LMFDB - Newform orbit 531.2.d.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [531,2,Mod(530,531)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("531.530"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(531, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	531.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$4.24005634733$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 10 x^{18} + 139 x^{16} - 476 x^{14} + 4681 x^{12} - 666 x^{10} + 82273 x^{8} + 168944 x^{6} + \cdots + 3374569 $ x^20 - 10x^18 + 139x^16 - 476x^14 + 4681x^12 - 666x^10 + 82273x^8 + 168944x^6 + 845867x^4 + 1613394*x^2 + 3374569
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{12} q^{2} + (\beta_{2} + 1) q^{4} + \beta_{13} q^{5} - \beta_{4} q^{7} + ( - \beta_{15} - \beta_{12}) q^{8} + (\beta_{6} + \beta_{3}) q^{10} - \beta_{18} q^{11} + \beta_{6} q^{13} + (\beta_{18} - \beta_{16}) q^{14}+ \cdots + (3 \beta_{19} - 2 \beta_{16} + \cdots - \beta_{12}) q^{98}+O(q^{100}) $ q - b12 * q^2 + (b2 + 1) * q^4 + b13 * q^5 - b4 * q^7 + (-b15 - b12) * q^8 + (b6 + b3) * q^10 - b18 * q^11 + b6 * q^13 + (b18 - b16) * q^14 + (b7 + b5 + b2) * q^16 + (-b14 + b11 + b10) * q^17 + (-b5 - b2) * q^19 + (b14 + b13 - b10) * q^20 + (b7 + b4 - 1) * q^22 + (-b19 + b18) * q^23 + b7 * q^25 + (b17 + b14 + 2b13 + b11) q^26 + (-b7 - b5 - b4 + 1) * q^28 + (-b17 - b13 - b11) * q^29 + (b9 - b6 - b3) * q^31 + (-b19 - b18 + b16 - b15 - b12) * q^32 + (b9 + b8 - b6 - b3 + 2b1) q^34 + (-b17 - b13 - 2b10) q^35 + (b6 - b3 - b1) * q^37 + (b19 - b16 + 2b15 + 2b12) * q^38 + (-b9 - b8 - b1) * q^40 + (-b17 - b14 - b13 - b11 + 2b10) q^41 + (-b8 + b6 + 2b3 - b1) q^43 + (b16 - b15) * q^44 + (-b7 + b5 - 2b4 - b2 + 1) q^46 + (-b19 + b16 + 2b12) q^47 + (-b7 - 3b5 + b4 + 2) q^49 + (-b18 - b15 - b12) * q^50 + (-b9 + b6 + b3 - b1) * q^52 + (-b17 - 2b14 - b13 - b11 - b10) q^53 + (-b9 - 2b3) q^55 + (b19 + 2b15) q^56 + (-b6 - b1) * q^58 + (b19 - b17 + b15 - b14 + b13 + b12 + b11) * q^59 + (b8 + 2b6 + b3 - b1) q^61 + (-3b14 - 3b13 + 2b10) q^62 + (b5 + 2b4 + 1) q^64 + (-b19 + b18 - b15 + 2b12) q^65 + (-b9 + b8 - 2b3 - b1) q^67 + (-4b14 - 3b13 + 4b11 + 2b10) * q^68 + (-2b8 - b6 + 2b3 + 2b1) q^70 + (2b17 + b14 + b13 + 3b11) * q^71 + (-b9 + 2b8 + b6 - b3 - b1) q^73 + (2b17 + 2b14 + b13 - b11 + b10) * q^74 + (-2b7 - 2b5 - b4 - 3b2 - 4) q^76 + (-2b19 + b16 - b15 + 2b12) * q^77 + (b7 + b5 + 2b4 + 2b2 - 1) * q^79 + (2b14 - 2b13 - 3b11 - b10) q^80 + (b9 + 2b8 - 2b6 - 2b3 - b1) q^82 + (b16 - b15) * q^83 + (b7 - b4 + 2b2 - 1) q^85 + (-b17 + 3b14 + 4b13 - 4b11 - 4b10) * q^86 + (-b7 + 2b5 + 2b2 + 1) * q^88 + (2b16 - b15 - 2b12) * q^89 + (b9 - 2b8 - 3b6 + b3 - b1) * q^91 + (b19 + b18 - b16 + b15 + 2b12) q^92 + (2b5 + b4 - 3b2 - 6) * q^94 + (b17 + 5b11) q^95 + (b9 - b8 + 4b1) q^97 + (3b19 - 2b16 + 4b15 - b12) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{4} - 8 q^{7} + 4 q^{16} - 8 q^{22} + 4 q^{25} + 8 q^{28} + 44 q^{49} + 36 q^{64} - 96 q^{76} - 24 q^{85} + 16 q^{88} - 112 q^{94}+O(q^{100}) $ 20 * q + 20 * q^4 - 8 * q^7 + 4 * q^16 - 8 * q^22 + 4 * q^25 + 8 * q^28 + 44 * q^49 + 36 * q^64 - 96 * q^76 - 24 * q^85 + 16 * q^88 - 112 * q^94

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} - 10 x^{18} + 139 x^{16} - 476 x^{14} + 4681 x^{12} - 666 x^{10} + 82273 x^{8} + 168944 x^{6} + \cdots + 3374569 $ x^20 - 10*x^18 + 139*x^16 - 476*x^14 + 4681*x^12 - 666*x^10 + 82273*x^8 + 168944*x^6 + 845867*x^4 + 1613394*x^2 + 3374569 :

