LMFDB - Newform orbit 3456.2.r.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3456,2,Mod(577,3456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3456, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3456.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3456 = 2^{7} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3456.r (of order $6$, degree $2$, not 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:	$27.5962989386$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
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} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 672 x^{12} - 1680 x^{11} + 3244 x^{10} - 4684 x^{9} + \cdots + 14641 $ x^16 - 8x^15 + 48x^14 - 196x^13 + 672x^12 - 1680x^11 + 3244x^10 - 4684x^9 + 5298x^8 - 5096x^7 + 9592x^6 - 26388x^5 + 27528x^4 - 704x^3 + 4356x^2 - 26620*x + 14641
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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_{10} q^{5} + ( - \beta_{15} - \beta_{3}) q^{7}+O(q^{10}) $ q + b10 * q^5 + (-b15 - b3) * q^7	$ q + \beta_{10} q^{5} + ( - \beta_{15} - \beta_{3}) q^{7} - \beta_{4} q^{11} + (\beta_{14} + \beta_{10}) q^{13} + (\beta_{2} - 3) q^{17} + (2 \beta_{8} + 2 \beta_{6} + \cdots - 2 \beta_{4}) q^{19}+ \cdots + (5 \beta_{11} - 10 \beta_{7}) q^{97}+O(q^{100}) $ q + b10 * q^5 + (-b15 - b3) * q^7 - b4 * q^11 + (b14 + b10) * q^13 + (b2 - 3) * q^17 + (2b8 + 2b6 - 4b5 - 2b4) * q^19 + (-b15 - b12 - b1) * q^23 + (-2b11 - 4b7) * q^25 + (-b13 - b9) * q^29 + (b15 - b12 - b1) * q^31 + (-5b8 - 5b6 + 7b5 + 5b4) * q^35 + (-b14 - b13 + 2b10 + 2b9) * q^37 + (b11 - 6b7 + b2 - 6) q^41 + (b8 + 4b4) q^43 + (-b15 + 2b12 - b3) q^47 + (5b11 - 5b7 + 5b2 - 5) q^49 + (-b14 - b13) * q^53 + (b3 + b1) * q^55 + (b6 + 3b5 + 3b4) * q^59 + (b13 + b9) * q^61 + (-9b11 - 6b7) * q^65 + (-2b6 - 5b5 - 5b4) q^67 + 3b1 q^71 + (7b2 + 1) q^73 - b10 * q^77 + (b15 + b12 + b3) * q^79 + (-3b8 - 4b4) * q^83 + (b14 - 2b10) q^85 + (3b2 - 3) q^89 + (12b8 + 12b6 - 3b5 - 12b4) * q^91 + (4b15 - 2b12 - 2b1) q^95 + (5b11 - 10b7) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 48 q^{17} + 32 q^{25} - 48 q^{41} - 40 q^{49} + 48 q^{65} + 16 q^{73} - 48 q^{89} + 80 q^{97}+O(q^{100}) $ 16 * q - 48 * q^17 + 32 * q^25 - 48 * q^41 - 40 * q^49 + 48 * q^65 + 16 * q^73 - 48 * q^89 + 80 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 672 x^{12} - 1680 x^{11} + 3244 x^{10} - 4684 x^{9} + \cdots + 14641 $ x^16 - 8*x^15 + 48*x^14 - 196*x^13 + 672*x^12 - 1680*x^11 + 3244*x^10 - 4684*x^9 + 5298*x^8 - 5096*x^7 + 9592*x^6 - 26388*x^5 + 27528*x^4 - 704*x^3 + 4356*x^2 - 26620*x + 14641 :

