087370666v^7 - 4992065728638574148v^6 + 9095599701847085878v^5 - 7133526464299110688v^4 + 3774840891023763161v^3 - 7465061177803119004v^2 + 14177921096540618264*v - 10707309166144657288) / 366523440079104800
$\beta_{9}$	$=$	$ ( - 58\!\cdots\!08 \nu^{15} + \cdots + 84\!\cdots\!28 ) / 10\!\cdots\!00 $ (-5801137088075990058597377208v^15 + 44374361179901173003936910609v^14 - 44883398995602467838053509377v^13 - 724304812614252267144934908173v^12 + 3450582661927746766625895654618v^11 - 3756949765890348309914803642458v^10 - 16833749373144736470106561408481v^9 + 70809200680807455050347273012897v^8 - 139299926660594083063273974071306v^7 + 285792422081724175202651611174818v^6 - 606525839556527507331836690060668v^5 + 671340658958474387624063398089298v^4 - 311123165279843762123603883071576v^3 + 469050913240786226818641810492469v^2 - 841493771606572273244589013367914*v + 841040759914296213499749035351828) / 10472574836347715713591894692800
$\beta_{10}$	$=$	$ ( 58391035384283 \nu^{15} - 340284802097573 \nu^{14} - 48306108152577 \nu^{13} + \cdots - 44\!\cdots\!44 ) / 98\!\cdots\!80 $ (58391035384283v^15 - 340284802097573v^14 - 48306108152577v^13 + 6649936334645056v^12 - 23442443101543350v^11 + 8018880147198549v^10 + 152074997410441361v^9 - 464651479228027440v^8 + 847789580697016670v^7 - 1907848441766773260v^6 + 3499946812108971074v^5 - 2957736086704258848v^4 + 1569763229381074715v^3 - 2654280103990674836v^2 + 5630202289892593608*v - 4491037380731800344) / 98008213974502880
$\beta_{11}$	$=$	$ ( - 129747753771601 \nu^{15} + 742677923572096 \nu^{14} + 152615443805654 \nu^{13} + \cdots + 96\!\cdots\!48 ) / 13\!\cdots\!80 $ (-129747753771601v^15 + 742677923572096v^14 + 152615443805654v^13 - 14667605669140197v^12 + 50877737255825240v^11 - 15248372178403533v^10 - 334913245560656472v^9 + 1007420739876044865v^8 - 1830563395597285560v^7 + 4135221539762614090v^6 - 7547470420479440998v^5 + 6354311637778988206v^4 - 3386141928544278285v^3 + 5772656403744448237v^2 - 12143280040176614426*v + 9602706660859587148) / 133441380923735680
$\beta_{12}$	$=$	$ ( 64\!\cdots\!61 \nu^{15} + \cdots - 48\!\cdots\!16 ) / 41\!\cdots\!20 $ (6495401683289560023231256361v^15 - 36731994542040480053631854668v^14 - 8916017759648770940655769866v^13 + 729017311695202525026913107801v^12 - 2505549508002572986239494438496v^11 + 705678060942340881443119030061v^10 + 16546048960741402398653376278292v^9 - 49482189584922906880304938042469v^8 + 90504373318898356145728652143392v^7 - 206004897347915019890446886537026v^6 + 375014009929425782037736229497526v^5 - 321077379179897551259792566310246v^4 + 186255104169812205162781390301077v^3 - 314657526816117049340947137495153v^2 + 628268830981466886602318219214498*v - 485242507977632831758890407727516) / 4189029934539086285436757877120
$\beta_{13}$	$=$	$ ( 98\!\cdots\!23 \nu^{15} + \cdots - 70\!\cdots\!88 ) / 41\!\cdots\!20 $ (9886410451601803380054743323v^15 - 56765053011922900737077595404v^14 - 13311434045385460842017073558v^13 + 1124530879532590541146607868403v^12 - 3862127973739119738137582016208v^11 + 999893867647730361523768490903v^10 + 25735324481995524284557469285956v^9 - 75822297126330780379825194275447v^8 + 136228883009466998583735477064016v^7 - 313766650554163404615481777818038v^6 + 574943361845736590525921220586018v^5 - 468110463704090733566190668090978v^4 + 270591868157004875851840483771391v^3 - 505307502525204141913801916517979v^2 + 940964504430551428600779815705334*v - 707060558216223711587654659366388) / 4189029934539086285436757877120
$\beta_{14}$	$=$	$ ( 20\!\cdots\!07 \nu^{15} + \cdots - 12\!\cdots\!16 ) / 83\!\cdots\!24 $ (2096030362470587214732357507v^15 - 11666564149750407272616872980v^14 - 4536256580487904242556157150v^13 + 236542101773868245936164472995v^12 - 779629403757712622319862648448v^11 + 104168440995576864871694408079v^10 + 5407989788793236239883566708844v^9 - 15177947660947706743129143444711v^8 + 26828643690815967246585069560640v^7 - 63262411693099497521982236735414v^6 + 113535380980664453553014554654418v^5 - 84494809563511629641301686742466v^4 + 49536617188283954366896124326759v^3 - 108402208483099788959453235922155v^2 + 179885483255873202080730641865494*v - 125269909874403727304614321123316) / 837805986907817257087351575424
$\beta_{15}$	$=$	$ ( 14\!\cdots\!31 \nu^{15} + \cdots - 11\!\cdots\!96 ) / 41\!\cdots\!20 $ (14296154769641482994832516731v^15 - 82769818937713497542951858908v^14 - 13408974053234193973161752006v^13 + 1621674946892391711069806635491v^12 - 5686351138702356708607555834416v^11 + 1883455371668741087786416420951v^10 + 36935199612116934272954923940052v^9 - 112377371348017334021239349852839v^8 + 205739132080234792794107623718192v^7 - 466624144102926510265160123473366v^6 + 857888076217355997412270905211106v^5 - 736906340273934190965508124713346v^4 + 425709744352383420216912276947487v^3 - 715863633441692442384129425674123v^2 + 1436643537905337999856641208557398*v - 1120890612014692344100001067958196) / 4189029934539086285436757877120

