LMFDB - Newform orbit 4000.2.c.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(1249,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4000.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4000.c (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:	$31.9401608085$
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} - 10 x^{14} + 40 x^{13} + 147 x^{12} - 378 x^{11} - 845 x^{10} + 1620 x^{9} + \cdots + 10324 $ x^16 - 4x^15 - 10x^14 + 40x^13 + 147x^12 - 378x^11 - 845x^10 + 1620x^9 + 4271x^8 - 6478x^7 - 14916x^6 + 16548x^5 + 51265x^4 + 19946x^3 - 18890x^2 - 2972*x + 10324
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

$f(q)$	$=$	$ q - \beta_{3} q^{3} - \beta_{15} q^{7} + (\beta_{4} - 2) q^{9}+O(q^{10}) $ q - b3 * q^3 - b15 * q^7 + (b4 - 2) * q^9	$ q - \beta_{3} q^{3} - \beta_{15} q^{7} + (\beta_{4} - 2) q^{9} + (\beta_{13} + \beta_{9} - \beta_{2}) q^{11} + (\beta_{8} - \beta_{5}) q^{13} + (\beta_{5} + \beta_1) q^{17} + ( - 2 \beta_{9} + \beta_{6} + \beta_{2}) q^{19} + ( - \beta_{12} + \beta_{7} - \beta_{4} + 3) q^{21} + (\beta_{15} + \beta_{11} + \beta_{10}) q^{23} + (2 \beta_{11} + \beta_{10} + 3 \beta_{3}) q^{27} + ( - \beta_{12} + \beta_{7} - 2) q^{29} - \beta_{6} q^{31} + (\beta_{14} - 3 \beta_{8} + \cdots + \beta_1) q^{33}+ \cdots + (\beta_{13} - 7 \beta_{9} + \cdots - \beta_{2}) q^{99}+O(q^{100}) $ q - b3 * q^3 - b15 * q^7 + (b4 - 2) * q^9 + (b13 + b9 - b2) * q^11 + (b8 - b5) * q^13 + (b5 + b1) * q^17 + (-2b9 + b6 + b2) q^19 + (-b12 + b7 - b4 + 3) * q^21 + (b15 + b11 + b10) * q^23 + (2b11 + b10 + 3b3) * q^27 + (-b12 + b7 - 2) * q^29 - b6 * q^31 + (b14 - 3b8 + b5 + b1) q^33 + (b14 - 3b8 - b5) q^37 + (4b9 - 3b6 - 2b2) q^39 + (b7 + b4 + 3) * q^41 + (2b11 + b10 - b3) q^43 + (-b15 - b11 + 2b10 + 2b3) * q^47 + (-b12 + 4b7 - b4 - 1) q^49 + (-3b9 + 4b6 + 2b2) q^51 + (-2b14 + 4b8 + b5) * q^53 + (-b14 + 7b8 - 2b5 - 4b1) q^57 + (2b13 - b9 + 2b6 + b2) * q^59 + (b12 + 5b7 + b4 + 5) q^61 + (-b15 - 2b11 + b10 - 4b3) * q^63 + (b10 + 2b3) q^67 + (2b12 + 6b7 - 2) * q^69 + (-2b13 + b9 - 5b6 + 2b2) q^71 + (b14 - 3b8 - 6b1) * q^73 + (-2b14 - 5b8) * q^77 + (-4b13 + 3b9 - b6) * q^79 + (b12 + 9b7 - 2b4 + 11) * q^81 + (-4b11 + 4b10) * q^83 + (2b15 + 2b10 + 5b3) q^87 + (2b12 + 5b7 - b4) * q^89 + (-3b13 + 2b9 + 2b6 - 2b2) * q^91 + (b5 + 5b1) q^93 + (b14 + 7b8 + 2b5) * q^97 + (b13 - 7b9 + 5b6 - b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 36 q^{9}+O(q^{10}) $ 16 * q - 36 * q^9	$ 16 q - 36 q^{9} + 36 q^{21} - 48 q^{29} + 36 q^{41} - 52 q^{49} + 44 q^{61} - 64 q^{69} + 120 q^{81} - 20 q^{89}+O(q^{100}) $ 16 * q - 36 * q^9 + 36 * q^21 - 48 * q^29 + 36 * q^41 - 52 * q^49 + 44 * q^61 - 64 * q^69 + 120 * q^81 - 20 * q^89

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4 x^{15} - 10 x^{14} + 40 x^{13} + 147 x^{12} - 378 x^{11} - 845 x^{10} + 1620 x^{9} + \cdots + 10324 $ x^16 - 4*x^15 - 10*x^14 + 40*x^13 + 147*x^12 - 378*x^11 - 845*x^10 + 1620*x^9 + 4271*x^8 - 6478*x^7 - 14916*x^6 + 16548*x^5 + 51265*x^4 + 19946*x^3 - 18890*x^2 - 2972*x + 10324 :