$\beta_{1}$	$=$	$ ( - 35\!\cdots\!41 \nu^{15} + \cdots + 47\!\cdots\!21 ) / 54\!\cdots\!60 $ (-3542765331747320841v^15 + 24534878448868799823v^14 - 144569705798207667999v^13 + 544837829173739081899v^12 - 1828610314207632467421v^11 + 4108396839281842001805v^10 - 7478169727907646469221v^9 + 9420796042103919530085v^8 - 10166015451929744711481v^7 + 9168711207946115797743v^6 - 26753192554640509996863v^5 + 68021624826747292165787v^4 - 32320287519502590624801v^3 - 15244081341253689782319v^2 - 26164925147230112974401*v + 47055550867160087747121) / 5472292173965809716260
$\beta_{2}$	$=$	$ ( - 763942000002 \nu^{15} + 5133556971680 \nu^{14} - 30093617256945 \nu^{13} + \cdots + 71\!\cdots\!82 ) / 641806760093410 $ (-763942000002v^15 + 5133556971680v^14 - 30093617256945v^13 + 111213454254400v^12 - 370935070238998v^11 + 808374348634909v^10 - 1440688308780473v^9 + 1723884698935294v^8 - 1789798761336670v^7 + 1464163253344806v^6 - 5114941575664177v^5 + 13084848716420246v^4 - 3563883675780494v^3 - 4340506041073433v^2 - 8844524677651729*v + 7194999162513882) / 641806760093410
$\beta_{3}$	$=$	$ ( - 11\!\cdots\!02 \nu^{15} + \cdots + 11\!\cdots\!42 ) / 54\!\cdots\!60 $ (-11130591326480295602v^15 + 73197694197690115323v^14 - 427337198197798344030v^13 + 1556069652609418637035v^12 - 5166333671914485938481v^11 + 11004017457270374454924v^10 - 19350708047654698205647v^9 + 22532157411020853560232v^8 - 23905505641415785758192v^7 + 21203509192382187565801v^6 - 77875485628221007325544v^5 + 189033940401929944667249v^4 - 31133680207361701945403v^3 - 68576283508535700585930v^2 - 159262717919621177467617*v + 115561991961034084519042) / 5472292173965809716260
$\beta_{4}$	$=$	$ ( 59\!\cdots\!56 \nu^{15} + \cdots - 85\!\cdots\!83 ) / 13\!\cdots\!40 $ (59090349116137330522156v^15 - 411547132157716902983199v^14 + 2411906047131857776991872v^13 - 9094965481906851252108405v^12 + 30353514257385579109440788v^11 - 68062559813357123822586529v^10 + 121945547863069262093532816v^9 - 151998545733187161806590267v^8 + 158268858938675445008328548v^7 - 139759507950121115100475571v^6 + 423809822818633076551156992v^5 - 1119093269634643894285386605v^4 + 471777263951946094048207636v^3 + 430615534826883954173995479v^2 + 701141455257919465355628128*v - 850183677927686712719993983) / 13538450838391413238027240
$\beta_{5}$	$=$	$ ( - 16\!\cdots\!67 \nu^{15} + \cdots + 24\!\cdots\!76 ) / 21\!\cdots\!40 $ (-16672966944053949767v^15 + 116549458437688470151v^14 - 683021780482278575707v^13 + 2580790927259582584863v^12 - 8613613116335313829002v^11 + 19368469211958363402434v^10 - 34719538026397935066932v^9 + 43445610541927565715862v^8 - 45246797037611188730001v^7 + 40223719680787074416137v^6 - 120422247305938877855061v^5 + 318940110168828589603045v^4 - 138990067210407790110996v^3 - 124334589151843049893496v^2 - 198322981475705469243146*v + 244205147062169704580576) / 2113401629471029228540
$\beta_{6}$	$=$	$ ( 83\!\cdots\!95 \nu^{15} + \cdots - 12\!\cdots\!08 ) / 76\!\cdots\!20 $ (83254441669016195v^15 - 583435770921430222v^14 + 3417621173901014885v^13 - 12929512163744633687v^12 + 43132172058070375979v^11 - 97127166903798011787v^10 + 173870633156026297485v^9 - 217807534629041998706v^8 + 225336475966123460783v^7 - 200606089666861008412v^6 + 598025696011359970233v^5 - 1600993988048435862117v^4 + 701282036612306368299v^3 + 631253437878301238363v^2 + 993170372900434791733*v - 1218134286703300247908) / 7602702041279092220
$\beta_{7}$	$=$	$ ( - 23425567614 \nu^{15} + 164299773841 \nu^{14} - 962588123140 \nu^{13} + \cdots + 347716407785219 ) / 1840970757560 $ (-23425567614v^15 + 164299773841v^14 - 962588123140v^13 + 3643425366295v^12 - 12157033806082v^11 + 27396834130473v^10 - 49082753599874v^9 + 61570024345219v^8 - 63879192629834v^7 + 57025494185977v^6 - 169189320269868v^5 + 451741206168243v^4 - 200580115826046v^3 - 178433447199915v^2 - 279470932674714*v + 347716407785219) / 1840970757560
$\beta_{8}$	$=$	$ ( 18\!\cdots\!52 \nu^{15} + \cdots - 28\!\cdots\!05 ) / 13\!\cdots\!40 $ (184500663932331760880552v^15 - 1297146149114261948860093v^14 + 7598913251343959577291528v^13 - 28798251025020622906512865v^12 + 96084888523170774893549536v^11 - 216891708747773392386771937v^10 + 388561943479684861962560932v^9 - 488361278600192943929997621v^8 + 506124012011507579128051960v^7 - 453336600590056638617506213v^6 + 1335920796289486676056794864v^5 - 3579443390670116520853478381v^4 + 1614334301256205591246709736v^3 + 1425234353706996762218178131v^2 + 2205652057493108393394671820*v - 2804618928601018723544179005) / 13538450838391413238027240
$\beta_{9}$	$=$	$ ( - 18\!\cdots\!56 \nu^{15} + \cdots + 28\!\cdots\!61 ) / 12\!\cdots\!40 $ (-18487855448241233125056v^15 + 130931633988755719691549v^14 - 766973758490093350612846v^13 + 2918114496837293949946861v^12 - 9736420296779932642088224v^11 + 22091831180446503803692749v^10 - 39589772201858054344502828v^9 + 50034018357660898455246681v^8 - 51546536574523245922707412v^7 + 46166377860366791323658853v^6 - 133661647799125012576257022v^5 + 363481082337770225657148841v^4 - 171834938966092833311884124v^3 - 149363263625953221265259807v^2 - 217234533363616704563723700*v + 288143156724360756560612761) / 1230768258035583021638840
$\beta_{10}$	$=$	$ ( - 22\!\cdots\!68 \nu^{15} + \cdots + 33\!\cdots\!93 ) / 12\!\cdots\!40 $ (-22607565394009246203568v^15 + 158218142665077846574389v^14 - 926466603948397590149686v^13 + 3501748320226825565836401v^12 - 11676823598838345238722552v^11 + 26257965362531128380791253v^10 - 46951947686744360689947596v^9 + 58717956653772548398968945v^8 - 60776192493876793464651892v^7 + 54409405004643403323311069v^6 - 162806490008746693519043574v^5 + 434507456516141767089620197v^4 - 186861605618419554101056228v^3 - 173559746437441196292042615v^2 - 272153061253147455603849924*v + 332425684600055608077812593) / 1230768258035583021638840
$\beta_{11}$	$=$	$ ( 21\!\cdots\!28 \nu^{15} + \cdots - 32\!\cdots\!93 ) / 99\!\cdots\!20 $ (21923332400030472528v^15 - 153983647990249267475v^14 + 901970243394928871736v^13 - 3416423381946416836951v^12 + 11397162373403652154408v^11 - 25706446743754811448853v^10 + 46028267326001952829694v^9 - 57781152001034842761061v^8 + 59775548196145346892556v^7 - 53423751710030692243359v^6 + 158042868009163662648568v^5 - 424100573179187504967503v^4 + 189167990616641667167636v^3 + 168506415806824291140539v^2 + 261949637284390436774470*v - 326982050033832535832193) / 994962213448329039320
$\beta_{12}$	$=$	$ ( - 23\!\cdots\!68 \nu^{15} + \cdots + 34\!\cdots\!85 ) / 54\!\cdots\!60 $ (-231361103294550671568v^15 + 1621113229861001672099v^14 - 9495678229796327931844v^13 + 35914935804516934374562v^12 - 119796654545764154840788v^11 + 269625546113908075943672v^10 - 482436349233063036903438v^9 + 603285277433045682728331v^8 - 623814721742830332479580v^7 + 554080785650065312509897v^6 - 1663369103551475665280584v^5 + 4448319863711578845024964v^4 - 1954194453852119980635120v^3 - 1786148388856897169147186v^2 - 2751195011699007423881210*v + 3433104270672875771941485) / 5472292173965809716260
$\beta_{13}$	$=$	$ ( 28\!\cdots\!25 \nu^{15} + \cdots - 42\!\cdots\!80 ) / 61\!\cdots\!20 $ (28199686213166864000725v^15 - 198245679428936313175460v^14 + 1161375692152327919528044v^13 - 4400965599302855665675549v^12 + 14682983946142850485802809v^11 - 33135447495306309200039803v^10 + 59340368274961885089329128v^9 - 74481632398451839070248890v^8 + 76995263560819415181199109v^7 - 68584704405261687216408350v^6 + 203309836170061821460423420v^5 - 546264332880225524077384851v^4 + 246460329399386866639028505v^3 + 219158010310650244986744023v^2 + 331518658670705150452466956*v - 424540327260797519033611580) / 615384129017791510819420
$\beta_{14}$	$=$	$ ( 29\!\cdots\!93 \nu^{15} + \cdots - 43\!\cdots\!78 ) / 61\!\cdots\!20 $ (29428109469504434946093v^15 - 206125255146324892207360v^14 + 1207134713919733135762354v^13 - 4564598219811960637243739v^12 + 15222529070855267135680541v^11 - 34249680495542493657721609v^10 + 61253533440913721976915850v^9 - 76593234648039892166573676v^8 + 79193376017605177311293789v^7 - 70554006262405001956795374v^6 + 211606701857774654022055658v^5 - 565626895613490553306070925v^4 + 246249662091608119549684981v^3 + 227586366727265053840138045v^2 + 352826744361123026182515382*v - 435495401105430723494619578) / 615384129017791510819420
$\beta_{15}$	$=$	$ ( - 60\!\cdots\!48 \nu^{15} + \cdots + 91\!\cdots\!73 ) / 10\!\cdots\!20 $ (-609571461065940385948v^15 + 4287089407564139953697v^14 - 25111417961019082918240v^13 + 95176671253912486935035v^12 - 317494819916684207566594v^11 + 716666882055458974079591v^10 - 1283032882393077537543458v^9 + 1611133181984210169216633v^8 - 1664294153299638139915368v^7 + 1486046183236873553379989v^6 - 4394902346345379746040196v^5 + 11819301731700019349667291v^4 - 5322335690997950298535582v^3 - 4770848664680200670586425v^2 - 7220492554563439100592038*v + 9193926842502155259511973) / 10944584347931619432520

