LMFDB - Newform orbit 192.8.f.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [192,8,Mod(95,192)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("192.95"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(192, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	192.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,15360] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$59.9779248930$
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} - 4372 x^{14} + 7613770 x^{12} - 6683530912 x^{10} + 3071754181219 x^{8} - 688859966054368 x^{6} + \cdots + 59\!\cdots\!21 $ x^16 - 4372x^14 + 7613770x^12 - 6683530912x^10 + 3071754181219x^8 - 688859966054368x^6 + 59012195426142490x^4 + 21745092566256572*x^2 + 5916741936370321
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{60}\cdot 3^{12} $
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_{5} q^{3} - \beta_{8} q^{5} + (\beta_{11} + 36 \beta_{4}) q^{7} + (\beta_{13} + 960) q^{9} + ( - 30 \beta_{7} + \beta_{6} + 61 \beta_{5}) q^{11} + ( - \beta_{10} - \beta_{9}) q^{13} + ( - 5 \beta_{11} + 132 \beta_{4} - \beta_{3}) q^{15}+ \cdots + (309 \beta_{15} + \cdots - 101988 \beta_{5}) q^{99}+O(q^{100}) $ q - b5 * q^3 - b8 * q^5 + (b11 + 36b4) q^7 + (b13 + 960) * q^9 + (-30b7 + b6 + 61b5) * q^11 + (-b10 - b9) * q^13 + (-5b11 + 132b4 - b3) * q^15 + (-b14 - 9b13 - 5b1) * q^17 + (b15 - 4b7 - 18b5) * q^19 + (21b12 + 3b10 - 46b9 - 45b8) * q^21 + (-b4 - 2b3 + 5b2) * q^23 + (-55b1 + 35545) q^25 + (-3b15 + 423b7 + 45b6 - 1080b5) * q^27 + (-91b12 + 182b9 + 243b8) q^29 + (17b11 + 17331b4) * q^31 + (-b14 - 88b13 + 7b1 + 111981) * q^33 + (-1185b7 - 245b6 + 2125b5) q^35 + (-63b12 + 5b10 - 247b9) q^37 + (-486b11 + 2667b4 + 15b3 + 21b2) * q^39 + (15b14 - 243b13 - 114b1) q^41 + (-7b15 - 1214b7 - 4842b5) q^43 + (-210b12 - 30b10 - 165b9 - 1737b8) * q^45 + (14b4 + 28b3 + 11b2) q^47 + (-287b1 - 116355) q^49 + (36b15 - 4347b7 + 1647b6 + 1845b5) * q^51 + (134b12 - 268b9 + 2975b8) q^53 + (710b11 + 3745b4) * q^55 + (15b14 + 288b13 + 624b1 + 41445) q^57 + (-3077b7 - 2484b6 + 3670b5) q^59 + (-963b12 + 27b10 - 3825b9) q^61 + (673b11 - 72712b4 - 90b3 - 133b2) * q^63 + (-91b14 + 1395b13 + 652b1) q^65 + (-12b15 - 12471b7 - 49860b5) q^67 + (-3015b12 + 90b10 + 585b9 - 5724b8) * q^69 + (-75b4 - 150b3 - 138b2) q^71 + (1126b1 - 1620866) q^73 + (-165b15 - 16830b7 + 2475b6 - 29440b5) * q^75 + (-4358b12 + 8716b9 + 7390b8) q^77 + (801b11 - 477037b4) * q^79 + (-90b14 + 816b13 - 1557b1 - 1224711) q^81 + (-47120b7 - 2665b6 + 91575b5) q^83 + (-4500b12 - 240b10 - 18240b9) q^85 + (3399b11 + 303168b4 + 243b3 + 273b2) * q^87 + (260b14 - 198b13 + 31b1) q^89 + (259b15 - 43381b7 - 174042b5) q^91 + (357b12 + 51b10 - 17501b9 - 765b8) * q^93 + (155b4 + 310b3 - 235b2) q^95 + (3143b1 + 2713832) q^97 + (309b15 - 34092b7 - 2448b6 - 101988b5) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 15360 q^{9} + 568720 q^{25} + 1791696 q^{33} - 1861680 q^{49} + 663120 q^{57} - 25933856 q^{73} - 19595376 q^{81} + 43421312 q^{97}+O(q^{100}) $ 16 * q + 15360 * q^9 + 568720 * q^25 + 1791696 * q^33 - 1861680 * q^49 + 663120 * q^57 - 25933856 * q^73 - 19595376 * q^81 + 43421312 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4372 x^{14} + 7613770 x^{12} - 6683530912 x^{10} + 3071754181219 x^{8} - 688859966054368 x^{6} + \cdots + 59\!\cdots\!21 $ x^16 - 4372*x^14 + 7613770*x^12 - 6683530912*x^10 + 3071754181219*x^8 - 688859966054368*x^6 + 59012195426142490*x^4 + 21745092566256572*x^2 + 5916741936370321 :

