LMFDB - Newform orbit 864.3.o.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,3,Mod(127,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 4]))

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

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

chi := DirichletCharacter("864.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	864.o (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:	$23.5422948407$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 10 x^{19} + 50 x^{18} - 130 x^{17} + 203 x^{16} - 296 x^{15} + 1260 x^{14} - 3380 x^{13} + \cdots + 64 $ x^20 - 10x^19 + 50x^18 - 130x^17 + 203x^16 - 296x^15 + 1260x^14 - 3380x^13 + 4843x^12 - 2370x^11 + 8978x^10 - 23570x^9 + 32213x^8 - 6940x^7 + 11856x^6 - 30488x^5 + 41556x^4 - 6096x^3 + 288x^2 + 192*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 288)
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 $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{6} - \beta_{3} - \beta_{2} - 1) q^{5} + ( - \beta_{12} + \beta_{10}) q^{7}+O(q^{10}) $ q + (b6 - b3 - b2 - 1) * q^5 + (-b12 + b10) * q^7	$ q + (\beta_{6} - \beta_{3} - \beta_{2} - 1) q^{5} + ( - \beta_{12} + \beta_{10}) q^{7} + (\beta_{18} + \beta_{8}) q^{11} + (\beta_{16} + \beta_{15} + 3 \beta_{3} + 3) q^{13} + (\beta_{14} + \beta_{6} - 3) q^{17} + (\beta_{12} + \beta_{11} + \cdots - 2 \beta_{4}) q^{19}+ \cdots + (5 \beta_{19} + \beta_{17} + \cdots - 12 \beta_{2}) q^{97}+O(q^{100}) $ q + (b6 - b3 - b2 - 1) * q^5 + (-b12 + b10) * q^7 + (b18 + b8) * q^11 + (b16 + b15 + 3b3 + 3) q^13 + (b14 + b6 - 3) * q^17 + (b12 + b11 + 5b9 + b8 - 2b4) * q^19 + (-b18 + 2b11 - b10 - b7 + b5 + b4 + 5b1) * q^23 + (b19 + b15 - 5b3 + 5b2) * q^25 + (-b19 + 2b17 + 2b14 - 14b3) q^29 + (5b11 - 2b10 + 4b7 - 4b4) * q^31 + (5b11 - 11b9 + 5b8 - b5 + 6b4) * q^35 + (-2b19 - b14 + 2b13 + 5) * q^37 + (b17 - b16 - b15 + b13 - 8b6 - b3 + 8b2 - 1) * q^41 + (-2b18 + 11b9 - 3b8 - 4b7 + 11b1) q^43 + (-b12 + b10 + 8b9 + 6b8 - 3b7 + 8b1) * q^47 + (-2b17 + 2b16 + 2b15 + 3b13 + 6b6 + 39b3 - 6b2 + 39) q^49 + (-5b19 + b16 + b14 + 5b13 + 2b6 + 12) q^53 + (2b12 + 7b11 - 6b9 + 7b8 - 2b5 + 11b4) * q^55 + (b18 + 6b11 - b10 + 6b7 - b5 - 6b4 + 10b1) * q^59 + (-b19 + 4b17 + 4b14 - 38b3 - 4b2) * q^61 + (-b19 + 2b15 - 6b3 - 10b2) q^65 + (-b18 + 11b11 + 2b10 + 2b7 + b5 - 2b4 + 10b1) q^67 + (5b12 + 13b11 + 2b9 + 13b8 - 9b4) q^71 + (-b19 - 6b16 + 8b14 + b13 - 9) * q^73 + (4b17 + 13b16 + 13b15 + b13 - 7b6 - 4b3 + 7b2 - 4) * q^77 + (5b18 + b12 - b10 + 9b9 - 4b8 - 7b7 + 9b1) q^79 + (-2b18 - b12 + b10 - 4b9 + 10b8 - 2b7 - 4b1) q^83 + (2b17 + 2b16 + 2b15 + b13 - 7b6 + 17b3 + 7b2 + 17) * q^85 + (2b19 + 5b16 - 4b14 - 2b13 + 13b6 + 18) q^89 + (-6b12 + 15b11 + 2b9 + 15b8 + 6b5 + 8b4) * q^91 + (-2b11 + 6b10 - 3b7 + 3b4) * q^95 + (5b19 + b17 + 7b15 + b14 - 29b3 - 12b2) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 14 q^{5}+O(q^{10}) $ 20 * q - 14 * q^5	$ 20 q - 14 q^{5} + 26 q^{13} - 72 q^{17} + 36 q^{25} + 134 q^{29} + 96 q^{37} + 26 q^{41} + 348 q^{49} + 192 q^{53} + 386 q^{61} + 106 q^{65} - 168 q^{73} - 58 q^{77} + 192 q^{85} + 240 q^{89} + 374 q^{97}+O(q^{100}) $ 20 * q - 14 * q^5 + 26 * q^13 - 72 * q^17 + 36 * q^25 + 134 * q^29 + 96 * q^37 + 26 * q^41 + 348 * q^49 + 192 * q^53 + 386 * q^61 + 106 * q^65 - 168 * q^73 - 58 * q^77 + 192 * q^85 + 240 * q^89 + 374 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} - 10 x^{19} + 50 x^{18} - 130 x^{17} + 203 x^{16} - 296 x^{15} + 1260 x^{14} - 3380 x^{13} + \cdots + 64 $ x^20 - 10*x^19 + 50*x^18 - 130*x^17 + 203*x^16 - 296*x^15 + 1260*x^14 - 3380*x^13 + 4843*x^12 - 2370*x^11 + 8978*x^10 - 23570*x^9 + 32213*x^8 - 6940*x^7 + 11856*x^6 - 30488*x^5 + 41556*x^4 - 6096*x^3 + 288*x^2 + 192*x + 64 :