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

$\nu$	$=$	$ ( -2\beta_{15} + \beta_{12} + 3\beta_{10} + 3\beta_{9} + 2\beta_{8} + \beta_{6} + \beta_{5} - \beta_{3} + 2\beta _1 + 3 ) / 6 $ (-2b15 + b12 + 3b10 + 3b9 + 2b8 + b6 + b5 - b3 + 2*b1 + 3) / 6
$\nu^{2}$	$=$	$ ( - 4 \beta_{15} - \beta_{14} + \beta_{13} + 2 \beta_{12} + 8 \beta_{11} + 2 \beta_{10} + 4 \beta_{9} + \cdots - 9 ) / 6 $ (-4b15 - b14 + b13 + 2b12 + 8b11 + 2b10 + 4b9 - 10b8 + 6b7 - 8b6 - 14b5 - 6b4 - 2b3 + b2 + 4b1 - 9) / 6
$\nu^{3}$	$=$	$ ( 6 \beta_{15} + 3 \beta_{14} - 6 \beta_{13} - 6 \beta_{12} + 39 \beta_{11} + 3 \beta_{10} - 15 \beta_{9} + \cdots - 6 ) / 6 $ (6b15 + 3b14 - 6b13 - 6b12 + 39b11 + 3b10 - 15b9 - 32b8 + 27b7 - 28b6 - 31b5 + 6b4 - 3b3 + 15b2 - 15*b1 - 6) / 6
$\nu^{4}$	$=$	$ 4 \beta_{14} - 4 \beta_{13} - 10 \beta_{11} + 7 \beta_{10} - 7 \beta_{9} + 4 \beta_{8} - 18 \beta_{7} + \cdots - 9 $ 4b14 - 4b13 - 10b11 + 7b10 - 7b9 + 4b8 - 18b7 + 4b6 + 12b5 - 4b4 - 4b3 - 5b2 - 10*b1 - 9
$\nu^{5}$	$=$	$ ( 6 \beta_{15} - 17 \beta_{14} - 7 \beta_{13} - 21 \beta_{12} - 470 \beta_{11} - 53 \beta_{10} + \cdots - 582 ) / 6 $ (6b15 - 17b14 - 7b13 - 21b12 - 470b11 - 53b10 + 17b9 + 303b8 - 600b7 + 264b6 + 279b5 - 585b4 + 45b3 - 355b2 + 21*b1 - 582) / 6
$\nu^{6}$	$=$	$ ( 480 \beta_{15} - 399 \beta_{14} - 63 \beta_{13} - 570 \beta_{12} - 182 \beta_{11} - 1134 \beta_{10} + \cdots - 273 ) / 6 $ (480b15 - 399b14 - 63b13 - 570b12 - 182b11 - 1134b10 - 336b9 + 130b8 + 260b6 - 130b5 - 390b4 + 570b3 - 91b2 + 240b1 - 273) / 6
$\nu^{7}$	$=$	$ ( 888 \beta_{15} - 410 \beta_{14} - 151 \beta_{13} - 840 \beta_{12} + 1883 \beta_{11} - 1469 \beta_{10} + \cdots + 8403 ) / 6 $ (888b15 - 410b14 - 151b13 - 840b12 + 1883b11 - 1469b10 - 979b9 - 1947b8 + 3087b7 + 117b6 + 1023b5 + 8403b4 + 609b3 + 4690b2 - 60*b1 + 8403) / 6
$\nu^{8}$	$=$	$ - 928 \beta_{15} + 578 \beta_{14} + 578 \beta_{13} + 1280 \beta_{12} + 1564 \beta_{10} + 1564 \beta_{9} + \cdots + 2839 $ -928b15 + 578b14 + 578b13 + 1280b12 + 1564b10 + 1564b9 - 292b8 + 292b6 + 1336b5 + 2964b4 - 464b3 + 1360b2 + 640*b1 + 2839
$\nu^{9}$	$=$	$ ( - 18450 \beta_{15} + 7218 \beta_{14} + 12618 \beta_{13} + 23247 \beta_{12} + 13416 \beta_{11} + \cdots - 36321 ) / 6 $ (-18450b15 + 7218b14 + 12618b13 + 23247b12 + 13416b11 + 22689b10 + 36297b9 + 4040b8 + 24624b7 - 15107b6 - 40361b5 - 49878b4 - 3411b3 - 23046b2 + 20268*b1 - 36321) / 6
$\nu^{10}$	$=$	$ ( 19372 \beta_{15} - 7175 \beta_{14} - 20869 \beta_{13} - 29522 \beta_{12} + 133988 \beta_{11} + \cdots - 89175 ) / 6 $ (19372b15 - 7175b14 - 20869b13 - 29522b12 + 133988b11 - 13694b10 - 55432b9 - 31378b8 + 223614b7 - 94772b6 - 252938b5 - 158166b4 - 10150b3 - 44813b2 - 39208*b1 - 89175) / 6
$\nu^{11}$	$=$	$ ( 104458 \beta_{15} + 26235 \beta_{14} - 152064 \beta_{13} - 141230 \beta_{12} - 54747 \beta_{11} + \cdots - 32964 ) / 6 $ (104458b15 + 26235b14 - 152064b13 - 141230b12 - 54747b11 + 67389b10 - 414741b9 - 890b8 - 133947b7 + 22310b6 + 110747b5 + 32964b4 - 162601b3 + 363b2 - 367135*b1 - 32964) / 6
$\nu^{12}$	$=$	$ 10296 \beta_{14} - 10296 \beta_{13} - 377850 \beta_{11} + 24453 \beta_{10} - 24453 \beta_{9} + \cdots - 328185 $ 10296b14 - 10296b13 - 377850b11 + 24453b10 - 24453b9 + 168700b8 - 656370b7 + 168700b6 + 464100b5 - 168700b4 - 20020b3 - 188925b2 - 31850*b1 - 328185
$\nu^{13}$	$=$	$ ( 1071774 \beta_{15} - 1580917 \beta_{14} + 248677 \beta_{13} - 1477971 \beta_{12} - 3174158 \beta_{11} + \cdots - 4621692 ) / 6 $ (1071774b15 - 1580917b14 + 248677b13 - 1477971b12 - 3174158b11 - 4325851b10 + 720151b9 + 1804359b8 - 5299164b7 + 1181946b6 + 2817333b5 - 3663777b4 + 2810211b3 - 2684539b2 + 2338737*b1 - 4621692) / 6
$\nu^{14}$	$=$	$ ( 5003752 \beta_{15} - 4779761 \beta_{14} - 1241605 \beta_{13} - 6759662 \beta_{12} + 13947562 \beta_{11} + \cdots + 33163401 ) / 6 $ (5003752b15 - 4779761b14 - 1241605b13 - 6759662b12 + 13947562b11 - 13097678b10 - 3538156b9 - 9862502b8 + 24484116b7 - 2411968b6 - 7450534b5 + 29587506b4 + 6759662b3 + 19215839b2 + 2501876*b1 + 33163401) / 6
$\nu^{15}$	$=$	$ ( - 17770032 \beta_{15} + 13745664 \beta_{14} + 8540961 \beta_{13} + 24644016 \beta_{12} + \cdots + 129694047 ) / 6 $ (-17770032b15 + 13745664b14 + 8540961b13 + 24644016b12 + 28916849b11 + 37209345b10 + 22773267b9 - 30965391b8 + 49952673b7 + 4945137b6 + 14042145b5 + 129694047b4 - 15412641b3 + 74756944b2 + 3537726*b1 + 129694047) / 6