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

$\nu$	$=$	$ ( \beta_{15} - \beta_{13} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \cdots + 7 ) / 16 $ (b15 - b13 + b11 - b10 + b9 + b8 - b7 + b6 - b5 + b4 + b3 - b2 - b1 + 7) / 16
$\nu^{2}$	$=$	$ ( 2 \beta_{15} - \beta_{14} - 2 \beta_{13} - \beta_{12} - 3 \beta_{11} - 11 \beta_{10} - \beta_{9} + \cdots + 37 ) / 16 $ (2b15 - b14 - 2b13 - b12 - 3b11 - 11b10 - b9 + 11b8 - 5b7 - 5b6 + 3b5 - 2b4 + 3b3 + 11b2 + 11b1 + 37) / 16
$\nu^{3}$	$=$	$ ( 2 \beta_{15} + 2 \beta_{14} - 5 \beta_{13} + \beta_{12} - \beta_{11} - 3 \beta_{10} + 2 \beta_{9} + \cdots - 64 ) / 8 $ (2b15 + 2b14 - 5b13 + b12 - b11 - 3b10 + 2b9 + 19b8 + 27b7 - 11b5 + 3b4 + 2b3 - 34*b1 - 64) / 8
$\nu^{4}$	$=$	$ ( 11 \beta_{15} + 14 \beta_{14} - 3 \beta_{13} - 42 \beta_{12} - 89 \beta_{11} - 47 \beta_{10} + \cdots + 217 ) / 16 $ (11b15 + 14b14 - 3b13 - 42b12 - 89b11 - 47b10 - 19b9 - 121b8 + 9b7 - 65b6 + 23b5 - 33b4 - 3b3 + 105b2 + 177*b1 + 217) / 16
$\nu^{5}$	$=$	$ ( - 33 \beta_{15} + 87 \beta_{14} - 81 \beta_{13} + 81 \beta_{12} + 30 \beta_{11} + 380 \beta_{10} + \cdots - 1248 ) / 16 $ (-33b15 + 87b14 - 81b13 + 81b12 + 30b11 + 380b10 + 86b9 + 204b8 + 1046b7 + 80b6 - 222b5 + 135b4 - 86b3 - 386b2 - 150*b1 - 1248) / 16
$\nu^{6}$	$=$	$ ( 19 \beta_{15} - 64 \beta_{14} + 176 \beta_{13} - 233 \beta_{12} - 435 \beta_{11} - 561 \beta_{10} + \cdots + 2762 ) / 8 $ (19b15 - 64b14 + 176b13 - 233b12 - 435b11 - 561b10 - 37b9 - 1045b8 - 1365b7 - 180b6 + 216b5 - 67b4 - 223b3 + 120b2 + 1808*b1 + 2762) / 8
$\nu^{7}$	$=$	$ ( - 423 \beta_{15} + 119 \beta_{14} - 1111 \beta_{13} + 2645 \beta_{12} + 4676 \beta_{11} + 4390 \beta_{10} + \cdots - 9856 ) / 16 $ (-423b15 + 119b14 - 1111b13 + 2645b12 + 4676b11 + 4390b10 + 954b9 + 8330b8 + 6544b7 + 1344b6 - 2022b5 + 1513b4 - 954b3 - 4270b2 - 1826*b1 - 9856) / 16
$\nu^{8}$	$=$	$ ( 1328 \beta_{15} - 4633 \beta_{14} + 6452 \beta_{13} - 6717 \beta_{12} - 8663 \beta_{11} - 24751 \beta_{10} + \cdots + 31353 ) / 16 $ (1328b15 - 4633b14 + 6452b13 - 6717b12 - 8663b11 - 24751b10 - 2071b9 - 28057b8 - 68633b7 - 4993b6 + 4189b5 - 3380b4 + 109b3 + 9639b2 + 17119*b1 + 31353) / 16
$\nu^{9}$	$=$	$ ( - 4454 \beta_{15} + 3465 \beta_{14} - 8911 \beta_{13} + 20022 \beta_{12} + 32525 \beta_{11} + \cdots - 77590 ) / 8 $ (-4454b15 + 3465b14 - 8911b13 + 20022b12 + 32525b11 + 44469b10 + 995b9 + 60083b8 + 80109b7 + 1494b6 - 8086b5 + 1654b4 + 4439b3 - 1206b2 - 39080*b1 - 77590) / 8
$\nu^{10}$	$=$	$ ( 11115 \beta_{15} - 33205 \beta_{14} + 100195 \beta_{13} - 139335 \beta_{12} - 242310 \beta_{11} + \cdots + 113372 ) / 16 $ (11115b15 - 33205b14 + 100195b13 - 139335b12 - 242310b11 - 285840b10 - 17606b9 - 595236b8 - 720810b7 - 30480b6 + 24822b5 - 27765b4 + 29866b3 + 102226b2 - 19150*b1 + 113372) / 16
$\nu^{11}$	$=$	$ ( - 133383 \beta_{15} + 203759 \beta_{14} - 311607 \beta_{13} + 522253 \beta_{12} + 741676 \beta_{11} + \cdots + 468904 ) / 16 $ (-133383b15 + 203759b14 - 311607b13 + 522253b12 + 741676b11 + 1527010b10 + 8626b9 + 1615230b8 + 3348456b7 + 42008b6 + 28582b5 + 16843b4 + 35046b3 + 38490b2 - 61722*b1 + 468904) / 16
$\nu^{12}$	$=$	$ ( 178955 \beta_{15} - 253481 \beta_{14} + 619966 \beta_{13} - 1057108 \beta_{12} - 1755745 \beta_{11} + \cdots + 291494 ) / 8 $ (178955b15 - 253481b14 + 619966b13 - 1057108b12 - 1755745b11 - 2451429b10 + 123214b9 - 3798865b8 - 5287455b7 + 283200b6 - 98301b5 + 203404b4 - 45296b3 - 446796b2 - 251772*b1 + 291494) / 8
$\nu^{13}$	$=$	$ ( - 944021 \beta_{15} + 2001617 \beta_{14} - 5179833 \beta_{13} + 7671535 \beta_{12} + 12608688 \beta_{11} + \cdots + 29076756 ) / 16 $ (-944021b15 + 2001617b14 - 5179833b13 + 7671535b12 + 12608688b11 + 16731950b10 - 744414b9 + 29838002b8 + 39925092b7 - 1266420b6 + 3173334b5 - 1189105b4 - 401578b3 + 3547090b2 + 11537566*b1 + 29076756) / 16
$\nu^{14}$	$=$	$ ( 6730869 \beta_{15} - 9138859 \beta_{14} + 15290033 \beta_{13} - 27361685 \beta_{12} - 41377686 \beta_{11} + \cdots - 119257066 ) / 16 $ (6730869b15 - 9138859b14 + 15290033b13 - 27361685b12 - 41377686b11 - 76562132b10 + 5843168b9 - 86458960b8 - 161934778b7 + 11281102b6 - 15894032b5 + 9461295b4 + 526848b3 - 24911936b2 - 53975632*b1 - 119257066) / 16
$\nu^{15}$	$=$	$ ( - 13819491 \beta_{15} + 27012609 \beta_{14} - 64884807 \beta_{13} + 98023347 \beta_{12} + \cdots + 504064602 ) / 16 $ (-13819491b15 + 27012609b14 - 64884807b13 + 98023347b12 + 159087420b11 + 223975650b10 - 33046246b9 + 371885022b8 + 525501336b7 - 65961090b6 + 74677882b5 - 53571015b4 + 4983206b3 + 142812660b2 + 235017360*b1 + 504064602) / 16

