LMFDB - Newform orbit 768.3.h.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,3,Mod(641,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("768.641");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	768.h (of order $2$, degree $1$, 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:	$20.9264843029$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 14 x^{14} - 28 x^{13} + 50 x^{12} - 104 x^{11} - 66 x^{10} + 640 x^{9} + 555 x^{8} + \cdots + 2401 $ x^16 - 4x^15 + 14x^14 - 28x^13 + 50x^12 - 104x^11 - 66x^10 + 640x^9 + 555x^8 - 7060x^7 + 17714x^6 - 25496x^5 + 24840x^4 - 17932x^3 + 11724x^2 - 7056*x + 2401
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{34}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

$f(q)$	$=$	$ q - \beta_{8} q^{3} - \beta_{11} q^{5} + ( - \beta_1 + 1) q^{7} + ( - \beta_{13} + \beta_1) q^{9}+O(q^{10}) $ q - b8 * q^3 - b11 * q^5 + (-b1 + 1) * q^7 + (-b13 + b1) * q^9	$ q - \beta_{8} q^{3} - \beta_{11} q^{5} + ( - \beta_1 + 1) q^{7} + ( - \beta_{13} + \beta_1) q^{9} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_{2}) q^{11}+ \cdots + ( - \beta_{14} - 4 \beta_{11} + \cdots - 10 \beta_{2}) q^{99}+O(q^{100}) $ q - b8 * q^3 - b11 * q^5 + (-b1 + 1) * q^7 + (-b13 + b1) * q^9 + (-b11 + b10 - b5 - b2) * q^11 + (b9 - 2b8 - b5 + b3) q^13 + (-b15 + b13 - b12 + b7 - b1 - 2) * q^15 + (-b15 + b13 - 2b12 - b6 - b1) q^17 + (-b14 - b9 + b8 - b3) * q^19 + (b14 - b10 + b9 - b8 + 2b5 - b3 + 2b2) * q^21 + (2b15 - 2b13 + b7 + b6 + b1) * q^23 + (-b13 + b12 - b6 - b4 - 3b1 + 5) q^25 + (-b14 + 3b11 + b10 + b8 + b5 + b3 + b2) q^27 + (b11 + 2b10 + 2b9 + 6b8 - b5) q^29 + (2b13 - b12 + b7 + b6 + b4 + 7) q^31 + (-b15 + 2b13 + 2b7 - b6 + 2b4 + 1) q^33 + (-7b11 - b10 - b9 - 3b8 + 3b5 + 5b2) * q^35 + (-2b14 + b9 - 6b8 - b3) * q^37 + (-b15 - 3b13 + 3b12 + b6 + 2b4 + b1 - 13) q^39 + (3b15 - 2b13 + b12 + 2b7 + 5b4) * q^41 + (-b14 + b9 - b8 - 6b5 + 3b3) * q^43 + (b14 + 2b11 + b10 + 4b9 + 3b8 - 2b5 - 6b2) q^45 + (2b13 - b12 - 2b6 + 3b4 - 2b1) * q^47 + (3b13 + b12 + 4b7 - b6 - b4 - 3b1 + 9) q^49 + (5b11 - b10 - b9 + 3b8 + 5b5 - 2b3 - 5b2) q^51 + (3b11 - 4b10 + 4b9 + 12b8 - 4b2) q^53 + (-2b13 + b12 - b7 - b6 - b4 + b1 + 24) q^55 + (3b15 + b13 - 3b12 + 2b7 - 2b6 - 3b4 - 3b1 + 5) * q^57 + (b9 + 3b8 - 4b5 - 8b2) q^59 + (-2b14 + 3b9 - 10b8 + 2b5 + b3) * q^61 + (-2b15 - 3b12 - 4b7 - 5b4 + 3b1 - 33) q^63 + (b15 - 7b12 + 2b7 - 2b6 + b4 - 2b1) * q^65 + (4b14 + 3b9 - 5b8 - 10b5) * q^67 + (-3b14 - 3b11 + 3b10 + 2b9 + b8 + 3b5 + 6b2) * q^69 + (2b15 + 4b13 - 3b12 - 3b7 - b6 - 7b4 - b1) q^71 + (2b12 + 2b7 - 2b6 - 2b4 + 2b1 + 10) q^73 + (5b14 + b11 - b10 + b9 - 2b8 + 17b5 - 3b3 + 9b2) q^75 + (2b11 - 4b10 + 4b9 + 12b8 + 8b5 + 12b2) * q^77 + (-2b13 + 3b12 + b7 - 3b6 - 3b4 + 51) * q^79 + (2b15 - 3b13 + 3b12 - 2b7 - 3b6 + 5b4 - 3b1 + 9) q^81 + (9b11 - b10 + 2b9 + 6b8 + 3b5 + 5b2) q^83 + (8b14 + 8b9 - 16b8 + 2b5) * q^85 + (-b15 + 7b13 + 2b12 - b7 - 4b6 + 3b4 - 5b1 - 46) q^87 + (-8b15 + b13 - b12 + 2b7 - 3b6 + 9b4 - 3b1) q^89 + (5b14 - 2b9 + 4b8 - 6b5 - 7b3) q^91 + (-2b14 - 5b11 + 3b9 - 10b8 - 12b5 + b3 - 8b2) * q^93 + (2b15 - 2b13 + 16b12 - 3b7 + 5b6 + 5b1) * q^95 + (-5b13 + 3b12 - 2b7 - 3b6 - 3b4 - b1 + 12) q^97 + (-b14 - 4b11 + b9 - b8 + 6b5 + 5b3 - 10b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{7}+O(q^{10}) $ 16 * q + 16 * q^7	$ 16 q + 16 q^{7} - 32 q^{15} + 80 q^{25} + 112 q^{31} + 16 q^{33} - 208 q^{39} + 144 q^{49} + 384 q^{55} + 80 q^{57} - 528 q^{63} + 160 q^{73} + 816 q^{79} + 144 q^{81} - 736 q^{87} + 192 q^{97}+O(q^{100}) $ 16 * q + 16 * q^7 - 32 * q^15 + 80 * q^25 + 112 * q^31 + 16 * q^33 - 208 * q^39 + 144 * q^49 + 384 * q^55 + 80 * q^57 - 528 * q^63 + 160 * q^73 + 816 * q^79 + 144 * q^81 - 736 * q^87 + 192 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4 x^{15} + 14 x^{14} - 28 x^{13} + 50 x^{12} - 104 x^{11} - 66 x^{10} + 640 x^{9} + 555 x^{8} + \cdots + 2401 $ x^16 - 4*x^15 + 14*x^14 - 28*x^13 + 50*x^12 - 104*x^11 - 66*x^10 + 640*x^9 + 555*x^8 - 7060*x^7 + 17714*x^6 - 25496*x^5 + 24840*x^4 - 17932*x^3 + 11724*x^2 - 7056*x + 2401 :