$\beta_{1}$	$=$	$ ( 188835 \nu^{14} - 3912579551 \nu^{12} + 11532772844222 \nu^{10} + \cdots + 90\!\cdots\!32 ) / 48\!\cdots\!80 $ (188835v^14 - 3912579551v^12 + 11532772844222v^10 - 13275553286089131v^8 + 6771772468779726774v^6 - 1312133465427264861931v^4 - 483197883212932960677*v^2 + 9042481370797842623321132) / 4882469313836870813280
$\beta_{2}$	$=$	$ ( - 12\!\cdots\!38 \nu^{14} + \cdots - 41\!\cdots\!42 ) / 23\!\cdots\!09 $ (-1215151519503738v^14 + 4656680089940146080v^12 - 6885469339497411467938v^10 + 4888047620193762659744406v^8 - 1672310241596461170189702018v^6 + 235219862512300539819427650150v^4 + 86636840704731993650692332296*v^2 - 4111131680642569975371164489015642) / 235005217946910205786145718109
$\beta_{3}$	$=$	$ ( 12\!\cdots\!00 \nu^{15} + \cdots - 20\!\cdots\!09 ) / 55\!\cdots\!00 $ (12032977124771053077929400v^15 - 275135405999448214368927066v^14 - 52572123316272644001980847000v^13 + 1058374614174785574294566655853v^12 + 91478159888030213226983798670000v^11 - 1653500083267057527101252638123005v^10 - 80218487644571764796071862226303000v^9 + 1344546748775771456933168551536022980v^8 + 36817322931766300549215010801377006000v^7 - 578568487526283620766866225612819438137v^6 - 8239419969386129720546140068454284341400v^5 + 104075164155331062844876928777672193374868v^4 + 703113593188890889217212305571158102891000v^3 + 38327525119127842596998605362801197773965v^2 + 482205703958316524422035680209009922880000*v - 201964801694716560644525553853236077894156309) / 55779488849910893750705676658263996391200
$\beta_{4}$	$=$	$ ( - 22\!\cdots\!61 \nu^{15} + \cdots - 89\!\cdots\!00 \nu ) / 51\!\cdots\!14 $ (-2228329097179824644061v^15 + 9735578391902341481848305v^13 - 16940399979264854301293296050v^11 + 14855275489735511999272567078945v^9 - 6818022765141907509113890889143890v^7 + 1525818512849283281582618531195237841v^5 - 130206220960905720225409686216881130165v^3 - 89297352584873430448525125964631467200v) / 5164767486102860532472747838728147814
$\beta_{5}$	$=$	$ ( - 38\!\cdots\!74 \nu^{15} + \cdots + 25\!\cdots\!64 ) / 11\!\cdots\!60 $ (-3839424292193902368277274v^15 + 2399065040446738979782819v^14 + 16787157161440772228528249860v^13 - 10518161684360285790917994107v^12 - 29237641174203371910518565261822v^11 + 18357459313060962373245277056366v^10 + 25669761307201356788293837670966768v^9 - 16134273363460971400340629275574843v^8 - 11801456775261507312390469524857938546v^7 + 7412725797597244884070801943147078702v^6 + 2648339243109190007342600683486899044532v^5 - 1657893492771842096835812672790570795195v^4 - 227364444051789876757059319945742234916776v^3 + 141459968878588081339009645112050723155019v^2 - 11632776654302061531316919697054410436548*v + 25980142740907273009780715269793669131264) / 1112117826066299976856278053499732581760
$\beta_{6}$	$=$	$ ( 57\!\cdots\!10 \nu^{15} + \cdots + 45\!\cdots\!92 ) / 16\!\cdots\!00 $ (57591364382908535524159110v^15 + 388787326593273671879838245v^14 - 251807357421611583427923747900v^13 - 1692121448826206070512489959237v^12 + 438564617613050578657778478927330v^11 + 2940682043049912050848396349409258v^10 - 385046419608020351824407565064501520v^9 - 2588679382743803499816072760530117053v^8 + 177021851628922609685857042872869078190v^7 + 1203878187160207998692952232946181644938v^6 - 39725088646637850110139010252303485667980v^5 - 277129377019270343151668954480422900277757v^4 + 3410466660776848151355889799186133523751640v^3 + 24668280324329404148050169378816065032860933v^2 + 174491649814530922969753795455816156548220*v + 4530894646876437618562747974998241822809192) / 16681767390994499652844170802495988726400
$\beta_{7}$	$=$	$ ( - 19\!\cdots\!37 \nu^{15} + \cdots - 25\!\cdots\!64 ) / 27\!\cdots\!40 $ (-1919712146096951184138637v^15 - 2399065040446738979782819v^14 + 8393578580720386114264124930v^13 + 10518161684360285790917994107v^12 - 14618820587101685955259282630911v^11 - 18357459313060962373245277056366v^10 + 12834880653600678394146918835483384v^9 + 16134273363460971400340629275574843v^8 - 5900728387630753656195234762428969273v^7 - 7412725797597244884070801943147078702v^6 + 1324169621554595003671300341743449522266v^5 + 1657893492771842096835812672790570795195v^4 - 113682222025894938378529659972871117458388v^3 - 141459968878588081339009645112050723155019v^2 - 5816388327151030765658459848527205218274*v - 25980142740907273009780715269793669131264) / 278029456516574994214069513374933145440
$\beta_{8}$	$=$	$ ( - 23\!\cdots\!26 \nu^{15} + \cdots - 71\!\cdots\!56 \nu ) / 21\!\cdots\!00 $ (-23666113767585966230913626v^15 + 103479922954232309078220351655v^13 - 180234684058120203893912359846143v^11 + 158247490974980328761494323223676063v^9 - 72758163997380951213574181210517635185v^7 + 16330333945593813529524103957292169312765v^5 - 1402663336600031863858237299254064223526723v^3 - 71765900872531967792720770724097232889756v) / 2104118649387101257815840796780003375200
$\beta_{9}$	$=$	$ ( - 19\!\cdots\!18 \nu^{15} + \cdots - 13\!\cdots\!08 ) / 70\!\cdots\!40 $ (-1906849316209964537360318v^15 - 1391734456473209033156533v^14 + 8337438365059578102886331620v^13 + 6001635196949380499560945229v^12 - 14521256731179036512681131390434v^11 - 10289397586787807971299227997042v^10 + 12749465378470343708663167085747096v^9 + 8869665406226886471056807384951101v^8 - 5861616487726104565534644189659843662v^7 - 3989502971162099619513548751260083634v^6 + 1315449629354033429909007321881854673844v^5 + 871559914101016910078967591169598799165v^4 - 112944295490495531632380147204816085633472v^3 - 72480257094481693266395086908435421291453v^2 - 5778643451534447523548865144861140234396*v - 13310806442409757662278227289553457949008) / 70137288312903375260528026559333445840
$\beta_{10}$	$=$	$ ( 38\!\cdots\!36 \nu^{15} + \cdots + 13\!\cdots\!69 ) / 14\!\cdots\!80 $ (3813698632419929074720636v^15 + 112688068962497567248605087v^14 - 16674876730119156205772663240v^13 - 495875255132320318732631592078v^12 + 29042513462358073025362262780868v^11 + 870480502661323882502100724171671v^10 - 25498930756940687417326334171494192v^9 - 771677530448432213093786204617771317v^8 + 11723232975452209131069288379319687324v^7 + 359022685579361456920842397937828391459v^6 - 2630899258708066859818014643763709347688v^5 - 81759877913656242813127877156703480318477v^4 + 225888590980991063264760294409632171266944v^3 + 7134918333815265317693167708974960775895760v^2 + 11557286903068895047097730289722280468792*v + 1310439334549755776336574383862757850349069) / 140274576625806750521056053118666891680
$\beta_{11}$	$=$	$ ( - 61\!\cdots\!42 \nu^{15} + \cdots - 24\!\cdots\!10 \nu ) / 12\!\cdots\!60 $ (-61103899295886916775478642v^15 + 267258222211327514983391458837v^13 - 465659339203408392978661200659415v^11 + 409008095385507579234744235386764599v^9 - 188106423259306782873146232919220364909v^7 + 42209091814210930054143733359800165725989v^5 - 3613383659439377620546986938231586286152851v^3 - 2478091584018184330830106096580223074283910v) / 1239544196664686527793459481294755475360
$\beta_{12}$	$=$	$ ( 38\!\cdots\!36 \nu^{15} + \cdots - 13\!\cdots\!08 ) / 35\!\cdots\!20 $ (3813698632419929074720636v^15 - 1391734456473209033156533v^14 - 16674876730119156205772663240v^13 + 6001635196949380499560945229v^12 + 29042513462358073025362262780868v^11 - 10289397586787807971299227997042v^10 - 25498930756940687417326334171494192v^9 + 8869665406226886471056807384951101v^8 + 11723232975452209131069288379319687324v^7 - 3989502971162099619513548751260083634v^6 - 2630899258708066859818014643763709347688v^5 + 871559914101016910078967591169598799165v^4 + 225888590980991063264760294409632171266944v^3 - 72480257094481693266395086908435421291453v^2 + 11557286903068895047097730289722280468792*v - 13310806442409757662278227289553457949008) / 35068644156451687630264013279666722920
$\beta_{13}$	$=$	$ ( 88\!\cdots\!00 \nu^{15} - 14525256369435 \nu^{14} + \cdots - 69\!\cdots\!52 ) / 75\!\cdots\!60 $ (88147776436545600v^15 - 14525256369435v^14 - 385378592519048698800v^13 + 300957031504137911v^12 + 671064666111716799813360v^11 - 887105050508553004142v^10 - 588925483425595671376593120v^9 + 1021160351240712234696291v^8 + 270523222349818820100429976560v^7 - 520887182908397812937445414v^6 - 60596461359948806007646724281440v^5 + 100929779840846232422347677091v^4 + 5176068417082776898851234095265840v^3 + 37167755611174643204073644397v^2 + 3549815309958042262559598443545680*v - 695550931377544912607048492368652) / 751122604383508796135722388160
$\beta_{14}$	$=$	$ ( 32\!\cdots\!48 \nu^{15} - 653636536624575 \nu^{14} + \cdots - 31\!\cdots\!40 ) / 33\!\cdots\!00 $ (32716857869949197948v^15 - 653636536624575v^14 - 143039077102939088305092v^13 + 13543066417686205995v^12 + 249135395679659490491456472v^11 - 39919727272884885186390v^10 - 218777958273002149982972814452v^9 + 45952215805832050561333095v^8 + 100625263849996346013009522667128v^7 - 23439923230877901582185043630v^6 - 22592294089931713607565315410705892v^5 + 4541840092838080459005645469095v^4 + 1936024423951248710336520039590033612v^3 + 1672549002502858944183313997865v^2 + 1327739772069446229661622924437737600*v - 31299791911989521067317182156589340) / 33800517197257895826107507467200
$\beta_{15}$	$=$	$ ( 15\!\cdots\!56 \nu^{15} + \cdots + 25\!\cdots\!64 ) / 55\!\cdots\!80 $ (1534172557933543800406711656v^15 + 2399065040446738979782819v^14 - 6707758515673193645964632816460v^13 - 10518161684360285790917994107v^12 + 11682545800177304039603433014735148v^11 + 18357459313060962373245277056366v^10 - 10256912476749442897766077750084590060v^9 - 16134273363460971400340629275574843v^8 + 4715620748120911566536830298002811504532v^7 + 7412725797597244884070801943147078702v^6 - 1058288607079738346438716485657157134448644v^5 - 1657893492771842096835812672790570795195v^4 + 90872228139240135862828054439493121661532636v^3 + 141459968878588081339009645112050723155019v^2 + 4649363003356243338961333355813737731213744*v + 25980142740907273009780715269793669131264) / 556058913033149988428139026749866290880

