LMFDB - Newform orbit 315.3.e.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [315,3,Mod(244,315)] 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("315.244"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	315.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,-32,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$8.58312832735$
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} - 76 x^{14} + 336 x^{13} + 2593 x^{12} - 8066 x^{11} - 46400 x^{10} + 130882 x^{9} + \cdots + 31397100 $ x^16 - 4x^15 - 76x^14 + 336x^13 + 2593x^12 - 8066x^11 - 46400x^10 + 130882x^9 + 482347x^8 - 1462818x^7 - 443896x^6 - 16494592x^5 - 6620606x^4 + 183639960x^3 + 285498360x^2 + 87465000*x + 31397100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{22}\cdot 5 $
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_{7} q^{2} + (\beta_1 - 2) q^{4} + \beta_{2} q^{5} - \beta_{4} q^{7} + ( - 2 \beta_{7} - \beta_{3}) q^{8} + \beta_{9} q^{10} - \beta_{11} q^{11} + ( - \beta_{10} + \beta_{9} + \cdots - \beta_{4}) q^{13}+ \cdots + ( - 2 \beta_{15} - \beta_{13} + \cdots - 13 \beta_{2}) q^{98}+O(q^{100}) $ q + b7 * q^2 + (b1 - 2) * q^4 + b2 * q^5 - b4 * q^7 + (-2b7 - b3) q^8 + b9 * q^10 - b11 * q^11 + (-b10 + b9 - b5 - b4) * q^13 + (-b14 + b11 - b8) * q^14 + q^16 + (-b13 - b2) * q^17 + (b10 - b6) * q^19 + (-b14 - b13 - 2b2) q^20 + (-b12 + b5 - b4) * q^22 + (b7 - b3) * q^23 + (b12 - b1 + 4) * q^25 + (b15 - 3b14 - 4b2) * q^26 + (b12 + b10 - b9 - 2b5 + b4) q^28 + (-b14 + b11 - 2b8) q^29 + (b10 + b9) * q^31 + (-7b7 - 4b3) * q^32 + (-3b10 - 2b9 + b6) * q^34 + (-b14 + b13 + b8 - b7 - 2b3) q^35 + (-b12 + 2b5 - 2b4) * q^37 + (-3b13 - 3b2) * q^38 + (-b10 - b9 + b6 + 3b5 + 3b4) * q^40 + (-b15 + b14 + 4b2) q^41 + (-b12 - 3b5 + 3b4) * q^43 + (-2b14 + 3b11 - 4b8) q^44 + (-b1 - 9) * q^46 + (b15 + b13 + 7b2) q^47 + (b10 + 2b9 + b6 + 5b1 - 14) * q^49 + (b14 - 5b11 + 2b8 + 8b7 + b3) q^50 + (7b10 - 7b9 + 5b5 + 5b4) * q^52 + (9b7 + 3b3) * q^53 + (-4b10 + b9 - b6 + 2b5 + 2b4) q^55 + (-b15 - 4b11 + b8 + 4b2) * q^56 + (b12 - 6b5 + 6b4) * q^58 + 5b14 q^59 + (-2b10 - b9 + b6) q^61 + (-b15 - 2b13 - 8b2) * q^62 + (-15b1 + 34) q^64 + (2b14 - 5b11 + 4b8 - 19b7 - 3b3) q^65 + (b12 + 3b5 - 3b4) * q^67 + (2b15 + 3b13 + 15b2) q^68 + (6b10 - 2b9 - b6 + 7b5 + 2b4 - 5b1) q^70 + (b14 + 6b11 + 2b8) * q^71 + (b10 - b9 - 10b5 - 10b4) * q^73 + (-3b14 + 9b11 - 6b8) q^74 + (-5b10 - 6b9 - b6) * q^76 + (-b15 + 3b13 + 18b7 + 8b3 - 3b2) * q^77 + (-6b1 + 8) q^79 + b2 * q^80 + (-7b10 + 7b9 - 3b5 - 3b4) * q^82 + (-2b15 + 4b13 - 8b2) q^83 + (5b5 - 5b4 + 15b1 - 25) q^85 + (2b14 - b11 + 4b8) * q^86 + (-b12 - 9b5 + 9b4) * q^88 + (3b15 + 3b14 - 12b2) q^89 + (6b10 + 5b9 - b6 - 5b1 + 35) q^91 + (-b7 - 3b3) q^92 + (8b10 + 7b9 - b6) * q^94 + (3b14 + 6b8 + 19b7 + 13b3) * q^95 + (-7b10 + 7b9 + 8b5 + 8b4) * q^97 + (-2b15 - b13 - 34b7 - 5b3 - 13b2) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} + 16 q^{16} + 64 q^{25} - 144 q^{46} - 224 q^{49} + 544 q^{64} + 128 q^{79} - 400 q^{85} + 560 q^{91}+O(q^{100}) $ 16 * q - 32 * q^4 + 16 * q^16 + 64 * q^25 - 144 * q^46 - 224 * q^49 + 544 * q^64 + 128 * q^79 - 400 * q^85 + 560 * q^91

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4 x^{15} - 76 x^{14} + 336 x^{13} + 2593 x^{12} - 8066 x^{11} - 46400 x^{10} + 130882 x^{9} + \cdots + 31397100 $ x^16 - 4*x^15 - 76*x^14 + 336*x^13 + 2593*x^12 - 8066*x^11 - 46400*x^10 + 130882*x^9 + 482347*x^8 - 1462818*x^7 - 443896*x^6 - 16494592*x^5 - 6620606*x^4 + 183639960*x^3 + 285498360*x^2 + 87465000*x + 31397100 :