$\beta_{1}$	$=$	$ ( - 73\!\cdots\!90 \nu^{15} + \cdots - 54\!\cdots\!89 ) / 43\!\cdots\!81 $ (-73605237118565390v^15 - 47633572835668896v^14 - 139688548619286130v^13 - 1233457767323771308v^12 + 529018760445806044v^11 - 1261831727497275602v^10 + 24100324820579854794v^9 + 7093423907199410714v^8 - 202037586301504503744v^7 + 61203334305725712986v^6 + 615363780535269695166v^5 - 1135776337355918990048v^4 + 1101140289291794144046v^3 - 795703763476158680914v^2 + 402147901328092287564*v - 544249893260957625489) / 43737612461167616781
$\beta_{2}$	$=$	$ ( - 99\!\cdots\!44 \nu^{15} + \cdots - 88\!\cdots\!24 ) / 53\!\cdots\!97 $ (-99362376719317588592644v^15 + 157643615961293517755580v^14 - 654015690334010596895884v^13 + 213271506955874822687428v^12 - 995377489052373457680032v^11 + 3156489507901330068044614v^10 + 22834363346293880559522940v^9 - 29651658206534911182424006v^8 - 185396532173326788843206856v^7 + 434141249588399178636341586v^6 - 249135055907033040587373936v^5 - 192281761863540202775765666v^4 + 394936329964996076653983336v^3 - 333975898645644202702191118v^2 - 104408253736098716785228332*v - 88262663720799637363681324) / 53820750425127815650841397
$\beta_{3}$	$=$	$ ( 18\!\cdots\!55 \nu^{15} + \cdots - 30\!\cdots\!38 ) / 53\!\cdots\!97 $ (189224219934601032264355v^15 - 640202263351630099843971v^14 + 1553120990278286891368081v^13 - 3621706423247909764322422v^12 + 1682269426299819388064648v^11 - 18840296358896891889846508v^10 - 40628742994151317163457289v^9 + 110448656540655743708472337v^8 + 320568277591574100193142280v^7 - 1216821803698994193762710484v^6 + 1574248391414505437517732471v^5 - 1486026968627483519601442321v^4 + 1132201301739918183217506765v^3 - 1417240917087734151396424793v^2 + 998815185235190819052860430*v - 303265474878736101234979238) / 53820750425127815650841397
$\beta_{4}$	$=$	$ ( 36\!\cdots\!58 \nu^{15} + \cdots - 16\!\cdots\!97 ) / 69\!\cdots\!61 $ (3640490453456954097858v^15 - 16696296122407843524494v^14 + 51123108622447450476286v^13 - 115611452811381285277960v^12 + 161505766591708053420364v^11 - 415306479428912573409582v^10 - 265512122939865771157942v^9 + 2864130346548477692928604v^8 + 2134124510400861462739424v^7 - 29166355503786286734932230v^6 + 69392925277420838243005046v^5 - 93214027840038103299879880v^4 + 78452027971271711503230598v^3 - 44569897710476397997173428v^2 + 27020900329861475554330964*v - 16533045201889393898502397) / 698970784741919683777161
$\beta_{5}$	$=$	$ ( 72684469693760 \nu^{15} - 224840478635056 \nu^{14} + 808003344635480 \nu^{13} + \cdots - 18\!\cdots\!80 ) / 11\!\cdots\!97 $ (72684469693760v^15 - 224840478635056v^14 + 808003344635480v^13 - 1282221816062488v^12 + 2428767128904824v^11 - 5282208866934684v^10 - 9590853089188752v^9 + 38143824965723228v^8 + 76037812918776760v^7 - 448764552639400932v^6 + 870266063458707304v^5 - 1015163612568399884v^4 + 845369308904828096v^3 - 555948001648106716v^2 + 399142132155581296*v - 187269373357533880) / 11795069338202997
$\beta_{6}$	$=$	$ ( - 30\!\cdots\!10 \nu^{15} + \cdots + 73\!\cdots\!59 ) / 48\!\cdots\!27 $ (-30964682511934783503810v^15 + 122223497851645985487134v^14 - 351087091130861083128466v^13 + 722317551925352307477868v^12 - 830174841462470676732352v^11 + 2767985707332668051783076v^10 + 4213575685143668590694674v^9 - 22382276527890369452376370v^8 - 32569021563057742201550024v^7 + 234941060542798650808228840v^6 - 435187087035768768804873974v^5 + 449020567409998748155182748v^4 - 249729491763515335269157846v^3 + 96943521582465721562442746v^2 - 112158128752047556736724332*v + 73635341491264992735750559) / 4892795493193437786440127
$\beta_{7}$	$=$	$ ( 35\!\cdots\!00 \nu^{15} + \cdots + 36\!\cdots\!72 ) / 48\!\cdots\!27 $ (35220176603425455380400v^15 - 32006703464375436767938v^14 + 238371309052252800044600v^13 + 36936950069573329739656v^12 + 521383090631562825694928v^11 - 932380116944940123230634v^10 - 8781276532560910172334860v^9 + 2724335024283234546960200v^8 + 59396498158646402591599588v^7 - 105822817363836738019230770v^6 + 79729091073366660116651896v^5 + 15786583481454523793225260v^4 - 103299492669661029430155556v^3 + 30926645260328445790809788v^2 + 21468729734104125727704124*v + 36707798999442881296523572) / 4892795493193437786440127
$\beta_{8}$	$=$	$ ( - 49\!\cdots\!07 \nu^{15} + \cdots + 94\!\cdots\!76 ) / 53\!\cdots\!97 $ (-496011314854612629237607v^15 + 1407423641116043044984739v^14 - 5236363327383127844285983v^13 + 7674579346797568635198524v^12 - 15170187401690208428739514v^11 + 33483168646838080662868536v^10 + 74147872428604737899472801v^9 - 233702847930081684199658665v^8 - 558017997097579641379188182v^7 + 2872720671982035201104284980v^6 - 5364246117903744242824857047v^5 + 6097981717619334251998696003v^4 - 4691038047271369875397782463v^3 + 2830934339193932474639480993v^2 - 1935825001234129541998511804*v + 948191375097728418362664476) / 53820750425127815650841397
$\beta_{9}$	$=$	$ ( 16\!\cdots\!93 \nu^{15} + \cdots - 53\!\cdots\!70 ) / 17\!\cdots\!99 $ (168205981222649866459993v^15 - 722283452909459301095925v^14 + 2200838562375756723362047v^13 - 4632646557071839349088766v^12 + 6619724633870462883487076v^11 - 16474709494587523434622366v^10 - 14166933236368567892115763v^9 + 130623547662795020307028957v^8 + 129084446355814662641230164v^7 - 1332938102861827597833812202v^6 + 2849430799189762299032510721v^5 - 3465449247703949276546491501v^4 + 2786498808998386784147595327v^3 - 1596024018956010006398663969v^2 + 1098557546541429389083110714*v - 531388868455846783559913470) / 17940250141709271883613799
$\beta_{10}$	$=$	$ ( 55\!\cdots\!36 \nu^{15} + \cdots - 79\!\cdots\!80 ) / 53\!\cdots\!97 $ (551433367755288482572036v^15 - 1408219547486122305150558v^14 + 5735315644397360890020964v^13 - 7688654266282923636617572v^12 + 17692316089595233177136120v^11 - 35945008775855893630086130v^10 - 85094055795357376986267568v^9 + 217705740186354083996814700v^8 + 624628548695658880225303860v^7 - 2887010849285275188911969358v^6 + 5532641255779921708628630832v^5 - 6902862737843796856509056704v^4 + 6125863483899854452295642124v^3 - 4068255658692241415900074508v^2 + 2284234538899424073865200552*v - 797051948299789731534756680) / 53820750425127815650841397
$\beta_{11}$	$=$	$ ( 11\!\cdots\!86 \nu^{15} + \cdots - 13\!\cdots\!90 ) / 10\!\cdots\!53 $ (11913187010808341876886v^15 - 23975944496966615836928v^14 + 106642177371262500986628v^13 - 98301957290644270684438v^12 + 290568605512493599763378v^11 - 573606376298445507057866v^10 - 2228428785917414061095450v^9 + 3750630518098402823426118v^8 + 16446624534719448243157154v^7 - 54982088196114388725677778v^6 + 84288446412857750839274966v^5 - 81148429500991178855466900v^4 + 52304577766421834673743632v^3 - 33176743048971708020631918v^2 + 26181132013428199701150194*v - 1355563446291770005494690) / 1098382661737302360221253
$\beta_{12}$	$=$	$ ( 55\!\cdots\!10 \nu^{15} + \cdots - 79\!\cdots\!29 ) / 48\!\cdots\!27 $ (55024520198138880566010v^15 - 137783157785429009305526v^14 + 563850788327923100286814v^13 - 679041821146738360614520v^12 + 1686037666090608468184996v^11 - 3052264204664955485878014v^10 - 8430821140772456286131806v^9 + 22863780092201702657803180v^8 + 63574381834947111771677288v^7 - 297132976451917827250801510v^6 + 537165476707658079331848830v^5 - 570509132045867792137860712v^4 + 415165850424849819875437414v^3 - 232158640560184879151842796v^2 + 190211240740289941355047052*v - 79874676786308848296228229) / 4892795493193437786440127
$\beta_{13}$	$=$	$ ( 31\!\cdots\!44 \nu^{15} + \cdots - 55\!\cdots\!90 ) / 16\!\cdots\!09 $ (31817579385645488535444v^15 - 88206778931937685372070v^14 + 340478365886124775238104v^13 - 477367827448243316342072v^12 + 1021272473055663921364308v^11 - 2064820988020967478333898v^10 - 4644466413137907833313952v^9 + 14475875407010959495107452v^8 + 34387091160570332254797960v^7 - 181898556717941278052465898v^6 + 347302706535448206066655268v^5 - 395686452807303374493405652v^4 + 306148472751434611559085668v^3 - 188198004344939422253821672v^2 + 121084587780806107656104808*v - 55028840081614189320893790) / 1630931831064479262146709
$\beta_{14}$	$=$	$ ( 19\!\cdots\!87 \nu^{15} + \cdots - 49\!\cdots\!50 ) / 53\!\cdots\!97 $ (1944509727790417716480887v^15 - 6114785714223635371831415v^14 + 21504052543851716703107285v^13 - 35505982914351192419445234v^12 + 63068059782007057113198960v^11 - 147950314564777996000852984v^10 - 265454012643265202130323149v^9 + 1029607431255999657920021909v^8 + 2064908151313945678083008888v^7 - 12029737929564727753564018440v^6 + 23399319641003821026377525075v^5 - 27921974802000853211099091929v^4 + 22502383378998811767386967937v^3 - 14520845490988459183442145385v^2 + 10403553731305134146927457278*v - 4957377610955501313794460850) / 53820750425127815650841397
$\beta_{15}$	$=$	$ ( 21\!\cdots\!08 \nu^{15} + \cdots - 51\!\cdots\!64 ) / 48\!\cdots\!27 $ (212457904015996007722308v^15 - 685391664748231303553084v^14 + 2377104280783034795722072v^13 - 3994202810179077315449392v^12 + 6956242868611039002966124v^11 - 16326080972690283552664140v^10 - 28253227622269491985415116v^9 + 116809191218979853253523988v^8 + 221342755050324522216294260v^7 - 1345616710449714098644922284v^6 + 2627341790001511721947146452v^5 - 3102265819152201974954549176v^4 + 2471016010147884355794089104v^3 - 1540230428741537897930332472v^2 + 1082580752062076026913195444*v - 518021587644632004999733864) / 4892795493193437786440127