$\beta_{1}$	$=$	$ ( - 726777222591367 \nu^{18} + \cdots - 37\!\cdots\!13 ) / 36\!\cdots\!36 $ (-726777222591367v^18 + 5941008833830113v^16 - 91602358590346650v^14 + 184616922793156367v^12 - 3231573008264319840v^10 - 5764563651169605570v^8 - 79965005644882419861v^6 - 345545294857588001933v^4 - 2626931079401957575386*v^2 - 3718258825763016078713) / 3682678142499621285836
$\beta_{2}$	$=$	$ ( - 726777222591367 \nu^{18} + \cdots - 55\!\cdots\!31 ) / 18\!\cdots\!18 $ (-726777222591367v^18 + 5941008833830113v^16 - 91602358590346650v^14 + 184616922793156367v^12 - 3231573008264319840v^10 - 5764563651169605570v^8 - 79965005644882419861v^6 - 345545294857588001933v^4 - 785592008152146932468*v^2 - 5559597897012826721631) / 1841339071249810642918
$\beta_{3}$	$=$	$ ( 11\!\cdots\!86 \nu^{18} + \cdots + 97\!\cdots\!39 ) / 22\!\cdots\!16 $ (11702514375950686v^18 + 191525605279787919v^16 - 1828324832041991813v^14 + 40947502777856166881v^12 - 153900951341063137884v^10 + 1633994733135824878086v^8 - 1329585349562152680146v^6 + 22289865192592561762665v^4 + 28952814023750531522531*v^2 + 97828433192122122178439) / 22096068854997727715016
$\beta_{4}$	$=$	$ ( - 93\!\cdots\!23 \nu^{18} + \cdots - 20\!\cdots\!11 ) / 55\!\cdots\!54 $ (-9362148199123223v^18 + 19191897951128837v^16 - 543021799440662688v^14 - 4287224488630087161v^12 - 21241652543593451848v^10 - 173791460355919676476v^8 - 875902073308331859993v^6 - 4327628291146543048391v^4 - 10028152012182697339842*v^2 - 20518677369002153331011) / 5524017213749431928754
$\beta_{5}$	$=$	$ ( - 22\!\cdots\!53 \nu^{18} + \cdots + 38\!\cdots\!41 ) / 55\!\cdots\!54 $ (-22400930441398253v^18 + 393624863282125412v^16 - 4589423775990728229v^14 + 29868406304144604483v^12 - 133974055694978085472v^10 + 431331440654920709576v^8 - 317152138506556243869v^6 + 3848613617077735072822v^4 + 10992841018618551425181*v^2 + 38541873287962536796141) / 5524017213749431928754
$\beta_{6}$	$=$	$ ( 15\!\cdots\!51 \nu^{18} + \cdots - 11\!\cdots\!65 ) / 22\!\cdots\!16 $ (153432456366075751v^18 - 1564699734951504606v^16 + 19796813498773297006v^14 - 61259897928339466411v^12 + 532099418306169203562v^10 + 13724608647346152510v^8 + 8776217525678006044873v^6 + 8161481150131271265696v^4 + 47465062693384228294784*v^2 - 11975032980151079951965) / 22096068854997727715016
$\beta_{7}$	$=$	$ ( 38\!\cdots\!22 \nu^{18} + \cdots - 60\!\cdots\!00 ) / 55\!\cdots\!54 $ (38423998434412022v^18 - 533160102699398963v^16 + 6644327398510161729v^14 - 34937937732203550072v^12 + 203124381279207904240v^10 - 362126019430061021786v^8 + 1619433462203003526372v^6 + 1005353064210506448539v^4 - 342389642582343308427*v^2 - 6080299787444652370000) / 5524017213749431928754
$\beta_{8}$	$=$	$ ( 92\!\cdots\!25 \nu^{18} + \cdots + 72\!\cdots\!28 ) / 11\!\cdots\!08 $ (92793018209944625v^18 - 687134536423822104v^16 + 8427164022449520308v^14 + 9008183684664779116v^12 + 74034372756108972030v^10 + 1635058748526631823514v^8 + 1842743426619832004243v^6 + 24925223384169194306136v^4 + 34988957025029982381592*v^2 + 72654097023721365828928) / 11048034427498863857508
$\beta_{9}$	$=$	$ ( 19\!\cdots\!31 \nu^{18} + \cdots + 71\!\cdots\!03 ) / 22\!\cdots\!16 $ (193439172704604631v^18 - 1900288265885171886v^16 + 24874594954774169956v^14 - 72421787808572572111v^12 + 707891653163121577050v^10 + 273160630360802824110v^8 + 12717344035655573182789v^6 + 18894425347970484922092v^4 + 117184153620937876755506*v^2 + 71376857991803719997903) / 22096068854997727715016
$\beta_{10}$	$=$	$ ( 14\!\cdots\!37 \nu^{19} + \cdots - 12\!\cdots\!33 \nu ) / 13\!\cdots\!64 $ (1453912374027885337v^19 - 200206497462219979484v^17 + 2485243350093145468378v^15 - 28085845778721192705159v^13 + 118957607460567016231670v^11 - 666984244627782686146770v^9 + 366868753568888535166035v^7 - 5527796614148455191857106v^5 - 11317687070878185333291868v^3 - 12798264250548058482872233v) / 13530159495543608604161464
$\beta_{11}$	$=$	$ ( 19\!\cdots\!07 \nu^{19} + \cdots + 80\!\cdots\!40 \nu ) / 67\!\cdots\!32 $ (1985355176503117307v^19 - 18518462007130831891v^17 + 265050736306187388092v^15 - 776755531285017042082v^13 + 8954306294040063867888v^11 + 4614153068630479419618v^9 + 173930629863639535630901v^7 + 482309560308791655598465v^5 + 2314113133736551486671090v^3 + 8028832522500466510394040v) / 6765079747771804302080732
$\beta_{12}$	$=$	$ ( - 19\!\cdots\!07 \nu^{19} + \cdots - 12\!\cdots\!08 \nu ) / 67\!\cdots\!32 $ (-1985355176503117307v^19 + 18518462007130831891v^17 - 265050736306187388092v^15 + 776755531285017042082v^13 - 8954306294040063867888v^11 - 4614153068630479419618v^9 - 173930629863639535630901v^7 - 482309560308791655598465v^5 - 2314113133736551486671090v^3 - 1263752774728662208313308v) / 6765079747771804302080732
$\beta_{13}$	$=$	$ ( - 37\!\cdots\!70 \nu^{19} + \cdots - 11\!\cdots\!47 \nu ) / 40\!\cdots\!92 $ (-37187856799063203570v^19 - 96076410145864892113v^17 + 245764276534215785165v^15 - 49029779750411193766913v^13 + 82710561737877519066524v^11 - 1654424832433422721121122v^9 - 2916993547352654756555014v^7 - 22292562500331931774882991v^5 - 59115491240349678772148291v^3 - 119602908757035216172794247v) / 40590478486630825812484392
$\beta_{14}$	$=$	$ ( 85\!\cdots\!56 \nu^{19} + \cdots + 19\!\cdots\!55 \nu ) / 67\!\cdots\!32 $ (8591686124615245356v^19 - 81678676807908241874v^17 + 1156980148800470144410v^15 - 3544636369254056964231v^13 + 38835131854018763793360v^11 + 12481261915953831666000v^9 + 722757433948548672869848v^7 + 1776781094890569118441404v^5 + 6744893275821361367371368v^3 + 19783561654032063411212955v) / 6765079747771804302080732
$\beta_{15}$	$=$	$ ( 98\!\cdots\!12 \nu^{19} + \cdots + 18\!\cdots\!77 \nu ) / 67\!\cdots\!32 $ (9892216961820797612v^19 - 96888334366678070494v^17 + 1350534555951911328494v^15 - 4419864857482034556037v^13 + 44870945209735780436976v^11 + 530561198817659641056v^9 + 776827262936529733562336v^7 + 1471866802201374110536492v^5 + 8486854505343476510826116v^3 + 1885104529864262452567277v) / 6765079747771804302080732
$\beta_{16}$	$=$	$ ( 50\!\cdots\!21 \nu^{19} + \cdots + 61\!\cdots\!49 \nu ) / 20\!\cdots\!96 $ (50537545069155461321v^19 - 375778886191321399266v^17 + 5085686413870215258452v^15 + 154428398823064316455v^13 + 83111686646821539471690v^11 + 897362386111149617256354v^9 + 848535058433236063982075v^7 + 18581808692851784878046496v^5 + 19461165530419986305223742v^3 + 61649815710789182072688049v) / 20295239243315412906242196
$\beta_{17}$	$=$	$ ( - 13\!\cdots\!83 \nu^{19} + \cdots - 72\!\cdots\!43 \nu ) / 40\!\cdots\!92 $ (-133039296209855075883v^19 + 1719044163565193243420v^17 - 22330275735783914425906v^15 + 114799996578738136528543v^13 - 776827519315239976072102v^11 + 1525766535586353527332754v^9 - 9200701991253304177230877v^7 - 4820165046650349930979850v^5 - 24783120529344677895857264v^3 - 72969910788043865490125443v) / 40590478486630825812484392
$\beta_{18}$	$=$	$ ( 74\!\cdots\!34 \nu^{19} + \cdots + 17\!\cdots\!86 \nu ) / 20\!\cdots\!96 $ (74523726599953098034v^19 - 808924802686748208939v^17 + 9732744956558897388850v^15 - 28537797399836103993346v^13 + 188184284786308469467020v^11 + 482603200706106553178454v^9 + 1158726816780825489341476v^7 + 11936951114800868504738151v^5 + 5043486507440718766497398v^3 + 17860049259417891672742586v) / 20295239243315412906242196
$\beta_{19}$	$=$	$ ( - 98\!\cdots\!07 \nu^{19} + \cdots + 77\!\cdots\!05 \nu ) / 20\!\cdots\!96 $ (-98627981057067316607v^19 + 1264987454461376683791v^17 - 15581695930441238455211v^15 + 74120767982285251088195v^13 - 461011267867857768796842v^11 + 826057583047452604194288v^9 - 5564157234602095533234653v^7 + 6013452215059329429102057v^5 - 24727191323555564063287903v^3 + 77994435824135538731214005v) / 20295239243315412906242196

