LMFDB - Newform orbit 1638.2.y.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1638,2,Mod(827,1638)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1638, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1638.827"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.y (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$13.0794958511$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} + 20 x^{14} - 12 x^{13} + 40 x^{12} + 40 x^{11} + 82 x^{10} + 104 x^{9} + 537 x^{8} + \cdots + 16 $ x^16 - 8x^15 + 20x^14 - 12x^13 + 40x^12 + 40x^11 + 82x^10 + 104x^9 + 537x^8 - 2528x^7 + 5494x^6 - 7284x^5 + 6002x^4 - 2984x^3 + 1000x^2 - 192*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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_1 q^{2} + \beta_{5} q^{4} + \beta_{7} q^{5} + \beta_1 q^{7} + \beta_{4} q^{8} + \beta_{11} q^{10} + ( - \beta_{14} - \beta_{12} + \beta_{6} + \cdots + 1) q^{11} + (\beta_{15} + \beta_{10} + \cdots + \beta_{4}) q^{13}+ \cdots + \beta_{4} q^{98}+O(q^{100}) $ q - b1 * q^2 + b5 * q^4 + b7 * q^5 + b1 * q^7 + b4 * q^8 + b11 * q^10 + (-b14 - b12 + b6 + b4 + 1) * q^11 + (b15 + b10 - b9 + b5 + b4) * q^13 - b5 * q^14 - q^16 + (-b12 + b9 - b2) * q^17 + (-b14 + b13 - b10 + b6 - b5 + 2b4 - b3 + 2) q^19 + b10 * q^20 + (b15 + b14 - b3 - b1 - 1) * q^22 + (-b15 - b14 - b3 + b1 - 1) * q^23 + (-b13 - b12 - b9 + b8 - b6 + 2b5 - b4 + 2b1) * q^25 + (b13 + b8 + b4 - b2) * q^26 - b4 * q^28 + (b15 - b14 + b13 + b10 + b8 + b7 - b6 + 3b5 + b4 + b1) q^29 + (2b12 + b11 + 2b4 - b2) * q^31 + b1 * q^32 + (-b13 - b7 - b3) * q^34 - b11 * q^35 + (-b13 + b11 + b9 + 2b7 - b3 + b2 - b1) q^37 + (b15 + b14 - b12 + b9 - b4 + b2 - 2b1 - 2) q^38 - b2 * q^40 + (-b15 - b14 + b11 + b8 + 2b7 - b6 + 2b5 + b2 + 2) * q^41 + (-b13 + b12 + b9 - b8 + b6 - b5 + b4 - 2b1) q^43 + (-b15 + b9 + b8 + b1 + 1) * q^44 + (b15 + b9 - b8 + b1 - 1) * q^46 + (-b14 - b13 - 2b12 - b11 + b6 + b5 + b3 + b2) q^47 + b5 * q^49 + (-b14 + b13 + b12 + b6 - b5 + 2b4 - b3 + 2) q^50 + (-b12 - b7 + b6) * q^52 + (b15 - b14 - 2b13 + b10 + b7 + b4) q^53 + (b15 + b14 - b12 + b9 - b4 - b2 - 2b1 - 2) q^55 + q^56 + (b15 - b14 - b12 + b11 + b8 + b6 + b5 + 3b4 - b2 + 1) q^58 + (-2b12 - 2b4) * q^59 + (-b10 + b7 - b2 - 2) * q^61 + (b10 - b7 + 2b3 - 2) q^62 - b5 * q^64 + (b15 - 2b12 + b10 + b9 + b7 - b6 + 2b5 - 2b4 - 2b1 - 5) * q^65 + (-b14 - b13 + b11 + 2b10 + b6 + b5 - 2b4 + b3 - b2) * q^67 + (b12 - b11 + b9) * q^68 - b10 * q^70 + (-2b13 + 2b11 + 2b7 + 2b5 - 2b3 + 2b2 + 2) * q^71 + (b15 + b14 + b13 - b11 - b8 - 3b7 + b6 - 2b5 + b3 - b2 - 2b1 - 2) q^73 + (-b13 + b12 + 2b11 + b10 + b9 + b7 + b5) q^74 + (-b15 - b13 + b8 + b7 + b5 - b3 + 2b1 + 2) q^76 + (-b15 - b14 + b3 + b1 + 1) * q^77 + (-b15 - b14 - b12 + b9 + b8 + b6 - 3b4 - 2b2 - 3b1 + 2) q^79 - b7 * q^80 + (b15 - b14 + 2b11 + b10 - b8 + b7 + b6 + 2b5 + 2b4 - 2b1) * q^82 + (-2b14 + 2b13 + 2b7 - 2b6 + 2b5 + 2b3) * q^83 + (-2b15 + b14 - b13 - b9 + 2b8 - 2b7 + b6 - b5 - b3 + 9b1 + 2) * q^85 + (b14 - b13 + b12 - b6 + b5 - b4 + b3 - 2) * q^86 + (-b13 - b8 + b6 - b5 - b1) * q^88 + (-2b13 - 2b12 + b11 + 2b10 + 2b5 - 2b4 + 2b3 - b2 - 2) * q^89 + (-b13 - b8 - b4 + b2) * q^91 + (-b13 + b8 - b6 - b5 + b1) * q^92 + (b15 + b14 + b12 - b10 - b9 + b7 + b4 - 2b3) q^94 + (b15 + b14 - 2b10 + 2b7 - b3 - 4b2 - b1 + 3) q^95 + (-b15 - 2b13 + b11 - b8 + b5 - 2b4 + 2b3 - b2 - 2) q^97 + b4 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{5} + 8 q^{11} + 4 q^{13} - 16 q^{16} + 8 q^{17} + 16 q^{19} + 4 q^{20} - 8 q^{22} - 24 q^{23} - 8 q^{26} - 8 q^{31} + 4 q^{34} - 4 q^{37} - 16 q^{38} + 16 q^{41} + 8 q^{44} - 4 q^{47} + 16 q^{50}+ \cdots - 28 q^{97}+O(q^{100}) $ 16 * q - 4 * q^5 + 8 * q^11 + 4 * q^13 - 16 * q^16 + 8 * q^17 + 16 * q^19 + 4 * q^20 - 8 * q^22 - 24 * q^23 - 8 * q^26 - 8 * q^31 + 4 * q^34 - 4 * q^37 - 16 * q^38 + 16 * q^41 + 8 * q^44 - 4 * q^47 + 16 * q^50 - 16 * q^55 + 16 * q^56 + 4 * q^58 + 8 * q^59 - 40 * q^61 - 24 * q^62 - 56 * q^65 - 4 * q^67 - 4 * q^70 + 24 * q^71 - 12 * q^73 + 16 * q^76 + 8 * q^77 + 16 * q^79 + 4 * q^80 + 8 * q^85 - 24 * q^86 - 16 * q^89 + 8 * q^91 - 8 * q^94 + 40 * q^95 - 28 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 8 x^{15} + 20 x^{14} - 12 x^{13} + 40 x^{12} + 40 x^{11} + 82 x^{10} + 104 x^{9} + 537 x^{8} + \cdots + 16 $ x^16 - 8*x^15 + 20*x^14 - 12*x^13 + 40*x^12 + 40*x^11 + 82*x^10 + 104*x^9 + 537*x^8 - 2528*x^7 + 5494*x^6 - 7284*x^5 + 6002*x^4 - 2984*x^3 + 1000*x^2 - 192*x + 16 :

