LMFDB - Newform orbit 1480.2.q.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1480,2,Mod(121,1480)] 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("1480.121"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1480, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1480 = 2^{3} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1480.q (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [18,0,2,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$11.8178594991$
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} - 2 x^{17} + 18 x^{16} - 22 x^{15} + 194 x^{14} - 224 x^{13} + 1053 x^{12} - 770 x^{11} + \cdots + 441 $ x^18 - 2x^17 + 18x^16 - 22x^15 + 194x^14 - 224x^13 + 1053x^12 - 770x^11 + 3419x^10 - 2128x^9 + 7324x^8 - 2915x^7 + 9758x^6 - 3440x^5 + 7942x^4 - 786x^3 + 2419x^2 - 483*x + 441
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
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_1 q^{3} + (\beta_{5} + 1) q^{5} - \beta_{8} q^{7} + ( - \beta_{16} + \beta_{15} + \cdots - \beta_{4}) q^{9} + ( - \beta_{13} + \beta_{6} - \beta_{2}) q^{11} + \beta_{17} q^{13} - \beta_{4} q^{15}+ \cdots + (2 \beta_{17} + \beta_{16} + \cdots - 2 \beta_{3}) q^{99}+O(q^{100}) $ q + b1 * q^3 + (b5 + 1) * q^5 - b8 * q^7 + (-b16 + b15 - b13 + b11 - b9 + b8 + b6 - b4) * q^9 + (-b13 + b6 - b2) * q^11 + b17 * q^13 - b4 * q^15 + (-b14 - b12 - b6) * q^17 + (-b15 - b14 + b11 + b7 - b5 - 1) * q^19 + (b11 - b9 + b6) * q^21 + (b16 + b9 - b3 + 1) * q^23 + b5 * q^25 + (-b16 - b13 + b12 - b10 - b9 - b4 - b3 - b1) * q^27 + (-b16 + b6 - b2 - 1) * q^29 + (-b16 - b13 + b6 - b3 - b2 + 2) * q^31 + (b15 + 2b11 + b8 - b7 + b5 + b2 + 1) q^33 + (b13 - b8) * q^35 + (-b14 + b13 + b9 - 2b8 - b7 - b6 + 2b5 + 2b4 + b3 + b1) q^37 + (-b13 + 2b11 - 2b9 + b8 + b7 + b6 - b5 - 2b4 - b3) q^39 + (b15 - b11 - b8 + b2 - 2b1) q^41 + (b12 - b9 - b4 + b3 - b1) * q^43 + (-b16 - b13 - b9 + b6 - b4 - b2 - b1) * q^45 + (-b16 + 2b10 - b9 + 3b6 - 2b4 - 3b2 - 2b1 - 1) q^47 + (-b17 + 2b13 - b11 + b10 + b9 - 2b8 - b7 + b3) * q^49 + (b16 - b13 - b6 - b4 + b2 - b1) * q^51 + (-b17 - 2b16 + 2b15 + b13 - b11 + b10 + b9 - b8 + 2b5) q^53 + (-b8 - b2) * q^55 + (b17 + 2b16 - 2b15 - b14 + 2b13 - b12 - b10 - 2b8 - b6 - 3b5) q^57 + (b17 - 2b13 + 2b11 - b10 - 2b9 + 2b8 + b7 + 2b6 - b5 - 2b4 - b3) * q^59 + (-b17 - b14 - b11 - b8 - 2b7 + 3b5 - b2 + 3) * q^61 + (b16 - b13 - b10 - b6 - b4 - b3 + b2 - b1 + 3) * q^63 + (b17 - b10) * q^65 + (-b17 + b15 - b14 + 2b11 - b7 - b2) q^67 + (b17 + b14 + b11 + b7 - 2b5 - 2b2 + 3b1 - 2) q^69 + (-b11 - b8 + 2b5 - b2 + 2b1 + 2) * q^71 + (2b16 + b13 - b12 + b9 - 2b6 + b4 - b3 + 2b2 + b1 + 3) q^73 + (-b4 - b1) * q^75 + (-b17 + 2b15 + b14 - b11 - b8 - 2b7 + 7b5 - 2b1 + 7) * q^77 + (-2b15 - 2b14 - 2b8 - b7 - 3b5 - b2 + b1 - 3) * q^79 + (-b17 - 2b15 - b14 - 3b11 - b8 + b7 - b5 - b2 - 3b1 - 1) q^81 + (-b17 - b13 + b10 + b8 - 3b5) q^83 + (-b12 - b6 + b2) * q^85 + (-b14 - b8 - b7 + b5 - b2 - 3b1 + 1) q^87 + (3b16 - 3b15 - 2b11 + 2b9 - 3b6 + 3b5 + 2b4) q^89 + (-b14 - 2b13 - b12 + 2b8 + 2b7 + b6 - b5 - b4 - 2b3) * q^91 + (-b15 - b14 - 2b8 + b5 - 2b2 - b1 + 1) * q^93 + (b16 - b15 - b14 - b12 + b11 - b9 + b7 - b5 - b3) * q^95 + (b16 + b13 - b12 + 2b10 - 2b9 + 2b6 - 2b4 - 2b2 - 2b1 + 3) * q^97 + (2b17 + b16 - b15 + b14 - b13 + b12 - 2b10 + b8 + 2b7 - b6 - 5b5 - 4b4 - 2b3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 2 q^{3} + 9 q^{5} - 5 q^{9} - 6 q^{11} + 2 q^{13} - 2 q^{15} + q^{17} - 5 q^{21} + 12 q^{23} - 9 q^{25} - 10 q^{27} - 20 q^{29} + 28 q^{31} + 11 q^{33} - 5 q^{39} - 5 q^{41} + 2 q^{43} - 10 q^{45}+ \cdots + 30 q^{99}+O(q^{100}) $ 18 * q + 2 * q^3 + 9 * q^5 - 5 * q^9 - 6 * q^11 + 2 * q^13 - 2 * q^15 + q^17 - 5 * q^21 + 12 * q^23 - 9 * q^25 - 10 * q^27 - 20 * q^29 + 28 * q^31 + 11 * q^33 - 5 * q^39 - 5 * q^41 + 2 * q^43 - 10 * q^45 - 36 * q^47 + 7 * q^49 - 2 * q^51 - 10 * q^53 - 3 * q^55 + 22 * q^57 - 10 * q^59 + 16 * q^61 + 42 * q^63 - 2 * q^65 - 4 * q^67 - 13 * q^69 + 17 * q^71 + 56 * q^73 - 4 * q^75 + 43 * q^77 - 23 * q^79 - 17 * q^81 + 29 * q^83 + 2 * q^85 - q^87 - 16 * q^89 - 4 * q^91 + 5 * q^93 + 26 * q^97 + 30 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{18} - 2 x^{17} + 18 x^{16} - 22 x^{15} + 194 x^{14} - 224 x^{13} + 1053 x^{12} - 770 x^{11} + \cdots + 441 $ x^18 - 2*x^17 + 18*x^16 - 22*x^15 + 194*x^14 - 224*x^13 + 1053*x^12 - 770*x^11 + 3419*x^10 - 2128*x^9 + 7324*x^8 - 2915*x^7 + 9758*x^6 - 3440*x^5 + 7942*x^4 - 786*x^3 + 2419*x^2 - 483*x + 441 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 12\!\cdots\!47 \nu^{17} + \cdots + 24\!\cdots\!26 ) / 42\!\cdots\!35 $ (12680671535375582589547v^17 - 94863536676533277513218v^16 + 479153006301364052562732v^15 - 1567463638010797924720558v^14 + 5406913685417593127671094v^13 - 15029889728807433230083886v^12 + 42584818285871773142484993v^11 - 68413640558360347586559986v^10 + 132652809600937454298172715v^9 - 125430771716727569133289801v^8 + 279047983588669187941201105v^7 - 139492745111501398810253975v^6 + 272694499340730051325877381v^5 + 60208349106461323659405313v^4 + 129712561413030203549322418v^3 + 73305388858880269592688312v^2 - 44288992522951642305319856*v + 24709257899243144154824526) / 42826887877547270447500935
$\beta_{3}$	$=$	$ ( 18\!\cdots\!50 \nu^{17} + \cdots - 48\!\cdots\!90 ) / 40\!\cdots\!47 $ (181600926925644441750v^17 + 624378093456778121973v^16 - 1282726640238060100053v^15 + 20794987748689678594360v^14 - 33065051984732062831298v^13 + 226683741635134835604622v^12 - 489094609494010091823804v^11 + 1628338065639933100804399v^10 - 2396227562291923143014694v^9 + 4824837344160572947122401v^8 - 5521272071706087517647735v^7 + 9303104961415135621827056v^6 - 7949106569914252995309973v^5 + 8314991050190498362530339v^4 - 3489373132247660923358185v^3 + 2879764816893023346900334v^2 - 841356661601286485972898*v - 489102531790320097642490) / 407875122643307337595247
$\beta_{4}$	$=$	$ ( - 10\!\cdots\!96 \nu^{17} + \cdots + 72\!\cdots\!37 ) / 20\!\cdots\!35 $ (-1037642949860136960596v^17 + 2622848317066160087574v^16 - 19552196099529772852926v^15 + 31781012042576880686364v^14 - 208690086099848986514302v^13 + 326520109527191144869628v^12 - 1167554956562423020389704v^11 + 1250407520868564542978838v^10 - 3686984357611538745344185v^9 + 3565660003496783107829358v^8 - 7987457317178262302808140v^7 + 5558485385378892190449045v^6 - 10057908869013073652171208v^5 + 6491969360534991905893876v^4 - 9164155469633511424817094v^3 + 2444534945700792213723374v^2 - 2935660261964297358201517*v + 722416291408921669368837) / 2039375613216536687976235
$\beta_{5}$	$=$	$ ( 34\!\cdots\!97 \nu^{17} + \cdots + 22\!\cdots\!71 ) / 42\!\cdots\!35 $ (34400775781377222350897v^17 - 47011049615691568529278v^16 + 564134149406400640477092v^15 - 346220949100173661808288v^14 + 6006349248693066641660374v^13 - 3323281966931669089800586v^12 + 29367094597719201093232353v^11 - 1969943263849577782006906v^10 + 91357694458288867815161245v^9 + 4221820647071584489519069v^8 + 177072421749374331233553110v^7 + 67458342258028905206106185v^6 + 218954576981722199700622981v^5 + 92877417561336901808509688v^4 + 136879604684463069887052578v^3 + 165408255098141243153353932v^2 + 31880242755434864378628989*v + 2206402921297775679247671) / 42826887877547270447500935
$\beta_{6}$	$=$	$ ( 41\!\cdots\!03 \nu^{17} + \cdots + 46\!\cdots\!44 ) / 42\!\cdots\!35 $ (41692589723432743719103v^17 - 40152045874674507025862v^16 + 666171508309303633978833v^15 + 3074153466271978631138v^14 + 6652890800949860698703996v^13 + 1737009147905958071659066v^12 + 28437225146174223838686477v^11 + 39102572296942621851893656v^10 + 49904176936452349208653625v^9 + 186068773884626904714302951v^8 + 27226324125028280862570745v^7 + 427323492599703794648713315v^6 - 118486054058919070625480701v^5 + 572641243915351886848386487v^4 - 185237072847574121642396693v^3 + 245319333294410890935427608v^2 - 89779861948274806574392124*v + 46181966509252861481906544) / 42826887877547270447500935