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

$\nu$	$=$	$ ( - 6 \beta_{15} + 6 \beta_{14} - 12 \beta_{13} + 9 \beta_{12} + 6 \beta_{11} + 6 \beta_{10} + 8 \beta_{9} + \cdots + 24 ) / 96 $ (-6b15 + 6b14 - 12b13 + 9b12 + 6b11 + 6b10 + 8b9 - 6b8 - 9b5 - 3b4 - 6b2 + 12b1 + 24) / 96
$\nu^{2}$	$=$	$ ( 6 \beta_{13} + 12 \beta_{12} + 2 \beta_{9} + 24 \beta_{8} + 6 \beta_{6} - 21 \beta_{5} + 12 \beta_{4} + \cdots - 36 ) / 48 $ (6b13 + 12b12 + 2b9 + 24b8 + 6b6 - 21b5 + 12b4 + 6b3 - 24b2 + 6b1 - 36) / 48
$\nu^{3}$	$=$	$ ( 21 \beta_{15} - 18 \beta_{14} + 18 \beta_{13} - 75 \beta_{11} + 9 \beta_{10} + 8 \beta_{9} + 21 \beta_{8} + \cdots - 60 ) / 48 $ (21b15 - 18b14 + 18b13 - 75b11 + 9b10 + 8b9 + 21b8 + 39b7 + 9b6 - 12b5 - 3b3 - 12b2 - 39*b1 - 60) / 48
$\nu^{4}$	$=$	$ ( - 6 \beta_{15} + 8 \beta_{13} - 29 \beta_{12} - 4 \beta_{9} + 4 \beta_{7} - 12 \beta_{6} + \cdots - 12 \beta_1 ) / 8 $ (-6b15 + 8b13 - 29b12 - 4b9 + 4b7 - 12b6 + 60b5 + 3b4 - 12b3 - 12b1) / 8
$\nu^{5}$	$=$	$ ( 336 \beta_{15} - 258 \beta_{14} - 816 \beta_{13} - 45 \beta_{12} + 1320 \beta_{11} - 96 \beta_{10} + \cdots + 1584 ) / 96 $ (336b15 - 258b14 - 816b13 - 45b12 + 1320b11 - 96b10 - 368b9 - 684b8 - 522b7 - 6b6 + 225b5 - 81b4 - 90b3 + 246b2 + 894*b1 + 1584) / 96
$\nu^{6}$	$=$	$ ( - 24 \beta_{15} + 102 \beta_{14} - 702 \beta_{13} + 1485 \beta_{12} + 90 \beta_{11} - 42 \beta_{10} + \cdots + 5724 ) / 48 $ (-24b15 + 102b14 - 702b13 + 1485b12 + 90b11 - 42b10 - 458b9 - 978b8 - 582b7 - 36b6 - 882b5 - 135b4 + 558b3 + 1350b2 + 168*b1 + 5724) / 48
$\nu^{7}$	$=$	$ ( - 5622 \beta_{15} + 4230 \beta_{14} + 2424 \beta_{13} + 3285 \beta_{12} - 4938 \beta_{11} + \cdots - 3168 ) / 96 $ (-5622b15 + 4230b14 + 2424b13 + 3285b12 - 4938b11 - 666b10 + 5732b9 - 6246b8 + 1404b7 + 2424b6 - 8445b5 - 207b4 + 2436b3 - 1602b2 - 1764*b1 - 3168) / 96
$\nu^{8}$	$=$	$ ( 711 \beta_{13} - 241 \beta_{12} - 204 \beta_{11} + 240 \beta_{10} + 664 \beta_{9} + 1992 \beta_{8} + \cdots - 7826 ) / 4 $ (711b13 - 241b12 - 204b11 + 240b10 + 664b9 + 1992b8 + 470b7 + 241b6 - 777b5 + 241b4 - 1314b2 + 15b1 - 7826) / 4
$\nu^{9}$	$=$	$ ( 25887 \beta_{15} - 19854 \beta_{14} + 17676 \beta_{13} - 25875 \beta_{12} - 22821 \beta_{11} + \cdots - 29760 ) / 48 $ (25887b15 - 19854b14 + 17676b13 - 25875b12 - 22821b11 - 1305b10 - 18458b9 + 49875b8 + 16473b7 - 4923b6 + 38136b5 + 7461b4 - 13011b3 - 8790b2 - 23115*b1 - 29760) / 48
$\nu^{10}$	$=$	$ ( 9960 \beta_{15} - 15954 \beta_{14} - 56760 \beta_{13} - 81057 \beta_{12} + 29322 \beta_{11} + \cdots + 559224 ) / 48 $ (9960b15 - 15954b14 - 56760b13 - 81057b12 + 29322b11 - 9402b10 - 66508b9 - 164586b8 - 21378b7 - 54186b6 + 221763b5 - 31053b4 - 50208b3 + 112110b2 - 22278*b1 + 559224) / 48
$\nu^{11}$	$=$	$ ( - 213516 \beta_{15} + 164130 \beta_{14} - 643848 \beta_{13} + 307989 \beta_{12} + 1017084 \beta_{11} + \cdots + 2317416 ) / 96 $ (-213516b15 + 164130b14 - 643848b13 + 307989b12 + 1017084b11 - 58836b10 - 118232b9 - 1040016b8 - 592170b7 - 59310b6 - 69249b5 - 120111b4 + 115278b3 + 418986b2 + 688350*b1 + 2317416) / 96
$\nu^{12}$	$=$	$ ( - 26632 \beta_{15} + 39942 \beta_{14} - 34232 \beta_{13} + 240323 \beta_{12} + 54690 \beta_{9} + \cdots + 76622 \beta_1 ) / 8 $ (-26632b15 + 39942b14 - 34232b13 + 240323b12 + 54690b9 - 11016b8 - 42390b7 + 76622b6 - 393759b5 + 2299b4 + 113112b3 + 76622b1) / 8
$\nu^{13}$	$=$	$ ( - 1932090 \beta_{15} + 1515762 \beta_{14} + 6297528 \beta_{13} + 952899 \beta_{12} - 9514074 \beta_{11} + \cdots - 25007400 ) / 96 $ (-1932090b15 + 1515762b14 + 6297528b13 + 952899b12 - 9514074b11 + 591966b10 + 4469908b9 + 6649074b8 + 4687188b7 + 1789152b6 - 6060903b5 + 1037199b4 + 1192080b3 - 4602438b2 - 5285364*b1 - 25007400) / 96
$\nu^{14}$	$=$	$ ( 1690992 \beta_{15} - 2026332 \beta_{14} + 7796130 \beta_{13} - 11968518 \beta_{12} - 4451004 \beta_{11} + \cdots - 52309692 ) / 48 $ (1690992b15 - 2026332b14 + 7796130b13 - 11968518b12 - 4451004b11 + 1069020b10 + 2404114b9 + 14662884b8 + 6394044b7 - 735954b6 + 9964353b5 + 1929762b4 - 4700778b3 - 10038756b2 - 4788618*b1 - 52309692) / 48
$\nu^{15}$	$=$	$ ( 20908563 \beta_{15} - 16832388 \beta_{14} - 10200522 \beta_{13} - 24458910 \beta_{12} + 18616767 \beta_{11} + \cdots + 52459200 ) / 48 $ (20908563b15 - 16832388b14 - 10200522b13 - 24458910b12 + 18616767b11 - 824673b10 - 25152404b9 + 7965531b8 - 4527519b7 - 11743989b6 + 52696743b5 - 224274b4 - 14701467b3 + 10462974b2 + 2479347*b1 + 52459200) / 48