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

$\nu$	$=$	$ ( \beta_{12} - 2\beta_{9} + 8\beta_{7} + 32\beta_{5} - 6\beta_{4} ) / 96 $ (b12 - 2b9 + 8b7 + 32b5 - 6b4) / 96
$\nu^{2}$	$=$	$ ( -\beta_{12} - 4\beta_{9} + 8\beta_{7} - 16\beta_{5} + 3\beta_{2} + 52464 ) / 96 $ (-b12 - 4b9 + 8b7 - 16b5 + 3b2 + 52464) / 96
$\nu^{3}$	$=$	$ ( - 12 \beta_{15} - 24 \beta_{13} + 759 \beta_{12} - 1518 \beta_{9} + 360 \beta_{8} + 4180 \beta_{7} + \cdots - 12 \beta_1 ) / 64 $ (-12b15 - 24b13 + 759b12 - 1518b9 + 360b8 + 4180b7 + 16744b5 - 6560b4 - 12*b1) / 64
$\nu^{4}$	$=$	$ ( - 1591 \beta_{12} + 72 \beta_{10} - 6292 \beta_{9} + 18764 \beta_{7} + 540 \beta_{6} - 36988 \beta_{5} + \cdots + 46642128 ) / 96 $ (-1591b12 + 72b10 - 6292b9 + 18764b7 + 540b6 - 36988b5 + 3276b2 + 5760b1 + 46642128) / 96
$\nu^{5}$	$=$	$ ( - 56556 \beta_{15} - 131160 \beta_{13} + 2449933 \beta_{12} - 57600 \beta_{11} - 4899866 \beta_{9} + \cdots - 65580 \beta_1 ) / 192 $ (-56556b15 - 131160b13 + 2449933b12 - 57600b11 - 4899866b9 + 659880b8 + 11385188b7 + 45653864b5 - 29191332b4 - 65580b1) / 192
$\nu^{6}$	$=$	$ ( - 730574 \beta_{12} + 56616 \beta_{10} - 2865680 \beta_{9} + 9980340 \beta_{7} + 165420 \beta_{6} + \cdots + 15025889056 ) / 32 $ (-730574b12 + 56616b10 - 2865680b9 + 9980340b7 + 165420b6 - 19795260b5 + 43200b4 + 86400b3 + 1173687b2 + 3142080b1 + 15025889056) / 32
$\nu^{7}$	$=$	$ ( - 36195264 \beta_{15} - 453600 \beta_{14} - 96551280 \beta_{13} + 1308101543 \beta_{12} + \cdots - 48502440 \beta_1 ) / 96 $ (-36195264b15 - 453600b14 - 96551280b13 + 1308101543b12 - 66084480b11 - 2616203086b9 - 45161280b8 + 5493851320b7 + 22047795808b5 - 19780823658b4 - 48502440*b1) / 96
$\nu^{8}$	$=$	$ ( - 2847645569 \beta_{12} + 290354904 \beta_{10} - 11100227372 \beta_{9} + 41908695364 \beta_{7} + \cdots + 45157691024592 ) / 96 $ (-2847645569b12 + 290354904b10 - 11100227372b9 + 41908695364b7 - 88005420b6 - 83905396148b5 + 282528000b4 + 565056000b3 + 3780106836b2 + 12427868160b1 + 45157691024592) / 96
$\nu^{9}$	$=$	$ ( - 28912543020 \beta_{15} - 849398400 \beta_{14} - 86871032856 \beta_{13} + 932611011327 \beta_{12} + \cdots - 43860215628 \beta_1 ) / 64 $ (-28912543020b15 - 849398400b14 - 86871032856b13 + 932611011327b12 - 74831627520b11 - 1865222022654b9 - 288711178200b8 + 3641259279156b7 + 14622862202664b5 - 17022619210960b4 - 43860215628*b1) / 64
$\nu^{10}$	$=$	$ ( - 3570580811393 \beta_{12} + 435866996376 \beta_{10} - 13846456249196 \beta_{9} + \cdots + 46\!\cdots\!64 ) / 96 $ (-3570580811393b12 + 435866996376b10 - 13846456249196b9 + 55188518657572b7 - 1083323482380b6 - 111460360797524b5 + 432895752000b4 + 865791504000b3 + 4065340001670b2 + 15186342888000b1 + 46127626095062064) / 96
$\nu^{11}$	$=$	$ ( - 100618611465852 \beta_{15} - 4785221707200 \beta_{14} - 336331732761816 \beta_{13} + \cdots - 170558477234508 \beta_1 ) / 192 $ (-100618611465852b15 - 4785221707200b14 - 336331732761816b13 + 3000560130844949b12 - 336158867784960b11 - 6001120261689898b9 - 1599764743202040b8 + 11065360242051700b7 + 44462678191138504b5 - 63935805851287116b4 - 170558477234508*b1) / 192
$\nu^{12}$	$=$	$ ( - 729582235098202 \beta_{12} + 101418500058168 \beta_{10} + \cdots + 79\!\cdots\!04 ) / 16 $ (-729582235098202b12 + 101418500058168b10 - 2816910440334640b9 + 11703593647199100b7 - 401927519045340b6 - 23809114813443540b5 + 96015680078400b4 + 192031360156800b3 + 730033326300261b2 + 2974417467399360b1 + 7957668256153553104) / 16
$\nu^{13}$	$=$	$ ( - 57\!\cdots\!84 \beta_{15} + \cdots - 10\!\cdots\!20 \beta_1 ) / 96 $ (-57219534806844384b15 - 3775745848680000b14 - 211236606040393440b13 + 1612948956531591529b12 - 234192273029694720b11 - 3225897913063183058b9 - 1142541744562198080b8 + 5683434007637137448b7 + 22848175100162238560b5 - 39238825424540178534b4 - 107506175944536720*b1) / 96
$\nu^{14}$	$=$	$ ( - 52\!\cdots\!05 \beta_{12} + \cdots + 49\!\cdots\!56 ) / 96 $ (-5289851547062286205b12 + 810317135007276624b10 - 20349089053241868196b9 + 87393116938072662512b7 - 4035507663337018920b6 - 178821741539482343944b5 + 712894074267542400b4 + 1425788148535084800b3 + 4725938647599262149b2 + 20474521048444538880b1 + 49917781538293241740656) / 96
$\nu^{15}$	$=$	$ ( - 42\!\cdots\!72 \beta_{15} + \cdots - 88\!\cdots\!40 \beta_1 ) / 64 $ (-42807693154344753372b15 - 3608593416719712000b14 - 173684732514981601080b13 + 1158122844997087974171b12 - 207482561006176627200b11 - 2316245689994175948342b9 - 976898104441145488440b8 + 3935720945004715543556b7 + 15828499166327551680968b5 - 31682964633778539886400b4 - 88646662965850656540*b1) / 64