$\beta_{1}$	$=$	$ ( - 19\!\cdots\!63 \nu^{19} + \cdots + 15\!\cdots\!96 ) / 65\!\cdots\!00 $ (-195688856698417563v^19 + 5433733490589803696v^18 - 41314325063191276826v^17 + 169263940036751944808v^16 - 352392253945037747325v^15 + 451043328586954776652v^14 - 878920211140236752452v^13 + 4454050253601822425736v^12 - 9159518528562322945241v^11 + 9028392773138913132440v^10 - 824928332195623447498v^9 + 36207057639011453922824v^8 - 61014989221804550515307v^7 + 56425964218804631157116v^6 + 36063155078538884263800v^5 + 70285547724458583369672v^4 - 69249007281323937609988v^3 + 75012466713507873161392v^2 + 63268204836087864567632*v + 15797130996705829506496) / 6520792123885532337600
$\beta_{2}$	$=$	$ ( - 17\!\cdots\!05 \nu^{19} + \cdots + 13\!\cdots\!08 ) / 40\!\cdots\!00 $ (-1795490609055194105v^19 + 56627636428100804982v^18 - 469525310244152399198v^17 + 2097333011770955542230v^16 - 5048825445209524976127v^15 + 7524421717139295497544v^14 - 12504467498891448049396v^13 + 53111842882354678842260v^12 - 130756840223930590369219v^11 + 167753587714389198092862v^10 - 78117160539997161935534v^9 + 383561041230390891628150v^8 - 910879771724906741914921v^7 + 1065734473271497126385780v^6 - 76810932136050561526544v^5 + 538446408407738562978048v^4 - 1193655896880592900466684v^3 + 1210777365077322585707440v^2 + 167131284654106563875632*v + 13757065250618537601408) / 40056294475296841502400
$\beta_{3}$	$=$	$ ( - 22981222561 \nu^{19} + 228756996538 \nu^{18} - 1139644587846 \nu^{17} + \cdots - 7487382052224 ) / 11835144571200 $ (-22981222561v^19 + 228756996538v^18 - 1139644587846v^17 + 2946486450646v^16 - 4587224505751v^15 + 6744453837936v^14 - 28881860254372v^13 + 76676420061652v^12 - 109181344833659v^11 + 53488917592626v^10 - 209544376738198v^9 + 534526288281398v^8 - 724627861422737v^7 + 155571870617652v^6 - 301910307851792v^5 + 693191571425888v^4 - 928477369658268v^3 + 135807725605424v^2 - 40685199696080*v - 7487382052224) / 11835144571200
$\beta_{4}$	$=$	$ ( 53\!\cdots\!38 \nu^{19} + \cdots + 72\!\cdots\!88 ) / 21\!\cdots\!00 $ (532645578137067458738v^19 - 5810711965331017368735v^18 + 31437696482743951146667v^17 - 93092185585167064427293v^16 + 169279535674757392238829v^15 - 251376722934263054849140v^14 + 807388718447594883807074v^13 - 2400199388122176081473666v^12 + 4171409842037725565063604v^11 - 3490602429011931396284259v^10 + 5762450317693294322953051v^9 - 16828177539691363188979989v^8 + 28254985796269972921885369v^7 - 18541462363624960137002626v^6 + 8562120764952124416437048v^5 - 21813912786320348832176348v^4 + 36523999288588418838359236v^3 - 23042667126345395296413752v^2 + 1671161387639704448550064*v + 72981831030879062833888) / 210295545995308417887600
$\beta_{5}$	$=$	$ ( 11\!\cdots\!80 \nu^{19} + \cdots + 30\!\cdots\!16 ) / 42\!\cdots\!00 $ (1164232051635459557480v^19 - 14941504594144704498381v^18 + 90953005074420034453339v^17 - 313916547816490229526835v^16 + 653653783580349539805801v^15 - 985749595547911796459242v^14 + 2399525604494185981681238v^13 - 8023051279785455738627990v^12 + 16484495891176518520571802v^11 - 18002530007242247328269661v^10 + 17253614642062115081881627v^9 - 56546262775619100196147395v^8 + 112664471209065157179035353v^7 - 109337028451827179946623740v^6 + 29641259283958177677684212v^5 - 73278251847599655430038884v^4 + 142248157526201449350833452v^3 - 137008946435455938729031520v^2 + 6670925547889210664662864*v + 309273404785466225178016) / 420591091990616835775200
$\beta_{6}$	$=$	$ ( - 32\!\cdots\!73 \nu^{19} + \cdots + 60\!\cdots\!56 ) / 10\!\cdots\!00 $ (-32656813285611366073v^19 + 324541966728653079726v^18 - 1612018079376409678866v^17 + 4138594232998671598958v^16 - 6340061195582629930205v^15 + 9198375638815291144602v^14 - 40502083130190229664032v^13 + 107829059781428226843316v^12 - 150855676201486806521991v^11 + 66113731117529908725370v^10 - 287286356177182701782858v^9 + 754735036124855679193374v^8 - 1005150610204735594796967v^7 + 151000262862936296148326v^6 - 361196033349336854633460v^5 + 999573010827362168218352v^4 - 1322881252831615024523868v^3 + 101507943883845028932792v^2 + 54280675166281697420592*v + 6046693251517943643856) / 10014073618824210375600
$\beta_{7}$	$=$	$ ( - 11\!\cdots\!21 \nu^{19} + \cdots - 24\!\cdots\!56 ) / 21\!\cdots\!00 $ (-1192614593540774190821v^19 + 11640884888843053346130v^18 - 56693293335468093987124v^17 + 139971629541105777941416v^16 - 201196264831046926382583v^15 + 286082421621385489465960v^14 - 1405743379736432241410528v^13 + 3652364711562108368497112v^12 - 4713009705437592553361703v^11 + 1206947031756011951464818v^10 - 9743992636129040874316732v^9 + 25464569423996837497639968v^8 - 30908738863183009520868913v^7 - 2592629176835489450750348v^6 - 10370694828213879887280896v^5 + 32915177148142136293264256v^4 - 39385016381495537794253932v^3 - 7090100907295834844140576v^2 + 3203301734554821570320912*v - 249812570141658952032256) / 210295545995308417887600
$\beta_{8}$	$=$	$ ( - 16\!\cdots\!45 \nu^{19} + \cdots + 12\!\cdots\!48 ) / 28\!\cdots\!00 $ (-1634459996038199826645v^19 + 16365683869957575382947v^18 - 81944111889313701865213v^17 + 213674593182883644563195v^16 - 335459139947101760671862v^15 + 491549187321625595061494v^14 - 2073538850677886750039546v^13 + 5562556312715673917144910v^12 - 8009561447817819213479589v^11 + 4056865587435527139667507v^10 - 14923474023943205309648549v^9 + 38937630834173102455472715v^8 - 53258106448117702484415856v^7 + 12465524512850807915820740v^6 - 21010525074543342234179444v^5 + 51485030902460675638819868v^4 - 68187554817961917926301744v^3 + 10881836903238666190879840v^2 - 3319167418512870504831488*v + 1253799601400382716877248) / 280394061327077890516800
$\beta_{9}$	$=$	$ ( - 19\!\cdots\!92 \nu^{19} + \cdots - 11\!\cdots\!92 ) / 32\!\cdots\!00 $ (-19552445072400701592v^19 + 191729567961353089965v^18 - 940700155585105187603v^17 + 2361815835891087576487v^16 - 3522727057290755170111v^15 + 5131970442379467484210v^14 - 23688200689222191901366v^13 + 61584387662491838297694v^12 - 83128047571001753198886v^11 + 31008664099825332277181v^10 - 170808095130022113372259v^9 + 428823745124372010582951v^8 - 550783558840762010271071v^7 + 34281404642883852738484v^6 - 236446806584115401736532v^5 + 556138808868527100071332v^4 - 715512701857500308581524v^3 - 14696052436321641092032v^2 - 31037922678698891446576*v - 1161771539726213664992) / 3260396061942766168800
$\beta_{10}$	$=$	$ ( 27\!\cdots\!17 \nu^{19} + \cdots - 97\!\cdots\!72 ) / 42\!\cdots\!00 $ (2708280417948315268117v^19 - 27719752727283886904046v^18 + 141262047855415325159297v^17 - 378952535760362297781167v^16 + 608735442965450967426147v^15 - 873159798878446635019352v^14 + 3515882710787767023758614v^13 - 9823806655490324341184974v^12 + 14646553184643517889149923v^11 - 8005318787685644309221902v^10 + 23898883188482622143979701v^9 - 68837866481636906425469631v^8 + 97386218638952836769088809v^7 - 29263242449821035734661164v^6 + 24528696441037430619008164v^5 - 90050771891766668637001636v^4 + 123674021518160691488028716v^3 - 31666707695948358686146768v^2 - 8275032063572343509402800*v - 974993336474972679148672) / 420591091990616835775200
$\beta_{11}$	$=$	$ ( - 90\!\cdots\!24 \nu^{19} + \cdots - 75\!\cdots\!32 ) / 14\!\cdots\!00 $ (-903036033673535895324v^19 + 9008561961721572712823v^18 - 44998025303397681599733v^17 + 116900408606535620832034v^16 - 183269230858736190498740v^15 + 269268386505247539349926v^14 - 1140035775685434011400366v^13 + 3037884058262869814256228v^12 - 4370801803599800959650848v^11 + 2199333272721864885343935v^10 - 8252869648400336082274809v^9 + 21112642487120958643816802v^8 - 29132770268358979770214656v^7 + 6638233929795180652088068v^6 - 11851504596412280996255780v^5 + 26882513955339929173977416v^4 - 37853401533091179162903904v^3 + 5727113396985685204584416v^2 - 1506003832492374535199424*v - 750677485991493929106432) / 140197030663538945258400
$\beta_{12}$	$=$	$ ( - 56\!\cdots\!35 \nu^{19} + \cdots - 20\!\cdots\!36 ) / 84\!\cdots\!00 $ (-5671712517398979118835v^19 + 53861426271653727260271v^18 - 255613223384412538963369v^17 + 600214388188840052024065v^16 - 808013613812302671285606v^15 + 1169744268664422640079422v^14 - 6410612689170486659272298v^13 + 15732519223783332949651850v^12 - 18512038321636378352576547v^11 + 1389878650135166193989391v^10 - 46729301872365400695103417v^9 + 109194348589216401261729585v^8 - 120306295412113361370579988v^7 - 41618170763843081336573900v^6 - 64536186399328990741214372v^5 + 141650188961808761523172724v^4 - 154574399293299629114828992v^3 - 76741333542423302307062560v^2 - 6427699346371399963036864*v - 206820432862051131478336) / 841182183981233671550400
$\beta_{13}$	$=$	$ ( - 15\!\cdots\!23 \nu^{19} + \cdots - 42\!\cdots\!16 ) / 20\!\cdots\!00 $ (-151762441316149564923v^19 + 1322414461600693380273v^18 - 5646746225793494898214v^17 + 10072850569413056475523v^16 - 5927686684303933043977v^15 + 6459301604343464796846v^14 - 134850630532163548225708v^13 + 268600573319174770425006v^12 - 85866569581400489626265v^11 - 556971364208446314090583v^10 - 930404485624213499608862v^9 + 1815356113179891025588859v^8 - 339333043259353490643263v^7 - 5051185482019541826447544v^6 - 604334775142806375463264v^5 + 2228019792902153266046260v^4 - 264267717940811445713572v^3 - 6933342064977201384507568v^2 + 1127680921587051623649104*v - 42253559183879094274816) / 20028147237648420751200
$\beta_{14}$	$=$	$ ( - 16\!\cdots\!35 \nu^{19} + \cdots - 22\!\cdots\!56 ) / 20\!\cdots\!00 $ (-161072688040075330335v^19 + 1602567903836264568991v^18 - 7968150454957323804449v^17 + 20489247569031910880665v^16 - 31410411491356583233376v^15 + 45367358138786093032962v^14 - 199428149418766287366458v^13 + 532706649247020179263650v^12 - 747253566001169732229787v^11 + 325379174472930510117311v^10 - 1397587862940570865592057v^9 + 3704502259170664966110385v^8 - 4963865101106795755385198v^7 + 736784482374956320679200v^6 - 1633210854667398484871612v^5 + 4704854607167868270216004v^4 - 6370420394320374687181032v^3 + 489581113650107131655440v^2 + 261620002650659593532256*v - 229654856322692625041056) / 20028147237648420751200
$\beta_{15}$	$=$	$ ( 38\!\cdots\!01 \nu^{19} + \cdots + 18\!\cdots\!60 ) / 40\!\cdots\!00 $ (381310343728584500101v^19 - 3628508398678781504294v^18 + 17277010573437836741280v^17 - 40929214119471838777386v^16 + 56383297021050425643917v^15 - 83205833341528382461488v^14 + 437626203424899270802840v^13 - 1072371773933196446403532v^12 + 1294759346731135721477211v^11 - 214858073837065801008222v^10 + 3269664554840972102531920v^9 - 7438169569959077012149098v^8 + 8390817812141268001946255v^7 + 1792234257086736267983908v^6 + 5113382364025393075155144v^5 - 9613984781208483818163072v^4 + 10546683668476806606180740v^3 + 3024315821044468732151856v^2 + 1426834826857183096866224*v + 187535560740338965925760) / 40056294475296841502400
$\beta_{16}$	$=$	$ ( 29\!\cdots\!77 \nu^{19} + \cdots - 15\!\cdots\!20 ) / 20\!\cdots\!00 $ (290259446075225659277v^19 - 2886061666610961290353v^18 + 14341820874486776209355v^17 - 36846737431411282730387v^16 + 56466118360853201605044v^15 - 81756942591237539926326v^14 + 359700707015266365602350v^13 - 959048628385169388860054v^12 + 1343379714121253603226217v^11 - 587096288219908666699969v^10 + 2537400117408653798235795v^9 - 6691462529303206157515771v^8 + 8935771882263122044464350v^7 - 1335977561472638164413944v^6 + 3098458850940207759176228v^5 - 8712756249521202745081484v^4 + 11639239106517292092405960v^3 - 893679830007381216470608v^2 - 477753952081820457751392*v - 153675940156486934324320) / 20028147237648420751200
$\beta_{17}$	$=$	$ ( - 68\!\cdots\!65 \nu^{19} + \cdots + 16\!\cdots\!36 ) / 40\!\cdots\!00 $ (-688266483162241317565v^19 + 6942045981266845592544v^18 - 35040287092946149677546v^17 + 92789488071962462340980v^16 - 149211575383655181942919v^15 + 220557177962229524794068v^14 - 892221225883253308087372v^13 + 2411161212833682425335120v^12 - 3576017719274414767970903v^11 + 2041605281031320135883704v^10 - 6496795076555547015691338v^9 + 16835104249305678074105540v^8 - 23831232079266034573890217v^7 + 7564278960458373862662180v^6 - 9702775901620181220863888v^5 + 21929706800830306239701976v^4 - 30692964776640207137752988v^3 + 7879976155881998299424720v^2 - 1898870896135485528079696*v + 164881461394568807327936) / 40056294475296841502400
$\beta_{18}$	$=$	$ ( - 17\!\cdots\!91 \nu^{19} + \cdots - 15\!\cdots\!20 ) / 84\!\cdots\!00 $ (-17893668084247384285291v^19 + 175040577193723657109304v^18 - 854584776979819877830460v^17 + 2120727827005605871130096v^16 - 3076395856194658042601607v^15 + 4394081584407841748813468v^14 - 21253965005259651619276840v^13 + 55368358665049651859959432v^12 - 72247375411495167074977821v^11 + 20592239243118307377642432v^10 - 148346508238254393104379020v^9 + 387206289088239249538981968v^8 - 475607715316133113492946165v^7 - 21817013421169560041622148v^6 - 166719112508325737731417984v^5 + 509830885791798957299179192v^4 - 616142697057956384465694140v^3 - 78738603886583370249392336v^2 + 37812375211108687145287216*v - 1564626894811594447694720) / 841182183981233671550400
$\beta_{19}$	$=$	$ ( 50\!\cdots\!12 \nu^{19} + \cdots + 94\!\cdots\!56 ) / 20\!\cdots\!00 $ (509023171278126786212v^19 - 5251848796375773452804v^18 + 27041826681911977259449v^17 - 73987683387335586357372v^16 + 122961263649113872218765v^15 - 179674330475796785421468v^14 + 682944305123550511707138v^13 - 1915377509349824295920624v^12 + 2977903228515612032534094v^11 - 1892325455977617229214420v^10 + 4804003694163665734181497v^9 - 13366861485676896109419756v^8 + 19994991854351184331934613v^7 - 8077832194964356133028524v^6 + 6189434643378605783491340v^5 - 17200869297138230285786168v^4 + 25945962997929648358390012v^3 - 8947132240399131926866608v^2 + 51438530949242276728592*v + 94260572163390623827456) / 20028147237648420751200