$\beta_{1}$	$=$	$ ( 25\!\cdots\!97 \nu^{15} + \cdots - 12\!\cdots\!51 ) / 26\!\cdots\!91 $ (25396070285577431070902429397v^15 - 150496199468231290408369584478v^14 - 1978213811491310409106442441644v^13 + 13853934480332831020862133764523v^12 + 70807228537864893104054063597245v^11 - 485005838324765778778933965109940v^10 - 1607967841918607503727043064541916v^9 + 11140839054482160318040137624837307v^8 + 24479032289640038105635798110484237v^7 - 166652167287651055852221462515452764v^6 - 157223793214034863322034478552282030v^5 + 794569743128795038488015044353844453v^4 + 971037964568308738400268481864992180v^3 - 71920323660577663404964212984001080v^2 - 1414542674073471350237035599359400*v - 122983656772106885197142153283124770951) / 26826029908929487795596271858119198291
$\beta_{2}$	$=$	$ ( 15\!\cdots\!15 \nu^{15} + \cdots + 37\!\cdots\!50 ) / 66\!\cdots\!90 $ (15059146135105574115065539961521404040715v^15 - 60580980085445978460112393076133253887831v^14 - 1113414120822286682915743949556031887308720v^13 + 4878062454182152532070433904979164797943602v^12 + 37092486730650344713937198660542941454202061v^11 - 106572298698195170214849228491422790554147873v^10 - 652469370249840696446245983743180545910568296v^9 + 1591585219218952709797862561891551751453352948v^8 + 6583536999249758310574110708555081124670179451v^7 - 16362792260047857144162309676424637001607901355v^6 - 4129991984460118646889982550073226372668499464v^5 - 311213666812876779339011188694643081271341819158v^4 - 21252006812540611314708321225884095826718506244v^3 + 2134478610496781704358616025488254407085309396850v^2 + 6187191059986053433172207495480959855850146166460*v + 3752526220154045126063687715428592954673137663450) / 664276568884065315025371035827147377185246412390
$\beta_{3}$	$=$	$ ( 25\!\cdots\!37 \nu^{15} + \cdots + 10\!\cdots\!50 ) / 67\!\cdots\!50 $ (25252258548421996477365241762484704056537v^15 - 114094055618032685519561401349985455254048v^14 - 1841611711331383718052814854552920354723552v^13 + 9315405946144199368445345814918665655101392v^12 + 59630560720667195300850710071563346377834131v^11 - 224643279554296550424163928218383085211153532v^10 - 1038342009060339909456857543235547903110585870v^9 + 3575311803314103579277120612489218078382754224v^8 + 10395643563431993888930441996003814341514702239v^7 - 37462444981000094858254011733648559037939208246v^6 + 3576606955445243390507173304784143326169589718v^5 - 463493732294739689550290726416489153462998251634v^4 + 210760079035482759401923820656908910886581620768v^3 + 4167667609754883676088552204277601101291448558350v^2 + 4911298507022301412667498932244073450405051999660*v + 1050003990579817244850832402521677169982951342050) / 675077813906570442098954304702385545919965866250
$\beta_{4}$	$=$	$ ( 22\!\cdots\!18 \nu^{15} + \cdots + 34\!\cdots\!30 ) / 53\!\cdots\!50 $ (2228150417880479556118532736844379818v^15 - 4395009601075870298885969907365979691v^14 - 193525759566042960776084850956916665682v^13 + 448935534364358888918773356676657386932v^12 + 7675721731648984395651784102001788078310v^11 - 9594499967810139616885268622836912821525v^10 - 148940694975000950260553214660871176364036v^9 + 168753143545267817884916189303700775833556v^8 + 1793878207337972436019200456391018914788808v^7 - 2382083628598100895618093054747012068135297v^6 - 7720844317608491041600189362193880460105656v^5 - 26146286667483152600664430176886174342150502v^4 - 112941186980512486037181861045572236340750460v^3 + 525681275184649261668224521394605190386682394v^2 + 1158263109914270348056342042374018387375005220*v + 340343484892067196825767058031695074998884630) / 53068593667339759231638324803324304019170750
$\beta_{5}$	$=$	$ ( - 22\!\cdots\!82 \nu^{15} + \cdots - 79\!\cdots\!70 ) / 53\!\cdots\!50 $ (-2264110330466569069517242498786414282v^15 + 4955838642926570478207418218824848709v^14 + 196058926757815574666862371891776859518v^13 - 499056057481424677074101818862453955818v^12 - 7747889500457030661400149096392750125890v^11 + 11541735267763761975140201113489298466375v^10 + 151560805835045363928347535304978488338764v^9 - 209408420001065618658514773797541488601494v^8 - 1844612928183624257134966235604284964506392v^7 + 2937620955360341700705995839684125165416603v^6 + 8240927238593646772685536721860703058396144v^5 + 22273252824598155626845103898195191293227648v^4 + 104277263667599123209215790663631295669728040v^3 - 531939040585955014149978477920856633368669106v^2 - 1160257767276978715235096214095688036180819780*v - 79910107295027025695125596513806995976733870) / 53068593667339759231638324803324304019170750
$\beta_{6}$	$=$	$ ( - 23\!\cdots\!22 \nu^{15} + \cdots + 24\!\cdots\!30 ) / 53\!\cdots\!50 $ (-2330981617775933733939634818011755422v^15 + 6356228856029793060425574867065660289v^14 + 200933116571903766834724939719697252228v^13 - 611138113786227063984640200170633301378v^12 - 7869148038388591923067380353935612646590v^11 + 15449177335747174598804329178878151703675v^10 + 156529333117608291534495497378026682293494v^9 - 268581854001520078324173589573829315593824v^8 - 1843677245090398206980102243891557290086932v^7 + 3682508916264854188594455844465887544814163v^6 + 6657980911485912970639672918462142746757974v^5 + 20876937229891025934035325579239444751561108v^4 + 123763898448635325006769249409711422633371840v^3 - 625756300160329760888660521128057939393810876v^2 - 1373287295256849493444112581501467082679767380*v + 245970816167757717076128290571889033531456230) / 53068593667339759231638324803324304019170750
$\beta_{7}$	$=$	$ ( 42\!\cdots\!41 \nu^{15} + \cdots + 17\!\cdots\!50 ) / 67\!\cdots\!50 $ (42834557272686275113394585719106538067941v^15 - 197038481244725827562139472486110194471714v^14 - 3099331644691283571379937420349030442520836v^13 + 16087819884395284155725585304618594777966206v^12 + 98847225490442699056237831899694790559644683v^11 - 391799588208228851935826330229942927765357726v^10 - 1676187798869664289523864917430471326784093910v^9 + 6325607444168388388389553597138569759378025082v^8 + 15785054248258530217763028180405406617828020227v^7 - 67263120542459122449731300876793992414768232128v^6 + 30152515487513901327655143972059863675057790774v^5 - 770163187796492125351381524194106007944600085212v^4 + 259828891270220309135529991425301590825183633924v^3 + 6990662681710673437352789743866934688787807221550v^2 + 8379455951493824191628942117723243899528757968380*v + 1784218438889712142859791636184828771043061885650) / 675077813906570442098954304702385545919965866250
$\beta_{8}$	$=$	$ ( - 65\!\cdots\!13 \nu^{15} + \cdots - 12\!\cdots\!50 ) / 86\!\cdots\!50 $ (-6573282701449904785533859191661042409513v^15 + 27855980509494121181801625201654281406177v^14 + 489270228629323424580840370049820451313048v^13 - 2268403493784721980145950919715420857662958v^12 - 16386294478855647173846869656078051217814319v^11 + 52798308796480376491144841721924187196159443v^10 + 295987934356231423897518389879536957900617680v^9 - 818104859788896768601177330650837516285986226v^8 - 3066763687018650560245484660307920751625355561v^7 + 8865882975165777364157139688834454638460282929v^6 + 3335840501754507638034906198742454047834945068v^5 + 118172300119798407285762151531515628763323081166v^4 - 36685513951602202980339766912319225924943103932v^3 - 972041999599603874741762987351050451198194427400v^2 - 2293247905410828981767141674071604868716949892840*v - 1264477697776324303591244573193145406529711112950) / 86046187679283071894478113449112354557674405750
$\beta_{9}$	$=$	$ ( 42\!\cdots\!38 \nu^{15} + \cdots - 62\!\cdots\!70 ) / 53\!\cdots\!50 $ (4204939360768620574658466347884031638v^15 - 9565101706083195540149590815104526231v^14 - 368867094826153313075310087820803144012v^13 + 952432424095286871704762144513968691312v^12 + 14765808276021640666039121616999426482210v^11 - 22559176903072712044283098631828993182625v^10 - 296736433117193945869754290810402414441726v^9 + 389901142198401474651396385622424781995846v^8 + 3625462068676167612515070349887969711780228v^7 - 5575423294565244922675051693218101464718477v^6 - 17974696865241370828680186374623269147312746v^5 - 42865047733756423421968723360367678870714482v^4 - 183474890804886810976151043147095344813764360v^3 + 1120892908320874714227823057068639763124804004v^2 + 2371817882508296504173623225255254035577032020*v - 62605781967836732494773378532022094888826170) / 53068593667339759231638324803324304019170750
$\beta_{10}$	$=$	$ ( 87\!\cdots\!68 \nu^{15} + \cdots + 18\!\cdots\!30 ) / 10\!\cdots\!50 $ (876999643168001088768161750620334168v^15 - 1868349156881169743110272433442552381v^14 - 76723223911883471179156241392188674672v^13 + 187754538652333398628779422026214572582v^12 + 3060913065589840070328444964058908279140v^11 - 4234524640409039469699687618478539305615v^10 - 60617800685243527034450068314865979538266v^9 + 72321341032971286560033002851516674217376v^8 + 735025476520256919160815904958176856713298v^7 - 1043549011579890242155914214343281180196767v^6 - 3564838633208123250275880889745389876046766v^5 - 9886285096080461040313850106002315714829352v^4 - 40888667931161899342646768211395230667335860v^3 + 219867917220528726723574487047051278258794664v^2 + 473026620493674104851652282650255168457165820*v + 184912640963341893271783174976097251376055530) / 10613718733467951846327664960664860803834150
$\beta_{11}$	$=$	$ ( - 19\!\cdots\!47 \nu^{15} + \cdots + 19\!\cdots\!00 ) / 21\!\cdots\!75 $ (-19497032640766695402997784781931061577347v^15 + 99190042019390881408453246957236170676288v^14 + 1387797270206058733251402173059873665549062v^13 - 8138023214752705690829499744184159379369227v^12 - 42690039326242081912507727389042645461684411v^11 + 210194151690266459809614641020776373213348942v^10 + 705884229923039166439298151034437035356489220v^9 - 3502278167219949418478750861575960927652782719v^8 - 6100913469059037060450860039987168403323947559v^7 + 38450269700647475943537351574807245065031483976v^6 - 28350704286091215288494424185847740166848666808v^5 + 312623806186302679524772991266692574300846803929v^4 - 207578852641872479573227877192532362734743120858v^3 - 3465713847216531606174141234772470749094163141350v^2 - 948525299447551994745597300259623562452675522960*v + 1963606579350570553811643832222381978692521222700) / 215115469198207679736195283622780886394186014375
$\beta_{12}$	$=$	$ ( 16\!\cdots\!58 \nu^{15} + \cdots + 15\!\cdots\!50 ) / 10\!\cdots\!15 $ (163070757114040056873912545825157758v^15 - 324158082338102207714077579820716387v^14 - 14187936124978253415701201522557080736v^13 + 33071452560828875316027229606507127890v^12 + 564567813591301955832674001680460714446v^11 - 714322296200576376877092759220897776297v^10 - 11064801872097954211144241537442768347804v^9 + 12699640344809584130667887972658833805274v^8 + 136053765529500363090512081122570483191256v^7 - 183016535569141502113467601221259043454007v^6 - 655248536781789521861033152223480023386974v^5 - 1789913689834683868139571078872109501341502v^4 - 7324443629810562146971713530053429871744730v^3 + 38563930091206502888127748589742871232137620v^2 + 83272558931809640612294598882312130370154000*v + 15129171217497324422960883564081541907683050) / 1061371873346795184632766496066486080383415
$\beta_{13}$	$=$	$ ( - 43\!\cdots\!77 \nu^{15} + \cdots + 28\!\cdots\!50 ) / 22\!\cdots\!30 $ (-43094136801509821579906952794840827202477v^15 + 220073206705695371913509923200824786095075v^14 + 3060027385264963641606812453645588757668272v^13 - 17974700950570053896902028193932786884061986v^12 - 93873644897503224970725791297135132179342443v^11 + 460049919454677820025031367142382681801829053v^10 + 1554983483707968722669314299933221259315272152v^9 - 7532834717594738202043681353076536950899310080v^8 - 13449993711440827353713216326282202921291607481v^7 + 80487260678433173213456106056844231055922764951v^6 - 61994268309859511367766828957497386432509333980v^5 + 762159750189664949815497419274827266578566976510v^4 - 505546473234419206806195797892404140644961493280v^3 - 8103414355647929015021784547391949393461404743150v^2 - 4089469155758563425699173265073197258123000657300*v + 2844613171705447224741087472412745084837018444750) / 221425522961355105008457011942382459061748804130
$\beta_{14}$	$=$	$ ( 17\!\cdots\!33 \nu^{15} + \cdots + 74\!\cdots\!00 ) / 57\!\cdots\!05 $ (175113994284306460268880234810307912233v^15 - 820687383519493011172034866739178536102v^14 - 12691070893039935003004482199884284743228v^13 + 66941364106223065136789995069838156560283v^12 + 405167546750500510356709136533232793226009v^11 - 1643417218432939071279288394087708088985348v^10 - 6959489316048894400456852389176128644695460v^9 + 26308541156840389824905283830993385943201811v^8 + 65909073246985808855533592537990475666683401v^7 - 279972410730008411259185306368215159250395884v^6 + 102069013005272053196896724896533248435260922v^5 - 3186702802301965030796063939774113338457302831v^4 + 1419305622871414077457505492608656621550336932v^3 + 29641439702082910417710281315748657520311769800v^2 + 35143397560812839845234913661105650743170060120*v + 7499948167258151186029224494944755651515135400) / 573641251195220479296520756327415697051162705
$\beta_{15}$	$=$	$ ( - 15\!\cdots\!85 \nu^{15} + \cdots - 24\!\cdots\!50 ) / 33\!\cdots\!95 $ (-150869907146716486574111341074200234428385v^15 + 734130164285866323597304689951720658402428v^14 + 10837583006090308099485303913630578559111700v^13 - 59941947029974124627271787478289784278724621v^12 - 339988178239908718641614106487966357478903253v^11 + 1501186470625982085312512692915420643139648634v^10 + 5738190912675891735326789880351537791015562788v^9 - 24356931375734617712949609745625519120561337549v^8 - 52076314437912975749719435445398618956644689433v^7 + 262298032798585748923958190109949773131563027710v^6 - 150833817686336423469066241899724814642924806518v^5 + 2641878534919677987434762003728987126640443911629v^4 - 1357930396052750027994354364187977160885666491428v^3 - 26218631152441790545697530679907137014773673713300v^2 - 23995374435771339268495100666587512662538394542480*v - 245099558610722797684923451958353313957097592350) / 332138284442032657512685517913573688592623206195

