LMFDB - Newform orbit 275.3.d.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [275,3,Mod(274,275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("275.274"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	275.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$7.49320726991$
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} - 8 x^{15} + 32 x^{14} - 54 x^{13} + 51 x^{12} - 118 x^{11} + 770 x^{10} - 1222 x^{9} + \cdots + 256 $ x^16 - 8x^15 + 32x^14 - 54x^13 + 51x^12 - 118x^11 + 770x^10 - 1222x^9 + 749x^8 + 724x^7 + 3920x^6 - 5016x^5 + 2916x^4 + 7088x^3 + 5408x^2 + 1664*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 55)
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_1 q^{2} + (\beta_{6} + \beta_{4}) q^{3} + ( - \beta_{2} + 3) q^{4} + (\beta_{8} - \beta_{5}) q^{6} - \beta_{10} q^{7} + ( - \beta_{13} + \beta_{11} + \cdots - 4 \beta_1) q^{8} + ( - \beta_{7} + \beta_{2} + 1) q^{9}+ \cdots + ( - 2 \beta_{15} - 3 \beta_{14} + \cdots + 23) q^{99}+O(q^{100}) $ q - b1 * q^2 + (b6 + b4) * q^3 + (-b2 + 3) * q^4 + (b8 - b5) * q^6 - b10 * q^7 + (-b13 + b11 + b10 - 4b1) q^8 + (-b7 + b2 + 1) * q^9 + (-b14 - 2b7 + b5 + b3 + 1) q^11 + (2b12 + b9 + 8b6 + 6b4) q^12 + (-b11 + 2b1) q^13 + (-3b7 + 2b3) * q^14 + (-4b3 - 4b2 + 9) * q^16 + (-b13 - 2b10 + b1) q^17 + (b13 - b11 - 2b10 + 4b1) * q^18 + (b15 + b14 + b5) * q^19 + (b14 + 2b5) q^21 + (b13 - 3b12 - 4b10 - b9 - 3b6 - 5b4) * q^22 + (-4b12 + b9 - b6 + 2b4) * q^23 + (-b15 + 3b14 + 4b8 - 8b5) q^24 + (-b7 - 5b3 + 5b2 - 10) * q^26 + (-b12 - b9 + 5b6 + 4b4) * q^27 + (2b13 - 3b10 + 2b1) q^28 + (-b15 + 2b14 - 3b8 - b5) * q^29 + (6b7 + 5b3 - b2 + 4) * q^31 + (-4b13 + 8b10 - 17b1) q^32 + (-2b13 + 2b12 + b11 + b10 - b9 + b6 - 10b4 - 4b1) * q^33 + (-10b7 - 3b3 - 10) * q^34 + (b7 + 6b3 + 4b2 - 25) * q^36 + (-7b12 + 2b9 - 2b6 + 8b4) * q^37 + (4b12 - 5b9 - 2b6 - 13b4) * q^38 + (-b14 - 3b8) q^39 + (b15 - 4b8 - 3b5) * q^41 + (3b12 - 2b9 - 19b4) q^42 + (-2b13 + b11 + 4b10 - 8b1) q^43 + (b15 - 3b8 + 8b5 + 11b3 + b2 - 1) q^44 + (-b15 - 3b14 - b8 - 2b5) * q^46 + (b9 + 7b6 + 4b4) * q^47 + (8b12 + 8b9 + 29b6 + 41b4) * q^48 + (3b7 + 2b3 + b2 - 21) * q^49 + (-b15 + 5b14 - 2b8 + b5) * q^51 + (-b11 + 4b10 + 22b1) * q^52 + (-9b12 - 4b9 + 6b6 - 2b4) * q^53 + (b15 - 2b14 + 5b8 + b5) * q^54 + (11b7 + 12b3 + 4b2 - 8) q^56 + (-4b13 + b11 + 7b10 - 6b1) q^57 + (-b12 + 5b9 - 25b6 - 2b4) q^58 + (21b7 + 7b3 + 4b2 + 4) q^59 + (-b14 + 2b8 - 12b5) * q^61 + (4b13 + b11 - 3b10 - 4b1) q^62 + (-b13 - b11 - b10 - 5b1) q^63 + (8b7 - 28b3 - 5b2 + 71) q^64 + (b15 + b14 + b8 - 4b7 + 12b5 - 11b3 - 9b2 + 18) * q^66 + (7b12 - 5b9 - 18b6 - 7b4) * q^67 + (b13 + 4b10 + 3b1) * q^68 + (-7b7 + 5b3 - 2b2 - 2) q^69 + (-4b7 + 9b3 - 5b2 + 32) q^71 + (6b13 - 7b10 + 35b1) q^72 + (3b13 + b11 - 7b10 + 11b1) q^73 + (-2b15 - 5b14 - 2b8 - 9b5) * q^74 + (b15 - 5b14 - 2b8 + 25b5) q^76 + (2b13 + 2b12 - b11 - 2b10 + 3b9 + 4b6 - 9b4 - 12b1) q^77 + (-9b12 - 35b6 - 3b4) q^78 + (b15 - 10b14 - 2b8 + 9b5) q^79 + (-18b7 - 8b3 + 10b2 - 35) q^81 + (-7b12 - b9 - 44b6 + 14b4) q^82 + (-6b13 - b11 + 10b10 - 8b1) q^83 + (2b15 - 3b14 + 16b5) q^84 + (5b7 - 17b3 - 13b2 + 46) q^86 + (7b13 - b11 - 4b10 - 21b1) q^87 + (8b13 + 7b12 - b11 - 7b10 - 8b9 - 32b6 - 52b4 + 17b1) q^88 + (-b7 - 4b3 - 5b2 - 58) * q^89 + (-9b7 - 15b3 + 6b2 - 2) q^91 + (4b12 + 2b9 - 8b6 + 17b4) * q^92 + (-4b12 - 4b9 + 9b6) q^93 + (-b15 + b14 + 7b8 - 8b5) * q^94 + (-4b15 + 4b14 + 13b8 - 49b5) * q^96 + (15b12 + 5b9 + 16b6 - 19b4) * q^97 + (3b13 - b11 - 2b10 + 28b1) q^98 + (-2b15 - 3b14 + 3b8 - 7b7 - 3b5 - 6b3 + 23) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 56 q^{4} + 8 q^{9} + 16 q^{11} + 176 q^{16} - 200 q^{26} + 72 q^{31} - 160 q^{34} - 432 q^{36} - 24 q^{44} - 344 q^{49} - 160 q^{56} + 32 q^{59} + 1176 q^{64} + 360 q^{66} - 16 q^{69} + 552 q^{71}+ \cdots + 368 q^{99}+O(q^{100}) $ 16 * q + 56 * q^4 + 8 * q^9 + 16 * q^11 + 176 * q^16 - 200 * q^26 + 72 * q^31 - 160 * q^34 - 432 * q^36 - 24 * q^44 - 344 * q^49 - 160 * q^56 + 32 * q^59 + 1176 * q^64 + 360 * q^66 - 16 * q^69 + 552 * q^71 - 640 * q^81 + 840 * q^86 - 888 * q^89 - 80 * q^91 + 368 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 8 x^{15} + 32 x^{14} - 54 x^{13} + 51 x^{12} - 118 x^{11} + 770 x^{10} - 1222 x^{9} + \cdots + 256 $ x^16 - 8*x^15 + 32*x^14 - 54*x^13 + 51*x^12 - 118*x^11 + 770*x^10 - 1222*x^9 + 749*x^8 + 724*x^7 + 3920*x^6 - 5016*x^5 + 2916*x^4 + 7088*x^3 + 5408*x^2 + 1664*x + 256 :