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

$\nu$	$=$	$ ( 2 \beta_{18} - 2 \beta_{17} - \beta_{16} - 2 \beta_{15} - \beta_{14} - \beta_{12} - \beta_{11} + \cdots + 13 ) / 24 $ (2b18 - 2b17 - b16 - 2b15 - b14 - b12 - b11 + 2b10 - 3b9 + b8 - 3b6 - b5 - 3b4 + 2b3 + 6b2 - 6b1 + 13) / 24
$\nu^{2}$	$=$	$ ( - 2 \beta_{17} - \beta_{16} - 2 \beta_{15} - \beta_{14} + 3 \beta_{12} - 3 \beta_{11} + \beta_{9} + \cdots + 1 ) / 12 $ (-2b17 - b16 - 2b15 - b14 + 3b12 - 3b11 + b9 - 3b8 - 3b6 + b5 - 11b4 + 2b3 + 6*b2 + 1) / 12
$\nu^{3}$	$=$	$ ( - \beta_{19} - 7 \beta_{18} - 6 \beta_{17} - 7 \beta_{16} - 8 \beta_{15} - 6 \beta_{14} - \beta_{13} + \cdots - 71 ) / 12 $ (-b19 - 7b18 - 6b17 - 7b16 - 8b15 - 6b14 - b13 + 8b12 + 3b11 - 7b10 + 14b9 - 12b8 - 14b6 + 5b5 - 15b4 - 4b3 + 22b2 + 25b1 - 71) / 12
$\nu^{4}$	$=$	$ ( - 6 \beta_{18} - 5 \beta_{16} - 5 \beta_{14} + 3 \beta_{12} + 7 \beta_{11} - 6 \beta_{10} + 11 \beta_{9} + \cdots - 98 ) / 3 $ (-6b18 - 5b16 - 5b14 + 3b12 + 7b11 - 6b10 + 11b9 - 7b8 - 8b6 + 3b5 + 22*b1 - 98) / 3
$\nu^{5}$	$=$	$ ( 13 \beta_{19} - 61 \beta_{18} + 50 \beta_{17} + 80 \beta_{15} - 15 \beta_{14} + 13 \beta_{13} + \cdots - 690 ) / 12 $ (13b19 - 61b18 + 50b17 + 80b15 - 15b14 + 13b13 - 35b12 + 152b11 - 65b10 + 113b9 - 17b8 - 14b7 + 29b6 + 18b5 + 158b4 + 128b3 - 198b2 + 251b1 - 690) / 12
$\nu^{6}$	$=$	$ ( 29 \beta_{19} + 102 \beta_{17} + 86 \beta_{16} + 172 \beta_{15} + 51 \beta_{14} + 29 \beta_{13} + \cdots + 166 ) / 6 $ (29b19 + 102b17 + 86b16 + 172b15 + 51b14 + 29b13 - 183b12 + 183b11 - 35b9 + 183b8 + 205b6 - 23b5 + 364b4 + 332b3 - 410*b2 + 166) / 6
$\nu^{7}$	$=$	$ ( 149 \beta_{19} + 587 \beta_{18} + 474 \beta_{17} + 910 \beta_{16} + 868 \beta_{15} + 671 \beta_{14} + \cdots + 9156 ) / 12 $ (149b19 + 587b18 + 474b17 + 910b16 + 868b15 + 671b14 + 149b13 - 1171b12 - 108b11 + 683b10 - 1511b9 + 1767b8 + 318b7 + 2007b6 - 374b5 + 1400b4 + 2144b3 - 1970b2 - 2693*b1 + 9156) / 12
$\nu^{8}$	$=$	$ ( - 3 \beta_{19} + 644 \beta_{18} + 591 \beta_{16} + 504 \beta_{14} + 3 \beta_{13} - 390 \beta_{12} + \cdots + 9564 ) / 3 $ (-3b19 + 644b18 + 591b16 + 504b14 + 3b13 - 390b12 - 1078b11 + 780b10 - 1534b9 + 1078b8 + 440b7 + 1341b6 - 322b5 - 220b4 - 3068*b1 + 9564) / 3
$\nu^{9}$	$=$	$ ( - 1711 \beta_{19} + 5935 \beta_{18} - 4674 \beta_{17} + 734 \beta_{16} - 9632 \beta_{15} + \cdots + 71440 ) / 12 $ (-1711b19 + 5935b18 - 4674b17 + 734b16 - 9632b15 + 2151b14 - 1603b13 + 5827b12 - 20250b11 + 7579b10 - 12053b9 + 585b8 + 5070b7 + 2647b6 - 2600b5 - 18936b4 - 30592b3 + 20710b2 - 29305*b1 + 71440) / 12
$\nu^{10}$	$=$	$ ( - 3857 \beta_{19} - 10370 \beta_{17} - 11280 \beta_{16} - 22560 \beta_{15} - 5185 \beta_{14} + \cdots - 39096 ) / 6 $ (-3857b19 - 10370b17 - 11280b16 - 22560b15 - 5185b14 - 3857b13 + 23133b12 - 24093b11 + 7397b9 - 24093b8 - 23851b6 + 329b5 - 38110b4 - 78192b3 + 47702*b2 - 39096) / 6
$\nu^{11}$	$=$	$ ( - 16531 \beta_{19} - 61629 \beta_{18} - 46546 \beta_{17} - 118166 \beta_{16} - 107500 \beta_{15} + \cdots - 1138536 ) / 12 $ (-16531b19 - 61629b18 - 46546b17 - 118166b16 - 107500b15 - 69077b14 - 19831b13 + 152225b12 + 120b11 - 86221b10 + 199573b9 - 231549b8 - 70250b7 - 268833b6 + 30270b5 - 139988b4 - 403872b3 + 224562b2 + 319187*b1 - 1138536) / 12
$\nu^{12}$	$=$	$ ( 2712 \beta_{19} - 70506 \beta_{18} - 77783 \beta_{16} - 51743 \beta_{14} - 2712 \beta_{13} + \cdots - 1093748 ) / 3 $ (2712b19 - 70506b18 - 77783b16 - 51743b14 - 2712b13 + 51483b12 + 136203b11 - 102966b10 + 185703b9 - 136203b8 - 89508b7 - 190616b6 + 35253b5 + 44754b4 + 371406*b1 - 1093748) / 3
$\nu^{13}$	$=$	$ ( 230865 \beta_{19} - 650297 \beta_{18} + 464186 \beta_{17} - 145626 \beta_{16} + 1201616 \beta_{15} + \cdots - 7631960 ) / 12 $ (230865b19 - 650297b18 + 464186b17 - 145626b16 + 1201616b15 - 233697b14 + 165345b13 - 731669b12 + 2646646b11 - 991517b10 + 1158019b9 + 70289b8 - 908466b7 - 597997b6 + 373112b5 + 2345296b4 + 5102752b3 - 2478222b2 + 3473727*b1 - 7631960) / 12
$\nu^{14}$	$=$	$ ( 461421 \beta_{19} + 1035746 \beta_{17} + 1419460 \beta_{16} + 2838920 \beta_{15} + 517873 \beta_{14} + \cdots + 6335792 ) / 6 $ (461421b19 + 1035746b17 + 1419460b16 + 2838920b15 + 517873b14 + 461421b13 - 2909385b12 + 3292137b11 - 1528889b9 + 3292137b8 + 2919999b6 + 156835b5 + 4412686b4 + 12671584b3 - 5839998*b2 + 6335792) / 6
$\nu^{15}$	$=$	$ ( 1614413 \beta_{19} + 6935267 \beta_{18} + 4624962 \beta_{17} + 15358378 \beta_{16} + 13441444 \beta_{15} + \cdots + 142752744 ) / 12 $ (1614413b19 + 6935267b18 + 4624962b17 + 15358378b16 + 13441444b15 + 7045979b14 + 2689493b13 - 19484455b12 + 1578960b11 + 11447843b10 - 26859155b9 + 30242427b8 + 11307606b7 + 35089131b6 - 2540714b5 + 14942636b4 + 62758400b3 - 27628682b2 - 37805357*b1 + 142752744) / 12
$\nu^{16}$	$=$	$ ( - 738045 \beta_{19} + 8029512 \beta_{18} + 10386267 \beta_{16} + 5336418 \beta_{14} + 738045 \beta_{13} + \cdots + 130500468 ) / 3 $ (-738045b19 + 8029512b18 + 10386267b16 + 5336418b14 + 738045b13 - 6876732b12 - 16900988b11 + 13753464b10 - 22048972b9 + 16900988b8 + 13968336b7 + 25529907b6 - 4014756b5 - 6984168b4 - 44097944*b1 + 130500468) / 3
$\nu^{17}$	$=$	$ ( - 31292783 \beta_{19} + 74533343 \beta_{18} - 46008394 \beta_{17} + 24561382 \beta_{16} + \cdots + 844714024 ) / 12 $ (-31292783b19 + 74533343b18 - 46008394b17 + 24561382b16 - 150474304b15 + 25115935b14 - 15429947b13 + 87970555b12 - 345418210b11 + 132276523b10 - 100840685b9 - 26250767b8 + 137529694b7 + 89659575b6 - 51388240b5 - 294520352b4 - 758378368b3 + 309868158b2 - 411877513*b1 + 844714024) / 12
$\nu^{18}$	$=$	$ ( - 54437221 \beta_{19} - 102534762 \beta_{17} - 178043536 \beta_{16} - 356087072 \beta_{15} + \cdots - 927457088 ) / 6 $ (-54437221b19 - 102534762b17 - 178043536b16 - 356087072b15 - 51267381b14 - 54437221b13 + 364712457b12 - 448645641b11 + 266984369b9 - 448645641b8 - 367386719b6 - 37424251b5 - 529550398b4 - 1854914176b3 + 734773438*b2 - 927457088) / 6
$\nu^{19}$	$=$	$ ( - 144330691 \beta_{19} - 805772845 \beta_{18} - 456832554 \beta_{17} - 1994153350 \beta_{16} + \cdots - 18028357304 ) / 12 $ (-144330691b19 - 805772845b18 - 456832554b17 - 1994153350b16 - 1686072140b15 - 717994293b14 - 363346351b13 + 2489751185b12 - 385093032b11 - 1527335389b10 + 3594893477b9 - 3943136589b8 - 1647767466b7 - 4544260265b6 + 207990398b5 - 1662577476b4 - 9053029600b3 + 3487821370b2 + 4495390627*b1 - 18028357304) / 12