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$1$	$1$	$1$	$1$	$-1$

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} $

1471.1

−0.825539	+	0.794836i
1.27200	−	0.118428i
−0.656837	−	1.70180i
2.51939	+	1.56062i
1.69504	−	2.74011i
0.754146	+	0.969255i
2.52830	−	0.173365i
−3.78650	+	0.542965i
2.51939	−	1.56062i
−0.656837	+	1.70180i
1.27200	+	0.118428i
−0.825539	−	0.794836i
−3.78650	−	0.542965i
2.52830	+	0.173365i
0.754146	−	0.969255i
1.69504	+	2.74011i

−

1.73205i

−2.23607

−

2.64575i

−3.00000

1471.2

−

1.73205i

−2.23607

−

2.64575i

−3.00000

1471.3

−

1.73205i

−2.23607

2.64575i

−3.00000

1471.4

−

1.73205i

−2.23607

2.64575i

−3.00000

1471.5

−

1.73205i

2.23607

−

2.64575i

−3.00000

1471.6

−

1.73205i

2.23607

−

2.64575i

−3.00000

1471.7

−

1.73205i

2.23607

2.64575i

−3.00000

1471.8

−

1.73205i

2.23607

2.64575i

−3.00000

1471.9

1.73205i

−2.23607

−

2.64575i

−3.00000

1471.10

1.73205i

−2.23607

−

2.64575i

−3.00000

1471.11

1.73205i

−2.23607

2.64575i

−3.00000

1471.12

1.73205i

−2.23607

2.64575i

−3.00000

1471.13

1.73205i

2.23607

−

2.64575i

−3.00000

1471.14

1.73205i

2.23607

−

2.64575i

−3.00000

1471.15

1.73205i

2.23607

2.64575i

−3.00000

1471.16

1.73205i

2.23607

2.64575i

−3.00000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1680.3.n.c	✓	16
4.b	odd	2	1	inner	1680.3.n.c	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1680.3.n.c	✓	16	1.a	even	1	1	trivial
1680.3.n.c	✓	16	4.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{11}^{16} + 1184 T_{11}^{14} + 549632 T_{11}^{12} + 129339392 T_{11}^{10} + 16624844800 T_{11}^{8} + \cdots + 470560911917056 $ T11^16 + 1184*T11^14 + 549632*T11^12 + 129339392*T11^10 + 16624844800*T11^8 + 1159167213568*T11^6 + 40011680448512*T11^4 + 514341392613376*T11^2 + 470560911917056 acting on $S_{3}^{\mathrm{new}}(1680, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{2} + 3)^{8} $ (T^2 + 3)^8
$5$	$ (T^{2} - 5)^{8} $ (T^2 - 5)^8
$7$	$ (T^{2} + 7)^{8} $ (T^2 + 7)^8
$11$	$ T^{16} + \cdots + 470560911917056 $ T^16 + 1184T^14 + 549632T^12 + 129339392T^10 + 16624844800T^8 + 1159167213568T^6 + 40011680448512T^4 + 514341392613376*T^2 + 470560911917056
$13$	$ (T^{8} - 24 T^{7} + \cdots - 91821824)^{2} $ (T^8 - 24T^7 - 304T^6 + 8096T^5 + 21984T^4 - 877696T^3 + 745728T^2 + 30563840*T - 91821824)^2
$17$	$ (T^{8} + 48 T^{7} + \cdots + 12887296)^{2} $ (T^8 + 48T^7 - 32T^6 - 24000T^5 - 174304T^4 + 1178880T^3 + 2259968T^2 - 14558208*T + 12887296)^2
$19$	$ T^{16} + \cdots + 21\!\cdots\!36 $ T^16 + 3232T^14 + 4240128T^12 + 2942095360T^10 + 1171157876736T^8 + 268507115683840T^6 + 32944394608836608T^4 + 1674087339065868288*T^2 + 21186390771367936
$23$	$ T^{16} + \cdots + 92\!\cdots\!76 $ T^16 + 3040T^14 + 3327744T^12 + 1730375680T^10 + 466284773376T^8 + 64932319068160T^6 + 4302347744313344T^4 + 111059462982205440*T^2 + 926080229888229376
$29$	$ (T^{8} - 56 T^{7} + \cdots + 258183424)^{2} $ (T^8 - 56T^7 - 3408T^6 + 231328T^5 + 50016T^4 - 139479680T^3 + 1086268160T^2 + 1469431296*T + 258183424)^2
$31$	$ T^{16} + \cdots + 83\!\cdots\!76 $ T^16 + 10656T^14 + 43235072T^12 + 82652459008T^10 + 72963404267520T^8 + 23359212111265792T^6 + 1320969520306716672T^4 + 9274074221347078144*T^2 + 8358957504293502976
$37$	$ (T^{8} + 112 T^{7} + \cdots - 198934225664)^{2} $ (T^8 + 112T^7 + 832T^6 - 304320T^5 - 10093664T^4 + 92483840T^3 + 7435335168T^2 + 68168641536*T - 198934225664)^2
$41$	$ (T^{8} + \cdots + 1175024800000)^{2} $ (T^8 - 80T^7 - 6480T^6 + 574400T^5 + 7234400T^4 - 1004896000T^3 + 5184480000T^2 + 268086400000*T + 1175024800000)^2
$43$	$ T^{16} + \cdots + 42\!\cdots\!56 $ T^16 + 20000T^14 + 164452608T^12 + 713626456064T^10 + 1740524121882624T^8 + 2340572055250927616T^6 + 1578777177292795805696T^4 + 444660835415470192459776*T^2 + 42538500391712303670624256
$47$	$ T^{16} + \cdots + 10\!\cdots\!96 $ T^16 + 8608T^14 + 27042048T^12 + 40772587520T^10 + 31782809501696T^8 + 12327481313853440T^6 + 1889051355767635968T^4 + 8418716766837932032*T^2 + 1025352149757853696
$53$	$ (T^{8} + 48 T^{7} + \cdots + 146263348480)^{2} $ (T^8 + 48T^7 - 9696T^6 - 442048T^5 + 15547936T^4 + 1065519360T^3 + 17321943040T^2 + 95558302720*T + 146263348480)^2