$\beta_{1}$	$=$	$ ( 88\!\cdots\!87 \nu^{15} + \cdots - 12\!\cdots\!20 ) / 29\!\cdots\!12 $ (8810453484481419787v^15 - 69972490944959614420v^14 + 172850373062827022431v^13 - 100481314110617395302v^12 + 354666884436870105751v^11 + 376243159872598807668v^10 + 772580391658572562209v^9 + 1020795990219764087478v^8 + 4903094374641295408884v^7 - 21808020956822433751512v^6 + 47667259736333919738010v^5 - 62649332902158867279196v^4 + 51391858403342645582656v^3 - 25336716035274686737392v^2 + 8241251957213875681648*v - 1241745401323893045120) / 291433150735989873312
$\beta_{2}$	$=$	$ ( - 11\!\cdots\!23 \nu^{15} + \cdots - 10\!\cdots\!28 ) / 29\!\cdots\!12 $ (-11698868362125004523v^15 + 83939786441538639147v^14 - 160527514035902012089v^13 - 24321857261037208927v^12 - 414765410197699453211v^11 - 832915315773135893517v^10 - 1496640349486741021843v^9 - 2232874028881130210033v^8 - 7715416262234299811216v^7 + 23777099458901783246312v^6 - 42222271916356105928370v^5 + 40583108581656169060702v^4 - 16870573462608118490784v^3 - 1803781446690608394440v^2 + 1899665583663956469552*v - 1085567858194490072528) / 291433150735989873312
$\beta_{3}$	$=$	$ ( - 70\!\cdots\!13 \nu^{15} + \cdots - 68\!\cdots\!36 ) / 14\!\cdots\!56 $ (-7056265369041347413v^15 + 50507927046357057783v^14 - 95940201065736020691v^13 - 16319571173558912787v^12 - 250853466954085937957v^11 - 506423030574247080325v^10 - 908623952155848342689v^9 - 1353648933950372385221v^8 - 4661033040459466556800v^7 + 14293572868649033061836v^6 - 25156031875780250361942v^5 + 24115377250520373698014v^4 - 9878164327266277702768v^3 - 1337614489088590409536v^2 + 1223669711029373195936*v - 688035310999430319936) / 145716575367994936656
$\beta_{4}$	$=$	$ ( - 15\!\cdots\!81 \nu^{15} + \cdots + 58\!\cdots\!56 ) / 29\!\cdots\!12 $ (-15512786429751648481v^15 + 118440678597716652792v^14 - 266427488701355331405v^13 + 84539613688524773590v^12 - 580973445915534642573v^11 - 832189061146464961192v^10 - 1554085951107517057027v^9 - 2140109490702079740406v^8 - 9040315704986997841364v^7 + 36016637613027621087280v^6 - 71714984825576732213630v^5 + 85522905553061971943036v^4 - 59666511517259564371568v^3 + 22015992179237618937056v^2 - 6345035029579124763184*v + 584542587322075416256) / 291433150735989873312
$\beta_{5}$	$=$	$ ( - 17629042972135 \nu^{15} + 138301296365753 \nu^{14} - 330704749144585 \nu^{13} + \cdots + 11\!\cdots\!76 ) / 211731643404256 $ (-17629042972135v^15 + 138301296365753v^14 - 330704749144585v^13 + 156692672910367v^12 - 672027705019791v^11 - 813094141556031v^10 - 1556179415520171v^9 - 2059136017414191v^8 - 9762784462391544v^7 + 43071159787870216v^6 - 89995923668790394v^5 + 113219568956294482v^4 - 85988580899868352v^3 + 36373121308786984v^2 - 9921841312383600*v + 1132557453324976) / 211731643404256
$\beta_{6}$	$=$	$ ( 27\!\cdots\!77 \nu^{15} + \cdots - 54\!\cdots\!56 ) / 29\!\cdots\!12 $ (27618690357517314377v^15 - 221460659791030258892v^14 + 555732503777147660849v^13 - 336668411993367414666v^12 + 1102498869243079260809v^11 + 1080922593807350558892v^10 + 2214609403385323639239v^9 + 2767834969348104265578v^8 + 14659355868512024837184v^7 - 70284854675750366215080v^6 + 152474456531607125759006v^5 - 202700550842959912371380v^4 + 167255862936333756869672v^3 - 83367849189249536007984v^2 + 28187891884784858482352*v - 5461217065603874041056) / 291433150735989873312
$\beta_{7}$	$=$	$ ( 29\!\cdots\!85 \nu^{15} + \cdots - 27\!\cdots\!16 ) / 22\!\cdots\!24 $ (2994205490140966885v^15 - 23725077653206080572v^14 + 57893279211948857965v^13 - 30177590547698697114v^12 + 114652279378357623829v^11 + 128940368648258272180v^10 + 248834741223746675003v^9 + 319025985360768953674v^8 + 1610616387006658622896v^7 - 7479537828009877711936v^6 + 15761322054866684640150v^5 - 20225042951737164171076v^4 + 15655604483093553031400v^3 - 6920281476952229423680v^2 + 1993104387746791628592*v - 279499445405065755616) / 22417934671999221024
$\beta_{8}$	$=$	$ ( - 21\!\cdots\!35 \nu^{15} + \cdots + 23\!\cdots\!88 ) / 14\!\cdots\!56 $ (-21021125277878065035v^15 + 158317828633569406855v^14 - 345810066853522286123v^13 + 87439838101676457197v^12 - 795843408701129124477v^11 - 1210140853074258222021v^10 - 2273883870064584412477v^9 - 3213503277998927522285v^8 - 12718504372712238057134v^7 + 47300775478873296839804v^6 - 92985688483113292755994v^5 + 108875122499736129093878v^4 - 74087120998872004208428v^3 + 27334688839069881823120v^2 - 8172884498812969815760*v + 230650006331699752288) / 145716575367994936656
$\beta_{9}$	$=$	$ ( - 49\!\cdots\!43 \nu^{15} + \cdots + 45\!\cdots\!12 ) / 29\!\cdots\!12 $ (-49126513370488191843v^15 + 388757174834211391657v^14 - 946327635507412920985v^13 + 488970196259846469679v^12 - 1884437999079307605855v^11 - 2132152285813122333791v^10 - 4119021926724842141523v^9 - 5298795931979829079119v^8 - 26518146320519299868868v^7 + 122377315584533125740504v^6 - 257602217727197625956850v^5 + 330245570373096907880674v^4 - 255893117550647532091576v^3 + 113290524649632008609896v^2 - 32704211824134297823056*v + 4594234067017961740912) / 291433150735989873312
$\beta_{10}$	$=$	$ ( 38\!\cdots\!03 \nu^{15} + \cdots - 17\!\cdots\!00 ) / 22\!\cdots\!24 $ (3864658061768228203v^15 - 29847215167948824266v^14 + 69052681399000190511v^13 - 27419531607829393332v^12 + 147208949701147260155v^11 + 195879950407577618886v^10 + 371518171538157562353v^9 + 505845176238109617604v^8 + 2216412847574252038128v^7 - 9156677325533603260864v^6 + 18702343634812872455386v^5 - 23031984120255279800080v^4 + 16843089874635801410280v^3 - 6839508571946680389328v^2 + 1877723782800634003472*v - 173800820357319824000) / 22417934671999221024
$\beta_{11}$	$=$	$ ( 12\!\cdots\!92 \nu^{15} + \cdots - 86\!\cdots\!40 ) / 63\!\cdots\!72 $ (1276526512941182592v^15 - 9928702582560774307v^14 + 23308379550483686890v^13 - 10035678268519997763v^12 + 48710885983548768572v^11 + 61656767642984327249v^10 + 117634991912270128330v^9 + 157144920416365985515v^8 + 716962068095369669060v^7 - 3073296023857682221012v^6 + 6315975651765399848848v^5 - 7873012308851057290458v^4 + 5881827248365393945584v^3 - 2492515748087927513296v^2 + 721447802675349777936*v - 86403333747699113040) / 6335503276869345072
$\beta_{12}$	$=$	$ ( - 63\!\cdots\!17 \nu^{15} + \cdots + 28\!\cdots\!88 ) / 29\!\cdots\!12 $ (-63339678856780924817v^15 + 489378408129875891831v^14 - 1132759458792787778755v^13 + 449310809676387503553v^12 - 2407748268319976601885v^11 - 3198697671069615739577v^10 - 6063214626278609704081v^9 - 8241672942777890208849v^8 - 36247032904767641034644v^7 + 150243396566568180601152v^6 - 306755660242939679338134v^5 + 377291053196996031008750v^4 - 275740924695600545790456v^3 + 111801051146031036422952v^2 - 30720251742910176984016*v + 2843022865035613822288) / 291433150735989873312
$\beta_{13}$	$=$	$ ( 37\!\cdots\!34 \nu^{15} + \cdots - 25\!\cdots\!84 ) / 14\!\cdots\!56 $ (37037873510832940534v^15 - 288250772557220061321v^14 + 677750823115831653832v^13 - 294961959206387202913v^12 + 1414642074489096799934v^11 + 1785615929598208319327v^10 + 3410112571569873689484v^9 + 4558720902396543677761v^8 + 20812646058759375633528v^7 - 89220721495115426880088v^6 + 183776514720788436149812v^5 - 229381109249059549524710v^4 + 171629244002356527144712v^3 - 72756860161951393888760v^2 + 21045328421921509411184*v - 2518978615690731247584) / 145716575367994936656
$\beta_{14}$	$=$	$ ( 97\!\cdots\!63 \nu^{15} + \cdots - 89\!\cdots\!60 ) / 29\!\cdots\!12 $ (97877540218991487563v^15 - 773428726504321982064v^14 + 1881186256855318965939v^13 - 985083990357572690098v^12 + 3803103338086808595731v^11 + 4303488155661520466960v^10 + 8423861562924163947573v^9 + 10998554188225265272386v^8 + 53615137411232141953176v^7 - 242208722605567502092176v^6 + 513739714253880450210650v^5 - 660948392250966820874916v^4 + 518214954251064184683624v^3 - 234920709364472643874720v^2 + 68990110593562938689168*v - 8959875000406315192960) / 291433150735989873312
$\beta_{15}$	$=$	$ ( - 25\!\cdots\!77 \nu^{15} + \cdots + 12\!\cdots\!92 ) / 63\!\cdots\!72 $ (-2506621258792392877v^15 + 19407561787926980496v^14 - 45070576058898156183v^13 + 17989365574802015890v^12 - 94589569346544940167v^11 - 124819562231765638096v^10 - 235321088511760626589v^9 - 317406292972740382954v^8 - 1420703207804796672278v^7 + 5981076301653901757392v^6 - 12191705274702342508766v^5 + 14975962328106194724620v^4 - 10909298670034740842996v^3 + 4360594470258801958640v^2 - 1189717155114541656976*v + 126368538107948144992) / 6335503276869345072