$\beta_{7}$	$=$	$ ( - 18\!\cdots\!87 \nu^{17} + \cdots - 92\!\cdots\!86 ) / 42\!\cdots\!35 $ (-182718739511839510543787v^17 + 505714578242012274681523v^16 - 3628211086669761563100552v^15 + 6487106118795336197105858v^14 - 38709330869712992026136734v^13 + 64800341116838836564381606v^12 - 221694525902415718087443183v^11 + 251368962682677702123649171v^10 - 660096434092972657890480655v^9 + 573149111295100952074286411v^8 - 1231369926399541216603384550v^7 + 680270543424319487647422895v^6 - 1261465516806643112211460801v^5 + 280012005882288794247191272v^4 - 564081183583974677497404608v^3 - 254561952946687195471342662v^2 - 93137916281150697186676274*v - 92371972217345299958107986) / 42826887877547270447500935
$\beta_{8}$	$=$	$ ( - 42\!\cdots\!77 \nu^{17} + \cdots - 13\!\cdots\!76 ) / 85\!\cdots\!70 $ (-425463218375596624509677v^17 + 1861856308294243261445443v^16 - 10684489322273948480883132v^15 + 28144790750337667474537013v^14 - 116319165706687798414500859v^13 + 278239489095834361082623426v^12 - 770613425146213735786787658v^11 + 1237351115384426929409164516v^10 - 2302832102437776811797937675v^9 + 2652134795986650333764042351v^8 - 4262703683913953791139410250v^7 + 3530943995475257300062720930v^6 - 4205528264644983060656542846v^5 + 1158435220616371524749730532v^4 - 1653252852048032036530335008v^3 - 105345172810787042975921382v^2 - 821827613828959945905333149*v - 132394609685758432922749476) / 85653775755094540895001870
$\beta_{9}$	$=$	$ ( 19\!\cdots\!32 \nu^{17} + \cdots + 25\!\cdots\!11 ) / 37\!\cdots\!70 $ (1972106707796190539632v^17 + 646296427317819391507v^16 + 14608993794857171640927v^15 + 70630860460756355462792v^14 + 67977275504098680427269v^13 + 787549245594823332446839v^12 - 1065594228267594378570262v^11 + 6535196672273923767763964v^10 - 7314361826772622281607080v^9 + 19428631770087858053737209v^8 - 17668682586436318396300095v^7 + 37273820781117138990395870v^6 - 27031594007422639477533874v^5 + 32940105845701404408673578v^4 - 10358521757576683369977292v^3 + 11177271259594608218339642v^2 - 3053869242921870348141516*v + 2540228724080437888043311) / 370795566039370306904770
$\beta_{10}$	$=$	$ ( - 27\!\cdots\!44 \nu^{17} + \cdots - 51\!\cdots\!77 ) / 40\!\cdots\!70 $ (-27622570560546571509344v^17 - 45319511682205423435529v^16 - 54869107870670027204469v^15 - 1849227475000633974002444v^14 + 1253329940432800659161387v^13 - 20289411663626711616053713v^12 + 36938575250809509419271194v^11 - 150783446291688691518319508v^10 + 199196475881432557368859410v^9 - 446589872067154385312156963v^8 + 467411472317104765460681115v^7 - 842088135020241268464825820v^6 + 687435646828946607577302888v^5 - 763326698583266223094550256v^4 + 310589780986326106963671474v^3 - 262141971037551650740297124v^2 + 74572018705902866115283302*v - 51663813636032526974250877) / 4078751226433073375952470
$\beta_{11}$	$=$	$ ( - 58\!\cdots\!73 \nu^{17} + \cdots + 27\!\cdots\!36 ) / 77\!\cdots\!70 $ (-58068945589908680473573v^17 + 295977268906634048472617v^16 - 1522024126916467486747968v^15 + 4553534709211293401296927v^14 - 16412614467286694855560421v^13 + 45550046693936614893636314v^12 - 110251923114307297969437882v^11 + 207236240501182099931730944v^10 - 321470454830978287479635675v^9 + 478407951241102319474111389v^8 - 584239730875203568262497810v^7 + 709951254088596610561375820v^6 - 518378896091010064954164104v^5 + 430736903112305844489209438v^4 - 143871374350096329576312142v^3 + 171267552995993129798366052v^2 - 17090066096111894064634771*v + 27890874812767028887765836) / 7786706886826776445000170
