LMFDB - Newform orbit 1340.2.i.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1340,2,Mod(841,1340)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1340, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1340.841");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1340 = 2^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1340.i (of order $3$, degree $2$, 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:	$10.6999538709$
Analytic rank:	$0$
Dimension:	$18$
Relative dimension:	$9$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} + 20 x^{16} - 19 x^{15} + 270 x^{14} - 260 x^{13} + 1914 x^{12} - 1982 x^{11} + \cdots + 2304 $ x^18 - x^17 + 20x^16 - 19x^15 + 270x^14 - 260x^13 + 1914x^12 - 1982x^11 + 9860x^10 - 9679x^9 + 30700x^8 - 26325x^7 + 65256x^6 - 44225x^5 + 62401x^4 - 8496x^3 + 13872x^2 - 2304x + 2304
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 $1,\beta_1,\ldots,\beta_{17}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{5} - \beta_1) q^{3} + q^{5} - \beta_{10} q^{7} + ( - \beta_{2} + 1) q^{9}+O(q^{10}) $ q + (b5 - b1) * q^3 + q^5 - b10 * q^7 + (-b2 + 1) * q^9	$ q + (\beta_{5} - \beta_1) q^{3} + q^{5} - \beta_{10} q^{7} + ( - \beta_{2} + 1) q^{9} + ( - \beta_{7} - 1) q^{11} + (\beta_{13} + \beta_{12} + \beta_{5}) q^{13} + (\beta_{5} - \beta_1) q^{15} + ( - \beta_{11} - \beta_{6}) q^{17} + ( - \beta_{17} - \beta_{4}) q^{19} + (\beta_{14} - \beta_{11} - \beta_1) q^{21} + (\beta_{15} + \beta_{9} + \beta_{5}) q^{23} + q^{25} + (\beta_{16} - \beta_{13} - \beta_{10} + \cdots + 1) q^{27}+ \cdots + ( - \beta_{15} - \beta_{7} + \beta_{2} - 1) q^{99}+O(q^{100}) $ q + (b5 - b1) * q^3 + q^5 - b10 * q^7 + (-b2 + 1) * q^9 + (-b7 - 1) * q^11 + (b13 + b12 + b5) * q^13 + (b5 - b1) * q^15 + (-b11 - b6) * q^17 + (-b17 - b4) * q^19 + (b14 - b11 - b1) * q^21 + (b15 + b9 + b5) * q^23 + q^25 + (b16 - b13 - b10 - b8 + b5 + b2 - b1 + 1) * q^27 + (-b15 + b14 + b12 + b4 + b2) * q^29 + (-b15 + b12 - b11 + b4 + b2) * q^31 + b1 * q^33 - b10 * q^35 + (b17 - b16 + b11 - b9 - 3b7 + b6 + b5 + b4) q^37 + (-b13 - b12 + b11 - 3b7 + b6 - b5 - b3) q^39 + (-b15 + b14 + b12 + b11 - b10 - b8 - b7 + b4 - b3 + b2 + b1 - 1) * q^41 + (-b17 + b16 - b13 - b10 + b8 - b5 + b2 + b1 + 2) * q^43 + (-b2 + 1) * q^45 + (b15 - b12 - b11 + b10 - b7 - b4 - b2 - 2b1 - 1) q^47 + (-b16 - b15 + 2b13 + 2b12 - 2b11 + b9 + 5b7 - 2b6 + b3) q^49 + (-b16 - b11 - b7 - b6 + b3) * q^51 + (-b17 - b16 + b13 + b10 - b5 + b1 - 1) * q^53 + (-b7 - 1) * q^55 + (-b13 - b12 - b9 - 2b7) q^57 + (b17 - b16 + b14 + b10 + b9 - b5 + b1 - 1) * q^59 + (-b17 - b16 + 2b13 + 2b12 - b11 - b6 - b4) * q^61 + (-b11 + b7 - b4 - 2b1 + 1) q^63 + (b13 + b12 + b5) * q^65 + (-b17 + b16 + b15 + b14 - 2b13 - 2b12 + b11 + b9 + 2b7 + b6 - b1 + 3) q^67 + (b17 + b16 - b15 + b13 + b12 - b7 - b5 + b4) * q^69 + (-2b15 + 3b12 - b10 + 2b7 + b4 + 2b2 + 2b1 + 2) q^71 + (b17 - 3b15 + 3b13 + 3b12 + b7 + b5 + b4) q^73 + (b5 - b1) * q^75 + b16 * q^77 + (-b14 + 2b12 + b10 + b8 + 2b7 + b3 + 2b1 + 2) q^79 + (-b17 + 2b14 + 2b9 - 2b6 - b5 - b2 + b1) q^81 + (-b16 - 2b15 + b13 + b12 - b9 - 2b5) * q^83 + (-b11 - b6) * q^85 + (b15 - b14 - b12 + b11 - b10 - b8 - b4 - b3 - b2) * q^87 + (-b17 - b16 + b14 + b13 + b10 + b9 + b8 + b6) * q^89 + (-b17 + 2b14 + 2b9 + 2b6 + 2b5 + b2 - 2b1 + 2) q^91 + (2b15 - b14 - b12 + b8 + b7 + b3 - 2b2 + 2b1 + 1) q^93 + (-b17 - b4) * q^95 + (-b15 + b9 - b7 + b5 - b3) * q^97 + (-b15 - b7 + b2 - 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 2 q^{3} + 18 q^{5} + 3 q^{7} + 24 q^{9}+O(q^{10}) $ 18 * q - 2 * q^3 + 18 * q^5 + 3 * q^7 + 24 * q^9	$ 18 q - 2 q^{3} + 18 q^{5} + 3 q^{7} + 24 q^{9} - 9 q^{11} + 3 q^{13} - 2 q^{15} + q^{19} + 3 q^{21} + 18 q^{25} + 10 q^{27} - 2 q^{29} - 6 q^{31} + q^{33} + 3 q^{35} + 18 q^{37} + 23 q^{39} - 6 q^{41} + 30 q^{43} + 24 q^{45} - 8 q^{47} - 32 q^{49} + 7 q^{51} - 12 q^{53} - 9 q^{55} + 10 q^{57} - 16 q^{59} + 6 q^{61} + 6 q^{63} + 3 q^{65} + 37 q^{67} + 19 q^{69} + 6 q^{71} + 10 q^{73} - 2 q^{75} + 3 q^{77} + 4 q^{79} + 26 q^{81} + 5 q^{83} + 6 q^{87} + 10 q^{89} + 44 q^{91} + 16 q^{93} + q^{95} + 14 q^{97} - 12 q^{99}+O(q^{100}) $ 18 * q - 2 * q^3 + 18 * q^5 + 3 * q^7 + 24 * q^9 - 9 * q^11 + 3 * q^13 - 2 * q^15 + q^19 + 3 * q^21 + 18 * q^25 + 10 * q^27 - 2 * q^29 - 6 * q^31 + q^33 + 3 * q^35 + 18 * q^37 + 23 * q^39 - 6 * q^41 + 30 * q^43 + 24 * q^45 - 8 * q^47 - 32 * q^49 + 7 * q^51 - 12 * q^53 - 9 * q^55 + 10 * q^57 - 16 * q^59 + 6 * q^61 + 6 * q^63 + 3 * q^65 + 37 * q^67 + 19 * q^69 + 6 * q^71 + 10 * q^73 - 2 * q^75 + 3 * q^77 + 4 * q^79 + 26 * q^81 + 5 * q^83 + 6 * q^87 + 10 * q^89 + 44 * q^91 + 16 * q^93 + q^95 + 14 * q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{18} - x^{17} + 20 x^{16} - 19 x^{15} + 270 x^{14} - 260 x^{13} + 1914 x^{12} - 1982 x^{11} + \cdots + 2304 $ x^18 - x^17 + 20*x^16 - 19*x^15 + 270*x^14 - 260*x^13 + 1914*x^12 - 1982*x^11 + 9860*x^10 - 9679*x^9 + 30700*x^8 - 26325*x^7 + 65256*x^6 - 44225*x^5 + 62401*x^4 - 8496*x^3 + 13872*x^2 - 2304*x + 2304 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( - 39\!\cdots\!15 \nu^{17} + \cdots + 54\!\cdots\!24 ) / 14\!\cdots\!31 $ (-3986208668095016768515v^17 + 3436759956191738550415v^16 - 74832651255741133210254v^15 + 59669954559407694918435v^14 - 984375681073361523742670v^13 + 805615116758713969518349v^12 - 6415931358012071954792800v^11 + 5762795442473889664528390v^10 - 31231052585047381760519837v^9 + 26375068288111831253593355v^8 - 82821491359031761707518020v^7 + 58095697423673463246918782v^6 - 156074292890669478995185710v^5 + 108007411846419217075329385v^4 - 56694289058291012566735152v^3 + 25424165950669705372935600v^2 - 7362196920454194545610240*v + 547673520269144522494785324) / 143436421741406860089548031
$\beta_{3}$	$=$	$ ( - 62\!\cdots\!76 \nu^{17} + \cdots + 18\!\cdots\!84 ) / 11\!\cdots\!96 $ (-6267245921760819460364976v^17 + 103070834785531579972246561v^16 - 150005550560819509099596401v^15 + 1936925564932121190876495044v^14 - 2336911190765252723592764147v^13 + 25353396075284347923977105278v^12 - 22147071130165210363286778500v^11 + 165239398402816141778749459338v^10 - 171412577801515605999608234686v^9 + 774193289515432938805301199396v^8 - 732666053150874365324463815711v^7 + 2098147678074979388905167709308v^6 - 1997013867777396288366564608885v^5 + 3894919385540810969312667964328v^4 - 2754143536254994274031643127169v^3 + 2329051984580832117681320278929v^2 - 646351075317757621045989382704*v + 183832131431610651206131881984) / 117044120140987997833071193296
$\beta_{4}$	$=$	$ ( - 74\!\cdots\!41 \nu^{17} + \cdots + 32\!\cdots\!12 ) / 11\!\cdots\!96 $ (-7433042275261155776642441v^17 + 38823525516737384227522656v^16 - 185265645832719653220103067v^15 + 705454850344799577172637639v^14 - 2617004923363615966545205107v^13 + 9325928134835198373196158198v^12 - 21907996087970970250425636934v^11 + 61092365995689803672524833428v^10 - 120057007186990247161345716662v^9 + 304460768188484659900983388099v^8 - 390164236761750307928845517365v^7 + 788052114583327870413025983169v^6 - 655142305353027901545511967051v^5 + 1455124858356941895060369428209v^4 - 125152112673790583696967482880v^3 + 191548666105572116887800066423v^2 + 1638503031659014987087737730320*v + 32603174201240922902418629712) / 117044120140987997833071193296
$\beta_{5}$	$=$	$ ( 18\!\cdots\!25 \nu^{17} + \cdots - 53\!\cdots\!40 ) / 22\!\cdots\!96 $ (181056713170020262940325v^17 - 244836051859540531236565v^16 + 3676122422699473075613140v^15 - 4637399970322243127230239v^14 + 49840031828855994112582710v^13 - 62824756321379052744367220v^12 + 359432390875558206780075634v^11 - 461509307231173312424408950v^10 + 1877423918935982027224058740v^9 - 2252144768133384233167723067v^8 + 5980442186929411372325471180v^7 - 6091461835945291609224343945v^6 + 12744568033401617690384548712v^5 - 10504421826194857792458844485v^4 + 13026238548065141900944490485v^3 - 2445366460025148355008763632v^2 + 2918405380305236373475158000*v - 534949817870993798544272640) / 2294982747862509761432768496
$\beta_{6}$	$=$	$ ( - 21\!\cdots\!07 \nu^{17} + \cdots - 56\!\cdots\!60 ) / 23\!\cdots\!92 $ (-21801661798943916263954507v^17 + 110447831149418918727785115v^16 - 561013073632190608565357564v^15 + 1969411229165157720290304065v^14 - 8316695459590167742144943370v^13 + 25651762489924926206970438780v^12 - 74415244032928719457258500574v^11 + 162848100540551622071246479370v^10 - 453467127867235333891501368524v^9 + 793350894487879324218751521925v^8 - 1721438420366942415903453639700v^7 + 2057151135762465875884027423063v^6 - 4067465960876955529726916001800v^5 + 3837746213246666818382970938107v^4 - 5059791448944736077228301271931v^3 + 886151100187174437860536825104v^2 - 204937319603304537715065673728*v - 563357086315782878361658966560) / 234088240281975995666142386592