$\beta_{1}$	$=$	$ ( - 27\!\cdots\!69 \nu^{15} + \cdots - 11\!\cdots\!44 ) / 79\!\cdots\!32 $ (-274223940304429169v^15 + 2311820544100620666v^14 - 9838419486534991212v^13 + 19522254988205748198v^12 - 24080867698733495687v^11 + 44606399946990703620v^10 - 231724362207142322226v^9 + 442893480642803976402v^8 - 446611853807793021913v^7 + 32460364067993845518v^6 - 1091316693169880830788v^5 + 1824016623936130884408v^4 - 2138634042363623579676v^3 - 981515064677011835976v^2 - 238225223087026630432*v - 1143599264828719472144) / 796923041584880842832
$\beta_{2}$	$=$	$ ( - 29\!\cdots\!93 \nu^{15} + \cdots + 15\!\cdots\!20 ) / 30\!\cdots\!36 $ (-2994297419615393v^15 + 25763923212930370v^14 - 111088409910698964v^13 + 226089326193995086v^12 - 277319409845813415v^11 + 497544489463531036v^10 - 2591726624411708810v^9 + 5222225978340301914v^8 - 5172598096612164777v^7 + 372665734690933846v^6 - 11754397194954160332v^5 + 24145142415053732880v^4 - 23670368447056532012v^3 - 10911220375354635944v^2 - 2655401805418001056*v + 15067874601472290320) / 3041690998415575736
$\beta_{3}$	$=$	$ ( 17\!\cdots\!99 \nu^{15} + \cdots - 96\!\cdots\!04 ) / 60\!\cdots\!72 $ (17340994225293299v^15 - 149805677524499814v^14 + 648824338305365020v^13 - 1333404800467108858v^12 + 1652801081022864581v^11 - 2910576203629004852v^10 + 15015697307741378334v^9 - 30695843189741642558v^8 + 31127112543300196683v^7 - 2575157522712897378v^6 + 64029287248153279140v^5 - 133054620565628886640v^4 + 132901095665974881860v^3 + 61469789341420665976v^2 + 14987333603488307680*v - 9674090539850689904) / 6083381996831151472
$\beta_{4}$	$=$	$ ( - 54\!\cdots\!83 \nu^{15} + \cdots - 48\!\cdots\!56 ) / 15\!\cdots\!64 $ (-5416815988184978583v^15 + 44431423666697545340v^14 - 182585393798321797320v^13 + 331861741308128808330v^12 - 354346635350256900525v^11 + 735507757400761455542v^10 - 4349373910690396323390v^9 + 7546246586390613117330v^8 - 5828769097721764864275v^7 - 2135327360214752406440v^6 - 21363760129957091427432v^5 + 31536015769415375895480v^4 - 23091501917291921085660v^3 - 29839855554800633877600v^2 - 25368080605396316832960*v - 4872988745652174469056) / 1593846083169761685664
$\beta_{5}$	$=$	$ ( 49\!\cdots\!89 \nu^{15} + \cdots + 47\!\cdots\!84 ) / 12\!\cdots\!44 $ (49722990453455689v^15 - 409761113306107084v^14 + 1694191387362303528v^13 - 3129395124129403062v^12 + 3440229817902220483v^11 - 6976590512891024962v^10 + 40276880607015004674v^9 - 71128400831769687198v^8 + 58131423762999518717v^7 + 15309052701853259728v^6 + 196404785516310036264v^5 - 296428108894350377352v^4 + 241572809822491720644v^3 + 257755082545867795584v^2 + 249590578858194428224*v + 47783920905553656384) / 12166763993662302944
$\beta_{6}$	$=$	$ ( 27\!\cdots\!93 \nu^{15} + \cdots + 20\!\cdots\!48 ) / 31\!\cdots\!28 $ (27334116808089729793v^15 - 221604450921266449408v^14 + 894683743467714619296v^13 - 1543252200504249975974v^12 + 1444292805281714785491v^11 - 3176101215200978060062v^10 + 21111982673537727750098v^9 - 35125548703612766659686v^8 + 21593961615290065133709v^7 + 21396760451854628255612v^6 + 100004730692355388775888v^5 - 147749005570357275072104v^4 + 84666176747761702235508v^3 + 194236293786131749570704v^2 + 107666046752342388406848*v + 20826719246177544522048) / 3187692166339523371328
$\beta_{7}$	$=$	$ ( 35\!\cdots\!07 \nu^{15} + \cdots - 12\!\cdots\!28 ) / 30\!\cdots\!36 $ (35643483073044507v^15 - 308941912993891862v^14 + 1343406444343004108v^13 - 2785646211061360546v^12 + 3489553021990883469v^11 - 6028019317204030532v^10 + 30705752950663967902v^9 - 63264265218595253662v^8 + 65089274801965665523v^7 - 5904728358813298322v^6 + 126574190669530792516v^5 - 264595169535503537840v^4 + 267343069629621435684v^3 + 123857067187115899576v^2 + 30224368028777734880*v - 12368343399849277528) / 3041690998415575736
$\beta_{8}$	$=$	$ ( 30\!\cdots\!27 \nu^{15} + \cdots + 24\!\cdots\!56 ) / 24\!\cdots\!88 $ (304056114626190127v^15 - 2467682657602190520v^14 + 9992309789873830656v^13 - 17314281466129378746v^12 + 16208287985411428493v^11 - 34453858188763086890v^10 + 234136233148448915070v^9 - 393980791616304365354v^8 + 249155661399020519411v^7 + 262574320587229276780v^6 + 1095709922680987041584v^5 - 1654673955324068538840v^4 + 1034799811680295021836v^3 + 2281817519997410601680v^2 + 1244092425554284615104*v + 240019216623817752256) / 24333527987324605888
$\beta_{9}$	$=$	$ ( 44\!\cdots\!65 \nu^{15} + \cdots + 41\!\cdots\!44 ) / 31\!\cdots\!28 $ (44113723981629103665v^15 - 363446351706889254696v^14 + 1504097057924230683792v^13 - 2790456097998261369478v^12 + 3131266206442145890867v^11 - 6407304989257831059606v^10 + 36071662280813543621746v^9 - 63435607815570953360502v^8 + 52863016091889954615533v^7 + 9017188416239806962836v^6 + 181626844036325797915552v^5 - 264570949218479893314088v^4 + 204396705416507215688916v^3 + 220568211611603028825392v^2 + 218173940823097434197056*v + 41848303621051479975744) / 3187692166339523371328
$\beta_{10}$	$=$	$ ( 30\!\cdots\!15 \nu^{15} + \cdots - 10\!\cdots\!36 ) / 16\!\cdots\!56 $ (300281369975679315v^15 - 2653134620407099054v^14 + 11667880961402859668v^13 - 24672010279833872194v^12 + 30694431509086347621v^11 - 51716859033558523644v^10 + 264496682741149751654v^9 - 564724215695594374454v^8 + 571399021030932352747v^7 - 49620855511184306410v^6 + 1071226833041605883084v^5 - 2391605660213682264792v^4 + 2305146553607721205844v^3 + 1071377634913537901784v^2 + 261966241461684096352*v - 102980796015921868336) / 16955809395422996656
$\beta_{11}$	$=$	$ ( 16\!\cdots\!65 \nu^{15} + \cdots - 16\!\cdots\!96 ) / 79\!\cdots\!32 $ (16046402982557990365v^15 - 142373215842824628642v^14 + 627204321188355031484v^13 - 1329606320506763213510v^12 + 1644677763340852169835v^11 - 2770551325651266243572v^10 + 14251643929143973739802v^9 - 30711298320872339045586v^8 + 30729690985913188588021v^7 - 2555118391959813965622v^6 + 58066978075596732372692v^5 - 134400242932010120233056v^4 + 124869642993712492103532v^3 + 58056760013298412459432v^2 + 14199543777096822981792*v - 16971713695243920586496) / 796923041584880842832
$\beta_{12}$	$=$	$ ( - 10\!\cdots\!03 \nu^{15} + \cdots - 81\!\cdots\!60 ) / 31\!\cdots\!28 $ (-108195488251500798403v^15 + 876501451954526123120v^14 - 3535822025821810783616v^13 + 6087152964706522075634v^12 - 5689891053346850028313v^11 + 12601133117466693445866v^10 - 83625381897508233679862v^9 + 138561748032173246836690v^8 - 84196728550733010396039v^7 - 85107251926187094846212v^6 - 397691113822270303948624v^5 + 583168674266171951445176v^4 - 322978418287719386068252v^3 - 785741364760782575428976v^2 - 419984921446004792935104*v - 81322996863684295808960) / 3187692166339523371328
$\beta_{13}$	$=$	$ ( 31\!\cdots\!23 \nu^{15} + \cdots - 10\!\cdots\!80 ) / 79\!\cdots\!32 $ (31681426257220138223v^15 - 279622940481143165302v^14 + 1228411769335929932132v^13 - 2591861428321274274674v^12 + 3217266996377624081049v^11 - 5445450772897847414636v^10 + 27927783008207710423022v^9 - 59506879529989506455174v^8 + 60015336959822514773815v^7 - 5109617645655150243058v^6 + 114051506511339794206812v^5 - 253951853188812724088176v^4 + 244314816164036742208356v^3 + 113500678537875924558520v^2 + 27745791726376950490336*v - 10407210472764190571680) / 796923041584880842832
$\beta_{14}$	$=$	$ ( - 11\!\cdots\!67 \nu^{15} + \cdots - 85\!\cdots\!12 ) / 24\!\cdots\!88 $ (-1117650522215340667v^15 + 9054509015984957984v^14 - 36594534393627143632v^13 + 63118046860550334658v^12 - 58880885215686803521v^11 + 127091670641734345706v^10 - 861762699571889234614v^9 + 1436560773489552860450v^8 - 885539892819063646783v^7 - 986524970164723772820v^6 - 4073690111361356177312v^5 + 6041464736065852934392v^4 - 3506725691883236205404v^3 - 8851110390182841182384v^2 - 4421896578817262813888*v - 855029220790410392512) / 24333527987324605888
$\beta_{15}$	$=$	$ ( - 94\!\cdots\!13 \nu^{15} + \cdots - 81\!\cdots\!48 ) / 12\!\cdots\!44 $ (-944272070273722513v^15 + 7716841179719169800v^14 - 31563688407854514496v^13 + 56513396384645377926v^12 - 58114010484886922803v^11 + 121369986843041505302v^10 - 748630663146153874050v^9 + 1285363546893930274774v^8 - 936581288862058778381v^7 - 533222219906166017620v^6 - 3628413529926065056848v^5 + 5380455426259980853800v^4 - 3737732722476812208116v^3 - 6238298761162618732080v^2 - 4230223033816736176704*v - 813802282047589552448) / 12166763993662302944