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$1$	$-1 - \beta_{3}$	$-1$

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} $

127.1

−0.804757	+	0.804757i
1.80476	+	1.80476i
2.24258	+	2.24258i
−1.24258	+	1.24258i
1.11023	−	1.11023i
−0.110232	−	0.110232i
0.189070	−	0.189070i
0.810930	+	0.810930i
−1.38201	−	1.38201i
2.38201	−	2.38201i
−0.804757	−	0.804757i
1.80476	−	1.80476i
2.24258	−	2.24258i
−1.24258	−	1.24258i
1.11023	+	1.11023i
−0.110232	+	0.110232i
0.189070	+	0.189070i
0.810930	−	0.810930i
−1.38201	+	1.38201i
2.38201	+	2.38201i

−4.49547

−

7.78638i

−8.46628

−

4.88801i

127.2

−4.49547

−

7.78638i

8.46628

4.88801i

127.3

−1.18939

−

2.06008i

−6.27165

−

3.62094i

127.4

−1.18939

−

2.06008i

6.27165

3.62094i

127.5

−0.704345

−

1.21996i

−0.768192

−

0.443516i

127.6

−0.704345

−

1.21996i

0.768192

0.443516i

127.7

0.966952

1.67481i

−8.96285

−

5.17470i

127.8

0.966952

1.67481i

8.96285

5.17470i

127.9

1.92225

3.32943i

−11.0596

−

6.38529i

127.10

1.92225

3.32943i

11.0596

6.38529i

415.1

−4.49547

7.78638i

−8.46628

4.88801i

415.2

−4.49547

7.78638i

8.46628

−

4.88801i

415.3

−1.18939

2.06008i

−6.27165

3.62094i

415.4

−1.18939

2.06008i

6.27165

−

3.62094i

415.5

−0.704345

1.21996i

−0.768192

0.443516i

415.6

−0.704345

1.21996i

0.768192

−

0.443516i

415.7

0.966952

−

1.67481i

−8.96285

5.17470i

415.8

0.966952

−

1.67481i

8.96285

−

5.17470i

415.9

1.92225

−

3.32943i

−11.0596

6.38529i

415.10

1.92225

−

3.32943i

11.0596

−

6.38529i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	864.3.o.b		20
3.b	odd	2	1		288.3.o.b	✓	20
4.b	odd	2	1	inner	864.3.o.b		20
8.b	even	2	1		1728.3.o.h		20
8.d	odd	2	1		1728.3.o.h		20
9.c	even	3	1	inner	864.3.o.b		20
9.c	even	3	1		2592.3.g.h		10
9.d	odd	6	1		288.3.o.b	✓	20
9.d	odd	6	1		2592.3.g.g		10
12.b	even	2	1		288.3.o.b	✓	20
24.f	even	2	1		576.3.o.h		20
24.h	odd	2	1		576.3.o.h		20
36.f	odd	6	1	inner	864.3.o.b		20
36.f	odd	6	1		2592.3.g.h		10
36.h	even	6	1		288.3.o.b	✓	20
36.h	even	6	1		2592.3.g.g		10
72.j	odd	6	1		576.3.o.h		20
72.l	even	6	1		576.3.o.h		20
72.n	even	6	1		1728.3.o.h		20
72.p	odd	6	1		1728.3.o.h		20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
288.3.o.b	✓	20	3.b	odd	2	1
288.3.o.b	✓	20	9.d	odd	6	1
288.3.o.b	✓	20	12.b	even	2	1
288.3.o.b	✓	20	36.h	even	6	1
576.3.o.h		20	24.f	even	2	1
576.3.o.h		20	24.h	odd	2	1
576.3.o.h		20	72.j	odd	6	1
576.3.o.h		20	72.l	even	6	1
864.3.o.b		20	1.a	even	1	1	trivial
864.3.o.b		20	4.b	odd	2	1	inner
864.3.o.b		20	9.c	even	3	1	inner
864.3.o.b		20	36.f	odd	6	1	inner
1728.3.o.h		20	8.b	even	2	1
1728.3.o.h		20	8.d	odd	2	1
1728.3.o.h		20	72.n	even	6	1
1728.3.o.h		20	72.p	odd	6	1
2592.3.g.g		10	9.d	odd	6	1
2592.3.g.g		10	36.h	even	6	1
2592.3.g.h		10	9.c	even	3	1
2592.3.g.h		10	36.f	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{10} + 7 T_{5}^{9} + 78 T_{5}^{8} - 21 T_{5}^{7} + 1374 T_{5}^{6} + 1407 T_{5}^{5} + 9729 T_{5}^{4} + \cdots + 50176 $ T5^10 + 7*T5^9 + 78*T5^8 - 21*T5^7 + 1374*T5^6 + 1407*T5^5 + 9729*T5^4 + 3528*T5^3 + 31200*T5^2 + 23296*T5 + 50176 acting on $S_{3}^{\mathrm{new}}(864, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} $ T^20
$3$	$ T^{20} $ T^20
$5$	$ (T^{10} + 7 T^{9} + \cdots + 50176)^{2} $ (T^10 + 7T^9 + 78T^8 - 21T^7 + 1374T^6 + 1407T^5 + 9729T^4 + 3528T^3 + 31200T^2 + 23296*T + 50176)^2
$7$	$ T^{20} + \cdots + 47\!\cdots\!00 $ T^20 - 419T^18 + 112758T^16 - 18336423T^14 + 2182164342T^12 - 174622784379T^10 + 10190026950057T^8 - 352968721999848T^6 + 7943406704940336T^4 - 6245177758160000*T^2 + 4745832100000000
$11$	$ T^{20} + \cdots + 43\!\cdots\!01 $ T^20 - 1049T^18 + 709275T^16 - 283447578T^14 + 82293754149T^12 - 16152091161699T^10 + 2327282843130405T^8 - 211853959519678698T^6 + 13079558068293507627T^4 - 274950858055666627625*T^2 + 4376133992795504068801
$13$	$ (T^{10} - 13 T^{9} + \cdots + 583696)^{2} $ (T^10 - 13T^9 + 390T^8 - 2625T^7 + 87426T^6 - 682341T^5 + 6917661T^4 - 8166840T^3 + 10211340T^2 + 2175872*T + 583696)^2
$17$	$ (T^{5} + 18 T^{4} + \cdots + 50832)^{4} $ (T^5 + 18T^4 - 159T^3 - 3288T^2 - 2760T + 50832)^4
$19$	$ (T^{10} + \cdots + 1691581169664)^{2} $ (T^10 + 2118T^8 + 1487049T^6 + 440018496T^4 + 52326033408T^2 + 1691581169664)^2
$23$	$ T^{20} + \cdots + 80\!\cdots\!56 $ T^20 - 2387T^18 + 4142790T^16 - 3318816615T^14 + 1941442550310T^12 - 269444100899115T^10 + 26612277730166217T^8 - 1208267977847932104T^6 + 39682580157303822000T^4 - 678487612291999065728*T^2 + 8034964736354453913856
$29$	$ (T^{10} + \cdots + 104284535520400)^{2} $ (T^10 - 67T^9 + 3942T^8 - 140751T^7 + 5181522T^6 - 148232187T^5 + 4156517373T^4 - 83777637816T^3 + 1414191079836T^2 - 14244650054080*T + 104284535520400)^2
$31$	$ T^{20} + \cdots + 15\!\cdots\!56 $ T^20 - 3755T^18 + 10130958T^16 - 12473311575T^14 + 11093172380958T^12 - 4096321891643259T^10 + 1087232410691317953T^8 - 107978579826050579424T^6 + 7892951713494543311616T^4 - 120856899062080181755904*T^2 + 1552673443527585662304256
$37$	$ (T^{5} - 24 T^{4} + \cdots - 14835456)^{4} $ (T^5 - 24T^4 - 2760T^3 + 76944T^2 + 610176T - 14835456)^4
$41$	$ (T^{10} + \cdots + 13190269049281)^{2} $ (T^10 - 13T^9 + 4311T^8 + 86298T^7 + 15454413T^6 + 102337389T^5 + 6391016589T^4 - 54276102582T^3 + 2163593148903T^2 - 5414395776733*T + 13190269049281)^2
$43$	$ T^{20} + \cdots + 95\!\cdots\!21 $ T^20 - 9689T^18 + 66155307T^16 - 215559984810T^14 + 505447981315557T^12 - 616952905843068627T^10 + 530714975525994479877T^8 - 144962377443603923108058T^6 + 29175665343108729215946171T^4 - 1883918433892802591849690057*T^2 + 95024971249211223373201069921
$47$	$ T^{20} + \cdots + 24\!\cdots\!36 $ T^20 - 9299T^18 + 55455222T^16 - 199403910519T^14 + 522502092157686T^12 - 908851499908897035T^10 + 1140289818219691982121T^8 - 834200634628441440968904T^6 + 436836217986744159428463792T^4 - 126499819494723327284461979264*T^2 + 24372613813792396710093710254336
$53$	$ (T^{5} - 48 T^{4} + \cdots - 2275179264)^{4} $ (T^5 - 48T^4 - 13080T^3 + 682128T^2 + 41355264T - 2275179264)^4
$59$	$ T^{20} + \cdots + 13\!\cdots\!25 $ T^20 - 11561T^18 + 86037387T^16 - 382728166026T^14 + 1239501119934117T^12 - 2662506590605817091T^10 + 4137260969983152880101T^8 - 3779065884199773374101050T^6 + 2412154849738275531771826875T^4 - 670684935779908592587920265625*T^2 + 133081290077299601763332062890625
$61$	$ (T^{10} + \cdots + 21\!\cdots\!00)^{2} $ (T^10 - 193T^9 + 27630T^8 - 2007453T^7 + 118484238T^6 - 3523353921T^5 + 143544079641T^4 - 3680173640064T^3 + 118640020665024T^2 - 1669675156111360*T + 21493061210214400)^2
$67$	$ T^{20} + \cdots + 37\!\cdots\!81 $ T^20 - 14297T^18 + 133541115T^16 - 698029026426T^14 + 2608530452708517T^12 - 6634603550914462275T^10 + 12574427096419014458085T^8 - 16665081480876951574757130T^6 + 16183275746776178378653393035T^4 - 9775982426440271570694615391913*T^2 + 3707625832742136744964684837716481
$71$	$ (T^{10} + \cdots + 10\!\cdots\!24)^{2} $ (T^10 + 27692T^8 + 261521776T^6 + 1014130874944T^4 + 1695361974622208T^2 + 1009811083007770624)^2
$73$	$ (T^{5} + 42 T^{4} + \cdots + 2701063440)^{4} $ (T^5 + 42T^4 - 19191T^3 - 808584T^2 + 70163688T + 2701063440)^4
$79$	$ T^{20} + \cdots + 10\!\cdots\!00 $ T^20 - 36587T^18 + 962730126T^16 - 10835992797207T^14 + 85825304463936414T^12 - 394657729918113310587T^10 + 1295987391929568471754881T^8 - 2319417393957644755428046752T^6 + 2983980832860714866018493831936T^4 - 2127871242974230853727551552921600*T^2 + 1016547461917106519338341236776960000
$83$	$ T^{20} + \cdots + 22\!\cdots\!76 $ T^20 - 13427T^18 + 123519870T^16 - 595004524863T^14 + 2063391178104270T^12 - 3769208391421223235T^10 + 4850216882526099578673T^8 - 2508043852353850748051808T^6 + 932630124364964804570155776T^4 - 174662356719838938413592633344*T^2 + 22893409779909552000960917733376
$89$	$ (T^{5} - 60 T^{4} + \cdots + 1531650240)^{4} $ (T^5 - 60T^4 - 15924T^3 + 204336T^2 + 65655264T + 1531650240)^4
$97$	$ (T^{10} + \cdots + 52\!\cdots\!25)^{2} $ (T^10 - 187T^9 + 52263T^8 - 5088810T^7 + 1209933165T^6 - 120860026245T^5 + 15158513925597T^4 - 526991127850074T^3 + 14581332811335591T^2 - 96281397637303915*T + 526670483608042225)^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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	864.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{6} - \beta_{3} - \beta_{2} - 1) q^{5} + ( - \beta_{12} + \beta_{10}) q^{7}+O(q^{10}) \) q + (b6 - b3 - b2 - 1) * q^5 + (-b12 + b10) * q^7	\( q + (\beta_{6} - \beta_{3} - \beta_{2} - 1) q^{5} + ( - \beta_{12} + \beta_{10}) q^{7} + (\beta_{18} + \beta_{8}) q^{11} + (\beta_{16} + \beta_{15} + 3 \beta_{3} + 3) q^{13} + (\beta_{14} + \beta_{6} - 3) q^{17} + (\beta_{12} + \beta_{11} + \cdots - 2 \beta_{4}) q^{19}+ \cdots + (5 \beta_{19} + \beta_{17} + \cdots - 12 \beta_{2}) q^{97}+O(q^{100}) \) q + (b6 - b3 - b2 - 1) * q^5 + (-b12 + b10) * q^7 + (b18 + b8) * q^11 + (b16 + b15 + 3b3 + 3) q^13 + (b14 + b6 - 3) * q^17 + (b12 + b11 + 5b9 + b8 - 2b4) * q^19 + (-b18 + 2b11 - b10 - b7 + b5 + b4 + 5b1) * q^23 + (b19 + b15 - 5b3 + 5b2) * q^25 + (-b19 + 2b17 + 2b14 - 14b3) q^29 + (5b11 - 2b10 + 4b7 - 4b4) * q^31 + (5b11 - 11b9 + 5b8 - b5 + 6b4) * q^35 + (-2b19 - b14 + 2b13 + 5) * q^37 + (b17 - b16 - b15 + b13 - 8b6 - b3 + 8b2 - 1) * q^41 + (-2b18 + 11b9 - 3b8 - 4b7 + 11b1) q^43 + (-b12 + b10 + 8b9 + 6b8 - 3b7 + 8b1) * q^47 + (-2b17 + 2b16 + 2b15 + 3b13 + 6b6 + 39b3 - 6b2 + 39) q^49 + (-5b19 + b16 + b14 + 5b13 + 2b6 + 12) q^53 + (2b12 + 7b11 - 6b9 + 7b8 - 2b5 + 11b4) * q^55 + (b18 + 6b11 - b10 + 6b7 - b5 - 6b4 + 10b1) * q^59 + (-b19 + 4b17 + 4b14 - 38b3 - 4b2) * q^61 + (-b19 + 2b15 - 6b3 - 10b2) q^65 + (-b18 + 11b11 + 2b10 + 2b7 + b5 - 2b4 + 10b1) q^67 + (5b12 + 13b11 + 2b9 + 13b8 - 9b4) q^71 + (-b19 - 6b16 + 8b14 + b13 - 9) * q^73 + (4b17 + 13b16 + 13b15 + b13 - 7b6 - 4b3 + 7b2 - 4) * q^77 + (5b18 + b12 - b10 + 9b9 - 4b8 - 7b7 + 9b1) q^79 + (-2b18 - b12 + b10 - 4b9 + 10b8 - 2b7 - 4b1) q^83 + (2b17 + 2b16 + 2b15 + b13 - 7b6 + 17b3 + 7b2 + 17) * q^85 + (2b19 + 5b16 - 4b14 - 2b13 + 13b6 + 18) q^89 + (-6b12 + 15b11 + 2b9 + 15b8 + 6b5 + 8b4) * q^91 + (-2b11 + 6b10 - 3b7 + 3b4) * q^95 + (5b19 + b17 + 7b15 + b14 - 29b3 - 12b2) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 14 q^{5}+O(q^{10}) \) 20 * q - 14 * q^5	\( 20 q - 14 q^{5} + 26 q^{13} - 72 q^{17} + 36 q^{25} + 134 q^{29} + 96 q^{37} + 26 q^{41} + 348 q^{49} + 192 q^{53} + 386 q^{61} + 106 q^{65} - 168 q^{73} - 58 q^{77} + 192 q^{85} + 240 q^{89} + 374 q^{97}+O(q^{100}) \) 20 * q - 14 * q^5 + 26 * q^13 - 72 * q^17 + 36 * q^25 + 134 * q^29 + 96 * q^37 + 26 * q^41 + 348 * q^49 + 192 * q^53 + 386 * q^61 + 106 * q^65 - 168 * q^73 - 58 * q^77 + 192 * q^85 + 240 * q^89 + 374 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	864.3.o.b