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

$\nu$	$=$	$ ( \beta_{12} + \beta_{10} - \beta_{4} + \beta_{3} - \beta_{2} + 1 ) / 2 $ (b12 + b10 - b4 + b3 - b2 + 1) / 2
$\nu^{2}$	$=$	$ ( 2 \beta_{15} - \beta_{13} + 2 \beta_{12} + 3 \beta_{11} + 4 \beta_{10} + 2 \beta_{8} - \beta_{5} + \cdots + 3 ) / 2 $ (2b15 - b13 + 2b12 + 3b11 + 4b10 + 2b8 - b5 - 6b4 + b3 - 3*b2 + 3) / 2
$\nu^{3}$	$=$	$ ( 6 \beta_{15} - 4 \beta_{14} - 12 \beta_{13} + 12 \beta_{12} + 20 \beta_{11} + 20 \beta_{10} + 3 \beta_{9} + \cdots + 9 ) / 2 $ (6b15 - 4b14 - 12b13 + 12b12 + 20b11 + 20b10 + 3b9 + 4b8 + 7b7 - 12b5 - 12b4 + 3b3 - 7b2 + 9b1 + 9) / 2
$\nu^{4}$	$=$	$ 16 \beta_{15} - 16 \beta_{14} - 24 \beta_{13} + 16 \beta_{12} + 47 \beta_{11} + 34 \beta_{10} + \cdots + 41 \beta_1 $ 16b15 - 16b14 - 24b13 + 16b12 + 47b11 + 34b10 + 16b9 + 5b8 + 34b7 - 5b6 - 36b5 - 20b4 + 41*b1
$\nu^{5}$	$=$	$ ( 100 \beta_{15} - 142 \beta_{14} - 220 \beta_{13} + 83 \beta_{12} + 414 \beta_{11} + 167 \beta_{10} + \cdots - 211 ) / 2 $ (100b15 - 142b14 - 220b13 + 83b12 + 414b11 + 167b10 + 220b9 + 414b7 - 100b6 - 262b5 - 111b4 - 83b3 + 167b2 + 504b1 - 211) / 2
$\nu^{6}$	$=$	$ ( 250 \beta_{15} - 634 \beta_{14} - 735 \beta_{13} + 1453 \beta_{11} + 1050 \beta_{9} - 250 \beta_{8} + \cdots - 1809 ) / 2 $ (250b15 - 634b14 - 735b13 + 1453b11 + 1050b9 - 250b8 + 2044b7 - 634b6 - 925b5 - 735b3 + 1453b2 + 2634b1 - 1809) / 2
$\nu^{7}$	$=$	$ ( - 2188 \beta_{14} - 1852 \beta_{13} - 1852 \beta_{12} + 3620 \beta_{11} - 3620 \beta_{10} + \cdots - 10939 ) / 2 $ (-2188b14 - 1852b13 - 1852b12 + 3620b11 - 3620b10 + 4549b9 - 2188b8 + 8781b7 - 3094b6 - 2358b5 + 2358b4 - 4549b3 + 8781b2 + 10939b1 - 10939) / 2
$\nu^{8}$	$=$	$ - 2736 \beta_{15} - 2736 \beta_{14} - 7955 \beta_{12} - 15504 \beta_{10} + 7955 \beta_{9} + \cdots - 27595 $ -2736b15 - 2736b14 - 7955b12 - 15504b10 + 7955b9 - 6665b8 + 15504b7 - 6665b6 + 10007b4 - 11269b3 + 21902b2 + 19408b1 - 27595
$\nu^{9}$	$=$	$ ( - 46918 \beta_{15} + 39848 \beta_{13} - 96507 \beta_{12} - 77484 \beta_{11} - 187229 \beta_{10} + \cdots - 234351 ) / 2 $ (-46918b15 + 39848b13 - 96507b12 - 77484b11 - 187229b10 + 39848b9 - 66342b8 + 77484b7 - 46918b6 + 50230b5 + 121091b4 - 96507b3 + 187229b2 + 97148b1 - 234351) / 2
$\nu^{10}$	$=$	$ ( - 283736 \beta_{15} + 117332 \beta_{14} + 340559 \beta_{13} - 481754 \beta_{12} - 661931 \beta_{11} + \cdots - 829369 ) / 2 $ (-283736b15 + 117332b14 + 340559b13 - 481754b12 - 661931b11 - 935908b10 - 283736b8 - 117332b6 + 428301b5 + 606606b4 - 340559b3 + 661931b2 - 829369) / 2
$\nu^{11}$	$=$	$ ( - 1417662 \beta_{15} + 1002490 \beta_{14} + 2057836 \beta_{13} - 2057836 \beta_{12} - 3996234 \beta_{11} + \cdots - 2074251 ) / 2 $ (-1417662b15 + 1002490b14 + 2057836b13 - 2057836b12 - 3996234b11 - 3996234b10 - 851889b9 - 1002490b8 - 1655015b7 + 2586654b5 + 2586654b4 - 851889b3 + 1655015b2 - 2074251b1 - 2074251) / 2
$\nu^{12}$	$=$	$ - 3027035 \beta_{15} + 3027035 \beta_{14} + 5143960 \beta_{13} - 3637174 \beta_{12} + \cdots - 8854439 \beta_1 $ -3027035b15 + 3027035b14 + 5143960b13 - 3637174b12 - 9992083b11 - 7065772b10 - 3637174b9 - 1253452b8 - 7065772b7 + 1253452b6 + 6470338b5 + 4573952b4 - 8854439*b1
$\nu^{13}$	$=$	$ ( - 21405892 \beta_{15} + 30272086 \beta_{14} + 43923136 \beta_{13} - 18191625 \beta_{12} + \cdots + 44284357 ) / 2 $ (-21405892b15 + 30272086b14 + 43923136b13 - 18191625b12 - 85313194b11 - 35336741b10 - 43923136b9 - 85313194b7 + 21405892b6 + 55229886b5 + 22878465b4 + 18191625b3 - 35336741b2 - 106907864b1 + 44284357) / 2
$\nu^{14}$	$=$	$ ( - 53528366 \beta_{15} + 129236330 \beta_{14} + 155317897 \beta_{13} - 301699343 \beta_{11} + \cdots + 378082599 ) / 2 $ (-53528366b15 + 129236330b14 + 155317897b13 - 301699343b11 - 219654094b9 + 53528366b8 - 426663732b7 + 129236330b6 + 195317903b5 + 155317897b3 - 301699343b2 - 534708278b1 + 378082599) / 2
$\nu^{15}$	$=$	$ ( 457017240 \beta_{14} + 388396692 \beta_{13} + 388396692 \beta_{12} - 754432232 \beta_{11} + \cdots + 2282489621 ) / 2 $ (457017240b14 + 388396692b13 + 388396692b12 - 754432232b11 + 754432232b10 - 937690835b9 + 457017240b8 - 1821371807b7 + 646317826b6 + 488428422b5 - 488428422b4 + 937690835b3 - 1821371807b2 - 2282489621b1 + 2282489621) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$.