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

$\nu$	$=$	$ ( \beta_{5} + \beta_{4} + 2\beta _1 + 2 ) / 4 $ (b5 + b4 + 2*b1 + 2) / 4
$\nu^{2}$	$=$	$ ( \beta_{9} + \beta_{6} + 2\beta_{5} + 9\beta_{4} ) / 4 $ (b9 + b6 + 2b5 + 9b4) / 4
$\nu^{3}$	$=$	$ ( \beta_{15} - \beta_{14} - 2 \beta_{13} + 2 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} - \beta_{8} + \cdots - 44 ) / 8 $ (b15 - b14 - 2b13 + 2b11 + 2b10 + 3b9 - b8 + 3b6 + 15b5 + 25b4 + 6b2 - 30*b1 - 44) / 8
$\nu^{4}$	$=$	$ ( -2\beta_{13} + 2\beta_{11} + 2\beta_{10} + 2\beta_{3} + 11\beta_{2} - 26\beta _1 - 60 ) / 2 $ (-2b13 + 2b11 + 2b10 + 2b3 + 11b2 - 26b1 - 60) / 2
$\nu^{5}$	$=$	$ ( - 13 \beta_{15} + 15 \beta_{14} - 30 \beta_{13} - 10 \beta_{12} + 26 \beta_{11} + 34 \beta_{10} + \cdots - 456 ) / 8 $ (-13b15 + 15b14 - 30b13 - 10b12 + 26b11 + 34b10 - 45b9 + 19b8 - 75b6 - 145b5 - 283b4 + 20b3 + 90b2 - 286b1 - 456) / 8
$\nu^{6}$	$=$	$ ( - 29 \beta_{15} + 35 \beta_{14} - 40 \beta_{12} - 121 \beta_{9} + 47 \beta_{8} - 249 \beta_{6} + \cdots - 731 \beta_{4} ) / 4 $ (-29b15 + 35b14 - 40b12 - 121b9 + 47b8 - 249b6 - 309b5 - 731b4) / 4
$\nu^{7}$	$=$	$ ( - 75 \beta_{15} + 98 \beta_{14} + 198 \beta_{13} - 105 \beta_{12} - 150 \beta_{11} - 248 \beta_{10} + \cdots + 2466 ) / 4 $ (-75b15 + 98b14 + 198b13 - 105b12 - 150b11 - 248b10 - 273b9 + 148b8 + 14b7 - 616b6 - 782b5 - 1625b4 - 224b3 - 546b2 + 1516*b1 + 2466) / 4
$\nu^{8}$	$=$	$ ( 484\beta_{13} - 348\beta_{11} - 628\beta_{10} + 58\beta_{7} - 678\beta_{3} - 1355\beta_{2} + 3468\beta _1 + 5942 ) / 2 $ (484b13 - 348b11 - 628b10 + 58b7 - 678b3 - 1355b2 + 3468*b1 + 5942) / 2
$\nu^{9}$	$=$	$ ( 1703 \beta_{15} - 2451 \beta_{14} + 5034 \beta_{13} + 3228 \beta_{12} - 3406 \beta_{11} - 6794 \beta_{10} + \cdots + 54748 ) / 8 $ (1703b15 - 2451b14 + 5034b13 + 3228b12 - 3406b11 - 6794b10 + 6351b9 - 4211b8 + 708b7 + 17451b6 + 17677b5 + 37661b4 - 7164b3 - 12702b2 + 33726*b1 + 54748) / 8
$\nu^{10}$	$=$	$ ( 4027 \beta_{15} - 6015 \beta_{14} + 8650 \beta_{12} + 15420 \beta_{9} - 10831 \beta_{8} + \cdots + 91776 \beta_{4} ) / 4 $ (4027b15 - 6015b14 + 8650b12 + 15420b9 - 10831b8 + 45730b6 + 41985b5 + 91776b4) / 4
$\nu^{11}$	$=$	$ ( 19447 \beta_{15} - 30085 \beta_{14} - 62770 \beta_{13} + 44330 \beta_{12} + 38894 \beta_{11} + \cdots - 618504 ) / 8 $ (19447b15 - 30085b14 - 62770b13 + 44330b12 + 38894b11 + 89246b10 + 73513b9 - 56561b8 - 12232b7 + 230967b6 + 204091b5 + 439327b4 + 100892b3 + 147026b2 - 384306*b1 - 618504) / 8
$\nu^{12}$	$=$	$ - 38836 \beta_{13} + 23240 \beta_{11} + 56286 \beta_{10} - 8473 \beta_{7} + 65538 \beta_{3} + \cdots - 370595 $ -38836b13 + 23240b11 + 56286b10 - 8473b7 + 65538b3 + 88848b2 - 229854*b1 - 370595
$\nu^{13}$	$=$	$ ( - 112088 \beta_{15} + 182895 \beta_{14} - 386444 \beta_{13} - 288535 \beta_{12} + 224176 \beta_{11} + \cdots - 3540210 ) / 4 $ (-112088b15 + 182895b14 - 386444b13 - 288535b12 + 224176b11 + 569366b10 - 427258b9 + 365817b8 - 90454b7 - 1473771b6 - 1191361b5 - 2576352b4 + 667524b3 + 854516b2 - 2220454*b1 - 3540210) / 4
$\nu^{14}$	$=$	$ ( - 539346 \beta_{15} + 898688 \beta_{14} - 1456766 \beta_{12} - 2066971 \beta_{9} + \cdots - 12514365 \beta_{4} ) / 4 $ (-539346b15 + 898688b14 - 1456766b12 - 2066971b9 + 1839588b8 - 7391173b6 - 5765280b5 - 12514365b4) / 4
$\nu^{15}$	$=$	$ ( - 2606317 \beta_{15} + 4422425 \beta_{14} + 9432910 \beta_{13} - 7292660 \beta_{12} - 5212634 \beta_{11} + \cdots + 81892508 ) / 8 $ (-2606317b15 + 4422425b14 + 9432910b13 - 7292660b12 - 5212634b11 - 14241246b10 - 9989635b9 + 9230761b8 + 2469860b7 - 36807335b6 - 28004295b5 - 60698409b4 - 17055180b3 - 19979270b2 + 51788314*b1 + 81892508) / 8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$-1$	$-1$