Level	$864$
Weight	$3$
Character orbit	864.o
Analytic conductor	$23.542$
Analytic rank	$0$
Dimension	$20$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(23.5422948407\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - 10 x^{19} + 50 x^{18} - 130 x^{17} + 203 x^{16} - 296 x^{15} + 1260 x^{14} - 3380 x^{13} + \cdots + 64 \) x^20 - 10x^19 + 50x^18 - 130x^17 + 203x^16 - 296x^15 + 1260x^14 - 3380x^13 + 4843x^12 - 2370x^11 + 8978x^10 - 23570x^9 + 32213x^8 - 6940x^7 + 11856x^6 - 30488x^5 + 41556x^4 - 6096x^3 + 288x^2 + 192*x + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{26}\cdot 3^{8} \)
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - 19\!\cdots\!63 \nu^{19} + \cdots + 15\!\cdots\!96 ) / 65\!\cdots\!00 \) (-195688856698417563v^19 + 5433733490589803696v^18 - 41314325063191276826v^17 + 169263940036751944808v^16 - 352392253945037747325v^15 + 451043328586954776652v^14 - 878920211140236752452v^13 + 4454050253601822425736v^12 - 9159518528562322945241v^11 + 9028392773138913132440v^10 - 824928332195623447498v^9 + 36207057639011453922824v^8 - 61014989221804550515307v^7 + 56425964218804631157116v^6 + 36063155078538884263800v^5 + 70285547724458583369672v^4 - 69249007281323937609988v^3 + 75012466713507873161392v^2 + 63268204836087864567632*v + 15797130996705829506496) / 6520792123885532337600
\(\beta_{2}\)	\(=\)	\( ( - 17\!\cdots\!05 \nu^{19} + \cdots + 13\!\cdots\!08 ) / 40\!\cdots\!00 \) (-1795490609055194105v^19 + 56627636428100804982v^18 - 469525310244152399198v^17 + 2097333011770955542230v^16 - 5048825445209524976127v^15 + 7524421717139295497544v^14 - 12504467498891448049396v^13 + 53111842882354678842260v^12 - 130756840223930590369219v^11 + 167753587714389198092862v^10 - 78117160539997161935534v^9 + 383561041230390891628150v^8 - 910879771724906741914921v^7 + 1065734473271497126385780v^6 - 76810932136050561526544v^5 + 538446408407738562978048v^4 - 1193655896880592900466684v^3 + 1210777365077322585707440v^2 + 167131284654106563875632*v + 13757065250618537601408) / 40056294475296841502400
\(\beta_{3}\)	\(=\)	\( ( - 22981222561 \nu^{19} + 228756996538 \nu^{18} - 1139644587846 \nu^{17} + \cdots - 7487382052224 ) / 11835144571200 \) (-22981222561v^19 + 228756996538v^18 - 1139644587846v^17 + 2946486450646v^16 - 4587224505751v^15 + 6744453837936v^14 - 28881860254372v^13 + 76676420061652v^12 - 109181344833659v^11 + 53488917592626v^10 - 209544376738198v^9 + 534526288281398v^8 - 724627861422737v^7 + 155571870617652v^6 - 301910307851792v^5 + 693191571425888v^4 - 928477369658268v^3 + 135807725605424v^2 - 40685199696080*v - 7487382052224) / 11835144571200
\(\beta_{4}\)	\(=\)	\( ( 53\!\cdots\!38 \nu^{19} + \cdots + 72\!\cdots\!88 ) / 21\!\cdots\!00 \) (532645578137067458738v^19 - 5810711965331017368735v^18 + 31437696482743951146667v^17 - 93092185585167064427293v^16 + 169279535674757392238829v^15 - 251376722934263054849140v^14 + 807388718447594883807074v^13 - 2400199388122176081473666v^12 + 4171409842037725565063604v^11 - 3490602429011931396284259v^10 + 5762450317693294322953051v^9 - 16828177539691363188979989v^8 + 28254985796269972921885369v^7 - 18541462363624960137002626v^6 + 8562120764952124416437048v^5 - 21813912786320348832176348v^4 + 36523999288588418838359236v^3 - 23042667126345395296413752v^2 + 1671161387639704448550064*v + 72981831030879062833888) / 210295545995308417887600
\(\beta_{5}\)	\(=\)	\( ( 11\!\cdots\!80 \nu^{19} + \cdots + 30\!\cdots\!16 ) / 42\!\cdots\!00 \) (1164232051635459557480v^19 - 14941504594144704498381v^18 + 90953005074420034453339v^17 - 313916547816490229526835v^16 + 653653783580349539805801v^15 - 985749595547911796459242v^14 + 2399525604494185981681238v^13 - 8023051279785455738627990v^12 + 16484495891176518520571802v^11 - 18002530007242247328269661v^10 + 17253614642062115081881627v^9 - 56546262775619100196147395v^8 + 112664471209065157179035353v^7 - 109337028451827179946623740v^6 + 29641259283958177677684212v^5 - 73278251847599655430038884v^4 + 142248157526201449350833452v^3 - 137008946435455938729031520v^2 + 6670925547889210664662864*v + 309273404785466225178016) / 420591091990616835775200
\(\beta_{6}\)	\(=\)	\( ( - 32\!\cdots\!73 \nu^{19} + \cdots + 60\!\cdots\!56 ) / 10\!\cdots\!00 \) (-32656813285611366073v^19 + 324541966728653079726v^18 - 1612018079376409678866v^17 + 4138594232998671598958v^16 - 6340061195582629930205v^15 + 9198375638815291144602v^14 - 40502083130190229664032v^13 + 107829059781428226843316v^12 - 150855676201486806521991v^11 + 66113731117529908725370v^10 - 287286356177182701782858v^9 + 754735036124855679193374v^8 - 1005150610204735594796967v^7 + 151000262862936296148326v^6 - 361196033349336854633460v^5 + 999573010827362168218352v^4 - 1322881252831615024523868v^3 + 101507943883845028932792v^2 + 54280675166281697420592*v + 6046693251517943643856) / 10014073618824210375600
\(\beta_{7}\)	\(=\)	\( ( - 11\!\cdots\!21 \nu^{19} + \cdots - 24\!\cdots\!56 ) / 21\!\cdots\!00 \) (-1192614593540774190821v^19 + 11640884888843053346130v^18 - 56693293335468093987124v^17 + 139971629541105777941416v^16 - 201196264831046926382583v^15 + 286082421621385489465960v^14 - 1405743379736432241410528v^13 + 3652364711562108368497112v^12 - 4713009705437592553361703v^11 + 1206947031756011951464818v^10 - 9743992636129040874316732v^9 + 25464569423996837497639968v^8 - 30908738863183009520868913v^7 - 2592629176835489450750348v^6 - 10370694828213879887280896v^5 + 32915177148142136293264256v^4 - 39385016381495537794253932v^3 - 7090100907295834844140576v^2 + 3203301734554821570320912*v - 249812570141658952032256) / 210295545995308417887600
\(\beta_{8}\)	\(=\)	\( ( - 16\!\cdots\!45 \nu^{19} + \cdots + 12\!\cdots\!48 ) / 28\!\cdots\!00 \) (-1634459996038199826645v^19 + 16365683869957575382947v^18 - 81944111889313701865213v^17 + 213674593182883644563195v^16 - 335459139947101760671862v^15 + 491549187321625595061494v^14 - 2073538850677886750039546v^13 + 5562556312715673917144910v^12 - 8009561447817819213479589v^11 + 4056865587435527139667507v^10 - 14923474023943205309648549v^9 + 38937630834173102455472715v^8 - 53258106448117702484415856v^7 + 12465524512850807915820740v^6 - 21010525074543342234179444v^5 + 51485030902460675638819868v^4 - 68187554817961917926301744v^3 + 10881836903238666190879840v^2 - 3319167418512870504831488*v + 1253799601400382716877248) / 280394061327077890516800
\(\beta_{9}\)	\(=\)	\( ( - 19\!\cdots\!92 \nu^{19} + \cdots - 11\!\cdots\!92 ) / 32\!\cdots\!00 \) (-19552445072400701592v^19 + 191729567961353089965v^18 - 940700155585105187603v^17 + 2361815835891087576487v^16 - 3522727057290755170111v^15 + 5131970442379467484210v^14 - 23688200689222191901366v^13 + 61584387662491838297694v^12 - 83128047571001753198886v^11 + 31008664099825332277181v^10 - 170808095130022113372259v^9 + 428823745124372010582951v^8 - 550783558840762010271071v^7 + 34281404642883852738484v^6 - 236446806584115401736532v^5 + 556138808868527100071332v^4 - 715512701857500308581524v^3 - 14696052436321641092032v^2 - 31037922678698891446576*v - 1161771539726213664992) / 3260396061942766168800
