LMFDB - Newform orbit 1470.3.f.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1470,3,Mod(391,1470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1470, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1470.391");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1470.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$40.0545988610$
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} + 92 x^{14} - 112 x^{13} + 5846 x^{12} - 7728 x^{11} + 197216 x^{10} - 298200 x^{9} + \cdots + 101626561 $ x^16 + 92x^14 - 112x^13 + 5846x^12 - 7728x^11 + 197216x^10 - 298200x^9 + 4836403x^8 - 6808704x^7 + 64376800x^6 - 91953512x^5 + 595763862x^4 - 630430976x^3 + 1087013404x^2 + 294123256x + 101626561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 210)
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} q^{3} + 2 q^{4} - \beta_{7} q^{5} - \beta_{8} q^{6} + 2 \beta_1 q^{8} - 3 q^{9}+O(q^{10}) $ q + b1 * q^2 + b6 * q^3 + 2 * q^4 - b7 * q^5 - b8 * q^6 + 2b1 q^8 - 3 * q^9	$ q + \beta_1 q^{2} + \beta_{6} q^{3} + 2 q^{4} - \beta_{7} q^{5} - \beta_{8} q^{6} + 2 \beta_1 q^{8} - 3 q^{9} + \beta_{10} q^{10} + ( - \beta_{12} + \beta_{2} - \beta_1 + 1) q^{11} + 2 \beta_{6} q^{12} + ( - \beta_{9} + 2 \beta_{6}) q^{13} - \beta_{2} q^{15} + 4 q^{16} + (5 \beta_{10} + \beta_{7} + \beta_{4}) q^{17} - 3 \beta_1 q^{18} + (\beta_{14} - 2 \beta_{8} + \cdots - 3 \beta_{6}) q^{19}+ \cdots + (3 \beta_{12} - 3 \beta_{2} + 3 \beta_1 - 3) q^{99}+O(q^{100}) $ q + b1 * q^2 + b6 * q^3 + 2 * q^4 - b7 * q^5 - b8 * q^6 + 2b1 q^8 - 3 * q^9 + b10 * q^10 + (-b12 + b2 - b1 + 1) * q^11 + 2b6 q^12 + (-b9 + 2b6) q^13 - b2 * q^15 + 4 * q^16 + (5b10 + b7 + b4) q^17 - 3b1 q^18 + (b14 - 2b8 - 5b7 - 3b6) q^19 - 2b7 q^20 + (b13 + b3 + b1 - 3) * q^22 + (-b13 + b12 + b11 - b3 - b2 - b1 + 1) * q^23 - 2b8 q^24 - 5 * q^25 + (b10 + b9 - b8 - b7 - b6 - b4) * q^26 - 3b6 q^27 + (b12 - b11 + b5 - 6b2 + 5b1 + 4) * q^29 - b3 * q^30 + (7b10 + b9 - b7 - 6b6 - b4) * q^31 + 4b1 q^32 + (-b14 - b9 + 2b8 - b7 + b6) q^33 + (-b10 + b9 - 11b7 - b6 + b4) q^34 - 6 * q^36 + (b13 + b12 - b11 - b5 - 5b3 - 2b2 + 6b1 - 6) q^37 + (-b15 + 5b10 + 3b8 - b7 + 3b6) q^38 + (b12 + b11 - b2 - 5) * q^39 + 2b10 q^40 + (b15 - b14 + 3b10 + b9 - 4b8 - 2b7 - 4b6 - 2b4) q^41 + (-2b13 - b12 + b11 - b5 - 2b3 + 5b1 - 3) q^43 + (-2b12 + 2b2 - 2b1 + 2) q^44 + 3b7 q^45 + (-b13 + 2b12 - b11 - b5 - b3 - 3b2 - b1 - 2) * q^46 + (12b10 + 2b9 - 6b7 - 2b6) * q^47 + 4b6 q^48 - 5b1 q^50 + (b13 - b12 + b5 - 5b3 + 2b2 + 3b1 + 2) q^51 + (-2b9 + 4b6) * q^52 + (2b13 - 2b11 + 2b3 - 14b1 - 4) * q^53 + 3b8 q^54 + (-b15 - b10 + b9 - b7 + 2b6) q^55 + (-2b12 + b11 - 4b2 - 3b1 + 10) q^57 + (-b13 + 2b5 - 7b3 + 2b2 + 5b1 + 11) * q^58 + (2b15 - 3b14 + 14b10 - b9 - 6b8 + 17b7 - 5b6) * q^59 - 2b2 q^60 + (-b14 + 9b10 - 3b9 - 4b8 - 4b7 + 7b6 + 3b4) * q^61 + (-2b9 + 5b8 - 12b7 + 2b6) * q^62 + 8 * q^64 + (-b12 - b11 - b5 - b2 - 3b1 - 2) q^65 + (b15 + 2b10 + b9 - 4b6 - b4) * q^66 + (-b13 + b12 + b11 + 2b5 - b3 + b2 + 29b1 + 20) * q^67 + (10b10 + 2b7 + 2b4) q^68 + (-b15 - 2b10 + 2b9 + b6 + b4) * q^69 + (2b13 + b12 - 2b5 + 2b3 - 9b2 - 3b1 - 9) q^71 - 6b1 q^72 + (3b14 + 7b10 + 2b9 + 12b8 + 2b7 + 13b6 - b4) * q^73 + (-b13 - 2b12 + 2b11 - b3 - 10b2 - 4b1 + 15) * q^74 - 5b6 q^75 + (2b14 - 4b8 - 10b7 - 6b6) * q^76 + (-b13 - b11 - b5 - b3 - b2 - 6b1) q^78 + (2b13 - 2b12 - b11 - 10b3 - 4b2 + b1 - 34) * q^79 - 4b7 q^80 + 9 * q^81 + (b15 - 2b14 + 2b10 - 3b9 + 4b8 - 2b7 + 12b6 - b4) * q^82 + (-b15 + 2b14 - 4b10 - 4b9 - 4b8 + 4b7 + 13b6 + 3b4) q^83 + (-b13 + b12 - b11 + b5 - b3 - 26b1 + 7) q^85 + (b13 + 4b12 - 2b5 + b3 - 6b2 - 6b1 + 9) * q^86 + (-b15 + 3b14 + b10 - b9 - 6b8 + 18b7 + 4b6 - 2b4) q^87 + (2b13 + 2b3 + 2b1 - 6) q^88 + (-b15 + 3b14 + 25b10 - 3b9 + 12b8 - 12b6 - 2b4) * q^89 - 3b10 q^90 + (-2b13 + 2b12 + 2b11 - 2b3 - 2b2 - 2b1 + 2) * q^92 + (-b13 - b11 - b5 - 7b3 - b2 - 3b1 + 15) * q^93 + (4b10 - 2b9 - 22b7 + 2b6 + 2b4) q^94 + (-2b13 + b11 - 2b3 + 3b2 - 3b1 - 25) * q^95 - 4b8 q^96 + (b14 - 17b10 + 3b9 + 28b8 - 13b6 - b4) * q^97 + (3b12 - 3b2 + 3b1 - 3) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4} - 48 q^{9}+O(q^{10}) $ 16 * q + 32 * q^4 - 48 * q^9	$ 16 q + 32 q^{4} - 48 q^{9} + 8 q^{11} + 64 q^{16} - 48 q^{22} + 24 q^{23} - 80 q^{25} + 72 q^{29} - 96 q^{36} - 88 q^{37} - 72 q^{39} - 56 q^{43} + 16 q^{44} - 16 q^{46} + 24 q^{51} - 64 q^{53} + 144 q^{57} + 176 q^{58} + 128 q^{64} - 40 q^{65} + 328 q^{67} - 136 q^{71} + 224 q^{74} - 560 q^{79} + 144 q^{81} + 120 q^{85} + 176 q^{86} - 96 q^{88} + 48 q^{92} + 240 q^{93} - 400 q^{95} - 24 q^{99}+O(q^{100}) $ 16 * q + 32 * q^4 - 48 * q^9 + 8 * q^11 + 64 * q^16 - 48 * q^22 + 24 * q^23 - 80 * q^25 + 72 * q^29 - 96 * q^36 - 88 * q^37 - 72 * q^39 - 56 * q^43 + 16 * q^44 - 16 * q^46 + 24 * q^51 - 64 * q^53 + 144 * q^57 + 176 * q^58 + 128 * q^64 - 40 * q^65 + 328 * q^67 - 136 * q^71 + 224 * q^74 - 560 * q^79 + 144 * q^81 + 120 * q^85 + 176 * q^86 - 96 * q^88 + 48 * q^92 + 240 * q^93 - 400 * q^95 - 24 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 92 x^{14} - 112 x^{13} + 5846 x^{12} - 7728 x^{11} + 197216 x^{10} - 298200 x^{9} + \cdots + 101626561 $ x^16 + 92*x^14 - 112*x^13 + 5846*x^12 - 7728*x^11 + 197216*x^10 - 298200*x^9 + 4836403*x^8 - 6808704*x^7 + 64376800*x^6 - 91953512*x^5 + 595763862*x^4 - 630430976*x^3 + 1087013404*x^2 + 294123256*x + 101626561 :