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} $

274.1

2.44303	−	2.44303i
2.44303	+	2.44303i
2.07977	+	2.07977i
2.07977	−	2.07977i
1.33385	−	1.33385i
1.33385	+	1.33385i
1.23052	−	1.23052i
1.23052	+	1.23052i
−0.230520	+	0.230520i
−0.230520	−	0.230520i
−0.333846	+	0.333846i
−0.333846	−	0.333846i
−1.07977	−	1.07977i
−1.07977	+	1.07977i
−1.44303	+	1.44303i
−1.44303	−	1.44303i

−3.88606

−

4.12151i

11.1014

16.0164i

3.44089

−27.5966

−7.98688

274.2

−3.88606

4.12151i

11.1014

−

16.0164i

3.44089

−27.5966

−7.98688

274.3

−3.15955

−

3.03112i

5.98274

9.57697i

−5.67155

−6.26455

−0.187686

274.4

−3.15955

3.03112i

5.98274

−

9.57697i

−5.67155

−6.26455

−0.187686

274.5

−1.66769

−

2.79505i

−1.21881

4.66128i

6.72266

8.70336

1.18769

274.6

−1.66769

2.79505i

−1.21881

−

4.66128i

6.72266

8.70336

1.18769

274.7

−1.46104

−

0.114554i

−1.86536

0.167368i

−4.56066

8.56953

8.98688

274.8

−1.46104

0.114554i

−1.86536

−

0.167368i

−4.56066

8.56953

8.98688

274.9

1.46104

−

0.114554i

−1.86536

−

0.167368i

4.56066

−8.56953

8.98688

274.10

1.46104

0.114554i

−1.86536

0.167368i

4.56066

−8.56953

8.98688

274.11

1.66769

−

2.79505i

−1.21881

−

4.66128i

−6.72266

−8.70336

1.18769

274.12

1.66769

2.79505i

−1.21881

4.66128i

−6.72266

−8.70336

1.18769

274.13

3.15955

−

3.03112i

5.98274

−

9.57697i

5.67155

6.26455

−0.187686

274.14

3.15955

3.03112i

5.98274

9.57697i

5.67155

6.26455

−0.187686

274.15

3.88606

−

4.12151i

11.1014

−

16.0164i

−3.44089

27.5966

−7.98688

274.16

3.88606

4.12151i

11.1014

16.0164i

−3.44089

27.5966

−7.98688

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	275.3.d.c		16
5.b	even	2	1	inner	275.3.d.c		16
5.c	odd	4	1		55.3.c.a	✓	8
5.c	odd	4	1		275.3.c.f		8
11.b	odd	2	1	inner	275.3.d.c		16
15.e	even	4	1		495.3.b.a		8
20.e	even	4	1		880.3.j.a		8
55.d	odd	2	1	inner	275.3.d.c		16
55.e	even	4	1		55.3.c.a	✓	8
55.e	even	4	1		275.3.c.f		8
165.l	odd	4	1		495.3.b.a		8
220.i	odd	4	1		880.3.j.a		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
55.3.c.a	✓	8	5.c	odd	4	1
55.3.c.a	✓	8	55.e	even	4	1
275.3.c.f		8	5.c	odd	4	1
275.3.c.f		8	55.e	even	4	1
275.3.d.c		16	1.a	even	1	1	trivial
275.3.d.c		16	5.b	even	2	1	inner
275.3.d.c		16	11.b	odd	2	1	inner
275.3.d.c		16	55.d	odd	2	1	inner
495.3.b.a		8	15.e	even	4	1
495.3.b.a		8	165.l	odd	4	1
880.3.j.a		8	20.e	even	4	1
880.3.j.a		8	220.i	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} - 30T_{2}^{6} + 280T_{2}^{4} - 890T_{2}^{2} + 895 $ T2^8 - 30*T2^6 + 280*T2^4 - 890*T2^2 + 895 acting on $S_{3}^{\mathrm{new}}(275, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} - 30 T^{6} + \cdots + 895)^{2} $ (T^8 - 30T^6 + 280T^4 - 890*T^2 + 895)^2
$3$	$ (T^{8} + 34 T^{6} + \cdots + 16)^{2} $ (T^8 + 34T^6 + 361T^4 + 1224*T^2 + 16)^2
$5$	$ T^{16} $ T^16
$7$	$ (T^{8} - 110 T^{6} + \cdots + 358000)^{2} $ (T^8 - 110T^6 + 4225T^4 - 66500*T^2 + 358000)^2
$11$	$ (T^{8} - 8 T^{7} + \cdots + 214358881)^{2} $ (T^8 - 8T^7 + 48T^6 - 1816T^5 + 23870T^4 - 219736T^3 + 702768T^2 - 14172488*T + 214358881)^2
$13$	$ (T^{8} - 620 T^{6} + \cdots + 27723520)^{2} $ (T^8 - 620T^6 + 109600T^4 - 4153280*T^2 + 27723520)^2
$17$	$ (T^{8} - 1270 T^{6} + \cdots + 1732720)^{2} $ (T^8 - 1270T^6 + 302185T^4 - 3536100*T^2 + 1732720)^2
$19$	$ (T^{8} + 2090 T^{6} + \cdots + 3931928320)^{2} $ (T^8 + 2090T^6 + 1199785T^4 + 217622960*T^2 + 3931928320)^2
$23$	$ (T^{8} + 1304 T^{6} + \cdots + 760877056)^{2} $ (T^8 + 1304T^6 + 431376T^4 + 45040384*T^2 + 760877056)^2
$29$	$ (T^{8} + 4850 T^{6} + \cdots + 27723520)^{2} $ (T^8 + 4850T^6 + 5909665T^4 + 404179520*T^2 + 27723520)^2
$31$	$ (T^{4} - 18 T^{3} + \cdots + 178076)^{4} $ (T^4 - 18T^3 - 951T^2 + 12488*T + 178076)^4
$37$	$ (T^{8} + \cdots + 1000408041616)^{2} $ (T^8 + 4794T^6 + 7829281T^4 + 4972958984*T^2 + 1000408041616)^2
$41$	$ (T^{8} + \cdots + 5420634603520)^{2} $ (T^8 + 9520T^6 + 25555200T^4 + 21158625280*T^2 + 5420634603520)^2
$43$	$ (T^{8} - 4180 T^{6} + \cdots + 91648000)^{2} $ (T^8 - 4180T^6 + 593200T^4 - 20512000*T^2 + 91648000)^2
$47$	$ (T^{8} + 1784 T^{6} + \cdots + 498896896)^{2} $ (T^8 + 1784T^6 + 766096T^4 + 95800064*T^2 + 498896896)^2
$53$	$ (T^{8} + \cdots + 198845082347536)^{2} $ (T^8 + 19114T^6 + 121691761T^4 + 292920887464*T^2 + 198845082347536)^2
$59$	$ (T^{4} - 8 T^{3} + \cdots - 465344)^{4} $ (T^4 - 8T^3 - 8316T^2 - 279472*T - 465344)^4