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times$.

$n$	$2053$	$2431$	$2945$
$\chi(n)$	$-1$	$1$	$-1 - \beta_{7}$

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} $

577.1

2.10176	−	2.65988i
−1.10176	−	0.870576i
0.0216957	−	2.36184i
0.978304	+	0.00898663i
−0.685411	+	0.698120i
1.68541	+	1.65473i
1.39465	+	0.163470i
−0.394650	+	3.36698i
2.10176	+	2.65988i
−1.10176	+	0.870576i
0.0216957	+	2.36184i
0.978304	−	0.00898663i
−0.685411	−	0.698120i
1.68541	−	1.65473i
1.39465	−	0.163470i
−0.394650	−	3.36698i

−3.05746

−

1.76523i

−0.913749

−

1.58266i

577.2

−3.05746

−

1.76523i

0.913749

1.58266i

577.3

−2.03763

−

1.17642i

−2.27268

−

3.93639i

577.4

−2.03763

−

1.17642i

2.27268

3.93639i

577.5

2.03763

1.17642i

−2.27268

−

3.93639i

577.6

2.03763

1.17642i

2.27268

3.93639i

577.7

3.05746

1.76523i

−0.913749

−

1.58266i

577.8

3.05746

1.76523i

0.913749

1.58266i

2881.1

−3.05746

1.76523i

−0.913749

1.58266i

2881.2

−3.05746

1.76523i

0.913749

−

1.58266i

2881.3

−2.03763

1.17642i

−2.27268

3.93639i

2881.4

−2.03763

1.17642i

2.27268

−

3.93639i

2881.5

2.03763

−

1.17642i

−2.27268

3.93639i

2881.6

2.03763

−

1.17642i

2.27268

−

3.93639i

2881.7

3.05746

−

1.76523i

−0.913749

1.58266i

2881.8

3.05746

−

1.76523i

0.913749

−

1.58266i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
9.c	even	3	1	inner
36.f	odd	6	1	inner
72.n	even	6	1	inner
72.p	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3456.2.r.f		16
3.b	odd	2	1		1152.2.r.e	✓	16
4.b	odd	2	1	inner	3456.2.r.f		16
8.b	even	2	1	inner	3456.2.r.f		16
8.d	odd	2	1	inner	3456.2.r.f		16
9.c	even	3	1	inner	3456.2.r.f		16
9.d	odd	6	1		1152.2.r.e	✓	16
12.b	even	2	1		1152.2.r.e	✓	16
24.f	even	2	1		1152.2.r.e	✓	16
24.h	odd	2	1		1152.2.r.e	✓	16
36.f	odd	6	1	inner	3456.2.r.f		16
36.h	even	6	1		1152.2.r.e	✓	16
72.j	odd	6	1		1152.2.r.e	✓	16
72.l	even	6	1		1152.2.r.e	✓	16
72.n	even	6	1	inner	3456.2.r.f		16
72.p	odd	6	1	inner	3456.2.r.f		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1152.2.r.e	✓	16	3.b	odd	2	1
1152.2.r.e	✓	16	9.d	odd	6	1
1152.2.r.e	✓	16	12.b	even	2	1
1152.2.r.e	✓	16	24.f	even	2	1
1152.2.r.e	✓	16	24.h	odd	2	1
1152.2.r.e	✓	16	36.h	even	6	1
1152.2.r.e	✓	16	72.j	odd	6	1
1152.2.r.e	✓	16	72.l	even	6	1
3456.2.r.f		16	1.a	even	1	1	trivial
3456.2.r.f		16	4.b	odd	2	1	inner
3456.2.r.f		16	8.b	even	2	1	inner
3456.2.r.f		16	8.d	odd	2	1	inner
3456.2.r.f		16	9.c	even	3	1	inner
3456.2.r.f		16	36.f	odd	6	1	inner
3456.2.r.f		16	72.n	even	6	1	inner
3456.2.r.f		16	72.p	odd	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3456, [\chi])$:

$ T_{5}^{8} - 18T_{5}^{6} + 255T_{5}^{4} - 1242T_{5}^{2} + 4761 $

$ T_{7}^{8} + 24T_{7}^{6} + 507T_{7}^{4} + 1656T_{7}^{2} + 4761 $

$ T_{11}^{8} - 4T_{11}^{6} + 15T_{11}^{4} - 4T_{11}^{2} + 1 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ (T^{8} - 18 T^{6} + \cdots + 4761)^{2} $ (T^8 - 18T^6 + 255T^4 - 1242*T^2 + 4761)^2
$7$	$ (T^{8} + 24 T^{6} + \cdots + 4761)^{2} $ (T^8 + 24T^6 + 507T^4 + 1656*T^2 + 4761)^2
$11$	$ (T^{8} - 4 T^{6} + 15 T^{4} + \cdots + 1)^{2} $ (T^8 - 4T^6 + 15T^4 - 4*T^2 + 1)^2
$13$	$ (T^{8} - 54 T^{6} + \cdots + 385641)^{2} $ (T^8 - 54T^6 + 2295T^4 - 33534*T^2 + 385641)^2
$17$	$ (T^{2} + 6 T + 6)^{8} $ (T^2 + 6*T + 6)^8
$19$	$ (T^{2} + 24)^{8} $ (T^2 + 24)^8
$23$	$ (T^{8} + 72 T^{6} + \cdots + 385641)^{2} $ (T^8 + 72T^6 + 4563T^4 + 44712*T^2 + 385641)^2
$29$	$ (T^{8} - 54 T^{6} + \cdots + 385641)^{2} $ (T^8 - 54T^6 + 2295T^4 - 33534*T^2 + 385641)^2
$31$	$ (T^{8} + 48 T^{6} + \cdots + 4761)^{2} $ (T^8 + 48T^6 + 2235T^4 + 3312*T^2 + 4761)^2
$37$	$ (T^{4} + 144 T^{2} + 2484)^{4} $ (T^4 + 144*T^2 + 2484)^4
$41$	$ (T^{4} + 12 T^{3} + \cdots + 1089)^{4} $ (T^4 + 12T^3 + 111T^2 + 396*T + 1089)^4
$43$	$ (T^{8} - 84 T^{6} + \cdots + 1185921)^{2} $ (T^8 - 84T^6 + 5967T^4 - 91476*T^2 + 1185921)^2
$47$	$ (T^{8} + 144 T^{6} + \cdots + 385641)^{2} $ (T^8 + 144T^6 + 20115T^4 + 89424*T^2 + 385641)^2
$53$	$ (T^{4} + 48 T^{2} + 276)^{4} $ (T^4 + 48*T^2 + 276)^4
$59$	$ (T^{8} - 52 T^{6} + 2703 T^{4} + \cdots + 1)^{2} $ (T^8 - 52T^6 + 2703T^4 - 52*T^2 + 1)^2
$61$	$ (T^{8} - 54 T^{6} + \cdots + 385641)^{2} $ (T^8 - 54T^6 + 2295T^4 - 33534*T^2 + 385641)^2
$67$	$ (T^{8} - 156 T^{6} + \cdots + 81)^{2} $ (T^8 - 156T^6 + 24327T^4 - 1404*T^2 + 81)^2
$71$	$ (T^{4} - 324 T^{2} + 22356)^{4} $ (T^4 - 324*T^2 + 22356)^4
$73$	$ (T^{2} - 2 T - 146)^{8} $ (T^2 - 2*T - 146)^8
$79$	$ (T^{8} + 72 T^{6} + \cdots + 385641)^{2} $ (T^8 + 72T^6 + 4563T^4 + 44712*T^2 + 385641)^2
$83$	$ (T^{8} - 148 T^{6} + \cdots + 28398241)^{2} $ (T^8 - 148T^6 + 16575T^4 - 788692*T^2 + 28398241)^2
$89$	$ (T^{2} + 6 T - 18)^{8} $ (T^2 + 6*T - 18)^8
$97$	$ (T^{4} - 20 T^{3} + \cdots + 625)^{4} $ (T^4 - 20T^3 + 375T^2 - 500*T + 625)^4
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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:	\( N \)	\(=\)	\( 3456 = 2^{7} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3456.r (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{10} q^{5} + ( - \beta_{15} - \beta_{3}) q^{7}+O(q^{10}) \) q + b10 * q^5 + (-b15 - b3) * q^7	\( q + \beta_{10} q^{5} + ( - \beta_{15} - \beta_{3}) q^{7} - \beta_{4} q^{11} + (\beta_{14} + \beta_{10}) q^{13} + (\beta_{2} - 3) q^{17} + (2 \beta_{8} + 2 \beta_{6} + \cdots - 2 \beta_{4}) q^{19}+ \cdots + (5 \beta_{11} - 10 \beta_{7}) q^{97}+O(q^{100}) \) q + b10 * q^5 + (-b15 - b3) * q^7 - b4 * q^11 + (b14 + b10) * q^13 + (b2 - 3) * q^17 + (2b8 + 2b6 - 4b5 - 2b4) * q^19 + (-b15 - b12 - b1) * q^23 + (-2b11 - 4b7) * q^25 + (-b13 - b9) * q^29 + (b15 - b12 - b1) * q^31 + (-5b8 - 5b6 + 7b5 + 5b4) * q^35 + (-b14 - b13 + 2b10 + 2b9) * q^37 + (b11 - 6b7 + b2 - 6) q^41 + (b8 + 4b4) q^43 + (-b15 + 2b12 - b3) q^47 + (5b11 - 5b7 + 5b2 - 5) q^49 + (-b14 - b13) * q^53 + (b3 + b1) * q^55 + (b6 + 3b5 + 3b4) * q^59 + (b13 + b9) * q^61 + (-9b11 - 6b7) * q^65 + (-2b6 - 5b5 - 5b4) q^67 + 3b1 q^71 + (7b2 + 1) q^73 - b10 * q^77 + (b15 + b12 + b3) * q^79 + (-3b8 - 4b4) * q^83 + (b14 - 2b10) q^85 + (3b2 - 3) q^89 + (12b8 + 12b6 - 3b5 - 12b4) * q^91 + (4b15 - 2b12 - 2b1) q^95 + (5b11 - 10b7) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 48 q^{17} + 32 q^{25} - 48 q^{41} - 40 q^{49} + 48 q^{65} + 16 q^{73} - 48 q^{89} + 80 q^{97}+O(q^{100}) \) 16 * q - 48 * q^17 + 32 * q^25 - 48 * q^41 - 40 * q^49 + 48 * q^65 + 16 * q^73 - 48 * q^89 + 80 * 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	3456.2.r.f