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

$\nu$	$=$	$ \beta_{12} + \beta_{11} $ b12 + b11
$\nu^{2}$	$=$	$ \beta_{2} - 2\beta _1 + 1 $ b2 - 2*b1 + 1
$\nu^{3}$	$=$	$ \beta_{15} - 3\beta_{14} - \beta_{12} + 7\beta_{11} $ b15 - 3b14 - b12 + 7b11
$\nu^{4}$	$=$	$ -4\beta_{9} + \beta_{7} + 4\beta_{6} + \beta_{5} - 5\beta_{2} - 12\beta _1 - 18 $ -4b9 + b7 + 4b6 + b5 - 5b2 - 12b1 - 18
$\nu^{5}$	$=$	$ \beta_{19} + \beta_{18} - 5 \beta_{17} - \beta_{16} - 11 \beta_{15} - 20 \beta_{14} + \cdots + 5 \beta_{10} $ b19 + b18 - 5b17 - b16 - 11b15 - 20b14 - 10b13 - 51b12 + 14b11 + 5*b10
$\nu^{6}$	$=$	$ - 8 \beta_{9} - 6 \beta_{8} - 20 \beta_{7} + 20 \beta_{6} - 19 \beta_{5} + 2 \beta_{4} + 12 \beta_{3} + \cdots - 171 $ -8b9 - 6b8 - 20b7 + 20b6 - 19b5 + 2b4 + 12b3 - 104b2 + 2*b1 - 171
$\nu^{7}$	$=$	$ - 31 \beta_{19} - 30 \beta_{18} + 14 \beta_{17} + 29 \beta_{16} - 171 \beta_{15} + 84 \beta_{14} + \cdots + 14 \beta_{10} $ -31b19 - 30b18 + 14b17 + 29b16 - 171b15 + 84b14 - 14b13 - 395b12 - 183b11 + 14b10
$\nu^{8}$	$=$	$ 224 \beta_{9} + 16 \beta_{8} - 201 \beta_{7} - 272 \beta_{6} - 245 \beta_{5} - 85 \beta_{4} - 16 \beta_{3} + \cdots - 590 $ 224b9 + 16b8 - 201b7 - 272b6 - 245b5 - 85b4 - 16b3 - 538b2 + 760*b1 - 590
$\nu^{9}$	$=$	$ - 197 \beta_{19} - 254 \beta_{18} + 372 \beta_{17} + 314 \beta_{16} - 488 \beta_{15} + \cdots - 558 \beta_{10} $ -197b19 - 254b18 + 372b17 + 314b16 - 488b15 + 2235b14 + 1050b13 - 347b12 - 2699b11 - 558b10
$\nu^{10}$	$=$	$ 2526 \beta_{9} + 812 \beta_{8} + 188 \beta_{7} - 3970 \beta_{6} + 51 \beta_{5} - 499 \beta_{4} + \cdots + 4913 $ 2526b9 + 812b8 + 188b7 - 3970b6 + 51b5 - 499b4 - 1864b3 + 2414b2 + 6958*b1 + 4913
$\nu^{11}$	$=$	$ 1495 \beta_{19} + 1313 \beta_{18} + 1419 \beta_{17} - 1416 \beta_{16} + 9149 \beta_{15} + \cdots - 6325 \beta_{10} $ 1495b19 + 1313b18 + 1419b17 - 1416b16 + 9149b15 + 14190b14 + 9702b13 + 23845b12 - 14354b11 - 6325b10
$\nu^{12}$	$=$	$ 6536 \beta_{9} + 5012 \beta_{8} + 18206 \beta_{7} - 13248 \beta_{6} + 21762 \beta_{5} + 7556 \beta_{4} + \cdots + 84875 $ 6536b9 + 5012b8 + 18206b7 - 13248b6 + 21762b5 + 7556b4 - 11776b3 + 61284b2 + 12564*b1 + 84875
$\nu^{13}$	$=$	$ 28474 \beta_{19} + 35786 \beta_{18} - 9178 \beta_{17} - 41094 \beta_{16} + 121036 \beta_{15} + \cdots - 2574 \beta_{10} $ 28474b19 + 35786b18 - 9178b17 - 41094b16 + 121036b15 - 35594b14 - 5252b13 + 250777b12 + 56211b11 - 2574b10
$\nu^{14}$	$=$	$ - 128156 \beta_{9} - 33212 \beta_{8} + 150218 \beta_{7} + 190356 \beta_{6} + 185352 \beta_{5} + \cdots + 471335 $ -128156b9 - 33212b8 + 150218b7 + 190356b6 + 185352b5 + 101252b4 + 75688b3 + 399791b2 - 380750*b1 + 471335
$\nu^{15}$	$=$	$ 123152 \beta_{19} + 185046 \beta_{18} - 163634 \beta_{17} - 210916 \beta_{16} + 416849 \beta_{15} + \cdots + 521738 \beta_{10} $ 123152b19 + 185046b18 - 163634b17 - 210916b16 + 416849b15 - 1410639b14 - 827104b13 + 659635b12 + 1600169b11 + 521738b10
$\nu^{16}$	$=$	$ - 1704336 \beta_{9} - 706784 \beta_{8} - 83477 \beta_{7} + 2777744 \beta_{6} - 76187 \beta_{5} + \cdots - 1465224 $ -1704336b9 - 706784b8 - 83477b7 + 2777744b6 - 76187b5 + 125622b4 + 1607040b3 - 765725b2 - 4559344*b1 - 1465224
$\nu^{17}$	$=$	$ - 1149595 \beta_{19} - 1371423 \beta_{18} - 961605 \beta_{17} + 1557605 \beta_{16} + \cdots + 4767089 \beta_{10} $ -1149595b19 - 1371423b18 - 961605b17 + 1557605b16 - 5407469b15 - 10603342b14 - 7010154b13 - 12198839b12 + 11167726b11 + 4767089b10
$\nu^{18}$	$=$	$ - 6345468 \beta_{9} - 3395666 \beta_{8} - 12507586 \beta_{7} + 11056432 \beta_{6} - 15124823 \beta_{5} + \cdots - 48347593 $ -6345468b9 - 3395666b8 - 12507586b7 + 11056432b6 - 15124823b5 - 7142522b4 + 7700428b3 - 37342172b2 - 15408482*b1 - 48347593
$\nu^{19}$	$=$	$ - 19835787 \beta_{19} - 26441440 \beta_{18} + 2009060 \beta_{17} + 29967773 \beta_{16} + \cdots + 1187120 \beta_{10} $ -19835787b19 - 26441440b18 + 2009060b17 + 29967773b16 - 82376481b15 + 7298242b14 + 436354b13 - 166356487b12 - 12620673b11 + 1187120b10