$\beta_{12}$	$=$	$ ( - 68\!\cdots\!12 \nu^{17} + \cdots - 11\!\cdots\!51 ) / 81\!\cdots\!94 $ (-6880022405497196588112v^17 - 828035755066547820007v^16 - 53321141454116422392129v^15 - 225021026579411701792696v^14 - 254952189580423516690329v^13 - 2531463856320105018919753v^12 + 3518484950648902478812670v^11 - 21906820457878676638523836v^10 + 25164935484208258669007562v^9 - 65332495179557461793660341v^8 + 59486752195992901765871035v^7 - 125647587976092066097856524v^6 + 91167867825542570455609890v^5 - 111063282647937597113419418v^4 + 28613525433197437099264044v^3 - 37695765406057135012145066v^2 + 10302608408675428387262532*v - 11432375237386254833618351) / 815750245286614675190494
$\beta_{13}$	$=$	$ ( 38\!\cdots\!68 \nu^{17} + \cdots - 54\!\cdots\!11 ) / 40\!\cdots\!70 $ (38464327690197992233568v^17 - 47719271646193873989507v^16 + 546356808679758718386493v^15 - 94598095685383722038392v^14 + 5226176975455428037844331v^13 - 533869794541566586232449v^12 + 18074806858936204994812262v^11 + 24077778044530247474383436v^10 + 29472804734868725728668020v^9 + 73903595624729446511712391v^8 + 43063882659263886740145095v^7 + 167330821566401465734628980v^6 + 16740704612782488283983874v^5 + 113243610760940364423385622v^4 + 74125469922776885904705892v^3 + 32774657246218626592072158v^2 - 3647832301872245690208684*v - 5408331602787815043021811) / 4078751226433073375952470
$\beta_{14}$	$=$	$ ( - 85\!\cdots\!31 \nu^{17} + \cdots + 88\!\cdots\!12 ) / 85\!\cdots\!70 $ (-853200470796681402282631v^17 + 5112581917057836613131569v^16 - 24191402264332952998668096v^15 + 80753106288984068440491349v^14 - 260703541997839112881971797v^13 + 810306955507781537862523838v^12 - 1807545004528587108068931534v^11 + 3766973851423442295040490708v^10 - 5212237472677083674740864025v^9 + 8844729779341457740747716463v^8 - 9321797322727245376146249820v^7 + 13638017404082309944647184340v^6 - 7794485167905590492211698888v^5 + 9140793858590157607757125946v^4 - 1535642827460068055625880474v^3 + 4522546121822366296104048684v^2 - 330187156344821857541870467*v + 885544067277092208962851812) / 85653775755094540895001870
$\beta_{15}$	$=$	$ ( 61\!\cdots\!82 \nu^{17} + \cdots - 33\!\cdots\!64 ) / 61\!\cdots\!05 $ (61103803116721773530182v^17 - 334467647236142053289603v^16 + 1675690302102244380311382v^15 - 5237977745840894181941203v^14 + 18139835836550408234257019v^13 - 52316931610359970967907176v^12 + 124355151294538438721834058v^11 - 241010907921662896202131061v^10 + 363040904534792510580373010v^9 - 550384112492047158799480711v^8 + 655373427550153255904018350v^7 - 818810077876761475133025185v^6 + 583665980058641253305534861v^5 - 472356896802152901929926922v^4 + 158825072687493090505856773v^3 - 203562837519947224009888203v^2 + 59339496643050164985072874*v - 33914575080637570937761164) / 6118126839649610063928705
$\beta_{16}$	$=$	$ ( - 27\!\cdots\!46 \nu^{17} + \cdots - 16\!\cdots\!53 ) / 20\!\cdots\!35 $ (-27112025457055013284946v^17 + 19412843239538261286329v^16 - 316117164013651135125686v^15 - 305551536681166919283026v^14 - 2624853084545112662822057v^13 - 3667599237353663966724422v^12 - 2231352679371820326026664v^11 - 44252341692903667601917217v^10 + 26596716764359814605076895v^9 - 132929381208444750869900727v^8 + 74175518805975953201810660v^7 - 267171250014182451768998770v^6 + 134656155816479175876565972v^5 - 220805985634736222849579734v^4 + 15704636130570202751380881v^3 - 72541317251617332887683501v^2 + 17571484504984304791838823*v - 16627524837408725657017153) / 2039375613216536687976235
$\beta_{17}$	$=$	$ ( 18\!\cdots\!17 \nu^{17} + \cdots - 53\!\cdots\!84 ) / 11\!\cdots\!10 $ (18658018221063834078317v^17 - 85675416180508022280943v^16 + 462475243370006272398612v^15 - 1287521317988838157152893v^14 + 4974381597658228193462149v^13 - 12883342214507701590236146v^12 + 32535127214087944080165558v^11 - 57585978583491122770099696v^10 + 95084646581380100414729395v^9 - 133276487102885346982313891v^8 + 173617133034043663767083480v^7 - 194510284704348049548340690v^6 + 157066574332528127585847766v^5 - 118163555603860300323810712v^4 + 47744064868065167602743308v^3 - 38811425817065760062730798v^2 + 1639987278188805714686879*v - 5331727741067987069277684) / 1112386698118110920714310