$59$	$ T^{16} + \cdots + 35\!\cdots\!96 $ T^16 + 24704T^14 + 213654528T^12 + 809190522880T^10 + 1562128375218176T^8 + 1653071283937607680T^6 + 967478325115713748992T^4 + 293320661101842287034368*T^2 + 35928326186141914702544896
$61$	$ (T^{8} + \cdots + 19981935575296)^{2} $ (T^8 - 40T^7 - 17936T^6 + 376800T^5 + 109312096T^4 - 75870080T^3 - 229498634496T^2 - 3939372971520*T + 19981935575296)^2
$67$	$ T^{16} + \cdots + 51\!\cdots\!36 $ T^16 + 42208T^14 + 736843008T^12 + 6867320934400T^10 + 36779241575940096T^8 + 113461103087001272320T^6 + 191935203452555569922048T^4 + 160170269865580472127455232*T^2 + 51053165288088816554906484736
$71$	$ T^{16} + \cdots + 92\!\cdots\!96 $ T^16 + 34720T^14 + 419543808T^12 + 2193472630784T^10 + 5178613813673984T^8 + 5164264191621595136T^6 + 1886094873684351123456T^4 + 126340388276843384406016*T^2 + 927206197096247455645696
$73$	$ (T^{8} + 24 T^{7} + \cdots - 356882398976)^{2} $ (T^8 + 24T^7 - 16912T^6 - 522400T^5 + 67373664T^4 + 3064839296T^3 + 8834443008T^2 - 434170146304*T - 356882398976)^2
$79$	$ T^{16} + \cdots + 92\!\cdots\!56 $ T^16 + 37504T^14 + 537817088T^12 + 3951481323520T^10 + 16438574454931456T^8 + 39722624843786485760T^6 + 54066790039313545428992T^4 + 37078203360845320209563648*T^2 + 9219237943109298733245792256
$83$	$ T^{16} + \cdots + 37\!\cdots\!36 $ T^16 + 53376T^14 + 1023024128T^12 + 9554471649280T^10 + 47796975934242816T^8 + 128695062935004774400T^6 + 172397164612188050030592T^4 + 90024825766146963241172992*T^2 + 3746500839301266403791732736
$89$	$ (T^{8} + \cdots + 83\!\cdots\!24)^{2} $ (T^8 + 80T^7 - 48720T^6 - 3074496T^5 + 770200416T^4 + 35395301120T^3 - 4615064202496T^2 - 129607533933568*T + 8318684410810624)^2
$97$	$ (T^{8} + \cdots + 134282266337536)^{2} $ (T^8 - 296T^7 + 4528T^6 + 5487840T^5 - 398804384T^4 - 18663041920T^3 + 2356315644672T^2 - 50475797386752*T + 134282266337536)^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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1680.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{7} q^{3} + \beta_1 q^{5} + \beta_{8} q^{7} - 3 q^{9} + ( - \beta_{13} + \beta_{10} + \beta_{7}) q^{11} + ( - \beta_{3} - \beta_1 + 3) q^{13} - \beta_{10} q^{15} + ( - \beta_{5} - 2 \beta_{2} - 6) q^{17}+ \cdots + (3 \beta_{13} - 3 \beta_{10} - 3 \beta_{7}) q^{99}+O(q^{100}) \) q - b7 * q^3 + b1 * q^5 + b8 * q^7 - 3 * q^9 + (-b13 + b10 + b7) * q^11 + (-b3 - b1 + 3) * q^13 - b10 * q^15 + (-b5 - 2b2 - 6) q^17 + (-b15 - b11 - b8) * q^19 + b2 * q^21 + (b13 + b12 - b11 + 3b10 - b8 + b7) q^23 + 5 * q^25 + 3b7 q^27 + (b9 + b6 + b4 + b3 - 3b2 - 3b1 + 7) * q^29 + (2b13 - b12 - 3b11 + 2b10 + b8) q^31 + (b5 - b4 + b3 + 3b1 + 3) q^33 + b11 * q^35 + (2b6 + b5 + b4 - 2b2 - 2b1 - 14) q^37 + (b15 - b13 - b12 + b10 - 3b7) q^39 + (-2b9 + 2b6 + 2b5 - b4 + 2b3 + 2b1 + 10) q^41 + (b15 - b14 - 5b11 - 5b10 + 5b8 - b7) q^43 - 3b1 q^45 + (b15 + b14 + b11 - b10 - 5b8 - 9b7) * q^47 - 7 * q^49 + (b14 - b13 + 6b8 + 6b7) * q^51 + (-b9 + 3b6 - 2b5 - 2b3 + b2 + 4b1 - 6) * q^53 + (b13 + 2b12 + b10 + 5b7) * q^55 + (b9 + b6 - 2b3 - b2) q^57 + (-2b14 - 4b11 - 2b10 - 8b8 - 6b7) q^59 + (b9 + 3b6 - 2b5 - 2b4 + b3 - 3b2 - 7b1 + 5) q^61 - 3b8 q^63 + (2b5 + b3 + 3b1 - 5) * q^65 + (b15 + 3b14 - 2b13 - 5b11 + b10 + 5b8 - 11b7) q^67 + (-b9 + b6 - b3 - b2 + 9b1 + 3) q^69 + (b15 - 3b13 - 3b12 - 2b11 - 7b10 - 10b8 + 19b7) * q^71 + (4b6 - b4 + b3 + 10b2 - 5b1 - 3) q^73 - 5b7 q^75 + (b9 + b6 - 2b5 - b2) q^77 + (-2b15 - 2b14 - 2b12 + 2b10 - 12b8 + 6b7) * q^79 + 9 * q^81 + (-4b15 - 2b14 + 4b13 - 6b10 + 4b8 - 2b7) * q^83 + (2b6 - b5 + 2b3 - 6b1) q^85 + (-b15 - b14 + 2b13 + 4b12 + 3b11 + 3b10 + 9b8 - 7b7) * q^87 + (2b9 + 6b6 - 2b5 + 5b4 - 2b3 + 8b2 + 10b1 - 10) q^89 + (-b15 - b14 + b13 + 2b12 - b11 + 3b8) * q^91 + (b9 + 3b6 - 3b5 + 3b4 - 2b3 + b2 + 6b1) q^93 + (b15 + 2b14 - b11 - 5b8) * q^95 + (-2b9 + 6b6 + 2b5 - b4 - 3b3 - 4b2 + 11b1 + 37) * q^97 + (3b13 - 3b10 - 3b7) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9} + 48 q^{13} - 96 q^{17} + 80 q^{25} + 112 q^{29} + 48 q^{33} - 224 q^{37} + 160 q^{41} - 112 q^{49} - 96 q^{53} + 80 q^{61} - 80 q^{65} + 48 q^{69} - 48 q^{73} + 144 q^{81} - 160 q^{89}+ \cdots + 592 q^{97}+O(q^{100}) \) 16 * q - 48 * q^9 + 48 * q^13 - 96 * q^17 + 80 * q^25 + 112 * q^29 + 48 * q^33 - 224 * q^37 + 160 * q^41 - 112 * q^49 - 96 * q^53 + 80 * q^61 - 80 * q^65 + 48 * q^69 - 48 * q^73 + 144 * q^81 - 160 * q^89 + 592 * q^97