$\beta_{7}$	$=$	$ ( 23\!\cdots\!85 \nu^{17} + \cdots + 88\!\cdots\!64 ) / 22\!\cdots\!96 $ (232183080673174391729285v^17 - 51126367503154128788960v^16 + 4398825561603947303349135v^15 - 735356110090840367243275v^14 + 58052031811434842639676711v^13 - 10527569146169347737031390v^12 + 381573660087076733025484270v^11 - 100754475018673437627367236v^10 + 1827815868206326190026341150v^9 - 369876118899672910323690775v^8 + 4875875808533069592921326433v^7 - 131777411791904489947956445v^6 + 9059877276463376497461878015v^5 + 2476271290630480216156919587v^4 + 3984034590891897425840268800v^3 + 11053611094665852268812485125v^2 + 775477235073126807059877888*v + 88472814571732813498116864) / 2294982747862509761432768496
$\beta_{8}$	$=$	$ ( 12\!\cdots\!85 \nu^{17} + \cdots + 18\!\cdots\!96 ) / 11\!\cdots\!96 $ (12038617769460385236886785v^17 - 74012834119064974774721537v^16 + 323230682701354981438661860v^15 - 1303682200607707993179137491v^14 + 4891404206139915918365875006v^13 - 16892773540165528154006443748v^12 + 45139398287917435661448730234v^11 - 104851195915452068920591957182v^10 + 282555694398684024742679535668v^9 - 508423047099148615241993838783v^8 + 1079866013623361000806682001196v^7 - 1302759071510768558034829753413v^6 + 2571863068700088021561096444712v^5 - 2454308658840852022230096358321v^4 + 3182360177588895409556970982785v^3 - 566312403005244880107366949680v^2 + 129898960325759829789667694592*v + 187231623364210445367954459696) / 117044120140987997833071193296
$\beta_{9}$	$=$	$ ( 25\!\cdots\!73 \nu^{17} + \cdots - 16\!\cdots\!24 ) / 23\!\cdots\!92 $ (25654584488604152443935673v^17 - 4632583135871865345210159v^16 + 582673423806837925767357226v^15 - 218507902408421320625013235v^14 + 8142921405069581876707194720v^13 - 4003274131302174762456588408v^12 + 64548812108533895918927326274v^11 - 50952146538390032631659274442v^10 + 339287754731204116085139254776v^9 - 335313533594697765280205087471v^8 + 1217171194055201380112240288150v^7 - 1330321963847294463472530169973v^6 + 2760101587528468178818391934838v^5 - 3073347067559907193609162547177v^4 + 3835854081008666026025666185167v^3 - 3205004488695275148229671255750v^2 + 866803353273340275141829293264*v - 161887351856786926511943815424) / 234088240281975995666142386592
$\beta_{10}$	$=$	$ ( 59\!\cdots\!21 \nu^{17} + \cdots - 52\!\cdots\!00 ) / 39\!\cdots\!32 $ (5947929331267131316376421v^17 - 16471157708368774843396912v^16 + 99288591602171090243847863v^15 - 321984490570685271755008723v^14 + 1252433960906998623222606759v^13 - 4307514065256656511149788022v^12 + 6985396510248809937451791006v^11 - 30386236055419230567781224548v^10 + 35364909890613612812074615694v^9 - 138121679968159563494780018343v^8 + 86903243774316790495472841841v^7 - 367346138227111504226114909301v^6 + 176634962338243707292064661663v^5 - 601702246652539981636055037365v^4 - 919500142415767632661255040v^3 - 260740591058652995561700422851v^2 - 166101814864688211582179394408*v - 52131237277121960344222112400) / 39014706713662665944357064432
$\beta_{11}$	$=$	$ ( 20\!\cdots\!15 \nu^{17} + \cdots + 43\!\cdots\!84 ) / 11\!\cdots\!96 $ (20409331414803779081114315v^17 + 18588999707577648652819008v^16 + 428609081184832544774300201v^15 + 421576610713446696253177507v^14 + 5828202721370244339652911801v^13 + 5576639082468394173983839398v^12 + 42394484390087272119551204194v^11 + 37434017614749950116041940084v^10 + 202996082614755523965471979586v^9 + 172935262814964239174130507767v^8 + 579431757366983354874524525023v^7 + 581759309984396807532878115989v^6 + 1019674432902514470752573505089v^5 + 1269754655321006021746095778853v^4 + 578251468397335132346447578752v^3 + 2029974855359410492898873480115v^2 - 364490066936937454386742352520*v + 431016747335423997821353117584) / 117044120140987997833071193296
$\beta_{12}$	$=$	$ ( - 32\!\cdots\!63 \nu^{17} + \cdots - 36\!\cdots\!60 ) / 11\!\cdots\!96 $ (-32787129605906051508177963v^17 - 6131162830961966186022016v^16 - 602657664106941515509466713v^15 - 118894044236123911908870371v^14 - 7876554753812290963783781065v^13 - 1444415084952095935304424310v^12 - 49861868677923454889535604738v^11 - 3193016286177087617599633140v^10 - 233331936669075203976710135474v^9 - 43521469254665062573335460791v^8 - 606078758974890949871305516975v^7 - 235548069405001811715923834565v^6 - 1133876650426019502923974669489v^5 - 927367160114874532973895378725v^4 - 551097352820282439294840726912v^3 - 1702740476400660898906716353043v^2 - 415957076293406604804833757624*v - 364341910018573190483534428560) / 117044120140987997833071193296
$\beta_{13}$	$=$	$ ( - 54\!\cdots\!09 \nu^{17} + \cdots + 64\!\cdots\!20 ) / 18\!\cdots\!84 $ (-5402049247467083249393409v^17 + 7733406801802485797285425v^16 - 107936354183833490728039220v^15 + 140498127340416030235682627v^14 - 1463007744116331045397840670v^13 + 1877079596405165366098757716v^12 - 10414327217130386578730302970v^11 + 13186654824644930119241842590v^10 - 54656232501508316705754998596v^9 + 63211606014002062715750314191v^8 - 168034376493870889731646714940v^7 + 166806468610770649478740509861v^6 - 353335413733999426163798512376v^5 + 288120594447036669201411288689v^4 - 292553654598458069009714718609v^3 + 67101568460145200243285665584v^2 - 17233957403040259550649673728*v + 64482404532068954438971420320) / 18006787713998153512780183584
$\beta_{14}$	$=$	$ ( 44\!\cdots\!83 \nu^{17} + \cdots + 23\!\cdots\!00 ) / 11\!\cdots\!96 $ (44012149600156441249923683v^17 - 40927172671979641174427376v^16 + 816917198279811989398153265v^15 - 774621171778576671313427189v^14 + 10708295507022898961802426849v^13 - 10442582645465579139772960362v^12 + 68874182032074896428055025778v^11 - 77868103675991626694500764332v^10 + 335530686742143414026012035010v^9 - 340169301708951867382264805329v^8 + 894091506797754781376644424599v^7 - 782439133796489351109710038531v^6 + 1674462921927677314628437343609v^5 - 794284971392576370869453355715v^4 + 573719355462396824895805849728v^3 + 1057032885843325884524786587131v^2 - 286250568971775615279402368616*v + 236422744389256649434326320400) / 117044120140987997833071193296
$\beta_{15}$	$=$	$ ( - 23\!\cdots\!85 \nu^{17} + \cdots - 88\!\cdots\!64 ) / 57\!\cdots\!24 $ (-232183080673174391729285v^17 + 51126367503154128788960v^16 - 4398825561603947303349135v^15 + 735356110090840367243275v^14 - 58052031811434842639676711v^13 + 10527569146169347737031390v^12 - 381573660087076733025484270v^11 + 100754475018673437627367236v^10 - 1827815868206326190026341150v^9 + 369876118899672910323690775v^8 - 4875875808533069592921326433v^7 + 131777411791904489947956445v^6 - 9059877276463376497461878015v^5 - 2476271290630480216156919587v^4 - 3984034590891897425840268800v^3 - 10479865407700224828454293001v^2 - 775477235073126807059877888*v - 88472814571732813498116864) / 573745686965627440358192124
$\beta_{16}$	$=$	$ ( - 44\!\cdots\!97 \nu^{17} + \cdots + 51\!\cdots\!84 ) / 78\!\cdots\!64 $ (-44410276449203974747752597v^17 + 7628607571432246247838703v^16 - 887868070777384373904248798v^15 + 166975507918336336970350519v^14 - 11900323370027311267913518548v^13 + 2771171808103525615198828064v^12 - 83119668431413384262212628394v^11 + 33173155919301642480263449946v^10 - 407580591560017410407272594208v^9 + 180387319058603345909325403779v^8 - 1212722150699119568886307751554v^7 + 537793564075105139588162596689v^6 - 2411577598987845564823074834378v^5 + 850208672661435022296833739365v^4 - 2139371719583354357635876319407v^3 - 40082849584685721371532149942v^2 - 464654356322019513190672309968*v + 51336608860418417386727779584) / 78029413427325331888714128864
$\beta_{17}$	$=$	$ ( 15\!\cdots\!35 \nu^{17} + \cdots - 40\!\cdots\!72 ) / 11\!\cdots\!96 $ (155230983958132420692235435v^17 - 208347588935531125493950011v^16 + 3134200799518344367696640140v^15 - 3920283350307666873118481473v^14 + 42493086473008940628849310218v^13 - 53023598845183265356320396492v^12 + 305036458803469662940229956862v^11 - 387483347586134982361001867146v^10 + 1594975632765095516464351062220v^9 - 1884906098584937046500182148549v^8 + 5024511675841185483207540739988v^7 - 5089286586000108607493072471783v^6 + 10663202857560602526111610192936v^5 - 8749322355702838719925412568443v^4 + 9806066112893432003559747154299v^3 - 2037287619719998772956171414800v^2 + 521059178879143087830060496896*v - 408260016383709152480234286672) / 117044120140987997833071193296