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ -\beta_{16} + \beta_{15} - \beta_{13} + \beta_{11} - \beta_{9} + \beta_{8} + \beta_{6} + 3\beta_{5} - \beta_{4} $ -b16 + b15 - b13 + b11 - b9 + b8 + b6 + 3*b5 - b4
$\nu^{3}$	$=$	$ -\beta_{16} - \beta_{13} + \beta_{12} - \beta_{10} - \beta_{9} - 7\beta_{4} - \beta_{3} - 7\beta_1 $ -b16 - b13 + b12 - b10 - b9 - 7b4 - b3 - 7b1
$\nu^{4}$	$=$	$ - \beta_{17} - 11 \beta_{15} - \beta_{14} - 12 \beta_{11} - 10 \beta_{8} + \beta_{7} - 19 \beta_{5} + \cdots - 19 $ -b17 - 11b15 - b14 - 12b11 - 10b8 + b7 - 19b5 - 10b2 - 12b1 - 19
$\nu^{5}$	$=$	$ - 12 \beta_{17} + 14 \beta_{16} - 14 \beta_{15} - 11 \beta_{14} + 12 \beta_{13} - 11 \beta_{12} + \cdots + 11 \beta_{3} $ -12b17 + 14b16 - 14b15 - 11b14 + 12b13 - 11b12 - 17b11 + 12b10 + 17b9 - 12b8 - 11b7 - b6 - 2b5 + 65b4 + 11b3
$\nu^{6}$	$=$	$ 111 \beta_{16} + 98 \beta_{13} - 14 \beta_{12} + 17 \beta_{10} + 129 \beta_{9} - 95 \beta_{6} + \cdots + 159 $ 111b16 + 98b13 - 14b12 + 17b10 + 129b9 - 95b6 + 136b4 - 9b3 + 95b2 + 136b1 + 159
$\nu^{7}$	$=$	$ 129 \beta_{17} + 171 \beta_{15} + 111 \beta_{14} + 222 \beta_{11} + 142 \beta_{8} + 107 \beta_{7} + \cdots + 52 $ 129b17 + 171b15 + 111b14 + 222b11 + 142b8 + 107b7 + 52b5 + 33b2 + 657*b1 + 52
$\nu^{8}$	$=$	$ 222 \beta_{17} - 1124 \beta_{16} + 1124 \beta_{15} + 171 \beta_{14} - 980 \beta_{13} + \cdots + 56 \beta_{3} $ 222b17 - 1124b16 + 1124b15 + 171b14 - 980b13 + 171b12 + 1351b11 - 222b10 - 1351b9 + 980b8 - 56b7 + 918b6 + 1476b5 - 1527b4 + 56*b3
$\nu^{9}$	$=$	$ - 2019 \beta_{16} - 1672 \beta_{13} + 1124 \beta_{12} - 1351 \beta_{10} - 2663 \beta_{9} + 601 \beta_{6} + \cdots - 910 $ -2019b16 - 1672b13 + 1124b12 - 1351b10 - 2663b9 + 601b6 - 6819b4 - 1025b3 - 601b2 - 6819b1 - 910
$\nu^{10}$	$=$	$ - 2663 \beta_{17} - 11510 \beta_{15} - 2019 \beta_{14} - 14079 \beta_{11} - 9969 \beta_{8} + \cdots - 14303 $ -2663b17 - 11510b15 - 2019b14 - 14079b11 - 9969b8 + 197b7 - 14303b5 - 9028b2 - 17033*b1 - 14303
$\nu^{11}$	$=$	$ - 14079 \beta_{17} + 23356 \beta_{16} - 23356 \beta_{15} - 11510 \beta_{14} + 19412 \beta_{13} + \cdots + 9869 \beta_{3} $ -14079b17 + 23356b16 - 23356b15 - 11510b14 + 19412b13 - 11510b12 - 30773b11 + 14079b10 + 30773b9 - 19412b8 - 9869b7 - 8783b6 - 13269b5 + 71545b4 + 9869*b3
$\nu^{12}$	$=$	$ 119007 \beta_{16} + 102649 \beta_{13} - 23356 \beta_{12} + 30773 \beta_{10} + 147016 \beta_{9} + \cdots + 141737 $ 119007b16 + 102649b13 - 23356b12 + 30773b10 + 147016b9 - 90043b6 + 188816b4 + 1483b3 + 90043b2 + 188816b1 + 141737
$\nu^{13}$	$=$	$ 147016 \beta_{17} + 265975 \beta_{15} + 119007 \beta_{14} + 348532 \beta_{11} + 222027 \beta_{8} + \cdots + 174956 $ 147016b17 + 265975b15 + 119007b14 + 348532b11 + 222027b8 + 95977b7 + 174956b5 + 115253b2 + 755256*b1 + 174956
$\nu^{14}$	$=$	$ 348532 \beta_{17} - 1239969 \beta_{16} + 1239969 \beta_{15} + 265975 \beta_{14} - 1066414 \beta_{13} + \cdots - 47285 \beta_{3} $ 348532b17 - 1239969b16 + 1239969b15 + 265975b14 - 1066414b13 + 265975b12 + 1541499b11 - 348532b10 - 1541499b9 + 1066414b8 + 47285b7 + 908649b6 + 1425442b5 - 2081696b4 - 47285*b3
$\nu^{15}$	$=$	$ - 2992251 \beta_{16} - 2507303 \beta_{13} + 1239969 \beta_{12} - 1541499 \beta_{10} - 3898425 \beta_{9} + \cdots - 2170884 $ -2992251b16 - 2507303b13 + 1239969b12 - 1541499b10 - 3898425b9 + 1422829b6 - 8006949b4 - 943921b3 - 1422829b2 - 8006949b1 - 2170884
$\nu^{16}$	$=$	$ - 3898425 \beta_{17} - 13000230 \beta_{15} - 2992251 \beta_{14} - 16234381 \beta_{11} - 11155617 \beta_{8} + \cdots - 14499638 $ -3898425b17 - 13000230b15 - 2992251b14 - 16234381b11 - 11155617b8 - 780438b7 - 14499638b5 - 9262782b2 - 22847710*b1 - 14499638
$\nu^{17}$	$=$	$ - 16234381 \beta_{17} + 33350098 \beta_{16} - 33350098 \beta_{15} - 13000230 \beta_{14} + \cdots + 9388518 \beta_{3} $ -16234381b17 + 33350098b16 - 33350098b15 - 13000230b14 + 28029870b13 - 13000230b12 - 43233461b11 + 16234381b10 + 43233461b9 - 28029870b8 - 9388518b7 - 16909494b6 - 25905412b5 + 85167226b4 + 9388518*b3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1480\mathbb{Z}\right)^\times$.