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

$\nu$	$=$	$ ( -\beta_{15} + 5\beta_{11} + 5\beta_{7} - 5\beta_{3} - 6\beta_{2} + 5\beta _1 + 5 ) / 20 $ (-b15 + 5b11 + 5b7 - 5b3 - 6b2 + 5*b1 + 5) / 20
$\nu^{2}$	$=$	$ ( - \beta_{15} + 5 \beta_{14} + 5 \beta_{13} + 10 \beta_{12} - 15 \beta_{10} + 10 \beta_{9} + 10 \beta_{8} + \cdots + 210 ) / 20 $ (-b15 + 5b14 + 5b13 + 10b12 - 15b10 + 10b9 + 10b8 + 5b7 - 5b6 + 25b5 - 5b4 - 15b3 - b2 - 10b1 + 210) / 20
$\nu^{3}$	$=$	$ ( 29 \beta_{15} + 60 \beta_{14} - 25 \beta_{13} + 75 \beta_{12} + 145 \beta_{11} + 30 \beta_{10} + \cdots - 80 ) / 40 $ (29b15 + 60b14 - 25b13 + 75b12 + 145b11 + 30b10 - 45b9 - 90b8 + 400b7 - 45b6 + 345b5 + 45b4 - 410b3 - 601b2 + 270*b1 - 80) / 40
$\nu^{4}$	$=$	$ ( 34 \beta_{15} + 89 \beta_{14} + 40 \beta_{13} + 74 \beta_{12} - 50 \beta_{11} - 48 \beta_{10} + \cdots + 231 ) / 4 $ (34b15 + 89b14 + 40b13 + 74b12 - 50b11 - 48b10 - 18b9 + 28b8 + 74b7 - 70b6 + 30b5 - 182b4 - 222b3 - 116b2 - 237*b1 + 231) / 4
$\nu^{5}$	$=$	$ ( 4138 \beta_{15} + 835 \beta_{14} - 4965 \beta_{13} + 3500 \beta_{12} + 390 \beta_{11} + 4875 \beta_{10} + \cdots - 63065 ) / 40 $ (4138b15 + 835b14 - 4965b13 + 3500b12 + 390b11 + 4875b10 - 6500b9 - 6530b8 + 14625b7 - 4625b6 + 5125b5 - 9625b4 - 13095b3 - 10887b2 - 2415*b1 - 63065) / 40
$\nu^{6}$	$=$	$ ( 16769 \beta_{15} + 11490 \beta_{14} - 10495 \beta_{13} + 9415 \beta_{12} - 22100 \beta_{11} + \cdots - 181110 ) / 20 $ (16769b15 + 11490b14 - 10495b13 + 9415b12 - 22100b11 + 1730b10 - 12745b9 - 9270b8 + 6265b7 - 9475b6 - 36805b5 - 59325b4 - 39245b3 + 18869b2 - 80940*b1 - 181110) / 20
$\nu^{7}$	$=$	$ ( 184559 \beta_{15} - 158725 \beta_{14} - 337950 \beta_{13} + 50925 \beta_{12} - 144095 \beta_{11} + \cdots - 5249255 ) / 40 $ (184559b15 - 158725b14 - 337950b13 + 50925b12 - 144095b11 + 307545b10 - 299005b9 - 366660b8 + 355575b7 - 83230b6 - 474320b5 - 772870b4 - 114915b3 + 304604b2 - 750155*b1 - 5249255) / 40
$\nu^{8}$	$=$	$ ( 104349 \beta_{15} - 90202 \beta_{14} - 179320 \beta_{13} - 21104 \beta_{12} - 243920 \beta_{11} + \cdots - 2543877 ) / 4 $ (104349b15 - 90202b14 - 179320b13 - 21104b12 - 243920b11 + 49324b10 - 72336b9 - 169064b8 - 194412b7 + 37068b6 - 615512b5 - 449160b4 + 61796b3 + 658244b2 - 803439*b1 - 2543877) / 4
$\nu^{9}$	$=$	$ ( 2180933 \beta_{15} - 16586550 \beta_{14} - 13595635 \beta_{13} - 6552075 \beta_{12} - 7505015 \beta_{11} + \cdots - 227425040 ) / 40 $ (2180933b15 - 16586550b14 - 13595635b13 - 6552075b12 - 7505015b11 + 10735050b10 - 4545375b9 - 14594790b8 - 2794410b7 + 6259125b6 - 37979025b5 - 19678725b4 + 24523200b3 + 38299613b2 - 36750690*b1 - 227425040) / 40
$\nu^{10}$	$=$	$ ( - 7258561 \beta_{15} - 50010350 \beta_{14} - 24749470 \beta_{13} - 31196515 \beta_{12} + \cdots - 353974215 ) / 20 $ (-7258561b15 - 50010350b14 - 24749470b13 - 31196515b12 - 35940625b11 - 1818200b10 + 14627175b9 - 23678950b8 - 104732725b7 + 37009725b6 - 138798485b5 - 9233915b4 + 122178175b3 + 182889164b2 - 114831435*b1 - 353974215) / 20
$\nu^{11}$	$=$	$ ( - 322597036 \beta_{15} - 853699735 \beta_{14} - 168932355 \beta_{13} - 588286050 \beta_{12} + \cdots - 2996560205 ) / 40 $ (-322597036b15 - 853699735b14 - 168932355b13 - 588286050b12 - 9414780b11 + 91489585b10 + 361786810b9 - 198340490b8 - 801728025b7 + 600015735b6 - 1198022485b5 + 698599165b4 + 2139514515b3 + 1602780779b2 - 251219055*b1 - 2996560205) / 40
$\nu^{12}$	$=$	$ ( - 349137935 \beta_{15} - 379396953 \beta_{14} + 152529160 \beta_{13} - 375753056 \beta_{12} + \cdots + 2501255046 ) / 4 $ (-349137935b15 - 379396953b14 + 152529160b13 - 375753056b12 + 59846150b11 - 250178244b10 + 432718786b9 + 168022644b8 - 1116447728b7 + 428097770b6 - 468748302b5 + 1013397826b4 + 1460358484b3 + 946370740b2 + 319772088*b1 + 2501255046) / 4
$\nu^{13}$	$=$	$ ( - 28201185657 \beta_{15} - 20032082775 \beta_{14} + 20923038000 \beta_{13} - 23850866975 \beta_{12} + \cdots + 333337402135 ) / 40 $ (-28201185657b15 - 20032082775b14 + 20923038000b13 - 23850866975b12 + 23838172915b11 - 15267044975b10 + 33404488325b9 + 21011338680b8 - 34142320245b7 + 24250096000b6 + 19017432200b5 + 96309290350b4 + 87404411565b3 + 3145811758b2 + 84938206135*b1 + 333337402135) / 40
$\nu^{14}$	$=$	$ ( - 99694763916 \beta_{15} + 7210488460 \beta_{14} + 119425725355 \beta_{13} - 46546206210 \beta_{12} + \cdots + 1911121200765 ) / 20 $ (-99694763916b15 + 7210488460b14 + 119425725355b13 - 46546206210b12 + 98920450475b11 - 98794126230b10 + 117975746670b9 + 125541177020b8 - 155541586040b7 + 54322378250b6 + 180774873300b5 + 372882955130b4 + 180538105070b3 - 111389348891b2 + 383658379335*b1 + 1911121200765) / 20
$\nu^{15}$	$=$	$ ( - 1115120752491 \beta_{15} + 689151779770 \beta_{14} + 1810404708455 \beta_{13} + \cdots + 28983534347170 ) / 40 $ (-1115120752491b15 + 689151779770b14 + 1810404708455b13 - 163161108875b12 + 1812488161855b11 - 1086338878100b10 + 1307346677775b9 + 1860202077090b8 + 289967159400b7 + 122643400025b6 + 4498678725475b5 + 4779032464475b4 + 564289020510b3 - 4237698674741b2 + 6682831973220*b1 + 28983534347170) / 40