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	1680.3.n.c

Level	$1680$
Weight	$3$
Character orbit	1680.n
Analytic conductor	$45.777$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(45.7766844125\)
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:	\( x^{16} - 7 x^{15} + 6 x^{14} + 115 x^{13} - 535 x^{12} + 605 x^{11} + 2453 x^{10} - 10997 x^{9} + \cdots + 92164 \) x^16 - 7x^15 + 6x^14 + 115x^13 - 535x^12 + 605x^11 + 2453x^10 - 10997x^9 + 23723x^8 - 49514x^7 + 98500x^6 - 121640x^5 + 87351x^4 - 81896x^3 + 155185x^2 - 191418*x + 92164
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{36} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 20\!\cdots\!92 \nu^{15} + \cdots - 37\!\cdots\!72 ) / 72\!\cdots\!60 \) (20164875834556192v^15 - 166629305861773791v^14 + 181175436945772383v^13 + 2742331780242735387v^12 - 13111368657365316102v^11 + 13032706252959764862v^10 + 69759297553716596239v^9 - 269345535648590571303v^8 + 468567568782019144854v^7 - 925421268323338367342v^6 + 2132809003228625893332v^5 - 2081777642093942931102v^4 + 227291479495220812064v^3 - 1574984849740851348531v^2 + 4061422187595623919846*v - 3733556287735812568972) / 728857387274279840960
\(\beta_{2}\)	\(=\)	\( ( - 13\!\cdots\!08 \nu^{15} + \cdots - 96\!\cdots\!44 ) / 33\!\cdots\!44 \) (-13315832128803808v^15 + 59073746351088553v^14 + 79143204792591959v^13 - 1347743712860600077v^12 + 3545289848794874698v^11 + 1846138178823774190v^10 - 28509207278162011385v^9 + 65612820364527807265v^8 - 123549104184020541754v^7 + 340451203286101107746v^6 - 545112784774331855884v^5 + 389791842556079290354v^4 - 219515096247192064736v^3 + 512035600332855575605v^2 - 905004298314197108714*v - 960792939111490724844) / 330001530540658574144
\(\beta_{3}\)	\(=\)	\( ( 58\!\cdots\!88 \nu^{15} + \cdots - 58\!\cdots\!68 ) / 10\!\cdots\!00 \) (582366636854799619277038288v^15 - 4837795869840362887322373939v^14 + 5062240346976378465338941107v^13 + 80363911153991564520095169583v^12 - 376636950071278654339502297518v^11 + 351500279045276145087660834998v^10 + 2047079366247964120042697573891v^9 - 7557931495378116458522850003147v^8 + 12859012329887969351220669033406v^7 - 27279004757497234835721254928918v^6 + 64987502795795310503445777258308v^5 - 59991302776270061945409663913478v^4 - 997146869759303676291781327824v^3 - 46203664582101451968299121982279v^2 + 131670842043932010093829929246094*v - 58763604128606214919009061815068) / 10472574836347715713591894692800
\(\beta_{4}\)	\(=\)	\( ( - 56\!\cdots\!91 \nu^{15} + \cdots - 13\!\cdots\!64 ) / 65\!\cdots\!00 \) (-56047855803089679899249191v^15 + 48410555193735353235158113v^14 + 1337623935499866783763930141v^13 - 5064124786041075996290780736v^12 - 5668881192238634124444656554v^11 + 73834859004659402732742386279v^10 - 134319870208491950394222572437v^9 - 151111300778750901285336057416v^8 + 800798629550237246394499874218v^7 - 1373523274579063075077336490604v^6 + 4173261952334018417225284446534v^5 - 8316080536691764063780097332504v^4 + 7277399878649258614896011736553v^3 - 2332235659508133320431484132252v^2 - 2123284095330733218023153318248*v - 13094510897902093659282518643064) / 654535927271732232099493418300
\(\beta_{5}\)	\(=\)	\( ( - 65\!\cdots\!12 \nu^{15} + \cdots + 66\!\cdots\!52 ) / 52\!\cdots\!00 \) (-653440116454697544763613412v^15 + 5244670371690778643109741691v^14 - 5385508743974265099941034963v^13 - 87105072614453014787413678427v^12 + 411611358134911033669080881422v^11 - 402896536170218454400510024922v^10 - 2204197160777032400557374664059v^9 + 8503938559849503377979882470663v^8 - 14891326630481649206094730660574v^7 + 29023689290482498109339464485822v^6 - 65525031293538861698177571836812v^5 + 64667881361699864845135100833822v^4 - 9990891041239390842983902051204v^3 + 49284056043705051458968734753411v^2 - 121930406334169990493494597303686*v + 66329118670334543223990843406252) / 5236287418173857856795947346400
\(\beta_{6}\)	\(=\)	\( ( 40\!\cdots\!28 \nu^{15} + \cdots - 49\!\cdots\!53 ) / 18\!\cdots\!15 \) (4045270610244428v^15 - 27614563320597354v^14 + 13877382408238632v^13 + 498297775623043408v^12 - 2057313093604515718v^11 + 1445319702582612078v^10 + 12145458709178910626v^9 - 42065974347296857402v^8 + 73528198040302600146v^7 - 149532072502703865648v^6 + 310385991217264058438v^5 - 289805693373966048238v^4 + 61493275908093068816v^3 - 247894433283840714764v^2 + 573474421636053123784*v - 492657714220523891453) / 18239310734541970015
\(\beta_{7}\)	\(=\)	\( ( - 44\!\cdots\!59 \nu^{15} + \cdots + 28\!\cdots\!04 ) / 14\!\cdots\!00 \) (-4451688264821559v^15 + 25127507241894272v^14 + 7533833085618394v^13 - 503095680267227459v^12 + 1700012156067937384v^11 - 365530518185361819v^10 - 11503770502154506088v^9 + 33404637660227258311v^8 - 59867045232295275528v^7 + 137863064513700905734v^6 - 248404394072046332874v^5 + 194104553896890261954v^4 - 104048739539033487963v^3 + 206618222870149974907v^2 - 388400377630040507062*v + 289965463667049565204) / 14621984778467900800