$\beta_{1}$	$=$	$ ( 57\!\cdots\!49 \nu^{15} + \cdots - 12\!\cdots\!09 ) / 85\!\cdots\!24 $ (57754008957283349v^15 + 676562680999836294v^14 + 8239103585403158159v^13 + 54078590776251682475v^12 + 455975368108340251668v^11 + 2793391292439344623383v^10 + 16545566601055515429559v^9 + 76261143494901216401580v^8 + 306572240997688184788012v^7 + 1467974756910134628134465v^6 + 3497231496721962844392765v^5 + 10591214719874231223793348v^4 - 4685579583089306764764583v^3 + 35874681437029057265441973v^2 + 9955887688266404429946106*v - 1202728311133779179554909109) / 852577600778545978990177524
$\beta_{2}$	$=$	$ ( 6718070224756 \nu^{15} - 35399436274870 \nu^{14} + 824042900206056 \nu^{13} + \cdots + 20\!\cdots\!64 ) / 53\!\cdots\!38 $ (6718070224756v^15 - 35399436274870v^14 + 824042900206056v^13 - 3468795000369922v^12 + 55447463307528888v^11 - 237157576567025547v^10 + 2160971883833844512v^9 - 7103633821675122284v^8 + 46638150914308904520v^7 - 152233236071607452553v^6 + 595135449699402838792v^5 - 1018214320678792586196v^4 + 1279768421429813362272v^3 - 459537667512276799289v^2 - 103279705437214175760*v + 206657596524136518657064) / 53375543816937984206838
$\beta_{3}$	$=$	$ ( 3871767397361 \nu^{15} + 79093222616710 \nu^{14} + 341679513333957 \nu^{13} + \cdots - 14\!\cdots\!75 ) / 27\!\cdots\!92 $ (3871767397361v^15 + 79093222616710v^14 + 341679513333957v^13 + 5895790087314937v^12 + 14035722749452236v^11 + 371621017202953791v^10 + 356604274247547817v^9 + 10456974871241558252v^8 + 4802805224198506740v^7 + 233091302320357629453v^6 + 23538017273856016799v^5 + 1467948791527369196388v^4 - 940733060256762782559v^3 + 3491995997388483650717v^2 + 957085277114840834430*v - 148193606010062737313275) / 27111518972679309429492
$\beta_{4}$	$=$	$ ( - 24\!\cdots\!58 \nu^{15} + \cdots - 49\!\cdots\!51 ) / 95\!\cdots\!44 $ (-241863822777540874882466993018758v^15 + 213469975304718887115010020250304v^14 - 23706868915498534505262085799802764v^13 + 33346620725345705792292386908536195v^12 - 1572246595585425326053498700187612372v^11 + 2238397531499978911345875836894047377v^10 - 56379450841845268013992185571629193172v^9 + 65194802114865894964792930629943731574v^8 - 1438089633101304793073636413061634509524v^7 + 1562153171319216137200389970915623241819v^6 - 21102866749248222462536154496131177234044v^5 + 16480923200844042141095313649474586060138v^4 - 199611409492269652994318983181462690900774v^3 + 235625405974736320182385020984337601858401v^2 - 450294361739753268802611711172967452983904*v - 49793140462485741740564019667302022211751) / 95852909214978668011338765982386839392644
$\beta_{5}$	$=$	$ ( - 15\!\cdots\!15 \nu^{15} + \cdots - 23\!\cdots\!33 ) / 31\!\cdots\!08 $ (-15676746441665707101942164015v^15 + 21207121467610615912805559174v^14 - 1369703339442836424119507090009v^13 + 3487555436126339212562826767107v^12 - 88569529078180963996479471688716v^11 + 205591363940722638333739187297079v^10 - 2919190798664141627791804403165965v^9 + 6428727908938342577118706461520812v^8 - 71687408890868806747434561888203188v^7 + 115859697731261714424380120491192081v^6 - 899631755687316031497385829078742651v^5 + 933885797263311750157099981268214596v^4 - 8608546324615197810872297777006685983v^3 - 714070897931272908462439708632495915v^2 - 229621811151920173894439902818460534*v - 23442301013525123996598271073372839033) / 3169424634294834110747570214012724908
$\beta_{6}$	$=$	$ ( 34\!\cdots\!04 \nu^{15} + \cdots + 35\!\cdots\!31 ) / 62\!\cdots\!01 $ (34800704278153535382807509504v^15 - 10894314337506543053670349736v^14 + 3184797771932765844822003594240v^13 - 4883872557229584935529507311852v^12 + 203150684489855051057790801147168v^11 - 329641444475618185980178937295216v^10 + 6850224385630247456211484006725040v^9 - 12335700651762730044495715881145318v^8 + 168345816581855321121206226303993216v^7 - 283581678836521920106992554468810640v^6 + 2236415316817892637807058734013008848v^5 - 3771284369594629070382322847180181060v^4 + 20757636283402814726509997166646534848v^3 - 27545780009134355888225371127509524184v^2 + 36052656915764843531341394646112567344*v + 3521207262416884848433762658587277531) / 6206482078151946905681090778450326301
$\beta_{7}$	$=$	$ ( - 536662839109972 \nu^{15} + 184355843936000 \nu^{14} + \cdots - 55\!\cdots\!08 ) / 75\!\cdots\!34 $ (-536662839109972v^15 + 184355843936000v^14 - 49316294058364856v^13 + 76985389887345138v^12 - 3154550976828630456v^11 + 5211291917675260305v^10 - 106964341125832723328v^9 + 195705915902593012444v^8 - 2639716566132467353984v^7 + 4517022991968883377451v^6 - 35333929028214918917480v^5 + 60517159199633559261440v^4 - 328552596355156909606256v^3 + 436496862559525516173889v^2 - 567954327233848070808304*v - 55329661987849902130608) / 75811879080713475788634
$\beta_{8}$	$=$	$ ( 2130173036201 \nu^{15} - 540413699312 \nu^{14} + 193661918213181 \nu^{13} + \cdots + 20\!\cdots\!93 ) / 25\!\cdots\!08 $ (2130173036201v^15 - 540413699312v^14 + 193661918213181v^13 - 293621077625441v^12 + 12285060311199138v^11 - 19810557668746935v^10 + 410965646117486839v^9 - 746713194852137098v^8 + 10036111646725656126v^7 - 17105922536607134721v^6 + 131897626701278295101v^5 - 228882972939831674094v^4 + 1220130262910161491531v^3 - 1620423665127954679375v^2 + 2091806790152256573456*v + 203021156853115775993) / 255239865906581664108
$\beta_{9}$	$=$	$ ( 27\!\cdots\!01 \nu^{15} + \cdots + 31\!\cdots\!96 ) / 31\!\cdots\!48 $ (273267982442506027837028536846301v^15 - 90824115373216270012966076556376v^14 + 24724502366652215915576790007891357v^13 - 35765374599523411150807746572903628v^12 + 1582514521052192986730127030508629594v^11 - 2345134475522779658037774090424825002v^10 + 52509790850976840228722541081735640975v^9 - 85155157364026490810922764150475232928v^8 + 1282216420561340630964640398485100256046v^7 - 1931951755527215176068822620639346594878v^6 + 16400400933464541644568104022630034167277v^5 - 25462330763896727659919529802689863003296v^4 + 153102783374700304960302234551110364381923v^3 - 184555318397999592675046544109329481337922v^2 + 298517316090639541206036201872067943971200*v + 31351166733895176216491140415459948534796) / 31950969738326222670446255327462279797548
$\beta_{10}$	$=$	$ ( - 44\!\cdots\!15 \nu^{15} + \cdots - 43\!\cdots\!65 ) / 42\!\cdots\!36 $ (-4448842632361015v^15 + 1527746136142088v^14 - 406573176070056149v^13 + 638852366768395737v^12 - 25956155317369110414v^11 + 43002282249586977093v^10 - 874597310627211904019v^9 + 1616020902079177673194v^8 - 21496564169263011322522v^7 + 37198980049890495166591v^6 - 284874104240500582055717v^5 + 500982390617084205830630v^4 - 2644373246682812474960417v^3 + 3517427264905839032095693v^2 - 4518691295624378534050984*v - 437660350242222703221165) / 427449074136574093101036
$\beta_{11}$	$=$	$ ( - 13\!\cdots\!03 \nu^{15} + \cdots - 91\!\cdots\!03 ) / 95\!\cdots\!24 $ (-139899057247192626960828150203v^15 - 205984038615707634647717699362v^14 - 12845894306918859627747662840373v^13 - 2722805495207821537930036165291v^12 - 791993126949886351938458511046812v^11 - 100975119829351085447600383013997v^10 - 25937879554799122322336137198622449v^9 + 2424669373205808452677178694538204v^8 - 609568656712704554220138175959467556v^7 - 17681141653075112811106269800634411v^6 - 7473449288344881015343436020856745167v^5 + 597458333330310777396600761626775892v^4 - 58619731313164471563160626236307114531v^3 - 22597125487299789997563421409019516695v^2 - 6470850633222513601969200095438569758*v - 91876261549562488474322233688611891103) / 9508273902884502332242710642038174724
$\beta_{12}$	$=$	$ ( 50\!\cdots\!76 \nu^{15} + \cdots + 54\!\cdots\!09 ) / 31\!\cdots\!08 $ (50541445469745635702521407476v^15 + 88189787602355302125069537998v^14 + 4686121529079582552930273937852v^13 + 2344084363837616559345483004641v^12 + 287669320037743266070478689305528v^11 + 102227182237608941304707315817543v^10 + 9383188325428496653322675718828448v^9 + 980369731111473431671427903471644v^8 + 220345273133399733933805748083108424v^7 + 33350653108110624529048868648542321v^6 + 2695217965796171147046350321069201188v^5 + 45053311281652415642974882019971172v^4 + 22564363014998968270435084775820227404v^3 + 8755731051360650982019965319468659565v^2 + 2499667994650377956679074524685699432*v + 54539131317226042274521458166636701909) / 3169424634294834110747570214012724908
$\beta_{13}$	$=$	$ ( 35\!\cdots\!52 \nu^{15} + \cdots + 40\!\cdots\!54 ) / 15\!\cdots\!54 $ (35618621022542461412963550752v^15 + 57402549257930764143801253636v^14 + 3269452438379247383903820031191v^13 + 1195103257893557548048710310882v^12 + 201097599940690016735522351078652v^11 + 50642128087521428839734928726662v^10 + 6545045094796181211847908243536239v^9 + 94213946227552098355377432246440v^8 + 154371956973564180787294432103580228v^7 + 13527137712560061152587689973211136v^6 + 1890245428081527900140741520716056325v^5 - 51743176237912170133092018183094008v^4 + 16011464587952105898188873235352187640v^3 + 5946720044570265499150836888229441082v^2 + 1699987857187158300731555890791555114*v + 40128729606389940427563972548234469854) / 1584712317147417055373785107006362454
$\beta_{14}$	$=$	$ ( 15\!\cdots\!08 \nu^{15} + \cdots + 24\!\cdots\!97 ) / 23\!\cdots\!61 $ (1524687532650925566384058512983608v^15 - 45409554837746644596280025048156v^14 + 141488245005809006513245332039530819v^13 - 172571033238879105571543248620219622v^12 + 9030723326987672595044485228224009498v^11 - 11987810186592915492952403117919938364v^10 + 307821302729523772843844588129480487590v^9 - 461578234521407040548049640832600487082v^8 + 7604888710910270137183375270238308341856v^7 - 10627157121886052664993396991945530374686v^6 + 103489208761170799566591481949888616327575v^5 - 144115666979851304912358383250150995962136v^4 + 971695144433049417614826274880961599060141v^3 - 1025012991098740975794373087499920956372100v^2 + 2150637685521930692819305889555961803356510*v + 241226071476552345162168621818414428421697) / 23963227303744667002834691495596709848161
$\beta_{15}$	$=$	$ ( 96\!\cdots\!17 \nu^{15} + \cdots + 14\!\cdots\!41 ) / 10\!\cdots\!52 $ (967177237679128037761186849117v^15 - 1952043673575989613570267656v^14 + 89527070939922271046261982843611v^13 - 109190857744315535774155928018049v^12 + 5700740767634372766157484444374170v^11 - 7597080890305333130374154566891551v^10 + 193818155261377055998617002044149413v^9 - 294775183761668393523411707373797314v^8 + 4779142678687522109036057214533659918v^7 - 6777773520474492153092935877383356565v^6 + 64784183508077896482320219799719599595v^5 - 92294600198557773363013468818806213150v^4 + 607562677390245683660668103797050433139v^3 - 644931194393471260333156135024330484311v^2 + 1331955896632363538983196299482134343112*v + 148851527006263660404650371701627502341) / 10365838565478389532966234019940179452