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$-1$	$-1$	$-1$

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} $

95.1

28.6975	−	0.500000i
16.4302	+	0.500000i
28.6975	+	0.500000i
16.4302	−	0.500000i
0.256837	+	0.500000i
33.0671	−	0.500000i
0.256837	−	0.500000i
33.0671	+	0.500000i
−33.0671	+	0.500000i
−0.256837	−	0.500000i
−33.0671	−	0.500000i
−0.256837	+	0.500000i
−16.4302	−	0.500000i
−28.6975	+	0.500000i
−16.4302	+	0.500000i
−28.6975	−	0.500000i

−45.1278

−

12.2673i

−464.255

−

639.029i

1886.03

1107.19i

95.2

−45.1278

−

12.2673i

464.255

639.029i

1886.03

1107.19i

95.3

−45.1278

12.2673i

−464.255

639.029i

1886.03

−

1107.19i

95.4

−45.1278

12.2673i

464.255

−

639.029i

1886.03

−

1107.19i

95.5

−33.3240

−

32.8103i

−108.659

−

1213.03i

33.9714

2186.74i

95.6

−33.3240

−

32.8103i

108.659

1213.03i

33.9714

2186.74i

95.7

−33.3240

32.8103i

−108.659

1213.03i

33.9714

−

2186.74i

95.8

−33.3240

32.8103i

108.659

−

1213.03i

33.9714

−

2186.74i

95.9

33.3240

−

32.8103i

−108.659

−

1213.03i

33.9714

−

2186.74i

95.10

33.3240

−

32.8103i

108.659

1213.03i

33.9714

−

2186.74i

95.11

33.3240

32.8103i

−108.659

1213.03i

33.9714

2186.74i

95.12

33.3240

32.8103i

108.659

−

1213.03i

33.9714

2186.74i

95.13

45.1278

−

12.2673i

−464.255

−

639.029i

1886.03

−

1107.19i

95.14

45.1278

−

12.2673i

464.255

639.029i

1886.03

−

1107.19i

95.15

45.1278

12.2673i

−464.255

639.029i

1886.03

1107.19i

95.16

45.1278

12.2673i

464.255

−

639.029i

1886.03

1107.19i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
12.b	even	2	1	inner
24.f	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	192.8.f.c	✓	16
3.b	odd	2	1	inner	192.8.f.c	✓	16
4.b	odd	2	1	inner	192.8.f.c	✓	16
8.b	even	2	1	inner	192.8.f.c	✓	16
8.d	odd	2	1	inner	192.8.f.c	✓	16
12.b	even	2	1	inner	192.8.f.c	✓	16
24.f	even	2	1	inner	192.8.f.c	✓	16
24.h	odd	2	1	inner	192.8.f.c	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
192.8.f.c	✓	16	1.a	even	1	1	trivial
192.8.f.c	✓	16	3.b	odd	2	1	inner
192.8.f.c	✓	16	4.b	odd	2	1	inner
192.8.f.c	✓	16	8.b	even	2	1	inner
192.8.f.c	✓	16	8.d	odd	2	1	inner
192.8.f.c	✓	16	12.b	even	2	1	inner
192.8.f.c	✓	16	24.f	even	2	1	inner
192.8.f.c	✓	16	24.h	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{4} - 227340T_{5}^{2} + 2544768000 $ T5^4 - 227340*T5^2 + 2544768000 acting on $S_{8}^{\mathrm{new}}(192, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + \cdots + 22876792454961)^{2} $ (T^8 - 3840T^6 + 9822222T^4 - 18366600960*T^2 + 22876792454961)^2
$5$	$ (T^{4} - 227340 T^{2} + 2544768000)^{4} $ (T^4 - 227340*T^2 + 2544768000)^4
$7$	$ (T^{4} + 1879796 T^{2} + 600873025600)^{4} $ (T^4 + 1879796*T^2 + 600873025600)^4
$11$	$ (T^{4} + \cdots + 73141191486720)^{4} $ (T^4 + 41773932*T^2 + 73141191486720)^4
$13$	$ (T^{4} + \cdots + 18\!\cdots\!80)^{4} $ (T^4 + 274488048*T^2 + 18055477928494080)^4
$17$	$ (T^{4} + \cdots + 99\!\cdots\!00)^{4} $ (T^4 + 2130105600*T^2 + 998883426277785600)^4
$19$	$ (T^{4} + \cdots + 25\!\cdots\!00)^{4} $ (T^4 - 1097727660*T^2 + 258151104952992000)^4
$23$	$ (T^{4} + \cdots + 47\!\cdots\!00)^{4} $ (T^4 - 13865022720*T^2 + 47813643183882240000)^4
$29$	$ (T^{4} + \cdots + 10\!\cdots\!80)^{4} $ (T^4 - 32312986668*T^2 + 101054825827448878080)^4
$31$	$ (T^{4} + \cdots + 35\!\cdots\!00)^{4} $ (T^4 + 38932760084*T^2 + 359888531473591758400)^4
$37$	$ (T^{4} + \cdots + 19\!\cdots\!20)^{4} $ (T^4 + 34324016112*T^2 + 193052057293970334720)^4
$41$	$ (T^{4} + \cdots + 44\!\cdots\!00)^{4} $ (T^4 + 730409944320*T^2 + 44417388888068751360000)^4
$43$	$ (T^{4} + \cdots + 67\!\cdots\!00)^{4} $ (T^4 - 219901068108*T^2 + 675047110862496672000)^4
$47$	$ (T^{4} + \cdots + 20\!\cdots\!00)^{4} $ (T^4 - 325636899840*T^2 + 20659359734181553766400)^4
$53$	$ (T^{4} + \cdots + 26\!\cdots\!00)^{4} $ (T^4 - 2144378795148*T^2 + 26849047832100404352000)^4
$59$	$ (T^{4} + \cdots + 98\!\cdots\!80)^{4} $ (T^4 + 5701817409228*T^2 + 988569467864808563969280)^4
$61$	$ (T^{4} + \cdots + 11\!\cdots\!00)^{4} $ (T^4 + 6815524984560*T^2 + 11588996540150527266432000)^4
$67$	$ (T^{4} + \cdots + 61\!\cdots\!80)^{4} $ (T^4 - 17764373594412*T^2 + 61587204030525295523508480)^4
$71$	$ (T^{4} + \cdots + 32\!\cdots\!00)^{4} $ (T^4 - 12433123226880*T^2 + 32249471256112736010240000)^4
$73$	$ (T^{2} + \cdots - 1721755163660)^{8} $ (T^2 + 3241732*T - 1721755163660)^8
$79$	$ (T^{4} + \cdots + 19\!\cdots\!00)^{4} $ (T^4 + 30240841942676*T^2 + 196561466740626506370510400)^4
$83$	$ (T^{4} + \cdots + 22\!\cdots\!00)^{4} $ (T^4 + 99809580948300*T^2 + 2205479299047380086528800000)^4
$89$	$ (T^{4} + \cdots + 17\!\cdots\!00)^{4} $ (T^4 + 112019174622720*T^2 + 1724631692924932125696000000)^4
$97$	$ (T^{2} + \cdots - 26519341845860)^{8} $ (T^2 - 5427664*T - 26519341845860)^8
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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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	192.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{5} q^{3} - \beta_{8} q^{5} + (\beta_{11} + 36 \beta_{4}) q^{7} + (\beta_{13} + 960) q^{9} + ( - 30 \beta_{7} + \beta_{6} + 61 \beta_{5}) q^{11} + ( - \beta_{10} - \beta_{9}) q^{13} + ( - 5 \beta_{11} + 132 \beta_{4} - \beta_{3}) q^{15}+ \cdots + (309 \beta_{15} + \cdots - 101988 \beta_{5}) q^{99}+O(q^{100}) \) q - b5 * q^3 - b8 * q^5 + (b11 + 36b4) q^7 + (b13 + 960) * q^9 + (-30b7 + b6 + 61b5) * q^11 + (-b10 - b9) * q^13 + (-5b11 + 132b4 - b3) * q^15 + (-b14 - 9b13 - 5b1) * q^17 + (b15 - 4b7 - 18b5) * q^19 + (21b12 + 3b10 - 46b9 - 45b8) * q^21 + (-b4 - 2b3 + 5b2) * q^23 + (-55b1 + 35545) q^25 + (-3b15 + 423b7 + 45b6 - 1080b5) * q^27 + (-91b12 + 182b9 + 243b8) q^29 + (17b11 + 17331b4) * q^31 + (-b14 - 88b13 + 7b1 + 111981) * q^33 + (-1185b7 - 245b6 + 2125b5) q^35 + (-63b12 + 5b10 - 247b9) q^37 + (-486b11 + 2667b4 + 15b3 + 21b2) * q^39 + (15b14 - 243b13 - 114b1) q^41 + (-7b15 - 1214b7 - 4842b5) q^43 + (-210b12 - 30b10 - 165b9 - 1737b8) * q^45 + (14b4 + 28b3 + 11b2) q^47 + (-287b1 - 116355) q^49 + (36b15 - 4347b7 + 1647b6 + 1845b5) * q^51 + (134b12 - 268b9 + 2975b8) q^53 + (710b11 + 3745b4) * q^55 + (15b14 + 288b13 + 624b1 + 41445) q^57 + (-3077b7 - 2484b6 + 3670b5) q^59 + (-963b12 + 27b10 - 3825b9) q^61 + (673b11 - 72712b4 - 90b3 - 133b2) * q^63 + (-91b14 + 1395b13 + 652b1) q^65 + (-12b15 - 12471b7 - 49860b5) q^67 + (-3015b12 + 90b10 + 585b9 - 5724b8) * q^69 + (-75b4 - 150b3 - 138b2) q^71 + (1126b1 - 1620866) q^73 + (-165b15 - 16830b7 + 2475b6 - 29440b5) * q^75 + (-4358b12 + 8716b9 + 7390b8) q^77 + (801b11 - 477037b4) * q^79 + (-90b14 + 816b13 - 1557b1 - 1224711) q^81 + (-47120b7 - 2665b6 + 91575b5) q^83 + (-4500b12 - 240b10 - 18240b9) q^85 + (3399b11 + 303168b4 + 243b3 + 273b2) * q^87 + (260b14 - 198b13 + 31b1) q^89 + (259b15 - 43381b7 - 174042b5) q^91 + (357b12 + 51b10 - 17501b9 - 765b8) * q^93 + (155b4 + 310b3 - 235b2) q^95 + (3143b1 + 2713832) q^97 + (309b15 - 34092b7 - 2448b6 - 101988b5) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 15360 q^{9} + 568720 q^{25} + 1791696 q^{33} - 1861680 q^{49} + 663120 q^{57} - 25933856 q^{73} - 19595376 q^{81} + 43421312 q^{97}+O(q^{100}) \) 16 * q + 15360 * q^9 + 568720 * q^25 + 1791696 * q^33 - 1861680 * q^49 + 663120 * q^57 - 25933856 * q^73 - 19595376 * q^81 + 43421312 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	192.8.f.c