\(\beta_{8}\)	\(=\)	\( ( 159676076414873 \nu^{15} - 914285067944359 \nu^{14} - 234643800713243 \nu^{13} + \cdots - 10\!\cdots\!88 ) / 36\!\cdots\!00 \) (159676076414873v^15 - 914285067944359v^14 - 234643800713243v^13 + 18191737755581248v^12 - 62099095508720498v^11 + 14598765205215543v^10 + 418108081976384011v^9 - 1220058645333238992v^8 + 2171271284087370666v^7 - 4992065728638574148v^6 + 9095599701847085878v^5 - 7133526464299110688v^4 + 3774840891023763161v^3 - 7465061177803119004v^2 + 14177921096540618264*v - 10707309166144657288) / 366523440079104800
\(\beta_{9}\)	\(=\)	\( ( - 58\!\cdots\!08 \nu^{15} + \cdots + 84\!\cdots\!28 ) / 10\!\cdots\!00 \) (-5801137088075990058597377208v^15 + 44374361179901173003936910609v^14 - 44883398995602467838053509377v^13 - 724304812614252267144934908173v^12 + 3450582661927746766625895654618v^11 - 3756949765890348309914803642458v^10 - 16833749373144736470106561408481v^9 + 70809200680807455050347273012897v^8 - 139299926660594083063273974071306v^7 + 285792422081724175202651611174818v^6 - 606525839556527507331836690060668v^5 + 671340658958474387624063398089298v^4 - 311123165279843762123603883071576v^3 + 469050913240786226818641810492469v^2 - 841493771606572273244589013367914*v + 841040759914296213499749035351828) / 10472574836347715713591894692800
\(\beta_{10}\)	\(=\)	\( ( 58391035384283 \nu^{15} - 340284802097573 \nu^{14} - 48306108152577 \nu^{13} + \cdots - 44\!\cdots\!44 ) / 98\!\cdots\!80 \) (58391035384283v^15 - 340284802097573v^14 - 48306108152577v^13 + 6649936334645056v^12 - 23442443101543350v^11 + 8018880147198549v^10 + 152074997410441361v^9 - 464651479228027440v^8 + 847789580697016670v^7 - 1907848441766773260v^6 + 3499946812108971074v^5 - 2957736086704258848v^4 + 1569763229381074715v^3 - 2654280103990674836v^2 + 5630202289892593608*v - 4491037380731800344) / 98008213974502880
\(\beta_{11}\)	\(=\)	\( ( - 129747753771601 \nu^{15} + 742677923572096 \nu^{14} + 152615443805654 \nu^{13} + \cdots + 96\!\cdots\!48 ) / 13\!\cdots\!80 \) (-129747753771601v^15 + 742677923572096v^14 + 152615443805654v^13 - 14667605669140197v^12 + 50877737255825240v^11 - 15248372178403533v^10 - 334913245560656472v^9 + 1007420739876044865v^8 - 1830563395597285560v^7 + 4135221539762614090v^6 - 7547470420479440998v^5 + 6354311637778988206v^4 - 3386141928544278285v^3 + 5772656403744448237v^2 - 12143280040176614426*v + 9602706660859587148) / 133441380923735680
\(\beta_{12}\)	\(=\)	\( ( 64\!\cdots\!61 \nu^{15} + \cdots - 48\!\cdots\!16 ) / 41\!\cdots\!20 \) (6495401683289560023231256361v^15 - 36731994542040480053631854668v^14 - 8916017759648770940655769866v^13 + 729017311695202525026913107801v^12 - 2505549508002572986239494438496v^11 + 705678060942340881443119030061v^10 + 16546048960741402398653376278292v^9 - 49482189584922906880304938042469v^8 + 90504373318898356145728652143392v^7 - 206004897347915019890446886537026v^6 + 375014009929425782037736229497526v^5 - 321077379179897551259792566310246v^4 + 186255104169812205162781390301077v^3 - 314657526816117049340947137495153v^2 + 628268830981466886602318219214498*v - 485242507977632831758890407727516) / 4189029934539086285436757877120
\(\beta_{13}\)	\(=\)	\( ( 98\!\cdots\!23 \nu^{15} + \cdots - 70\!\cdots\!88 ) / 41\!\cdots\!20 \) (9886410451601803380054743323v^15 - 56765053011922900737077595404v^14 - 13311434045385460842017073558v^13 + 1124530879532590541146607868403v^12 - 3862127973739119738137582016208v^11 + 999893867647730361523768490903v^10 + 25735324481995524284557469285956v^9 - 75822297126330780379825194275447v^8 + 136228883009466998583735477064016v^7 - 313766650554163404615481777818038v^6 + 574943361845736590525921220586018v^5 - 468110463704090733566190668090978v^4 + 270591868157004875851840483771391v^3 - 505307502525204141913801916517979v^2 + 940964504430551428600779815705334*v - 707060558216223711587654659366388) / 4189029934539086285436757877120
\(\beta_{14}\)	\(=\)	\( ( 20\!\cdots\!07 \nu^{15} + \cdots - 12\!\cdots\!16 ) / 83\!\cdots\!24 \) (2096030362470587214732357507v^15 - 11666564149750407272616872980v^14 - 4536256580487904242556157150v^13 + 236542101773868245936164472995v^12 - 779629403757712622319862648448v^11 + 104168440995576864871694408079v^10 + 5407989788793236239883566708844v^9 - 15177947660947706743129143444711v^8 + 26828643690815967246585069560640v^7 - 63262411693099497521982236735414v^6 + 113535380980664453553014554654418v^5 - 84494809563511629641301686742466v^4 + 49536617188283954366896124326759v^3 - 108402208483099788959453235922155v^2 + 179885483255873202080730641865494*v - 125269909874403727304614321123316) / 837805986907817257087351575424
\(\beta_{15}\)	\(=\)	\( ( 14\!\cdots\!31 \nu^{15} + \cdots - 11\!\cdots\!96 ) / 41\!\cdots\!20 \) (14296154769641482994832516731v^15 - 82769818937713497542951858908v^14 - 13408974053234193973161752006v^13 + 1621674946892391711069806635491v^12 - 5686351138702356708607555834416v^11 + 1883455371668741087786416420951v^10 + 36935199612116934272954923940052v^9 - 112377371348017334021239349852839v^8 + 205739132080234792794107623718192v^7 - 466624144102926510265160123473366v^6 + 857888076217355997412270905211106v^5 - 736906340273934190965508124713346v^4 + 425709744352383420216912276947487v^3 - 715863633441692442384129425674123v^2 + 1436643537905337999856641208557398*v - 1120890612014692344100001067958196) / 4189029934539086285436757877120