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$.

$n$	$127$	$136$	$281$
$\chi(n)$	$-1$	$-1$	$1$

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} $

244.1

−1.65864	−	0.339479i
3.96507	−	0.339479i
−0.132647	−	0.339479i
−5.75636	−	0.339479i
−0.225314	−	2.85215i
6.51070	−	2.85215i
3.01660	−	2.85215i
−3.71941	−	2.85215i
6.51070	+	2.85215i
−0.225314	+	2.85215i
−3.71941	+	2.85215i
3.01660	+	2.85215i
3.96507	+	0.339479i
−1.65864	+	0.339479i
−5.75636	+	0.339479i
−0.132647	+	0.339479i

−

3.25309i

−6.58258

−4.09772

−

2.86509i

−2.45837

−

6.55412i

8.40134i

−9.32037

13.3302i

244.2

−

3.25309i

−6.58258

−4.09772

2.86509i

2.45837

6.55412i

8.40134i

9.32037

13.3302i

244.3

−

3.25309i

−6.58258

4.09772

−

2.86509i

−2.45837

6.55412i

8.40134i

−9.32037

−

13.3302i

244.4

−

3.25309i

−6.58258

4.09772

2.86509i

2.45837

−

6.55412i

8.40134i

9.32037

−

13.3302i

244.5

−

1.19056i

2.58258

−3.49410

−

3.57649i

5.38112

4.47700i

−

7.83693i

−4.25801

4.15992i

244.6

−

1.19056i

2.58258

−3.49410

3.57649i

−5.38112

−

4.47700i

−

7.83693i

4.25801

4.15992i

244.7

−

1.19056i

2.58258

3.49410

−

3.57649i

5.38112

−

4.47700i

−

7.83693i

−4.25801

−

4.15992i

244.8

−

1.19056i

2.58258

3.49410

3.57649i

−5.38112

4.47700i

−

7.83693i

4.25801

−

4.15992i

244.9

1.19056i

2.58258

−3.49410

−

3.57649i

−5.38112

4.47700i

7.83693i

4.25801

−

4.15992i

244.10

1.19056i

2.58258

−3.49410

3.57649i

5.38112

−

4.47700i

7.83693i

−4.25801

−

4.15992i

244.11

1.19056i

2.58258

3.49410

−

3.57649i

−5.38112

−

4.47700i

7.83693i

4.25801

4.15992i

244.12

1.19056i

2.58258

3.49410

3.57649i

5.38112

4.47700i

7.83693i

−4.25801

4.15992i

244.13

3.25309i

−6.58258

−4.09772

−

2.86509i

2.45837

−

6.55412i

−

8.40134i

9.32037

−

13.3302i

244.14

3.25309i

−6.58258

−4.09772

2.86509i

−2.45837

6.55412i

−

8.40134i

−9.32037

−

13.3302i

244.15

3.25309i

−6.58258

4.09772

−

2.86509i

2.45837

6.55412i

−

8.40134i

9.32037

13.3302i

244.16

3.25309i

−6.58258

4.09772

2.86509i

−2.45837

−

6.55412i

−

8.40134i

−9.32037

13.3302i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
7.b	odd	2	1	inner
15.d	odd	2	1	inner
21.c	even	2	1	inner
35.c	odd	2	1	inner
105.g	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	315.3.e.d	✓	16
3.b	odd	2	1	inner	315.3.e.d	✓	16
5.b	even	2	1	inner	315.3.e.d	✓	16
7.b	odd	2	1	inner	315.3.e.d	✓	16
15.d	odd	2	1	inner	315.3.e.d	✓	16
21.c	even	2	1	inner	315.3.e.d	✓	16
35.c	odd	2	1	inner	315.3.e.d	✓	16
105.g	even	2	1	inner	315.3.e.d	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
315.3.e.d	✓	16	1.a	even	1	1	trivial
315.3.e.d	✓	16	3.b	odd	2	1	inner
315.3.e.d	✓	16	5.b	even	2	1	inner
315.3.e.d	✓	16	7.b	odd	2	1	inner
315.3.e.d	✓	16	15.d	odd	2	1	inner
315.3.e.d	✓	16	21.c	even	2	1	inner
315.3.e.d	✓	16	35.c	odd	2	1	inner
315.3.e.d	✓	16	105.g	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(315, [\chi])$:

$ T_{2}^{4} + 12T_{2}^{2} + 15 $

T2^4 + 12*T2^2 + 15

$ T_{13}^{4} - 560T_{13}^{2} + 2800 $

T13^4 - 560*T13^2 + 2800

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} + 12 T^{2} + 15)^{4} $ (T^4 + 12*T^2 + 15)^4
$3$	$ T^{16} $ T^16
$5$	$ (T^{8} - 16 T^{6} + \cdots + 390625)^{2} $ (T^8 - 16T^6 + 1230T^4 - 10000*T^2 + 390625)^2
$7$	$ (T^{8} + 56 T^{6} + \cdots + 5764801)^{2} $ (T^8 + 56T^6 + 3486T^4 + 134456*T^2 + 5764801)^2
$11$	$ (T^{4} - 308 T^{2} + 22960)^{4} $ (T^4 - 308*T^2 + 22960)^4
$13$	$ (T^{4} - 560 T^{2} + 2800)^{4} $ (T^4 - 560*T^2 + 2800)^4
$17$	$ (T^{4} - 680 T^{2} + 82000)^{4} $ (T^4 - 680*T^2 + 82000)^4
$19$	$ (T^{4} + 1440 T^{2} + 442800)^{4} $ (T^4 + 1440*T^2 + 442800)^4
$23$	$ (T^{4} + 132 T^{2} + 240)^{4} $ (T^4 + 132*T^2 + 240)^4
$29$	$ (T^{4} - 1568 T^{2} + 574000)^{4} $ (T^4 - 1568*T^2 + 574000)^4
$31$	$ (T^{4} + 780 T^{2} + 49200)^{4} $ (T^4 + 780*T^2 + 49200)^4
$37$	$ (T^{4} + 2520 T^{2} + 123984)^{4} $ (T^4 + 2520*T^2 + 123984)^4
$41$	$ (T^{4} + 2520 T^{2} + 1050000)^{4} $ (T^4 + 2520*T^2 + 1050000)^4
$43$	$ (T^{4} + 2940 T^{2} + 675024)^{4} $ (T^4 + 2940*T^2 + 675024)^4
$47$	$ (T^{4} - 4580 T^{2} + 2050000)^{4} $ (T^4 - 4580*T^2 + 2050000)^4
$53$	$ (T^{4} + 1620 T^{2} + 486000)^{4} $ (T^4 + 1620*T^2 + 486000)^4
$59$	$ (T^{4} + 10500 T^{2} + 26250000)^{4} $ (T^4 + 10500*T^2 + 26250000)^4
$61$	$ (T^{4} + 2820 T^{2} + 1230000)^{4} $ (T^4 + 2820*T^2 + 1230000)^4
$67$	$ (T^{4} + 2940 T^{2} + 675024)^{4} $ (T^4 + 2940*T^2 + 675024)^4
$71$	$ (T^{4} - 16268 T^{2} + 31432240)^{4} $ (T^4 - 16268*T^2 + 31432240)^4
$73$	$ (T^{4} - 14420 T^{2} + 12569200)^{4} $ (T^4 - 14420*T^2 + 12569200)^4
$79$	$ (T^{2} - 16 T - 692)^{8} $ (T^2 - 16*T - 692)^8
$83$	$ (T^{4} - 14480 T^{2} + 1312000)^{4} $ (T^4 - 14480*T^2 + 1312000)^4
$89$	$ (T^{4} + 22680 T^{2} + 30618000)^{4} $ (T^4 + 22680*T^2 + 30618000)^4
$97$	$ (T^{4} - 29540 T^{2} + 176402800)^{4} $ (T^4 - 29540*T^2 + 176402800)^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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	315.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{7} q^{2} + (\beta_1 - 2) q^{4} + \beta_{2} q^{5} - \beta_{4} q^{7} + ( - 2 \beta_{7} - \beta_{3}) q^{8} + \beta_{9} q^{10} - \beta_{11} q^{11} + ( - \beta_{10} + \beta_{9} + \cdots - \beta_{4}) q^{13}+ \cdots + ( - 2 \beta_{15} - \beta_{13} + \cdots - 13 \beta_{2}) q^{98}+O(q^{100}) \) q + b7 * q^2 + (b1 - 2) * q^4 + b2 * q^5 - b4 * q^7 + (-2b7 - b3) q^8 + b9 * q^10 - b11 * q^11 + (-b10 + b9 - b5 - b4) * q^13 + (-b14 + b11 - b8) * q^14 + q^16 + (-b13 - b2) * q^17 + (b10 - b6) * q^19 + (-b14 - b13 - 2b2) q^20 + (-b12 + b5 - b4) * q^22 + (b7 - b3) * q^23 + (b12 - b1 + 4) * q^25 + (b15 - 3b14 - 4b2) * q^26 + (b12 + b10 - b9 - 2b5 + b4) q^28 + (-b14 + b11 - 2b8) q^29 + (b10 + b9) * q^31 + (-7b7 - 4b3) * q^32 + (-3b10 - 2b9 + b6) * q^34 + (-b14 + b13 + b8 - b7 - 2b3) q^35 + (-b12 + 2b5 - 2b4) * q^37 + (-3b13 - 3b2) * q^38 + (-b10 - b9 + b6 + 3b5 + 3b4) * q^40 + (-b15 + b14 + 4b2) q^41 + (-b12 - 3b5 + 3b4) * q^43 + (-2b14 + 3b11 - 4b8) q^44 + (-b1 - 9) * q^46 + (b15 + b13 + 7b2) q^47 + (b10 + 2b9 + b6 + 5b1 - 14) * q^49 + (b14 - 5b11 + 2b8 + 8b7 + b3) q^50 + (7b10 - 7b9 + 5b5 + 5b4) * q^52 + (9b7 + 3b3) * q^53 + (-4b10 + b9 - b6 + 2b5 + 2b4) q^55 + (-b15 - 4b11 + b8 + 4b2) * q^56 + (b12 - 6b5 + 6b4) * q^58 + 5b14 q^59 + (-2b10 - b9 + b6) q^61 + (-b15 - 2b13 - 8b2) * q^62 + (-15b1 + 34) q^64 + (2b14 - 5b11 + 4b8 - 19b7 - 3b3) q^65 + (b12 + 3b5 - 3b4) * q^67 + (2b15 + 3b13 + 15b2) q^68 + (6b10 - 2b9 - b6 + 7b5 + 2b4 - 5b1) q^70 + (b14 + 6b11 + 2b8) * q^71 + (b10 - b9 - 10b5 - 10b4) * q^73 + (-3b14 + 9b11 - 6b8) q^74 + (-5b10 - 6b9 - b6) * q^76 + (-b15 + 3b13 + 18b7 + 8b3 - 3b2) * q^77 + (-6b1 + 8) q^79 + b2 * q^80 + (-7b10 + 7b9 - 3b5 - 3b4) * q^82 + (-2b15 + 4b13 - 8b2) q^83 + (5b5 - 5b4 + 15b1 - 25) q^85 + (2b14 - b11 + 4b8) * q^86 + (-b12 - 9b5 + 9b4) * q^88 + (3b15 + 3b14 - 12b2) q^89 + (6b10 + 5b9 - b6 - 5b1 + 35) q^91 + (-b7 - 3b3) q^92 + (8b10 + 7b9 - b6) * q^94 + (3b14 + 6b8 + 19b7 + 13b3) * q^95 + (-7b10 + 7b9 + 8b5 + 8b4) * q^97 + (-2b15 - b13 - 34b7 - 5b3 - 13b2) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} + 16 q^{16} + 64 q^{25} - 144 q^{46} - 224 q^{49} + 544 q^{64} + 128 q^{79} - 400 q^{85} + 560 q^{91}+O(q^{100}) \) 16 * q - 32 * q^4 + 16 * q^16 + 64 * q^25 - 144 * q^46 - 224 * q^49 + 544 * q^64 + 128 * q^79 - 400 * q^85 + 560 * q^91

Introduction

L-functions

Modular forms

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Fields

Representations

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Database

Properties

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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	315.3.e.d