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/531\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$-1$	$-1$

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} $

530.1

2.59182	+	1.41421i
2.59182	−	1.41421i
2.08812	−	1.41421i
2.08812	+	1.41421i
1.45273	−	1.41421i
1.45273	+	1.41421i
1.23509	+	1.41421i
1.23509	−	1.41421i
0.535103	+	1.41421i
0.535103	−	1.41421i
−0.535103	+	1.41421i
−0.535103	−	1.41421i
−1.23509	+	1.41421i
−1.23509	−	1.41421i
−1.45273	−	1.41421i
−1.45273	+	1.41421i
−2.08812	−	1.41421i
−2.08812	+	1.41421i
−2.59182	+	1.41421i
−2.59182	−	1.41421i

−2.59182

4.71754

−

1.33755i

−1.10356

−7.04337

3.46670i

530.2

−2.59182

4.71754

1.33755i

−1.10356

−7.04337

−

3.46670i

530.3

−2.08812

2.36026

−

2.45675i

4.05068

−0.752258

5.12999i

530.4

−2.08812

2.36026

2.45675i

4.05068

−0.752258

−

5.12999i

530.5

−1.45273

0.110435

−

2.26259i

−4.85531

2.74503

3.28694i

530.6

−1.45273

0.110435

2.26259i

−4.85531

2.74503

−

3.28694i

530.7

−1.23509

−0.474562

−

3.32457i

−1.59444

3.05630

4.10613i

530.8

−1.23509

−0.474562

3.32457i

−1.59444

3.05630

−

4.10613i

530.9

−0.535103

−1.71366

−

0.0572139i

1.50263

1.98719

0.0306154i

530.10

−0.535103

−1.71366

0.0572139i

1.50263

1.98719

−

0.0306154i

530.11

0.535103

−1.71366

−

0.0572139i

1.50263

−1.98719

−

0.0306154i

530.12

0.535103

−1.71366

0.0572139i

1.50263

−1.98719

0.0306154i

530.13

1.23509

−0.474562

−

3.32457i

−1.59444

−3.05630

−

4.10613i

530.14

1.23509

−0.474562

3.32457i

−1.59444

−3.05630

4.10613i

530.15

1.45273

0.110435

−

2.26259i

−4.85531

−2.74503

−

3.28694i

530.16

1.45273

0.110435

2.26259i

−4.85531

−2.74503

3.28694i

530.17

2.08812

2.36026

−

2.45675i

4.05068

0.752258

−

5.12999i

530.18

2.08812

2.36026

2.45675i

4.05068

0.752258

5.12999i

530.19

2.59182

4.71754

−

1.33755i

−1.10356

7.04337

−

3.46670i

530.20

2.59182

4.71754

1.33755i

−1.10356

7.04337

3.46670i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
59.b	odd	2	1	inner
177.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	531.2.d.a	✓	20
3.b	odd	2	1	inner	531.2.d.a	✓	20
59.b	odd	2	1	inner	531.2.d.a	✓	20
177.d	even	2	1	inner	531.2.d.a	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
531.2.d.a	✓	20	1.a	even	1	1	trivial
531.2.d.a	✓	20	3.b	odd	2	1	inner
531.2.d.a	✓	20	59.b	odd	2	1	inner
531.2.d.a	✓	20	177.d	even	2	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{2}^{\mathrm{new}}(531, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{10} - 15 T^{8} + \cdots - 27)^{2} $ (T^10 - 15T^8 + 77T^6 - 163T^4 + 135T^2 - 27)^2
$3$	$ T^{20} $ T^20
$5$	$ (T^{10} + 24 T^{8} + \cdots + 2)^{2} $ (T^10 + 24T^8 + 194T^6 + 618T^4 + 613T^2 + 2)^2
$7$	$ (T^{5} + 2 T^{4} - 21 T^{3} + \cdots + 52)^{4} $ (T^5 + 2T^4 - 21T^3 - 28T^2 + 43T + 52)^4
$11$	$ (T^{10} - 58 T^{8} + \cdots - 38988)^{2} $ (T^10 - 58T^8 + 1196T^6 - 10534T^4 + 38115T^2 - 38988)^2
$13$	$ (T^{10} + 64 T^{8} + \cdots + 864)^{2} $ (T^10 + 64T^8 + 1048T^6 + 5048T^4 + 5931T^2 + 864)^2
$17$	$ (T^{10} + 70 T^{8} + \cdots + 34322)^{2} $ (T^10 + 70T^8 + 1467T^6 + 12136T^4 + 37477T^2 + 34322)^2
$19$	$ (T^{5} - 35 T^{3} + \cdots - 412)^{4} $ (T^5 - 35T^3 + 38T^2 + 251*T - 412)^4
$23$	$ (T^{10} - 146 T^{8} + \cdots - 1581228)^{2} $ (T^10 - 146T^8 + 6787T^6 - 109876T^4 + 708939T^2 - 1581228)^2
$29$	$ (T^{10} + 76 T^{8} + \cdots + 3698)^{2} $ (T^10 + 76T^8 + 1695T^6 + 9562T^4 + 16681T^2 + 3698)^2
$31$	$ (T^{10} + 154 T^{8} + \cdots + 1888326)^{2} $ (T^10 + 154T^8 + 8221T^6 + 173066T^4 + 1155555T^2 + 1888326)^2
$37$	$ (T^{10} + 196 T^{8} + \cdots + 32350104)^{2} $ (T^10 + 196T^8 + 14083T^6 + 456260T^4 + 6563835T^2 + 32350104)^2
$41$	$ (T^{10} + 308 T^{8} + \cdots + 5126402)^{2} $ (T^10 + 308T^8 + 33059T^6 + 1386342T^4 + 15970721T^2 + 5126402)^2
$43$	$ (T^{10} + 266 T^{8} + \cdots + 68830614)^{2} $ (T^10 + 266T^8 + 24104T^6 + 875072T^4 + 13181715T^2 + 68830614)^2