Level	$192$
Weight	$8$
Character orbit	192.f
Analytic conductor	$59.978$
Analytic rank	$0$
Dimension	$16$
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(59.9779248930\)
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:	\( x^{16} - 4372 x^{14} + 7613770 x^{12} - 6683530912 x^{10} + 3071754181219 x^{8} - 688859966054368 x^{6} + \cdots + 59\!\cdots\!21 \) x^16 - 4372x^14 + 7613770x^12 - 6683530912x^10 + 3071754181219x^8 - 688859966054368x^6 + 59012195426142490x^4 + 21745092566256572*x^2 + 5916741936370321
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{60}\cdot 3^{12} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 188835 \nu^{14} - 3912579551 \nu^{12} + 11532772844222 \nu^{10} + \cdots + 90\!\cdots\!32 ) / 48\!\cdots\!80 \) (188835v^14 - 3912579551v^12 + 11532772844222v^10 - 13275553286089131v^8 + 6771772468779726774v^6 - 1312133465427264861931v^4 - 483197883212932960677*v^2 + 9042481370797842623321132) / 4882469313836870813280
\(\beta_{2}\)	\(=\)	\( ( - 12\!\cdots\!38 \nu^{14} + \cdots - 41\!\cdots\!42 ) / 23\!\cdots\!09 \) (-1215151519503738v^14 + 4656680089940146080v^12 - 6885469339497411467938v^10 + 4888047620193762659744406v^8 - 1672310241596461170189702018v^6 + 235219862512300539819427650150v^4 + 86636840704731993650692332296*v^2 - 4111131680642569975371164489015642) / 235005217946910205786145718109
\(\beta_{3}\)	\(=\)	\( ( 12\!\cdots\!00 \nu^{15} + \cdots - 20\!\cdots\!09 ) / 55\!\cdots\!00 \) (12032977124771053077929400v^15 - 275135405999448214368927066v^14 - 52572123316272644001980847000v^13 + 1058374614174785574294566655853v^12 + 91478159888030213226983798670000v^11 - 1653500083267057527101252638123005v^10 - 80218487644571764796071862226303000v^9 + 1344546748775771456933168551536022980v^8 + 36817322931766300549215010801377006000v^7 - 578568487526283620766866225612819438137v^6 - 8239419969386129720546140068454284341400v^5 + 104075164155331062844876928777672193374868v^4 + 703113593188890889217212305571158102891000v^3 + 38327525119127842596998605362801197773965v^2 + 482205703958316524422035680209009922880000*v - 201964801694716560644525553853236077894156309) / 55779488849910893750705676658263996391200
\(\beta_{4}\)	\(=\)	\( ( - 22\!\cdots\!61 \nu^{15} + \cdots - 89\!\cdots\!00 \nu ) / 51\!\cdots\!14 \) (-2228329097179824644061v^15 + 9735578391902341481848305v^13 - 16940399979264854301293296050v^11 + 14855275489735511999272567078945v^9 - 6818022765141907509113890889143890v^7 + 1525818512849283281582618531195237841v^5 - 130206220960905720225409686216881130165v^3 - 89297352584873430448525125964631467200v) / 5164767486102860532472747838728147814
\(\beta_{5}\)	\(=\)	\( ( - 38\!\cdots\!74 \nu^{15} + \cdots + 25\!\cdots\!64 ) / 11\!\cdots\!60 \) (-3839424292193902368277274v^15 + 2399065040446738979782819v^14 + 16787157161440772228528249860v^13 - 10518161684360285790917994107v^12 - 29237641174203371910518565261822v^11 + 18357459313060962373245277056366v^10 + 25669761307201356788293837670966768v^9 - 16134273363460971400340629275574843v^8 - 11801456775261507312390469524857938546v^7 + 7412725797597244884070801943147078702v^6 + 2648339243109190007342600683486899044532v^5 - 1657893492771842096835812672790570795195v^4 - 227364444051789876757059319945742234916776v^3 + 141459968878588081339009645112050723155019v^2 - 11632776654302061531316919697054410436548*v + 25980142740907273009780715269793669131264) / 1112117826066299976856278053499732581760
\(\beta_{6}\)	\(=\)	\( ( 57\!\cdots\!10 \nu^{15} + \cdots + 45\!\cdots\!92 ) / 16\!\cdots\!00 \) (57591364382908535524159110v^15 + 388787326593273671879838245v^14 - 251807357421611583427923747900v^13 - 1692121448826206070512489959237v^12 + 438564617613050578657778478927330v^11 + 2940682043049912050848396349409258v^10 - 385046419608020351824407565064501520v^9 - 2588679382743803499816072760530117053v^8 + 177021851628922609685857042872869078190v^7 + 1203878187160207998692952232946181644938v^6 - 39725088646637850110139010252303485667980v^5 - 277129377019270343151668954480422900277757v^4 + 3410466660776848151355889799186133523751640v^3 + 24668280324329404148050169378816065032860933v^2 + 174491649814530922969753795455816156548220*v + 4530894646876437618562747974998241822809192) / 16681767390994499652844170802495988726400
\(\beta_{7}\)	\(=\)	\( ( - 19\!\cdots\!37 \nu^{15} + \cdots - 25\!\cdots\!64 ) / 27\!\cdots\!40 \) (-1919712146096951184138637v^15 - 2399065040446738979782819v^14 + 8393578580720386114264124930v^13 + 10518161684360285790917994107v^12 - 14618820587101685955259282630911v^11 - 18357459313060962373245277056366v^10 + 12834880653600678394146918835483384v^9 + 16134273363460971400340629275574843v^8 - 5900728387630753656195234762428969273v^7 - 7412725797597244884070801943147078702v^6 + 1324169621554595003671300341743449522266v^5 + 1657893492771842096835812672790570795195v^4 - 113682222025894938378529659972871117458388v^3 - 141459968878588081339009645112050723155019v^2 - 5816388327151030765658459848527205218274*v - 25980142740907273009780715269793669131264) / 278029456516574994214069513374933145440
\(\beta_{8}\)	\(=\)	\( ( - 23\!\cdots\!26 \nu^{15} + \cdots - 71\!\cdots\!56 \nu ) / 21\!\cdots\!00 \) (-23666113767585966230913626v^15 + 103479922954232309078220351655v^13 - 180234684058120203893912359846143v^11 + 158247490974980328761494323223676063v^9 - 72758163997380951213574181210517635185v^7 + 16330333945593813529524103957292169312765v^5 - 1402663336600031863858237299254064223526723v^3 - 71765900872531967792720770724097232889756v) / 2104118649387101257815840796780003375200