$\beta_{1}$	$=$	$ ( 45\!\cdots\!09 \nu^{15} + \cdots - 24\!\cdots\!74 ) / 40\!\cdots\!50 $ (4553146742589561609v^15 - 35003350064632260575v^14 + 66132042853184049580v^13 + 132846648325346150840v^12 - 302282814989119069337v^11 - 1801009158637322208865v^10 + 6696559837483912130845v^9 - 1484553970464502618465v^8 - 25777283212027163197571v^7 + 15754101362919904682979v^6 + 104530076813519722570872v^5 - 106569838624833177191210v^4 - 249001050074540677976305v^3 + 500999094078610501852839v^2 + 387633944031079750802176*v - 248935498443206167575574) / 401243618600469885016150
$\beta_{2}$	$=$	$ ( 12\!\cdots\!91 \nu^{15} + \cdots + 77\!\cdots\!50 ) / 53\!\cdots\!30 $ (12993718820769899817891v^15 - 91250836180416749552589v^14 + 21162990903317801830866v^13 + 1049621753840401157626858v^12 - 323678283685472167517981v^11 - 10710205062452658341598809v^10 + 8581272959007786371578121v^9 + 62311740327731523409285695v^8 - 70519843389439761714797893v^7 - 242541969095747514869226897v^6 + 287031226818437031682356864v^5 + 873192950358100340466533126v^4 - 1005239860473666436685285447v^3 - 1516774414102009987524317505v^2 + 1779454718276742162433608138*v + 776354981045748956177974050) / 534536748699545980818515030
$\beta_{3}$	$=$	$ ( - 13\!\cdots\!39 \nu^{15} + \cdots - 99\!\cdots\!58 ) / 24\!\cdots\!65 $ (-1311038151101863689439v^15 + 5122328393837034019115v^14 + 13637871747935978383446v^13 - 52115489057285685715871v^12 - 195912761726302884317413v^11 + 488032130052759695433292v^10 + 1147015390754989775110387v^9 - 2156870583898610105229588v^8 - 5770683654586137885862353v^7 + 8722273042382048194194645v^6 + 20689274396237999970726812v^5 - 22777550961648719802522182v^4 - 72965072319707884617340575v^3 - 19090768175113280254141961v^2 + 12411074579881469258415432*v - 9905241666900988299992258) / 24297124940888453673568865
$\beta_{4}$	$=$	$ ( 24\!\cdots\!50 \nu^{15} + \cdots + 19\!\cdots\!85 ) / 36\!\cdots\!65 $ (246067797342682350v^15 - 1102676723525897023v^14 - 1892288492541076784v^13 + 10704765621086870266v^12 + 30145991661117717682v^11 - 107498434678991443161v^10 - 146326308205337755980v^9 + 491578721315056906446v^8 + 722899163649992667868v^7 - 2143512681687973913205v^6 - 2478895939446290884646v^5 + 7149229417491996876198v^4 + 8081117069025905783110v^3 - 3307389003946460610478v^2 - 2583189664446146998972*v + 19005424346496903568985) / 3647669260004271681965
$\beta_{5}$	$=$	$ ( 56\!\cdots\!17 \nu^{15} + \cdots + 22\!\cdots\!46 ) / 80\!\cdots\!30 $ (5655136380292235417v^15 - 30025132372266538353v^14 - 33395243216370538522v^13 + 327321177973776075162v^12 + 597525861253352408937v^11 - 3557456941604559869099v^10 - 2673638515221631319791v^9 + 18296225479303557712351v^8 + 16021663287109609602041v^7 - 85460746784674281482995v^6 - 45826768654254086069740v^5 + 261668681294833872913472v^4 + 216286541806714749110113v^3 - 527572031531995126092451v^2 - 306948934872479477622072*v + 228623949114822642112646) / 80248723720093977003230
$\beta_{6}$	$=$	$ ( - 28\!\cdots\!48 \nu^{15} + \cdots - 31\!\cdots\!04 ) / 26\!\cdots\!15 $ (-28137172598833749956648v^15 + 132514624610895065323131v^14 + 218943508413841771517608v^13 - 1477472313434802507634670v^12 - 2999025117878042435054986v^11 + 14215927975563623429401404v^10 + 14425776814396117786815835v^9 - 72555827860052116985253098v^8 - 55266748450888620586455568v^7 + 280565845377796017214215992v^6 + 179520905780801922511803096v^5 - 836848980486307207225606874v^4 - 669405663830621047465885043v^3 + 465910828757296689893534159v^2 + 84172183300363922940099506*v - 310448627901461723572321504) / 267268374349772990409257515
$\beta_{7}$	$=$	$ ( 48\!\cdots\!55 \nu^{15} + \cdots + 70\!\cdots\!90 ) / 38\!\cdots\!70 $ (48667644918788355v^15 - 224735491670846423v^14 - 376060964252748004v^13 + 2364061609531810406v^12 + 5599928941355807877v^11 - 23203239341861238921v^10 - 27934853920944657815v^9 + 113162353409449299761v^8 + 127928054524090638423v^7 - 453034986869913019355v^6 - 437865836061024537596v^5 + 1371537003907454552688v^4 + 1453663217298342141295v^3 - 626222435942202362463v^2 - 454717829086694624842*v + 706653361471456567090) / 383965185263607545470
$\beta_{8}$	$=$	$ ( 4530761383 \nu^{15} - 27061517780 \nu^{14} + 673052840 \nu^{13} + 219162261900 \nu^{12} + \cdots + 36471622088842 ) / 31993673048150 $ (4530761383v^15 - 27061517780v^14 + 673052840v^13 + 219162261900v^12 + 256097835661v^11 - 2539908528070v^10 + 534090179205v^9 + 9826563951280v^8 + 1457030165813v^7 - 45305643547872v^6 + 9091502139004v^5 + 114845640447800v^4 + 34581525395645v^3 - 111909802243762v^2 - 27513409935788*v + 36471622088842) / 31993673048150
$\beta_{9}$	$=$	$ ( - 41\!\cdots\!21 \nu^{15} + \cdots - 30\!\cdots\!46 ) / 26\!\cdots\!15 $ (-41095200129351444276321v^15 + 186976239185517904803085v^14 + 341274862299954350725433v^13 - 2024954059708509548397907v^12 - 4903791618621220081401270v^11 + 19643655237023328560151402v^10 + 25543983471189715662637856v^9 - 96443809841607360729572977v^8 - 116096606533990113794339145v^7 + 378695959331700999634483396v^6 + 400241743734662063169853498v^5 - 1104391378955500593199619565v^4 - 1381267695700002391848627919v^3 + 353419540619788317277188471v^2 + 619358458883099279135401522*v - 305196864302239768392307846) / 267268374349772990409257515
$\beta_{10}$	$=$	$ ( - 37\!\cdots\!36 \nu^{15} + \cdots - 94\!\cdots\!32 ) / 24\!\cdots\!65 $ (-3749752452972790309836v^15 + 20732863114036149974334v^14 + 6726224940442947017928v^13 - 167417531331478374374848v^12 - 286090670248330300561116v^11 + 1905024152577373142561956v^10 + 232462116125913050336682v^9 - 7137595468514527773208299v^8 - 4028888727294004533956076v^7 + 32889253045836149416553118v^6 + 2436987371392293579493658v^5 - 77908154756014835500535970v^4 - 49812830489980484585115494v^3 + 18598144603966802173558229v^2 - 5380312806834899188364748*v - 9430008346102970811232632) / 24297124940888453673568865
$\beta_{11}$	$=$	$ ( 16\!\cdots\!81 \nu^{15} + \cdots + 96\!\cdots\!62 ) / 48\!\cdots\!30 $ (16470168550969741223681v^15 - 71870035431308235598686v^14 - 133219295569144022286550v^13 + 681714957641911390025460v^12 + 2150231631382714052837073v^11 - 6829871286638493035687430v^10 - 10784007288541905867329869v^9 + 28448063520026534344879210v^8 + 57696866090796810100912583v^7 - 121930752524425552830611882v^6 - 180236067483033699789933788v^5 + 306149779122753280644865106v^4 + 677804783426085602871352221v^3 + 126020251843693812822151512v^2 - 94441083020796753052449404*v + 96075909657405141265191762) / 48594249881776907347137730
$\beta_{12}$	$=$	$ ( - 33\!\cdots\!85 \nu^{15} + \cdots - 52\!\cdots\!70 ) / 72\!\cdots\!30 $ (-3370703395126322785v^15 + 15495604019523482067v^14 + 26695308022051658876v^13 - 166083683807058573164v^12 - 385689652854120766883v^11 + 1616760037176248284719v^10 + 1922330337103219298065v^9 - 7975610094237435056969v^8 - 8454760380058935675037v^7 + 31616505138858977844375v^6 + 28654696031137831024664v^5 - 95002610577203785398772v^4 - 98318929568311644650865v^3 + 43113590494733483003697v^2 + 30496848408833185816198*v - 52254785449777176496870) / 7295338520008543363930
$\beta_{13}$	$=$	$ ( 13\!\cdots\!93 \nu^{15} + \cdots + 95\!\cdots\!01 ) / 26\!\cdots\!15 $ (138618733944064656701793v^15 - 625376582509086943959378v^14 - 1146301403278371270047241v^13 + 6603515949494172204097708v^12 + 16897708024989469539486008v^11 - 64328753553874720078640970v^10 - 88497859789696090763202104v^9 + 309959434243517322190375252v^8 + 415682678321866356204315949v^7 - 1237446239076968380407908805v^6 - 1442844248297700239579562720v^5 + 3661556063559093025491808457v^4 + 4955013187588874822634303335v^3 - 994812573754178634125868511v^2 - 2416361124053653891907438426*v + 950094218834732222293940101) / 267268374349772990409257515
$\beta_{14}$	$=$	$ ( 45\!\cdots\!53 \nu^{15} + \cdots + 40\!\cdots\!26 ) / 80\!\cdots\!30 $ (45369637291229841153v^15 - 267993407964008848704v^14 - 13270316485414436876v^13 + 2197624416856760704236v^12 + 2776551364098304275875v^11 - 25465020052003625319132v^10 + 3367524945449549513277v^9 + 99846803912414683282538v^8 + 23356535407857782368875v^7 - 463358241274880771798218v^6 + 69639623478464722886616v^5 + 1183791263381713459560396v^4 + 391398737429391284496229v^3 - 1212652331560182825821726v^2 - 336158884782390398850508*v + 407047977908653791587226) / 80248723720093977003230
$\beta_{15}$	$=$	$ ( 30\!\cdots\!23 \nu^{15} + \cdots + 95\!\cdots\!26 ) / 48\!\cdots\!30 $ (30353919795772637896923v^15 - 161631294061460498897621v^14 - 85896039654266511675098v^13 + 1332224814969152169245508v^12 + 2548863960122219957310067v^11 - 14685057386595343054569511v^10 - 4857623912696445584359515v^9 + 55230792071365679146196749v^8 + 43837096800234031754716047v^7 - 245372001766665950912435157v^6 - 81938323654947809862639504v^5 + 588191260269603661689902272v^4 + 578735350354413485984124941v^3 - 50490861676791213669388245v^2 - 6200720418189019219603360*v + 95123918714823557352504726) / 48594249881776907347137730