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

$\nu$	$=$	$ ( 4 \beta_{15} - 5 \beta_{14} - 3 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + 5 \beta_{10} + 5 \beta_{9} + \cdots - 1 ) / 28 $ (4b15 - 5b14 - 3b13 + 2b12 - 2b11 + 5b10 + 5b9 - 4b8 - 2b7 - 3b6 + b5 - b4 - 3b3 - b2 - 10b1 - 1) / 28
$\nu^{2}$	$=$	$ ( - 5 \beta_{15} + 8 \beta_{14} - 2 \beta_{13} + 6 \beta_{12} + \beta_{11} - \beta_{10} + 6 \beta_{9} + \cdots - 325 ) / 28 $ (-5b15 + 8b14 - 2b13 + 6b12 + b11 - b10 + 6b9 + 96b8 - 127b7 - 327b6 + 3b5 - 4b4 - 2b3 + 39b2 - 100*b1 - 325) / 28
$\nu^{3}$	$=$	$ ( 17\beta_{13} - 18\beta_{12} + 8\beta_{11} - 3\beta_{5} + 5\beta_{3} + 9\beta_{2} + 88\beta _1 + 51 ) / 2 $ (17b13 - 18b12 + 8b11 - 3b5 + 5b3 + 9b2 + 88*b1 + 51) / 2
$\nu^{4}$	$=$	$ ( 198 \beta_{15} - 272 \beta_{14} - 124 \beta_{13} + 232 \beta_{12} + 34 \beta_{11} - 834 \beta_{10} + \cdots - 5765 ) / 14 $ (198b15 - 272b14 - 124b13 + 232b12 + 34b11 - 834b10 - 176b9 - 2228b8 + 3100b7 + 5749b6 + 74b5 + 24b4 + 212b3 + 892b2 - 2476*b1 - 5765) / 14
$\nu^{5}$	$=$	$ ( - 5990 \beta_{15} + 8667 \beta_{14} - 5413 \beta_{13} + 6530 \beta_{12} - 1560 \beta_{11} + \cdots - 25805 ) / 28 $ (-5990b15 + 8667b14 - 5413b13 + 6530b12 - 1560b11 + 9155b10 - 2003b9 + 20200b8 - 9948b7 - 19409b6 + 577b5 + 4259b4 + 1055b3 - 325b2 - 33126*b1 - 25805) / 28
$\nu^{6}$	$=$	$ ( 2622 \beta_{13} - 4310 \beta_{12} - 373 \beta_{11} - 811 \beta_{5} - 4770 \beta_{3} - 11579 \beta_{2} + \cdots + 67649 ) / 2 $ (2622b13 - 4310b12 - 373b11 - 811b5 - 4770b3 - 11579b2 + 35956*b1 + 67649) / 2
$\nu^{7}$	$=$	$ ( 279262 \beta_{15} - 400797 \beta_{14} - 259745 \beta_{13} + 334798 \beta_{12} - 46482 \beta_{11} + \cdots - 1632009 ) / 28 $ (279262b15 - 400797b14 - 259745b13 + 334798b12 - 46482b11 - 699547b10 + 17911b9 - 1109420b8 + 965760b7 + 1345299b6 + 19517b5 - 220711b4 + 140095b3 + 120721b2 - 1730928*b1 - 1632009) / 28
$\nu^{8}$	$=$	$ ( - 691284 \beta_{15} + 964464 \beta_{14} - 585096 \beta_{13} + 900336 \beta_{12} + 42060 \beta_{11} + \cdots - 10697503 ) / 14 $ (-691284b15 + 964464b14 - 585096b13 + 900336b12 + 42060b11 + 3656460b10 + 399360b9 + 5207880b8 - 7456656b7 - 10260655b6 + 106188b5 + 372720b4 + 988392b3 + 1906704b2 - 6545064*b1 - 10697503) / 14
$\nu^{9}$	$=$	$ ( 1846509 \beta_{13} - 2471702 \beta_{12} + 211658 \beta_{11} - 112891 \beta_{5} - 1436811 \beta_{3} + \cdots + 13889443 ) / 2 $ (1846509b13 - 2471702b12 + 211658b11 - 112891b5 - 1436811b3 - 1653581b2 + 13011958*b1 + 13889443) / 2
$\nu^{10}$	$=$	$ ( 77787413 \beta_{15} - 109133120 \beta_{14} - 69460730 \beta_{13} + 103071894 \beta_{12} + \cdots - 1030326949 ) / 28 $ (77787413b15 - 109133120b14 - 69460730b13 + 103071894b12 + 2265457b11 - 403264199b10 - 38142078b9 - 527802720b8 + 757264471b7 + 971458359b6 + 8326683b5 - 52807364b4 + 108492262b3 + 185727303b2 - 689743204*b1 - 1030326949) / 28
$\nu^{11}$	$=$	$ ( - 696502112 \beta_{15} + 990432227 \beta_{14} - 660025487 \beta_{13} + 903385734 \beta_{12} + \cdots - 5586803001 ) / 28 $ (-696502112b15 + 990432227b14 - 660025487b13 + 903385734b12 - 50569868b11 + 2558370023b10 + 142220511b9 + 3268336104b8 - 4115827654b7 - 4912684271b6 + 36476625b5 + 587072237b4 + 620622637b3 + 798700893b2 - 4845840568*b1 - 5586803001) / 28
$\nu^{12}$	$=$	$ 283303276 \beta_{13} - 411884296 \beta_{12} - 2844970 \beta_{11} - 25153442 \beta_{5} + \cdots + 3695031179 $ 283303276b13 - 411884296b12 - 2844970b11 - 25153442b5 - 415417460b3 - 664916932b2 + 2616526588*b1 + 3695031179
$\nu^{13}$	$=$	$ ( 36136952786 \beta_{15} - 51274366995 \beta_{14} - 34314451069 \beta_{13} + 47593238282 \beta_{12} + \cdots - 313758094217 ) / 28 $ (36136952786b15 - 51274366995b14 - 34314451069b13 + 47593238282b12 - 1858626996b11 - 144106197839b10 - 9561533221b9 - 177814191664b8 + 238342892184b7 + 279407517869b6 + 1822501717b5 - 30669447635b4 + 35989748351b3 + 48945848111b2 - 259758006294*b1 - 313758094217) / 28
$\nu^{14}$	$=$	$ ( - 237558015007 \beta_{15} + 335044352924 \beta_{14} - 221473349274 \beta_{13} + 318262664810 \beta_{12} + \cdots - 2667438845879 ) / 28 $ (-237558015007b15 + 335044352924b14 - 221473349274b13 + 318262664810b12 - 697022381b11 + 1167123223825b10 + 95395270774b9 + 1430841773632b8 - 2050689201961b7 - 2461353139957b6 + 16084665733b5 + 189304017808b4 + 309855069606b3 + 476981853485b2 - 1955190144364*b1 - 2667438845879) / 28
$\nu^{15}$	$=$	$ ( 257998257311 \beta_{13} - 360676661674 \beta_{12} + 10562294418 \beta_{11} - 13560958727 \beta_{5} + \cdots + 2480567013939 ) / 2 $ (257998257311b13 - 360676661674b12 + 10562294418b11 - 13560958727b5 - 289138472617b3 - 405091585843b2 + 1997234018400*b1 + 2480567013939) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times$.