\(\beta_{9}\)	\(=\)	\( ( - 19\!\cdots\!18 \nu^{15} + \cdots - 13\!\cdots\!08 ) / 70\!\cdots\!40 \) (-1906849316209964537360318v^15 - 1391734456473209033156533v^14 + 8337438365059578102886331620v^13 + 6001635196949380499560945229v^12 - 14521256731179036512681131390434v^11 - 10289397586787807971299227997042v^10 + 12749465378470343708663167085747096v^9 + 8869665406226886471056807384951101v^8 - 5861616487726104565534644189659843662v^7 - 3989502971162099619513548751260083634v^6 + 1315449629354033429909007321881854673844v^5 + 871559914101016910078967591169598799165v^4 - 112944295490495531632380147204816085633472v^3 - 72480257094481693266395086908435421291453v^2 - 5778643451534447523548865144861140234396*v - 13310806442409757662278227289553457949008) / 70137288312903375260528026559333445840
\(\beta_{10}\)	\(=\)	\( ( 38\!\cdots\!36 \nu^{15} + \cdots + 13\!\cdots\!69 ) / 14\!\cdots\!80 \) (3813698632419929074720636v^15 + 112688068962497567248605087v^14 - 16674876730119156205772663240v^13 - 495875255132320318732631592078v^12 + 29042513462358073025362262780868v^11 + 870480502661323882502100724171671v^10 - 25498930756940687417326334171494192v^9 - 771677530448432213093786204617771317v^8 + 11723232975452209131069288379319687324v^7 + 359022685579361456920842397937828391459v^6 - 2630899258708066859818014643763709347688v^5 - 81759877913656242813127877156703480318477v^4 + 225888590980991063264760294409632171266944v^3 + 7134918333815265317693167708974960775895760v^2 + 11557286903068895047097730289722280468792*v + 1310439334549755776336574383862757850349069) / 140274576625806750521056053118666891680
\(\beta_{11}\)	\(=\)	\( ( - 61\!\cdots\!42 \nu^{15} + \cdots - 24\!\cdots\!10 \nu ) / 12\!\cdots\!60 \) (-61103899295886916775478642v^15 + 267258222211327514983391458837v^13 - 465659339203408392978661200659415v^11 + 409008095385507579234744235386764599v^9 - 188106423259306782873146232919220364909v^7 + 42209091814210930054143733359800165725989v^5 - 3613383659439377620546986938231586286152851v^3 - 2478091584018184330830106096580223074283910v) / 1239544196664686527793459481294755475360
\(\beta_{12}\)	\(=\)	\( ( 38\!\cdots\!36 \nu^{15} + \cdots - 13\!\cdots\!08 ) / 35\!\cdots\!20 \) (3813698632419929074720636v^15 - 1391734456473209033156533v^14 - 16674876730119156205772663240v^13 + 6001635196949380499560945229v^12 + 29042513462358073025362262780868v^11 - 10289397586787807971299227997042v^10 - 25498930756940687417326334171494192v^9 + 8869665406226886471056807384951101v^8 + 11723232975452209131069288379319687324v^7 - 3989502971162099619513548751260083634v^6 - 2630899258708066859818014643763709347688v^5 + 871559914101016910078967591169598799165v^4 + 225888590980991063264760294409632171266944v^3 - 72480257094481693266395086908435421291453v^2 + 11557286903068895047097730289722280468792*v - 13310806442409757662278227289553457949008) / 35068644156451687630264013279666722920
\(\beta_{13}\)	\(=\)	\( ( 88\!\cdots\!00 \nu^{15} - 14525256369435 \nu^{14} + \cdots - 69\!\cdots\!52 ) / 75\!\cdots\!60 \) (88147776436545600v^15 - 14525256369435v^14 - 385378592519048698800v^13 + 300957031504137911v^12 + 671064666111716799813360v^11 - 887105050508553004142v^10 - 588925483425595671376593120v^9 + 1021160351240712234696291v^8 + 270523222349818820100429976560v^7 - 520887182908397812937445414v^6 - 60596461359948806007646724281440v^5 + 100929779840846232422347677091v^4 + 5176068417082776898851234095265840v^3 + 37167755611174643204073644397v^2 + 3549815309958042262559598443545680*v - 695550931377544912607048492368652) / 751122604383508796135722388160
\(\beta_{14}\)	\(=\)	\( ( 32\!\cdots\!48 \nu^{15} - 653636536624575 \nu^{14} + \cdots - 31\!\cdots\!40 ) / 33\!\cdots\!00 \) (32716857869949197948v^15 - 653636536624575v^14 - 143039077102939088305092v^13 + 13543066417686205995v^12 + 249135395679659490491456472v^11 - 39919727272884885186390v^10 - 218777958273002149982972814452v^9 + 45952215805832050561333095v^8 + 100625263849996346013009522667128v^7 - 23439923230877901582185043630v^6 - 22592294089931713607565315410705892v^5 + 4541840092838080459005645469095v^4 + 1936024423951248710336520039590033612v^3 + 1672549002502858944183313997865v^2 + 1327739772069446229661622924437737600*v - 31299791911989521067317182156589340) / 33800517197257895826107507467200
\(\beta_{15}\)	\(=\)	\( ( 15\!\cdots\!56 \nu^{15} + \cdots + 25\!\cdots\!64 ) / 55\!\cdots\!80 \) (1534172557933543800406711656v^15 + 2399065040446738979782819v^14 - 6707758515673193645964632816460v^13 - 10518161684360285790917994107v^12 + 11682545800177304039603433014735148v^11 + 18357459313060962373245277056366v^10 - 10256912476749442897766077750084590060v^9 - 16134273363460971400340629275574843v^8 + 4715620748120911566536830298002811504532v^7 + 7412725797597244884070801943147078702v^6 - 1058288607079738346438716485657157134448644v^5 - 1657893492771842096835812672790570795195v^4 + 90872228139240135862828054439493121661532636v^3 + 141459968878588081339009645112050723155019v^2 + 4649363003356243338961333355813737731213744*v + 25980142740907273009780715269793669131264) / 556058913033149988428139026749866290880