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

$\nu$	$=$	$ ( -\beta_{9} - \beta_{7} + \beta_{6} - \beta_{3} + \beta_{2} ) / 2 $ (-b9 - b7 + b6 - b3 + b2) / 2
$\nu^{2}$	$=$	$ ( \beta_{13} + \beta_{10} + 2\beta_{8} - 3\beta_{7} + \beta_{6} - 2\beta_{5} + \beta_{2} - 4\beta _1 + 3 ) / 2 $ (b13 + b10 + 2b8 - 3b7 + b6 - 2b5 + b2 - 4b1 + 3) / 2
$\nu^{3}$	$=$	$ ( 3 \beta_{14} + 2 \beta_{13} - \beta_{11} + 4 \beta_{10} + 2 \beta_{9} - 6 \beta_{8} - 8 \beta_{7} + \cdots + 4 ) / 2 $ (3b14 + 2b13 - b11 + 4b10 + 2b9 - 6b8 - 8b7 - b6 - 3b5 - 9b3 + 2b2 - 3b1 + 4) / 2
$\nu^{4}$	$=$	$ ( 2 \beta_{15} + 8 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} + 24 \beta_{10} - 9 \beta_{9} - 8 \beta_{8} + \cdots - 22 ) / 2 $ (2b15 + 8b14 + 2b13 + 2b12 + 24b10 - 9b9 - 8b8 - 13b7 + 3b6 - 16b5 + 8b4 - 5b3 + 5b2 - 24b1 - 22) / 2
$\nu^{5}$	$=$	$ ( 3 \beta_{15} + 30 \beta_{14} + 2 \beta_{13} + 25 \beta_{12} + 3 \beta_{11} + 67 \beta_{10} + 19 \beta_{9} + \cdots + 13 ) / 2 $ (3b15 + 30b14 + 2b13 + 25b12 + 3b11 + 67b10 + 19b9 - 45b8 + 55b7 - 58b6 - 45b5 + 5b4 - 19b3 - 12b2 - 60*b1 + 13) / 2
$\nu^{6}$	$=$	$ ( - 10 \beta_{15} + 94 \beta_{14} - 51 \beta_{13} + 90 \beta_{12} - 4 \beta_{11} + 132 \beta_{10} + \cdots - 240 ) / 2 $ (-10b15 + 94b14 - 51b13 + 90b12 - 4b11 + 132b10 - 113b9 - 186b8 + 246b7 - 154b6 - 90b5 + 75b4 - 135b3 - 56b2 - 18*b1 - 240) / 2
$\nu^{7}$	$=$	$ ( - 83 \beta_{15} + 70 \beta_{14} - 200 \beta_{13} + 385 \beta_{12} + 116 \beta_{11} + 290 \beta_{10} + \cdots - 678 ) / 2 $ (-83b15 + 70b14 - 200b13 + 385b12 + 116b11 + 290b10 - 207b9 + 280b8 + 1303b7 - 734b6 - 252b5 + 210b4 + 30b3 - 304b2 - 175*b1 - 678) / 2
$\nu^{8}$	$=$	$ ( - 698 \beta_{15} + 160 \beta_{14} - 681 \beta_{13} + 1242 \beta_{12} + 290 \beta_{11} - 787 \beta_{10} + \cdots - 1963 ) / 2 $ (-698b15 + 160b14 - 681b13 + 1242b12 + 290b11 - 787b10 - 238b9 + 368b8 + 4685b7 - 2761b6 + 40b5 + 560b4 - 854b3 - 1197b2 + 1944*b1 - 1963) / 2
$\nu^{9}$	$=$	$ ( - 2952 \beta_{15} - 1371 \beta_{14} - 2480 \beta_{13} + 2817 \beta_{12} + 1861 \beta_{11} - 5017 \beta_{10} + \cdots - 9942 ) / 2 $ (-2952b15 - 1371b14 - 2480b13 + 2817b12 + 1861b11 - 5017b10 - 2542b9 + 7092b8 + 11465b7 - 6331b6 + 1707b5 + 2376b4 + 314b3 - 3322b2 + 9171*b1 - 9942) / 2
$\nu^{10}$	$=$	$ ( - 11816 \beta_{15} - 7636 \beta_{14} - 4614 \beta_{13} + 6245 \beta_{12} + 7584 \beta_{11} + \cdots - 12633 ) / 2 $ (-11816b15 - 7636b14 - 4614b13 + 6245b12 + 7584b11 - 25324b10 + 3547b9 + 29996b8 + 31980b7 - 17689b6 + 10766b5 + 2275b4 + 5118b3 - 9301b2 + 41378*b1 - 12633) / 2
$\nu^{11}$	$=$	$ ( - 42431 \beta_{15} - 26191 \beta_{14} - 9882 \beta_{13} - 2475 \beta_{12} + 23489 \beta_{11} + \cdots - 32285 ) / 2 $ (-42431b15 - 26191b14 - 9882b13 - 2475b12 + 23489b11 - 105509b10 + 3228b9 + 85866b8 + 29479b7 - 9818b6 + 49346b5 + 1595b4 + 10307b3 - 14238b2 + 170313*b1 - 32285) / 2
$\nu^{12}$	$=$	$ ( - 120806 \beta_{15} - 107668 \beta_{14} + 3095 \beta_{13} - 72761 \beta_{12} + 83294 \beta_{11} + \cdots + 49902 ) / 2 $ (-120806b15 - 107668b14 + 3095b13 - 72761b12 + 83294b11 - 307593b10 + 54407b9 + 353724b8 - 105345b7 + 102034b6 + 153868b5 - 41593b4 + 108658b3 + 5964b2 + 480732*b1 + 49902) / 2
$\nu^{13}$	$=$	$ ( - 315749 \beta_{15} - 250744 \beta_{14} + 126942 \beta_{13} - 426270 \beta_{12} + 182242 \beta_{11} + \cdots + 704120 ) / 2 $ (-315749b15 - 250744b14 + 126942b13 - 426270b12 + 182242b11 - 890843b10 + 338826b9 + 717054b8 - 943916b7 + 716523b6 + 451373b5 - 290264b4 + 175991b3 + 178844b2 + 1365832*b1 + 704120) / 2
$\nu^{14}$	$=$	$ ( - 513110 \beta_{15} - 545228 \beta_{14} + 598989 \beta_{13} - 1953441 \beta_{12} + 324840 \beta_{11} + \cdots + 2738175 ) / 2 $ (-513110b15 - 545228b14 + 598989b13 - 1953441b12 + 324840b11 - 1619050b10 + 879678b9 + 1467944b8 - 5111964b7 + 3614513b6 + 879088b5 - 1113756b4 + 522719b3 + 1070017b2 + 2361172*b1 + 2738175) / 2
$\nu^{15}$	$=$	$ ( 332066 \beta_{15} - 365277 \beta_{14} + 2659498 \beta_{13} - 6794375 \beta_{12} - 116159 \beta_{11} + \cdots + 12385763 ) / 2 $ (332066b15 - 365277b14 + 2659498b13 - 6794375b12 - 116159b11 - 115346b10 + 4299151b9 + 331212b8 - 18307195b7 + 12694888b6 + 349242b5 - 4437065b4 + 809562b3 + 4105062b2 - 552834*b1 + 12385763) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$.