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{15} + 4\beta_{7} $ b15 + 4*b7
$\nu^{3}$	$=$	$ \beta_{16} - \beta_{13} - \beta_{10} - \beta_{8} + 7\beta_{5} + \beta_{2} - 7\beta _1 + 1 $ b16 - b13 - b10 - b8 + 7b5 + b2 - 7b1 + 1
$\nu^{4}$	$=$	$ -10\beta_{15} - 2\beta_{14} - 2\beta_{11} - 27\beta_{7} - \beta_{4} + 10\beta_{2} - \beta _1 - 27 $ -10b15 - 2b14 - 2b11 - 27b7 - b4 + 10*b2 - b1 - 27
$\nu^{5}$	$=$	$ - 2 \beta_{17} - 12 \beta_{16} - 13 \beta_{15} + 11 \beta_{13} + 11 \beta_{12} + 2 \beta_{11} + \cdots - 14 \beta_{3} $ -2b17 - 12b16 - 13b15 + 11b13 + 11b12 + 2b11 + b9 + 6b7 + 2b6 - 61b5 - 2b4 - 14*b3
$\nu^{6}$	$=$	$ - 15 \beta_{17} - 3 \beta_{16} + 28 \beta_{14} - 2 \beta_{13} + 3 \beta_{10} + 28 \beta_{9} + \cdots + 228 $ -15b17 - 3b16 + 28b14 - 2b13 + 3b10 + 28b9 + b8 - 29b6 - 26b5 - 100b2 + 26b1 + 228
$\nu^{7}$	$=$	$ 153 \beta_{15} + 19 \beta_{14} - 111 \beta_{12} - 32 \beta_{11} + 126 \beta_{10} + 159 \beta_{8} + \cdots - 2 $ 153b15 + 19b14 - 111b12 - 32b11 + 126b10 + 159b8 - 2b7 + 29b4 + 159b3 - 153b2 + 584*b1 - 2
$\nu^{8}$	$=$	$ 178 \beta_{17} + 64 \beta_{16} + 1026 \beta_{15} + 25 \beta_{13} + 25 \beta_{12} + 335 \beta_{11} + \cdots + 29 \beta_{3} $ 178b17 + 64b16 + 1026b15 + 25b13 + 25b12 + 335b11 - 314b9 + 2147b7 + 335b6 + 417b5 + 178b4 + 29b3
$\nu^{9}$	$=$	$ 343 \beta_{17} + 1290 \beta_{16} - 271 \beta_{14} - 1121 \beta_{13} - 1290 \beta_{10} - 271 \beta_{9} + \cdots - 525 $ 343b17 + 1290b16 - 271b14 - 1121b13 - 1290b10 - 271b9 - 1700b8 - 366b6 + 5843b5 + 1761b2 - 5843*b1 - 525
$\nu^{10}$	$=$	$ - 10696 \beta_{15} - 3333 \beta_{14} - 175 \beta_{12} - 3597 \beta_{11} - 969 \beta_{10} - 545 \beta_{8} + \cdots - 21254 $ -10696b15 - 3333b14 - 175b12 - 3597b11 - 969b10 - 545b8 - 21254b7 - 1971b4 - 545b3 + 10696b2 - 5639*b1 - 21254
$\nu^{11}$	$=$	$ - 3878 \beta_{17} - 13220 \beta_{16} - 20038 \beta_{15} + 11402 \beta_{13} + 11402 \beta_{12} + \cdots - 17801 \beta_{3} $ -3878b17 - 13220b16 - 20038b15 + 11402b13 + 11402b12 + 3651b11 + 3485b9 - 10267b7 + 3651b6 - 59759b5 - 3878b4 - 17801b3
$\nu^{12}$	$=$	$ - 21286 \beta_{17} - 12858 \beta_{16} + 34899 \beta_{14} - 284 \beta_{13} + 12858 \beta_{10} + \cdots + 215464 $ -21286b17 - 12858b16 + 34899b14 - 284b13 + 12858b10 + 34899b9 + 8470b8 - 37435b6 - 70530b5 - 112485b2 + 70530*b1 + 215464
$\nu^{13}$	$=$	$ 226100 \beta_{15} + 42614 \beta_{14} - 116547 \beta_{12} - 33637 \beta_{11} + 136378 \beta_{10} + \cdots + 150672 $ 226100b15 + 42614b14 - 116547b12 - 33637b11 + 136378b10 + 185103b8 + 150672b7 + 43369b4 + 185103b3 - 226100b2 + 618613*b1 + 150672
$\nu^{14}$	$=$	$ 227717 \beta_{17} + 159862 \beta_{16} + 1188977 \beta_{15} - 15810 \beta_{13} - 15810 \beta_{12} + \cdots + 118530 \beta_{3} $ 227717b17 + 159862b16 + 1188977b15 - 15810b13 - 15810b12 + 384012b11 - 364850b9 + 2211923b7 + 384012b6 + 846378b5 + 227717b4 + 118530b3
$\nu^{15}$	$=$	$ 483380 \beta_{17} + 1416135 \beta_{16} - 506109 \beta_{14} - 1195444 \beta_{13} - 1416135 \beta_{10} + \cdots - 1984069 $ 483380b17 + 1416135b16 - 506109b14 - 1195444b13 - 1416135b10 - 506109b9 - 1922029b8 - 291004b6 + 6451412b5 + 2534545b2 - 6451412*b1 - 1984069
$\nu^{16}$	$=$	$ - 12609539 \beta_{15} - 3821544 \beta_{14} + 369311 \beta_{12} - 3914371 \beta_{11} - 1915738 \beta_{10} + \cdots - 22880980 $ -12609539b15 - 3821544b14 + 369311b12 - 3914371b11 - 1915738b10 - 1554206b8 - 22880980b7 - 2428138b4 - 1554206b3 + 12609539b2 - 9910372*b1 - 22880980
$\nu^{17}$	$=$	$ - 5375750 \beta_{17} - 14784050 \beta_{16} - 28264972 \beta_{15} + 12298576 \beta_{13} + \cdots - 19976143 \beta_{3} $ -5375750b17 - 14784050b16 - 28264972b15 + 12298576b13 + 12298576b12 + 2352386b11 + 5898082b9 - 24719495b7 + 2352386b6 - 67621842b5 - 5375750b4 - 19976143b3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1340\mathbb{Z}\right)^\times$.