$n$	$379$	$703$	$911$
$\chi(n)$	$-\beta_{5}$	$1$	$-1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

827.1

−0.777611	+	1.87732i
0.162053	−	0.391230i
0.743579	−	1.79516i
0.457766	−	1.10514i
0.213522	+	0.0884436i
4.26883	+	1.76821i
−1.90660	−	0.789738i
0.838459	+	0.347301i
−0.777611	−	1.87732i
0.162053	+	0.391230i
0.743579	+	1.79516i
0.457766	+	1.10514i
0.213522	−	0.0884436i
4.26883	−	1.76821i
−1.90660	+	0.789738i
0.838459	−	0.347301i

−0.707107

0.707107i

−

1.00000i

−3.00337

3.00337i

0.707107

−

0.707107i

0.707107

0.707107i

−

4.24741i

827.2

−0.707107

0.707107i

−

1.00000i

−0.509458

0.509458i

0.707107

−

0.707107i

0.707107

0.707107i

−

0.720483i

827.3

−0.707107

0.707107i

−

1.00000i

0.162918

−

0.162918i

0.707107

−

0.707107i

0.707107

0.707107i

0.230401i

827.4

−0.707107

0.707107i

−

1.00000i

2.34991

−

2.34991i

0.707107

−

0.707107i

0.707107

0.707107i

3.32328i

827.5

0.707107

−

0.707107i

−

1.00000i

−1.98814

1.98814i

−0.707107

0.707107i

−0.707107

−

0.707107i

2.81166i

827.6

0.707107

−

0.707107i

−

1.00000i

−1.40096

1.40096i

−0.707107

0.707107i

−0.707107

−

0.707107i

1.98125i

827.7

0.707107

−

0.707107i

−

1.00000i

0.746053

−

0.746053i

−0.707107

0.707107i

−0.707107

−

0.707107i

−

1.05508i

827.8

0.707107

−

0.707107i

−

1.00000i

1.64304

−

1.64304i

−0.707107

0.707107i

−0.707107

−

0.707107i

−

2.32362i

1331.1

−0.707107

−

0.707107i

1.00000i

−3.00337

−

3.00337i

0.707107

0.707107i

0.707107

−

0.707107i

4.24741i

1331.2

−0.707107

−

0.707107i

1.00000i

−0.509458

−

0.509458i

0.707107

0.707107i

0.707107

−

0.707107i

0.720483i

1331.3

−0.707107

−

0.707107i

1.00000i

0.162918

0.162918i

0.707107

0.707107i

0.707107

−

0.707107i

−

0.230401i

1331.4

−0.707107

−

0.707107i

1.00000i

2.34991

2.34991i

0.707107

0.707107i

0.707107

−

0.707107i

−

3.32328i

1331.5

0.707107

0.707107i

1.00000i

−1.98814

−

1.98814i

−0.707107

−

0.707107i

−0.707107

0.707107i

−

2.81166i

1331.6

0.707107

0.707107i

1.00000i

−1.40096

−

1.40096i

−0.707107

−

0.707107i

−0.707107

0.707107i

−

1.98125i

1331.7

0.707107

0.707107i

1.00000i

0.746053

0.746053i

−0.707107

−

0.707107i

−0.707107

0.707107i

1.05508i

1331.8

0.707107

0.707107i

1.00000i

1.64304

1.64304i

−0.707107

−

0.707107i

−0.707107

0.707107i

2.32362i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
39.f	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1638.2.y.c	✓	16
3.b	odd	2	1		1638.2.y.d	yes	16
13.d	odd	4	1		1638.2.y.d	yes	16
39.f	even	4	1	inner	1638.2.y.c	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1638.2.y.c	✓	16	1.a	even	1	1	trivial
1638.2.y.c	✓	16	39.f	even	4	1	inner
1638.2.y.d	yes	16	3.b	odd	2	1
1638.2.y.d	yes	16	13.d	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{16} + 4 T_{5}^{15} + 8 T_{5}^{14} - 16 T_{5}^{13} + 182 T_{5}^{12} + 592 T_{5}^{11} + 1040 T_{5}^{10} + \cdots + 1024 $ T5^16 + 4*T5^15 + 8*T5^14 - 16*T5^13 + 182*T5^12 + 592*T5^11 + 1040*T5^10 - 1464*T5^9 + 5625*T5^8 + 14284*T5^7 + 21512*T5^6 - 20328*T5^5 + 17988*T5^4 + 17216*T5^3 + 12800*T5^2 - 5120*T5 + 1024 acting on $S_{2}^{\mathrm{new}}(1638, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} + 1)^{4} $ (T^4 + 1)^4
$3$	$ T^{16} $ T^16
$5$	$ T^{16} + 4 T^{15} + \cdots + 1024 $ T^16 + 4T^15 + 8T^14 - 16T^13 + 182T^12 + 592T^11 + 1040T^10 - 1464T^9 + 5625T^8 + 14284T^7 + 21512T^6 - 20328T^5 + 17988T^4 + 17216T^3 + 12800T^2 - 5120*T + 1024
$7$	$ (T^{4} + 1)^{4} $ (T^4 + 1)^4
$11$	$ T^{16} - 8 T^{15} + \cdots + 65536 $ T^16 - 8T^15 + 32T^14 + 40T^13 + 210T^12 - 2264T^11 + 12192T^10 + 22568T^9 + 3313T^8 - 4432T^7 + 834688T^6 + 2543360T^5 + 3863040T^4 + 3190784T^3 + 1605632T^2 + 458752*T + 65536
$13$	$ T^{16} + \cdots + 815730721 $ T^16 - 4T^15 + 10T^14 - 116T^13 + 368T^12 - 636T^11 + 7334T^10 - 26700T^9 + 44414T^8 - 347100T^7 + 1239446T^6 - 1397292T^5 + 10510448T^4 - 43069988T^3 + 48268090T^2 - 250994068*T + 815730721
$17$	$ (T^{8} - 4 T^{7} + \cdots - 2312)^{2} $ (T^8 - 4T^7 - 84T^6 + 312T^5 + 1573T^4 - 5964T^3 + 2594T^2 + 5032*T - 2312)^2
$19$	$ T^{16} - 16 T^{15} + \cdots + 12845056 $ T^16 - 16T^15 + 128T^14 - 168T^13 - 734T^12 - 328T^11 + 113312T^10 + 46680T^9 - 414847T^8 + 4641032T^7 + 54226464T^6 + 191160128T^5 + 372654144T^4 + 394622976T^3 + 248020992T^2 + 79822848*T + 12845056
$23$	$ (T^{8} + 12 T^{7} + \cdots + 4096)^{2} $ (T^8 + 12T^7 - 30T^6 - 820T^5 - 1847T^4 + 11896T^3 + 57288T^2 + 68096*T + 4096)^2
$29$	$ T^{16} + \cdots + 359548941376 $ T^16 + 312T^14 + 39258T^12 + 2596884T^10 + 98553369T^8 + 2182571692T^6 + 27126963700T^4 + 167244176032*T^2 + 359548941376
$31$	$ T^{16} + 8 T^{15} + \cdots + 262144 $ T^16 + 8T^15 + 32T^14 + 128T^13 + 6872T^12 + 60160T^11 + 269568T^10 + 437376T^9 + 3734928T^8 + 30703488T^7 + 139235840T^6 + 197810688T^5 + 152588544T^4 + 55685120T^3 + 7372800T^2 - 1966080*T + 262144
$37$	$ T^{16} + 4 T^{15} + \cdots + 64 $ T^16 + 4T^15 + 8T^14 - 104T^13 + 10806T^12 + 31960T^11 + 46800T^10 + 45024T^9 + 1276297T^8 + 4262868T^7 + 7158600T^6 + 1618120T^5 + 4289556T^4 + 17059840T^3 + 33948800T^2 + 65920*T + 64
$41$	$ T^{16} + \cdots + 685182976 $ T^16 - 16T^15 + 128T^14 - 256T^13 + 6440T^12 - 83328T^11 + 541696T^10 - 1907968T^9 + 8975888T^8 - 67028224T^7 + 395233280T^6 - 1475185664T^5 + 3593976320T^4 - 5613473792T^3 + 5427986432T^2 - 2727329792*T + 685182976
$43$	$ T^{16} + \cdots + 2842169344 $ T^16 + 220T^14 + 17942T^12 + 697204T^10 + 14239777T^8 + 157505528T^6 + 921533456T^4 + 2625534464*T^2 + 2842169344
$47$	$ T^{16} + \cdots + 27518828544 $ T^16 + 4T^15 + 8T^14 + 216T^13 + 20036T^12 + 118848T^11 + 338432T^10 - 1206784T^9 + 15795712T^8 + 67055616T^7 + 162791424T^6 - 396361728T^5 + 755122176T^4 + 1911029760T^3 + 6115295232T^2 - 18345885696*T + 27518828544