$n$	$297$	$741$	$1001$	$1111$
$\chi(n)$	$1$	$1$	$-1 - \beta_{5}$	$1$

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} $

121.1

−1.47098	−	2.54781i
−0.727131	−	1.25943i
−0.725451	−	1.25652i
−0.306255	−	0.530449i
0.234802	+	0.406690i
0.579516	+	1.00375i
0.769214	+	1.33232i
1.00426	+	1.73943i
1.64202	+	2.84407i
−1.47098	+	2.54781i
−0.727131	+	1.25943i
−0.725451	+	1.25652i
−0.306255	+	0.530449i
0.234802	−	0.406690i
0.579516	−	1.00375i
0.769214	−	1.33232i
1.00426	−	1.73943i
1.64202	−	2.84407i

−1.47098

−

2.54781i

0.500000

0.866025i

−1.07613

−

1.86391i

−2.82755

4.89745i

121.2

−0.727131

−

1.25943i

0.500000

0.866025i

0.958244

1.65973i

0.442561

−

0.766538i

121.3

−0.725451

−

1.25652i

0.500000

0.866025i

−0.516766

−

0.895065i

0.447441

−

0.774991i

121.4

−0.306255

−

0.530449i

0.500000

0.866025i

−0.678169

−

1.17462i

1.31242

−

2.27317i

121.5

0.234802

0.406690i

0.500000

0.866025i

2.24809

3.89381i

1.38974

−

2.40709i

121.6

0.579516

1.00375i

0.500000

0.866025i

−1.05730

−

1.83130i

0.828323

−

1.43470i

121.7

0.769214

1.33232i

0.500000

0.866025i

1.55566

2.69448i

0.316621

−

0.548403i

121.8

1.00426

1.73943i

0.500000

0.866025i

−1.60485

−

2.77968i

−0.517073

0.895596i

121.9

1.64202

2.84407i

0.500000

0.866025i

0.171219

0.296561i

−3.89248

6.74197i

1321.1

−1.47098

2.54781i

0.500000

−

0.866025i

−1.07613

1.86391i

−2.82755

−

4.89745i

1321.2

−0.727131

1.25943i

0.500000

−

0.866025i

0.958244

−

1.65973i

0.442561

0.766538i

1321.3

−0.725451

1.25652i

0.500000

−

0.866025i

−0.516766

0.895065i

0.447441

0.774991i

1321.4

−0.306255

0.530449i

0.500000

−

0.866025i

−0.678169

1.17462i

1.31242

2.27317i

1321.5

0.234802

−

0.406690i

0.500000

−

0.866025i

2.24809

−

3.89381i

1.38974

2.40709i

1321.6

0.579516

−

1.00375i

0.500000

−

0.866025i

−1.05730

1.83130i

0.828323

1.43470i

1321.7

0.769214

−

1.33232i

0.500000

−

0.866025i

1.55566

−

2.69448i

0.316621

0.548403i

1321.8

1.00426

−

1.73943i

0.500000

−

0.866025i

−1.60485

2.77968i

−0.517073

−

0.895596i

1321.9

1.64202

−

2.84407i

0.500000

−

0.866025i

0.171219

−

0.296561i

−3.89248

−

6.74197i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1480.2.q.b	✓	18
37.c	even	3	1	inner	1480.2.q.b	✓	18