$n$	$537$	$671$	$1141$
$\chi(n)$	$1$	$1$	$-1 - \beta_{7}$

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} $

841.1

1.64560	−	2.85027i
0.967235	−	1.67530i
0.875672	−	1.51671i
0.716763	−	1.24147i
0.222497	−	0.385375i
−0.242344	+	0.419751i
−0.938909	+	1.62624i
−1.19375	+	2.06764i
−1.55276	+	2.68946i
1.64560	+	2.85027i
0.967235	+	1.67530i
0.875672	+	1.51671i
0.716763	+	1.24147i
0.222497	+	0.385375i
−0.242344	−	0.419751i
−0.938909	−	1.62624i
−1.19375	−	2.06764i
−1.55276	−	2.68946i

−3.29120

1.00000

−0.519571

0.899923i

7.83201

841.2

−1.93447

1.00000

0.186507

−

0.323039i

0.742177

841.3

−1.75134

1.00000

2.48460

−

4.30346i

0.0672025

841.4

−1.43353

1.00000

−1.92448

3.33329i

−0.945003

841.5

−0.444993

1.00000

1.34485

−

2.32935i

−2.80198

841.6

0.484687

1.00000

−0.720789

1.24844i

−2.76508

841.7

1.87782

1.00000

−2.08487

3.61111i

0.526199

841.8

2.38751

1.00000

2.62323

−

4.54357i

2.70018

841.9

3.10553

1.00000

0.110519

−

0.191425i

6.64429

1101.1

−3.29120

1.00000

−0.519571

−

0.899923i

7.83201

1101.2

−1.93447

1.00000

0.186507

0.323039i

0.742177

1101.3

−1.75134

1.00000

2.48460

4.30346i

0.0672025

1101.4

−1.43353

1.00000

−1.92448

−

3.33329i

−0.945003

1101.5

−0.444993

1.00000

1.34485

2.32935i

−2.80198

1101.6

0.484687

1.00000

−0.720789

−

1.24844i

−2.76508

1101.7

1.87782

1.00000

−2.08487

−

3.61111i

0.526199

1101.8

2.38751

1.00000

2.62323

4.54357i

2.70018

1101.9

3.10553

1.00000

0.110519

0.191425i

6.64429

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
67.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1340.2.i.d	✓	18
67.c	even	3	1	inner	1340.2.i.d	✓	18

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1340.2.i.d	✓	18	1.a	even	1	1	trivial
1340.2.i.d	✓	18	67.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{9} + T_{3}^{8} - 19T_{3}^{7} - 19T_{3}^{6} + 110T_{3}^{5} + 119T_{3}^{4} - 209T_{3}^{3} - 241T_{3}^{2} + 48T_{3} + 48 $ T3^9 + T3^8 - 19*T3^7 - 19*T3^6 + 110*T3^5 + 119*T3^4 - 209*T3^3 - 241*T3^2 + 48*T3 + 48 acting on $S_{2}^{\mathrm{new}}(1340, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{18} $ T^18
$3$	$ (T^{9} + T^{8} - 19 T^{7} + \cdots + 48)^{2} $ (T^9 + T^8 - 19T^7 - 19T^6 + 110T^5 + 119T^4 - 209T^3 - 241T^2 + 48*T + 48)^2
$5$	$ (T - 1)^{18} $ (T - 1)^18
$7$	$ T^{18} - 3 T^{17} + \cdots + 19321 $ T^18 - 3T^17 + 52T^16 - 45T^15 + 1526T^14 - 763T^13 + 26703T^12 + 11867T^11 + 293240T^10 + 86585T^9 + 1518106T^8 + 1589713T^7 + 4183512T^6 + 1439840T^5 + 1797511T^4 - 858040T^3 + 610469T^2 - 112312*T + 19321
$11$	$ (T^{2} + T + 1)^{9} $ (T^2 + T + 1)^9
$13$	$ T^{18} + \cdots + 629959801 $ T^18 - 3T^17 + 56T^16 - 115T^15 + 1803T^14 - 3212T^13 + 36798T^12 - 52438T^11 + 531074T^10 - 686362T^9 + 5298739T^8 - 5772187T^7 + 37108196T^6 - 38727602T^5 + 163007280T^4 - 127660724T^3 + 449494880T^2 - 303999088*T + 629959801
$17$	$ T^{18} + \cdots + 124344801 $ T^18 + 68T^16 - 178T^15 + 3312T^14 - 9191T^13 + 85647T^12 - 283763T^11 + 1640097T^10 - 4911424T^9 + 19129276T^8 - 54601203T^7 + 157189854T^6 - 325486449T^5 + 557946828T^4 - 664172676T^3 + 608436441T^2 - 334998342T + 124344801
$19$	$ T^{18} - T^{17} + \cdots + 17330569 $ T^18 - T^17 + 78T^16 - 147T^15 + 4537T^14 - 8013T^13 + 116933T^12 - 189668T^11 + 2166274T^10 - 3091482T^9 + 11205860T^8 - 8021926T^7 + 29252628T^6 - 17237695T^5 + 49723117T^4 - 12575879T^3 + 41827862T^2 - 15973431T + 17330569
$23$	$ T^{18} + \cdots + 20826241969 $ T^18 + 100T^16 + 166T^15 + 6934T^14 + 11204T^13 + 240049T^12 + 292878T^11 + 5825441T^10 + 5863918T^9 + 89451484T^8 + 72064827T^7 + 980182316T^6 + 834483162T^5 + 5985150024T^4 + 5784183252T^3 + 25547142781T^2 + 20773423411T + 20826241969
$29$	$ T^{18} + \cdots + 4149034569 $ T^18 + 2T^17 + 105T^16 + 26T^15 + 7330T^14 + 528T^13 + 267453T^12 - 139453T^11 + 6811136T^10 - 413011T^9 + 70626735T^8 + 39886971T^7 + 540160791T^6 + 403871172T^5 + 2326054519T^4 + 2392689426T^3 + 7127315901T^2 + 5048948592*T + 4149034569
$31$	$ T^{18} + \cdots + 39887678961 $ T^18 + 6T^17 + 148T^16 + 316T^15 + 11796T^14 + 23000T^13 + 501384T^12 + 540731T^11 + 13657785T^10 + 15513532T^9 + 228494280T^8 + 194707750T^7 + 2504558872T^6 + 2572089062T^5 + 13642088971T^4 + 5185982025T^3 + 33563480544T^2 + 22121475597*T + 39887678961
$37$	$ T^{18} + \cdots + 13790876950449 $ T^18 - 18T^17 + 379T^16 - 4392T^15 + 62386T^14 - 622243T^13 + 6603222T^12 - 52557347T^11 + 429909552T^10 - 2851569786T^9 + 18416225575T^8 - 91785658383T^7 + 397655366859T^6 - 1202792774661T^5 + 2979857076894T^4 - 4183831051362T^3 + 5999867045784T^2 - 2765775658176*T + 13790876950449
$41$	$ T^{18} + \cdots + 8157521761 $ T^18 + 6T^17 + 284T^16 + 568T^15 + 50371T^14 + 86859T^13 + 4738688T^12 + 366664T^11 + 289856946T^10 + 385426655T^9 + 4446731274T^8 + 4386937879T^7 + 48361767680T^6 + 67419910103T^5 + 104099259448T^4 + 46224527028T^3 + 31976992751T^2 - 3378382195*T + 8157521761
$43$	$ (T^{9} - 15 T^{8} + \cdots + 1087728)^{2} $ (T^9 - 15T^8 - 134T^7 + 2866T^6 - 2963T^5 - 122066T^4 + 443961T^3 + 821983T^2 - 4398048T + 1087728)^2
$47$	$ T^{18} + 8 T^{17} + \cdots + 6305121 $ T^18 + 8T^17 + 222T^16 + 694T^15 + 27962T^14 + 84640T^13 + 1619305T^12 - 2603918T^11 + 26976003T^10 - 8035198T^9 + 156480884T^8 - 164950355T^7 + 443302732T^6 - 128675638T^5 + 227329216T^4 + 101857140T^3 + 123403905T^2 + 27638577*T + 6305121
$53$	$ (T^{9} + 6 T^{8} + \cdots - 6441444)^{2} $ (T^9 + 6T^8 - 210T^7 - 834T^6 + 16561T^5 + 34629T^4 - 573957T^3 - 260793T^2 + 7245288T - 6441444)^2
$59$	$ (T^{9} + 8 T^{8} + \cdots - 18465136)^{2} $ (T^9 + 8T^8 - 250T^7 - 2145T^6 + 16839T^5 + 186637T^4 - 37992T^3 - 4993483T^2 - 17455016T - 18465136)^2
$61$	$ T^{18} + \cdots + 920154407398209 $ T^18 - 6T^17 + 355T^16 - 1192T^15 + 77742T^14 - 212597T^13 + 9755958T^12 - 12410321T^11 + 832874472T^10 - 821974978T^9 + 43509264417T^8 - 7537245835T^7 + 1456271202685T^6 - 1486342215167T^5 + 22304532934252T^4 - 39386856683610T^3 + 274355855742036T^2 - 415754019411138*T + 920154407398209