$53$	$ T^{16} + \cdots + 67051995136 $ T^16 + 476T^14 + 82452T^12 + 6493600T^10 + 241206976T^8 + 4127602688T^6 + 28770413568T^4 + 76843917312*T^2 + 67051995136
$59$	$ T^{16} + \cdots + 268435456 $ T^16 - 8T^15 + 32T^14 + 320T^13 + 2656T^12 - 13568T^11 + 74752T^10 + 755712T^9 + 3033344T^8 - 673792T^7 + 43655168T^6 + 417677312T^5 + 1712603136T^4 + 3393716224T^3 + 3699376128T^2 + 1409286144*T + 268435456
$61$	$ (T^{8} + 20 T^{7} + \cdots - 5768)^{2} $ (T^8 + 20T^7 + 88T^6 - 328T^5 - 3947T^4 - 13556T^3 - 22414T^2 - 18168*T - 5768)^2
$67$	$ T^{16} + \cdots + 268435456 $ T^16 + 4T^15 + 8T^14 + 568T^13 + 18564T^12 + 149632T^11 + 611328T^10 + 305664T^9 + 655360T^8 + 8232960T^7 + 47480832T^6 + 7143424T^5 - 26148864T^4 - 71303168T^3 + 536870912T^2 - 536870912*T + 268435456
$71$	$ T^{16} + \cdots + 112810000384 $ T^16 - 24T^15 + 288T^14 - 1088T^13 + 31328T^12 - 692480T^11 + 8188928T^10 - 31752192T^9 + 185395456T^8 - 2771843072T^7 + 30898266112T^6 - 128592363520T^5 + 279206707200T^4 - 130470117376T^3 + 4060086272T^2 + 30266097664*T + 112810000384
$73$	$ T^{16} + \cdots + 31085921344 $ T^16 + 12T^15 + 72T^14 + 576T^13 + 24822T^12 + 322400T^11 + 2247504T^10 + 9330776T^9 + 92019497T^8 + 1039168884T^7 + 7402485512T^6 + 27023031128T^5 + 48299316436T^4 - 6669802656T^3 + 8733132800T^2 + 23301393920*T + 31085921344
$79$	$ (T^{8} - 8 T^{7} + \cdots - 210944)^{2} $ (T^8 - 8T^7 - 236T^6 + 2128T^5 + 12004T^4 - 128512T^3 + 57056T^2 + 612864*T - 210944)^2
$83$	$ T^{16} + \cdots + 53396107165696 $ T^16 - 320T^13 + 54816T^12 - 89344T^11 + 51200T^10 + 392192T^9 + 752914688T^8 - 1538445312T^7 + 1059094528T^6 + 122763673600T^5 + 1644926402560T^4 + 2958508949504T^3 + 1638832013312T^2 - 13229304578048*T + 53396107165696
$89$	$ T^{16} + \cdots + 2699475288064 $ T^16 + 16T^15 + 128T^14 + 96T^13 + 38488T^12 + 622400T^11 + 5036544T^10 + 2168960T^9 + 162127248T^8 + 2549594112T^7 + 20093812736T^6 + 12822735872T^5 + 133923213568T^4 + 2103308570624T^3 + 17029634752512T^2 + 9588647264256*T + 2699475288064
$97$	$ T^{16} + \cdots + 41943040000 $ T^16 + 28T^15 + 392T^14 + 968T^13 - 1068T^12 + 73440T^11 + 2943488T^10 - 3982080T^9 + 1719360T^8 - 9449472T^7 + 1047134208T^6 - 2376466432T^5 + 2690514944T^4 + 3334471680T^3 + 30198988800T^2 - 50331648000*T + 41943040000
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.y (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{2} + \beta_{5} q^{4} + \beta_{7} q^{5} + \beta_1 q^{7} + \beta_{4} q^{8} + \beta_{11} q^{10} + ( - \beta_{14} - \beta_{12} + \beta_{6} + \cdots + 1) q^{11} + (\beta_{15} + \beta_{10} + \cdots + \beta_{4}) q^{13}+ \cdots + \beta_{4} q^{98}+O(q^{100}) \) q - b1 * q^2 + b5 * q^4 + b7 * q^5 + b1 * q^7 + b4 * q^8 + b11 * q^10 + (-b14 - b12 + b6 + b4 + 1) * q^11 + (b15 + b10 - b9 + b5 + b4) * q^13 - b5 * q^14 - q^16 + (-b12 + b9 - b2) * q^17 + (-b14 + b13 - b10 + b6 - b5 + 2b4 - b3 + 2) q^19 + b10 * q^20 + (b15 + b14 - b3 - b1 - 1) * q^22 + (-b15 - b14 - b3 + b1 - 1) * q^23 + (-b13 - b12 - b9 + b8 - b6 + 2b5 - b4 + 2b1) * q^25 + (b13 + b8 + b4 - b2) * q^26 - b4 * q^28 + (b15 - b14 + b13 + b10 + b8 + b7 - b6 + 3b5 + b4 + b1) q^29 + (2b12 + b11 + 2b4 - b2) * q^31 + b1 * q^32 + (-b13 - b7 - b3) * q^34 - b11 * q^35 + (-b13 + b11 + b9 + 2b7 - b3 + b2 - b1) q^37 + (b15 + b14 - b12 + b9 - b4 + b2 - 2b1 - 2) q^38 - b2 * q^40 + (-b15 - b14 + b11 + b8 + 2b7 - b6 + 2b5 + b2 + 2) * q^41 + (-b13 + b12 + b9 - b8 + b6 - b5 + b4 - 2b1) q^43 + (-b15 + b9 + b8 + b1 + 1) * q^44 + (b15 + b9 - b8 + b1 - 1) * q^46 + (-b14 - b13 - 2b12 - b11 + b6 + b5 + b3 + b2) q^47 + b5 * q^49 + (-b14 + b13 + b12 + b6 - b5 + 2b4 - b3 + 2) q^50 + (-b12 - b7 + b6) * q^52 + (b15 - b14 - 2b13 + b10 + b7 + b4) q^53 + (b15 + b14 - b12 + b9 - b4 - b2 - 2b1 - 2) q^55 + q^56 + (b15 - b14 - b12 + b11 + b8 + b6 + b5 + 3b4 - b2 + 1) q^58 + (-2b12 - 2b4) * q^59 + (-b10 + b7 - b2 - 2) * q^61 + (b10 - b7 + 2b3 - 2) q^62 - b5 * q^64 + (b15 - 2b12 + b10 + b9 + b7 - b6 + 2b5 - 2b4 - 2b1 - 5) * q^65 + (-b14 - b13 + b11 + 2b10 + b6 + b5 - 2b4 + b3 - b2) * q^67 + (b12 - b11 + b9) * q^68 - b10 * q^70 + (-2b13 + 2b11 + 2b7 + 2b5 - 2b3 + 2b2 + 2) * q^71 + (b15 + b14 + b13 - b11 - b8 - 3b7 + b6 - 2b5 + b3 - b2 - 2b1 - 2) q^73 + (-b13 + b12 + 2b11 + b10 + b9 + b7 + b5) q^74 + (-b15 - b13 + b8 + b7 + b5 - b3 + 2b1 + 2) q^76 + (-b15 - b14 + b3 + b1 + 1) * q^77 + (-b15 - b14 - b12 + b9 + b8 + b6 - 3b4 - 2b2 - 3b1 + 2) q^79 - b7 * q^80 + (b15 - b14 + 2b11 + b10 - b8 + b7 + b6 + 2b5 + 2b4 - 2b1) * q^82 + (-2b14 + 2b13 + 2b7 - 2b6 + 2b5 + 2b3) * q^83 + (-2b15 + b14 - b13 - b9 + 2b8 - 2b7 + b6 - b5 - b3 + 9b1 + 2) * q^85 + (b14 - b13 + b12 - b6 + b5 - b4 + b3 - 2) * q^86 + (-b13 - b8 + b6 - b5 - b1) * q^88 + (-2b13 - 2b12 + b11 + 2b10 + 2b5 - 2b4 + 2b3 - b2 - 2) * q^89 + (-b13 - b8 - b4 + b2) * q^91 + (-b13 + b8 - b6 - b5 + b1) * q^92 + (b15 + b14 + b12 - b10 - b9 + b7 + b4 - 2b3) q^94 + (b15 + b14 - 2b10 + 2b7 - b3 - 4b2 - b1 + 3) q^95 + (-b15 - 2b13 + b11 - b8 + b5 - 2b4 + 2b3 - b2 - 2) q^97 + b4 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{5} + 8 q^{11} + 4 q^{13} - 16 q^{16} + 8 q^{17} + 16 q^{19} + 4 q^{20} - 8 q^{22} - 24 q^{23} - 8 q^{26} - 8 q^{31} + 4 q^{34} - 4 q^{37} - 16 q^{38} + 16 q^{41} + 8 q^{44} - 4 q^{47} + 16 q^{50}+ \cdots - 28 q^{97}+O(q^{100}) \) 16 * q - 4 * q^5 + 8 * q^11 + 4 * q^13 - 16 * q^16 + 8 * q^17 + 16 * q^19 + 4 * q^20 - 8 * q^22 - 24 * q^23 - 8 * q^26 - 8 * q^31 + 4 * q^34 - 4 * q^37 - 16 * q^38 + 16 * q^41 + 8 * q^44 - 4 * q^47 + 16 * q^50 - 16 * q^55 + 16 * q^56 + 4 * q^58 + 8 * q^59 - 40 * q^61 - 24 * q^62 - 56 * q^65 - 4 * q^67 - 4 * q^70 + 24 * q^71 - 12 * q^73 + 16 * q^76 + 8 * q^77 + 16 * q^79 + 4 * q^80 + 8 * q^85 - 24 * q^86 - 16 * q^89 + 8 * q^91 - 8 * q^94 + 40 * q^95 - 28 * q^97