$47$	$ (T^{10} - 196 T^{8} + \cdots - 3345408)^{2} $ (T^10 - 196T^8 + 12005T^6 - 265882T^4 + 2032731T^2 - 3345408)^2
$53$	$ (T^{10} + 332 T^{8} + \cdots + 211233458)^{2} $ (T^10 + 332T^8 + 38522T^6 + 1871382T^4 + 36686453T^2 + 211233458)^2
$59$	$ T^{20} + \cdots + 51\!\cdots\!01 $ T^20 + 66T^18 - 1643T^16 + 74616T^14 - 5923198T^12 - 2067604916T^10 - 20618652238T^8 + 904149008376T^6 - 69302616772163T^4 + 9690808881885186*T^2 + 511116753300641401
$61$	$ (T^{10} + 328 T^{8} + \cdots + 2032344)^{2} $ (T^10 + 328T^8 + 34485T^6 + 1171256T^4 + 4384539T^2 + 2032344)^2
$67$	$ (T^{10} + 346 T^{8} + \cdots + 196379046)^{2} $ (T^10 + 346T^8 + 36256T^6 + 1607168T^4 + 30699315T^2 + 196379046)^2
$71$	$ (T^{10} + 298 T^{8} + \cdots + 1492992)^{2} $ (T^10 + 298T^8 + 27070T^6 + 731178T^4 + 2165049T^2 + 1492992)^2
$73$	$ (T^{10} + 568 T^{8} + \cdots + 1932353496)^{2} $ (T^10 + 568T^8 + 113665T^6 + 9213248T^4 + 248601339T^2 + 1932353496)^2
$79$	$ (T^{5} - 172 T^{3} + \cdots + 4972)^{4} $ (T^5 - 172T^3 - 420T^2 + 2215*T + 4972)^4
$83$	$ (T^{10} - 152 T^{8} + \cdots - 38988)^{2} $ (T^10 - 152T^8 + 7725T^6 - 141586T^4 + 494739T^2 - 38988)^2
$89$	$ (T^{10} - 564 T^{8} + \cdots - 12064006188)^{2} $ (T^10 - 564T^8 + 122633T^6 - 12778210T^4 + 635233995T^2 - 12064006188)^2
$97$	$ (T^{10} + 720 T^{8} + \cdots + 11933889624)^{2} $ (T^10 + 720T^8 + 182912T^6 + 19644248T^4 + 858204531T^2 + 11933889624)^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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	531.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{12} q^{2} + (\beta_{2} + 1) q^{4} + \beta_{13} q^{5} - \beta_{4} q^{7} + ( - \beta_{15} - \beta_{12}) q^{8} + (\beta_{6} + \beta_{3}) q^{10} - \beta_{18} q^{11} + \beta_{6} q^{13} + (\beta_{18} - \beta_{16}) q^{14}+ \cdots + (3 \beta_{19} - 2 \beta_{16} + \cdots - \beta_{12}) q^{98}+O(q^{100}) \) q - b12 * q^2 + (b2 + 1) * q^4 + b13 * q^5 - b4 * q^7 + (-b15 - b12) * q^8 + (b6 + b3) * q^10 - b18 * q^11 + b6 * q^13 + (b18 - b16) * q^14 + (b7 + b5 + b2) * q^16 + (-b14 + b11 + b10) * q^17 + (-b5 - b2) * q^19 + (b14 + b13 - b10) * q^20 + (b7 + b4 - 1) * q^22 + (-b19 + b18) * q^23 + b7 * q^25 + (b17 + b14 + 2b13 + b11) q^26 + (-b7 - b5 - b4 + 1) * q^28 + (-b17 - b13 - b11) * q^29 + (b9 - b6 - b3) * q^31 + (-b19 - b18 + b16 - b15 - b12) * q^32 + (b9 + b8 - b6 - b3 + 2b1) q^34 + (-b17 - b13 - 2b10) q^35 + (b6 - b3 - b1) * q^37 + (b19 - b16 + 2b15 + 2b12) * q^38 + (-b9 - b8 - b1) * q^40 + (-b17 - b14 - b13 - b11 + 2b10) q^41 + (-b8 + b6 + 2b3 - b1) q^43 + (b16 - b15) * q^44 + (-b7 + b5 - 2b4 - b2 + 1) q^46 + (-b19 + b16 + 2b12) q^47 + (-b7 - 3b5 + b4 + 2) q^49 + (-b18 - b15 - b12) * q^50 + (-b9 + b6 + b3 - b1) * q^52 + (-b17 - 2b14 - b13 - b11 - b10) q^53 + (-b9 - 2b3) q^55 + (b19 + 2b15) q^56 + (-b6 - b1) * q^58 + (b19 - b17 + b15 - b14 + b13 + b12 + b11) * q^59 + (b8 + 2b6 + b3 - b1) q^61 + (-3b14 - 3b13 + 2b10) q^62 + (b5 + 2b4 + 1) q^64 + (-b19 + b18 - b15 + 2b12) q^65 + (-b9 + b8 - 2b3 - b1) q^67 + (-4b14 - 3b13 + 4b11 + 2b10) * q^68 + (-2b8 - b6 + 2b3 + 2b1) q^70 + (2b17 + b14 + b13 + 3b11) * q^71 + (-b9 + 2b8 + b6 - b3 - b1) q^73 + (2b17 + 2b14 + b13 - b11 + b10) * q^74 + (-2b7 - 2b5 - b4 - 3b2 - 4) q^76 + (-2b19 + b16 - b15 + 2b12) * q^77 + (b7 + b5 + 2b4 + 2b2 - 1) * q^79 + (2b14 - 2b13 - 3b11 - b10) q^80 + (b9 + 2b8 - 2b6 - 2b3 - b1) q^82 + (b16 - b15) * q^83 + (b7 - b4 + 2b2 - 1) q^85 + (-b17 + 3b14 + 4b13 - 4b11 - 4b10) * q^86 + (-b7 + 2b5 + 2b2 + 1) * q^88 + (2b16 - b15 - 2b12) * q^89 + (b9 - 2b8 - 3b6 + b3 - b1) * q^91 + (b19 + b18 - b16 + b15 + 2b12) q^92 + (2b5 + b4 - 3b2 - 6) * q^94 + (b17 + 5b11) q^95 + (b9 - b8 + 4b1) q^97 + (3b19 - 2b16 + 4b15 - b12) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{4} - 8 q^{7} + 4 q^{16} - 8 q^{22} + 4 q^{25} + 8 q^{28} + 44 q^{49} + 36 q^{64} - 96 q^{76} - 24 q^{85} + 16 q^{88} - 112 q^{94}+O(q^{100}) \) 20 * q + 20 * q^4 - 8 * q^7 + 4 * q^16 - 8 * q^22 + 4 * q^25 + 8 * q^28 + 44 * q^49 + 36 * q^64 - 96 * q^76 - 24 * q^85 + 16 * q^88 - 112 * q^94