$61$	$ (T^{8} + \cdots + 44785710872320)^{2} $ (T^8 + 17650T^6 + 79305745T^4 + 120983959920*T^2 + 44785710872320)^2
$67$	$ (T^{8} + \cdots + 12232198471936)^{2} $ (T^8 + 23504T^6 + 149240416T^4 + 162396781824*T^2 + 12232198471936)^2
$71$	$ (T^{4} - 138 T^{3} + \cdots - 22924)^{4} $ (T^4 - 138T^3 + 3169T^2 - 14172*T - 22924)^4
$73$	$ (T^{8} + \cdots + 15327220913920)^{2} $ (T^8 - 13900T^6 + 45636000T^4 - 48397789120*T^2 + 15327220913920)^2
$79$	$ (T^{8} + \cdots + 67909027102720)^{2} $ (T^8 + 27760T^6 + 226274560T^4 + 453908807680*T^2 + 67909027102720)^2
$83$	$ (T^{8} + \cdots + 26187838846720)^{2} $ (T^8 - 30700T^6 + 259522080T^4 - 400109149120*T^2 + 26187838846720)^2
$89$	$ (T^{4} + 222 T^{3} + \cdots - 1268884)^{4} $ (T^4 + 222T^3 + 16109T^2 + 364488*T - 1268884)^4
$97$	$ (T^{8} + \cdots + 444790799524096)^{2} $ (T^8 + 38864T^6 + 379489376T^4 + 1091635473664*T^2 + 444790799524096)^2
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	275.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{2} + (\beta_{6} + \beta_{4}) q^{3} + ( - \beta_{2} + 3) q^{4} + (\beta_{8} - \beta_{5}) q^{6} - \beta_{10} q^{7} + ( - \beta_{13} + \beta_{11} + \cdots - 4 \beta_1) q^{8} + ( - \beta_{7} + \beta_{2} + 1) q^{9}+ \cdots + ( - 2 \beta_{15} - 3 \beta_{14} + \cdots + 23) q^{99}+O(q^{100}) \) q - b1 * q^2 + (b6 + b4) * q^3 + (-b2 + 3) * q^4 + (b8 - b5) * q^6 - b10 * q^7 + (-b13 + b11 + b10 - 4b1) q^8 + (-b7 + b2 + 1) * q^9 + (-b14 - 2b7 + b5 + b3 + 1) q^11 + (2b12 + b9 + 8b6 + 6b4) q^12 + (-b11 + 2b1) q^13 + (-3b7 + 2b3) * q^14 + (-4b3 - 4b2 + 9) * q^16 + (-b13 - 2b10 + b1) q^17 + (b13 - b11 - 2b10 + 4b1) * q^18 + (b15 + b14 + b5) * q^19 + (b14 + 2b5) q^21 + (b13 - 3b12 - 4b10 - b9 - 3b6 - 5b4) * q^22 + (-4b12 + b9 - b6 + 2b4) * q^23 + (-b15 + 3b14 + 4b8 - 8b5) q^24 + (-b7 - 5b3 + 5b2 - 10) * q^26 + (-b12 - b9 + 5b6 + 4b4) * q^27 + (2b13 - 3b10 + 2b1) q^28 + (-b15 + 2b14 - 3b8 - b5) * q^29 + (6b7 + 5b3 - b2 + 4) * q^31 + (-4b13 + 8b10 - 17b1) q^32 + (-2b13 + 2b12 + b11 + b10 - b9 + b6 - 10b4 - 4b1) * q^33 + (-10b7 - 3b3 - 10) * q^34 + (b7 + 6b3 + 4b2 - 25) * q^36 + (-7b12 + 2b9 - 2b6 + 8b4) * q^37 + (4b12 - 5b9 - 2b6 - 13b4) * q^38 + (-b14 - 3b8) q^39 + (b15 - 4b8 - 3b5) * q^41 + (3b12 - 2b9 - 19b4) q^42 + (-2b13 + b11 + 4b10 - 8b1) q^43 + (b15 - 3b8 + 8b5 + 11b3 + b2 - 1) q^44 + (-b15 - 3b14 - b8 - 2b5) * q^46 + (b9 + 7b6 + 4b4) * q^47 + (8b12 + 8b9 + 29b6 + 41b4) * q^48 + (3b7 + 2b3 + b2 - 21) * q^49 + (-b15 + 5b14 - 2b8 + b5) * q^51 + (-b11 + 4b10 + 22b1) * q^52 + (-9b12 - 4b9 + 6b6 - 2b4) * q^53 + (b15 - 2b14 + 5b8 + b5) * q^54 + (11b7 + 12b3 + 4b2 - 8) q^56 + (-4b13 + b11 + 7b10 - 6b1) q^57 + (-b12 + 5b9 - 25b6 - 2b4) q^58 + (21b7 + 7b3 + 4b2 + 4) q^59 + (-b14 + 2b8 - 12b5) * q^61 + (4b13 + b11 - 3b10 - 4b1) q^62 + (-b13 - b11 - b10 - 5b1) q^63 + (8b7 - 28b3 - 5b2 + 71) q^64 + (b15 + b14 + b8 - 4b7 + 12b5 - 11b3 - 9b2 + 18) * q^66 + (7b12 - 5b9 - 18b6 - 7b4) * q^67 + (b13 + 4b10 + 3b1) * q^68 + (-7b7 + 5b3 - 2b2 - 2) q^69 + (-4b7 + 9b3 - 5b2 + 32) q^71 + (6b13 - 7b10 + 35b1) q^72 + (3b13 + b11 - 7b10 + 11b1) q^73 + (-2b15 - 5b14 - 2b8 - 9b5) * q^74 + (b15 - 5b14 - 2b8 + 25b5) q^76 + (2b13 + 2b12 - b11 - 2b10 + 3b9 + 4b6 - 9b4 - 12b1) q^77 + (-9b12 - 35b6 - 3b4) q^78 + (b15 - 10b14 - 2b8 + 9b5) q^79 + (-18b7 - 8b3 + 10b2 - 35) q^81 + (-7b12 - b9 - 44b6 + 14b4) q^82 + (-6b13 - b11 + 10b10 - 8b1) q^83 + (2b15 - 3b14 + 16b5) q^84 + (5b7 - 17b3 - 13b2 + 46) q^86 + (7b13 - b11 - 4b10 - 21b1) q^87 + (8b13 + 7b12 - b11 - 7b10 - 8b9 - 32b6 - 52b4 + 17b1) q^88 + (-b7 - 4b3 - 5b2 - 58) * q^89 + (-9b7 - 15b3 + 6b2 - 2) q^91 + (4b12 + 2b9 - 8b6 + 17b4) * q^92 + (-4b12 - 4b9 + 9b6) q^93 + (-b15 + b14 + 7b8 - 8b5) * q^94 + (-4b15 + 4b14 + 13b8 - 49b5) * q^96 + (15b12 + 5b9 + 16b6 - 19b4) * q^97 + (3b13 - b11 - 2b10 + 28b1) q^98 + (-2b15 - 3b14 + 3b8 - 7b7 - 3b5 - 6b3 + 23) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 56 q^{4} + 8 q^{9} + 16 q^{11} + 176 q^{16} - 200 q^{26} + 72 q^{31} - 160 q^{34} - 432 q^{36} - 24 q^{44} - 344 q^{49} - 160 q^{56} + 32 q^{59} + 1176 q^{64} + 360 q^{66} - 16 q^{69} + 552 q^{71}+ \cdots + 368 q^{99}+O(q^{100}) \) 16 * q + 56 * q^4 + 8 * q^9 + 16 * q^11 + 176 * q^16 - 200 * q^26 + 72 * q^31 - 160 * q^34 - 432 * q^36 - 24 * q^44 - 344 * q^49 - 160 * q^56 + 32 * q^59 + 1176 * q^64 + 360 * q^66 - 16 * q^69 + 552 * q^71 - 640 * q^81 + 840 * q^86 - 888 * q^89 - 80 * q^91 + 368 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	275.3.d.c