$n$	$491$	$1081$	$1177$
$\chi(n)$	$1$	$-1$	$1$

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} $

391.1

−0.141814	−	0.245629i
0.848921	+	1.47037i
−2.10711	−	3.64962i
2.81422	+	4.87437i
−2.10711	+	3.64962i
2.81422	−	4.87437i
−0.141814	+	0.245629i
0.848921	−	1.47037i
−2.63284	−	4.56021i
1.92573	+	3.33546i
2.96377	+	5.13339i
−3.67087	−	6.35814i
2.96377	−	5.13339i
−3.67087	+	6.35814i
−2.63284	+	4.56021i
1.92573	−	3.33546i

−1.41421

−

1.73205i

2.00000

−

2.23607i

2.44949i

−2.82843

−3.00000

3.16228i

391.2

−1.41421

−

1.73205i

2.00000

−

2.23607i

2.44949i

−2.82843

−3.00000

3.16228i

391.3

−1.41421

−

1.73205i

2.00000

2.23607i

2.44949i

−2.82843

−3.00000

−

3.16228i

391.4

−1.41421

−

1.73205i

2.00000

2.23607i

2.44949i

−2.82843

−3.00000

−

3.16228i

391.5

−1.41421

1.73205i

2.00000

−

2.23607i

−

2.44949i

−2.82843

−3.00000

3.16228i

391.6

−1.41421

1.73205i

2.00000

−

2.23607i

−

2.44949i

−2.82843

−3.00000

3.16228i

391.7

−1.41421

1.73205i

2.00000

2.23607i

−

2.44949i

−2.82843

−3.00000

−

3.16228i

391.8

−1.41421

1.73205i

2.00000

2.23607i

−

2.44949i

−2.82843

−3.00000

−

3.16228i

391.9

1.41421

−

1.73205i

2.00000

−

2.23607i

−

2.44949i

2.82843

−3.00000

−

3.16228i

391.10

1.41421

−

1.73205i

2.00000

−

2.23607i

−

2.44949i

2.82843

−3.00000

−

3.16228i

391.11

1.41421

−

1.73205i

2.00000

2.23607i

−

2.44949i

2.82843

−3.00000

3.16228i

391.12

1.41421

−

1.73205i

2.00000

2.23607i

−

2.44949i

2.82843

−3.00000

3.16228i

391.13

1.41421

1.73205i

2.00000

−

2.23607i

2.44949i

2.82843

−3.00000

−

3.16228i

391.14

1.41421

1.73205i

2.00000

−

2.23607i

2.44949i

2.82843

−3.00000

−

3.16228i

391.15

1.41421

1.73205i

2.00000

2.23607i

2.44949i

2.82843

−3.00000

3.16228i

391.16

1.41421

1.73205i

2.00000

2.23607i

2.44949i

2.82843

−3.00000

3.16228i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1470.3.f.d		16
7.b	odd	2	1	inner	1470.3.f.d		16
7.c	even	3	1		210.3.o.b	✓	16
7.d	odd	6	1		210.3.o.b	✓	16
21.g	even	6	1		630.3.v.c		16
21.h	odd	6	1		630.3.v.c		16
35.i	odd	6	1		1050.3.p.i		16
35.j	even	6	1		1050.3.p.i		16
35.k	even	12	2		1050.3.q.e		32
35.l	odd	12	2		1050.3.q.e		32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
210.3.o.b	✓	16	7.c	even	3	1
210.3.o.b	✓	16	7.d	odd	6	1
630.3.v.c		16	21.g	even	6	1
630.3.v.c		16	21.h	odd	6	1
1050.3.p.i		16	35.i	odd	6	1
1050.3.p.i		16	35.j	even	6	1
1050.3.q.e		32	35.k	even	12	2
1050.3.q.e		32	35.l	odd	12	2
1470.3.f.d		16	1.a	even	1	1	trivial
1470.3.f.d		16	7.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{11}^{8} - 4 T_{11}^{7} - 588 T_{11}^{6} + 1352 T_{11}^{5} + 99088 T_{11}^{4} - 127344 T_{11}^{3} + \cdots + 3111696 $ T11^8 - 4*T11^7 - 588*T11^6 + 1352*T11^5 + 99088*T11^4 - 127344*T11^3 - 5129712*T11^2 + 3259872*T11 + 3111696 acting on $S_{3}^{\mathrm{new}}(1470, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} - 2)^{8} $ (T^2 - 2)^8
$3$	$ (T^{2} + 3)^{8} $ (T^2 + 3)^8
$5$	$ (T^{2} + 5)^{8} $ (T^2 + 5)^8
$7$	$ T^{16} $ T^16
$11$	$ (T^{8} - 4 T^{7} + \cdots + 3111696)^{2} $ (T^8 - 4T^7 - 588T^6 + 1352T^5 + 99088T^4 - 127344T^3 - 5129712T^2 + 3259872*T + 3111696)^2
$13$	$ T^{16} + \cdots + 17613726190641 $ T^16 + 1368T^14 + 631916T^12 + 121166856T^10 + 9820643542T^8 + 360016963752T^6 + 5708126314764T^4 + 32826023199864*T^2 + 17613726190641
$17$	$ T^{16} + \cdots + 13\!\cdots\!76 $ T^16 + 3928T^14 + 6101232T^12 + 4784547424T^10 + 2017746863776T^8 + 454984069961088T^6 + 50870437865807616T^4 + 2283763344946398720*T^2 + 13699042635136594176
$19$	$ T^{16} + \cdots + 10\!\cdots\!41 $ T^16 + 3336T^14 + 4484108T^12 + 3115002840T^10 + 1183982767030T^8 + 236106754605432T^6 + 20360865271560300T^4 + 290484186498030888*T^2 + 1088507693685749841
$23$	$ (T^{8} - 12 T^{7} + \cdots + 44064727056)^{2} $ (T^8 - 12T^7 - 2284T^6 + 12120T^5 + 1742608T^4 - 2584848T^3 - 510960240T^2 - 176329440*T + 44064727056)^2
$29$	$ (T^{8} - 36 T^{7} + \cdots - 235474074864)^{2} $ (T^8 - 36T^7 - 5284T^6 + 231048T^5 + 5584336T^4 - 368051952T^3 + 4047381936T^2 + 13878381024*T - 235474074864)^2
$31$	$ T^{16} + \cdots + 10\!\cdots\!01 $ T^16 + 6600T^14 + 17957324T^12 + 26107228824T^10 + 21930849063286T^8 + 10661944260564408T^6 + 2784936550544458668T^4 + 304492676211707065704*T^2 + 1054350189971139482001
$37$	$ (T^{8} + \cdots - 3600469848911)^{2} $ (T^8 + 44T^7 - 7060T^6 - 341892T^5 + 13499750T^4 + 757931316T^3 - 4253125684T^2 - 467761445788*T - 3600469848911)^2
$41$	$ T^{16} + \cdots + 26\!\cdots\!96 $ T^16 + 15480T^14 + 91433264T^12 + 258539104224T^10 + 362479500954016T^8 + 240256669708752768T^6 + 61910849899642860288T^4 + 2537276667253552479744*T^2 + 26127497467342672240896
$43$	$ (T^{8} + 28 T^{7} + \cdots + 563400737569)^{2} $ (T^8 + 28T^7 - 9060T^6 - 168116T^5 + 26540822T^4 + 190593380T^3 - 25761062852T^2 + 106450443444*T + 563400737569)^2
$47$	$ T^{16} + \cdots + 10\!\cdots\!16 $ T^16 + 15840T^14 + 98872256T^12 + 311314002432T^10 + 534824656938496T^8 + 522197239921975296T^6 + 289031601087775752192T^4 + 84415991942485643231232*T^2 + 10101978488064049707810816
$53$	$ (T^{8} + 32 T^{7} + \cdots - 110361636864)^{2} $ (T^8 + 32T^7 - 9376T^6 - 247040T^5 + 24835456T^4 + 352598016T^3 - 19146074112T^2 + 105598107648*T - 110361636864)^2
$59$	$ T^{16} + \cdots + 95\!\cdots\!56 $ T^16 + 53800T^14 + 1178179056T^12 + 13442394615328T^10 + 84876758291333536T^8 + 291079798534197684864T^6 + 499911605964802885531392T^4 + 371413100775188382417658368*T^2 + 95659041749658506515012813056
$61$	$ T^{16} + \cdots + 13\!\cdots\!96 $ T^16 + 33504T^14 + 400428608T^12 + 2089220190720T^10 + 4845303856514560T^8 + 4395513661648969728T^6 + 952576928404071137280T^4 + 65997272344738048376832*T^2 + 1392598099630469949751296