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$1$	$-1$

Embeddings

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

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

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

641.1

1.38361	−	0.573111i
1.38361	+	0.573111i
0.463678	+	1.11942i
0.463678	−	1.11942i
−0.280847	+	0.678024i
−0.280847	−	0.678024i
0.926870	−	0.383922i
0.926870	+	0.383922i
−1.09921	−	2.65372i
−1.09921	+	2.65372i
−2.18318	−	0.904303i
−2.18318	+	0.904303i
1.57980	−	0.654376i
1.57980	+	0.654376i
1.20927	+	2.91944i
1.20927	−	2.91944i

−2.98985

−

0.246559i

6.63641

0.578158

8.87842

1.47435i

641.2

−2.98985

0.246559i

6.63641

0.578158

8.87842

−

1.47435i

641.3

−2.55118

−

1.57844i

−1.31534

−10.2329

4.01705

8.05378i

641.4

−2.55118

1.57844i

−1.31534

−10.2329

4.01705

−

8.05378i

641.5

−1.32750

−

2.69031i

−0.640013

2.72077

−5.47550

7.14275i

641.6

−1.32750

2.69031i

−0.640013

2.72077

−5.47550

−

7.14275i

641.7

−0.888828

−

2.86531i

−8.59176

10.9340

−7.41997

5.09353i

641.8

−0.888828

2.86531i

−8.59176

10.9340

−7.41997

−

5.09353i

641.9

0.888828

−

2.86531i

8.59176

10.9340

−7.41997

−

5.09353i

641.10

0.888828

2.86531i

8.59176

10.9340

−7.41997

5.09353i

641.11

1.32750

−

2.69031i

0.640013

2.72077

−5.47550

−

7.14275i

641.12

1.32750

2.69031i

0.640013

2.72077

−5.47550

7.14275i

641.13

2.55118

−

1.57844i

1.31534

−10.2329

4.01705

−

8.05378i

641.14

2.55118

1.57844i

1.31534

−10.2329

4.01705

8.05378i

641.15

2.98985

−

0.246559i

−6.63641

0.578158

8.87842

−

1.47435i

641.16

2.98985

0.246559i

−6.63641

0.578158

8.87842

1.47435i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
8.b	even	2	1	inner
24.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	768.3.h.h		16
3.b	odd	2	1	inner	768.3.h.h		16
4.b	odd	2	1		768.3.h.g		16
8.b	even	2	1	inner	768.3.h.h		16
8.d	odd	2	1		768.3.h.g		16
12.b	even	2	1		768.3.h.g		16
16.e	even	4	1		384.3.e.a	✓	8
16.e	even	4	1		384.3.e.c	yes	8
16.f	odd	4	1		384.3.e.b	yes	8
16.f	odd	4	1		384.3.e.d	yes	8
24.f	even	2	1		768.3.h.g		16
24.h	odd	2	1	inner	768.3.h.h		16
48.i	odd	4	1		384.3.e.a	✓	8
48.i	odd	4	1		384.3.e.c	yes	8
48.k	even	4	1		384.3.e.b	yes	8
48.k	even	4	1		384.3.e.d	yes	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
384.3.e.a	✓	8	16.e	even	4	1
384.3.e.a	✓	8	48.i	odd	4	1
384.3.e.b	yes	8	16.f	odd	4	1
384.3.e.b	yes	8	48.k	even	4	1
384.3.e.c	yes	8	16.e	even	4	1
384.3.e.c	yes	8	48.i	odd	4	1
384.3.e.d	yes	8	16.f	odd	4	1
384.3.e.d	yes	8	48.k	even	4	1
768.3.h.g		16	4.b	odd	2	1
768.3.h.g		16	8.d	odd	2	1
768.3.h.g		16	12.b	even	2	1
768.3.h.g		16	24.f	even	2	1
768.3.h.h		16	1.a	even	1	1	trivial
768.3.h.h		16	3.b	odd	2	1	inner
768.3.h.h		16	8.b	even	2	1	inner
768.3.h.h		16	24.h	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(768, [\chi])$:

$ T_{5}^{8} - 120T_{5}^{6} + 3504T_{5}^{4} - 7040T_{5}^{2} + 2304 $

$ T_{7}^{4} - 4T_{7}^{3} - 108T_{7}^{2} + 368T_{7} - 176 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} - 36 T^{12} + \cdots + 43046721 $ T^16 - 36T^12 - 512T^10 + 4230T^8 - 41472T^6 - 236196*T^4 + 43046721
$5$	$ (T^{8} - 120 T^{6} + \cdots + 2304)^{2} $ (T^8 - 120T^6 + 3504T^4 - 7040*T^2 + 2304)^2
$7$	$ (T^{4} - 4 T^{3} + \cdots - 176)^{4} $ (T^4 - 4T^3 - 108T^2 + 368*T - 176)^4
$11$	$ (T^{8} - 488 T^{6} + \cdots + 20214016)^{2} $ (T^8 - 488T^6 + 69552T^4 - 3167360*T^2 + 20214016)^2
$13$	$ (T^{8} + 720 T^{6} + \cdots + 61214976)^{2} $ (T^8 + 720T^6 + 145248T^4 + 5698816*T^2 + 61214976)^2
$17$	$ (T^{8} + 1248 T^{6} + \cdots + 991494144)^{2} $ (T^8 + 1248T^6 + 480000T^4 + 60416000*T^2 + 991494144)^2
$19$	$ (T^{8} + 1704 T^{6} + \cdots + 9673115904)^{2} $ (T^8 + 1704T^6 + 833712T^4 + 155767936*T^2 + 9673115904)^2
$23$	$ (T^{8} + 2336 T^{6} + \cdots + 48132849664)^{2} $ (T^8 + 2336T^6 + 1671936T^4 + 478846976*T^2 + 48132849664)^2
$29$	$ (T^{8} - 3704 T^{6} + \cdots + 78767790336)^{2} $ (T^8 - 3704T^6 + 3074736T^4 - 896411520*T^2 + 78767790336)^2
$31$	$ (T^{4} - 28 T^{3} + \cdots + 50256)^{4} $ (T^4 - 28T^3 - 828T^2 + 17520*T + 50256)^4
$37$	$ (T^{8} + 7312 T^{6} + \cdots + 238565818624)^{2} $ (T^8 + 7312T^6 + 14380128T^4 + 4824434944*T^2 + 238565818624)^2
$41$	$ (T^{8} + \cdots + 3916979306496)^{2} $ (T^8 + 10976T^6 + 34597632T^4 + 31405105152*T^2 + 3916979306496)^2
$43$	$ (T^{8} + 5800 T^{6} + \cdots + 841666795776)^{2} $ (T^8 + 5800T^6 + 11090352T^4 + 7476619392*T^2 + 841666795776)^2
$47$	$ (T^{8} + 5376 T^{6} + \cdots + 424144797696)^{2} $ (T^8 + 5376T^6 + 6905856T^4 + 3025141760*T^2 + 424144797696)^2
$53$	$ (T^{8} + \cdots + 870911869798656)^{2} $ (T^8 - 22776T^6 + 188831664T^4 - 673606710144*T^2 + 870911869798656)^2
$59$	$ (T^{8} + \cdots + 12745356964096)^{2} $ (T^8 - 8840T^6 + 26455344T^4 - 31548679808*T^2 + 12745356964096)^2
$61$	$ (T^{8} + \cdots + 21019364057344)^{2} $ (T^8 + 10768T^6 + 37553760T^4 + 48727118080*T^2 + 21019364057344)^2
$67$	$ (T^{8} + \cdots + 46033728153856)^{2} $ (T^8 + 17800T^6 + 81127728T^4 + 124646806144*T^2 + 46033728153856)^2
$71$	$ (T^{8} + \cdots + 1706597351424)^{2} $ (T^8 + 22688T^6 + 77642496T^4 + 36124434432*T^2 + 1706597351424)^2
$73$	$ (T^{4} - 40 T^{3} + \cdots - 899312)^{4} $ (T^4 - 40T^3 - 5544T^2 + 192608*T - 899312)^4
$79$	$ (T^{4} - 204 T^{3} + \cdots - 38762928)^{4} $ (T^4 - 204T^3 + 7524T^2 + 676400*T - 38762928)^4
$83$	$ (T^{8} + \cdots + 56638749360384)^{2} $ (T^8 - 13416T^6 + 59473584T^4 - 102766133376*T^2 + 56638749360384)^2
$89$	$ (T^{8} + \cdots + 92\!\cdots\!56)^{2} $ (T^8 + 45632T^6 + 730308096T^4 + 4688908992512*T^2 + 9213001971859456)^2
$97$	$ (T^{4} - 48 T^{3} + \cdots + 8416272)^{4} $ (T^4 - 48T^3 - 8136T^2 + 274112*T + 8416272)^4
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	768.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{8} q^{3} - \beta_{11} q^{5} + ( - \beta_1 + 1) q^{7} + ( - \beta_{13} + \beta_1) q^{9}+O(q^{10}) \) q - b8 * q^3 - b11 * q^5 + (-b1 + 1) * q^7 + (-b13 + b1) * q^9	\( q - \beta_{8} q^{3} - \beta_{11} q^{5} + ( - \beta_1 + 1) q^{7} + ( - \beta_{13} + \beta_1) q^{9} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_{2}) q^{11}+ \cdots + ( - \beta_{14} - 4 \beta_{11} + \cdots - 10 \beta_{2}) q^{99}+O(q^{100}) \) q - b8 * q^3 - b11 * q^5 + (-b1 + 1) * q^7 + (-b13 + b1) * q^9 + (-b11 + b10 - b5 - b2) * q^11 + (b9 - 2b8 - b5 + b3) q^13 + (-b15 + b13 - b12 + b7 - b1 - 2) * q^15 + (-b15 + b13 - 2b12 - b6 - b1) q^17 + (-b14 - b9 + b8 - b3) * q^19 + (b14 - b10 + b9 - b8 + 2b5 - b3 + 2b2) * q^21 + (2b15 - 2b13 + b7 + b6 + b1) * q^23 + (-b13 + b12 - b6 - b4 - 3b1 + 5) q^25 + (-b14 + 3b11 + b10 + b8 + b5 + b3 + b2) q^27 + (b11 + 2b10 + 2b9 + 6b8 - b5) q^29 + (2b13 - b12 + b7 + b6 + b4 + 7) q^31 + (-b15 + 2b13 + 2b7 - b6 + 2b4 + 1) q^33 + (-7b11 - b10 - b9 - 3b8 + 3b5 + 5b2) * q^35 + (-2b14 + b9 - 6b8 - b3) * q^37 + (-b15 - 3b13 + 3b12 + b6 + 2b4 + b1 - 13) q^39 + (3b15 - 2b13 + b12 + 2b7 + 5b4) * q^41 + (-b14 + b9 - b8 - 6b5 + 3b3) * q^43 + (b14 + 2b11 + b10 + 4b9 + 3b8 - 2b5 - 6b2) q^45 + (2b13 - b12 - 2b6 + 3b4 - 2b1) * q^47 + (3b13 + b12 + 4b7 - b6 - b4 - 3b1 + 9) q^49 + (5b11 - b10 - b9 + 3b8 + 5b5 - 2b3 - 5b2) q^51 + (3b11 - 4b10 + 4b9 + 12b8 - 4b2) q^53 + (-2b13 + b12 - b7 - b6 - b4 + b1 + 24) q^55 + (3b15 + b13 - 3b12 + 2b7 - 2b6 - 3b4 - 3b1 + 5) * q^57 + (b9 + 3b8 - 4b5 - 8b2) q^59 + (-2b14 + 3b9 - 10b8 + 2b5 + b3) * q^61 + (-2b15 - 3b12 - 4b7 - 5b4 + 3b1 - 33) q^63 + (b15 - 7b12 + 2b7 - 2b6 + b4 - 2b1) * q^65 + (4b14 + 3b9 - 5b8 - 10b5) * q^67 + (-3b14 - 3b11 + 3b10 + 2b9 + b8 + 3b5 + 6b2) * q^69 + (2b15 + 4b13 - 3b12 - 3b7 - b6 - 7b4 - b1) q^71 + (2b12 + 2b7 - 2b6 - 2b4 + 2b1 + 10) q^73 + (5b14 + b11 - b10 + b9 - 2b8 + 17b5 - 3b3 + 9b2) q^75 + (2b11 - 4b10 + 4b9 + 12b8 + 8b5 + 12b2) * q^77 + (-2b13 + 3b12 + b7 - 3b6 - 3b4 + 51) * q^79 + (2b15 - 3b13 + 3b12 - 2b7 - 3b6 + 5b4 - 3b1 + 9) q^81 + (9b11 - b10 + 2b9 + 6b8 + 3b5 + 5b2) q^83 + (8b14 + 8b9 - 16b8 + 2b5) * q^85 + (-b15 + 7b13 + 2b12 - b7 - 4b6 + 3b4 - 5b1 - 46) q^87 + (-8b15 + b13 - b12 + 2b7 - 3b6 + 9b4 - 3b1) q^89 + (5b14 - 2b9 + 4b8 - 6b5 - 7b3) q^91 + (-2b14 - 5b11 + 3b9 - 10b8 - 12b5 + b3 - 8b2) * q^93 + (2b15 - 2b13 + 16b12 - 3b7 + 5b6 + 5b1) * q^95 + (-5b13 + 3b12 - 2b7 - 3b6 - 3b4 - b1 + 12) q^97 + (-b14 - 4b11 + b9 - b8 + 6b5 + 5b3 - 10b2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{7}+O(q^{10}) \) 16 * q + 16 * q^7	\( 16 q + 16 q^{7} - 32 q^{15} + 80 q^{25} + 112 q^{31} + 16 q^{33} - 208 q^{39} + 144 q^{49} + 384 q^{55} + 80 q^{57} - 528 q^{63} + 160 q^{73} + 816 q^{79} + 144 q^{81} - 736 q^{87} + 192 q^{97}+O(q^{100}) \) 16 * q + 16 * q^7 - 32 * q^15 + 80 * q^25 + 112 * q^31 + 16 * q^33 - 208 * q^39 + 144 * q^49 + 384 * q^55 + 80 * q^57 - 528 * q^63 + 160 * q^73 + 816 * q^79 + 144 * q^81 - 736 * q^87 + 192 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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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	768.3.h.h