Level	$3456$
Weight	$2$
Character orbit	3456.r
Analytic conductor	$27.596$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(27.5962989386\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 672 x^{12} - 1680 x^{11} + 3244 x^{10} - 4684 x^{9} + \cdots + 14641 \) x^16 - 8x^15 + 48x^14 - 196x^13 + 672x^12 - 1680x^11 + 3244x^10 - 4684x^9 + 5298x^8 - 5096x^7 + 9592x^6 - 26388x^5 + 27528x^4 - 704x^3 + 4356x^2 - 26620*x + 14641
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{6}\cdot 3^{4} \)
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - 35\!\cdots\!41 \nu^{15} + \cdots + 47\!\cdots\!21 ) / 54\!\cdots\!60 \) (-3542765331747320841v^15 + 24534878448868799823v^14 - 144569705798207667999v^13 + 544837829173739081899v^12 - 1828610314207632467421v^11 + 4108396839281842001805v^10 - 7478169727907646469221v^9 + 9420796042103919530085v^8 - 10166015451929744711481v^7 + 9168711207946115797743v^6 - 26753192554640509996863v^5 + 68021624826747292165787v^4 - 32320287519502590624801v^3 - 15244081341253689782319v^2 - 26164925147230112974401*v + 47055550867160087747121) / 5472292173965809716260
\(\beta_{2}\)	\(=\)	\( ( - 763942000002 \nu^{15} + 5133556971680 \nu^{14} - 30093617256945 \nu^{13} + \cdots + 71\!\cdots\!82 ) / 641806760093410 \) (-763942000002v^15 + 5133556971680v^14 - 30093617256945v^13 + 111213454254400v^12 - 370935070238998v^11 + 808374348634909v^10 - 1440688308780473v^9 + 1723884698935294v^8 - 1789798761336670v^7 + 1464163253344806v^6 - 5114941575664177v^5 + 13084848716420246v^4 - 3563883675780494v^3 - 4340506041073433v^2 - 8844524677651729*v + 7194999162513882) / 641806760093410
\(\beta_{3}\)	\(=\)	\( ( - 11\!\cdots\!02 \nu^{15} + \cdots + 11\!\cdots\!42 ) / 54\!\cdots\!60 \) (-11130591326480295602v^15 + 73197694197690115323v^14 - 427337198197798344030v^13 + 1556069652609418637035v^12 - 5166333671914485938481v^11 + 11004017457270374454924v^10 - 19350708047654698205647v^9 + 22532157411020853560232v^8 - 23905505641415785758192v^7 + 21203509192382187565801v^6 - 77875485628221007325544v^5 + 189033940401929944667249v^4 - 31133680207361701945403v^3 - 68576283508535700585930v^2 - 159262717919621177467617*v + 115561991961034084519042) / 5472292173965809716260
\(\beta_{4}\)	\(=\)	\( ( 59\!\cdots\!56 \nu^{15} + \cdots - 85\!\cdots\!83 ) / 13\!\cdots\!40 \) (59090349116137330522156v^15 - 411547132157716902983199v^14 + 2411906047131857776991872v^13 - 9094965481906851252108405v^12 + 30353514257385579109440788v^11 - 68062559813357123822586529v^10 + 121945547863069262093532816v^9 - 151998545733187161806590267v^8 + 158268858938675445008328548v^7 - 139759507950121115100475571v^6 + 423809822818633076551156992v^5 - 1119093269634643894285386605v^4 + 471777263951946094048207636v^3 + 430615534826883954173995479v^2 + 701141455257919465355628128*v - 850183677927686712719993983) / 13538450838391413238027240
\(\beta_{5}\)	\(=\)	\( ( - 16\!\cdots\!67 \nu^{15} + \cdots + 24\!\cdots\!76 ) / 21\!\cdots\!40 \) (-16672966944053949767v^15 + 116549458437688470151v^14 - 683021780482278575707v^13 + 2580790927259582584863v^12 - 8613613116335313829002v^11 + 19368469211958363402434v^10 - 34719538026397935066932v^9 + 43445610541927565715862v^8 - 45246797037611188730001v^7 + 40223719680787074416137v^6 - 120422247305938877855061v^5 + 318940110168828589603045v^4 - 138990067210407790110996v^3 - 124334589151843049893496v^2 - 198322981475705469243146*v + 244205147062169704580576) / 2113401629471029228540
\(\beta_{6}\)	\(=\)	\( ( 83\!\cdots\!95 \nu^{15} + \cdots - 12\!\cdots\!08 ) / 76\!\cdots\!20 \) (83254441669016195v^15 - 583435770921430222v^14 + 3417621173901014885v^13 - 12929512163744633687v^12 + 43132172058070375979v^11 - 97127166903798011787v^10 + 173870633156026297485v^9 - 217807534629041998706v^8 + 225336475966123460783v^7 - 200606089666861008412v^6 + 598025696011359970233v^5 - 1600993988048435862117v^4 + 701282036612306368299v^3 + 631253437878301238363v^2 + 993170372900434791733*v - 1218134286703300247908) / 7602702041279092220
\(\beta_{7}\)	\(=\)	\( ( - 23425567614 \nu^{15} + 164299773841 \nu^{14} - 962588123140 \nu^{13} + \cdots + 347716407785219 ) / 1840970757560 \) (-23425567614v^15 + 164299773841v^14 - 962588123140v^13 + 3643425366295v^12 - 12157033806082v^11 + 27396834130473v^10 - 49082753599874v^9 + 61570024345219v^8 - 63879192629834v^7 + 57025494185977v^6 - 169189320269868v^5 + 451741206168243v^4 - 200580115826046v^3 - 178433447199915v^2 - 279470932674714*v + 347716407785219) / 1840970757560
\(\beta_{8}\)	\(=\)	\( ( 18\!\cdots\!52 \nu^{15} + \cdots - 28\!\cdots\!05 ) / 13\!\cdots\!40 \) (184500663932331760880552v^15 - 1297146149114261948860093v^14 + 7598913251343959577291528v^13 - 28798251025020622906512865v^12 + 96084888523170774893549536v^11 - 216891708747773392386771937v^10 + 388561943479684861962560932v^9 - 488361278600192943929997621v^8 + 506124012011507579128051960v^7 - 453336600590056638617506213v^6 + 1335920796289486676056794864v^5 - 3579443390670116520853478381v^4 + 1614334301256205591246709736v^3 + 1425234353706996762218178131v^2 + 2205652057493108393394671820*v - 2804618928601018723544179005) / 13538450838391413238027240
\(\beta_{9}\)	\(=\)	\( ( - 18\!\cdots\!56 \nu^{15} + \cdots + 28\!\cdots\!61 ) / 12\!\cdots\!40 \) (-18487855448241233125056v^15 + 130931633988755719691549v^14 - 766973758490093350612846v^13 + 2918114496837293949946861v^12 - 9736420296779932642088224v^11 + 22091831180446503803692749v^10 - 39589772201858054344502828v^9 + 50034018357660898455246681v^8 - 51546536574523245922707412v^7 + 46166377860366791323658853v^6 - 133661647799125012576257022v^5 + 363481082337770225657148841v^4 - 171834938966092833311884124v^3 - 149363263625953221265259807v^2 - 217234533363616704563723700*v + 288143156724360756560612761) / 1230768258035583021638840
\(\beta_{10}\)	\(=\)	\( ( - 22\!\cdots\!68 \nu^{15} + \cdots + 33\!\cdots\!93 ) / 12\!\cdots\!40 \) (-22607565394009246203568v^15 + 158218142665077846574389v^14 - 926466603948397590149686v^13 + 3501748320226825565836401v^12 - 11676823598838345238722552v^11 + 26257965362531128380791253v^10 - 46951947686744360689947596v^9 + 58717956653772548398968945v^8 - 60776192493876793464651892v^7 + 54409405004643403323311069v^6 - 162806490008746693519043574v^5 + 434507456516141767089620197v^4 - 186861605618419554101056228v^3 - 173559746437441196292042615v^2 - 272153061253147455603849924*v + 332425684600055608077812593) / 1230768258035583021638840
\(\beta_{11}\)	\(=\)	\( ( 21\!\cdots\!28 \nu^{15} + \cdots - 32\!\cdots\!93 ) / 99\!\cdots\!20 \) (21923332400030472528v^15 - 153983647990249267475v^14 + 901970243394928871736v^13 - 3416423381946416836951v^12 + 11397162373403652154408v^11 - 25706446743754811448853v^10 + 46028267326001952829694v^9 - 57781152001034842761061v^8 + 59775548196145346892556v^7 - 53423751710030692243359v^6 + 158042868009163662648568v^5 - 424100573179187504967503v^4 + 189167990616641667167636v^3 + 168506415806824291140539v^2 + 261949637284390436774470*v - 326982050033832535832193) / 994962213448329039320
\(\beta_{12}\)	\(=\)	\( ( - 23\!\cdots\!68 \nu^{15} + \cdots + 34\!\cdots\!85 ) / 54\!\cdots\!60 \) (-231361103294550671568v^15 + 1621113229861001672099v^14 - 9495678229796327931844v^13 + 35914935804516934374562v^12 - 119796654545764154840788v^11 + 269625546113908075943672v^10 - 482436349233063036903438v^9 + 603285277433045682728331v^8 - 623814721742830332479580v^7 + 554080785650065312509897v^6 - 1663369103551475665280584v^5 + 4448319863711578845024964v^4 - 1954194453852119980635120v^3 - 1786148388856897169147186v^2 - 2751195011699007423881210*v + 3433104270672875771941485) / 5472292173965809716260
\(\beta_{13}\)	\(=\)	\( ( 28\!\cdots\!25 \nu^{15} + \cdots - 42\!\cdots\!80 ) / 61\!\cdots\!20 \) (28199686213166864000725v^15 - 198245679428936313175460v^14 + 1161375692152327919528044v^13 - 4400965599302855665675549v^12 + 14682983946142850485802809v^11 - 33135447495306309200039803v^10 + 59340368274961885089329128v^9 - 74481632398451839070248890v^8 + 76995263560819415181199109v^7 - 68584704405261687216408350v^6 + 203309836170061821460423420v^5 - 546264332880225524077384851v^4 + 246460329399386866639028505v^3 + 219158010310650244986744023v^2 + 331518658670705150452466956*v - 424540327260797519033611580) / 615384129017791510819420
\(\beta_{14}\)	\(=\)	\( ( 29\!\cdots\!93 \nu^{15} + \cdots - 43\!\cdots\!78 ) / 61\!\cdots\!20 \) (29428109469504434946093v^15 - 206125255146324892207360v^14 + 1207134713919733135762354v^13 - 4564598219811960637243739v^12 + 15222529070855267135680541v^11 - 34249680495542493657721609v^10 + 61253533440913721976915850v^9 - 76593234648039892166573676v^8 + 79193376017605177311293789v^7 - 70554006262405001956795374v^6 + 211606701857774654022055658v^5 - 565626895613490553306070925v^4 + 246249662091608119549684981v^3 + 227586366727265053840138045v^2 + 352826744361123026182515382*v - 435495401105430723494619578) / 615384129017791510819420
\(\beta_{15}\)	\(=\)	\( ( - 60\!\cdots\!48 \nu^{15} + \cdots + 91\!\cdots\!73 ) / 10\!\cdots\!20 \) (-609571461065940385948v^15 + 4287089407564139953697v^14 - 25111417961019082918240v^13 + 95176671253912486935035v^12 - 317494819916684207566594v^11 + 716666882055458974079591v^10 - 1283032882393077537543458v^9 + 1611133181984210169216633v^8 - 1664294153299638139915368v^7 + 1486046183236873553379989v^6 - 4394902346345379746040196v^5 + 11819301731700019349667291v^4 - 5322335690997950298535582v^3 - 4770848664680200670586425v^2 - 7220492554563439100592038*v + 9193926842502155259511973) / 10944584347931619432520