$67$	$ (T^{8} + \cdots + 1596721546161)^{2} $ (T^8 - 164T^7 - 2676T^6 + 1129036T^5 - 15599930T^4 - 1195387164T^3 + 39775201164T^2 - 434983961484*T + 1596721546161)^2
$71$	$ (T^{8} + \cdots - 2334473020656)^{2} $ (T^8 + 68T^7 - 14236T^6 - 204488T^5 + 70875760T^4 - 2359994064T^3 + 12080564496T^2 + 365315438880*T - 2334473020656)^2
$73$	$ T^{16} + \cdots + 99\!\cdots\!61 $ T^16 + 40952T^14 + 596855436T^12 + 3948236075048T^10 + 13051748831699830T^8 + 22350077858102322312T^6 + 19300556035717437028716T^4 + 7468937247147188121338328*T^2 + 995049780300698330601190161
$79$	$ (T^{8} + \cdots + 101515662800761)^{2} $ (T^8 + 280T^7 + 13940T^6 - 2479368T^5 - 299642482T^4 - 8650431480T^3 + 139189714436T^2 + 9296406570088*T + 101515662800761)^2
$83$	$ T^{16} + \cdots + 13\!\cdots\!16 $ T^16 + 44040T^14 + 666154928T^12 + 3974136166944T^10 + 7403270734587808T^8 + 2722620180898843776T^6 + 307745806772397964032T^4 + 7581543315742352208384*T^2 + 1312045200775282934016
$89$	$ T^{16} + \cdots + 26\!\cdots\!56 $ T^16 + 107400T^14 + 4691236208T^12 + 106830987964320T^10 + 1349344150954861984T^8 + 9283143638587347607680T^6 + 32015173752969153782038272T^4 + 48633874512679923981639252480*T^2 + 26119520169155990714944934297856
$97$	$ T^{16} + \cdots + 18\!\cdots\!36 $ T^16 + 80096T^14 + 2531519808T^12 + 40307196119552T^10 + 345608150837679616T^8 + 1605606437907707437056T^6 + 3776343128542461157392384T^4 + 3526977654558045584073031680*T^2 + 185911339063854590759980302336
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1470.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + \beta_{6} q^{3} + 2 q^{4} - \beta_{7} q^{5} - \beta_{8} q^{6} + 2 \beta_1 q^{8} - 3 q^{9}+O(q^{10}) \) q + b1 * q^2 + b6 * q^3 + 2 * q^4 - b7 * q^5 - b8 * q^6 + 2b1 q^8 - 3 * q^9	\( q + \beta_1 q^{2} + \beta_{6} q^{3} + 2 q^{4} - \beta_{7} q^{5} - \beta_{8} q^{6} + 2 \beta_1 q^{8} - 3 q^{9} + \beta_{10} q^{10} + ( - \beta_{12} + \beta_{2} - \beta_1 + 1) q^{11} + 2 \beta_{6} q^{12} + ( - \beta_{9} + 2 \beta_{6}) q^{13} - \beta_{2} q^{15} + 4 q^{16} + (5 \beta_{10} + \beta_{7} + \beta_{4}) q^{17} - 3 \beta_1 q^{18} + (\beta_{14} - 2 \beta_{8} + \cdots - 3 \beta_{6}) q^{19}+ \cdots + (3 \beta_{12} - 3 \beta_{2} + 3 \beta_1 - 3) q^{99}+O(q^{100}) \) q + b1 * q^2 + b6 * q^3 + 2 * q^4 - b7 * q^5 - b8 * q^6 + 2b1 q^8 - 3 * q^9 + b10 * q^10 + (-b12 + b2 - b1 + 1) * q^11 + 2b6 q^12 + (-b9 + 2b6) q^13 - b2 * q^15 + 4 * q^16 + (5b10 + b7 + b4) q^17 - 3b1 q^18 + (b14 - 2b8 - 5b7 - 3b6) q^19 - 2b7 q^20 + (b13 + b3 + b1 - 3) * q^22 + (-b13 + b12 + b11 - b3 - b2 - b1 + 1) * q^23 - 2b8 q^24 - 5 * q^25 + (b10 + b9 - b8 - b7 - b6 - b4) * q^26 - 3b6 q^27 + (b12 - b11 + b5 - 6b2 + 5b1 + 4) * q^29 - b3 * q^30 + (7b10 + b9 - b7 - 6b6 - b4) * q^31 + 4b1 q^32 + (-b14 - b9 + 2b8 - b7 + b6) q^33 + (-b10 + b9 - 11b7 - b6 + b4) q^34 - 6 * q^36 + (b13 + b12 - b11 - b5 - 5b3 - 2b2 + 6b1 - 6) q^37 + (-b15 + 5b10 + 3b8 - b7 + 3b6) q^38 + (b12 + b11 - b2 - 5) * q^39 + 2b10 q^40 + (b15 - b14 + 3b10 + b9 - 4b8 - 2b7 - 4b6 - 2b4) q^41 + (-2b13 - b12 + b11 - b5 - 2b3 + 5b1 - 3) q^43 + (-2b12 + 2b2 - 2b1 + 2) q^44 + 3b7 q^45 + (-b13 + 2b12 - b11 - b5 - b3 - 3b2 - b1 - 2) * q^46 + (12b10 + 2b9 - 6b7 - 2b6) * q^47 + 4b6 q^48 - 5b1 q^50 + (b13 - b12 + b5 - 5b3 + 2b2 + 3b1 + 2) q^51 + (-2b9 + 4b6) * q^52 + (2b13 - 2b11 + 2b3 - 14b1 - 4) * q^53 + 3b8 q^54 + (-b15 - b10 + b9 - b7 + 2b6) q^55 + (-2b12 + b11 - 4b2 - 3b1 + 10) q^57 + (-b13 + 2b5 - 7b3 + 2b2 + 5b1 + 11) * q^58 + (2b15 - 3b14 + 14b10 - b9 - 6b8 + 17b7 - 5b6) * q^59 - 2b2 q^60 + (-b14 + 9b10 - 3b9 - 4b8 - 4b7 + 7b6 + 3b4) * q^61 + (-2b9 + 5b8 - 12b7 + 2b6) * q^62 + 8 * q^64 + (-b12 - b11 - b5 - b2 - 3b1 - 2) q^65 + (b15 + 2b10 + b9 - 4b6 - b4) * q^66 + (-b13 + b12 + b11 + 2b5 - b3 + b2 + 29b1 + 20) * q^67 + (10b10 + 2b7 + 2b4) q^68 + (-b15 - 2b10 + 2b9 + b6 + b4) * q^69 + (2b13 + b12 - 2b5 + 2b3 - 9b2 - 3b1 - 9) q^71 - 6b1 q^72 + (3b14 + 7b10 + 2b9 + 12b8 + 2b7 + 13b6 - b4) * q^73 + (-b13 - 2b12 + 2b11 - b3 - 10b2 - 4b1 + 15) * q^74 - 5b6 q^75 + (2b14 - 4b8 - 10b7 - 6b6) * q^76 + (-b13 - b11 - b5 - b3 - b2 - 6b1) q^78 + (2b13 - 2b12 - b11 - 10b3 - 4b2 + b1 - 34) * q^79 - 4b7 q^80 + 9 * q^81 + (b15 - 2b14 + 2b10 - 3b9 + 4b8 - 2b7 + 12b6 - b4) * q^82 + (-b15 + 2b14 - 4b10 - 4b9 - 4b8 + 4b7 + 13b6 + 3b4) q^83 + (-b13 + b12 - b11 + b5 - b3 - 26b1 + 7) q^85 + (b13 + 4b12 - 2b5 + b3 - 6b2 - 6b1 + 9) * q^86 + (-b15 + 3b14 + b10 - b9 - 6b8 + 18b7 + 4b6 - 2b4) q^87 + (2b13 + 2b3 + 2b1 - 6) q^88 + (-b15 + 3b14 + 25b10 - 3b9 + 12b8 - 12b6 - 2b4) * q^89 - 3b10 q^90 + (-2b13 + 2b12 + 2b11 - 2b3 - 2b2 - 2b1 + 2) * q^92 + (-b13 - b11 - b5 - 7b3 - b2 - 3b1 + 15) * q^93 + (4b10 - 2b9 - 22b7 + 2b6 + 2b4) q^94 + (-2b13 + b11 - 2b3 + 3b2 - 3b1 - 25) * q^95 - 4b8 q^96 + (b14 - 17b10 + 3b9 + 28b8 - 13b6 - b4) * q^97 + (3b12 - 3b2 + 3b1 - 3) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} - 48 q^{9}+O(q^{10}) \) 16 * q + 32 * q^4 - 48 * q^9	\( 16 q + 32 q^{4} - 48 q^{9} + 8 q^{11} + 64 q^{16} - 48 q^{22} + 24 q^{23} - 80 q^{25} + 72 q^{29} - 96 q^{36} - 88 q^{37} - 72 q^{39} - 56 q^{43} + 16 q^{44} - 16 q^{46} + 24 q^{51} - 64 q^{53} + 144 q^{57} + 176 q^{58} + 128 q^{64} - 40 q^{65} + 328 q^{67} - 136 q^{71} + 224 q^{74} - 560 q^{79} + 144 q^{81} + 120 q^{85} + 176 q^{86} - 96 q^{88} + 48 q^{92} + 240 q^{93} - 400 q^{95} - 24 q^{99}+O(q^{100}) \) 16 * q + 32 * q^4 - 48 * q^9 + 8 * q^11 + 64 * q^16 - 48 * q^22 + 24 * q^23 - 80 * q^25 + 72 * q^29 - 96 * q^36 - 88 * q^37 - 72 * q^39 - 56 * q^43 + 16 * q^44 - 16 * q^46 + 24 * q^51 - 64 * q^53 + 144 * q^57 + 176 * q^58 + 128 * q^64 - 40 * q^65 + 328 * q^67 - 136 * q^71 + 224 * q^74 - 560 * q^79 + 144 * q^81 + 120 * q^85 + 176 * q^86 - 96 * q^88 + 48 * q^92 + 240 * q^93 - 400 * q^95 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	1470.3.f.d