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	1638.2.y.c

Level	$1638$
Weight	$2$
Character orbit	1638.y
Analytic conductor	$13.079$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(13.0794958511\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 8 x^{15} + 20 x^{14} - 12 x^{13} + 40 x^{12} + 40 x^{11} + 82 x^{10} + 104 x^{9} + 537 x^{8} + \cdots + 16 \) x^16 - 8x^15 + 20x^14 - 12x^13 + 40x^12 + 40x^11 + 82x^10 + 104x^9 + 537x^8 - 2528x^7 + 5494x^6 - 7284x^5 + 6002x^4 - 2984x^3 + 1000x^2 - 192*x + 16
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{5} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( 88\!\cdots\!87 \nu^{15} + \cdots - 12\!\cdots\!20 ) / 29\!\cdots\!12 \) (8810453484481419787v^15 - 69972490944959614420v^14 + 172850373062827022431v^13 - 100481314110617395302v^12 + 354666884436870105751v^11 + 376243159872598807668v^10 + 772580391658572562209v^9 + 1020795990219764087478v^8 + 4903094374641295408884v^7 - 21808020956822433751512v^6 + 47667259736333919738010v^5 - 62649332902158867279196v^4 + 51391858403342645582656v^3 - 25336716035274686737392v^2 + 8241251957213875681648*v - 1241745401323893045120) / 291433150735989873312
\(\beta_{2}\)	\(=\)	\( ( - 11\!\cdots\!23 \nu^{15} + \cdots - 10\!\cdots\!28 ) / 29\!\cdots\!12 \) (-11698868362125004523v^15 + 83939786441538639147v^14 - 160527514035902012089v^13 - 24321857261037208927v^12 - 414765410197699453211v^11 - 832915315773135893517v^10 - 1496640349486741021843v^9 - 2232874028881130210033v^8 - 7715416262234299811216v^7 + 23777099458901783246312v^6 - 42222271916356105928370v^5 + 40583108581656169060702v^4 - 16870573462608118490784v^3 - 1803781446690608394440v^2 + 1899665583663956469552*v - 1085567858194490072528) / 291433150735989873312
\(\beta_{3}\)	\(=\)	\( ( - 70\!\cdots\!13 \nu^{15} + \cdots - 68\!\cdots\!36 ) / 14\!\cdots\!56 \) (-7056265369041347413v^15 + 50507927046357057783v^14 - 95940201065736020691v^13 - 16319571173558912787v^12 - 250853466954085937957v^11 - 506423030574247080325v^10 - 908623952155848342689v^9 - 1353648933950372385221v^8 - 4661033040459466556800v^7 + 14293572868649033061836v^6 - 25156031875780250361942v^5 + 24115377250520373698014v^4 - 9878164327266277702768v^3 - 1337614489088590409536v^2 + 1223669711029373195936*v - 688035310999430319936) / 145716575367994936656
\(\beta_{4}\)	\(=\)	\( ( - 15\!\cdots\!81 \nu^{15} + \cdots + 58\!\cdots\!56 ) / 29\!\cdots\!12 \) (-15512786429751648481v^15 + 118440678597716652792v^14 - 266427488701355331405v^13 + 84539613688524773590v^12 - 580973445915534642573v^11 - 832189061146464961192v^10 - 1554085951107517057027v^9 - 2140109490702079740406v^8 - 9040315704986997841364v^7 + 36016637613027621087280v^6 - 71714984825576732213630v^5 + 85522905553061971943036v^4 - 59666511517259564371568v^3 + 22015992179237618937056v^2 - 6345035029579124763184*v + 584542587322075416256) / 291433150735989873312
\(\beta_{5}\)	\(=\)	\( ( - 17629042972135 \nu^{15} + 138301296365753 \nu^{14} - 330704749144585 \nu^{13} + \cdots + 11\!\cdots\!76 ) / 211731643404256 \) (-17629042972135v^15 + 138301296365753v^14 - 330704749144585v^13 + 156692672910367v^12 - 672027705019791v^11 - 813094141556031v^10 - 1556179415520171v^9 - 2059136017414191v^8 - 9762784462391544v^7 + 43071159787870216v^6 - 89995923668790394v^5 + 113219568956294482v^4 - 85988580899868352v^3 + 36373121308786984v^2 - 9921841312383600*v + 1132557453324976) / 211731643404256
\(\beta_{6}\)	\(=\)	\( ( 27\!\cdots\!77 \nu^{15} + \cdots - 54\!\cdots\!56 ) / 29\!\cdots\!12 \) (27618690357517314377v^15 - 221460659791030258892v^14 + 555732503777147660849v^13 - 336668411993367414666v^12 + 1102498869243079260809v^11 + 1080922593807350558892v^10 + 2214609403385323639239v^9 + 2767834969348104265578v^8 + 14659355868512024837184v^7 - 70284854675750366215080v^6 + 152474456531607125759006v^5 - 202700550842959912371380v^4 + 167255862936333756869672v^3 - 83367849189249536007984v^2 + 28187891884784858482352*v - 5461217065603874041056) / 291433150735989873312
\(\beta_{7}\)	\(=\)	\( ( 29\!\cdots\!85 \nu^{15} + \cdots - 27\!\cdots\!16 ) / 22\!\cdots\!24 \) (2994205490140966885v^15 - 23725077653206080572v^14 + 57893279211948857965v^13 - 30177590547698697114v^12 + 114652279378357623829v^11 + 128940368648258272180v^10 + 248834741223746675003v^9 + 319025985360768953674v^8 + 1610616387006658622896v^7 - 7479537828009877711936v^6 + 15761322054866684640150v^5 - 20225042951737164171076v^4 + 15655604483093553031400v^3 - 6920281476952229423680v^2 + 1993104387746791628592*v - 279499445405065755616) / 22417934671999221024
\(\beta_{8}\)	\(=\)	\( ( - 21\!\cdots\!35 \nu^{15} + \cdots + 23\!\cdots\!88 ) / 14\!\cdots\!56 \) (-21021125277878065035v^15 + 158317828633569406855v^14 - 345810066853522286123v^13 + 87439838101676457197v^12 - 795843408701129124477v^11 - 1210140853074258222021v^10 - 2273883870064584412477v^9 - 3213503277998927522285v^8 - 12718504372712238057134v^7 + 47300775478873296839804v^6 - 92985688483113292755994v^5 + 108875122499736129093878v^4 - 74087120998872004208428v^3 + 27334688839069881823120v^2 - 8172884498812969815760*v + 230650006331699752288) / 145716575367994936656
\(\beta_{9}\)	\(=\)	\( ( - 49\!\cdots\!43 \nu^{15} + \cdots + 45\!\cdots\!12 ) / 29\!\cdots\!12 \) (-49126513370488191843v^15 + 388757174834211391657v^14 - 946327635507412920985v^13 + 488970196259846469679v^12 - 1884437999079307605855v^11 - 2132152285813122333791v^10 - 4119021926724842141523v^9 - 5298795931979829079119v^8 - 26518146320519299868868v^7 + 122377315584533125740504v^6 - 257602217727197625956850v^5 + 330245570373096907880674v^4 - 255893117550647532091576v^3 + 113290524649632008609896v^2 - 32704211824134297823056*v + 4594234067017961740912) / 291433150735989873312
\(\beta_{10}\)	\(=\)	\( ( 38\!\cdots\!03 \nu^{15} + \cdots - 17\!\cdots\!00 ) / 22\!\cdots\!24 \) (3864658061768228203v^15 - 29847215167948824266v^14 + 69052681399000190511v^13 - 27419531607829393332v^12 + 147208949701147260155v^11 + 195879950407577618886v^10 + 371518171538157562353v^9 + 505845176238109617604v^8 + 2216412847574252038128v^7 - 9156677325533603260864v^6 + 18702343634812872455386v^5 - 23031984120255279800080v^4 + 16843089874635801410280v^3 - 6839508571946680389328v^2 + 1877723782800634003472*v - 173800820357319824000) / 22417934671999221024
\(\beta_{11}\)	\(=\)	\( ( 12\!\cdots\!92 \nu^{15} + \cdots - 86\!\cdots\!40 ) / 63\!\cdots\!72 \) (1276526512941182592v^15 - 9928702582560774307v^14 + 23308379550483686890v^13 - 10035678268519997763v^12 + 48710885983548768572v^11 + 61656767642984327249v^10 + 117634991912270128330v^9 + 157144920416365985515v^8 + 716962068095369669060v^7 - 3073296023857682221012v^6 + 6315975651765399848848v^5 - 7873012308851057290458v^4 + 5881827248365393945584v^3 - 2492515748087927513296v^2 + 721447802675349777936*v - 86403333747699113040) / 6335503276869345072
\(\beta_{12}\)	\(=\)	\( ( - 63\!\cdots\!17 \nu^{15} + \cdots + 28\!\cdots\!88 ) / 29\!\cdots\!12 \) (-63339678856780924817v^15 + 489378408129875891831v^14 - 1132759458792787778755v^13 + 449310809676387503553v^12 - 2407748268319976601885v^11 - 3198697671069615739577v^10 - 6063214626278609704081v^9 - 8241672942777890208849v^8 - 36247032904767641034644v^7 + 150243396566568180601152v^6 - 306755660242939679338134v^5 + 377291053196996031008750v^4 - 275740924695600545790456v^3 + 111801051146031036422952v^2 - 30720251742910176984016*v + 2843022865035613822288) / 291433150735989873312
\(\beta_{13}\)	\(=\)	\( ( 37\!\cdots\!34 \nu^{15} + \cdots - 25\!\cdots\!84 ) / 14\!\cdots\!56 \) (37037873510832940534v^15 - 288250772557220061321v^14 + 677750823115831653832v^13 - 294961959206387202913v^12 + 1414642074489096799934v^11 + 1785615929598208319327v^10 + 3410112571569873689484v^9 + 4558720902396543677761v^8 + 20812646058759375633528v^7 - 89220721495115426880088v^6 + 183776514720788436149812v^5 - 229381109249059549524710v^4 + 171629244002356527144712v^3 - 72756860161951393888760v^2 + 21045328421921509411184*v - 2518978615690731247584) / 145716575367994936656
\(\beta_{14}\)	\(=\)	\( ( 97\!\cdots\!63 \nu^{15} + \cdots - 89\!\cdots\!60 ) / 29\!\cdots\!12 \) (97877540218991487563v^15 - 773428726504321982064v^14 + 1881186256855318965939v^13 - 985083990357572690098v^12 + 3803103338086808595731v^11 + 4303488155661520466960v^10 + 8423861562924163947573v^9 + 10998554188225265272386v^8 + 53615137411232141953176v^7 - 242208722605567502092176v^6 + 513739714253880450210650v^5 - 660948392250966820874916v^4 + 518214954251064184683624v^3 - 234920709364472643874720v^2 + 68990110593562938689168*v - 8959875000406315192960) / 291433150735989873312
\(\beta_{15}\)	\(=\)	\( ( - 25\!\cdots\!77 \nu^{15} + \cdots + 12\!\cdots\!92 ) / 63\!\cdots\!72 \) (-2506621258792392877v^15 + 19407561787926980496v^14 - 45070576058898156183v^13 + 17989365574802015890v^12 - 94589569346544940167v^11 - 124819562231765638096v^10 - 235321088511760626589v^9 - 317406292972740382954v^8 - 1420703207804796672278v^7 + 5981076301653901757392v^6 - 12191705274702342508766v^5 + 14975962328106194724620v^4 - 10909298670034740842996v^3 + 4360594470258801958640v^2 - 1189717155114541656976*v + 126368538107948144992) / 6335503276869345072