$67$	$ T^{18} + \cdots + 27\!\cdots\!47 $ T^18 - 37T^17 + 683T^16 - 9643T^15 + 131270T^14 - 1649392T^13 + 17700245T^12 - 171361186T^11 + 1604327069T^10 - 13947825268T^9 + 107489913623T^8 - 769240363954T^7 + 5323578786935T^6 - 33237097768432T^5 + 177230922795890T^4 - 872290179255667T^3 + 4139466026435609T^2 - 15024504069595717*T + 27206534396294947
$71$	$ T^{18} + \cdots + 3332074207609 $ T^18 - 6T^17 + 310T^16 - 428T^15 + 57445T^14 - 38359T^13 + 5987986T^12 + 6516654T^11 + 419117366T^10 + 446373007T^9 + 16072615546T^8 + 33874129413T^7 + 380547734198T^6 + 277351815267T^5 + 2442885038826T^4 - 2635128134538T^3 + 13321849042849T^2 - 6678041511785*T + 3332074207609
$73$	$ T^{18} + \cdots + 8276778009721 $ T^18 - 10T^17 + 508T^16 - 1508T^15 + 135245T^14 - 100145T^13 + 23648556T^12 + 77002100T^11 + 2593699326T^10 + 12776886697T^9 + 223641294416T^8 + 1206556369741T^7 + 9854996409178T^6 + 24844146858263T^5 + 97104116183570T^4 - 86909945381774T^3 + 378687908171153T^2 + 54780914628271*T + 8276778009721
$79$	$ T^{18} + \cdots + 232201002088449 $ T^18 - 4T^17 + 426T^16 - 1274T^15 + 121848T^14 - 327035T^13 + 18294039T^12 - 30322863T^11 + 1923554025T^10 - 2859796190T^9 + 115438168244T^8 - 100293370677T^7 + 4704076331188T^6 - 4549637118615T^5 + 71751002275624T^4 + 35560011785928T^3 + 602736540271227T^2 - 351812668423662*T + 232201002088449
$83$	$ T^{18} + \cdots + 2205732899241 $ T^18 - 5T^17 + 298T^16 + 503T^15 + 51619T^14 + 172295T^13 + 5883426T^12 + 32874024T^11 + 459243370T^10 + 2333955304T^9 + 21505754156T^8 + 110098698250T^7 + 647097484715T^6 + 2088086330903T^5 + 5172972399619T^4 + 7917485443977T^3 + 8954511468750T^2 + 5361367803543*T + 2205732899241
$89$	$ (T^{9} - 5 T^{8} + \cdots - 1881036)^{2} $ (T^9 - 5T^8 - 438T^7 + 2927T^6 + 45676T^5 - 395237T^4 + 69378T^3 + 3858377T^2 - 4098564T - 1881036)^2
$97$	$ T^{18} + \cdots + 78020458060561 $ T^18 - 14T^17 + 522T^16 - 2718T^15 + 129698T^14 - 568429T^13 + 18249465T^12 + 14440513T^11 + 981377899T^10 + 2093172274T^9 + 40290536488T^8 + 120002027793T^7 + 823889143268T^6 + 1980373445067T^5 + 9865891709204T^4 + 23746948300228T^3 + 62070359370679T^2 + 73853078043442*T + 78020458060561
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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:	\( N \)	\(=\)	\( 1340 = 2^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1340.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{5} - \beta_1) q^{3} + q^{5} - \beta_{10} q^{7} + ( - \beta_{2} + 1) q^{9}+O(q^{10}) \) q + (b5 - b1) * q^3 + q^5 - b10 * q^7 + (-b2 + 1) * q^9	\( q + (\beta_{5} - \beta_1) q^{3} + q^{5} - \beta_{10} q^{7} + ( - \beta_{2} + 1) q^{9} + ( - \beta_{7} - 1) q^{11} + (\beta_{13} + \beta_{12} + \beta_{5}) q^{13} + (\beta_{5} - \beta_1) q^{15} + ( - \beta_{11} - \beta_{6}) q^{17} + ( - \beta_{17} - \beta_{4}) q^{19} + (\beta_{14} - \beta_{11} - \beta_1) q^{21} + (\beta_{15} + \beta_{9} + \beta_{5}) q^{23} + q^{25} + (\beta_{16} - \beta_{13} - \beta_{10} + \cdots + 1) q^{27}+ \cdots + ( - \beta_{15} - \beta_{7} + \beta_{2} - 1) q^{99}+O(q^{100}) \) q + (b5 - b1) * q^3 + q^5 - b10 * q^7 + (-b2 + 1) * q^9 + (-b7 - 1) * q^11 + (b13 + b12 + b5) * q^13 + (b5 - b1) * q^15 + (-b11 - b6) * q^17 + (-b17 - b4) * q^19 + (b14 - b11 - b1) * q^21 + (b15 + b9 + b5) * q^23 + q^25 + (b16 - b13 - b10 - b8 + b5 + b2 - b1 + 1) * q^27 + (-b15 + b14 + b12 + b4 + b2) * q^29 + (-b15 + b12 - b11 + b4 + b2) * q^31 + b1 * q^33 - b10 * q^35 + (b17 - b16 + b11 - b9 - 3b7 + b6 + b5 + b4) q^37 + (-b13 - b12 + b11 - 3b7 + b6 - b5 - b3) q^39 + (-b15 + b14 + b12 + b11 - b10 - b8 - b7 + b4 - b3 + b2 + b1 - 1) * q^41 + (-b17 + b16 - b13 - b10 + b8 - b5 + b2 + b1 + 2) * q^43 + (-b2 + 1) * q^45 + (b15 - b12 - b11 + b10 - b7 - b4 - b2 - 2b1 - 1) q^47 + (-b16 - b15 + 2b13 + 2b12 - 2b11 + b9 + 5b7 - 2b6 + b3) q^49 + (-b16 - b11 - b7 - b6 + b3) * q^51 + (-b17 - b16 + b13 + b10 - b5 + b1 - 1) * q^53 + (-b7 - 1) * q^55 + (-b13 - b12 - b9 - 2b7) q^57 + (b17 - b16 + b14 + b10 + b9 - b5 + b1 - 1) * q^59 + (-b17 - b16 + 2b13 + 2b12 - b11 - b6 - b4) * q^61 + (-b11 + b7 - b4 - 2b1 + 1) q^63 + (b13 + b12 + b5) * q^65 + (-b17 + b16 + b15 + b14 - 2b13 - 2b12 + b11 + b9 + 2b7 + b6 - b1 + 3) q^67 + (b17 + b16 - b15 + b13 + b12 - b7 - b5 + b4) * q^69 + (-2b15 + 3b12 - b10 + 2b7 + b4 + 2b2 + 2b1 + 2) q^71 + (b17 - 3b15 + 3b13 + 3b12 + b7 + b5 + b4) q^73 + (b5 - b1) * q^75 + b16 * q^77 + (-b14 + 2b12 + b10 + b8 + 2b7 + b3 + 2b1 + 2) q^79 + (-b17 + 2b14 + 2b9 - 2b6 - b5 - b2 + b1) q^81 + (-b16 - 2b15 + b13 + b12 - b9 - 2b5) * q^83 + (-b11 - b6) * q^85 + (b15 - b14 - b12 + b11 - b10 - b8 - b4 - b3 - b2) * q^87 + (-b17 - b16 + b14 + b13 + b10 + b9 + b8 + b6) * q^89 + (-b17 + 2b14 + 2b9 + 2b6 + 2b5 + b2 - 2b1 + 2) q^91 + (2b15 - b14 - b12 + b8 + b7 + b3 - 2b2 + 2b1 + 1) q^93 + (-b17 - b4) * q^95 + (-b15 + b9 - b7 + b5 - b3) * q^97 + (-b15 - b7 + b2 - 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 2 q^{3} + 18 q^{5} + 3 q^{7} + 24 q^{9}+O(q^{10}) \) 18 * q - 2 * q^3 + 18 * q^5 + 3 * q^7 + 24 * q^9	\( 18 q - 2 q^{3} + 18 q^{5} + 3 q^{7} + 24 q^{9} - 9 q^{11} + 3 q^{13} - 2 q^{15} + q^{19} + 3 q^{21} + 18 q^{25} + 10 q^{27} - 2 q^{29} - 6 q^{31} + q^{33} + 3 q^{35} + 18 q^{37} + 23 q^{39} - 6 q^{41} + 30 q^{43} + 24 q^{45} - 8 q^{47} - 32 q^{49} + 7 q^{51} - 12 q^{53} - 9 q^{55} + 10 q^{57} - 16 q^{59} + 6 q^{61} + 6 q^{63} + 3 q^{65} + 37 q^{67} + 19 q^{69} + 6 q^{71} + 10 q^{73} - 2 q^{75} + 3 q^{77} + 4 q^{79} + 26 q^{81} + 5 q^{83} + 6 q^{87} + 10 q^{89} + 44 q^{91} + 16 q^{93} + q^{95} + 14 q^{97} - 12 q^{99}+O(q^{100}) \) 18 * q - 2 * q^3 + 18 * q^5 + 3 * q^7 + 24 * q^9 - 9 * q^11 + 3 * q^13 - 2 * q^15 + q^19 + 3 * q^21 + 18 * q^25 + 10 * q^27 - 2 * q^29 - 6 * q^31 + q^33 + 3 * q^35 + 18 * q^37 + 23 * q^39 - 6 * q^41 + 30 * q^43 + 24 * q^45 - 8 * q^47 - 32 * q^49 + 7 * q^51 - 12 * q^53 - 9 * q^55 + 10 * q^57 - 16 * q^59 + 6 * q^61 + 6 * q^63 + 3 * q^65 + 37 * q^67 + 19 * q^69 + 6 * q^71 + 10 * q^73 - 2 * q^75 + 3 * q^77 + 4 * q^79 + 26 * q^81 + 5 * q^83 + 6 * q^87 + 10 * q^89 + 44 * q^91 + 16 * q^93 + q^95 + 14 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	1340.2.i.d