Level	$275$
Weight	$3$
Character orbit	275.d
Analytic conductor	$7.493$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(7.49320726991\)
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:	\( x^{16} - 8 x^{15} + 32 x^{14} - 54 x^{13} + 51 x^{12} - 118 x^{11} + 770 x^{10} - 1222 x^{9} + \cdots + 256 \) x^16 - 8x^15 + 32x^14 - 54x^13 + 51x^12 - 118x^11 + 770x^10 - 1222x^9 + 749x^8 + 724x^7 + 3920x^6 - 5016x^5 + 2916x^4 + 7088x^3 + 5408x^2 + 1664*x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{22} \)
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 27\!\cdots\!69 \nu^{15} + \cdots - 11\!\cdots\!44 ) / 79\!\cdots\!32 \) (-274223940304429169v^15 + 2311820544100620666v^14 - 9838419486534991212v^13 + 19522254988205748198v^12 - 24080867698733495687v^11 + 44606399946990703620v^10 - 231724362207142322226v^9 + 442893480642803976402v^8 - 446611853807793021913v^7 + 32460364067993845518v^6 - 1091316693169880830788v^5 + 1824016623936130884408v^4 - 2138634042363623579676v^3 - 981515064677011835976v^2 - 238225223087026630432*v - 1143599264828719472144) / 796923041584880842832
\(\beta_{2}\)	\(=\)	\( ( - 29\!\cdots\!93 \nu^{15} + \cdots + 15\!\cdots\!20 ) / 30\!\cdots\!36 \) (-2994297419615393v^15 + 25763923212930370v^14 - 111088409910698964v^13 + 226089326193995086v^12 - 277319409845813415v^11 + 497544489463531036v^10 - 2591726624411708810v^9 + 5222225978340301914v^8 - 5172598096612164777v^7 + 372665734690933846v^6 - 11754397194954160332v^5 + 24145142415053732880v^4 - 23670368447056532012v^3 - 10911220375354635944v^2 - 2655401805418001056*v + 15067874601472290320) / 3041690998415575736
\(\beta_{3}\)	\(=\)	\( ( 17\!\cdots\!99 \nu^{15} + \cdots - 96\!\cdots\!04 ) / 60\!\cdots\!72 \) (17340994225293299v^15 - 149805677524499814v^14 + 648824338305365020v^13 - 1333404800467108858v^12 + 1652801081022864581v^11 - 2910576203629004852v^10 + 15015697307741378334v^9 - 30695843189741642558v^8 + 31127112543300196683v^7 - 2575157522712897378v^6 + 64029287248153279140v^5 - 133054620565628886640v^4 + 132901095665974881860v^3 + 61469789341420665976v^2 + 14987333603488307680*v - 9674090539850689904) / 6083381996831151472
\(\beta_{4}\)	\(=\)	\( ( - 54\!\cdots\!83 \nu^{15} + \cdots - 48\!\cdots\!56 ) / 15\!\cdots\!64 \) (-5416815988184978583v^15 + 44431423666697545340v^14 - 182585393798321797320v^13 + 331861741308128808330v^12 - 354346635350256900525v^11 + 735507757400761455542v^10 - 4349373910690396323390v^9 + 7546246586390613117330v^8 - 5828769097721764864275v^7 - 2135327360214752406440v^6 - 21363760129957091427432v^5 + 31536015769415375895480v^4 - 23091501917291921085660v^3 - 29839855554800633877600v^2 - 25368080605396316832960*v - 4872988745652174469056) / 1593846083169761685664
\(\beta_{5}\)	\(=\)	\( ( 49\!\cdots\!89 \nu^{15} + \cdots + 47\!\cdots\!84 ) / 12\!\cdots\!44 \) (49722990453455689v^15 - 409761113306107084v^14 + 1694191387362303528v^13 - 3129395124129403062v^12 + 3440229817902220483v^11 - 6976590512891024962v^10 + 40276880607015004674v^9 - 71128400831769687198v^8 + 58131423762999518717v^7 + 15309052701853259728v^6 + 196404785516310036264v^5 - 296428108894350377352v^4 + 241572809822491720644v^3 + 257755082545867795584v^2 + 249590578858194428224*v + 47783920905553656384) / 12166763993662302944
\(\beta_{6}\)	\(=\)	\( ( 27\!\cdots\!93 \nu^{15} + \cdots + 20\!\cdots\!48 ) / 31\!\cdots\!28 \) (27334116808089729793v^15 - 221604450921266449408v^14 + 894683743467714619296v^13 - 1543252200504249975974v^12 + 1444292805281714785491v^11 - 3176101215200978060062v^10 + 21111982673537727750098v^9 - 35125548703612766659686v^8 + 21593961615290065133709v^7 + 21396760451854628255612v^6 + 100004730692355388775888v^5 - 147749005570357275072104v^4 + 84666176747761702235508v^3 + 194236293786131749570704v^2 + 107666046752342388406848*v + 20826719246177544522048) / 3187692166339523371328
\(\beta_{7}\)	\(=\)	\( ( 35\!\cdots\!07 \nu^{15} + \cdots - 12\!\cdots\!28 ) / 30\!\cdots\!36 \) (35643483073044507v^15 - 308941912993891862v^14 + 1343406444343004108v^13 - 2785646211061360546v^12 + 3489553021990883469v^11 - 6028019317204030532v^10 + 30705752950663967902v^9 - 63264265218595253662v^8 + 65089274801965665523v^7 - 5904728358813298322v^6 + 126574190669530792516v^5 - 264595169535503537840v^4 + 267343069629621435684v^3 + 123857067187115899576v^2 + 30224368028777734880*v - 12368343399849277528) / 3041690998415575736
\(\beta_{8}\)	\(=\)	\( ( 30\!\cdots\!27 \nu^{15} + \cdots + 24\!\cdots\!56 ) / 24\!\cdots\!88 \) (304056114626190127v^15 - 2467682657602190520v^14 + 9992309789873830656v^13 - 17314281466129378746v^12 + 16208287985411428493v^11 - 34453858188763086890v^10 + 234136233148448915070v^9 - 393980791616304365354v^8 + 249155661399020519411v^7 + 262574320587229276780v^6 + 1095709922680987041584v^5 - 1654673955324068538840v^4 + 1034799811680295021836v^3 + 2281817519997410601680v^2 + 1244092425554284615104*v + 240019216623817752256) / 24333527987324605888
\(\beta_{9}\)	\(=\)	\( ( 44\!\cdots\!65 \nu^{15} + \cdots + 41\!\cdots\!44 ) / 31\!\cdots\!28 \) (44113723981629103665v^15 - 363446351706889254696v^14 + 1504097057924230683792v^13 - 2790456097998261369478v^12 + 3131266206442145890867v^11 - 6407304989257831059606v^10 + 36071662280813543621746v^9 - 63435607815570953360502v^8 + 52863016091889954615533v^7 + 9017188416239806962836v^6 + 181626844036325797915552v^5 - 264570949218479893314088v^4 + 204396705416507215688916v^3 + 220568211611603028825392v^2 + 218173940823097434197056*v + 41848303621051479975744) / 3187692166339523371328
\(\beta_{10}\)	\(=\)	\( ( 30\!\cdots\!15 \nu^{15} + \cdots - 10\!\cdots\!36 ) / 16\!\cdots\!56 \) (300281369975679315v^15 - 2653134620407099054v^14 + 11667880961402859668v^13 - 24672010279833872194v^12 + 30694431509086347621v^11 - 51716859033558523644v^10 + 264496682741149751654v^9 - 564724215695594374454v^8 + 571399021030932352747v^7 - 49620855511184306410v^6 + 1071226833041605883084v^5 - 2391605660213682264792v^4 + 2305146553607721205844v^3 + 1071377634913537901784v^2 + 261966241461684096352*v - 102980796015921868336) / 16955809395422996656
\(\beta_{11}\)	\(=\)	\( ( 16\!\cdots\!65 \nu^{15} + \cdots - 16\!\cdots\!96 ) / 79\!\cdots\!32 \) (16046402982557990365v^15 - 142373215842824628642v^14 + 627204321188355031484v^13 - 1329606320506763213510v^12 + 1644677763340852169835v^11 - 2770551325651266243572v^10 + 14251643929143973739802v^9 - 30711298320872339045586v^8 + 30729690985913188588021v^7 - 2555118391959813965622v^6 + 58066978075596732372692v^5 - 134400242932010120233056v^4 + 124869642993712492103532v^3 + 58056760013298412459432v^2 + 14199543777096822981792*v - 16971713695243920586496) / 796923041584880842832
\(\beta_{12}\)	\(=\)	\( ( - 10\!\cdots\!03 \nu^{15} + \cdots - 81\!\cdots\!60 ) / 31\!\cdots\!28 \) (-108195488251500798403v^15 + 876501451954526123120v^14 - 3535822025821810783616v^13 + 6087152964706522075634v^12 - 5689891053346850028313v^11 + 12601133117466693445866v^10 - 83625381897508233679862v^9 + 138561748032173246836690v^8 - 84196728550733010396039v^7 - 85107251926187094846212v^6 - 397691113822270303948624v^5 + 583168674266171951445176v^4 - 322978418287719386068252v^3 - 785741364760782575428976v^2 - 419984921446004792935104*v - 81322996863684295808960) / 3187692166339523371328
\(\beta_{13}\)	\(=\)	\( ( 31\!\cdots\!23 \nu^{15} + \cdots - 10\!\cdots\!80 ) / 79\!\cdots\!32 \) (31681426257220138223v^15 - 279622940481143165302v^14 + 1228411769335929932132v^13 - 2591861428321274274674v^12 + 3217266996377624081049v^11 - 5445450772897847414636v^10 + 27927783008207710423022v^9 - 59506879529989506455174v^8 + 60015336959822514773815v^7 - 5109617645655150243058v^6 + 114051506511339794206812v^5 - 253951853188812724088176v^4 + 244314816164036742208356v^3 + 113500678537875924558520v^2 + 27745791726376950490336*v - 10407210472764190571680) / 796923041584880842832
\(\beta_{14}\)	\(=\)	\( ( - 11\!\cdots\!67 \nu^{15} + \cdots - 85\!\cdots\!12 ) / 24\!\cdots\!88 \) (-1117650522215340667v^15 + 9054509015984957984v^14 - 36594534393627143632v^13 + 63118046860550334658v^12 - 58880885215686803521v^11 + 127091670641734345706v^10 - 861762699571889234614v^9 + 1436560773489552860450v^8 - 885539892819063646783v^7 - 986524970164723772820v^6 - 4073690111361356177312v^5 + 6041464736065852934392v^4 - 3506725691883236205404v^3 - 8851110390182841182384v^2 - 4421896578817262813888*v - 855029220790410392512) / 24333527987324605888
\(\beta_{15}\)	\(=\)	\( ( - 94\!\cdots\!13 \nu^{15} + \cdots - 81\!\cdots\!48 ) / 12\!\cdots\!44 \) (-944272070273722513v^15 + 7716841179719169800v^14 - 31563688407854514496v^13 + 56513396384645377926v^12 - 58114010484886922803v^11 + 121369986843041505302v^10 - 748630663146153874050v^9 + 1285363546893930274774v^8 - 936581288862058778381v^7 - 533222219906166017620v^6 - 3628413529926065056848v^5 + 5380455426259980853800v^4 - 3737732722476812208116v^3 - 6238298761162618732080v^2 - 4230223033816736176704*v - 813802282047589552448) / 12166763993662302944