Level	$768$
Weight	$3$
Character orbit	768.h
Analytic conductor	$20.926$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(20.9264843029\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4 x^{15} + 14 x^{14} - 28 x^{13} + 50 x^{12} - 104 x^{11} - 66 x^{10} + 640 x^{9} + 555 x^{8} + \cdots + 2401 \) x^16 - 4x^15 + 14x^14 - 28x^13 + 50x^12 - 104x^11 - 66x^10 + 640x^9 + 555x^8 - 7060x^7 + 17714x^6 - 25496x^5 + 24840x^4 - 17932x^3 + 11724x^2 - 7056*x + 2401
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{34}\cdot 3^{2} \)
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 73\!\cdots\!90 \nu^{15} + \cdots - 54\!\cdots\!89 ) / 43\!\cdots\!81 \) (-73605237118565390v^15 - 47633572835668896v^14 - 139688548619286130v^13 - 1233457767323771308v^12 + 529018760445806044v^11 - 1261831727497275602v^10 + 24100324820579854794v^9 + 7093423907199410714v^8 - 202037586301504503744v^7 + 61203334305725712986v^6 + 615363780535269695166v^5 - 1135776337355918990048v^4 + 1101140289291794144046v^3 - 795703763476158680914v^2 + 402147901328092287564*v - 544249893260957625489) / 43737612461167616781
\(\beta_{2}\)	\(=\)	\( ( - 99\!\cdots\!44 \nu^{15} + \cdots - 88\!\cdots\!24 ) / 53\!\cdots\!97 \) (-99362376719317588592644v^15 + 157643615961293517755580v^14 - 654015690334010596895884v^13 + 213271506955874822687428v^12 - 995377489052373457680032v^11 + 3156489507901330068044614v^10 + 22834363346293880559522940v^9 - 29651658206534911182424006v^8 - 185396532173326788843206856v^7 + 434141249588399178636341586v^6 - 249135055907033040587373936v^5 - 192281761863540202775765666v^4 + 394936329964996076653983336v^3 - 333975898645644202702191118v^2 - 104408253736098716785228332*v - 88262663720799637363681324) / 53820750425127815650841397
\(\beta_{3}\)	\(=\)	\( ( 18\!\cdots\!55 \nu^{15} + \cdots - 30\!\cdots\!38 ) / 53\!\cdots\!97 \) (189224219934601032264355v^15 - 640202263351630099843971v^14 + 1553120990278286891368081v^13 - 3621706423247909764322422v^12 + 1682269426299819388064648v^11 - 18840296358896891889846508v^10 - 40628742994151317163457289v^9 + 110448656540655743708472337v^8 + 320568277591574100193142280v^7 - 1216821803698994193762710484v^6 + 1574248391414505437517732471v^5 - 1486026968627483519601442321v^4 + 1132201301739918183217506765v^3 - 1417240917087734151396424793v^2 + 998815185235190819052860430*v - 303265474878736101234979238) / 53820750425127815650841397
\(\beta_{4}\)	\(=\)	\( ( 36\!\cdots\!58 \nu^{15} + \cdots - 16\!\cdots\!97 ) / 69\!\cdots\!61 \) (3640490453456954097858v^15 - 16696296122407843524494v^14 + 51123108622447450476286v^13 - 115611452811381285277960v^12 + 161505766591708053420364v^11 - 415306479428912573409582v^10 - 265512122939865771157942v^9 + 2864130346548477692928604v^8 + 2134124510400861462739424v^7 - 29166355503786286734932230v^6 + 69392925277420838243005046v^5 - 93214027840038103299879880v^4 + 78452027971271711503230598v^3 - 44569897710476397997173428v^2 + 27020900329861475554330964*v - 16533045201889393898502397) / 698970784741919683777161
\(\beta_{5}\)	\(=\)	\( ( 72684469693760 \nu^{15} - 224840478635056 \nu^{14} + 808003344635480 \nu^{13} + \cdots - 18\!\cdots\!80 ) / 11\!\cdots\!97 \) (72684469693760v^15 - 224840478635056v^14 + 808003344635480v^13 - 1282221816062488v^12 + 2428767128904824v^11 - 5282208866934684v^10 - 9590853089188752v^9 + 38143824965723228v^8 + 76037812918776760v^7 - 448764552639400932v^6 + 870266063458707304v^5 - 1015163612568399884v^4 + 845369308904828096v^3 - 555948001648106716v^2 + 399142132155581296*v - 187269373357533880) / 11795069338202997
\(\beta_{6}\)	\(=\)	\( ( - 30\!\cdots\!10 \nu^{15} + \cdots + 73\!\cdots\!59 ) / 48\!\cdots\!27 \) (-30964682511934783503810v^15 + 122223497851645985487134v^14 - 351087091130861083128466v^13 + 722317551925352307477868v^12 - 830174841462470676732352v^11 + 2767985707332668051783076v^10 + 4213575685143668590694674v^9 - 22382276527890369452376370v^8 - 32569021563057742201550024v^7 + 234941060542798650808228840v^6 - 435187087035768768804873974v^5 + 449020567409998748155182748v^4 - 249729491763515335269157846v^3 + 96943521582465721562442746v^2 - 112158128752047556736724332*v + 73635341491264992735750559) / 4892795493193437786440127
\(\beta_{7}\)	\(=\)	\( ( 35\!\cdots\!00 \nu^{15} + \cdots + 36\!\cdots\!72 ) / 48\!\cdots\!27 \) (35220176603425455380400v^15 - 32006703464375436767938v^14 + 238371309052252800044600v^13 + 36936950069573329739656v^12 + 521383090631562825694928v^11 - 932380116944940123230634v^10 - 8781276532560910172334860v^9 + 2724335024283234546960200v^8 + 59396498158646402591599588v^7 - 105822817363836738019230770v^6 + 79729091073366660116651896v^5 + 15786583481454523793225260v^4 - 103299492669661029430155556v^3 + 30926645260328445790809788v^2 + 21468729734104125727704124*v + 36707798999442881296523572) / 4892795493193437786440127
\(\beta_{8}\)	\(=\)	\( ( - 49\!\cdots\!07 \nu^{15} + \cdots + 94\!\cdots\!76 ) / 53\!\cdots\!97 \) (-496011314854612629237607v^15 + 1407423641116043044984739v^14 - 5236363327383127844285983v^13 + 7674579346797568635198524v^12 - 15170187401690208428739514v^11 + 33483168646838080662868536v^10 + 74147872428604737899472801v^9 - 233702847930081684199658665v^8 - 558017997097579641379188182v^7 + 2872720671982035201104284980v^6 - 5364246117903744242824857047v^5 + 6097981717619334251998696003v^4 - 4691038047271369875397782463v^3 + 2830934339193932474639480993v^2 - 1935825001234129541998511804*v + 948191375097728418362664476) / 53820750425127815650841397
\(\beta_{9}\)	\(=\)	\( ( 16\!\cdots\!93 \nu^{15} + \cdots - 53\!\cdots\!70 ) / 17\!\cdots\!99 \) (168205981222649866459993v^15 - 722283452909459301095925v^14 + 2200838562375756723362047v^13 - 4632646557071839349088766v^12 + 6619724633870462883487076v^11 - 16474709494587523434622366v^10 - 14166933236368567892115763v^9 + 130623547662795020307028957v^8 + 129084446355814662641230164v^7 - 1332938102861827597833812202v^6 + 2849430799189762299032510721v^5 - 3465449247703949276546491501v^4 + 2786498808998386784147595327v^3 - 1596024018956010006398663969v^2 + 1098557546541429389083110714*v - 531388868455846783559913470) / 17940250141709271883613799
\(\beta_{10}\)	\(=\)	\( ( 55\!\cdots\!36 \nu^{15} + \cdots - 79\!\cdots\!80 ) / 53\!\cdots\!97 \) (551433367755288482572036v^15 - 1408219547486122305150558v^14 + 5735315644397360890020964v^13 - 7688654266282923636617572v^12 + 17692316089595233177136120v^11 - 35945008775855893630086130v^10 - 85094055795357376986267568v^9 + 217705740186354083996814700v^8 + 624628548695658880225303860v^7 - 2887010849285275188911969358v^6 + 5532641255779921708628630832v^5 - 6902862737843796856509056704v^4 + 6125863483899854452295642124v^3 - 4068255658692241415900074508v^2 + 2284234538899424073865200552*v - 797051948299789731534756680) / 53820750425127815650841397
\(\beta_{11}\)	\(=\)	\( ( 11\!\cdots\!86 \nu^{15} + \cdots - 13\!\cdots\!90 ) / 10\!\cdots\!53 \) (11913187010808341876886v^15 - 23975944496966615836928v^14 + 106642177371262500986628v^13 - 98301957290644270684438v^12 + 290568605512493599763378v^11 - 573606376298445507057866v^10 - 2228428785917414061095450v^9 + 3750630518098402823426118v^8 + 16446624534719448243157154v^7 - 54982088196114388725677778v^6 + 84288446412857750839274966v^5 - 81148429500991178855466900v^4 + 52304577766421834673743632v^3 - 33176743048971708020631918v^2 + 26181132013428199701150194*v - 1355563446291770005494690) / 1098382661737302360221253
\(\beta_{12}\)	\(=\)	\( ( 55\!\cdots\!10 \nu^{15} + \cdots - 79\!\cdots\!29 ) / 48\!\cdots\!27 \) (55024520198138880566010v^15 - 137783157785429009305526v^14 + 563850788327923100286814v^13 - 679041821146738360614520v^12 + 1686037666090608468184996v^11 - 3052264204664955485878014v^10 - 8430821140772456286131806v^9 + 22863780092201702657803180v^8 + 63574381834947111771677288v^7 - 297132976451917827250801510v^6 + 537165476707658079331848830v^5 - 570509132045867792137860712v^4 + 415165850424849819875437414v^3 - 232158640560184879151842796v^2 + 190211240740289941355047052*v - 79874676786308848296228229) / 4892795493193437786440127
\(\beta_{13}\)	\(=\)	\( ( 31\!\cdots\!44 \nu^{15} + \cdots - 55\!\cdots\!90 ) / 16\!\cdots\!09 \) (31817579385645488535444v^15 - 88206778931937685372070v^14 + 340478365886124775238104v^13 - 477367827448243316342072v^12 + 1021272473055663921364308v^11 - 2064820988020967478333898v^10 - 4644466413137907833313952v^9 + 14475875407010959495107452v^8 + 34387091160570332254797960v^7 - 181898556717941278052465898v^6 + 347302706535448206066655268v^5 - 395686452807303374493405652v^4 + 306148472751434611559085668v^3 - 188198004344939422253821672v^2 + 121084587780806107656104808*v - 55028840081614189320893790) / 1630931831064479262146709
\(\beta_{14}\)	\(=\)	\( ( 19\!\cdots\!87 \nu^{15} + \cdots - 49\!\cdots\!50 ) / 53\!\cdots\!97 \) (1944509727790417716480887v^15 - 6114785714223635371831415v^14 + 21504052543851716703107285v^13 - 35505982914351192419445234v^12 + 63068059782007057113198960v^11 - 147950314564777996000852984v^10 - 265454012643265202130323149v^9 + 1029607431255999657920021909v^8 + 2064908151313945678083008888v^7 - 12029737929564727753564018440v^6 + 23399319641003821026377525075v^5 - 27921974802000853211099091929v^4 + 22502383378998811767386967937v^3 - 14520845490988459183442145385v^2 + 10403553731305134146927457278*v - 4957377610955501313794460850) / 53820750425127815650841397
\(\beta_{15}\)	\(=\)	\( ( 21\!\cdots\!08 \nu^{15} + \cdots - 51\!\cdots\!64 ) / 48\!\cdots\!27 \) (212457904015996007722308v^15 - 685391664748231303553084v^14 + 2377104280783034795722072v^13 - 3994202810179077315449392v^12 + 6956242868611039002966124v^11 - 16326080972690283552664140v^10 - 28253227622269491985415116v^9 + 116809191218979853253523988v^8 + 221342755050324522216294260v^7 - 1345616710449714098644922284v^6 + 2627341790001511721947146452v^5 - 3102265819152201974954549176v^4 + 2471016010147884355794089104v^3 - 1540230428741537897930332472v^2 + 1082580752062076026913195444*v - 518021587644632004999733864) / 4892795493193437786440127