Level	$1340$
Weight	$2$
Character orbit	1340.i
Analytic conductor	$10.700$
Analytic rank	$0$
Dimension	$18$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(10.6999538709\)
Analytic rank:	\(0\)
Dimension:	\(18\)
Relative dimension:	\(9\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{18} - x^{17} + 20 x^{16} - 19 x^{15} + 270 x^{14} - 260 x^{13} + 1914 x^{12} - 1982 x^{11} + \cdots + 2304 \) x^18 - x^17 + 20x^16 - 19x^15 + 270x^14 - 260x^13 + 1914x^12 - 1982x^11 + 9860x^10 - 9679x^9 + 30700x^8 - 26325x^7 + 65256x^6 - 44225x^5 + 62401x^4 - 8496x^3 + 13872x^2 - 2304x + 2304
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{2}\cdot 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( - 39\!\cdots\!15 \nu^{17} + \cdots + 54\!\cdots\!24 ) / 14\!\cdots\!31 \) (-3986208668095016768515v^17 + 3436759956191738550415v^16 - 74832651255741133210254v^15 + 59669954559407694918435v^14 - 984375681073361523742670v^13 + 805615116758713969518349v^12 - 6415931358012071954792800v^11 + 5762795442473889664528390v^10 - 31231052585047381760519837v^9 + 26375068288111831253593355v^8 - 82821491359031761707518020v^7 + 58095697423673463246918782v^6 - 156074292890669478995185710v^5 + 108007411846419217075329385v^4 - 56694289058291012566735152v^3 + 25424165950669705372935600v^2 - 7362196920454194545610240*v + 547673520269144522494785324) / 143436421741406860089548031
\(\beta_{3}\)	\(=\)	\( ( - 62\!\cdots\!76 \nu^{17} + \cdots + 18\!\cdots\!84 ) / 11\!\cdots\!96 \) (-6267245921760819460364976v^17 + 103070834785531579972246561v^16 - 150005550560819509099596401v^15 + 1936925564932121190876495044v^14 - 2336911190765252723592764147v^13 + 25353396075284347923977105278v^12 - 22147071130165210363286778500v^11 + 165239398402816141778749459338v^10 - 171412577801515605999608234686v^9 + 774193289515432938805301199396v^8 - 732666053150874365324463815711v^7 + 2098147678074979388905167709308v^6 - 1997013867777396288366564608885v^5 + 3894919385540810969312667964328v^4 - 2754143536254994274031643127169v^3 + 2329051984580832117681320278929v^2 - 646351075317757621045989382704*v + 183832131431610651206131881984) / 117044120140987997833071193296
\(\beta_{4}\)	\(=\)	\( ( - 74\!\cdots\!41 \nu^{17} + \cdots + 32\!\cdots\!12 ) / 11\!\cdots\!96 \) (-7433042275261155776642441v^17 + 38823525516737384227522656v^16 - 185265645832719653220103067v^15 + 705454850344799577172637639v^14 - 2617004923363615966545205107v^13 + 9325928134835198373196158198v^12 - 21907996087970970250425636934v^11 + 61092365995689803672524833428v^10 - 120057007186990247161345716662v^9 + 304460768188484659900983388099v^8 - 390164236761750307928845517365v^7 + 788052114583327870413025983169v^6 - 655142305353027901545511967051v^5 + 1455124858356941895060369428209v^4 - 125152112673790583696967482880v^3 + 191548666105572116887800066423v^2 + 1638503031659014987087737730320*v + 32603174201240922902418629712) / 117044120140987997833071193296
\(\beta_{5}\)	\(=\)	\( ( 18\!\cdots\!25 \nu^{17} + \cdots - 53\!\cdots\!40 ) / 22\!\cdots\!96 \) (181056713170020262940325v^17 - 244836051859540531236565v^16 + 3676122422699473075613140v^15 - 4637399970322243127230239v^14 + 49840031828855994112582710v^13 - 62824756321379052744367220v^12 + 359432390875558206780075634v^11 - 461509307231173312424408950v^10 + 1877423918935982027224058740v^9 - 2252144768133384233167723067v^8 + 5980442186929411372325471180v^7 - 6091461835945291609224343945v^6 + 12744568033401617690384548712v^5 - 10504421826194857792458844485v^4 + 13026238548065141900944490485v^3 - 2445366460025148355008763632v^2 + 2918405380305236373475158000*v - 534949817870993798544272640) / 2294982747862509761432768496
\(\beta_{6}\)	\(=\)	\( ( - 21\!\cdots\!07 \nu^{17} + \cdots - 56\!\cdots\!60 ) / 23\!\cdots\!92 \) (-21801661798943916263954507v^17 + 110447831149418918727785115v^16 - 561013073632190608565357564v^15 + 1969411229165157720290304065v^14 - 8316695459590167742144943370v^13 + 25651762489924926206970438780v^12 - 74415244032928719457258500574v^11 + 162848100540551622071246479370v^10 - 453467127867235333891501368524v^9 + 793350894487879324218751521925v^8 - 1721438420366942415903453639700v^7 + 2057151135762465875884027423063v^6 - 4067465960876955529726916001800v^5 + 3837746213246666818382970938107v^4 - 5059791448944736077228301271931v^3 + 886151100187174437860536825104v^2 - 204937319603304537715065673728*v - 563357086315782878361658966560) / 234088240281975995666142386592
\(\beta_{7}\)	\(=\)	\( ( 23\!\cdots\!85 \nu^{17} + \cdots + 88\!\cdots\!64 ) / 22\!\cdots\!96 \) (232183080673174391729285v^17 - 51126367503154128788960v^16 + 4398825561603947303349135v^15 - 735356110090840367243275v^14 + 58052031811434842639676711v^13 - 10527569146169347737031390v^12 + 381573660087076733025484270v^11 - 100754475018673437627367236v^10 + 1827815868206326190026341150v^9 - 369876118899672910323690775v^8 + 4875875808533069592921326433v^7 - 131777411791904489947956445v^6 + 9059877276463376497461878015v^5 + 2476271290630480216156919587v^4 + 3984034590891897425840268800v^3 + 11053611094665852268812485125v^2 + 775477235073126807059877888*v + 88472814571732813498116864) / 2294982747862509761432768496
\(\beta_{8}\)	\(=\)	\( ( 12\!\cdots\!85 \nu^{17} + \cdots + 18\!\cdots\!96 ) / 11\!\cdots\!96 \) (12038617769460385236886785v^17 - 74012834119064974774721537v^16 + 323230682701354981438661860v^15 - 1303682200607707993179137491v^14 + 4891404206139915918365875006v^13 - 16892773540165528154006443748v^12 + 45139398287917435661448730234v^11 - 104851195915452068920591957182v^10 + 282555694398684024742679535668v^9 - 508423047099148615241993838783v^8 + 1079866013623361000806682001196v^7 - 1302759071510768558034829753413v^6 + 2571863068700088021561096444712v^5 - 2454308658840852022230096358321v^4 + 3182360177588895409556970982785v^3 - 566312403005244880107366949680v^2 + 129898960325759829789667694592*v + 187231623364210445367954459696) / 117044120140987997833071193296
\(\beta_{9}\)	\(=\)	\( ( 25\!\cdots\!73 \nu^{17} + \cdots - 16\!\cdots\!24 ) / 23\!\cdots\!92 \) (25654584488604152443935673v^17 - 4632583135871865345210159v^16 + 582673423806837925767357226v^15 - 218507902408421320625013235v^14 + 8142921405069581876707194720v^13 - 4003274131302174762456588408v^12 + 64548812108533895918927326274v^11 - 50952146538390032631659274442v^10 + 339287754731204116085139254776v^9 - 335313533594697765280205087471v^8 + 1217171194055201380112240288150v^7 - 1330321963847294463472530169973v^6 + 2760101587528468178818391934838v^5 - 3073347067559907193609162547177v^4 + 3835854081008666026025666185167v^3 - 3205004488695275148229671255750v^2 + 866803353273340275141829293264*v - 161887351856786926511943815424) / 234088240281975995666142386592
\(\beta_{10}\)	\(=\)	\( ( 59\!\cdots\!21 \nu^{17} + \cdots - 52\!\cdots\!00 ) / 39\!\cdots\!32 \) (5947929331267131316376421v^17 - 16471157708368774843396912v^16 + 99288591602171090243847863v^15 - 321984490570685271755008723v^14 + 1252433960906998623222606759v^13 - 4307514065256656511149788022v^12 + 6985396510248809937451791006v^11 - 30386236055419230567781224548v^10 + 35364909890613612812074615694v^9 - 138121679968159563494780018343v^8 + 86903243774316790495472841841v^7 - 367346138227111504226114909301v^6 + 176634962338243707292064661663v^5 - 601702246652539981636055037365v^4 - 919500142415767632661255040v^3 - 260740591058652995561700422851v^2 - 166101814864688211582179394408*v - 52131237277121960344222112400) / 39014706713662665944357064432
\(\beta_{11}\)	\(=\)	\( ( 20\!\cdots\!15 \nu^{17} + \cdots + 43\!\cdots\!84 ) / 11\!\cdots\!96 \) (20409331414803779081114315v^17 + 18588999707577648652819008v^16 + 428609081184832544774300201v^15 + 421576610713446696253177507v^14 + 5828202721370244339652911801v^13 + 5576639082468394173983839398v^12 + 42394484390087272119551204194v^11 + 37434017614749950116041940084v^10 + 202996082614755523965471979586v^9 + 172935262814964239174130507767v^8 + 579431757366983354874524525023v^7 + 581759309984396807532878115989v^6 + 1019674432902514470752573505089v^5 + 1269754655321006021746095778853v^4 + 578251468397335132346447578752v^3 + 2029974855359410492898873480115v^2 - 364490066936937454386742352520*v + 431016747335423997821353117584) / 117044120140987997833071193296
\(\beta_{12}\)	\(=\)	\( ( - 32\!\cdots\!63 \nu^{17} + \cdots - 36\!\cdots\!60 ) / 11\!\cdots\!96 \) (-32787129605906051508177963v^17 - 6131162830961966186022016v^16 - 602657664106941515509466713v^15 - 118894044236123911908870371v^14 - 7876554753812290963783781065v^13 - 1444415084952095935304424310v^12 - 49861868677923454889535604738v^11 - 3193016286177087617599633140v^10 - 233331936669075203976710135474v^9 - 43521469254665062573335460791v^8 - 606078758974890949871305516975v^7 - 235548069405001811715923834565v^6 - 1133876650426019502923974669489v^5 - 927367160114874532973895378725v^4 - 551097352820282439294840726912v^3 - 1702740476400660898906716353043v^2 - 415957076293406604804833757624*v - 364341910018573190483534428560) / 117044120140987997833071193296
\(\beta_{13}\)	\(=\)	\( ( - 54\!\cdots\!09 \nu^{17} + \cdots + 64\!\cdots\!20 ) / 18\!\cdots\!84 \) (-5402049247467083249393409v^17 + 7733406801802485797285425v^16 - 107936354183833490728039220v^15 + 140498127340416030235682627v^14 - 1463007744116331045397840670v^13 + 1877079596405165366098757716v^12 - 10414327217130386578730302970v^11 + 13186654824644930119241842590v^10 - 54656232501508316705754998596v^9 + 63211606014002062715750314191v^8 - 168034376493870889731646714940v^7 + 166806468610770649478740509861v^6 - 353335413733999426163798512376v^5 + 288120594447036669201411288689v^4 - 292553654598458069009714718609v^3 + 67101568460145200243285665584v^2 - 17233957403040259550649673728*v + 64482404532068954438971420320) / 18006787713998153512780183584
\(\beta_{14}\)	\(=\)	\( ( 44\!\cdots\!83 \nu^{17} + \cdots + 23\!\cdots\!00 ) / 11\!\cdots\!96 \) (44012149600156441249923683v^17 - 40927172671979641174427376v^16 + 816917198279811989398153265v^15 - 774621171778576671313427189v^14 + 10708295507022898961802426849v^13 - 10442582645465579139772960362v^12 + 68874182032074896428055025778v^11 - 77868103675991626694500764332v^10 + 335530686742143414026012035010v^9 - 340169301708951867382264805329v^8 + 894091506797754781376644424599v^7 - 782439133796489351109710038531v^6 + 1674462921927677314628437343609v^5 - 794284971392576370869453355715v^4 + 573719355462396824895805849728v^3 + 1057032885843325884524786587131v^2 - 286250568971775615279402368616*v + 236422744389256649434326320400) / 117044120140987997833071193296
\(\beta_{15}\)	\(=\)	\( ( - 23\!\cdots\!85 \nu^{17} + \cdots - 88\!\cdots\!64 ) / 57\!\cdots\!24 \) (-232183080673174391729285v^17 + 51126367503154128788960v^16 - 4398825561603947303349135v^15 + 735356110090840367243275v^14 - 58052031811434842639676711v^13 + 10527569146169347737031390v^12 - 381573660087076733025484270v^11 + 100754475018673437627367236v^10 - 1827815868206326190026341150v^9 + 369876118899672910323690775v^8 - 4875875808533069592921326433v^7 + 131777411791904489947956445v^6 - 9059877276463376497461878015v^5 - 2476271290630480216156919587v^4 - 3984034590891897425840268800v^3 - 10479865407700224828454293001v^2 - 775477235073126807059877888*v - 88472814571732813498116864) / 573745686965627440358192124
\(\beta_{16}\)	\(=\)	\( ( - 44\!\cdots\!97 \nu^{17} + \cdots + 51\!\cdots\!84 ) / 78\!\cdots\!64 \) (-44410276449203974747752597v^17 + 7628607571432246247838703v^16 - 887868070777384373904248798v^15 + 166975507918336336970350519v^14 - 11900323370027311267913518548v^13 + 2771171808103525615198828064v^12 - 83119668431413384262212628394v^11 + 33173155919301642480263449946v^10 - 407580591560017410407272594208v^9 + 180387319058603345909325403779v^8 - 1212722150699119568886307751554v^7 + 537793564075105139588162596689v^6 - 2411577598987845564823074834378v^5 + 850208672661435022296833739365v^4 - 2139371719583354357635876319407v^3 - 40082849584685721371532149942v^2 - 464654356322019513190672309968*v + 51336608860418417386727779584) / 78029413427325331888714128864
\(\beta_{17}\)	\(=\)	\( ( 15\!\cdots\!35 \nu^{17} + \cdots - 40\!\cdots\!72 ) / 11\!\cdots\!96 \) (155230983958132420692235435v^17 - 208347588935531125493950011v^16 + 3134200799518344367696640140v^15 - 3920283350307666873118481473v^14 + 42493086473008940628849310218v^13 - 53023598845183265356320396492v^12 + 305036458803469662940229956862v^11 - 387483347586134982361001867146v^10 + 1594975632765095516464351062220v^9 - 1884906098584937046500182148549v^8 + 5024511675841185483207540739988v^7 - 5089286586000108607493072471783v^6 + 10663202857560602526111610192936v^5 - 8749322355702838719925412568443v^4 + 9806066112893432003559747154299v^3 - 2037287619719998772956171414800v^2 + 521059178879143087830060496896*v - 408260016383709152480234286672) / 117044120140987997833071193296