\(\nu\)	\(=\)	\( ( \beta_{5} + \beta_{4} + 2\beta _1 + 2 ) / 4 \) (b5 + b4 + 2*b1 + 2) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{9} + \beta_{6} + 2\beta_{5} + 9\beta_{4} ) / 4 \) (b9 + b6 + 2b5 + 9b4) / 4
\(\nu^{3}\)	\(=\)	\( ( \beta_{15} - \beta_{14} - 2 \beta_{13} + 2 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} - \beta_{8} + \cdots - 44 ) / 8 \) (b15 - b14 - 2b13 + 2b11 + 2b10 + 3b9 - b8 + 3b6 + 15b5 + 25b4 + 6b2 - 30*b1 - 44) / 8
\(\nu^{4}\)	\(=\)	\( ( -2\beta_{13} + 2\beta_{11} + 2\beta_{10} + 2\beta_{3} + 11\beta_{2} - 26\beta _1 - 60 ) / 2 \) (-2b13 + 2b11 + 2b10 + 2b3 + 11b2 - 26b1 - 60) / 2
\(\nu^{5}\)	\(=\)	\( ( - 13 \beta_{15} + 15 \beta_{14} - 30 \beta_{13} - 10 \beta_{12} + 26 \beta_{11} + 34 \beta_{10} + \cdots - 456 ) / 8 \) (-13b15 + 15b14 - 30b13 - 10b12 + 26b11 + 34b10 - 45b9 + 19b8 - 75b6 - 145b5 - 283b4 + 20b3 + 90b2 - 286b1 - 456) / 8
\(\nu^{6}\)	\(=\)	\( ( - 29 \beta_{15} + 35 \beta_{14} - 40 \beta_{12} - 121 \beta_{9} + 47 \beta_{8} - 249 \beta_{6} + \cdots - 731 \beta_{4} ) / 4 \) (-29b15 + 35b14 - 40b12 - 121b9 + 47b8 - 249b6 - 309b5 - 731b4) / 4
\(\nu^{7}\)	\(=\)	\( ( - 75 \beta_{15} + 98 \beta_{14} + 198 \beta_{13} - 105 \beta_{12} - 150 \beta_{11} - 248 \beta_{10} + \cdots + 2466 ) / 4 \) (-75b15 + 98b14 + 198b13 - 105b12 - 150b11 - 248b10 - 273b9 + 148b8 + 14b7 - 616b6 - 782b5 - 1625b4 - 224b3 - 546b2 + 1516*b1 + 2466) / 4
\(\nu^{8}\)	\(=\)	\( ( 484\beta_{13} - 348\beta_{11} - 628\beta_{10} + 58\beta_{7} - 678\beta_{3} - 1355\beta_{2} + 3468\beta _1 + 5942 ) / 2 \) (484b13 - 348b11 - 628b10 + 58b7 - 678b3 - 1355b2 + 3468*b1 + 5942) / 2
\(\nu^{9}\)	\(=\)	\( ( 1703 \beta_{15} - 2451 \beta_{14} + 5034 \beta_{13} + 3228 \beta_{12} - 3406 \beta_{11} - 6794 \beta_{10} + \cdots + 54748 ) / 8 \) (1703b15 - 2451b14 + 5034b13 + 3228b12 - 3406b11 - 6794b10 + 6351b9 - 4211b8 + 708b7 + 17451b6 + 17677b5 + 37661b4 - 7164b3 - 12702b2 + 33726*b1 + 54748) / 8
\(\nu^{10}\)	\(=\)	\( ( 4027 \beta_{15} - 6015 \beta_{14} + 8650 \beta_{12} + 15420 \beta_{9} - 10831 \beta_{8} + \cdots + 91776 \beta_{4} ) / 4 \) (4027b15 - 6015b14 + 8650b12 + 15420b9 - 10831b8 + 45730b6 + 41985b5 + 91776b4) / 4
\(\nu^{11}\)	\(=\)	\( ( 19447 \beta_{15} - 30085 \beta_{14} - 62770 \beta_{13} + 44330 \beta_{12} + 38894 \beta_{11} + \cdots - 618504 ) / 8 \) (19447b15 - 30085b14 - 62770b13 + 44330b12 + 38894b11 + 89246b10 + 73513b9 - 56561b8 - 12232b7 + 230967b6 + 204091b5 + 439327b4 + 100892b3 + 147026b2 - 384306*b1 - 618504) / 8
\(\nu^{12}\)	\(=\)	\( - 38836 \beta_{13} + 23240 \beta_{11} + 56286 \beta_{10} - 8473 \beta_{7} + 65538 \beta_{3} + \cdots - 370595 \) -38836b13 + 23240b11 + 56286b10 - 8473b7 + 65538b3 + 88848b2 - 229854*b1 - 370595
\(\nu^{13}\)	\(=\)	\( ( - 112088 \beta_{15} + 182895 \beta_{14} - 386444 \beta_{13} - 288535 \beta_{12} + 224176 \beta_{11} + \cdots - 3540210 ) / 4 \) (-112088b15 + 182895b14 - 386444b13 - 288535b12 + 224176b11 + 569366b10 - 427258b9 + 365817b8 - 90454b7 - 1473771b6 - 1191361b5 - 2576352b4 + 667524b3 + 854516b2 - 2220454*b1 - 3540210) / 4
\(\nu^{14}\)	\(=\)	\( ( - 539346 \beta_{15} + 898688 \beta_{14} - 1456766 \beta_{12} - 2066971 \beta_{9} + \cdots - 12514365 \beta_{4} ) / 4 \) (-539346b15 + 898688b14 - 1456766b12 - 2066971b9 + 1839588b8 - 7391173b6 - 5765280b5 - 12514365b4) / 4
\(\nu^{15}\)	\(=\)	\( ( - 2606317 \beta_{15} + 4422425 \beta_{14} + 9432910 \beta_{13} - 7292660 \beta_{12} - 5212634 \beta_{11} + \cdots + 81892508 ) / 8 \) (-2606317b15 + 4422425b14 + 9432910b13 - 7292660b12 - 5212634b11 - 14241246b10 - 9989635b9 + 9230761b8 + 2469860b7 - 36807335b6 - 28004295b5 - 60698409b4 - 17055180b3 - 19979270b2 + 51788314*b1 + 81892508) / 8