\(\beta_{10}\)	\(=\)	\( ( 27\!\cdots\!17 \nu^{19} + \cdots - 97\!\cdots\!72 ) / 42\!\cdots\!00 \) (2708280417948315268117v^19 - 27719752727283886904046v^18 + 141262047855415325159297v^17 - 378952535760362297781167v^16 + 608735442965450967426147v^15 - 873159798878446635019352v^14 + 3515882710787767023758614v^13 - 9823806655490324341184974v^12 + 14646553184643517889149923v^11 - 8005318787685644309221902v^10 + 23898883188482622143979701v^9 - 68837866481636906425469631v^8 + 97386218638952836769088809v^7 - 29263242449821035734661164v^6 + 24528696441037430619008164v^5 - 90050771891766668637001636v^4 + 123674021518160691488028716v^3 - 31666707695948358686146768v^2 - 8275032063572343509402800*v - 974993336474972679148672) / 420591091990616835775200
\(\beta_{11}\)	\(=\)	\( ( - 90\!\cdots\!24 \nu^{19} + \cdots - 75\!\cdots\!32 ) / 14\!\cdots\!00 \) (-903036033673535895324v^19 + 9008561961721572712823v^18 - 44998025303397681599733v^17 + 116900408606535620832034v^16 - 183269230858736190498740v^15 + 269268386505247539349926v^14 - 1140035775685434011400366v^13 + 3037884058262869814256228v^12 - 4370801803599800959650848v^11 + 2199333272721864885343935v^10 - 8252869648400336082274809v^9 + 21112642487120958643816802v^8 - 29132770268358979770214656v^7 + 6638233929795180652088068v^6 - 11851504596412280996255780v^5 + 26882513955339929173977416v^4 - 37853401533091179162903904v^3 + 5727113396985685204584416v^2 - 1506003832492374535199424*v - 750677485991493929106432) / 140197030663538945258400
\(\beta_{12}\)	\(=\)	\( ( - 56\!\cdots\!35 \nu^{19} + \cdots - 20\!\cdots\!36 ) / 84\!\cdots\!00 \) (-5671712517398979118835v^19 + 53861426271653727260271v^18 - 255613223384412538963369v^17 + 600214388188840052024065v^16 - 808013613812302671285606v^15 + 1169744268664422640079422v^14 - 6410612689170486659272298v^13 + 15732519223783332949651850v^12 - 18512038321636378352576547v^11 + 1389878650135166193989391v^10 - 46729301872365400695103417v^9 + 109194348589216401261729585v^8 - 120306295412113361370579988v^7 - 41618170763843081336573900v^6 - 64536186399328990741214372v^5 + 141650188961808761523172724v^4 - 154574399293299629114828992v^3 - 76741333542423302307062560v^2 - 6427699346371399963036864*v - 206820432862051131478336) / 841182183981233671550400
\(\beta_{13}\)	\(=\)	\( ( - 15\!\cdots\!23 \nu^{19} + \cdots - 42\!\cdots\!16 ) / 20\!\cdots\!00 \) (-151762441316149564923v^19 + 1322414461600693380273v^18 - 5646746225793494898214v^17 + 10072850569413056475523v^16 - 5927686684303933043977v^15 + 6459301604343464796846v^14 - 134850630532163548225708v^13 + 268600573319174770425006v^12 - 85866569581400489626265v^11 - 556971364208446314090583v^10 - 930404485624213499608862v^9 + 1815356113179891025588859v^8 - 339333043259353490643263v^7 - 5051185482019541826447544v^6 - 604334775142806375463264v^5 + 2228019792902153266046260v^4 - 264267717940811445713572v^3 - 6933342064977201384507568v^2 + 1127680921587051623649104*v - 42253559183879094274816) / 20028147237648420751200
\(\beta_{14}\)	\(=\)	\( ( - 16\!\cdots\!35 \nu^{19} + \cdots - 22\!\cdots\!56 ) / 20\!\cdots\!00 \) (-161072688040075330335v^19 + 1602567903836264568991v^18 - 7968150454957323804449v^17 + 20489247569031910880665v^16 - 31410411491356583233376v^15 + 45367358138786093032962v^14 - 199428149418766287366458v^13 + 532706649247020179263650v^12 - 747253566001169732229787v^11 + 325379174472930510117311v^10 - 1397587862940570865592057v^9 + 3704502259170664966110385v^8 - 4963865101106795755385198v^7 + 736784482374956320679200v^6 - 1633210854667398484871612v^5 + 4704854607167868270216004v^4 - 6370420394320374687181032v^3 + 489581113650107131655440v^2 + 261620002650659593532256*v - 229654856322692625041056) / 20028147237648420751200
\(\beta_{15}\)	\(=\)	\( ( 38\!\cdots\!01 \nu^{19} + \cdots + 18\!\cdots\!60 ) / 40\!\cdots\!00 \) (381310343728584500101v^19 - 3628508398678781504294v^18 + 17277010573437836741280v^17 - 40929214119471838777386v^16 + 56383297021050425643917v^15 - 83205833341528382461488v^14 + 437626203424899270802840v^13 - 1072371773933196446403532v^12 + 1294759346731135721477211v^11 - 214858073837065801008222v^10 + 3269664554840972102531920v^9 - 7438169569959077012149098v^8 + 8390817812141268001946255v^7 + 1792234257086736267983908v^6 + 5113382364025393075155144v^5 - 9613984781208483818163072v^4 + 10546683668476806606180740v^3 + 3024315821044468732151856v^2 + 1426834826857183096866224*v + 187535560740338965925760) / 40056294475296841502400
\(\beta_{16}\)	\(=\)	\( ( 29\!\cdots\!77 \nu^{19} + \cdots - 15\!\cdots\!20 ) / 20\!\cdots\!00 \) (290259446075225659277v^19 - 2886061666610961290353v^18 + 14341820874486776209355v^17 - 36846737431411282730387v^16 + 56466118360853201605044v^15 - 81756942591237539926326v^14 + 359700707015266365602350v^13 - 959048628385169388860054v^12 + 1343379714121253603226217v^11 - 587096288219908666699969v^10 + 2537400117408653798235795v^9 - 6691462529303206157515771v^8 + 8935771882263122044464350v^7 - 1335977561472638164413944v^6 + 3098458850940207759176228v^5 - 8712756249521202745081484v^4 + 11639239106517292092405960v^3 - 893679830007381216470608v^2 - 477753952081820457751392*v - 153675940156486934324320) / 20028147237648420751200
\(\beta_{17}\)	\(=\)	\( ( - 68\!\cdots\!65 \nu^{19} + \cdots + 16\!\cdots\!36 ) / 40\!\cdots\!00 \) (-688266483162241317565v^19 + 6942045981266845592544v^18 - 35040287092946149677546v^17 + 92789488071962462340980v^16 - 149211575383655181942919v^15 + 220557177962229524794068v^14 - 892221225883253308087372v^13 + 2411161212833682425335120v^12 - 3576017719274414767970903v^11 + 2041605281031320135883704v^10 - 6496795076555547015691338v^9 + 16835104249305678074105540v^8 - 23831232079266034573890217v^7 + 7564278960458373862662180v^6 - 9702775901620181220863888v^5 + 21929706800830306239701976v^4 - 30692964776640207137752988v^3 + 7879976155881998299424720v^2 - 1898870896135485528079696*v + 164881461394568807327936) / 40056294475296841502400
\(\beta_{18}\)	\(=\)	\( ( - 17\!\cdots\!91 \nu^{19} + \cdots - 15\!\cdots\!20 ) / 84\!\cdots\!00 \) (-17893668084247384285291v^19 + 175040577193723657109304v^18 - 854584776979819877830460v^17 + 2120727827005605871130096v^16 - 3076395856194658042601607v^15 + 4394081584407841748813468v^14 - 21253965005259651619276840v^13 + 55368358665049651859959432v^12 - 72247375411495167074977821v^11 + 20592239243118307377642432v^10 - 148346508238254393104379020v^9 + 387206289088239249538981968v^8 - 475607715316133113492946165v^7 - 21817013421169560041622148v^6 - 166719112508325737731417984v^5 + 509830885791798957299179192v^4 - 616142697057956384465694140v^3 - 78738603886583370249392336v^2 + 37812375211108687145287216*v - 1564626894811594447694720) / 841182183981233671550400
\(\beta_{19}\)	\(=\)	\( ( 50\!\cdots\!12 \nu^{19} + \cdots + 94\!\cdots\!56 ) / 20\!\cdots\!00 \) (509023171278126786212v^19 - 5251848796375773452804v^18 + 27041826681911977259449v^17 - 73987683387335586357372v^16 + 122961263649113872218765v^15 - 179674330475796785421468v^14 + 682944305123550511707138v^13 - 1915377509349824295920624v^12 + 2977903228515612032534094v^11 - 1892325455977617229214420v^10 + 4804003694163665734181497v^9 - 13366861485676896109419756v^8 + 19994991854351184331934613v^7 - 8077832194964356133028524v^6 + 6189434643378605783491340v^5 - 17200869297138230285786168v^4 + 25945962997929648358390012v^3 - 8947132240399131926866608v^2 + 51438530949242276728592*v + 94260572163390623827456) / 20028147237648420751200