Level	$315$
Weight	$3$
Character orbit	315.e
Analytic conductor	$8.583$
Analytic rank	$0$
Dimension	$16$
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(8.58312832735\)
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:	\( x^{16} - 4 x^{15} - 76 x^{14} + 336 x^{13} + 2593 x^{12} - 8066 x^{11} - 46400 x^{10} + 130882 x^{9} + \cdots + 31397100 \) x^16 - 4x^15 - 76x^14 + 336x^13 + 2593x^12 - 8066x^11 - 46400x^10 + 130882x^9 + 482347x^8 - 1462818x^7 - 443896x^6 - 16494592x^5 - 6620606x^4 + 183639960x^3 + 285498360x^2 + 87465000*x + 31397100
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{22}\cdot 5 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 25\!\cdots\!97 \nu^{15} + \cdots - 12\!\cdots\!51 ) / 26\!\cdots\!91 \) (25396070285577431070902429397v^15 - 150496199468231290408369584478v^14 - 1978213811491310409106442441644v^13 + 13853934480332831020862133764523v^12 + 70807228537864893104054063597245v^11 - 485005838324765778778933965109940v^10 - 1607967841918607503727043064541916v^9 + 11140839054482160318040137624837307v^8 + 24479032289640038105635798110484237v^7 - 166652167287651055852221462515452764v^6 - 157223793214034863322034478552282030v^5 + 794569743128795038488015044353844453v^4 + 971037964568308738400268481864992180v^3 - 71920323660577663404964212984001080v^2 - 1414542674073471350237035599359400*v - 122983656772106885197142153283124770951) / 26826029908929487795596271858119198291
\(\beta_{2}\)	\(=\)	\( ( 15\!\cdots\!15 \nu^{15} + \cdots + 37\!\cdots\!50 ) / 66\!\cdots\!90 \) (15059146135105574115065539961521404040715v^15 - 60580980085445978460112393076133253887831v^14 - 1113414120822286682915743949556031887308720v^13 + 4878062454182152532070433904979164797943602v^12 + 37092486730650344713937198660542941454202061v^11 - 106572298698195170214849228491422790554147873v^10 - 652469370249840696446245983743180545910568296v^9 + 1591585219218952709797862561891551751453352948v^8 + 6583536999249758310574110708555081124670179451v^7 - 16362792260047857144162309676424637001607901355v^6 - 4129991984460118646889982550073226372668499464v^5 - 311213666812876779339011188694643081271341819158v^4 - 21252006812540611314708321225884095826718506244v^3 + 2134478610496781704358616025488254407085309396850v^2 + 6187191059986053433172207495480959855850146166460*v + 3752526220154045126063687715428592954673137663450) / 664276568884065315025371035827147377185246412390
\(\beta_{3}\)	\(=\)	\( ( 25\!\cdots\!37 \nu^{15} + \cdots + 10\!\cdots\!50 ) / 67\!\cdots\!50 \) (25252258548421996477365241762484704056537v^15 - 114094055618032685519561401349985455254048v^14 - 1841611711331383718052814854552920354723552v^13 + 9315405946144199368445345814918665655101392v^12 + 59630560720667195300850710071563346377834131v^11 - 224643279554296550424163928218383085211153532v^10 - 1038342009060339909456857543235547903110585870v^9 + 3575311803314103579277120612489218078382754224v^8 + 10395643563431993888930441996003814341514702239v^7 - 37462444981000094858254011733648559037939208246v^6 + 3576606955445243390507173304784143326169589718v^5 - 463493732294739689550290726416489153462998251634v^4 + 210760079035482759401923820656908910886581620768v^3 + 4167667609754883676088552204277601101291448558350v^2 + 4911298507022301412667498932244073450405051999660*v + 1050003990579817244850832402521677169982951342050) / 675077813906570442098954304702385545919965866250
\(\beta_{4}\)	\(=\)	\( ( 22\!\cdots\!18 \nu^{15} + \cdots + 34\!\cdots\!30 ) / 53\!\cdots\!50 \) (2228150417880479556118532736844379818v^15 - 4395009601075870298885969907365979691v^14 - 193525759566042960776084850956916665682v^13 + 448935534364358888918773356676657386932v^12 + 7675721731648984395651784102001788078310v^11 - 9594499967810139616885268622836912821525v^10 - 148940694975000950260553214660871176364036v^9 + 168753143545267817884916189303700775833556v^8 + 1793878207337972436019200456391018914788808v^7 - 2382083628598100895618093054747012068135297v^6 - 7720844317608491041600189362193880460105656v^5 - 26146286667483152600664430176886174342150502v^4 - 112941186980512486037181861045572236340750460v^3 + 525681275184649261668224521394605190386682394v^2 + 1158263109914270348056342042374018387375005220*v + 340343484892067196825767058031695074998884630) / 53068593667339759231638324803324304019170750
\(\beta_{5}\)	\(=\)	\( ( - 22\!\cdots\!82 \nu^{15} + \cdots - 79\!\cdots\!70 ) / 53\!\cdots\!50 \) (-2264110330466569069517242498786414282v^15 + 4955838642926570478207418218824848709v^14 + 196058926757815574666862371891776859518v^13 - 499056057481424677074101818862453955818v^12 - 7747889500457030661400149096392750125890v^11 + 11541735267763761975140201113489298466375v^10 + 151560805835045363928347535304978488338764v^9 - 209408420001065618658514773797541488601494v^8 - 1844612928183624257134966235604284964506392v^7 + 2937620955360341700705995839684125165416603v^6 + 8240927238593646772685536721860703058396144v^5 + 22273252824598155626845103898195191293227648v^4 + 104277263667599123209215790663631295669728040v^3 - 531939040585955014149978477920856633368669106v^2 - 1160257767276978715235096214095688036180819780*v - 79910107295027025695125596513806995976733870) / 53068593667339759231638324803324304019170750
\(\beta_{6}\)	\(=\)	\( ( - 23\!\cdots\!22 \nu^{15} + \cdots + 24\!\cdots\!30 ) / 53\!\cdots\!50 \) (-2330981617775933733939634818011755422v^15 + 6356228856029793060425574867065660289v^14 + 200933116571903766834724939719697252228v^13 - 611138113786227063984640200170633301378v^12 - 7869148038388591923067380353935612646590v^11 + 15449177335747174598804329178878151703675v^10 + 156529333117608291534495497378026682293494v^9 - 268581854001520078324173589573829315593824v^8 - 1843677245090398206980102243891557290086932v^7 + 3682508916264854188594455844465887544814163v^6 + 6657980911485912970639672918462142746757974v^5 + 20876937229891025934035325579239444751561108v^4 + 123763898448635325006769249409711422633371840v^3 - 625756300160329760888660521128057939393810876v^2 - 1373287295256849493444112581501467082679767380*v + 245970816167757717076128290571889033531456230) / 53068593667339759231638324803324304019170750
\(\beta_{7}\)	\(=\)	\( ( 42\!\cdots\!41 \nu^{15} + \cdots + 17\!\cdots\!50 ) / 67\!\cdots\!50 \) (42834557272686275113394585719106538067941v^15 - 197038481244725827562139472486110194471714v^14 - 3099331644691283571379937420349030442520836v^13 + 16087819884395284155725585304618594777966206v^12 + 98847225490442699056237831899694790559644683v^11 - 391799588208228851935826330229942927765357726v^10 - 1676187798869664289523864917430471326784093910v^9 + 6325607444168388388389553597138569759378025082v^8 + 15785054248258530217763028180405406617828020227v^7 - 67263120542459122449731300876793992414768232128v^6 + 30152515487513901327655143972059863675057790774v^5 - 770163187796492125351381524194106007944600085212v^4 + 259828891270220309135529991425301590825183633924v^3 + 6990662681710673437352789743866934688787807221550v^2 + 8379455951493824191628942117723243899528757968380*v + 1784218438889712142859791636184828771043061885650) / 675077813906570442098954304702385545919965866250
\(\beta_{8}\)	\(=\)	\( ( - 65\!\cdots\!13 \nu^{15} + \cdots - 12\!\cdots\!50 ) / 86\!\cdots\!50 \) (-6573282701449904785533859191661042409513v^15 + 27855980509494121181801625201654281406177v^14 + 489270228629323424580840370049820451313048v^13 - 2268403493784721980145950919715420857662958v^12 - 16386294478855647173846869656078051217814319v^11 + 52798308796480376491144841721924187196159443v^10 + 295987934356231423897518389879536957900617680v^9 - 818104859788896768601177330650837516285986226v^8 - 3066763687018650560245484660307920751625355561v^7 + 8865882975165777364157139688834454638460282929v^6 + 3335840501754507638034906198742454047834945068v^5 + 118172300119798407285762151531515628763323081166v^4 - 36685513951602202980339766912319225924943103932v^3 - 972041999599603874741762987351050451198194427400v^2 - 2293247905410828981767141674071604868716949892840*v - 1264477697776324303591244573193145406529711112950) / 86046187679283071894478113449112354557674405750
\(\beta_{9}\)	\(=\)	\( ( 42\!\cdots\!38 \nu^{15} + \cdots - 62\!\cdots\!70 ) / 53\!\cdots\!50 \) (4204939360768620574658466347884031638v^15 - 9565101706083195540149590815104526231v^14 - 368867094826153313075310087820803144012v^13 + 952432424095286871704762144513968691312v^12 + 14765808276021640666039121616999426482210v^11 - 22559176903072712044283098631828993182625v^10 - 296736433117193945869754290810402414441726v^9 + 389901142198401474651396385622424781995846v^8 + 3625462068676167612515070349887969711780228v^7 - 5575423294565244922675051693218101464718477v^6 - 17974696865241370828680186374623269147312746v^5 - 42865047733756423421968723360367678870714482v^4 - 183474890804886810976151043147095344813764360v^3 + 1120892908320874714227823057068639763124804004v^2 + 2371817882508296504173623225255254035577032020*v - 62605781967836732494773378532022094888826170) / 53068593667339759231638324803324304019170750
\(\beta_{10}\)	\(=\)	\( ( 87\!\cdots\!68 \nu^{15} + \cdots + 18\!\cdots\!30 ) / 10\!\cdots\!50 \) (876999643168001088768161750620334168v^15 - 1868349156881169743110272433442552381v^14 - 76723223911883471179156241392188674672v^13 + 187754538652333398628779422026214572582v^12 + 3060913065589840070328444964058908279140v^11 - 4234524640409039469699687618478539305615v^10 - 60617800685243527034450068314865979538266v^9 + 72321341032971286560033002851516674217376v^8 + 735025476520256919160815904958176856713298v^7 - 1043549011579890242155914214343281180196767v^6 - 3564838633208123250275880889745389876046766v^5 - 9886285096080461040313850106002315714829352v^4 - 40888667931161899342646768211395230667335860v^3 + 219867917220528726723574487047051278258794664v^2 + 473026620493674104851652282650255168457165820*v + 184912640963341893271783174976097251376055530) / 10613718733467951846327664960664860803834150
\(\beta_{11}\)	\(=\)	\( ( - 19\!\cdots\!47 \nu^{15} + \cdots + 19\!\cdots\!00 ) / 21\!\cdots\!75 \) (-19497032640766695402997784781931061577347v^15 + 99190042019390881408453246957236170676288v^14 + 1387797270206058733251402173059873665549062v^13 - 8138023214752705690829499744184159379369227v^12 - 42690039326242081912507727389042645461684411v^11 + 210194151690266459809614641020776373213348942v^10 + 705884229923039166439298151034437035356489220v^9 - 3502278167219949418478750861575960927652782719v^8 - 6100913469059037060450860039987168403323947559v^7 + 38450269700647475943537351574807245065031483976v^6 - 28350704286091215288494424185847740166848666808v^5 + 312623806186302679524772991266692574300846803929v^4 - 207578852641872479573227877192532362734743120858v^3 - 3465713847216531606174141234772470749094163141350v^2 - 948525299447551994745597300259623562452675522960*v + 1963606579350570553811643832222381978692521222700) / 215115469198207679736195283622780886394186014375
\(\beta_{12}\)	\(=\)	\( ( 16\!\cdots\!58 \nu^{15} + \cdots + 15\!\cdots\!50 ) / 10\!\cdots\!15 \) (163070757114040056873912545825157758v^15 - 324158082338102207714077579820716387v^14 - 14187936124978253415701201522557080736v^13 + 33071452560828875316027229606507127890v^12 + 564567813591301955832674001680460714446v^11 - 714322296200576376877092759220897776297v^10 - 11064801872097954211144241537442768347804v^9 + 12699640344809584130667887972658833805274v^8 + 136053765529500363090512081122570483191256v^7 - 183016535569141502113467601221259043454007v^6 - 655248536781789521861033152223480023386974v^5 - 1789913689834683868139571078872109501341502v^4 - 7324443629810562146971713530053429871744730v^3 + 38563930091206502888127748589742871232137620v^2 + 83272558931809640612294598882312130370154000*v + 15129171217497324422960883564081541907683050) / 1061371873346795184632766496066486080383415
\(\beta_{13}\)	\(=\)	\( ( - 43\!\cdots\!77 \nu^{15} + \cdots + 28\!\cdots\!50 ) / 22\!\cdots\!30 \) (-43094136801509821579906952794840827202477v^15 + 220073206705695371913509923200824786095075v^14 + 3060027385264963641606812453645588757668272v^13 - 17974700950570053896902028193932786884061986v^12 - 93873644897503224970725791297135132179342443v^11 + 460049919454677820025031367142382681801829053v^10 + 1554983483707968722669314299933221259315272152v^9 - 7532834717594738202043681353076536950899310080v^8 - 13449993711440827353713216326282202921291607481v^7 + 80487260678433173213456106056844231055922764951v^6 - 61994268309859511367766828957497386432509333980v^5 + 762159750189664949815497419274827266578566976510v^4 - 505546473234419206806195797892404140644961493280v^3 - 8103414355647929015021784547391949393461404743150v^2 - 4089469155758563425699173265073197258123000657300*v + 2844613171705447224741087472412745084837018444750) / 221425522961355105008457011942382459061748804130
\(\beta_{14}\)	\(=\)	\( ( 17\!\cdots\!33 \nu^{15} + \cdots + 74\!\cdots\!00 ) / 57\!\cdots\!05 \) (175113994284306460268880234810307912233v^15 - 820687383519493011172034866739178536102v^14 - 12691070893039935003004482199884284743228v^13 + 66941364106223065136789995069838156560283v^12 + 405167546750500510356709136533232793226009v^11 - 1643417218432939071279288394087708088985348v^10 - 6959489316048894400456852389176128644695460v^9 + 26308541156840389824905283830993385943201811v^8 + 65909073246985808855533592537990475666683401v^7 - 279972410730008411259185306368215159250395884v^6 + 102069013005272053196896724896533248435260922v^5 - 3186702802301965030796063939774113338457302831v^4 + 1419305622871414077457505492608656621550336932v^3 + 29641439702082910417710281315748657520311769800v^2 + 35143397560812839845234913661105650743170060120*v + 7499948167258151186029224494944755651515135400) / 573641251195220479296520756327415697051162705
\(\beta_{15}\)	\(=\)	\( ( - 15\!\cdots\!85 \nu^{15} + \cdots - 24\!\cdots\!50 ) / 33\!\cdots\!95 \) (-150869907146716486574111341074200234428385v^15 + 734130164285866323597304689951720658402428v^14 + 10837583006090308099485303913630578559111700v^13 - 59941947029974124627271787478289784278724621v^12 - 339988178239908718641614106487966357478903253v^11 + 1501186470625982085312512692915420643139648634v^10 + 5738190912675891735326789880351537791015562788v^9 - 24356931375734617712949609745625519120561337549v^8 - 52076314437912975749719435445398618956644689433v^7 + 262298032798585748923958190109949773131563027710v^6 - 150833817686336423469066241899724814642924806518v^5 + 2641878534919677987434762003728987126640443911629v^4 - 1357930396052750027994354364187977160885666491428v^3 - 26218631152441790545697530679907137014773673713300v^2 - 23995374435771339268495100666587512662538394542480*v - 245099558610722797684923451958353313957097592350) / 332138284442032657512685517913573688592623206195