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$-1$	$1$	$1$

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} $

1249.1

1.49589	−	1.66741i
−2.11393	−	1.66741i
3.07429	−	1.36342i
−1.45625	−	1.36342i
2.74786	−	0.697553i
−1.12983	−	0.697553i
0.466692	−	0.352515i
−1.08473	−	0.352515i
0.466692	+	0.352515i
−1.08473	+	0.352515i
2.74786	+	0.697553i
−1.12983	+	0.697553i
3.07429	+	1.36342i
−1.45625	+	1.36342i
1.49589	+	1.66741i
−2.11393	+	1.66741i

−

3.33481i

1.78602i

−8.12097

1249.2

−

3.33481i

1.78602i

−8.12097

1249.3

−

2.72684i

2.84526i

−4.43565

1249.4

−

2.72684i

2.84526i

−4.43565

1249.5

−

1.39511i

−

4.73991i

1.05368

1249.6

−

1.39511i

−

4.73991i

1.05368

1249.7

−

0.705030i

2.69218i

2.50293

1249.8

−

0.705030i

2.69218i

2.50293

1249.9

0.705030i

−

2.69218i

2.50293

1249.10

0.705030i

−

2.69218i

2.50293

1249.11

1.39511i

4.73991i

1.05368

1249.12

1.39511i

4.73991i

1.05368

1249.13

2.72684i

−

2.84526i

−4.43565

1249.14

2.72684i

−

2.84526i

−4.43565

1249.15

3.33481i

−

1.78602i

−8.12097

1249.16

3.33481i

−

1.78602i

−8.12097

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4000.2.c.h		16
4.b	odd	2	1	inner	4000.2.c.h		16
5.b	even	2	1	inner	4000.2.c.h		16
5.c	odd	4	1		4000.2.a.q	✓	8
5.c	odd	4	1		4000.2.a.r	yes	8
20.d	odd	2	1	inner	4000.2.c.h		16
20.e	even	4	1		4000.2.a.q	✓	8
20.e	even	4	1		4000.2.a.r	yes	8
40.i	odd	4	1		8000.2.a.ca		8
40.i	odd	4	1		8000.2.a.cb		8
40.k	even	4	1		8000.2.a.ca		8
40.k	even	4	1		8000.2.a.cb		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4000.2.a.q	✓	8	5.c	odd	4	1
4000.2.a.q	✓	8	20.e	even	4	1
4000.2.a.r	yes	8	5.c	odd	4	1
4000.2.a.r	yes	8	20.e	even	4	1
4000.2.c.h		16	1.a	even	1	1	trivial
4000.2.c.h		16	4.b	odd	2	1	inner
4000.2.c.h		16	5.b	even	2	1	inner
4000.2.c.h		16	20.d	odd	2	1	inner
8000.2.a.ca		8	40.i	odd	4	1
8000.2.a.ca		8	40.k	even	4	1
8000.2.a.cb		8	40.i	odd	4	1
8000.2.a.cb		8	40.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4000, [\chi])$:

$ T_{3}^{8} + 21T_{3}^{6} + 129T_{3}^{4} + 220T_{3}^{2} + 80 $

$ T_{7}^{8} + 41T_{7}^{6} + 524T_{7}^{4} + 2605T_{7}^{2} + 4205 $

$ T_{11}^{8} - 49T_{11}^{6} + 684T_{11}^{4} - 2445T_{11}^{2} + 5 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 21 T^{6} + \cdots + 80)^{2} $ (T^8 + 21T^6 + 129T^4 + 220*T^2 + 80)^2
$5$	$ T^{16} $ T^16
$7$	$ (T^{8} + 41 T^{6} + \cdots + 4205)^{2} $ (T^8 + 41T^6 + 524T^4 + 2605*T^2 + 4205)^2
$11$	$ (T^{8} - 49 T^{6} + 684 T^{4} + \cdots + 5)^{2} $ (T^8 - 49T^6 + 684T^4 - 2445*T^2 + 5)^2
$13$	$ (T^{8} + 70 T^{6} + \cdots + 22201)^{2} $ (T^8 + 70T^6 + 1523T^4 + 11150*T^2 + 22201)^2
$17$	$ (T^{8} + 66 T^{6} + \cdots + 30976)^{2} $ (T^8 + 66T^6 + 1441T^4 + 12016*T^2 + 30976)^2
$19$	$ (T^{8} - 80 T^{6} + \cdots + 3125)^{2} $ (T^8 - 80T^6 + 1645T^4 - 5500*T^2 + 3125)^2
$23$	$ (T^{8} + 91 T^{6} + \cdots + 20480)^{2} $ (T^8 + 91T^6 + 2329T^4 + 15680*T^2 + 20480)^2
$29$	$ (T^{4} + 12 T^{3} + \cdots + 20)^{4} $ (T^4 + 12T^3 + 21T^2 - 70*T + 20)^4
$31$	$ (T^{8} - 29 T^{6} + \cdots + 80)^{2} $ (T^8 - 29T^6 + 209T^4 - 460*T^2 + 80)^2
$37$	$ (T^{8} + 143 T^{6} + \cdots + 102400)^{2} $ (T^8 + 143T^6 + 5641T^4 + 71360*T^2 + 102400)^2
$41$	$ (T^{4} - 9 T^{3} - 6 T^{2} + \cdots + 95)^{4} $ (T^4 - 9T^3 - 6T^2 + 95*T + 95)^4
$43$	$ (T^{8} + 234 T^{6} + \cdots + 3872000)^{2} $ (T^8 + 234T^6 + 17109T^4 + 473200*T^2 + 3872000)^2
$47$	$ (T^{8} + 176 T^{6} + \cdots + 17405)^{2} $ (T^8 + 176T^6 + 8069T^4 + 51460*T^2 + 17405)^2
$53$	$ (T^{8} + 374 T^{6} + \cdots + 6255001)^{2} $ (T^8 + 374T^6 + 43171T^4 + 1470174*T^2 + 6255001)^2
$59$	$ (T^{8} - 196 T^{6} + \cdots + 85805)^{2} $ (T^8 - 196T^6 + 11809T^4 - 202760*T^2 + 85805)^2
$61$	$ (T^{4} - 11 T^{3} + \cdots - 2620)^{4} $ (T^4 - 11T^3 - 61T^2 + 980*T - 2620)^4
$67$	$ (T^{8} + 81 T^{6} + \cdots + 80)^{2} $ (T^8 + 81T^6 + 369T^4 + 380*T^2 + 80)^2
$71$	$ (T^{8} - 694 T^{6} + \cdots + 849686480)^{2} $ (T^8 - 694T^6 + 178669T^4 - 20224700*T^2 + 849686480)^2
$73$	$ (T^{8} + 379 T^{6} + \cdots + 17007376)^{2} $ (T^8 + 379T^6 + 40381T^4 + 1463404*T^2 + 17007376)^2
$79$	$ (T^{8} - 546 T^{6} + \cdots + 3494480)^{2} $ (T^8 - 546T^6 + 78069T^4 - 1154440*T^2 + 3494480)^2
$83$	$ (T^{8} + 576 T^{6} + \cdots + 8192000)^{2} $ (T^8 + 576T^6 + 90624T^4 + 2867200*T^2 + 8192000)^2
$89$	$ (T^{4} + 5 T^{3} + \cdots - 2000)^{4} $ (T^4 + 5T^3 - 245T^2 - 1900*T - 2000)^4
$97$	$ (T^{8} + 491 T^{6} + \cdots + 59536)^{2} $ (T^8 + 491T^6 + 65101T^4 + 2019996*T^2 + 59536)^2
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4000.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{3} q^{3} - \beta_{15} q^{7} + (\beta_{4} - 2) q^{9}+O(q^{10}) \) q - b3 * q^3 - b15 * q^7 + (b4 - 2) * q^9	\( q - \beta_{3} q^{3} - \beta_{15} q^{7} + (\beta_{4} - 2) q^{9} + (\beta_{13} + \beta_{9} - \beta_{2}) q^{11} + (\beta_{8} - \beta_{5}) q^{13} + (\beta_{5} + \beta_1) q^{17} + ( - 2 \beta_{9} + \beta_{6} + \beta_{2}) q^{19} + ( - \beta_{12} + \beta_{7} - \beta_{4} + 3) q^{21} + (\beta_{15} + \beta_{11} + \beta_{10}) q^{23} + (2 \beta_{11} + \beta_{10} + 3 \beta_{3}) q^{27} + ( - \beta_{12} + \beta_{7} - 2) q^{29} - \beta_{6} q^{31} + (\beta_{14} - 3 \beta_{8} + \cdots + \beta_1) q^{33}+ \cdots + (\beta_{13} - 7 \beta_{9} + \cdots - \beta_{2}) q^{99}+O(q^{100}) \) q - b3 * q^3 - b15 * q^7 + (b4 - 2) * q^9 + (b13 + b9 - b2) * q^11 + (b8 - b5) * q^13 + (b5 + b1) * q^17 + (-2b9 + b6 + b2) q^19 + (-b12 + b7 - b4 + 3) * q^21 + (b15 + b11 + b10) * q^23 + (2b11 + b10 + 3b3) * q^27 + (-b12 + b7 - 2) * q^29 - b6 * q^31 + (b14 - 3b8 + b5 + b1) q^33 + (b14 - 3b8 - b5) q^37 + (4b9 - 3b6 - 2b2) q^39 + (b7 + b4 + 3) * q^41 + (2b11 + b10 - b3) q^43 + (-b15 - b11 + 2b10 + 2b3) * q^47 + (-b12 + 4b7 - b4 - 1) q^49 + (-3b9 + 4b6 + 2b2) q^51 + (-2b14 + 4b8 + b5) * q^53 + (-b14 + 7b8 - 2b5 - 4b1) q^57 + (2b13 - b9 + 2b6 + b2) * q^59 + (b12 + 5b7 + b4 + 5) q^61 + (-b15 - 2b11 + b10 - 4b3) * q^63 + (b10 + 2b3) q^67 + (2b12 + 6b7 - 2) * q^69 + (-2b13 + b9 - 5b6 + 2b2) q^71 + (b14 - 3b8 - 6b1) * q^73 + (-2b14 - 5b8) * q^77 + (-4b13 + 3b9 - b6) * q^79 + (b12 + 9b7 - 2b4 + 11) * q^81 + (-4b11 + 4b10) * q^83 + (2b15 + 2b10 + 5b3) q^87 + (2b12 + 5b7 - b4) * q^89 + (-3b13 + 2b9 + 2b6 - 2b2) * q^91 + (b5 + 5b1) q^93 + (b14 + 7b8 + 2b5) * q^97 + (b13 - 7b9 + 5b6 - b2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 36 q^{9}+O(q^{10}) \) 16 * q - 36 * q^9	\( 16 q - 36 q^{9} + 36 q^{21} - 48 q^{29} + 36 q^{41} - 52 q^{49} + 44 q^{61} - 64 q^{69} + 120 q^{81} - 20 q^{89}+O(q^{100}) \) 16 * q - 36 * q^9 + 36 * q^21 - 48 * q^29 + 36 * q^41 - 52 * q^49 + 44 * q^61 - 64 * q^69 + 120 * q^81 - 20 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4000.2.c.h