\(\nu\)	\(=\)	\( ( \beta_{12} + \beta_{10} - \beta_{4} + \beta_{3} - \beta_{2} + 1 ) / 2 \) (b12 + b10 - b4 + b3 - b2 + 1) / 2
\(\nu^{2}\)	\(=\)	\( ( 2 \beta_{15} - \beta_{13} + 2 \beta_{12} + 3 \beta_{11} + 4 \beta_{10} + 2 \beta_{8} - \beta_{5} + \cdots + 3 ) / 2 \) (2b15 - b13 + 2b12 + 3b11 + 4b10 + 2b8 - b5 - 6b4 + b3 - 3*b2 + 3) / 2
\(\nu^{3}\)	\(=\)	\( ( 6 \beta_{15} - 4 \beta_{14} - 12 \beta_{13} + 12 \beta_{12} + 20 \beta_{11} + 20 \beta_{10} + 3 \beta_{9} + \cdots + 9 ) / 2 \) (6b15 - 4b14 - 12b13 + 12b12 + 20b11 + 20b10 + 3b9 + 4b8 + 7b7 - 12b5 - 12b4 + 3b3 - 7b2 + 9b1 + 9) / 2
\(\nu^{4}\)	\(=\)	\( 16 \beta_{15} - 16 \beta_{14} - 24 \beta_{13} + 16 \beta_{12} + 47 \beta_{11} + 34 \beta_{10} + \cdots + 41 \beta_1 \) 16b15 - 16b14 - 24b13 + 16b12 + 47b11 + 34b10 + 16b9 + 5b8 + 34b7 - 5b6 - 36b5 - 20b4 + 41*b1
\(\nu^{5}\)	\(=\)	\( ( 100 \beta_{15} - 142 \beta_{14} - 220 \beta_{13} + 83 \beta_{12} + 414 \beta_{11} + 167 \beta_{10} + \cdots - 211 ) / 2 \) (100b15 - 142b14 - 220b13 + 83b12 + 414b11 + 167b10 + 220b9 + 414b7 - 100b6 - 262b5 - 111b4 - 83b3 + 167b2 + 504b1 - 211) / 2
\(\nu^{6}\)	\(=\)	\( ( 250 \beta_{15} - 634 \beta_{14} - 735 \beta_{13} + 1453 \beta_{11} + 1050 \beta_{9} - 250 \beta_{8} + \cdots - 1809 ) / 2 \) (250b15 - 634b14 - 735b13 + 1453b11 + 1050b9 - 250b8 + 2044b7 - 634b6 - 925b5 - 735b3 + 1453b2 + 2634b1 - 1809) / 2
\(\nu^{7}\)	\(=\)	\( ( - 2188 \beta_{14} - 1852 \beta_{13} - 1852 \beta_{12} + 3620 \beta_{11} - 3620 \beta_{10} + \cdots - 10939 ) / 2 \) (-2188b14 - 1852b13 - 1852b12 + 3620b11 - 3620b10 + 4549b9 - 2188b8 + 8781b7 - 3094b6 - 2358b5 + 2358b4 - 4549b3 + 8781b2 + 10939b1 - 10939) / 2
\(\nu^{8}\)	\(=\)	\( - 2736 \beta_{15} - 2736 \beta_{14} - 7955 \beta_{12} - 15504 \beta_{10} + 7955 \beta_{9} + \cdots - 27595 \) -2736b15 - 2736b14 - 7955b12 - 15504b10 + 7955b9 - 6665b8 + 15504b7 - 6665b6 + 10007b4 - 11269b3 + 21902b2 + 19408b1 - 27595
\(\nu^{9}\)	\(=\)	\( ( - 46918 \beta_{15} + 39848 \beta_{13} - 96507 \beta_{12} - 77484 \beta_{11} - 187229 \beta_{10} + \cdots - 234351 ) / 2 \) (-46918b15 + 39848b13 - 96507b12 - 77484b11 - 187229b10 + 39848b9 - 66342b8 + 77484b7 - 46918b6 + 50230b5 + 121091b4 - 96507b3 + 187229b2 + 97148b1 - 234351) / 2
\(\nu^{10}\)	\(=\)	\( ( - 283736 \beta_{15} + 117332 \beta_{14} + 340559 \beta_{13} - 481754 \beta_{12} - 661931 \beta_{11} + \cdots - 829369 ) / 2 \) (-283736b15 + 117332b14 + 340559b13 - 481754b12 - 661931b11 - 935908b10 - 283736b8 - 117332b6 + 428301b5 + 606606b4 - 340559b3 + 661931b2 - 829369) / 2
\(\nu^{11}\)	\(=\)	\( ( - 1417662 \beta_{15} + 1002490 \beta_{14} + 2057836 \beta_{13} - 2057836 \beta_{12} - 3996234 \beta_{11} + \cdots - 2074251 ) / 2 \) (-1417662b15 + 1002490b14 + 2057836b13 - 2057836b12 - 3996234b11 - 3996234b10 - 851889b9 - 1002490b8 - 1655015b7 + 2586654b5 + 2586654b4 - 851889b3 + 1655015b2 - 2074251b1 - 2074251) / 2
\(\nu^{12}\)	\(=\)	\( - 3027035 \beta_{15} + 3027035 \beta_{14} + 5143960 \beta_{13} - 3637174 \beta_{12} + \cdots - 8854439 \beta_1 \) -3027035b15 + 3027035b14 + 5143960b13 - 3637174b12 - 9992083b11 - 7065772b10 - 3637174b9 - 1253452b8 - 7065772b7 + 1253452b6 + 6470338b5 + 4573952b4 - 8854439*b1
\(\nu^{13}\)	\(=\)	\( ( - 21405892 \beta_{15} + 30272086 \beta_{14} + 43923136 \beta_{13} - 18191625 \beta_{12} + \cdots + 44284357 ) / 2 \) (-21405892b15 + 30272086b14 + 43923136b13 - 18191625b12 - 85313194b11 - 35336741b10 - 43923136b9 - 85313194b7 + 21405892b6 + 55229886b5 + 22878465b4 + 18191625b3 - 35336741b2 - 106907864b1 + 44284357) / 2
\(\nu^{14}\)	\(=\)	\( ( - 53528366 \beta_{15} + 129236330 \beta_{14} + 155317897 \beta_{13} - 301699343 \beta_{11} + \cdots + 378082599 ) / 2 \) (-53528366b15 + 129236330b14 + 155317897b13 - 301699343b11 - 219654094b9 + 53528366b8 - 426663732b7 + 129236330b6 + 195317903b5 + 155317897b3 - 301699343b2 - 534708278b1 + 378082599) / 2
\(\nu^{15}\)	\(=\)	\( ( 457017240 \beta_{14} + 388396692 \beta_{13} + 388396692 \beta_{12} - 754432232 \beta_{11} + \cdots + 2282489621 ) / 2 \) (457017240b14 + 388396692b13 + 388396692b12 - 754432232b11 + 754432232b10 - 937690835b9 + 457017240b8 - 1821371807b7 + 646317826b6 + 488428422b5 - 488428422b4 + 937690835b3 - 1821371807b2 - 2282489621b1 + 2282489621) / 2