Level	$1470$
Weight	$3$
Character orbit	1470.f
Analytic conductor	$40.055$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(40.0545988610\)
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:	\( x^{16} + 92 x^{14} - 112 x^{13} + 5846 x^{12} - 7728 x^{11} + 197216 x^{10} - 298200 x^{9} + \cdots + 101626561 \) x^16 + 92x^14 - 112x^13 + 5846x^12 - 7728x^11 + 197216x^10 - 298200x^9 + 4836403x^8 - 6808704x^7 + 64376800x^6 - 91953512x^5 + 595763862x^4 - 630430976x^3 + 1087013404x^2 + 294123256x + 101626561
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{16}\cdot 7^{2} \)
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 57\!\cdots\!49 \nu^{15} + \cdots - 12\!\cdots\!09 ) / 85\!\cdots\!24 \) (57754008957283349v^15 + 676562680999836294v^14 + 8239103585403158159v^13 + 54078590776251682475v^12 + 455975368108340251668v^11 + 2793391292439344623383v^10 + 16545566601055515429559v^9 + 76261143494901216401580v^8 + 306572240997688184788012v^7 + 1467974756910134628134465v^6 + 3497231496721962844392765v^5 + 10591214719874231223793348v^4 - 4685579583089306764764583v^3 + 35874681437029057265441973v^2 + 9955887688266404429946106*v - 1202728311133779179554909109) / 852577600778545978990177524
\(\beta_{2}\)	\(=\)	\( ( 6718070224756 \nu^{15} - 35399436274870 \nu^{14} + 824042900206056 \nu^{13} + \cdots + 20\!\cdots\!64 ) / 53\!\cdots\!38 \) (6718070224756v^15 - 35399436274870v^14 + 824042900206056v^13 - 3468795000369922v^12 + 55447463307528888v^11 - 237157576567025547v^10 + 2160971883833844512v^9 - 7103633821675122284v^8 + 46638150914308904520v^7 - 152233236071607452553v^6 + 595135449699402838792v^5 - 1018214320678792586196v^4 + 1279768421429813362272v^3 - 459537667512276799289v^2 - 103279705437214175760*v + 206657596524136518657064) / 53375543816937984206838
\(\beta_{3}\)	\(=\)	\( ( 3871767397361 \nu^{15} + 79093222616710 \nu^{14} + 341679513333957 \nu^{13} + \cdots - 14\!\cdots\!75 ) / 27\!\cdots\!92 \) (3871767397361v^15 + 79093222616710v^14 + 341679513333957v^13 + 5895790087314937v^12 + 14035722749452236v^11 + 371621017202953791v^10 + 356604274247547817v^9 + 10456974871241558252v^8 + 4802805224198506740v^7 + 233091302320357629453v^6 + 23538017273856016799v^5 + 1467948791527369196388v^4 - 940733060256762782559v^3 + 3491995997388483650717v^2 + 957085277114840834430*v - 148193606010062737313275) / 27111518972679309429492
\(\beta_{4}\)	\(=\)	\( ( - 24\!\cdots\!58 \nu^{15} + \cdots - 49\!\cdots\!51 ) / 95\!\cdots\!44 \) (-241863822777540874882466993018758v^15 + 213469975304718887115010020250304v^14 - 23706868915498534505262085799802764v^13 + 33346620725345705792292386908536195v^12 - 1572246595585425326053498700187612372v^11 + 2238397531499978911345875836894047377v^10 - 56379450841845268013992185571629193172v^9 + 65194802114865894964792930629943731574v^8 - 1438089633101304793073636413061634509524v^7 + 1562153171319216137200389970915623241819v^6 - 21102866749248222462536154496131177234044v^5 + 16480923200844042141095313649474586060138v^4 - 199611409492269652994318983181462690900774v^3 + 235625405974736320182385020984337601858401v^2 - 450294361739753268802611711172967452983904*v - 49793140462485741740564019667302022211751) / 95852909214978668011338765982386839392644
\(\beta_{5}\)	\(=\)	\( ( - 15\!\cdots\!15 \nu^{15} + \cdots - 23\!\cdots\!33 ) / 31\!\cdots\!08 \) (-15676746441665707101942164015v^15 + 21207121467610615912805559174v^14 - 1369703339442836424119507090009v^13 + 3487555436126339212562826767107v^12 - 88569529078180963996479471688716v^11 + 205591363940722638333739187297079v^10 - 2919190798664141627791804403165965v^9 + 6428727908938342577118706461520812v^8 - 71687408890868806747434561888203188v^7 + 115859697731261714424380120491192081v^6 - 899631755687316031497385829078742651v^5 + 933885797263311750157099981268214596v^4 - 8608546324615197810872297777006685983v^3 - 714070897931272908462439708632495915v^2 - 229621811151920173894439902818460534*v - 23442301013525123996598271073372839033) / 3169424634294834110747570214012724908
\(\beta_{6}\)	\(=\)	\( ( 34\!\cdots\!04 \nu^{15} + \cdots + 35\!\cdots\!31 ) / 62\!\cdots\!01 \) (34800704278153535382807509504v^15 - 10894314337506543053670349736v^14 + 3184797771932765844822003594240v^13 - 4883872557229584935529507311852v^12 + 203150684489855051057790801147168v^11 - 329641444475618185980178937295216v^10 + 6850224385630247456211484006725040v^9 - 12335700651762730044495715881145318v^8 + 168345816581855321121206226303993216v^7 - 283581678836521920106992554468810640v^6 + 2236415316817892637807058734013008848v^5 - 3771284369594629070382322847180181060v^4 + 20757636283402814726509997166646534848v^3 - 27545780009134355888225371127509524184v^2 + 36052656915764843531341394646112567344*v + 3521207262416884848433762658587277531) / 6206482078151946905681090778450326301
\(\beta_{7}\)	\(=\)	\( ( - 536662839109972 \nu^{15} + 184355843936000 \nu^{14} + \cdots - 55\!\cdots\!08 ) / 75\!\cdots\!34 \) (-536662839109972v^15 + 184355843936000v^14 - 49316294058364856v^13 + 76985389887345138v^12 - 3154550976828630456v^11 + 5211291917675260305v^10 - 106964341125832723328v^9 + 195705915902593012444v^8 - 2639716566132467353984v^7 + 4517022991968883377451v^6 - 35333929028214918917480v^5 + 60517159199633559261440v^4 - 328552596355156909606256v^3 + 436496862559525516173889v^2 - 567954327233848070808304*v - 55329661987849902130608) / 75811879080713475788634
\(\beta_{8}\)	\(=\)	\( ( 2130173036201 \nu^{15} - 540413699312 \nu^{14} + 193661918213181 \nu^{13} + \cdots + 20\!\cdots\!93 ) / 25\!\cdots\!08 \) (2130173036201v^15 - 540413699312v^14 + 193661918213181v^13 - 293621077625441v^12 + 12285060311199138v^11 - 19810557668746935v^10 + 410965646117486839v^9 - 746713194852137098v^8 + 10036111646725656126v^7 - 17105922536607134721v^6 + 131897626701278295101v^5 - 228882972939831674094v^4 + 1220130262910161491531v^3 - 1620423665127954679375v^2 + 2091806790152256573456*v + 203021156853115775993) / 255239865906581664108
\(\beta_{9}\)	\(=\)	\( ( 27\!\cdots\!01 \nu^{15} + \cdots + 31\!\cdots\!96 ) / 31\!\cdots\!48 \) (273267982442506027837028536846301v^15 - 90824115373216270012966076556376v^14 + 24724502366652215915576790007891357v^13 - 35765374599523411150807746572903628v^12 + 1582514521052192986730127030508629594v^11 - 2345134475522779658037774090424825002v^10 + 52509790850976840228722541081735640975v^9 - 85155157364026490810922764150475232928v^8 + 1282216420561340630964640398485100256046v^7 - 1931951755527215176068822620639346594878v^6 + 16400400933464541644568104022630034167277v^5 - 25462330763896727659919529802689863003296v^4 + 153102783374700304960302234551110364381923v^3 - 184555318397999592675046544109329481337922v^2 + 298517316090639541206036201872067943971200*v + 31351166733895176216491140415459948534796) / 31950969738326222670446255327462279797548
\(\beta_{10}\)	\(=\)	\( ( - 44\!\cdots\!15 \nu^{15} + \cdots - 43\!\cdots\!65 ) / 42\!\cdots\!36 \) (-4448842632361015v^15 + 1527746136142088v^14 - 406573176070056149v^13 + 638852366768395737v^12 - 25956155317369110414v^11 + 43002282249586977093v^10 - 874597310627211904019v^9 + 1616020902079177673194v^8 - 21496564169263011322522v^7 + 37198980049890495166591v^6 - 284874104240500582055717v^5 + 500982390617084205830630v^4 - 2644373246682812474960417v^3 + 3517427264905839032095693v^2 - 4518691295624378534050984*v - 437660350242222703221165) / 427449074136574093101036
\(\beta_{11}\)	\(=\)	\( ( - 13\!\cdots\!03 \nu^{15} + \cdots - 91\!\cdots\!03 ) / 95\!\cdots\!24 \) (-139899057247192626960828150203v^15 - 205984038615707634647717699362v^14 - 12845894306918859627747662840373v^13 - 2722805495207821537930036165291v^12 - 791993126949886351938458511046812v^11 - 100975119829351085447600383013997v^10 - 25937879554799122322336137198622449v^9 + 2424669373205808452677178694538204v^8 - 609568656712704554220138175959467556v^7 - 17681141653075112811106269800634411v^6 - 7473449288344881015343436020856745167v^5 + 597458333330310777396600761626775892v^4 - 58619731313164471563160626236307114531v^3 - 22597125487299789997563421409019516695v^2 - 6470850633222513601969200095438569758*v - 91876261549562488474322233688611891103) / 9508273902884502332242710642038174724
\(\beta_{12}\)	\(=\)	\( ( 50\!\cdots\!76 \nu^{15} + \cdots + 54\!\cdots\!09 ) / 31\!\cdots\!08 \) (50541445469745635702521407476v^15 + 88189787602355302125069537998v^14 + 4686121529079582552930273937852v^13 + 2344084363837616559345483004641v^12 + 287669320037743266070478689305528v^11 + 102227182237608941304707315817543v^10 + 9383188325428496653322675718828448v^9 + 980369731111473431671427903471644v^8 + 220345273133399733933805748083108424v^7 + 33350653108110624529048868648542321v^6 + 2695217965796171147046350321069201188v^5 + 45053311281652415642974882019971172v^4 + 22564363014998968270435084775820227404v^3 + 8755731051360650982019965319468659565v^2 + 2499667994650377956679074524685699432*v + 54539131317226042274521458166636701909) / 3169424634294834110747570214012724908
\(\beta_{13}\)	\(=\)	\( ( 35\!\cdots\!52 \nu^{15} + \cdots + 40\!\cdots\!54 ) / 15\!\cdots\!54 \) (35618621022542461412963550752v^15 + 57402549257930764143801253636v^14 + 3269452438379247383903820031191v^13 + 1195103257893557548048710310882v^12 + 201097599940690016735522351078652v^11 + 50642128087521428839734928726662v^10 + 6545045094796181211847908243536239v^9 + 94213946227552098355377432246440v^8 + 154371956973564180787294432103580228v^7 + 13527137712560061152587689973211136v^6 + 1890245428081527900140741520716056325v^5 - 51743176237912170133092018183094008v^4 + 16011464587952105898188873235352187640v^3 + 5946720044570265499150836888229441082v^2 + 1699987857187158300731555890791555114*v + 40128729606389940427563972548234469854) / 1584712317147417055373785107006362454
\(\beta_{14}\)	\(=\)	\( ( 15\!\cdots\!08 \nu^{15} + \cdots + 24\!\cdots\!97 ) / 23\!\cdots\!61 \) (1524687532650925566384058512983608v^15 - 45409554837746644596280025048156v^14 + 141488245005809006513245332039530819v^13 - 172571033238879105571543248620219622v^12 + 9030723326987672595044485228224009498v^11 - 11987810186592915492952403117919938364v^10 + 307821302729523772843844588129480487590v^9 - 461578234521407040548049640832600487082v^8 + 7604888710910270137183375270238308341856v^7 - 10627157121886052664993396991945530374686v^6 + 103489208761170799566591481949888616327575v^5 - 144115666979851304912358383250150995962136v^4 + 971695144433049417614826274880961599060141v^3 - 1025012991098740975794373087499920956372100v^2 + 2150637685521930692819305889555961803356510*v + 241226071476552345162168621818414428421697) / 23963227303744667002834691495596709848161
\(\beta_{15}\)	\(=\)	\( ( 96\!\cdots\!17 \nu^{15} + \cdots + 14\!\cdots\!41 ) / 10\!\cdots\!52 \) (967177237679128037761186849117v^15 - 1952043673575989613570267656v^14 + 89527070939922271046261982843611v^13 - 109190857744315535774155928018049v^12 + 5700740767634372766157484444374170v^11 - 7597080890305333130374154566891551v^10 + 193818155261377055998617002044149413v^9 - 294775183761668393523411707373797314v^8 + 4779142678687522109036057214533659918v^7 - 6777773520474492153092935877383356565v^6 + 64784183508077896482320219799719599595v^5 - 92294600198557773363013468818806213150v^4 + 607562677390245683660668103797050433139v^3 - 644931194393471260333156135024330484311v^2 + 1331955896632363538983196299482134343112*v + 148851527006263660404650371701627502341) / 10365838565478389532966234019940179452