Level	$4000$
Weight	$2$
Character orbit	4000.c
Analytic conductor	$31.940$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(31.9401608085\)
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:	\( x^{16} - 4 x^{15} - 10 x^{14} + 40 x^{13} + 147 x^{12} - 378 x^{11} - 845 x^{10} + 1620 x^{9} + \cdots + 10324 \) x^16 - 4x^15 - 10x^14 + 40x^13 + 147x^12 - 378x^11 - 845x^10 + 1620x^9 + 4271x^8 - 6478x^7 - 14916x^6 + 16548x^5 + 51265x^4 + 19946x^3 - 18890x^2 - 2972*x + 10324
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 45\!\cdots\!09 \nu^{15} + \cdots - 24\!\cdots\!74 ) / 40\!\cdots\!50 \) (4553146742589561609v^15 - 35003350064632260575v^14 + 66132042853184049580v^13 + 132846648325346150840v^12 - 302282814989119069337v^11 - 1801009158637322208865v^10 + 6696559837483912130845v^9 - 1484553970464502618465v^8 - 25777283212027163197571v^7 + 15754101362919904682979v^6 + 104530076813519722570872v^5 - 106569838624833177191210v^4 - 249001050074540677976305v^3 + 500999094078610501852839v^2 + 387633944031079750802176*v - 248935498443206167575574) / 401243618600469885016150
\(\beta_{2}\)	\(=\)	\( ( 12\!\cdots\!91 \nu^{15} + \cdots + 77\!\cdots\!50 ) / 53\!\cdots\!30 \) (12993718820769899817891v^15 - 91250836180416749552589v^14 + 21162990903317801830866v^13 + 1049621753840401157626858v^12 - 323678283685472167517981v^11 - 10710205062452658341598809v^10 + 8581272959007786371578121v^9 + 62311740327731523409285695v^8 - 70519843389439761714797893v^7 - 242541969095747514869226897v^6 + 287031226818437031682356864v^5 + 873192950358100340466533126v^4 - 1005239860473666436685285447v^3 - 1516774414102009987524317505v^2 + 1779454718276742162433608138*v + 776354981045748956177974050) / 534536748699545980818515030
\(\beta_{3}\)	\(=\)	\( ( - 13\!\cdots\!39 \nu^{15} + \cdots - 99\!\cdots\!58 ) / 24\!\cdots\!65 \) (-1311038151101863689439v^15 + 5122328393837034019115v^14 + 13637871747935978383446v^13 - 52115489057285685715871v^12 - 195912761726302884317413v^11 + 488032130052759695433292v^10 + 1147015390754989775110387v^9 - 2156870583898610105229588v^8 - 5770683654586137885862353v^7 + 8722273042382048194194645v^6 + 20689274396237999970726812v^5 - 22777550961648719802522182v^4 - 72965072319707884617340575v^3 - 19090768175113280254141961v^2 + 12411074579881469258415432*v - 9905241666900988299992258) / 24297124940888453673568865
\(\beta_{4}\)	\(=\)	\( ( 24\!\cdots\!50 \nu^{15} + \cdots + 19\!\cdots\!85 ) / 36\!\cdots\!65 \) (246067797342682350v^15 - 1102676723525897023v^14 - 1892288492541076784v^13 + 10704765621086870266v^12 + 30145991661117717682v^11 - 107498434678991443161v^10 - 146326308205337755980v^9 + 491578721315056906446v^8 + 722899163649992667868v^7 - 2143512681687973913205v^6 - 2478895939446290884646v^5 + 7149229417491996876198v^4 + 8081117069025905783110v^3 - 3307389003946460610478v^2 - 2583189664446146998972*v + 19005424346496903568985) / 3647669260004271681965
\(\beta_{5}\)	\(=\)	\( ( 56\!\cdots\!17 \nu^{15} + \cdots + 22\!\cdots\!46 ) / 80\!\cdots\!30 \) (5655136380292235417v^15 - 30025132372266538353v^14 - 33395243216370538522v^13 + 327321177973776075162v^12 + 597525861253352408937v^11 - 3557456941604559869099v^10 - 2673638515221631319791v^9 + 18296225479303557712351v^8 + 16021663287109609602041v^7 - 85460746784674281482995v^6 - 45826768654254086069740v^5 + 261668681294833872913472v^4 + 216286541806714749110113v^3 - 527572031531995126092451v^2 - 306948934872479477622072*v + 228623949114822642112646) / 80248723720093977003230
\(\beta_{6}\)	\(=\)	\( ( - 28\!\cdots\!48 \nu^{15} + \cdots - 31\!\cdots\!04 ) / 26\!\cdots\!15 \) (-28137172598833749956648v^15 + 132514624610895065323131v^14 + 218943508413841771517608v^13 - 1477472313434802507634670v^12 - 2999025117878042435054986v^11 + 14215927975563623429401404v^10 + 14425776814396117786815835v^9 - 72555827860052116985253098v^8 - 55266748450888620586455568v^7 + 280565845377796017214215992v^6 + 179520905780801922511803096v^5 - 836848980486307207225606874v^4 - 669405663830621047465885043v^3 + 465910828757296689893534159v^2 + 84172183300363922940099506*v - 310448627901461723572321504) / 267268374349772990409257515
\(\beta_{7}\)	\(=\)	\( ( 48\!\cdots\!55 \nu^{15} + \cdots + 70\!\cdots\!90 ) / 38\!\cdots\!70 \) (48667644918788355v^15 - 224735491670846423v^14 - 376060964252748004v^13 + 2364061609531810406v^12 + 5599928941355807877v^11 - 23203239341861238921v^10 - 27934853920944657815v^9 + 113162353409449299761v^8 + 127928054524090638423v^7 - 453034986869913019355v^6 - 437865836061024537596v^5 + 1371537003907454552688v^4 + 1453663217298342141295v^3 - 626222435942202362463v^2 - 454717829086694624842*v + 706653361471456567090) / 383965185263607545470
\(\beta_{8}\)	\(=\)	\( ( 4530761383 \nu^{15} - 27061517780 \nu^{14} + 673052840 \nu^{13} + 219162261900 \nu^{12} + \cdots + 36471622088842 ) / 31993673048150 \) (4530761383v^15 - 27061517780v^14 + 673052840v^13 + 219162261900v^12 + 256097835661v^11 - 2539908528070v^10 + 534090179205v^9 + 9826563951280v^8 + 1457030165813v^7 - 45305643547872v^6 + 9091502139004v^5 + 114845640447800v^4 + 34581525395645v^3 - 111909802243762v^2 - 27513409935788*v + 36471622088842) / 31993673048150
\(\beta_{9}\)	\(=\)	\( ( - 41\!\cdots\!21 \nu^{15} + \cdots - 30\!\cdots\!46 ) / 26\!\cdots\!15 \) (-41095200129351444276321v^15 + 186976239185517904803085v^14 + 341274862299954350725433v^13 - 2024954059708509548397907v^12 - 4903791618621220081401270v^11 + 19643655237023328560151402v^10 + 25543983471189715662637856v^9 - 96443809841607360729572977v^8 - 116096606533990113794339145v^7 + 378695959331700999634483396v^6 + 400241743734662063169853498v^5 - 1104391378955500593199619565v^4 - 1381267695700002391848627919v^3 + 353419540619788317277188471v^2 + 619358458883099279135401522*v - 305196864302239768392307846) / 267268374349772990409257515
\(\beta_{10}\)	\(=\)	\( ( - 37\!\cdots\!36 \nu^{15} + \cdots - 94\!\cdots\!32 ) / 24\!\cdots\!65 \) (-3749752452972790309836v^15 + 20732863114036149974334v^14 + 6726224940442947017928v^13 - 167417531331478374374848v^12 - 286090670248330300561116v^11 + 1905024152577373142561956v^10 + 232462116125913050336682v^9 - 7137595468514527773208299v^8 - 4028888727294004533956076v^7 + 32889253045836149416553118v^6 + 2436987371392293579493658v^5 - 77908154756014835500535970v^4 - 49812830489980484585115494v^3 + 18598144603966802173558229v^2 - 5380312806834899188364748*v - 9430008346102970811232632) / 24297124940888453673568865
\(\beta_{11}\)	\(=\)	\( ( 16\!\cdots\!81 \nu^{15} + \cdots + 96\!\cdots\!62 ) / 48\!\cdots\!30 \) (16470168550969741223681v^15 - 71870035431308235598686v^14 - 133219295569144022286550v^13 + 681714957641911390025460v^12 + 2150231631382714052837073v^11 - 6829871286638493035687430v^10 - 10784007288541905867329869v^9 + 28448063520026534344879210v^8 + 57696866090796810100912583v^7 - 121930752524425552830611882v^6 - 180236067483033699789933788v^5 + 306149779122753280644865106v^4 + 677804783426085602871352221v^3 + 126020251843693812822151512v^2 - 94441083020796753052449404*v + 96075909657405141265191762) / 48594249881776907347137730
\(\beta_{12}\)	\(=\)	\( ( - 33\!\cdots\!85 \nu^{15} + \cdots - 52\!\cdots\!70 ) / 72\!\cdots\!30 \) (-3370703395126322785v^15 + 15495604019523482067v^14 + 26695308022051658876v^13 - 166083683807058573164v^12 - 385689652854120766883v^11 + 1616760037176248284719v^10 + 1922330337103219298065v^9 - 7975610094237435056969v^8 - 8454760380058935675037v^7 + 31616505138858977844375v^6 + 28654696031137831024664v^5 - 95002610577203785398772v^4 - 98318929568311644650865v^3 + 43113590494733483003697v^2 + 30496848408833185816198*v - 52254785449777176496870) / 7295338520008543363930
\(\beta_{13}\)	\(=\)	\( ( 13\!\cdots\!93 \nu^{15} + \cdots + 95\!\cdots\!01 ) / 26\!\cdots\!15 \) (138618733944064656701793v^15 - 625376582509086943959378v^14 - 1146301403278371270047241v^13 + 6603515949494172204097708v^12 + 16897708024989469539486008v^11 - 64328753553874720078640970v^10 - 88497859789696090763202104v^9 + 309959434243517322190375252v^8 + 415682678321866356204315949v^7 - 1237446239076968380407908805v^6 - 1442844248297700239579562720v^5 + 3661556063559093025491808457v^4 + 4955013187588874822634303335v^3 - 994812573754178634125868511v^2 - 2416361124053653891907438426*v + 950094218834732222293940101) / 267268374349772990409257515
\(\beta_{14}\)	\(=\)	\( ( 45\!\cdots\!53 \nu^{15} + \cdots + 40\!\cdots\!26 ) / 80\!\cdots\!30 \) (45369637291229841153v^15 - 267993407964008848704v^14 - 13270316485414436876v^13 + 2197624416856760704236v^12 + 2776551364098304275875v^11 - 25465020052003625319132v^10 + 3367524945449549513277v^9 + 99846803912414683282538v^8 + 23356535407857782368875v^7 - 463358241274880771798218v^6 + 69639623478464722886616v^5 + 1183791263381713459560396v^4 + 391398737429391284496229v^3 - 1212652331560182825821726v^2 - 336158884782390398850508*v + 407047977908653791587226) / 80248723720093977003230
\(\beta_{15}\)	\(=\)	\( ( 30\!\cdots\!23 \nu^{15} + \cdots + 95\!\cdots\!26 ) / 48\!\cdots\!30 \) (30353919795772637896923v^15 - 161631294061460498897621v^14 - 85896039654266511675098v^13 + 1332224814969152169245508v^12 + 2548863960122219957310067v^11 - 14685057386595343054569511v^10 - 4857623912696445584359515v^9 + 55230792071365679146196749v^8 + 43837096800234031754716047v^7 - 245372001766665950912435157v^6 - 81938323654947809862639504v^5 + 588191260269603661689902272v^4 + 578735350354413485984124941v^3 - 50490861676791213669388245v^2 - 6200720418189019219603360*v + 95123918714823557352504726) / 48594249881776907347137730