\(n\)	\(379\)	\(703\)	\(911\)
\(\chi(n)\)	\(-\beta_{5}\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 827.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} + 1)^{4} \) (T^4 + 1)^4
$3$	\( T^{16} \) T^16
$5$	\( T^{16} + 4 T^{15} + \cdots + 1024 \) T^16 + 4T^15 + 8T^14 - 16T^13 + 182T^12 + 592T^11 + 1040T^10 - 1464T^9 + 5625T^8 + 14284T^7 + 21512T^6 - 20328T^5 + 17988T^4 + 17216T^3 + 12800T^2 - 5120*T + 1024
$7$	\( (T^{4} + 1)^{4} \) (T^4 + 1)^4
$11$	\( T^{16} - 8 T^{15} + \cdots + 65536 \) T^16 - 8T^15 + 32T^14 + 40T^13 + 210T^12 - 2264T^11 + 12192T^10 + 22568T^9 + 3313T^8 - 4432T^7 + 834688T^6 + 2543360T^5 + 3863040T^4 + 3190784T^3 + 1605632T^2 + 458752*T + 65536
$13$	\( T^{16} + \cdots + 815730721 \) T^16 - 4T^15 + 10T^14 - 116T^13 + 368T^12 - 636T^11 + 7334T^10 - 26700T^9 + 44414T^8 - 347100T^7 + 1239446T^6 - 1397292T^5 + 10510448T^4 - 43069988T^3 + 48268090T^2 - 250994068*T + 815730721
$17$	\( (T^{8} - 4 T^{7} + \cdots - 2312)^{2} \) (T^8 - 4T^7 - 84T^6 + 312T^5 + 1573T^4 - 5964T^3 + 2594T^2 + 5032*T - 2312)^2
$19$	\( T^{16} - 16 T^{15} + \cdots + 12845056 \) T^16 - 16T^15 + 128T^14 - 168T^13 - 734T^12 - 328T^11 + 113312T^10 + 46680T^9 - 414847T^8 + 4641032T^7 + 54226464T^6 + 191160128T^5 + 372654144T^4 + 394622976T^3 + 248020992T^2 + 79822848*T + 12845056
$23$	\( (T^{8} + 12 T^{7} + \cdots + 4096)^{2} \) (T^8 + 12T^7 - 30T^6 - 820T^5 - 1847T^4 + 11896T^3 + 57288T^2 + 68096*T + 4096)^2
$29$	\( T^{16} + \cdots + 359548941376 \) T^16 + 312T^14 + 39258T^12 + 2596884T^10 + 98553369T^8 + 2182571692T^6 + 27126963700T^4 + 167244176032*T^2 + 359548941376
$31$	\( T^{16} + 8 T^{15} + \cdots + 262144 \) T^16 + 8T^15 + 32T^14 + 128T^13 + 6872T^12 + 60160T^11 + 269568T^10 + 437376T^9 + 3734928T^8 + 30703488T^7 + 139235840T^6 + 197810688T^5 + 152588544T^4 + 55685120T^3 + 7372800T^2 - 1966080*T + 262144
$37$	\( T^{16} + 4 T^{15} + \cdots + 64 \) T^16 + 4T^15 + 8T^14 - 104T^13 + 10806T^12 + 31960T^11 + 46800T^10 + 45024T^9 + 1276297T^8 + 4262868T^7 + 7158600T^6 + 1618120T^5 + 4289556T^4 + 17059840T^3 + 33948800T^2 + 65920*T + 64
$41$	\( T^{16} + \cdots + 685182976 \) T^16 - 16T^15 + 128T^14 - 256T^13 + 6440T^12 - 83328T^11 + 541696T^10 - 1907968T^9 + 8975888T^8 - 67028224T^7 + 395233280T^6 - 1475185664T^5 + 3593976320T^4 - 5613473792T^3 + 5427986432T^2 - 2727329792*T + 685182976
$43$	\( T^{16} + \cdots + 2842169344 \) T^16 + 220T^14 + 17942T^12 + 697204T^10 + 14239777T^8 + 157505528T^6 + 921533456T^4 + 2625534464*T^2 + 2842169344
$47$	\( T^{16} + \cdots + 27518828544 \) T^16 + 4T^15 + 8T^14 + 216T^13 + 20036T^12 + 118848T^11 + 338432T^10 - 1206784T^9 + 15795712T^8 + 67055616T^7 + 162791424T^6 - 396361728T^5 + 755122176T^4 + 1911029760T^3 + 6115295232T^2 - 18345885696*T + 27518828544
$53$	\( T^{16} + \cdots + 67051995136 \) T^16 + 476T^14 + 82452T^12 + 6493600T^10 + 241206976T^8 + 4127602688T^6 + 28770413568T^4 + 76843917312*T^2 + 67051995136
$59$	\( T^{16} + \cdots + 268435456 \) T^16 - 8T^15 + 32T^14 + 320T^13 + 2656T^12 - 13568T^11 + 74752T^10 + 755712T^9 + 3033344T^8 - 673792T^7 + 43655168T^6 + 417677312T^5 + 1712603136T^4 + 3393716224T^3 + 3699376128T^2 + 1409286144*T + 268435456
$61$	\( (T^{8} + 20 T^{7} + \cdots - 5768)^{2} \) (T^8 + 20T^7 + 88T^6 - 328T^5 - 3947T^4 - 13556T^3 - 22414T^2 - 18168*T - 5768)^2
$67$	\( T^{16} + \cdots + 268435456 \) T^16 + 4T^15 + 8T^14 + 568T^13 + 18564T^12 + 149632T^11 + 611328T^10 + 305664T^9 + 655360T^8 + 8232960T^7 + 47480832T^6 + 7143424T^5 - 26148864T^4 - 71303168T^3 + 536870912T^2 - 536870912*T + 268435456
$71$	\( T^{16} + \cdots + 112810000384 \) T^16 - 24T^15 + 288T^14 - 1088T^13 + 31328T^12 - 692480T^11 + 8188928T^10 - 31752192T^9 + 185395456T^8 - 2771843072T^7 + 30898266112T^6 - 128592363520T^5 + 279206707200T^4 - 130470117376T^3 + 4060086272T^2 + 30266097664*T + 112810000384
$73$	\( T^{16} + \cdots + 31085921344 \) T^16 + 12T^15 + 72T^14 + 576T^13 + 24822T^12 + 322400T^11 + 2247504T^10 + 9330776T^9 + 92019497T^8 + 1039168884T^7 + 7402485512T^6 + 27023031128T^5 + 48299316436T^4 - 6669802656T^3 + 8733132800T^2 + 23301393920*T + 31085921344
$79$	\( (T^{8} - 8 T^{7} + \cdots - 210944)^{2} \) (T^8 - 8T^7 - 236T^6 + 2128T^5 + 12004T^4 - 128512T^3 + 57056T^2 + 612864*T - 210944)^2
$83$	\( T^{16} + \cdots + 53396107165696 \) T^16 - 320T^13 + 54816T^12 - 89344T^11 + 51200T^10 + 392192T^9 + 752914688T^8 - 1538445312T^7 + 1059094528T^6 + 122763673600T^5 + 1644926402560T^4 + 2958508949504T^3 + 1638832013312T^2 - 13229304578048*T + 53396107165696
$89$	\( T^{16} + \cdots + 2699475288064 \) T^16 + 16T^15 + 128T^14 + 96T^13 + 38488T^12 + 622400T^11 + 5036544T^10 + 2168960T^9 + 162127248T^8 + 2549594112T^7 + 20093812736T^6 + 12822735872T^5 + 133923213568T^4 + 2103308570624T^3 + 17029634752512T^2 + 9588647264256*T + 2699475288064
$97$	\( T^{16} + \cdots + 41943040000 \) T^16 + 28T^15 + 392T^14 + 968T^13 - 1068T^12 + 73440T^11 + 2943488T^10 - 3982080T^9 + 1719360T^8 - 9449472T^7 + 1047134208T^6 - 2376466432T^5 + 2690514944T^4 + 3334471680T^3 + 30198988800T^2 - 50331648000*T + 41943040000
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