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 274.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} - 30 T^{6} + \cdots + 895)^{2} \) (T^8 - 30T^6 + 280T^4 - 890*T^2 + 895)^2
$3$	\( (T^{8} + 34 T^{6} + \cdots + 16)^{2} \) (T^8 + 34T^6 + 361T^4 + 1224*T^2 + 16)^2
$5$	\( T^{16} \) T^16
$7$	\( (T^{8} - 110 T^{6} + \cdots + 358000)^{2} \) (T^8 - 110T^6 + 4225T^4 - 66500*T^2 + 358000)^2
$11$	\( (T^{8} - 8 T^{7} + \cdots + 214358881)^{2} \) (T^8 - 8T^7 + 48T^6 - 1816T^5 + 23870T^4 - 219736T^3 + 702768T^2 - 14172488*T + 214358881)^2
$13$	\( (T^{8} - 620 T^{6} + \cdots + 27723520)^{2} \) (T^8 - 620T^6 + 109600T^4 - 4153280*T^2 + 27723520)^2
$17$	\( (T^{8} - 1270 T^{6} + \cdots + 1732720)^{2} \) (T^8 - 1270T^6 + 302185T^4 - 3536100*T^2 + 1732720)^2
$19$	\( (T^{8} + 2090 T^{6} + \cdots + 3931928320)^{2} \) (T^8 + 2090T^6 + 1199785T^4 + 217622960*T^2 + 3931928320)^2
$23$	\( (T^{8} + 1304 T^{6} + \cdots + 760877056)^{2} \) (T^8 + 1304T^6 + 431376T^4 + 45040384*T^2 + 760877056)^2
$29$	\( (T^{8} + 4850 T^{6} + \cdots + 27723520)^{2} \) (T^8 + 4850T^6 + 5909665T^4 + 404179520*T^2 + 27723520)^2
$31$	\( (T^{4} - 18 T^{3} + \cdots + 178076)^{4} \) (T^4 - 18T^3 - 951T^2 + 12488*T + 178076)^4
$37$	\( (T^{8} + \cdots + 1000408041616)^{2} \) (T^8 + 4794T^6 + 7829281T^4 + 4972958984*T^2 + 1000408041616)^2
$41$	\( (T^{8} + \cdots + 5420634603520)^{2} \) (T^8 + 9520T^6 + 25555200T^4 + 21158625280*T^2 + 5420634603520)^2
$43$	\( (T^{8} - 4180 T^{6} + \cdots + 91648000)^{2} \) (T^8 - 4180T^6 + 593200T^4 - 20512000*T^2 + 91648000)^2
$47$	\( (T^{8} + 1784 T^{6} + \cdots + 498896896)^{2} \) (T^8 + 1784T^6 + 766096T^4 + 95800064*T^2 + 498896896)^2
$53$	\( (T^{8} + \cdots + 198845082347536)^{2} \) (T^8 + 19114T^6 + 121691761T^4 + 292920887464*T^2 + 198845082347536)^2
$59$	\( (T^{4} - 8 T^{3} + \cdots - 465344)^{4} \) (T^4 - 8T^3 - 8316T^2 - 279472*T - 465344)^4
$61$	\( (T^{8} + \cdots + 44785710872320)^{2} \) (T^8 + 17650T^6 + 79305745T^4 + 120983959920*T^2 + 44785710872320)^2
$67$	\( (T^{8} + \cdots + 12232198471936)^{2} \) (T^8 + 23504T^6 + 149240416T^4 + 162396781824*T^2 + 12232198471936)^2
$71$	\( (T^{4} - 138 T^{3} + \cdots - 22924)^{4} \) (T^4 - 138T^3 + 3169T^2 - 14172*T - 22924)^4
$73$	\( (T^{8} + \cdots + 15327220913920)^{2} \) (T^8 - 13900T^6 + 45636000T^4 - 48397789120*T^2 + 15327220913920)^2
$79$	\( (T^{8} + \cdots + 67909027102720)^{2} \) (T^8 + 27760T^6 + 226274560T^4 + 453908807680*T^2 + 67909027102720)^2
$83$	\( (T^{8} + \cdots + 26187838846720)^{2} \) (T^8 - 30700T^6 + 259522080T^4 - 400109149120*T^2 + 26187838846720)^2
$89$	\( (T^{4} + 222 T^{3} + \cdots - 1268884)^{4} \) (T^4 + 222T^3 + 16109T^2 + 364488*T - 1268884)^4
$97$	\( (T^{8} + \cdots + 444790799524096)^{2} \) (T^8 + 38864T^6 + 379489376T^4 + 1091635473664*T^2 + 444790799524096)^2
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\(n\)	\(101\)	\(177\)
\(\chi(n)\)	\(-1\)	\(-1\)