\(\nu\)	\(=\)	\( ( -\beta_{9} - \beta_{7} + \beta_{6} - \beta_{3} + \beta_{2} ) / 2 \) (-b9 - b7 + b6 - b3 + b2) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{13} + \beta_{10} + 2\beta_{8} - 3\beta_{7} + \beta_{6} - 2\beta_{5} + \beta_{2} - 4\beta _1 + 3 ) / 2 \) (b13 + b10 + 2b8 - 3b7 + b6 - 2b5 + b2 - 4b1 + 3) / 2
\(\nu^{3}\)	\(=\)	\( ( 3 \beta_{14} + 2 \beta_{13} - \beta_{11} + 4 \beta_{10} + 2 \beta_{9} - 6 \beta_{8} - 8 \beta_{7} + \cdots + 4 ) / 2 \) (3b14 + 2b13 - b11 + 4b10 + 2b9 - 6b8 - 8b7 - b6 - 3b5 - 9b3 + 2b2 - 3b1 + 4) / 2
\(\nu^{4}\)	\(=\)	\( ( 2 \beta_{15} + 8 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} + 24 \beta_{10} - 9 \beta_{9} - 8 \beta_{8} + \cdots - 22 ) / 2 \) (2b15 + 8b14 + 2b13 + 2b12 + 24b10 - 9b9 - 8b8 - 13b7 + 3b6 - 16b5 + 8b4 - 5b3 + 5b2 - 24b1 - 22) / 2
\(\nu^{5}\)	\(=\)	\( ( 3 \beta_{15} + 30 \beta_{14} + 2 \beta_{13} + 25 \beta_{12} + 3 \beta_{11} + 67 \beta_{10} + 19 \beta_{9} + \cdots + 13 ) / 2 \) (3b15 + 30b14 + 2b13 + 25b12 + 3b11 + 67b10 + 19b9 - 45b8 + 55b7 - 58b6 - 45b5 + 5b4 - 19b3 - 12b2 - 60*b1 + 13) / 2
\(\nu^{6}\)	\(=\)	\( ( - 10 \beta_{15} + 94 \beta_{14} - 51 \beta_{13} + 90 \beta_{12} - 4 \beta_{11} + 132 \beta_{10} + \cdots - 240 ) / 2 \) (-10b15 + 94b14 - 51b13 + 90b12 - 4b11 + 132b10 - 113b9 - 186b8 + 246b7 - 154b6 - 90b5 + 75b4 - 135b3 - 56b2 - 18*b1 - 240) / 2
\(\nu^{7}\)	\(=\)	\( ( - 83 \beta_{15} + 70 \beta_{14} - 200 \beta_{13} + 385 \beta_{12} + 116 \beta_{11} + 290 \beta_{10} + \cdots - 678 ) / 2 \) (-83b15 + 70b14 - 200b13 + 385b12 + 116b11 + 290b10 - 207b9 + 280b8 + 1303b7 - 734b6 - 252b5 + 210b4 + 30b3 - 304b2 - 175*b1 - 678) / 2
\(\nu^{8}\)	\(=\)	\( ( - 698 \beta_{15} + 160 \beta_{14} - 681 \beta_{13} + 1242 \beta_{12} + 290 \beta_{11} - 787 \beta_{10} + \cdots - 1963 ) / 2 \) (-698b15 + 160b14 - 681b13 + 1242b12 + 290b11 - 787b10 - 238b9 + 368b8 + 4685b7 - 2761b6 + 40b5 + 560b4 - 854b3 - 1197b2 + 1944*b1 - 1963) / 2
\(\nu^{9}\)	\(=\)	\( ( - 2952 \beta_{15} - 1371 \beta_{14} - 2480 \beta_{13} + 2817 \beta_{12} + 1861 \beta_{11} - 5017 \beta_{10} + \cdots - 9942 ) / 2 \) (-2952b15 - 1371b14 - 2480b13 + 2817b12 + 1861b11 - 5017b10 - 2542b9 + 7092b8 + 11465b7 - 6331b6 + 1707b5 + 2376b4 + 314b3 - 3322b2 + 9171*b1 - 9942) / 2
\(\nu^{10}\)	\(=\)	\( ( - 11816 \beta_{15} - 7636 \beta_{14} - 4614 \beta_{13} + 6245 \beta_{12} + 7584 \beta_{11} + \cdots - 12633 ) / 2 \) (-11816b15 - 7636b14 - 4614b13 + 6245b12 + 7584b11 - 25324b10 + 3547b9 + 29996b8 + 31980b7 - 17689b6 + 10766b5 + 2275b4 + 5118b3 - 9301b2 + 41378*b1 - 12633) / 2
\(\nu^{11}\)	\(=\)	\( ( - 42431 \beta_{15} - 26191 \beta_{14} - 9882 \beta_{13} - 2475 \beta_{12} + 23489 \beta_{11} + \cdots - 32285 ) / 2 \) (-42431b15 - 26191b14 - 9882b13 - 2475b12 + 23489b11 - 105509b10 + 3228b9 + 85866b8 + 29479b7 - 9818b6 + 49346b5 + 1595b4 + 10307b3 - 14238b2 + 170313*b1 - 32285) / 2
\(\nu^{12}\)	\(=\)	\( ( - 120806 \beta_{15} - 107668 \beta_{14} + 3095 \beta_{13} - 72761 \beta_{12} + 83294 \beta_{11} + \cdots + 49902 ) / 2 \) (-120806b15 - 107668b14 + 3095b13 - 72761b12 + 83294b11 - 307593b10 + 54407b9 + 353724b8 - 105345b7 + 102034b6 + 153868b5 - 41593b4 + 108658b3 + 5964b2 + 480732*b1 + 49902) / 2
\(\nu^{13}\)	\(=\)	\( ( - 315749 \beta_{15} - 250744 \beta_{14} + 126942 \beta_{13} - 426270 \beta_{12} + 182242 \beta_{11} + \cdots + 704120 ) / 2 \) (-315749b15 - 250744b14 + 126942b13 - 426270b12 + 182242b11 - 890843b10 + 338826b9 + 717054b8 - 943916b7 + 716523b6 + 451373b5 - 290264b4 + 175991b3 + 178844b2 + 1365832*b1 + 704120) / 2
\(\nu^{14}\)	\(=\)	\( ( - 513110 \beta_{15} - 545228 \beta_{14} + 598989 \beta_{13} - 1953441 \beta_{12} + 324840 \beta_{11} + \cdots + 2738175 ) / 2 \) (-513110b15 - 545228b14 + 598989b13 - 1953441b12 + 324840b11 - 1619050b10 + 879678b9 + 1467944b8 - 5111964b7 + 3614513b6 + 879088b5 - 1113756b4 + 522719b3 + 1070017b2 + 2361172*b1 + 2738175) / 2
\(\nu^{15}\)	\(=\)	\( ( 332066 \beta_{15} - 365277 \beta_{14} + 2659498 \beta_{13} - 6794375 \beta_{12} - 116159 \beta_{11} + \cdots + 12385763 ) / 2 \) (332066b15 - 365277b14 + 2659498b13 - 6794375b12 - 116159b11 - 115346b10 + 4299151b9 + 331212b8 - 18307195b7 + 12694888b6 + 349242b5 - 4437065b4 + 809562b3 + 4105062b2 - 552834*b1 + 12385763) / 2