\(\nu\)	\(=\)	\( ( -2\beta_{15} + \beta_{12} + 3\beta_{10} + 3\beta_{9} + 2\beta_{8} + \beta_{6} + \beta_{5} - \beta_{3} + 2\beta _1 + 3 ) / 6 \) (-2b15 + b12 + 3b10 + 3b9 + 2b8 + b6 + b5 - b3 + 2*b1 + 3) / 6
\(\nu^{2}\)	\(=\)	\( ( - 4 \beta_{15} - \beta_{14} + \beta_{13} + 2 \beta_{12} + 8 \beta_{11} + 2 \beta_{10} + 4 \beta_{9} + \cdots - 9 ) / 6 \) (-4b15 - b14 + b13 + 2b12 + 8b11 + 2b10 + 4b9 - 10b8 + 6b7 - 8b6 - 14b5 - 6b4 - 2b3 + b2 + 4b1 - 9) / 6
\(\nu^{3}\)	\(=\)	\( ( 6 \beta_{15} + 3 \beta_{14} - 6 \beta_{13} - 6 \beta_{12} + 39 \beta_{11} + 3 \beta_{10} - 15 \beta_{9} + \cdots - 6 ) / 6 \) (6b15 + 3b14 - 6b13 - 6b12 + 39b11 + 3b10 - 15b9 - 32b8 + 27b7 - 28b6 - 31b5 + 6b4 - 3b3 + 15b2 - 15*b1 - 6) / 6
\(\nu^{4}\)	\(=\)	\( 4 \beta_{14} - 4 \beta_{13} - 10 \beta_{11} + 7 \beta_{10} - 7 \beta_{9} + 4 \beta_{8} - 18 \beta_{7} + \cdots - 9 \) 4b14 - 4b13 - 10b11 + 7b10 - 7b9 + 4b8 - 18b7 + 4b6 + 12b5 - 4b4 - 4b3 - 5b2 - 10*b1 - 9
\(\nu^{5}\)	\(=\)	\( ( 6 \beta_{15} - 17 \beta_{14} - 7 \beta_{13} - 21 \beta_{12} - 470 \beta_{11} - 53 \beta_{10} + \cdots - 582 ) / 6 \) (6b15 - 17b14 - 7b13 - 21b12 - 470b11 - 53b10 + 17b9 + 303b8 - 600b7 + 264b6 + 279b5 - 585b4 + 45b3 - 355b2 + 21*b1 - 582) / 6
\(\nu^{6}\)	\(=\)	\( ( 480 \beta_{15} - 399 \beta_{14} - 63 \beta_{13} - 570 \beta_{12} - 182 \beta_{11} - 1134 \beta_{10} + \cdots - 273 ) / 6 \) (480b15 - 399b14 - 63b13 - 570b12 - 182b11 - 1134b10 - 336b9 + 130b8 + 260b6 - 130b5 - 390b4 + 570b3 - 91b2 + 240b1 - 273) / 6
\(\nu^{7}\)	\(=\)	\( ( 888 \beta_{15} - 410 \beta_{14} - 151 \beta_{13} - 840 \beta_{12} + 1883 \beta_{11} - 1469 \beta_{10} + \cdots + 8403 ) / 6 \) (888b15 - 410b14 - 151b13 - 840b12 + 1883b11 - 1469b10 - 979b9 - 1947b8 + 3087b7 + 117b6 + 1023b5 + 8403b4 + 609b3 + 4690b2 - 60*b1 + 8403) / 6
\(\nu^{8}\)	\(=\)	\( - 928 \beta_{15} + 578 \beta_{14} + 578 \beta_{13} + 1280 \beta_{12} + 1564 \beta_{10} + 1564 \beta_{9} + \cdots + 2839 \) -928b15 + 578b14 + 578b13 + 1280b12 + 1564b10 + 1564b9 - 292b8 + 292b6 + 1336b5 + 2964b4 - 464b3 + 1360b2 + 640*b1 + 2839
\(\nu^{9}\)	\(=\)	\( ( - 18450 \beta_{15} + 7218 \beta_{14} + 12618 \beta_{13} + 23247 \beta_{12} + 13416 \beta_{11} + \cdots - 36321 ) / 6 \) (-18450b15 + 7218b14 + 12618b13 + 23247b12 + 13416b11 + 22689b10 + 36297b9 + 4040b8 + 24624b7 - 15107b6 - 40361b5 - 49878b4 - 3411b3 - 23046b2 + 20268*b1 - 36321) / 6
\(\nu^{10}\)	\(=\)	\( ( 19372 \beta_{15} - 7175 \beta_{14} - 20869 \beta_{13} - 29522 \beta_{12} + 133988 \beta_{11} + \cdots - 89175 ) / 6 \) (19372b15 - 7175b14 - 20869b13 - 29522b12 + 133988b11 - 13694b10 - 55432b9 - 31378b8 + 223614b7 - 94772b6 - 252938b5 - 158166b4 - 10150b3 - 44813b2 - 39208*b1 - 89175) / 6
\(\nu^{11}\)	\(=\)	\( ( 104458 \beta_{15} + 26235 \beta_{14} - 152064 \beta_{13} - 141230 \beta_{12} - 54747 \beta_{11} + \cdots - 32964 ) / 6 \) (104458b15 + 26235b14 - 152064b13 - 141230b12 - 54747b11 + 67389b10 - 414741b9 - 890b8 - 133947b7 + 22310b6 + 110747b5 + 32964b4 - 162601b3 + 363b2 - 367135*b1 - 32964) / 6
\(\nu^{12}\)	\(=\)	\( 10296 \beta_{14} - 10296 \beta_{13} - 377850 \beta_{11} + 24453 \beta_{10} - 24453 \beta_{9} + \cdots - 328185 \) 10296b14 - 10296b13 - 377850b11 + 24453b10 - 24453b9 + 168700b8 - 656370b7 + 168700b6 + 464100b5 - 168700b4 - 20020b3 - 188925b2 - 31850*b1 - 328185
\(\nu^{13}\)	\(=\)	\( ( 1071774 \beta_{15} - 1580917 \beta_{14} + 248677 \beta_{13} - 1477971 \beta_{12} - 3174158 \beta_{11} + \cdots - 4621692 ) / 6 \) (1071774b15 - 1580917b14 + 248677b13 - 1477971b12 - 3174158b11 - 4325851b10 + 720151b9 + 1804359b8 - 5299164b7 + 1181946b6 + 2817333b5 - 3663777b4 + 2810211b3 - 2684539b2 + 2338737*b1 - 4621692) / 6
\(\nu^{14}\)	\(=\)	\( ( 5003752 \beta_{15} - 4779761 \beta_{14} - 1241605 \beta_{13} - 6759662 \beta_{12} + 13947562 \beta_{11} + \cdots + 33163401 ) / 6 \) (5003752b15 - 4779761b14 - 1241605b13 - 6759662b12 + 13947562b11 - 13097678b10 - 3538156b9 - 9862502b8 + 24484116b7 - 2411968b6 - 7450534b5 + 29587506b4 + 6759662b3 + 19215839b2 + 2501876*b1 + 33163401) / 6
\(\nu^{15}\)	\(=\)	\( ( - 17770032 \beta_{15} + 13745664 \beta_{14} + 8540961 \beta_{13} + 24644016 \beta_{12} + \cdots + 129694047 ) / 6 \) (-17770032b15 + 13745664b14 + 8540961b13 + 24644016b12 + 28916849b11 + 37209345b10 + 22773267b9 - 30965391b8 + 49952673b7 + 4945137b6 + 14042145b5 + 129694047b4 - 15412641b3 + 74756944b2 + 3537726*b1 + 129694047) / 6