\(\nu\)	\(=\)	\( ( \beta_{12} - 2\beta_{9} + 8\beta_{7} + 32\beta_{5} - 6\beta_{4} ) / 96 \) (b12 - 2b9 + 8b7 + 32b5 - 6b4) / 96
\(\nu^{2}\)	\(=\)	\( ( -\beta_{12} - 4\beta_{9} + 8\beta_{7} - 16\beta_{5} + 3\beta_{2} + 52464 ) / 96 \) (-b12 - 4b9 + 8b7 - 16b5 + 3b2 + 52464) / 96
\(\nu^{3}\)	\(=\)	\( ( - 12 \beta_{15} - 24 \beta_{13} + 759 \beta_{12} - 1518 \beta_{9} + 360 \beta_{8} + 4180 \beta_{7} + \cdots - 12 \beta_1 ) / 64 \) (-12b15 - 24b13 + 759b12 - 1518b9 + 360b8 + 4180b7 + 16744b5 - 6560b4 - 12*b1) / 64
\(\nu^{4}\)	\(=\)	\( ( - 1591 \beta_{12} + 72 \beta_{10} - 6292 \beta_{9} + 18764 \beta_{7} + 540 \beta_{6} - 36988 \beta_{5} + \cdots + 46642128 ) / 96 \) (-1591b12 + 72b10 - 6292b9 + 18764b7 + 540b6 - 36988b5 + 3276b2 + 5760b1 + 46642128) / 96
\(\nu^{5}\)	\(=\)	\( ( - 56556 \beta_{15} - 131160 \beta_{13} + 2449933 \beta_{12} - 57600 \beta_{11} - 4899866 \beta_{9} + \cdots - 65580 \beta_1 ) / 192 \) (-56556b15 - 131160b13 + 2449933b12 - 57600b11 - 4899866b9 + 659880b8 + 11385188b7 + 45653864b5 - 29191332b4 - 65580b1) / 192
\(\nu^{6}\)	\(=\)	\( ( - 730574 \beta_{12} + 56616 \beta_{10} - 2865680 \beta_{9} + 9980340 \beta_{7} + 165420 \beta_{6} + \cdots + 15025889056 ) / 32 \) (-730574b12 + 56616b10 - 2865680b9 + 9980340b7 + 165420b6 - 19795260b5 + 43200b4 + 86400b3 + 1173687b2 + 3142080b1 + 15025889056) / 32
\(\nu^{7}\)	\(=\)	\( ( - 36195264 \beta_{15} - 453600 \beta_{14} - 96551280 \beta_{13} + 1308101543 \beta_{12} + \cdots - 48502440 \beta_1 ) / 96 \) (-36195264b15 - 453600b14 - 96551280b13 + 1308101543b12 - 66084480b11 - 2616203086b9 - 45161280b8 + 5493851320b7 + 22047795808b5 - 19780823658b4 - 48502440*b1) / 96
\(\nu^{8}\)	\(=\)	\( ( - 2847645569 \beta_{12} + 290354904 \beta_{10} - 11100227372 \beta_{9} + 41908695364 \beta_{7} + \cdots + 45157691024592 ) / 96 \) (-2847645569b12 + 290354904b10 - 11100227372b9 + 41908695364b7 - 88005420b6 - 83905396148b5 + 282528000b4 + 565056000b3 + 3780106836b2 + 12427868160b1 + 45157691024592) / 96
\(\nu^{9}\)	\(=\)	\( ( - 28912543020 \beta_{15} - 849398400 \beta_{14} - 86871032856 \beta_{13} + 932611011327 \beta_{12} + \cdots - 43860215628 \beta_1 ) / 64 \) (-28912543020b15 - 849398400b14 - 86871032856b13 + 932611011327b12 - 74831627520b11 - 1865222022654b9 - 288711178200b8 + 3641259279156b7 + 14622862202664b5 - 17022619210960b4 - 43860215628*b1) / 64
\(\nu^{10}\)	\(=\)	\( ( - 3570580811393 \beta_{12} + 435866996376 \beta_{10} - 13846456249196 \beta_{9} + \cdots + 46\!\cdots\!64 ) / 96 \) (-3570580811393b12 + 435866996376b10 - 13846456249196b9 + 55188518657572b7 - 1083323482380b6 - 111460360797524b5 + 432895752000b4 + 865791504000b3 + 4065340001670b2 + 15186342888000b1 + 46127626095062064) / 96
\(\nu^{11}\)	\(=\)	\( ( - 100618611465852 \beta_{15} - 4785221707200 \beta_{14} - 336331732761816 \beta_{13} + \cdots - 170558477234508 \beta_1 ) / 192 \) (-100618611465852b15 - 4785221707200b14 - 336331732761816b13 + 3000560130844949b12 - 336158867784960b11 - 6001120261689898b9 - 1599764743202040b8 + 11065360242051700b7 + 44462678191138504b5 - 63935805851287116b4 - 170558477234508*b1) / 192
\(\nu^{12}\)	\(=\)	\( ( - 729582235098202 \beta_{12} + 101418500058168 \beta_{10} + \cdots + 79\!\cdots\!04 ) / 16 \) (-729582235098202b12 + 101418500058168b10 - 2816910440334640b9 + 11703593647199100b7 - 401927519045340b6 - 23809114813443540b5 + 96015680078400b4 + 192031360156800b3 + 730033326300261b2 + 2974417467399360b1 + 7957668256153553104) / 16
\(\nu^{13}\)	\(=\)	\( ( - 57\!\cdots\!84 \beta_{15} + \cdots - 10\!\cdots\!20 \beta_1 ) / 96 \) (-57219534806844384b15 - 3775745848680000b14 - 211236606040393440b13 + 1612948956531591529b12 - 234192273029694720b11 - 3225897913063183058b9 - 1142541744562198080b8 + 5683434007637137448b7 + 22848175100162238560b5 - 39238825424540178534b4 - 107506175944536720*b1) / 96
\(\nu^{14}\)	\(=\)	\( ( - 52\!\cdots\!05 \beta_{12} + \cdots + 49\!\cdots\!56 ) / 96 \) (-5289851547062286205b12 + 810317135007276624b10 - 20349089053241868196b9 + 87393116938072662512b7 - 4035507663337018920b6 - 178821741539482343944b5 + 712894074267542400b4 + 1425788148535084800b3 + 4725938647599262149b2 + 20474521048444538880b1 + 49917781538293241740656) / 96
\(\nu^{15}\)	\(=\)	\( ( - 42\!\cdots\!72 \beta_{15} + \cdots - 88\!\cdots\!40 \beta_1 ) / 64 \) (-42807693154344753372b15 - 3608593416719712000b14 - 173684732514981601080b13 + 1158122844997087974171b12 - 207482561006176627200b11 - 2316245689994175948342b9 - 976898104441145488440b8 + 3935720945004715543556b7 + 15828499166327551680968b5 - 31682964633778539886400b4 - 88646662965850656540*b1) / 64