\(\nu\)	\(=\)	\( ( 4 \beta_{15} - 5 \beta_{14} - 3 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + 5 \beta_{10} + 5 \beta_{9} + \cdots - 1 ) / 28 \) (4b15 - 5b14 - 3b13 + 2b12 - 2b11 + 5b10 + 5b9 - 4b8 - 2b7 - 3b6 + b5 - b4 - 3b3 - b2 - 10b1 - 1) / 28
\(\nu^{2}\)	\(=\)	\( ( - 5 \beta_{15} + 8 \beta_{14} - 2 \beta_{13} + 6 \beta_{12} + \beta_{11} - \beta_{10} + 6 \beta_{9} + \cdots - 325 ) / 28 \) (-5b15 + 8b14 - 2b13 + 6b12 + b11 - b10 + 6b9 + 96b8 - 127b7 - 327b6 + 3b5 - 4b4 - 2b3 + 39b2 - 100*b1 - 325) / 28
\(\nu^{3}\)	\(=\)	\( ( 17\beta_{13} - 18\beta_{12} + 8\beta_{11} - 3\beta_{5} + 5\beta_{3} + 9\beta_{2} + 88\beta _1 + 51 ) / 2 \) (17b13 - 18b12 + 8b11 - 3b5 + 5b3 + 9b2 + 88*b1 + 51) / 2
\(\nu^{4}\)	\(=\)	\( ( 198 \beta_{15} - 272 \beta_{14} - 124 \beta_{13} + 232 \beta_{12} + 34 \beta_{11} - 834 \beta_{10} + \cdots - 5765 ) / 14 \) (198b15 - 272b14 - 124b13 + 232b12 + 34b11 - 834b10 - 176b9 - 2228b8 + 3100b7 + 5749b6 + 74b5 + 24b4 + 212b3 + 892b2 - 2476*b1 - 5765) / 14
\(\nu^{5}\)	\(=\)	\( ( - 5990 \beta_{15} + 8667 \beta_{14} - 5413 \beta_{13} + 6530 \beta_{12} - 1560 \beta_{11} + \cdots - 25805 ) / 28 \) (-5990b15 + 8667b14 - 5413b13 + 6530b12 - 1560b11 + 9155b10 - 2003b9 + 20200b8 - 9948b7 - 19409b6 + 577b5 + 4259b4 + 1055b3 - 325b2 - 33126*b1 - 25805) / 28
\(\nu^{6}\)	\(=\)	\( ( 2622 \beta_{13} - 4310 \beta_{12} - 373 \beta_{11} - 811 \beta_{5} - 4770 \beta_{3} - 11579 \beta_{2} + \cdots + 67649 ) / 2 \) (2622b13 - 4310b12 - 373b11 - 811b5 - 4770b3 - 11579b2 + 35956*b1 + 67649) / 2
\(\nu^{7}\)	\(=\)	\( ( 279262 \beta_{15} - 400797 \beta_{14} - 259745 \beta_{13} + 334798 \beta_{12} - 46482 \beta_{11} + \cdots - 1632009 ) / 28 \) (279262b15 - 400797b14 - 259745b13 + 334798b12 - 46482b11 - 699547b10 + 17911b9 - 1109420b8 + 965760b7 + 1345299b6 + 19517b5 - 220711b4 + 140095b3 + 120721b2 - 1730928*b1 - 1632009) / 28
\(\nu^{8}\)	\(=\)	\( ( - 691284 \beta_{15} + 964464 \beta_{14} - 585096 \beta_{13} + 900336 \beta_{12} + 42060 \beta_{11} + \cdots - 10697503 ) / 14 \) (-691284b15 + 964464b14 - 585096b13 + 900336b12 + 42060b11 + 3656460b10 + 399360b9 + 5207880b8 - 7456656b7 - 10260655b6 + 106188b5 + 372720b4 + 988392b3 + 1906704b2 - 6545064*b1 - 10697503) / 14
\(\nu^{9}\)	\(=\)	\( ( 1846509 \beta_{13} - 2471702 \beta_{12} + 211658 \beta_{11} - 112891 \beta_{5} - 1436811 \beta_{3} + \cdots + 13889443 ) / 2 \) (1846509b13 - 2471702b12 + 211658b11 - 112891b5 - 1436811b3 - 1653581b2 + 13011958*b1 + 13889443) / 2
\(\nu^{10}\)	\(=\)	\( ( 77787413 \beta_{15} - 109133120 \beta_{14} - 69460730 \beta_{13} + 103071894 \beta_{12} + \cdots - 1030326949 ) / 28 \) (77787413b15 - 109133120b14 - 69460730b13 + 103071894b12 + 2265457b11 - 403264199b10 - 38142078b9 - 527802720b8 + 757264471b7 + 971458359b6 + 8326683b5 - 52807364b4 + 108492262b3 + 185727303b2 - 689743204*b1 - 1030326949) / 28
\(\nu^{11}\)	\(=\)	\( ( - 696502112 \beta_{15} + 990432227 \beta_{14} - 660025487 \beta_{13} + 903385734 \beta_{12} + \cdots - 5586803001 ) / 28 \) (-696502112b15 + 990432227b14 - 660025487b13 + 903385734b12 - 50569868b11 + 2558370023b10 + 142220511b9 + 3268336104b8 - 4115827654b7 - 4912684271b6 + 36476625b5 + 587072237b4 + 620622637b3 + 798700893b2 - 4845840568*b1 - 5586803001) / 28
\(\nu^{12}\)	\(=\)	\( 283303276 \beta_{13} - 411884296 \beta_{12} - 2844970 \beta_{11} - 25153442 \beta_{5} + \cdots + 3695031179 \) 283303276b13 - 411884296b12 - 2844970b11 - 25153442b5 - 415417460b3 - 664916932b2 + 2616526588*b1 + 3695031179
\(\nu^{13}\)	\(=\)	\( ( 36136952786 \beta_{15} - 51274366995 \beta_{14} - 34314451069 \beta_{13} + 47593238282 \beta_{12} + \cdots - 313758094217 ) / 28 \) (36136952786b15 - 51274366995b14 - 34314451069b13 + 47593238282b12 - 1858626996b11 - 144106197839b10 - 9561533221b9 - 177814191664b8 + 238342892184b7 + 279407517869b6 + 1822501717b5 - 30669447635b4 + 35989748351b3 + 48945848111b2 - 259758006294*b1 - 313758094217) / 28
\(\nu^{14}\)	\(=\)	\( ( - 237558015007 \beta_{15} + 335044352924 \beta_{14} - 221473349274 \beta_{13} + 318262664810 \beta_{12} + \cdots - 2667438845879 ) / 28 \) (-237558015007b15 + 335044352924b14 - 221473349274b13 + 318262664810b12 - 697022381b11 + 1167123223825b10 + 95395270774b9 + 1430841773632b8 - 2050689201961b7 - 2461353139957b6 + 16084665733b5 + 189304017808b4 + 309855069606b3 + 476981853485b2 - 1955190144364*b1 - 2667438845879) / 28
\(\nu^{15}\)	\(=\)	\( ( 257998257311 \beta_{13} - 360676661674 \beta_{12} + 10562294418 \beta_{11} - 13560958727 \beta_{5} + \cdots + 2480567013939 ) / 2 \) (257998257311b13 - 360676661674b12 + 10562294418b11 - 13560958727b5 - 289138472617b3 - 405091585843b2 + 1997234018400*b1 + 2480567013939) / 2