\(\nu\)	\(=\)	\( ( 2 \beta_{18} - 2 \beta_{17} - \beta_{16} - 2 \beta_{15} - \beta_{14} - \beta_{12} - \beta_{11} + \cdots + 13 ) / 24 \) (2b18 - 2b17 - b16 - 2b15 - b14 - b12 - b11 + 2b10 - 3b9 + b8 - 3b6 - b5 - 3b4 + 2b3 + 6b2 - 6b1 + 13) / 24
\(\nu^{2}\)	\(=\)	\( ( - 2 \beta_{17} - \beta_{16} - 2 \beta_{15} - \beta_{14} + 3 \beta_{12} - 3 \beta_{11} + \beta_{9} + \cdots + 1 ) / 12 \) (-2b17 - b16 - 2b15 - b14 + 3b12 - 3b11 + b9 - 3b8 - 3b6 + b5 - 11b4 + 2b3 + 6*b2 + 1) / 12
\(\nu^{3}\)	\(=\)	\( ( - \beta_{19} - 7 \beta_{18} - 6 \beta_{17} - 7 \beta_{16} - 8 \beta_{15} - 6 \beta_{14} - \beta_{13} + \cdots - 71 ) / 12 \) (-b19 - 7b18 - 6b17 - 7b16 - 8b15 - 6b14 - b13 + 8b12 + 3b11 - 7b10 + 14b9 - 12b8 - 14b6 + 5b5 - 15b4 - 4b3 + 22b2 + 25b1 - 71) / 12
\(\nu^{4}\)	\(=\)	\( ( - 6 \beta_{18} - 5 \beta_{16} - 5 \beta_{14} + 3 \beta_{12} + 7 \beta_{11} - 6 \beta_{10} + 11 \beta_{9} + \cdots - 98 ) / 3 \) (-6b18 - 5b16 - 5b14 + 3b12 + 7b11 - 6b10 + 11b9 - 7b8 - 8b6 + 3b5 + 22*b1 - 98) / 3
\(\nu^{5}\)	\(=\)	\( ( 13 \beta_{19} - 61 \beta_{18} + 50 \beta_{17} + 80 \beta_{15} - 15 \beta_{14} + 13 \beta_{13} + \cdots - 690 ) / 12 \) (13b19 - 61b18 + 50b17 + 80b15 - 15b14 + 13b13 - 35b12 + 152b11 - 65b10 + 113b9 - 17b8 - 14b7 + 29b6 + 18b5 + 158b4 + 128b3 - 198b2 + 251b1 - 690) / 12
\(\nu^{6}\)	\(=\)	\( ( 29 \beta_{19} + 102 \beta_{17} + 86 \beta_{16} + 172 \beta_{15} + 51 \beta_{14} + 29 \beta_{13} + \cdots + 166 ) / 6 \) (29b19 + 102b17 + 86b16 + 172b15 + 51b14 + 29b13 - 183b12 + 183b11 - 35b9 + 183b8 + 205b6 - 23b5 + 364b4 + 332b3 - 410*b2 + 166) / 6
\(\nu^{7}\)	\(=\)	\( ( 149 \beta_{19} + 587 \beta_{18} + 474 \beta_{17} + 910 \beta_{16} + 868 \beta_{15} + 671 \beta_{14} + \cdots + 9156 ) / 12 \) (149b19 + 587b18 + 474b17 + 910b16 + 868b15 + 671b14 + 149b13 - 1171b12 - 108b11 + 683b10 - 1511b9 + 1767b8 + 318b7 + 2007b6 - 374b5 + 1400b4 + 2144b3 - 1970b2 - 2693*b1 + 9156) / 12
\(\nu^{8}\)	\(=\)	\( ( - 3 \beta_{19} + 644 \beta_{18} + 591 \beta_{16} + 504 \beta_{14} + 3 \beta_{13} - 390 \beta_{12} + \cdots + 9564 ) / 3 \) (-3b19 + 644b18 + 591b16 + 504b14 + 3b13 - 390b12 - 1078b11 + 780b10 - 1534b9 + 1078b8 + 440b7 + 1341b6 - 322b5 - 220b4 - 3068*b1 + 9564) / 3
\(\nu^{9}\)	\(=\)	\( ( - 1711 \beta_{19} + 5935 \beta_{18} - 4674 \beta_{17} + 734 \beta_{16} - 9632 \beta_{15} + \cdots + 71440 ) / 12 \) (-1711b19 + 5935b18 - 4674b17 + 734b16 - 9632b15 + 2151b14 - 1603b13 + 5827b12 - 20250b11 + 7579b10 - 12053b9 + 585b8 + 5070b7 + 2647b6 - 2600b5 - 18936b4 - 30592b3 + 20710b2 - 29305*b1 + 71440) / 12
\(\nu^{10}\)	\(=\)	\( ( - 3857 \beta_{19} - 10370 \beta_{17} - 11280 \beta_{16} - 22560 \beta_{15} - 5185 \beta_{14} + \cdots - 39096 ) / 6 \) (-3857b19 - 10370b17 - 11280b16 - 22560b15 - 5185b14 - 3857b13 + 23133b12 - 24093b11 + 7397b9 - 24093b8 - 23851b6 + 329b5 - 38110b4 - 78192b3 + 47702*b2 - 39096) / 6
\(\nu^{11}\)	\(=\)	\( ( - 16531 \beta_{19} - 61629 \beta_{18} - 46546 \beta_{17} - 118166 \beta_{16} - 107500 \beta_{15} + \cdots - 1138536 ) / 12 \) (-16531b19 - 61629b18 - 46546b17 - 118166b16 - 107500b15 - 69077b14 - 19831b13 + 152225b12 + 120b11 - 86221b10 + 199573b9 - 231549b8 - 70250b7 - 268833b6 + 30270b5 - 139988b4 - 403872b3 + 224562b2 + 319187*b1 - 1138536) / 12
\(\nu^{12}\)	\(=\)	\( ( 2712 \beta_{19} - 70506 \beta_{18} - 77783 \beta_{16} - 51743 \beta_{14} - 2712 \beta_{13} + \cdots - 1093748 ) / 3 \) (2712b19 - 70506b18 - 77783b16 - 51743b14 - 2712b13 + 51483b12 + 136203b11 - 102966b10 + 185703b9 - 136203b8 - 89508b7 - 190616b6 + 35253b5 + 44754b4 + 371406*b1 - 1093748) / 3
\(\nu^{13}\)	\(=\)	\( ( 230865 \beta_{19} - 650297 \beta_{18} + 464186 \beta_{17} - 145626 \beta_{16} + 1201616 \beta_{15} + \cdots - 7631960 ) / 12 \) (230865b19 - 650297b18 + 464186b17 - 145626b16 + 1201616b15 - 233697b14 + 165345b13 - 731669b12 + 2646646b11 - 991517b10 + 1158019b9 + 70289b8 - 908466b7 - 597997b6 + 373112b5 + 2345296b4 + 5102752b3 - 2478222b2 + 3473727*b1 - 7631960) / 12
\(\nu^{14}\)	\(=\)	\( ( 461421 \beta_{19} + 1035746 \beta_{17} + 1419460 \beta_{16} + 2838920 \beta_{15} + 517873 \beta_{14} + \cdots + 6335792 ) / 6 \) (461421b19 + 1035746b17 + 1419460b16 + 2838920b15 + 517873b14 + 461421b13 - 2909385b12 + 3292137b11 - 1528889b9 + 3292137b8 + 2919999b6 + 156835b5 + 4412686b4 + 12671584b3 - 5839998*b2 + 6335792) / 6
\(\nu^{15}\)	\(=\)	\( ( 1614413 \beta_{19} + 6935267 \beta_{18} + 4624962 \beta_{17} + 15358378 \beta_{16} + 13441444 \beta_{15} + \cdots + 142752744 ) / 12 \) (1614413b19 + 6935267b18 + 4624962b17 + 15358378b16 + 13441444b15 + 7045979b14 + 2689493b13 - 19484455b12 + 1578960b11 + 11447843b10 - 26859155b9 + 30242427b8 + 11307606b7 + 35089131b6 - 2540714b5 + 14942636b4 + 62758400b3 - 27628682b2 - 37805357*b1 + 142752744) / 12
\(\nu^{16}\)	\(=\)	\( ( - 738045 \beta_{19} + 8029512 \beta_{18} + 10386267 \beta_{16} + 5336418 \beta_{14} + 738045 \beta_{13} + \cdots + 130500468 ) / 3 \) (-738045b19 + 8029512b18 + 10386267b16 + 5336418b14 + 738045b13 - 6876732b12 - 16900988b11 + 13753464b10 - 22048972b9 + 16900988b8 + 13968336b7 + 25529907b6 - 4014756b5 - 6984168b4 - 44097944*b1 + 130500468) / 3
\(\nu^{17}\)	\(=\)	\( ( - 31292783 \beta_{19} + 74533343 \beta_{18} - 46008394 \beta_{17} + 24561382 \beta_{16} + \cdots + 844714024 ) / 12 \) (-31292783b19 + 74533343b18 - 46008394b17 + 24561382b16 - 150474304b15 + 25115935b14 - 15429947b13 + 87970555b12 - 345418210b11 + 132276523b10 - 100840685b9 - 26250767b8 + 137529694b7 + 89659575b6 - 51388240b5 - 294520352b4 - 758378368b3 + 309868158b2 - 411877513*b1 + 844714024) / 12
\(\nu^{18}\)	\(=\)	\( ( - 54437221 \beta_{19} - 102534762 \beta_{17} - 178043536 \beta_{16} - 356087072 \beta_{15} + \cdots - 927457088 ) / 6 \) (-54437221b19 - 102534762b17 - 178043536b16 - 356087072b15 - 51267381b14 - 54437221b13 + 364712457b12 - 448645641b11 + 266984369b9 - 448645641b8 - 367386719b6 - 37424251b5 - 529550398b4 - 1854914176b3 + 734773438*b2 - 927457088) / 6
\(\nu^{19}\)	\(=\)	\( ( - 144330691 \beta_{19} - 805772845 \beta_{18} - 456832554 \beta_{17} - 1994153350 \beta_{16} + \cdots - 18028357304 ) / 12 \) (-144330691b19 - 805772845b18 - 456832554b17 - 1994153350b16 - 1686072140b15 - 717994293b14 - 363346351b13 + 2489751185b12 - 385093032b11 - 1527335389b10 + 3594893477b9 - 3943136589b8 - 1647767466b7 - 4544260265b6 + 207990398b5 - 1662577476b4 - 9053029600b3 + 3487821370b2 + 4495390627*b1 - 18028357304) / 12