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	531.2.d.a

Level	$531$
Weight	$2$
Character orbit	531.d
Analytic conductor	$4.240$
Analytic rank	$0$
Dimension	$20$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(4.24005634733\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - 10 x^{18} + 139 x^{16} - 476 x^{14} + 4681 x^{12} - 666 x^{10} + 82273 x^{8} + 168944 x^{6} + \cdots + 3374569 \) x^20 - 10x^18 + 139x^16 - 476x^14 + 4681x^12 - 666x^10 + 82273x^8 + 168944x^6 + 845867x^4 + 1613394*x^2 + 3374569
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{6}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 726777222591367 \nu^{18} + \cdots - 37\!\cdots\!13 ) / 36\!\cdots\!36 \) (-726777222591367v^18 + 5941008833830113v^16 - 91602358590346650v^14 + 184616922793156367v^12 - 3231573008264319840v^10 - 5764563651169605570v^8 - 79965005644882419861v^6 - 345545294857588001933v^4 - 2626931079401957575386*v^2 - 3718258825763016078713) / 3682678142499621285836
\(\beta_{2}\)	\(=\)	\( ( - 726777222591367 \nu^{18} + \cdots - 55\!\cdots\!31 ) / 18\!\cdots\!18 \) (-726777222591367v^18 + 5941008833830113v^16 - 91602358590346650v^14 + 184616922793156367v^12 - 3231573008264319840v^10 - 5764563651169605570v^8 - 79965005644882419861v^6 - 345545294857588001933v^4 - 785592008152146932468*v^2 - 5559597897012826721631) / 1841339071249810642918
\(\beta_{3}\)	\(=\)	\( ( 11\!\cdots\!86 \nu^{18} + \cdots + 97\!\cdots\!39 ) / 22\!\cdots\!16 \) (11702514375950686v^18 + 191525605279787919v^16 - 1828324832041991813v^14 + 40947502777856166881v^12 - 153900951341063137884v^10 + 1633994733135824878086v^8 - 1329585349562152680146v^6 + 22289865192592561762665v^4 + 28952814023750531522531*v^2 + 97828433192122122178439) / 22096068854997727715016
\(\beta_{4}\)	\(=\)	\( ( - 93\!\cdots\!23 \nu^{18} + \cdots - 20\!\cdots\!11 ) / 55\!\cdots\!54 \) (-9362148199123223v^18 + 19191897951128837v^16 - 543021799440662688v^14 - 4287224488630087161v^12 - 21241652543593451848v^10 - 173791460355919676476v^8 - 875902073308331859993v^6 - 4327628291146543048391v^4 - 10028152012182697339842*v^2 - 20518677369002153331011) / 5524017213749431928754
\(\beta_{5}\)	\(=\)	\( ( - 22\!\cdots\!53 \nu^{18} + \cdots + 38\!\cdots\!41 ) / 55\!\cdots\!54 \) (-22400930441398253v^18 + 393624863282125412v^16 - 4589423775990728229v^14 + 29868406304144604483v^12 - 133974055694978085472v^10 + 431331440654920709576v^8 - 317152138506556243869v^6 + 3848613617077735072822v^4 + 10992841018618551425181*v^2 + 38541873287962536796141) / 5524017213749431928754
\(\beta_{6}\)	\(=\)	\( ( 15\!\cdots\!51 \nu^{18} + \cdots - 11\!\cdots\!65 ) / 22\!\cdots\!16 \) (153432456366075751v^18 - 1564699734951504606v^16 + 19796813498773297006v^14 - 61259897928339466411v^12 + 532099418306169203562v^10 + 13724608647346152510v^8 + 8776217525678006044873v^6 + 8161481150131271265696v^4 + 47465062693384228294784*v^2 - 11975032980151079951965) / 22096068854997727715016
\(\beta_{7}\)	\(=\)	\( ( 38\!\cdots\!22 \nu^{18} + \cdots - 60\!\cdots\!00 ) / 55\!\cdots\!54 \) (38423998434412022v^18 - 533160102699398963v^16 + 6644327398510161729v^14 - 34937937732203550072v^12 + 203124381279207904240v^10 - 362126019430061021786v^8 + 1619433462203003526372v^6 + 1005353064210506448539v^4 - 342389642582343308427*v^2 - 6080299787444652370000) / 5524017213749431928754
\(\beta_{8}\)	\(=\)	\( ( 92\!\cdots\!25 \nu^{18} + \cdots + 72\!\cdots\!28 ) / 11\!\cdots\!08 \) (92793018209944625v^18 - 687134536423822104v^16 + 8427164022449520308v^14 + 9008183684664779116v^12 + 74034372756108972030v^10 + 1635058748526631823514v^8 + 1842743426619832004243v^6 + 24925223384169194306136v^4 + 34988957025029982381592*v^2 + 72654097023721365828928) / 11048034427498863857508
\(\beta_{9}\)	\(=\)	\( ( 19\!\cdots\!31 \nu^{18} + \cdots + 71\!\cdots\!03 ) / 22\!\cdots\!16 \) (193439172704604631v^18 - 1900288265885171886v^16 + 24874594954774169956v^14 - 72421787808572572111v^12 + 707891653163121577050v^10 + 273160630360802824110v^8 + 12717344035655573182789v^6 + 18894425347970484922092v^4 + 117184153620937876755506*v^2 + 71376857991803719997903) / 22096068854997727715016
\(\beta_{10}\)	\(=\)	\( ( 14\!\cdots\!37 \nu^{19} + \cdots - 12\!\cdots\!33 \nu ) / 13\!\cdots\!64 \) (1453912374027885337v^19 - 200206497462219979484v^17 + 2485243350093145468378v^15 - 28085845778721192705159v^13 + 118957607460567016231670v^11 - 666984244627782686146770v^9 + 366868753568888535166035v^7 - 5527796614148455191857106v^5 - 11317687070878185333291868v^3 - 12798264250548058482872233v) / 13530159495543608604161464
\(\beta_{11}\)	\(=\)	\( ( 19\!\cdots\!07 \nu^{19} + \cdots + 80\!\cdots\!40 \nu ) / 67\!\cdots\!32 \) (1985355176503117307v^19 - 18518462007130831891v^17 + 265050736306187388092v^15 - 776755531285017042082v^13 + 8954306294040063867888v^11 + 4614153068630479419618v^9 + 173930629863639535630901v^7 + 482309560308791655598465v^5 + 2314113133736551486671090v^3 + 8028832522500466510394040v) / 6765079747771804302080732
\(\beta_{12}\)	\(=\)	\( ( - 19\!\cdots\!07 \nu^{19} + \cdots - 12\!\cdots\!08 \nu ) / 67\!\cdots\!32 \) (-1985355176503117307v^19 + 18518462007130831891v^17 - 265050736306187388092v^15 + 776755531285017042082v^13 - 8954306294040063867888v^11 - 4614153068630479419618v^9 - 173930629863639535630901v^7 - 482309560308791655598465v^5 - 2314113133736551486671090v^3 - 1263752774728662208313308v) / 6765079747771804302080732
\(\beta_{13}\)	\(=\)	\( ( - 37\!\cdots\!70 \nu^{19} + \cdots - 11\!\cdots\!47 \nu ) / 40\!\cdots\!92 \) (-37187856799063203570v^19 - 96076410145864892113v^17 + 245764276534215785165v^15 - 49029779750411193766913v^13 + 82710561737877519066524v^11 - 1654424832433422721121122v^9 - 2916993547352654756555014v^7 - 22292562500331931774882991v^5 - 59115491240349678772148291v^3 - 119602908757035216172794247v) / 40590478486630825812484392
\(\beta_{14}\)	\(=\)	\( ( 85\!\cdots\!56 \nu^{19} + \cdots + 19\!\cdots\!55 \nu ) / 67\!\cdots\!32 \) (8591686124615245356v^19 - 81678676807908241874v^17 + 1156980148800470144410v^15 - 3544636369254056964231v^13 + 38835131854018763793360v^11 + 12481261915953831666000v^9 + 722757433948548672869848v^7 + 1776781094890569118441404v^5 + 6744893275821361367371368v^3 + 19783561654032063411212955v) / 6765079747771804302080732
\(\beta_{15}\)	\(=\)	\( ( 98\!\cdots\!12 \nu^{19} + \cdots + 18\!\cdots\!77 \nu ) / 67\!\cdots\!32 \) (9892216961820797612v^19 - 96888334366678070494v^17 + 1350534555951911328494v^15 - 4419864857482034556037v^13 + 44870945209735780436976v^11 + 530561198817659641056v^9 + 776827262936529733562336v^7 + 1471866802201374110536492v^5 + 8486854505343476510826116v^3 + 1885104529864262452567277v) / 6765079747771804302080732
\(\beta_{16}\)	\(=\)	\( ( 50\!\cdots\!21 \nu^{19} + \cdots + 61\!\cdots\!49 \nu ) / 20\!\cdots\!96 \) (50537545069155461321v^19 - 375778886191321399266v^17 + 5085686413870215258452v^15 + 154428398823064316455v^13 + 83111686646821539471690v^11 + 897362386111149617256354v^9 + 848535058433236063982075v^7 + 18581808692851784878046496v^5 + 19461165530419986305223742v^3 + 61649815710789182072688049v) / 20295239243315412906242196
\(\beta_{17}\)	\(=\)	\( ( - 13\!\cdots\!83 \nu^{19} + \cdots - 72\!\cdots\!43 \nu ) / 40\!\cdots\!92 \) (-133039296209855075883v^19 + 1719044163565193243420v^17 - 22330275735783914425906v^15 + 114799996578738136528543v^13 - 776827519315239976072102v^11 + 1525766535586353527332754v^9 - 9200701991253304177230877v^7 - 4820165046650349930979850v^5 - 24783120529344677895857264v^3 - 72969910788043865490125443v) / 40590478486630825812484392
\(\beta_{18}\)	\(=\)	\( ( 74\!\cdots\!34 \nu^{19} + \cdots + 17\!\cdots\!86 \nu ) / 20\!\cdots\!96 \) (74523726599953098034v^19 - 808924802686748208939v^17 + 9732744956558897388850v^15 - 28537797399836103993346v^13 + 188184284786308469467020v^11 + 482603200706106553178454v^9 + 1158726816780825489341476v^7 + 11936951114800868504738151v^5 + 5043486507440718766497398v^3 + 17860049259417891672742586v) / 20295239243315412906242196
\(\beta_{19}\)	\(=\)	\( ( - 98\!\cdots\!07 \nu^{19} + \cdots + 77\!\cdots\!05 \nu ) / 20\!\cdots\!96 \) (-98627981057067316607v^19 + 1264987454461376683791v^17 - 15581695930441238455211v^15 + 74120767982285251088195v^13 - 461011267867857768796842v^11 + 826057583047452604194288v^9 - 5564157234602095533234653v^7 + 6013452215059329429102057v^5 - 24727191323555564063287903v^3 + 77994435824135538731214005v) / 20295239243315412906242196