\(\nu\)	\(=\)	\( ( \beta_{15} - \beta_{13} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \cdots + 7 ) / 16 \) (b15 - b13 + b11 - b10 + b9 + b8 - b7 + b6 - b5 + b4 + b3 - b2 - b1 + 7) / 16
\(\nu^{2}\)	\(=\)	\( ( 2 \beta_{15} - \beta_{14} - 2 \beta_{13} - \beta_{12} - 3 \beta_{11} - 11 \beta_{10} - \beta_{9} + \cdots + 37 ) / 16 \) (2b15 - b14 - 2b13 - b12 - 3b11 - 11b10 - b9 + 11b8 - 5b7 - 5b6 + 3b5 - 2b4 + 3b3 + 11b2 + 11b1 + 37) / 16
\(\nu^{3}\)	\(=\)	\( ( 2 \beta_{15} + 2 \beta_{14} - 5 \beta_{13} + \beta_{12} - \beta_{11} - 3 \beta_{10} + 2 \beta_{9} + \cdots - 64 ) / 8 \) (2b15 + 2b14 - 5b13 + b12 - b11 - 3b10 + 2b9 + 19b8 + 27b7 - 11b5 + 3b4 + 2b3 - 34*b1 - 64) / 8
\(\nu^{4}\)	\(=\)	\( ( 11 \beta_{15} + 14 \beta_{14} - 3 \beta_{13} - 42 \beta_{12} - 89 \beta_{11} - 47 \beta_{10} + \cdots + 217 ) / 16 \) (11b15 + 14b14 - 3b13 - 42b12 - 89b11 - 47b10 - 19b9 - 121b8 + 9b7 - 65b6 + 23b5 - 33b4 - 3b3 + 105b2 + 177*b1 + 217) / 16
\(\nu^{5}\)	\(=\)	\( ( - 33 \beta_{15} + 87 \beta_{14} - 81 \beta_{13} + 81 \beta_{12} + 30 \beta_{11} + 380 \beta_{10} + \cdots - 1248 ) / 16 \) (-33b15 + 87b14 - 81b13 + 81b12 + 30b11 + 380b10 + 86b9 + 204b8 + 1046b7 + 80b6 - 222b5 + 135b4 - 86b3 - 386b2 - 150*b1 - 1248) / 16
\(\nu^{6}\)	\(=\)	\( ( 19 \beta_{15} - 64 \beta_{14} + 176 \beta_{13} - 233 \beta_{12} - 435 \beta_{11} - 561 \beta_{10} + \cdots + 2762 ) / 8 \) (19b15 - 64b14 + 176b13 - 233b12 - 435b11 - 561b10 - 37b9 - 1045b8 - 1365b7 - 180b6 + 216b5 - 67b4 - 223b3 + 120b2 + 1808*b1 + 2762) / 8
\(\nu^{7}\)	\(=\)	\( ( - 423 \beta_{15} + 119 \beta_{14} - 1111 \beta_{13} + 2645 \beta_{12} + 4676 \beta_{11} + 4390 \beta_{10} + \cdots - 9856 ) / 16 \) (-423b15 + 119b14 - 1111b13 + 2645b12 + 4676b11 + 4390b10 + 954b9 + 8330b8 + 6544b7 + 1344b6 - 2022b5 + 1513b4 - 954b3 - 4270b2 - 1826*b1 - 9856) / 16
\(\nu^{8}\)	\(=\)	\( ( 1328 \beta_{15} - 4633 \beta_{14} + 6452 \beta_{13} - 6717 \beta_{12} - 8663 \beta_{11} - 24751 \beta_{10} + \cdots + 31353 ) / 16 \) (1328b15 - 4633b14 + 6452b13 - 6717b12 - 8663b11 - 24751b10 - 2071b9 - 28057b8 - 68633b7 - 4993b6 + 4189b5 - 3380b4 + 109b3 + 9639b2 + 17119*b1 + 31353) / 16
\(\nu^{9}\)	\(=\)	\( ( - 4454 \beta_{15} + 3465 \beta_{14} - 8911 \beta_{13} + 20022 \beta_{12} + 32525 \beta_{11} + \cdots - 77590 ) / 8 \) (-4454b15 + 3465b14 - 8911b13 + 20022b12 + 32525b11 + 44469b10 + 995b9 + 60083b8 + 80109b7 + 1494b6 - 8086b5 + 1654b4 + 4439b3 - 1206b2 - 39080*b1 - 77590) / 8
\(\nu^{10}\)	\(=\)	\( ( 11115 \beta_{15} - 33205 \beta_{14} + 100195 \beta_{13} - 139335 \beta_{12} - 242310 \beta_{11} + \cdots + 113372 ) / 16 \) (11115b15 - 33205b14 + 100195b13 - 139335b12 - 242310b11 - 285840b10 - 17606b9 - 595236b8 - 720810b7 - 30480b6 + 24822b5 - 27765b4 + 29866b3 + 102226b2 - 19150*b1 + 113372) / 16
\(\nu^{11}\)	\(=\)	\( ( - 133383 \beta_{15} + 203759 \beta_{14} - 311607 \beta_{13} + 522253 \beta_{12} + 741676 \beta_{11} + \cdots + 468904 ) / 16 \) (-133383b15 + 203759b14 - 311607b13 + 522253b12 + 741676b11 + 1527010b10 + 8626b9 + 1615230b8 + 3348456b7 + 42008b6 + 28582b5 + 16843b4 + 35046b3 + 38490b2 - 61722*b1 + 468904) / 16
\(\nu^{12}\)	\(=\)	\( ( 178955 \beta_{15} - 253481 \beta_{14} + 619966 \beta_{13} - 1057108 \beta_{12} - 1755745 \beta_{11} + \cdots + 291494 ) / 8 \) (178955b15 - 253481b14 + 619966b13 - 1057108b12 - 1755745b11 - 2451429b10 + 123214b9 - 3798865b8 - 5287455b7 + 283200b6 - 98301b5 + 203404b4 - 45296b3 - 446796b2 - 251772*b1 + 291494) / 8
\(\nu^{13}\)	\(=\)	\( ( - 944021 \beta_{15} + 2001617 \beta_{14} - 5179833 \beta_{13} + 7671535 \beta_{12} + 12608688 \beta_{11} + \cdots + 29076756 ) / 16 \) (-944021b15 + 2001617b14 - 5179833b13 + 7671535b12 + 12608688b11 + 16731950b10 - 744414b9 + 29838002b8 + 39925092b7 - 1266420b6 + 3173334b5 - 1189105b4 - 401578b3 + 3547090b2 + 11537566*b1 + 29076756) / 16
\(\nu^{14}\)	\(=\)	\( ( 6730869 \beta_{15} - 9138859 \beta_{14} + 15290033 \beta_{13} - 27361685 \beta_{12} - 41377686 \beta_{11} + \cdots - 119257066 ) / 16 \) (6730869b15 - 9138859b14 + 15290033b13 - 27361685b12 - 41377686b11 - 76562132b10 + 5843168b9 - 86458960b8 - 161934778b7 + 11281102b6 - 15894032b5 + 9461295b4 + 526848b3 - 24911936b2 - 53975632*b1 - 119257066) / 16
\(\nu^{15}\)	\(=\)	\( ( - 13819491 \beta_{15} + 27012609 \beta_{14} - 64884807 \beta_{13} + 98023347 \beta_{12} + \cdots + 504064602 ) / 16 \) (-13819491b15 + 27012609b14 - 64884807b13 + 98023347b12 + 159087420b11 + 223975650b10 - 33046246b9 + 371885022b8 + 525501336b7 - 65961090b6 + 74677882b5 - 53571015b4 + 4983206b3 + 142812660b2 + 235017360*b1 + 504064602) / 16