\(n\)	\(325\)	\(353\)	\(703\)
\(\chi(n)\)	\(1\)	\(-1 - \beta_{3}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 127.10
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{20} \) T^20
$3$	\( T^{20} \) T^20
$5$	\( (T^{10} + 7 T^{9} + \cdots + 50176)^{2} \) (T^10 + 7T^9 + 78T^8 - 21T^7 + 1374T^6 + 1407T^5 + 9729T^4 + 3528T^3 + 31200T^2 + 23296*T + 50176)^2
$7$	\( T^{20} + \cdots + 47\!\cdots\!00 \) T^20 - 419T^18 + 112758T^16 - 18336423T^14 + 2182164342T^12 - 174622784379T^10 + 10190026950057T^8 - 352968721999848T^6 + 7943406704940336T^4 - 6245177758160000*T^2 + 4745832100000000
$11$	\( T^{20} + \cdots + 43\!\cdots\!01 \) T^20 - 1049T^18 + 709275T^16 - 283447578T^14 + 82293754149T^12 - 16152091161699T^10 + 2327282843130405T^8 - 211853959519678698T^6 + 13079558068293507627T^4 - 274950858055666627625*T^2 + 4376133992795504068801
$13$	\( (T^{10} - 13 T^{9} + \cdots + 583696)^{2} \) (T^10 - 13T^9 + 390T^8 - 2625T^7 + 87426T^6 - 682341T^5 + 6917661T^4 - 8166840T^3 + 10211340T^2 + 2175872*T + 583696)^2
$17$	\( (T^{5} + 18 T^{4} + \cdots + 50832)^{4} \) (T^5 + 18T^4 - 159T^3 - 3288T^2 - 2760T + 50832)^4
$19$	\( (T^{10} + \cdots + 1691581169664)^{2} \) (T^10 + 2118T^8 + 1487049T^6 + 440018496T^4 + 52326033408T^2 + 1691581169664)^2
$23$	\( T^{20} + \cdots + 80\!\cdots\!56 \) T^20 - 2387T^18 + 4142790T^16 - 3318816615T^14 + 1941442550310T^12 - 269444100899115T^10 + 26612277730166217T^8 - 1208267977847932104T^6 + 39682580157303822000T^4 - 678487612291999065728*T^2 + 8034964736354453913856
$29$	\( (T^{10} + \cdots + 104284535520400)^{2} \) (T^10 - 67T^9 + 3942T^8 - 140751T^7 + 5181522T^6 - 148232187T^5 + 4156517373T^4 - 83777637816T^3 + 1414191079836T^2 - 14244650054080*T + 104284535520400)^2
$31$	\( T^{20} + \cdots + 15\!\cdots\!56 \) T^20 - 3755T^18 + 10130958T^16 - 12473311575T^14 + 11093172380958T^12 - 4096321891643259T^10 + 1087232410691317953T^8 - 107978579826050579424T^6 + 7892951713494543311616T^4 - 120856899062080181755904*T^2 + 1552673443527585662304256
$37$	\( (T^{5} - 24 T^{4} + \cdots - 14835456)^{4} \) (T^5 - 24T^4 - 2760T^3 + 76944T^2 + 610176T - 14835456)^4
$41$	\( (T^{10} + \cdots + 13190269049281)^{2} \) (T^10 - 13T^9 + 4311T^8 + 86298T^7 + 15454413T^6 + 102337389T^5 + 6391016589T^4 - 54276102582T^3 + 2163593148903T^2 - 5414395776733*T + 13190269049281)^2
$43$	\( T^{20} + \cdots + 95\!\cdots\!21 \) T^20 - 9689T^18 + 66155307T^16 - 215559984810T^14 + 505447981315557T^12 - 616952905843068627T^10 + 530714975525994479877T^8 - 144962377443603923108058T^6 + 29175665343108729215946171T^4 - 1883918433892802591849690057*T^2 + 95024971249211223373201069921
$47$	\( T^{20} + \cdots + 24\!\cdots\!36 \) T^20 - 9299T^18 + 55455222T^16 - 199403910519T^14 + 522502092157686T^12 - 908851499908897035T^10 + 1140289818219691982121T^8 - 834200634628441440968904T^6 + 436836217986744159428463792T^4 - 126499819494723327284461979264*T^2 + 24372613813792396710093710254336
$53$	\( (T^{5} - 48 T^{4} + \cdots - 2275179264)^{4} \) (T^5 - 48T^4 - 13080T^3 + 682128T^2 + 41355264T - 2275179264)^4
$59$	\( T^{20} + \cdots + 13\!\cdots\!25 \) T^20 - 11561T^18 + 86037387T^16 - 382728166026T^14 + 1239501119934117T^12 - 2662506590605817091T^10 + 4137260969983152880101T^8 - 3779065884199773374101050T^6 + 2412154849738275531771826875T^4 - 670684935779908592587920265625*T^2 + 133081290077299601763332062890625
$61$	\( (T^{10} + \cdots + 21\!\cdots\!00)^{2} \) (T^10 - 193T^9 + 27630T^8 - 2007453T^7 + 118484238T^6 - 3523353921T^5 + 143544079641T^4 - 3680173640064T^3 + 118640020665024T^2 - 1669675156111360*T + 21493061210214400)^2
$67$	\( T^{20} + \cdots + 37\!\cdots\!81 \) T^20 - 14297T^18 + 133541115T^16 - 698029026426T^14 + 2608530452708517T^12 - 6634603550914462275T^10 + 12574427096419014458085T^8 - 16665081480876951574757130T^6 + 16183275746776178378653393035T^4 - 9775982426440271570694615391913*T^2 + 3707625832742136744964684837716481
$71$	\( (T^{10} + \cdots + 10\!\cdots\!24)^{2} \) (T^10 + 27692T^8 + 261521776T^6 + 1014130874944T^4 + 1695361974622208T^2 + 1009811083007770624)^2
$73$	\( (T^{5} + 42 T^{4} + \cdots + 2701063440)^{4} \) (T^5 + 42T^4 - 19191T^3 - 808584T^2 + 70163688T + 2701063440)^4
$79$	\( T^{20} + \cdots + 10\!\cdots\!00 \) T^20 - 36587T^18 + 962730126T^16 - 10835992797207T^14 + 85825304463936414T^12 - 394657729918113310587T^10 + 1295987391929568471754881T^8 - 2319417393957644755428046752T^6 + 2983980832860714866018493831936T^4 - 2127871242974230853727551552921600*T^2 + 1016547461917106519338341236776960000
$83$	\( T^{20} + \cdots + 22\!\cdots\!76 \) T^20 - 13427T^18 + 123519870T^16 - 595004524863T^14 + 2063391178104270T^12 - 3769208391421223235T^10 + 4850216882526099578673T^8 - 2508043852353850748051808T^6 + 932630124364964804570155776T^4 - 174662356719838938413592633344*T^2 + 22893409779909552000960917733376
$89$	\( (T^{5} - 60 T^{4} + \cdots + 1531650240)^{4} \) (T^5 - 60T^4 - 15924T^3 + 204336T^2 + 65655264T + 1531650240)^4
$97$	\( (T^{10} + \cdots + 52\!\cdots\!25)^{2} \) (T^10 - 187T^9 + 52263T^8 - 5088810T^7 + 1209933165T^6 - 120860026245T^5 + 15158513925597T^4 - 526991127850074T^3 + 14581332811335591T^2 - 96281397637303915*T + 526670483608042225)^2
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