\(\nu\)	\(=\)	\( ( - 6 \beta_{15} + 6 \beta_{14} - 12 \beta_{13} + 9 \beta_{12} + 6 \beta_{11} + 6 \beta_{10} + 8 \beta_{9} + \cdots + 24 ) / 96 \) (-6b15 + 6b14 - 12b13 + 9b12 + 6b11 + 6b10 + 8b9 - 6b8 - 9b5 - 3b4 - 6b2 + 12b1 + 24) / 96
\(\nu^{2}\)	\(=\)	\( ( 6 \beta_{13} + 12 \beta_{12} + 2 \beta_{9} + 24 \beta_{8} + 6 \beta_{6} - 21 \beta_{5} + 12 \beta_{4} + \cdots - 36 ) / 48 \) (6b13 + 12b12 + 2b9 + 24b8 + 6b6 - 21b5 + 12b4 + 6b3 - 24b2 + 6b1 - 36) / 48
\(\nu^{3}\)	\(=\)	\( ( 21 \beta_{15} - 18 \beta_{14} + 18 \beta_{13} - 75 \beta_{11} + 9 \beta_{10} + 8 \beta_{9} + 21 \beta_{8} + \cdots - 60 ) / 48 \) (21b15 - 18b14 + 18b13 - 75b11 + 9b10 + 8b9 + 21b8 + 39b7 + 9b6 - 12b5 - 3b3 - 12b2 - 39*b1 - 60) / 48
\(\nu^{4}\)	\(=\)	\( ( - 6 \beta_{15} + 8 \beta_{13} - 29 \beta_{12} - 4 \beta_{9} + 4 \beta_{7} - 12 \beta_{6} + \cdots - 12 \beta_1 ) / 8 \) (-6b15 + 8b13 - 29b12 - 4b9 + 4b7 - 12b6 + 60b5 + 3b4 - 12b3 - 12b1) / 8
\(\nu^{5}\)	\(=\)	\( ( 336 \beta_{15} - 258 \beta_{14} - 816 \beta_{13} - 45 \beta_{12} + 1320 \beta_{11} - 96 \beta_{10} + \cdots + 1584 ) / 96 \) (336b15 - 258b14 - 816b13 - 45b12 + 1320b11 - 96b10 - 368b9 - 684b8 - 522b7 - 6b6 + 225b5 - 81b4 - 90b3 + 246b2 + 894*b1 + 1584) / 96
\(\nu^{6}\)	\(=\)	\( ( - 24 \beta_{15} + 102 \beta_{14} - 702 \beta_{13} + 1485 \beta_{12} + 90 \beta_{11} - 42 \beta_{10} + \cdots + 5724 ) / 48 \) (-24b15 + 102b14 - 702b13 + 1485b12 + 90b11 - 42b10 - 458b9 - 978b8 - 582b7 - 36b6 - 882b5 - 135b4 + 558b3 + 1350b2 + 168*b1 + 5724) / 48
\(\nu^{7}\)	\(=\)	\( ( - 5622 \beta_{15} + 4230 \beta_{14} + 2424 \beta_{13} + 3285 \beta_{12} - 4938 \beta_{11} + \cdots - 3168 ) / 96 \) (-5622b15 + 4230b14 + 2424b13 + 3285b12 - 4938b11 - 666b10 + 5732b9 - 6246b8 + 1404b7 + 2424b6 - 8445b5 - 207b4 + 2436b3 - 1602b2 - 1764*b1 - 3168) / 96
\(\nu^{8}\)	\(=\)	\( ( 711 \beta_{13} - 241 \beta_{12} - 204 \beta_{11} + 240 \beta_{10} + 664 \beta_{9} + 1992 \beta_{8} + \cdots - 7826 ) / 4 \) (711b13 - 241b12 - 204b11 + 240b10 + 664b9 + 1992b8 + 470b7 + 241b6 - 777b5 + 241b4 - 1314b2 + 15b1 - 7826) / 4
\(\nu^{9}\)	\(=\)	\( ( 25887 \beta_{15} - 19854 \beta_{14} + 17676 \beta_{13} - 25875 \beta_{12} - 22821 \beta_{11} + \cdots - 29760 ) / 48 \) (25887b15 - 19854b14 + 17676b13 - 25875b12 - 22821b11 - 1305b10 - 18458b9 + 49875b8 + 16473b7 - 4923b6 + 38136b5 + 7461b4 - 13011b3 - 8790b2 - 23115*b1 - 29760) / 48
\(\nu^{10}\)	\(=\)	\( ( 9960 \beta_{15} - 15954 \beta_{14} - 56760 \beta_{13} - 81057 \beta_{12} + 29322 \beta_{11} + \cdots + 559224 ) / 48 \) (9960b15 - 15954b14 - 56760b13 - 81057b12 + 29322b11 - 9402b10 - 66508b9 - 164586b8 - 21378b7 - 54186b6 + 221763b5 - 31053b4 - 50208b3 + 112110b2 - 22278*b1 + 559224) / 48
\(\nu^{11}\)	\(=\)	\( ( - 213516 \beta_{15} + 164130 \beta_{14} - 643848 \beta_{13} + 307989 \beta_{12} + 1017084 \beta_{11} + \cdots + 2317416 ) / 96 \) (-213516b15 + 164130b14 - 643848b13 + 307989b12 + 1017084b11 - 58836b10 - 118232b9 - 1040016b8 - 592170b7 - 59310b6 - 69249b5 - 120111b4 + 115278b3 + 418986b2 + 688350*b1 + 2317416) / 96
\(\nu^{12}\)	\(=\)	\( ( - 26632 \beta_{15} + 39942 \beta_{14} - 34232 \beta_{13} + 240323 \beta_{12} + 54690 \beta_{9} + \cdots + 76622 \beta_1 ) / 8 \) (-26632b15 + 39942b14 - 34232b13 + 240323b12 + 54690b9 - 11016b8 - 42390b7 + 76622b6 - 393759b5 + 2299b4 + 113112b3 + 76622b1) / 8
\(\nu^{13}\)	\(=\)	\( ( - 1932090 \beta_{15} + 1515762 \beta_{14} + 6297528 \beta_{13} + 952899 \beta_{12} - 9514074 \beta_{11} + \cdots - 25007400 ) / 96 \) (-1932090b15 + 1515762b14 + 6297528b13 + 952899b12 - 9514074b11 + 591966b10 + 4469908b9 + 6649074b8 + 4687188b7 + 1789152b6 - 6060903b5 + 1037199b4 + 1192080b3 - 4602438b2 - 5285364*b1 - 25007400) / 96
\(\nu^{14}\)	\(=\)	\( ( 1690992 \beta_{15} - 2026332 \beta_{14} + 7796130 \beta_{13} - 11968518 \beta_{12} - 4451004 \beta_{11} + \cdots - 52309692 ) / 48 \) (1690992b15 - 2026332b14 + 7796130b13 - 11968518b12 - 4451004b11 + 1069020b10 + 2404114b9 + 14662884b8 + 6394044b7 - 735954b6 + 9964353b5 + 1929762b4 - 4700778b3 - 10038756b2 - 4788618*b1 - 52309692) / 48
\(\nu^{15}\)	\(=\)	\( ( 20908563 \beta_{15} - 16832388 \beta_{14} - 10200522 \beta_{13} - 24458910 \beta_{12} + 18616767 \beta_{11} + \cdots + 52459200 ) / 48 \) (20908563b15 - 16832388b14 - 10200522b13 - 24458910b12 + 18616767b11 - 824673b10 - 25152404b9 + 7965531b8 - 4527519b7 - 11743989b6 + 52696743b5 - 224274b4 - 14701467b3 + 10462974b2 + 2479347*b1 + 52459200) / 48