\(n\)	\(2053\)	\(2431\)	\(2945\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1 - \beta_{7}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 577.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( (T^{8} - 18 T^{6} + \cdots + 4761)^{2} \) (T^8 - 18T^6 + 255T^4 - 1242*T^2 + 4761)^2
$7$	\( (T^{8} + 24 T^{6} + \cdots + 4761)^{2} \) (T^8 + 24T^6 + 507T^4 + 1656*T^2 + 4761)^2
$11$	\( (T^{8} - 4 T^{6} + 15 T^{4} + \cdots + 1)^{2} \) (T^8 - 4T^6 + 15T^4 - 4*T^2 + 1)^2
$13$	\( (T^{8} - 54 T^{6} + \cdots + 385641)^{2} \) (T^8 - 54T^6 + 2295T^4 - 33534*T^2 + 385641)^2
$17$	\( (T^{2} + 6 T + 6)^{8} \) (T^2 + 6*T + 6)^8
$19$	\( (T^{2} + 24)^{8} \) (T^2 + 24)^8
$23$	\( (T^{8} + 72 T^{6} + \cdots + 385641)^{2} \) (T^8 + 72T^6 + 4563T^4 + 44712*T^2 + 385641)^2
$29$	\( (T^{8} - 54 T^{6} + \cdots + 385641)^{2} \) (T^8 - 54T^6 + 2295T^4 - 33534*T^2 + 385641)^2
$31$	\( (T^{8} + 48 T^{6} + \cdots + 4761)^{2} \) (T^8 + 48T^6 + 2235T^4 + 3312*T^2 + 4761)^2
$37$	\( (T^{4} + 144 T^{2} + 2484)^{4} \) (T^4 + 144*T^2 + 2484)^4
$41$	\( (T^{4} + 12 T^{3} + \cdots + 1089)^{4} \) (T^4 + 12T^3 + 111T^2 + 396*T + 1089)^4
$43$	\( (T^{8} - 84 T^{6} + \cdots + 1185921)^{2} \) (T^8 - 84T^6 + 5967T^4 - 91476*T^2 + 1185921)^2
$47$	\( (T^{8} + 144 T^{6} + \cdots + 385641)^{2} \) (T^8 + 144T^6 + 20115T^4 + 89424*T^2 + 385641)^2
$53$	\( (T^{4} + 48 T^{2} + 276)^{4} \) (T^4 + 48*T^2 + 276)^4
$59$	\( (T^{8} - 52 T^{6} + 2703 T^{4} + \cdots + 1)^{2} \) (T^8 - 52T^6 + 2703T^4 - 52*T^2 + 1)^2
$61$	\( (T^{8} - 54 T^{6} + \cdots + 385641)^{2} \) (T^8 - 54T^6 + 2295T^4 - 33534*T^2 + 385641)^2
$67$	\( (T^{8} - 156 T^{6} + \cdots + 81)^{2} \) (T^8 - 156T^6 + 24327T^4 - 1404*T^2 + 81)^2
$71$	\( (T^{4} - 324 T^{2} + 22356)^{4} \) (T^4 - 324*T^2 + 22356)^4
$73$	\( (T^{2} - 2 T - 146)^{8} \) (T^2 - 2*T - 146)^8
$79$	\( (T^{8} + 72 T^{6} + \cdots + 385641)^{2} \) (T^8 + 72T^6 + 4563T^4 + 44712*T^2 + 385641)^2
$83$	\( (T^{8} - 148 T^{6} + \cdots + 28398241)^{2} \) (T^8 - 148T^6 + 16575T^4 - 788692*T^2 + 28398241)^2
$89$	\( (T^{2} + 6 T - 18)^{8} \) (T^2 + 6*T - 18)^8
$97$	\( (T^{4} - 20 T^{3} + \cdots + 625)^{4} \) (T^4 - 20T^3 + 375T^2 - 500*T + 625)^4
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