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 95.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + \cdots + 22876792454961)^{2} \) (T^8 - 3840T^6 + 9822222T^4 - 18366600960*T^2 + 22876792454961)^2
$5$	\( (T^{4} - 227340 T^{2} + 2544768000)^{4} \) (T^4 - 227340*T^2 + 2544768000)^4
$7$	\( (T^{4} + 1879796 T^{2} + 600873025600)^{4} \) (T^4 + 1879796*T^2 + 600873025600)^4
$11$	\( (T^{4} + \cdots + 73141191486720)^{4} \) (T^4 + 41773932*T^2 + 73141191486720)^4
$13$	\( (T^{4} + \cdots + 18\!\cdots\!80)^{4} \) (T^4 + 274488048*T^2 + 18055477928494080)^4
$17$	\( (T^{4} + \cdots + 99\!\cdots\!00)^{4} \) (T^4 + 2130105600*T^2 + 998883426277785600)^4
$19$	\( (T^{4} + \cdots + 25\!\cdots\!00)^{4} \) (T^4 - 1097727660*T^2 + 258151104952992000)^4
$23$	\( (T^{4} + \cdots + 47\!\cdots\!00)^{4} \) (T^4 - 13865022720*T^2 + 47813643183882240000)^4
$29$	\( (T^{4} + \cdots + 10\!\cdots\!80)^{4} \) (T^4 - 32312986668*T^2 + 101054825827448878080)^4
$31$	\( (T^{4} + \cdots + 35\!\cdots\!00)^{4} \) (T^4 + 38932760084*T^2 + 359888531473591758400)^4
$37$	\( (T^{4} + \cdots + 19\!\cdots\!20)^{4} \) (T^4 + 34324016112*T^2 + 193052057293970334720)^4
$41$	\( (T^{4} + \cdots + 44\!\cdots\!00)^{4} \) (T^4 + 730409944320*T^2 + 44417388888068751360000)^4
$43$	\( (T^{4} + \cdots + 67\!\cdots\!00)^{4} \) (T^4 - 219901068108*T^2 + 675047110862496672000)^4
$47$	\( (T^{4} + \cdots + 20\!\cdots\!00)^{4} \) (T^4 - 325636899840*T^2 + 20659359734181553766400)^4
$53$	\( (T^{4} + \cdots + 26\!\cdots\!00)^{4} \) (T^4 - 2144378795148*T^2 + 26849047832100404352000)^4
$59$	\( (T^{4} + \cdots + 98\!\cdots\!80)^{4} \) (T^4 + 5701817409228*T^2 + 988569467864808563969280)^4
$61$	\( (T^{4} + \cdots + 11\!\cdots\!00)^{4} \) (T^4 + 6815524984560*T^2 + 11588996540150527266432000)^4
$67$	\( (T^{4} + \cdots + 61\!\cdots\!80)^{4} \) (T^4 - 17764373594412*T^2 + 61587204030525295523508480)^4
$71$	\( (T^{4} + \cdots + 32\!\cdots\!00)^{4} \) (T^4 - 12433123226880*T^2 + 32249471256112736010240000)^4
$73$	\( (T^{2} + \cdots - 1721755163660)^{8} \) (T^2 + 3241732*T - 1721755163660)^8
$79$	\( (T^{4} + \cdots + 19\!\cdots\!00)^{4} \) (T^4 + 30240841942676*T^2 + 196561466740626506370510400)^4
$83$	\( (T^{4} + \cdots + 22\!\cdots\!00)^{4} \) (T^4 + 99809580948300*T^2 + 2205479299047380086528800000)^4
$89$	\( (T^{4} + \cdots + 17\!\cdots\!00)^{4} \) (T^4 + 112019174622720*T^2 + 1724631692924932125696000000)^4
$97$	\( (T^{2} + \cdots - 26519341845860)^{8} \) (T^2 - 5427664*T - 26519341845860)^8
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\(n\)	\(65\)	\(127\)	\(133\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)