\(\nu\)	\(=\)	\( ( -\beta_{15} + 5\beta_{11} + 5\beta_{7} - 5\beta_{3} - 6\beta_{2} + 5\beta _1 + 5 ) / 20 \) (-b15 + 5b11 + 5b7 - 5b3 - 6b2 + 5*b1 + 5) / 20
\(\nu^{2}\)	\(=\)	\( ( - \beta_{15} + 5 \beta_{14} + 5 \beta_{13} + 10 \beta_{12} - 15 \beta_{10} + 10 \beta_{9} + 10 \beta_{8} + \cdots + 210 ) / 20 \) (-b15 + 5b14 + 5b13 + 10b12 - 15b10 + 10b9 + 10b8 + 5b7 - 5b6 + 25b5 - 5b4 - 15b3 - b2 - 10b1 + 210) / 20
\(\nu^{3}\)	\(=\)	\( ( 29 \beta_{15} + 60 \beta_{14} - 25 \beta_{13} + 75 \beta_{12} + 145 \beta_{11} + 30 \beta_{10} + \cdots - 80 ) / 40 \) (29b15 + 60b14 - 25b13 + 75b12 + 145b11 + 30b10 - 45b9 - 90b8 + 400b7 - 45b6 + 345b5 + 45b4 - 410b3 - 601b2 + 270*b1 - 80) / 40
\(\nu^{4}\)	\(=\)	\( ( 34 \beta_{15} + 89 \beta_{14} + 40 \beta_{13} + 74 \beta_{12} - 50 \beta_{11} - 48 \beta_{10} + \cdots + 231 ) / 4 \) (34b15 + 89b14 + 40b13 + 74b12 - 50b11 - 48b10 - 18b9 + 28b8 + 74b7 - 70b6 + 30b5 - 182b4 - 222b3 - 116b2 - 237*b1 + 231) / 4
\(\nu^{5}\)	\(=\)	\( ( 4138 \beta_{15} + 835 \beta_{14} - 4965 \beta_{13} + 3500 \beta_{12} + 390 \beta_{11} + 4875 \beta_{10} + \cdots - 63065 ) / 40 \) (4138b15 + 835b14 - 4965b13 + 3500b12 + 390b11 + 4875b10 - 6500b9 - 6530b8 + 14625b7 - 4625b6 + 5125b5 - 9625b4 - 13095b3 - 10887b2 - 2415*b1 - 63065) / 40
\(\nu^{6}\)	\(=\)	\( ( 16769 \beta_{15} + 11490 \beta_{14} - 10495 \beta_{13} + 9415 \beta_{12} - 22100 \beta_{11} + \cdots - 181110 ) / 20 \) (16769b15 + 11490b14 - 10495b13 + 9415b12 - 22100b11 + 1730b10 - 12745b9 - 9270b8 + 6265b7 - 9475b6 - 36805b5 - 59325b4 - 39245b3 + 18869b2 - 80940*b1 - 181110) / 20
\(\nu^{7}\)	\(=\)	\( ( 184559 \beta_{15} - 158725 \beta_{14} - 337950 \beta_{13} + 50925 \beta_{12} - 144095 \beta_{11} + \cdots - 5249255 ) / 40 \) (184559b15 - 158725b14 - 337950b13 + 50925b12 - 144095b11 + 307545b10 - 299005b9 - 366660b8 + 355575b7 - 83230b6 - 474320b5 - 772870b4 - 114915b3 + 304604b2 - 750155*b1 - 5249255) / 40
\(\nu^{8}\)	\(=\)	\( ( 104349 \beta_{15} - 90202 \beta_{14} - 179320 \beta_{13} - 21104 \beta_{12} - 243920 \beta_{11} + \cdots - 2543877 ) / 4 \) (104349b15 - 90202b14 - 179320b13 - 21104b12 - 243920b11 + 49324b10 - 72336b9 - 169064b8 - 194412b7 + 37068b6 - 615512b5 - 449160b4 + 61796b3 + 658244b2 - 803439*b1 - 2543877) / 4
\(\nu^{9}\)	\(=\)	\( ( 2180933 \beta_{15} - 16586550 \beta_{14} - 13595635 \beta_{13} - 6552075 \beta_{12} - 7505015 \beta_{11} + \cdots - 227425040 ) / 40 \) (2180933b15 - 16586550b14 - 13595635b13 - 6552075b12 - 7505015b11 + 10735050b10 - 4545375b9 - 14594790b8 - 2794410b7 + 6259125b6 - 37979025b5 - 19678725b4 + 24523200b3 + 38299613b2 - 36750690*b1 - 227425040) / 40
\(\nu^{10}\)	\(=\)	\( ( - 7258561 \beta_{15} - 50010350 \beta_{14} - 24749470 \beta_{13} - 31196515 \beta_{12} + \cdots - 353974215 ) / 20 \) (-7258561b15 - 50010350b14 - 24749470b13 - 31196515b12 - 35940625b11 - 1818200b10 + 14627175b9 - 23678950b8 - 104732725b7 + 37009725b6 - 138798485b5 - 9233915b4 + 122178175b3 + 182889164b2 - 114831435*b1 - 353974215) / 20
\(\nu^{11}\)	\(=\)	\( ( - 322597036 \beta_{15} - 853699735 \beta_{14} - 168932355 \beta_{13} - 588286050 \beta_{12} + \cdots - 2996560205 ) / 40 \) (-322597036b15 - 853699735b14 - 168932355b13 - 588286050b12 - 9414780b11 + 91489585b10 + 361786810b9 - 198340490b8 - 801728025b7 + 600015735b6 - 1198022485b5 + 698599165b4 + 2139514515b3 + 1602780779b2 - 251219055*b1 - 2996560205) / 40
\(\nu^{12}\)	\(=\)	\( ( - 349137935 \beta_{15} - 379396953 \beta_{14} + 152529160 \beta_{13} - 375753056 \beta_{12} + \cdots + 2501255046 ) / 4 \) (-349137935b15 - 379396953b14 + 152529160b13 - 375753056b12 + 59846150b11 - 250178244b10 + 432718786b9 + 168022644b8 - 1116447728b7 + 428097770b6 - 468748302b5 + 1013397826b4 + 1460358484b3 + 946370740b2 + 319772088*b1 + 2501255046) / 4
\(\nu^{13}\)	\(=\)	\( ( - 28201185657 \beta_{15} - 20032082775 \beta_{14} + 20923038000 \beta_{13} - 23850866975 \beta_{12} + \cdots + 333337402135 ) / 40 \) (-28201185657b15 - 20032082775b14 + 20923038000b13 - 23850866975b12 + 23838172915b11 - 15267044975b10 + 33404488325b9 + 21011338680b8 - 34142320245b7 + 24250096000b6 + 19017432200b5 + 96309290350b4 + 87404411565b3 + 3145811758b2 + 84938206135*b1 + 333337402135) / 40
\(\nu^{14}\)	\(=\)	\( ( - 99694763916 \beta_{15} + 7210488460 \beta_{14} + 119425725355 \beta_{13} - 46546206210 \beta_{12} + \cdots + 1911121200765 ) / 20 \) (-99694763916b15 + 7210488460b14 + 119425725355b13 - 46546206210b12 + 98920450475b11 - 98794126230b10 + 117975746670b9 + 125541177020b8 - 155541586040b7 + 54322378250b6 + 180774873300b5 + 372882955130b4 + 180538105070b3 - 111389348891b2 + 383658379335*b1 + 1911121200765) / 20
\(\nu^{15}\)	\(=\)	\( ( - 1115120752491 \beta_{15} + 689151779770 \beta_{14} + 1810404708455 \beta_{13} + \cdots + 28983534347170 ) / 40 \) (-1115120752491b15 + 689151779770b14 + 1810404708455b13 - 163161108875b12 + 1812488161855b11 - 1086338878100b10 + 1307346677775b9 + 1860202077090b8 + 289967159400b7 + 122643400025b6 + 4498678725475b5 + 4779032464475b4 + 564289020510b3 - 4237698674741b2 + 6682831973220*b1 + 28983534347170) / 40