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{15} + 4\beta_{7} \) b15 + 4*b7
\(\nu^{3}\)	\(=\)	\( \beta_{16} - \beta_{13} - \beta_{10} - \beta_{8} + 7\beta_{5} + \beta_{2} - 7\beta _1 + 1 \) b16 - b13 - b10 - b8 + 7b5 + b2 - 7b1 + 1
\(\nu^{4}\)	\(=\)	\( -10\beta_{15} - 2\beta_{14} - 2\beta_{11} - 27\beta_{7} - \beta_{4} + 10\beta_{2} - \beta _1 - 27 \) -10b15 - 2b14 - 2b11 - 27b7 - b4 + 10*b2 - b1 - 27
\(\nu^{5}\)	\(=\)	\( - 2 \beta_{17} - 12 \beta_{16} - 13 \beta_{15} + 11 \beta_{13} + 11 \beta_{12} + 2 \beta_{11} + \cdots - 14 \beta_{3} \) -2b17 - 12b16 - 13b15 + 11b13 + 11b12 + 2b11 + b9 + 6b7 + 2b6 - 61b5 - 2b4 - 14*b3
\(\nu^{6}\)	\(=\)	\( - 15 \beta_{17} - 3 \beta_{16} + 28 \beta_{14} - 2 \beta_{13} + 3 \beta_{10} + 28 \beta_{9} + \cdots + 228 \) -15b17 - 3b16 + 28b14 - 2b13 + 3b10 + 28b9 + b8 - 29b6 - 26b5 - 100b2 + 26b1 + 228
\(\nu^{7}\)	\(=\)	\( 153 \beta_{15} + 19 \beta_{14} - 111 \beta_{12} - 32 \beta_{11} + 126 \beta_{10} + 159 \beta_{8} + \cdots - 2 \) 153b15 + 19b14 - 111b12 - 32b11 + 126b10 + 159b8 - 2b7 + 29b4 + 159b3 - 153b2 + 584*b1 - 2
\(\nu^{8}\)	\(=\)	\( 178 \beta_{17} + 64 \beta_{16} + 1026 \beta_{15} + 25 \beta_{13} + 25 \beta_{12} + 335 \beta_{11} + \cdots + 29 \beta_{3} \) 178b17 + 64b16 + 1026b15 + 25b13 + 25b12 + 335b11 - 314b9 + 2147b7 + 335b6 + 417b5 + 178b4 + 29b3
\(\nu^{9}\)	\(=\)	\( 343 \beta_{17} + 1290 \beta_{16} - 271 \beta_{14} - 1121 \beta_{13} - 1290 \beta_{10} - 271 \beta_{9} + \cdots - 525 \) 343b17 + 1290b16 - 271b14 - 1121b13 - 1290b10 - 271b9 - 1700b8 - 366b6 + 5843b5 + 1761b2 - 5843*b1 - 525
\(\nu^{10}\)	\(=\)	\( - 10696 \beta_{15} - 3333 \beta_{14} - 175 \beta_{12} - 3597 \beta_{11} - 969 \beta_{10} - 545 \beta_{8} + \cdots - 21254 \) -10696b15 - 3333b14 - 175b12 - 3597b11 - 969b10 - 545b8 - 21254b7 - 1971b4 - 545b3 + 10696b2 - 5639*b1 - 21254
\(\nu^{11}\)	\(=\)	\( - 3878 \beta_{17} - 13220 \beta_{16} - 20038 \beta_{15} + 11402 \beta_{13} + 11402 \beta_{12} + \cdots - 17801 \beta_{3} \) -3878b17 - 13220b16 - 20038b15 + 11402b13 + 11402b12 + 3651b11 + 3485b9 - 10267b7 + 3651b6 - 59759b5 - 3878b4 - 17801b3
\(\nu^{12}\)	\(=\)	\( - 21286 \beta_{17} - 12858 \beta_{16} + 34899 \beta_{14} - 284 \beta_{13} + 12858 \beta_{10} + \cdots + 215464 \) -21286b17 - 12858b16 + 34899b14 - 284b13 + 12858b10 + 34899b9 + 8470b8 - 37435b6 - 70530b5 - 112485b2 + 70530*b1 + 215464
\(\nu^{13}\)	\(=\)	\( 226100 \beta_{15} + 42614 \beta_{14} - 116547 \beta_{12} - 33637 \beta_{11} + 136378 \beta_{10} + \cdots + 150672 \) 226100b15 + 42614b14 - 116547b12 - 33637b11 + 136378b10 + 185103b8 + 150672b7 + 43369b4 + 185103b3 - 226100b2 + 618613*b1 + 150672
\(\nu^{14}\)	\(=\)	\( 227717 \beta_{17} + 159862 \beta_{16} + 1188977 \beta_{15} - 15810 \beta_{13} - 15810 \beta_{12} + \cdots + 118530 \beta_{3} \) 227717b17 + 159862b16 + 1188977b15 - 15810b13 - 15810b12 + 384012b11 - 364850b9 + 2211923b7 + 384012b6 + 846378b5 + 227717b4 + 118530b3
\(\nu^{15}\)	\(=\)	\( 483380 \beta_{17} + 1416135 \beta_{16} - 506109 \beta_{14} - 1195444 \beta_{13} - 1416135 \beta_{10} + \cdots - 1984069 \) 483380b17 + 1416135b16 - 506109b14 - 1195444b13 - 1416135b10 - 506109b9 - 1922029b8 - 291004b6 + 6451412b5 + 2534545b2 - 6451412*b1 - 1984069
\(\nu^{16}\)	\(=\)	\( - 12609539 \beta_{15} - 3821544 \beta_{14} + 369311 \beta_{12} - 3914371 \beta_{11} - 1915738 \beta_{10} + \cdots - 22880980 \) -12609539b15 - 3821544b14 + 369311b12 - 3914371b11 - 1915738b10 - 1554206b8 - 22880980b7 - 2428138b4 - 1554206b3 + 12609539b2 - 9910372*b1 - 22880980
\(\nu^{17}\)	\(=\)	\( - 5375750 \beta_{17} - 14784050 \beta_{16} - 28264972 \beta_{15} + 12298576 \beta_{13} + \cdots - 19976143 \beta_{3} \) -5375750b17 - 14784050b16 - 28264972b15 + 12298576b13 + 12298576b12 + 2352386b11 + 5898082b9 - 24719495b7 + 2352386b6 - 67621842b5 - 5375750b4 - 19976143b3