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1480.2.q.b	✓	18	1.a	even	1	1	trivial
1480.2.q.b	✓	18	37.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{18} - 2 T_{3}^{17} + 18 T_{3}^{16} - 22 T_{3}^{15} + 194 T_{3}^{14} - 224 T_{3}^{13} + 1053 T_{3}^{12} + \cdots + 441 $ T3^18 - 2*T3^17 + 18*T3^16 - 22*T3^15 + 194*T3^14 - 224*T3^13 + 1053*T3^12 - 770*T3^11 + 3419*T3^10 - 2128*T3^9 + 7324*T3^8 - 2915*T3^7 + 9758*T3^6 - 3440*T3^5 + 7942*T3^4 - 786*T3^3 + 2419*T3^2 - 483*T3 + 441 acting on $S_{2}^{\mathrm{new}}(1480, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{18} $ T^18
$3$	$ T^{18} - 2 T^{17} + \cdots + 441 $ T^18 - 2T^17 + 18T^16 - 22T^15 + 194T^14 - 224T^13 + 1053T^12 - 770T^11 + 3419T^10 - 2128T^9 + 7324T^8 - 2915T^7 + 9758T^6 - 3440T^5 + 7942T^4 - 786T^3 + 2419T^2 - 483*T + 441
$5$	$ (T^{2} - T + 1)^{9} $ (T^2 - T + 1)^9
$7$	$ T^{18} + 28 T^{16} + \cdots + 35344 $ T^18 + 28T^16 + 48T^15 + 559T^14 + 1000T^13 + 5984T^12 + 12037T^11 + 46203T^10 + 78084T^9 + 196454T^8 + 272054T^7 + 538910T^6 + 584790T^5 + 718573T^4 + 348310T^3 + 212664T^2 - 36472T + 35344
$11$	$ (T^{9} + 3 T^{8} + \cdots - 7164)^{2} $ (T^9 + 3T^8 - 49T^7 - 110T^6 + 788T^5 + 1117T^4 - 5240T^3 - 2225T^2 + 13222T - 7164)^2
$13$	$ T^{18} - 2 T^{17} + \cdots + 790321 $ T^18 - 2T^17 + 67T^16 - 110T^15 + 3016T^14 - 5049T^13 + 70021T^12 - 129872T^11 + 1189502T^10 - 2076747T^9 + 11914103T^8 - 17421889T^7 + 80280765T^6 - 89615619T^5 + 228227723T^4 + 28670404T^3 + 20953977T^2 - 2523871*T + 790321
$17$	$ T^{18} - T^{17} + \cdots + 26584336 $ T^18 - T^17 + 88T^16 - 199T^15 + 5779T^14 - 11640T^13 + 165215T^12 - 238774T^11 + 3254150T^10 - 3737676T^9 + 24450270T^8 + 2852226T^7 + 90697721T^6 + 51439058T^5 + 281849449T^4 + 263415690T^3 + 357027768T^2 + 102862200T + 26584336
$19$	$ T^{18} + \cdots + 44522688016 $ T^18 + 124T^16 - 106T^15 + 10402T^14 - 10386T^13 + 480073T^12 - 608757T^11 + 16100070T^10 - 20267824T^9 + 309752819T^8 - 428665272T^7 + 4163833518T^6 - 4703877092T^5 + 18059638817T^4 - 15253147076T^3 + 52757860988T^2 - 40724616016T + 44522688016
$23$	$ (T^{9} - 6 T^{8} + \cdots + 6572)^{2} $ (T^9 - 6T^8 - 89T^7 + 420T^6 + 2484T^5 - 7657T^4 - 24571T^3 + 33225T^2 + 35178T + 6572)^2
$29$	$ (T^{9} + 10 T^{8} + \cdots + 3548)^{2} $ (T^9 + 10T^8 - 8T^7 - 350T^6 - 698T^5 + 2844T^4 + 8243T^3 - 5969T^2 - 21472T + 3548)^2
$31$	$ (T^{9} - 14 T^{8} + \cdots - 4123)^{2} $ (T^9 - 14T^8 - 30T^7 + 818T^6 + 336T^5 - 13652T^4 - 13034T^3 + 14767T^2 + 10359T - 4123)^2
$37$	$ T^{18} + \cdots + 129961739795077 $ T^18 + 90T^16 - 470T^15 + 1449T^14 - 41670T^13 + 30338T^12 - 670437T^11 + 8329023T^10 + 16966273T^9 + 308173851T^8 - 917828253T^7 + 1536710714T^6 - 78096288870T^5 + 100479393693T^4 - 1205891412230T^3 + 8543868941970*T^2 + 129961739795077
$41$	$ T^{18} + 5 T^{17} + \cdots + 23843689 $ T^18 + 5T^17 + 192T^16 + 359T^15 + 23229T^14 + 38029T^13 + 1308964T^12 - 320528T^11 + 45197524T^10 - 22666670T^9 + 967136620T^8 - 1120481528T^7 + 12808153293T^6 - 6379987935T^5 + 67291962879T^4 - 32587293713T^3 + 252454285680T^2 + 2451856843*T + 23843689
$43$	$ (T^{9} - T^{8} - 214 T^{7} + \cdots + 25)^{2} $ (T^9 - T^8 - 214T^7 + 192T^6 + 12697T^5 - 9465T^4 - 228054T^3 + 144583T^2 + 5160*T + 25)^2
$47$	$ (T^{9} + 18 T^{8} + \cdots - 107144572)^{2} $ (T^9 + 18T^8 - 223T^7 - 5569T^6 + 283T^5 + 490161T^4 + 1847571T^3 - 11529243T^2 - 77088590T - 107144572)^2
$53$	$ T^{18} + \cdots + 7650341105625 $ T^18 + 10T^17 + 369T^16 + 1928T^15 + 76648T^14 + 340151T^13 + 8368942T^12 + 1727177T^11 + 411602712T^10 - 453833160T^9 + 15268608045T^8 - 34980080529T^7 + 281223309319T^6 - 514599120597T^5 + 3011659264646T^4 - 5831372123350T^3 + 16712584180450T^2 - 11915051715000*T + 7650341105625
$59$	$ T^{18} + \cdots + 17956541850256 $ T^18 + 10T^17 + 304T^16 + 1878T^15 + 50258T^14 + 276894T^13 + 4720061T^12 + 14368439T^11 + 237860776T^10 + 597467856T^9 + 8050337137T^8 + 11480710402T^7 + 155577268834T^6 + 167386850200T^5 + 2005759145137T^4 + 324346155854T^3 + 7125683148864T^2 + 3745396316856*T + 17956541850256