\(\nu\)	\(=\)	\( \beta_{12} + \beta_{11} \) b12 + b11
\(\nu^{2}\)	\(=\)	\( \beta_{2} - 2\beta _1 + 1 \) b2 - 2*b1 + 1
\(\nu^{3}\)	\(=\)	\( \beta_{15} - 3\beta_{14} - \beta_{12} + 7\beta_{11} \) b15 - 3b14 - b12 + 7b11
\(\nu^{4}\)	\(=\)	\( -4\beta_{9} + \beta_{7} + 4\beta_{6} + \beta_{5} - 5\beta_{2} - 12\beta _1 - 18 \) -4b9 + b7 + 4b6 + b5 - 5b2 - 12b1 - 18
\(\nu^{5}\)	\(=\)	\( \beta_{19} + \beta_{18} - 5 \beta_{17} - \beta_{16} - 11 \beta_{15} - 20 \beta_{14} + \cdots + 5 \beta_{10} \) b19 + b18 - 5b17 - b16 - 11b15 - 20b14 - 10b13 - 51b12 + 14b11 + 5*b10
\(\nu^{6}\)	\(=\)	\( - 8 \beta_{9} - 6 \beta_{8} - 20 \beta_{7} + 20 \beta_{6} - 19 \beta_{5} + 2 \beta_{4} + 12 \beta_{3} + \cdots - 171 \) -8b9 - 6b8 - 20b7 + 20b6 - 19b5 + 2b4 + 12b3 - 104b2 + 2*b1 - 171
\(\nu^{7}\)	\(=\)	\( - 31 \beta_{19} - 30 \beta_{18} + 14 \beta_{17} + 29 \beta_{16} - 171 \beta_{15} + 84 \beta_{14} + \cdots + 14 \beta_{10} \) -31b19 - 30b18 + 14b17 + 29b16 - 171b15 + 84b14 - 14b13 - 395b12 - 183b11 + 14b10
\(\nu^{8}\)	\(=\)	\( 224 \beta_{9} + 16 \beta_{8} - 201 \beta_{7} - 272 \beta_{6} - 245 \beta_{5} - 85 \beta_{4} - 16 \beta_{3} + \cdots - 590 \) 224b9 + 16b8 - 201b7 - 272b6 - 245b5 - 85b4 - 16b3 - 538b2 + 760*b1 - 590
\(\nu^{9}\)	\(=\)	\( - 197 \beta_{19} - 254 \beta_{18} + 372 \beta_{17} + 314 \beta_{16} - 488 \beta_{15} + \cdots - 558 \beta_{10} \) -197b19 - 254b18 + 372b17 + 314b16 - 488b15 + 2235b14 + 1050b13 - 347b12 - 2699b11 - 558b10
\(\nu^{10}\)	\(=\)	\( 2526 \beta_{9} + 812 \beta_{8} + 188 \beta_{7} - 3970 \beta_{6} + 51 \beta_{5} - 499 \beta_{4} + \cdots + 4913 \) 2526b9 + 812b8 + 188b7 - 3970b6 + 51b5 - 499b4 - 1864b3 + 2414b2 + 6958*b1 + 4913
\(\nu^{11}\)	\(=\)	\( 1495 \beta_{19} + 1313 \beta_{18} + 1419 \beta_{17} - 1416 \beta_{16} + 9149 \beta_{15} + \cdots - 6325 \beta_{10} \) 1495b19 + 1313b18 + 1419b17 - 1416b16 + 9149b15 + 14190b14 + 9702b13 + 23845b12 - 14354b11 - 6325b10
\(\nu^{12}\)	\(=\)	\( 6536 \beta_{9} + 5012 \beta_{8} + 18206 \beta_{7} - 13248 \beta_{6} + 21762 \beta_{5} + 7556 \beta_{4} + \cdots + 84875 \) 6536b9 + 5012b8 + 18206b7 - 13248b6 + 21762b5 + 7556b4 - 11776b3 + 61284b2 + 12564*b1 + 84875
\(\nu^{13}\)	\(=\)	\( 28474 \beta_{19} + 35786 \beta_{18} - 9178 \beta_{17} - 41094 \beta_{16} + 121036 \beta_{15} + \cdots - 2574 \beta_{10} \) 28474b19 + 35786b18 - 9178b17 - 41094b16 + 121036b15 - 35594b14 - 5252b13 + 250777b12 + 56211b11 - 2574b10
\(\nu^{14}\)	\(=\)	\( - 128156 \beta_{9} - 33212 \beta_{8} + 150218 \beta_{7} + 190356 \beta_{6} + 185352 \beta_{5} + \cdots + 471335 \) -128156b9 - 33212b8 + 150218b7 + 190356b6 + 185352b5 + 101252b4 + 75688b3 + 399791b2 - 380750*b1 + 471335
\(\nu^{15}\)	\(=\)	\( 123152 \beta_{19} + 185046 \beta_{18} - 163634 \beta_{17} - 210916 \beta_{16} + 416849 \beta_{15} + \cdots + 521738 \beta_{10} \) 123152b19 + 185046b18 - 163634b17 - 210916b16 + 416849b15 - 1410639b14 - 827104b13 + 659635b12 + 1600169b11 + 521738b10
\(\nu^{16}\)	\(=\)	\( - 1704336 \beta_{9} - 706784 \beta_{8} - 83477 \beta_{7} + 2777744 \beta_{6} - 76187 \beta_{5} + \cdots - 1465224 \) -1704336b9 - 706784b8 - 83477b7 + 2777744b6 - 76187b5 + 125622b4 + 1607040b3 - 765725b2 - 4559344*b1 - 1465224
\(\nu^{17}\)	\(=\)	\( - 1149595 \beta_{19} - 1371423 \beta_{18} - 961605 \beta_{17} + 1557605 \beta_{16} + \cdots + 4767089 \beta_{10} \) -1149595b19 - 1371423b18 - 961605b17 + 1557605b16 - 5407469b15 - 10603342b14 - 7010154b13 - 12198839b12 + 11167726b11 + 4767089b10
\(\nu^{18}\)	\(=\)	\( - 6345468 \beta_{9} - 3395666 \beta_{8} - 12507586 \beta_{7} + 11056432 \beta_{6} - 15124823 \beta_{5} + \cdots - 48347593 \) -6345468b9 - 3395666b8 - 12507586b7 + 11056432b6 - 15124823b5 - 7142522b4 + 7700428b3 - 37342172b2 - 15408482*b1 - 48347593
\(\nu^{19}\)	\(=\)	\( - 19835787 \beta_{19} - 26441440 \beta_{18} + 2009060 \beta_{17} + 29967773 \beta_{16} + \cdots + 1187120 \beta_{10} \) -19835787b19 - 26441440b18 + 2009060b17 + 29967773b16 - 82376481b15 + 7298242b14 + 436354b13 - 166356487b12 - 12620673b11 + 1187120b10