\(n\)	\(537\)	\(671\)	\(1141\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1 - \beta_{7}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 841.9
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{18} \) T^18
$3$	\( (T^{9} + T^{8} - 19 T^{7} + \cdots + 48)^{2} \) (T^9 + T^8 - 19T^7 - 19T^6 + 110T^5 + 119T^4 - 209T^3 - 241T^2 + 48*T + 48)^2
$5$	\( (T - 1)^{18} \) (T - 1)^18
$7$	\( T^{18} - 3 T^{17} + \cdots + 19321 \) T^18 - 3T^17 + 52T^16 - 45T^15 + 1526T^14 - 763T^13 + 26703T^12 + 11867T^11 + 293240T^10 + 86585T^9 + 1518106T^8 + 1589713T^7 + 4183512T^6 + 1439840T^5 + 1797511T^4 - 858040T^3 + 610469T^2 - 112312*T + 19321
$11$	\( (T^{2} + T + 1)^{9} \) (T^2 + T + 1)^9
$13$	\( T^{18} + \cdots + 629959801 \) T^18 - 3T^17 + 56T^16 - 115T^15 + 1803T^14 - 3212T^13 + 36798T^12 - 52438T^11 + 531074T^10 - 686362T^9 + 5298739T^8 - 5772187T^7 + 37108196T^6 - 38727602T^5 + 163007280T^4 - 127660724T^3 + 449494880T^2 - 303999088*T + 629959801
$17$	\( T^{18} + \cdots + 124344801 \) T^18 + 68T^16 - 178T^15 + 3312T^14 - 9191T^13 + 85647T^12 - 283763T^11 + 1640097T^10 - 4911424T^9 + 19129276T^8 - 54601203T^7 + 157189854T^6 - 325486449T^5 + 557946828T^4 - 664172676T^3 + 608436441T^2 - 334998342T + 124344801
$19$	\( T^{18} - T^{17} + \cdots + 17330569 \) T^18 - T^17 + 78T^16 - 147T^15 + 4537T^14 - 8013T^13 + 116933T^12 - 189668T^11 + 2166274T^10 - 3091482T^9 + 11205860T^8 - 8021926T^7 + 29252628T^6 - 17237695T^5 + 49723117T^4 - 12575879T^3 + 41827862T^2 - 15973431T + 17330569
$23$	\( T^{18} + \cdots + 20826241969 \) T^18 + 100T^16 + 166T^15 + 6934T^14 + 11204T^13 + 240049T^12 + 292878T^11 + 5825441T^10 + 5863918T^9 + 89451484T^8 + 72064827T^7 + 980182316T^6 + 834483162T^5 + 5985150024T^4 + 5784183252T^3 + 25547142781T^2 + 20773423411T + 20826241969
$29$	\( T^{18} + \cdots + 4149034569 \) T^18 + 2T^17 + 105T^16 + 26T^15 + 7330T^14 + 528T^13 + 267453T^12 - 139453T^11 + 6811136T^10 - 413011T^9 + 70626735T^8 + 39886971T^7 + 540160791T^6 + 403871172T^5 + 2326054519T^4 + 2392689426T^3 + 7127315901T^2 + 5048948592*T + 4149034569
$31$	\( T^{18} + \cdots + 39887678961 \) T^18 + 6T^17 + 148T^16 + 316T^15 + 11796T^14 + 23000T^13 + 501384T^12 + 540731T^11 + 13657785T^10 + 15513532T^9 + 228494280T^8 + 194707750T^7 + 2504558872T^6 + 2572089062T^5 + 13642088971T^4 + 5185982025T^3 + 33563480544T^2 + 22121475597*T + 39887678961
$37$	\( T^{18} + \cdots + 13790876950449 \) T^18 - 18T^17 + 379T^16 - 4392T^15 + 62386T^14 - 622243T^13 + 6603222T^12 - 52557347T^11 + 429909552T^10 - 2851569786T^9 + 18416225575T^8 - 91785658383T^7 + 397655366859T^6 - 1202792774661T^5 + 2979857076894T^4 - 4183831051362T^3 + 5999867045784T^2 - 2765775658176*T + 13790876950449
$41$	\( T^{18} + \cdots + 8157521761 \) T^18 + 6T^17 + 284T^16 + 568T^15 + 50371T^14 + 86859T^13 + 4738688T^12 + 366664T^11 + 289856946T^10 + 385426655T^9 + 4446731274T^8 + 4386937879T^7 + 48361767680T^6 + 67419910103T^5 + 104099259448T^4 + 46224527028T^3 + 31976992751T^2 - 3378382195*T + 8157521761
$43$	\( (T^{9} - 15 T^{8} + \cdots + 1087728)^{2} \) (T^9 - 15T^8 - 134T^7 + 2866T^6 - 2963T^5 - 122066T^4 + 443961T^3 + 821983T^2 - 4398048T + 1087728)^2
$47$	\( T^{18} + 8 T^{17} + \cdots + 6305121 \) T^18 + 8T^17 + 222T^16 + 694T^15 + 27962T^14 + 84640T^13 + 1619305T^12 - 2603918T^11 + 26976003T^10 - 8035198T^9 + 156480884T^8 - 164950355T^7 + 443302732T^6 - 128675638T^5 + 227329216T^4 + 101857140T^3 + 123403905T^2 + 27638577*T + 6305121
$53$	\( (T^{9} + 6 T^{8} + \cdots - 6441444)^{2} \) (T^9 + 6T^8 - 210T^7 - 834T^6 + 16561T^5 + 34629T^4 - 573957T^3 - 260793T^2 + 7245288T - 6441444)^2
$59$	\( (T^{9} + 8 T^{8} + \cdots - 18465136)^{2} \) (T^9 + 8T^8 - 250T^7 - 2145T^6 + 16839T^5 + 186637T^4 - 37992T^3 - 4993483T^2 - 17455016T - 18465136)^2
$61$	\( T^{18} + \cdots + 920154407398209 \) T^18 - 6T^17 + 355T^16 - 1192T^15 + 77742T^14 - 212597T^13 + 9755958T^12 - 12410321T^11 + 832874472T^10 - 821974978T^9 + 43509264417T^8 - 7537245835T^7 + 1456271202685T^6 - 1486342215167T^5 + 22304532934252T^4 - 39386856683610T^3 + 274355855742036T^2 - 415754019411138*T + 920154407398209
$67$	\( T^{18} + \cdots + 27\!\cdots\!47 \) T^18 - 37T^17 + 683T^16 - 9643T^15 + 131270T^14 - 1649392T^13 + 17700245T^12 - 171361186T^11 + 1604327069T^10 - 13947825268T^9 + 107489913623T^8 - 769240363954T^7 + 5323578786935T^6 - 33237097768432T^5 + 177230922795890T^4 - 872290179255667T^3 + 4139466026435609T^2 - 15024504069595717*T + 27206534396294947
$71$	\( T^{18} + \cdots + 3332074207609 \) T^18 - 6T^17 + 310T^16 - 428T^15 + 57445T^14 - 38359T^13 + 5987986T^12 + 6516654T^11 + 419117366T^10 + 446373007T^9 + 16072615546T^8 + 33874129413T^7 + 380547734198T^6 + 277351815267T^5 + 2442885038826T^4 - 2635128134538T^3 + 13321849042849T^2 - 6678041511785*T + 3332074207609
$73$	\( T^{18} + \cdots + 8276778009721 \) T^18 - 10T^17 + 508T^16 - 1508T^15 + 135245T^14 - 100145T^13 + 23648556T^12 + 77002100T^11 + 2593699326T^10 + 12776886697T^9 + 223641294416T^8 + 1206556369741T^7 + 9854996409178T^6 + 24844146858263T^5 + 97104116183570T^4 - 86909945381774T^3 + 378687908171153T^2 + 54780914628271*T + 8276778009721
$79$	\( T^{18} + \cdots + 232201002088449 \) T^18 - 4T^17 + 426T^16 - 1274T^15 + 121848T^14 - 327035T^13 + 18294039T^12 - 30322863T^11 + 1923554025T^10 - 2859796190T^9 + 115438168244T^8 - 100293370677T^7 + 4704076331188T^6 - 4549637118615T^5 + 71751002275624T^4 + 35560011785928T^3 + 602736540271227T^2 - 351812668423662*T + 232201002088449
$83$	\( T^{18} + \cdots + 2205732899241 \) T^18 - 5T^17 + 298T^16 + 503T^15 + 51619T^14 + 172295T^13 + 5883426T^12 + 32874024T^11 + 459243370T^10 + 2333955304T^9 + 21505754156T^8 + 110098698250T^7 + 647097484715T^6 + 2088086330903T^5 + 5172972399619T^4 + 7917485443977T^3 + 8954511468750T^2 + 5361367803543*T + 2205732899241
$89$	\( (T^{9} - 5 T^{8} + \cdots - 1881036)^{2} \) (T^9 - 5T^8 - 438T^7 + 2927T^6 + 45676T^5 - 395237T^4 + 69378T^3 + 3858377T^2 - 4098564T - 1881036)^2
$97$	\( T^{18} + \cdots + 78020458060561 \) T^18 - 14T^17 + 522T^16 - 2718T^15 + 129698T^14 - 568429T^13 + 18249465T^12 + 14440513T^11 + 981377899T^10 + 2093172274T^9 + 40290536488T^8 + 120002027793T^7 + 823889143268T^6 + 1980373445067T^5 + 9865891709204T^4 + 23746948300228T^3 + 62070359370679T^2 + 73853078043442*T + 78020458060561
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