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 641.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} - 36 T^{12} + \cdots + 43046721 \) T^16 - 36T^12 - 512T^10 + 4230T^8 - 41472T^6 - 236196*T^4 + 43046721
$5$	\( (T^{8} - 120 T^{6} + \cdots + 2304)^{2} \) (T^8 - 120T^6 + 3504T^4 - 7040*T^2 + 2304)^2
$7$	\( (T^{4} - 4 T^{3} + \cdots - 176)^{4} \) (T^4 - 4T^3 - 108T^2 + 368*T - 176)^4
$11$	\( (T^{8} - 488 T^{6} + \cdots + 20214016)^{2} \) (T^8 - 488T^6 + 69552T^4 - 3167360*T^2 + 20214016)^2
$13$	\( (T^{8} + 720 T^{6} + \cdots + 61214976)^{2} \) (T^8 + 720T^6 + 145248T^4 + 5698816*T^2 + 61214976)^2
$17$	\( (T^{8} + 1248 T^{6} + \cdots + 991494144)^{2} \) (T^8 + 1248T^6 + 480000T^4 + 60416000*T^2 + 991494144)^2
$19$	\( (T^{8} + 1704 T^{6} + \cdots + 9673115904)^{2} \) (T^8 + 1704T^6 + 833712T^4 + 155767936*T^2 + 9673115904)^2
$23$	\( (T^{8} + 2336 T^{6} + \cdots + 48132849664)^{2} \) (T^8 + 2336T^6 + 1671936T^4 + 478846976*T^2 + 48132849664)^2
$29$	\( (T^{8} - 3704 T^{6} + \cdots + 78767790336)^{2} \) (T^8 - 3704T^6 + 3074736T^4 - 896411520*T^2 + 78767790336)^2
$31$	\( (T^{4} - 28 T^{3} + \cdots + 50256)^{4} \) (T^4 - 28T^3 - 828T^2 + 17520*T + 50256)^4
$37$	\( (T^{8} + 7312 T^{6} + \cdots + 238565818624)^{2} \) (T^8 + 7312T^6 + 14380128T^4 + 4824434944*T^2 + 238565818624)^2
$41$	\( (T^{8} + \cdots + 3916979306496)^{2} \) (T^8 + 10976T^6 + 34597632T^4 + 31405105152*T^2 + 3916979306496)^2
$43$	\( (T^{8} + 5800 T^{6} + \cdots + 841666795776)^{2} \) (T^8 + 5800T^6 + 11090352T^4 + 7476619392*T^2 + 841666795776)^2
$47$	\( (T^{8} + 5376 T^{6} + \cdots + 424144797696)^{2} \) (T^8 + 5376T^6 + 6905856T^4 + 3025141760*T^2 + 424144797696)^2
$53$	\( (T^{8} + \cdots + 870911869798656)^{2} \) (T^8 - 22776T^6 + 188831664T^4 - 673606710144*T^2 + 870911869798656)^2
$59$	\( (T^{8} + \cdots + 12745356964096)^{2} \) (T^8 - 8840T^6 + 26455344T^4 - 31548679808*T^2 + 12745356964096)^2
$61$	\( (T^{8} + \cdots + 21019364057344)^{2} \) (T^8 + 10768T^6 + 37553760T^4 + 48727118080*T^2 + 21019364057344)^2
$67$	\( (T^{8} + \cdots + 46033728153856)^{2} \) (T^8 + 17800T^6 + 81127728T^4 + 124646806144*T^2 + 46033728153856)^2
$71$	\( (T^{8} + \cdots + 1706597351424)^{2} \) (T^8 + 22688T^6 + 77642496T^4 + 36124434432*T^2 + 1706597351424)^2
$73$	\( (T^{4} - 40 T^{3} + \cdots - 899312)^{4} \) (T^4 - 40T^3 - 5544T^2 + 192608*T - 899312)^4
$79$	\( (T^{4} - 204 T^{3} + \cdots - 38762928)^{4} \) (T^4 - 204T^3 + 7524T^2 + 676400*T - 38762928)^4
$83$	\( (T^{8} + \cdots + 56638749360384)^{2} \) (T^8 - 13416T^6 + 59473584T^4 - 102766133376*T^2 + 56638749360384)^2
$89$	\( (T^{8} + \cdots + 92\!\cdots\!56)^{2} \) (T^8 + 45632T^6 + 730308096T^4 + 4688908992512*T^2 + 9213001971859456)^2
$97$	\( (T^{4} - 48 T^{3} + \cdots + 8416272)^{4} \) (T^4 - 48T^3 - 8136T^2 + 274112*T + 8416272)^4
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\(n\)	\(257\)	\(511\)	\(517\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)