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 530.20
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{10} - 15 T^{8} + \cdots - 27)^{2} \) (T^10 - 15T^8 + 77T^6 - 163T^4 + 135T^2 - 27)^2
$3$	\( T^{20} \) T^20
$5$	\( (T^{10} + 24 T^{8} + \cdots + 2)^{2} \) (T^10 + 24T^8 + 194T^6 + 618T^4 + 613T^2 + 2)^2
$7$	\( (T^{5} + 2 T^{4} - 21 T^{3} + \cdots + 52)^{4} \) (T^5 + 2T^4 - 21T^3 - 28T^2 + 43T + 52)^4
$11$	\( (T^{10} - 58 T^{8} + \cdots - 38988)^{2} \) (T^10 - 58T^8 + 1196T^6 - 10534T^4 + 38115T^2 - 38988)^2
$13$	\( (T^{10} + 64 T^{8} + \cdots + 864)^{2} \) (T^10 + 64T^8 + 1048T^6 + 5048T^4 + 5931T^2 + 864)^2
$17$	\( (T^{10} + 70 T^{8} + \cdots + 34322)^{2} \) (T^10 + 70T^8 + 1467T^6 + 12136T^4 + 37477T^2 + 34322)^2
$19$	\( (T^{5} - 35 T^{3} + \cdots - 412)^{4} \) (T^5 - 35T^3 + 38T^2 + 251*T - 412)^4
$23$	\( (T^{10} - 146 T^{8} + \cdots - 1581228)^{2} \) (T^10 - 146T^8 + 6787T^6 - 109876T^4 + 708939T^2 - 1581228)^2
$29$	\( (T^{10} + 76 T^{8} + \cdots + 3698)^{2} \) (T^10 + 76T^8 + 1695T^6 + 9562T^4 + 16681T^2 + 3698)^2
$31$	\( (T^{10} + 154 T^{8} + \cdots + 1888326)^{2} \) (T^10 + 154T^8 + 8221T^6 + 173066T^4 + 1155555T^2 + 1888326)^2
$37$	\( (T^{10} + 196 T^{8} + \cdots + 32350104)^{2} \) (T^10 + 196T^8 + 14083T^6 + 456260T^4 + 6563835T^2 + 32350104)^2
$41$	\( (T^{10} + 308 T^{8} + \cdots + 5126402)^{2} \) (T^10 + 308T^8 + 33059T^6 + 1386342T^4 + 15970721T^2 + 5126402)^2
$43$	\( (T^{10} + 266 T^{8} + \cdots + 68830614)^{2} \) (T^10 + 266T^8 + 24104T^6 + 875072T^4 + 13181715T^2 + 68830614)^2
$47$	\( (T^{10} - 196 T^{8} + \cdots - 3345408)^{2} \) (T^10 - 196T^8 + 12005T^6 - 265882T^4 + 2032731T^2 - 3345408)^2
$53$	\( (T^{10} + 332 T^{8} + \cdots + 211233458)^{2} \) (T^10 + 332T^8 + 38522T^6 + 1871382T^4 + 36686453T^2 + 211233458)^2
$59$	\( T^{20} + \cdots + 51\!\cdots\!01 \) T^20 + 66T^18 - 1643T^16 + 74616T^14 - 5923198T^12 - 2067604916T^10 - 20618652238T^8 + 904149008376T^6 - 69302616772163T^4 + 9690808881885186*T^2 + 511116753300641401
$61$	\( (T^{10} + 328 T^{8} + \cdots + 2032344)^{2} \) (T^10 + 328T^8 + 34485T^6 + 1171256T^4 + 4384539T^2 + 2032344)^2
$67$	\( (T^{10} + 346 T^{8} + \cdots + 196379046)^{2} \) (T^10 + 346T^8 + 36256T^6 + 1607168T^4 + 30699315T^2 + 196379046)^2
$71$	\( (T^{10} + 298 T^{8} + \cdots + 1492992)^{2} \) (T^10 + 298T^8 + 27070T^6 + 731178T^4 + 2165049T^2 + 1492992)^2
$73$	\( (T^{10} + 568 T^{8} + \cdots + 1932353496)^{2} \) (T^10 + 568T^8 + 113665T^6 + 9213248T^4 + 248601339T^2 + 1932353496)^2
$79$	\( (T^{5} - 172 T^{3} + \cdots + 4972)^{4} \) (T^5 - 172T^3 - 420T^2 + 2215*T + 4972)^4
$83$	\( (T^{10} - 152 T^{8} + \cdots - 38988)^{2} \) (T^10 - 152T^8 + 7725T^6 - 141586T^4 + 494739T^2 - 38988)^2
$89$	\( (T^{10} - 564 T^{8} + \cdots - 12064006188)^{2} \) (T^10 - 564T^8 + 122633T^6 - 12778210T^4 + 635233995T^2 - 12064006188)^2
$97$	\( (T^{10} + 720 T^{8} + \cdots + 11933889624)^{2} \) (T^10 + 720T^8 + 182912T^6 + 19644248T^4 + 858204531T^2 + 11933889624)^2
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\(n\)	\(119\)	\(415\)
\(\chi(n)\)	\(-1\)	\(-1\)