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 244.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} + 12 T^{2} + 15)^{4} \) (T^4 + 12*T^2 + 15)^4
$3$	\( T^{16} \) T^16
$5$	\( (T^{8} - 16 T^{6} + \cdots + 390625)^{2} \) (T^8 - 16T^6 + 1230T^4 - 10000*T^2 + 390625)^2
$7$	\( (T^{8} + 56 T^{6} + \cdots + 5764801)^{2} \) (T^8 + 56T^6 + 3486T^4 + 134456*T^2 + 5764801)^2
$11$	\( (T^{4} - 308 T^{2} + 22960)^{4} \) (T^4 - 308*T^2 + 22960)^4
$13$	\( (T^{4} - 560 T^{2} + 2800)^{4} \) (T^4 - 560*T^2 + 2800)^4
$17$	\( (T^{4} - 680 T^{2} + 82000)^{4} \) (T^4 - 680*T^2 + 82000)^4
$19$	\( (T^{4} + 1440 T^{2} + 442800)^{4} \) (T^4 + 1440*T^2 + 442800)^4
$23$	\( (T^{4} + 132 T^{2} + 240)^{4} \) (T^4 + 132*T^2 + 240)^4
$29$	\( (T^{4} - 1568 T^{2} + 574000)^{4} \) (T^4 - 1568*T^2 + 574000)^4
$31$	\( (T^{4} + 780 T^{2} + 49200)^{4} \) (T^4 + 780*T^2 + 49200)^4
$37$	\( (T^{4} + 2520 T^{2} + 123984)^{4} \) (T^4 + 2520*T^2 + 123984)^4
$41$	\( (T^{4} + 2520 T^{2} + 1050000)^{4} \) (T^4 + 2520*T^2 + 1050000)^4
$43$	\( (T^{4} + 2940 T^{2} + 675024)^{4} \) (T^4 + 2940*T^2 + 675024)^4
$47$	\( (T^{4} - 4580 T^{2} + 2050000)^{4} \) (T^4 - 4580*T^2 + 2050000)^4
$53$	\( (T^{4} + 1620 T^{2} + 486000)^{4} \) (T^4 + 1620*T^2 + 486000)^4
$59$	\( (T^{4} + 10500 T^{2} + 26250000)^{4} \) (T^4 + 10500*T^2 + 26250000)^4
$61$	\( (T^{4} + 2820 T^{2} + 1230000)^{4} \) (T^4 + 2820*T^2 + 1230000)^4
$67$	\( (T^{4} + 2940 T^{2} + 675024)^{4} \) (T^4 + 2940*T^2 + 675024)^4
$71$	\( (T^{4} - 16268 T^{2} + 31432240)^{4} \) (T^4 - 16268*T^2 + 31432240)^4
$73$	\( (T^{4} - 14420 T^{2} + 12569200)^{4} \) (T^4 - 14420*T^2 + 12569200)^4
$79$	\( (T^{2} - 16 T - 692)^{8} \) (T^2 - 16*T - 692)^8
$83$	\( (T^{4} - 14480 T^{2} + 1312000)^{4} \) (T^4 - 14480*T^2 + 1312000)^4
$89$	\( (T^{4} + 22680 T^{2} + 30618000)^{4} \) (T^4 + 22680*T^2 + 30618000)^4
$97$	\( (T^{4} - 29540 T^{2} + 176402800)^{4} \) (T^4 - 29540*T^2 + 176402800)^4
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\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)