\(n\)	\(241\)	\(337\)	\(421\)	\(1121\)	\(1471\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1471.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{2} + 3)^{8} \) (T^2 + 3)^8
$5$	\( (T^{2} - 5)^{8} \) (T^2 - 5)^8
$7$	\( (T^{2} + 7)^{8} \) (T^2 + 7)^8
$11$	\( T^{16} + \cdots + 470560911917056 \) T^16 + 1184T^14 + 549632T^12 + 129339392T^10 + 16624844800T^8 + 1159167213568T^6 + 40011680448512T^4 + 514341392613376*T^2 + 470560911917056
$13$	\( (T^{8} - 24 T^{7} + \cdots - 91821824)^{2} \) (T^8 - 24T^7 - 304T^6 + 8096T^5 + 21984T^4 - 877696T^3 + 745728T^2 + 30563840*T - 91821824)^2
$17$	\( (T^{8} + 48 T^{7} + \cdots + 12887296)^{2} \) (T^8 + 48T^7 - 32T^6 - 24000T^5 - 174304T^4 + 1178880T^3 + 2259968T^2 - 14558208*T + 12887296)^2
$19$	\( T^{16} + \cdots + 21\!\cdots\!36 \) T^16 + 3232T^14 + 4240128T^12 + 2942095360T^10 + 1171157876736T^8 + 268507115683840T^6 + 32944394608836608T^4 + 1674087339065868288*T^2 + 21186390771367936
$23$	\( T^{16} + \cdots + 92\!\cdots\!76 \) T^16 + 3040T^14 + 3327744T^12 + 1730375680T^10 + 466284773376T^8 + 64932319068160T^6 + 4302347744313344T^4 + 111059462982205440*T^2 + 926080229888229376
$29$	\( (T^{8} - 56 T^{7} + \cdots + 258183424)^{2} \) (T^8 - 56T^7 - 3408T^6 + 231328T^5 + 50016T^4 - 139479680T^3 + 1086268160T^2 + 1469431296*T + 258183424)^2
$31$	\( T^{16} + \cdots + 83\!\cdots\!76 \) T^16 + 10656T^14 + 43235072T^12 + 82652459008T^10 + 72963404267520T^8 + 23359212111265792T^6 + 1320969520306716672T^4 + 9274074221347078144*T^2 + 8358957504293502976
$37$	\( (T^{8} + 112 T^{7} + \cdots - 198934225664)^{2} \) (T^8 + 112T^7 + 832T^6 - 304320T^5 - 10093664T^4 + 92483840T^3 + 7435335168T^2 + 68168641536*T - 198934225664)^2
$41$	\( (T^{8} + \cdots + 1175024800000)^{2} \) (T^8 - 80T^7 - 6480T^6 + 574400T^5 + 7234400T^4 - 1004896000T^3 + 5184480000T^2 + 268086400000*T + 1175024800000)^2
$43$	\( T^{16} + \cdots + 42\!\cdots\!56 \) T^16 + 20000T^14 + 164452608T^12 + 713626456064T^10 + 1740524121882624T^8 + 2340572055250927616T^6 + 1578777177292795805696T^4 + 444660835415470192459776*T^2 + 42538500391712303670624256
$47$	\( T^{16} + \cdots + 10\!\cdots\!96 \) T^16 + 8608T^14 + 27042048T^12 + 40772587520T^10 + 31782809501696T^8 + 12327481313853440T^6 + 1889051355767635968T^4 + 8418716766837932032*T^2 + 1025352149757853696
$53$	\( (T^{8} + 48 T^{7} + \cdots + 146263348480)^{2} \) (T^8 + 48T^7 - 9696T^6 - 442048T^5 + 15547936T^4 + 1065519360T^3 + 17321943040T^2 + 95558302720*T + 146263348480)^2
$59$	\( T^{16} + \cdots + 35\!\cdots\!96 \) T^16 + 24704T^14 + 213654528T^12 + 809190522880T^10 + 1562128375218176T^8 + 1653071283937607680T^6 + 967478325115713748992T^4 + 293320661101842287034368*T^2 + 35928326186141914702544896
$61$	\( (T^{8} + \cdots + 19981935575296)^{2} \) (T^8 - 40T^7 - 17936T^6 + 376800T^5 + 109312096T^4 - 75870080T^3 - 229498634496T^2 - 3939372971520*T + 19981935575296)^2
$67$	\( T^{16} + \cdots + 51\!\cdots\!36 \) T^16 + 42208T^14 + 736843008T^12 + 6867320934400T^10 + 36779241575940096T^8 + 113461103087001272320T^6 + 191935203452555569922048T^4 + 160170269865580472127455232*T^2 + 51053165288088816554906484736
$71$	\( T^{16} + \cdots + 92\!\cdots\!96 \) T^16 + 34720T^14 + 419543808T^12 + 2193472630784T^10 + 5178613813673984T^8 + 5164264191621595136T^6 + 1886094873684351123456T^4 + 126340388276843384406016*T^2 + 927206197096247455645696
$73$	\( (T^{8} + 24 T^{7} + \cdots - 356882398976)^{2} \) (T^8 + 24T^7 - 16912T^6 - 522400T^5 + 67373664T^4 + 3064839296T^3 + 8834443008T^2 - 434170146304*T - 356882398976)^2
$79$	\( T^{16} + \cdots + 92\!\cdots\!56 \) T^16 + 37504T^14 + 537817088T^12 + 3951481323520T^10 + 16438574454931456T^8 + 39722624843786485760T^6 + 54066790039313545428992T^4 + 37078203360845320209563648*T^2 + 9219237943109298733245792256
$83$	\( T^{16} + \cdots + 37\!\cdots\!36 \) T^16 + 53376T^14 + 1023024128T^12 + 9554471649280T^10 + 47796975934242816T^8 + 128695062935004774400T^6 + 172397164612188050030592T^4 + 90024825766146963241172992*T^2 + 3746500839301266403791732736
$89$	\( (T^{8} + \cdots + 83\!\cdots\!24)^{2} \) (T^8 + 80T^7 - 48720T^6 - 3074496T^5 + 770200416T^4 + 35395301120T^3 - 4615064202496T^2 - 129607533933568*T + 8318684410810624)^2
$97$	\( (T^{8} + \cdots + 134282266337536)^{2} \) (T^8 - 296T^7 + 4528T^6 + 5487840T^5 - 398804384T^4 - 18663041920T^3 + 2356315644672T^2 - 50475797386752*T + 134282266337536)^2
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