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 391.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} - 2)^{8} \) (T^2 - 2)^8
$3$	\( (T^{2} + 3)^{8} \) (T^2 + 3)^8
$5$	\( (T^{2} + 5)^{8} \) (T^2 + 5)^8
$7$	\( T^{16} \) T^16
$11$	\( (T^{8} - 4 T^{7} + \cdots + 3111696)^{2} \) (T^8 - 4T^7 - 588T^6 + 1352T^5 + 99088T^4 - 127344T^3 - 5129712T^2 + 3259872*T + 3111696)^2
$13$	\( T^{16} + \cdots + 17613726190641 \) T^16 + 1368T^14 + 631916T^12 + 121166856T^10 + 9820643542T^8 + 360016963752T^6 + 5708126314764T^4 + 32826023199864*T^2 + 17613726190641
$17$	\( T^{16} + \cdots + 13\!\cdots\!76 \) T^16 + 3928T^14 + 6101232T^12 + 4784547424T^10 + 2017746863776T^8 + 454984069961088T^6 + 50870437865807616T^4 + 2283763344946398720*T^2 + 13699042635136594176
$19$	\( T^{16} + \cdots + 10\!\cdots\!41 \) T^16 + 3336T^14 + 4484108T^12 + 3115002840T^10 + 1183982767030T^8 + 236106754605432T^6 + 20360865271560300T^4 + 290484186498030888*T^2 + 1088507693685749841
$23$	\( (T^{8} - 12 T^{7} + \cdots + 44064727056)^{2} \) (T^8 - 12T^7 - 2284T^6 + 12120T^5 + 1742608T^4 - 2584848T^3 - 510960240T^2 - 176329440*T + 44064727056)^2
$29$	\( (T^{8} - 36 T^{7} + \cdots - 235474074864)^{2} \) (T^8 - 36T^7 - 5284T^6 + 231048T^5 + 5584336T^4 - 368051952T^3 + 4047381936T^2 + 13878381024*T - 235474074864)^2
$31$	\( T^{16} + \cdots + 10\!\cdots\!01 \) T^16 + 6600T^14 + 17957324T^12 + 26107228824T^10 + 21930849063286T^8 + 10661944260564408T^6 + 2784936550544458668T^4 + 304492676211707065704*T^2 + 1054350189971139482001
$37$	\( (T^{8} + \cdots - 3600469848911)^{2} \) (T^8 + 44T^7 - 7060T^6 - 341892T^5 + 13499750T^4 + 757931316T^3 - 4253125684T^2 - 467761445788*T - 3600469848911)^2
$41$	\( T^{16} + \cdots + 26\!\cdots\!96 \) T^16 + 15480T^14 + 91433264T^12 + 258539104224T^10 + 362479500954016T^8 + 240256669708752768T^6 + 61910849899642860288T^4 + 2537276667253552479744*T^2 + 26127497467342672240896
$43$	\( (T^{8} + 28 T^{7} + \cdots + 563400737569)^{2} \) (T^8 + 28T^7 - 9060T^6 - 168116T^5 + 26540822T^4 + 190593380T^3 - 25761062852T^2 + 106450443444*T + 563400737569)^2
$47$	\( T^{16} + \cdots + 10\!\cdots\!16 \) T^16 + 15840T^14 + 98872256T^12 + 311314002432T^10 + 534824656938496T^8 + 522197239921975296T^6 + 289031601087775752192T^4 + 84415991942485643231232*T^2 + 10101978488064049707810816
$53$	\( (T^{8} + 32 T^{7} + \cdots - 110361636864)^{2} \) (T^8 + 32T^7 - 9376T^6 - 247040T^5 + 24835456T^4 + 352598016T^3 - 19146074112T^2 + 105598107648*T - 110361636864)^2
$59$	\( T^{16} + \cdots + 95\!\cdots\!56 \) T^16 + 53800T^14 + 1178179056T^12 + 13442394615328T^10 + 84876758291333536T^8 + 291079798534197684864T^6 + 499911605964802885531392T^4 + 371413100775188382417658368*T^2 + 95659041749658506515012813056
$61$	\( T^{16} + \cdots + 13\!\cdots\!96 \) T^16 + 33504T^14 + 400428608T^12 + 2089220190720T^10 + 4845303856514560T^8 + 4395513661648969728T^6 + 952576928404071137280T^4 + 65997272344738048376832*T^2 + 1392598099630469949751296
$67$	\( (T^{8} + \cdots + 1596721546161)^{2} \) (T^8 - 164T^7 - 2676T^6 + 1129036T^5 - 15599930T^4 - 1195387164T^3 + 39775201164T^2 - 434983961484*T + 1596721546161)^2
$71$	\( (T^{8} + \cdots - 2334473020656)^{2} \) (T^8 + 68T^7 - 14236T^6 - 204488T^5 + 70875760T^4 - 2359994064T^3 + 12080564496T^2 + 365315438880*T - 2334473020656)^2
$73$	\( T^{16} + \cdots + 99\!\cdots\!61 \) T^16 + 40952T^14 + 596855436T^12 + 3948236075048T^10 + 13051748831699830T^8 + 22350077858102322312T^6 + 19300556035717437028716T^4 + 7468937247147188121338328*T^2 + 995049780300698330601190161
$79$	\( (T^{8} + \cdots + 101515662800761)^{2} \) (T^8 + 280T^7 + 13940T^6 - 2479368T^5 - 299642482T^4 - 8650431480T^3 + 139189714436T^2 + 9296406570088*T + 101515662800761)^2
$83$	\( T^{16} + \cdots + 13\!\cdots\!16 \) T^16 + 44040T^14 + 666154928T^12 + 3974136166944T^10 + 7403270734587808T^8 + 2722620180898843776T^6 + 307745806772397964032T^4 + 7581543315742352208384*T^2 + 1312045200775282934016
$89$	\( T^{16} + \cdots + 26\!\cdots\!56 \) T^16 + 107400T^14 + 4691236208T^12 + 106830987964320T^10 + 1349344150954861984T^8 + 9283143638587347607680T^6 + 32015173752969153782038272T^4 + 48633874512679923981639252480*T^2 + 26119520169155990714944934297856
$97$	\( T^{16} + \cdots + 18\!\cdots\!36 \) T^16 + 80096T^14 + 2531519808T^12 + 40307196119552T^10 + 345608150837679616T^8 + 1605606437907707437056T^6 + 3776343128542461157392384T^4 + 3526977654558045584073031680*T^2 + 185911339063854590759980302336
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\(n\)	\(491\)	\(1081\)	\(1177\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)