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1249.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 21 T^{6} + \cdots + 80)^{2} \) (T^8 + 21T^6 + 129T^4 + 220*T^2 + 80)^2
$5$	\( T^{16} \) T^16
$7$	\( (T^{8} + 41 T^{6} + \cdots + 4205)^{2} \) (T^8 + 41T^6 + 524T^4 + 2605*T^2 + 4205)^2
$11$	\( (T^{8} - 49 T^{6} + 684 T^{4} + \cdots + 5)^{2} \) (T^8 - 49T^6 + 684T^4 - 2445*T^2 + 5)^2
$13$	\( (T^{8} + 70 T^{6} + \cdots + 22201)^{2} \) (T^8 + 70T^6 + 1523T^4 + 11150*T^2 + 22201)^2
$17$	\( (T^{8} + 66 T^{6} + \cdots + 30976)^{2} \) (T^8 + 66T^6 + 1441T^4 + 12016*T^2 + 30976)^2
$19$	\( (T^{8} - 80 T^{6} + \cdots + 3125)^{2} \) (T^8 - 80T^6 + 1645T^4 - 5500*T^2 + 3125)^2
$23$	\( (T^{8} + 91 T^{6} + \cdots + 20480)^{2} \) (T^8 + 91T^6 + 2329T^4 + 15680*T^2 + 20480)^2
$29$	\( (T^{4} + 12 T^{3} + \cdots + 20)^{4} \) (T^4 + 12T^3 + 21T^2 - 70*T + 20)^4
$31$	\( (T^{8} - 29 T^{6} + \cdots + 80)^{2} \) (T^8 - 29T^6 + 209T^4 - 460*T^2 + 80)^2
$37$	\( (T^{8} + 143 T^{6} + \cdots + 102400)^{2} \) (T^8 + 143T^6 + 5641T^4 + 71360*T^2 + 102400)^2
$41$	\( (T^{4} - 9 T^{3} - 6 T^{2} + \cdots + 95)^{4} \) (T^4 - 9T^3 - 6T^2 + 95*T + 95)^4
$43$	\( (T^{8} + 234 T^{6} + \cdots + 3872000)^{2} \) (T^8 + 234T^6 + 17109T^4 + 473200*T^2 + 3872000)^2
$47$	\( (T^{8} + 176 T^{6} + \cdots + 17405)^{2} \) (T^8 + 176T^6 + 8069T^4 + 51460*T^2 + 17405)^2
$53$	\( (T^{8} + 374 T^{6} + \cdots + 6255001)^{2} \) (T^8 + 374T^6 + 43171T^4 + 1470174*T^2 + 6255001)^2
$59$	\( (T^{8} - 196 T^{6} + \cdots + 85805)^{2} \) (T^8 - 196T^6 + 11809T^4 - 202760*T^2 + 85805)^2
$61$	\( (T^{4} - 11 T^{3} + \cdots - 2620)^{4} \) (T^4 - 11T^3 - 61T^2 + 980*T - 2620)^4
$67$	\( (T^{8} + 81 T^{6} + \cdots + 80)^{2} \) (T^8 + 81T^6 + 369T^4 + 380*T^2 + 80)^2
$71$	\( (T^{8} - 694 T^{6} + \cdots + 849686480)^{2} \) (T^8 - 694T^6 + 178669T^4 - 20224700*T^2 + 849686480)^2
$73$	\( (T^{8} + 379 T^{6} + \cdots + 17007376)^{2} \) (T^8 + 379T^6 + 40381T^4 + 1463404*T^2 + 17007376)^2
$79$	\( (T^{8} - 546 T^{6} + \cdots + 3494480)^{2} \) (T^8 - 546T^6 + 78069T^4 - 1154440*T^2 + 3494480)^2
$83$	\( (T^{8} + 576 T^{6} + \cdots + 8192000)^{2} \) (T^8 + 576T^6 + 90624T^4 + 2867200*T^2 + 8192000)^2
$89$	\( (T^{4} + 5 T^{3} + \cdots - 2000)^{4} \) (T^4 + 5T^3 - 245T^2 - 1900*T - 2000)^4
$97$	\( (T^{8} + 491 T^{6} + \cdots + 59536)^{2} \) (T^8 + 491T^6 + 65101T^4 + 2019996*T^2 + 59536)^2
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\(n\)	\(1377\)	\(2501\)	\(2751\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)