$61$	$ T^{18} + \cdots + 111805161849616 $ T^18 - 16T^17 + 414T^16 - 3178T^15 + 62284T^14 - 329976T^13 + 6330369T^12 - 13575009T^11 + 336663738T^10 + 493588334T^9 + 14729419001T^8 + 46340086040T^7 + 475159674870T^6 + 2155664346324T^5 + 11174523000393T^4 + 32654085617640T^3 + 81255334314852T^2 + 108942158549472*T + 111805161849616
$67$	$ T^{18} + \cdots + 44472039837696 $ T^18 + 4T^17 + 391T^16 - 888T^15 + 97080T^14 - 335980T^13 + 14359273T^12 - 83064743T^11 + 1482461658T^10 - 7220437123T^9 + 76753719079T^8 - 387098549916T^7 + 2734319029663T^6 - 9163115612591T^5 + 30557404291033T^4 - 31824846627712T^3 + 43622397509440T^2 + 14275576430592*T + 44472039837696
$71$	$ T^{18} + \cdots + 1911870351616 $ T^18 - 17T^17 + 339T^16 - 4342T^15 + 63225T^14 - 682664T^13 + 6531114T^12 - 46180186T^11 + 276750881T^10 - 1273231372T^9 + 5246422541T^8 - 17440578264T^7 + 57216889839T^6 - 144702999437T^5 + 359986870685T^4 - 499761018852T^3 + 833403576000T^2 - 222775737664*T + 1911870351616
$73$	$ (T^{9} - 28 T^{8} + \cdots - 1989644)^{2} $ (T^9 - 28T^8 + 152T^7 + 1686T^6 - 14739T^5 - 22418T^4 + 307580T^3 + 181447T^2 - 2018062T - 1989644)^2
$79$	$ T^{18} + \cdots + 362504540360704 $ T^18 + 23T^17 + 670T^16 + 10183T^15 + 205024T^14 + 2754321T^13 + 39879945T^12 + 422926123T^11 + 4681110596T^10 + 41437926723T^9 + 356553072525T^8 + 2301434798952T^7 + 12682948813400T^6 + 45960562726514T^5 + 130053280908985T^4 + 165800132200544T^3 + 214519223749184T^2 - 16327710529536*T + 362504540360704
$83$	$ T^{18} + \cdots + 6249851136 $ T^18 - 29T^17 + 564T^16 - 6913T^15 + 66259T^14 - 462053T^13 + 2732425T^12 - 12561315T^11 + 57503976T^10 - 211184053T^9 + 840505909T^8 - 2159579790T^7 + 6916505697T^6 - 14620434822T^5 + 37518690105T^4 - 45689431536T^3 + 43880166288T^2 - 19227684096*T + 6249851136
$89$	$ T^{18} + \cdots + 149023837531024 $ T^18 + 16T^17 + 586T^16 + 5128T^15 + 165190T^14 + 1234430T^13 + 27848590T^12 + 87791497T^11 + 2150074186T^10 + 1888533482T^9 + 130195895226T^8 - 114229331582T^7 + 3018830458915T^6 - 2842659176951T^5 + 49707564230077T^4 - 57281834115372T^3 + 242763325201092T^2 + 156522973268752*T + 149023837531024
$97$	$ (T^{9} - 13 T^{8} + \cdots - 4302148)^{2} $ (T^9 - 13T^8 - 459T^7 + 4494T^6 + 70948T^5 - 392235T^4 - 4038204T^3 + 8486767T^2 + 68621228T - 4302148)^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:	\( N \)	\(=\)	\( 1480 = 2^{3} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1480.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + (\beta_{5} + 1) q^{5} - \beta_{8} q^{7} + ( - \beta_{16} + \beta_{15} + \cdots - \beta_{4}) q^{9} + ( - \beta_{13} + \beta_{6} - \beta_{2}) q^{11} + \beta_{17} q^{13} - \beta_{4} q^{15}+ \cdots + (2 \beta_{17} + \beta_{16} + \cdots - 2 \beta_{3}) q^{99}+O(q^{100}) \) q + b1 * q^3 + (b5 + 1) * q^5 - b8 * q^7 + (-b16 + b15 - b13 + b11 - b9 + b8 + b6 - b4) * q^9 + (-b13 + b6 - b2) * q^11 + b17 * q^13 - b4 * q^15 + (-b14 - b12 - b6) * q^17 + (-b15 - b14 + b11 + b7 - b5 - 1) * q^19 + (b11 - b9 + b6) * q^21 + (b16 + b9 - b3 + 1) * q^23 + b5 * q^25 + (-b16 - b13 + b12 - b10 - b9 - b4 - b3 - b1) * q^27 + (-b16 + b6 - b2 - 1) * q^29 + (-b16 - b13 + b6 - b3 - b2 + 2) * q^31 + (b15 + 2b11 + b8 - b7 + b5 + b2 + 1) q^33 + (b13 - b8) * q^35 + (-b14 + b13 + b9 - 2b8 - b7 - b6 + 2b5 + 2b4 + b3 + b1) q^37 + (-b13 + 2b11 - 2b9 + b8 + b7 + b6 - b5 - 2b4 - b3) q^39 + (b15 - b11 - b8 + b2 - 2b1) q^41 + (b12 - b9 - b4 + b3 - b1) * q^43 + (-b16 - b13 - b9 + b6 - b4 - b2 - b1) * q^45 + (-b16 + 2b10 - b9 + 3b6 - 2b4 - 3b2 - 2b1 - 1) q^47 + (-b17 + 2b13 - b11 + b10 + b9 - 2b8 - b7 + b3) * q^49 + (b16 - b13 - b6 - b4 + b2 - b1) * q^51 + (-b17 - 2b16 + 2b15 + b13 - b11 + b10 + b9 - b8 + 2b5) q^53 + (-b8 - b2) * q^55 + (b17 + 2b16 - 2b15 - b14 + 2b13 - b12 - b10 - 2b8 - b6 - 3b5) q^57 + (b17 - 2b13 + 2b11 - b10 - 2b9 + 2b8 + b7 + 2b6 - b5 - 2b4 - b3) * q^59 + (-b17 - b14 - b11 - b8 - 2b7 + 3b5 - b2 + 3) * q^61 + (b16 - b13 - b10 - b6 - b4 - b3 + b2 - b1 + 3) * q^63 + (b17 - b10) * q^65 + (-b17 + b15 - b14 + 2b11 - b7 - b2) q^67 + (b17 + b14 + b11 + b7 - 2b5 - 2b2 + 3b1 - 2) q^69 + (-b11 - b8 + 2b5 - b2 + 2b1 + 2) * q^71 + (2b16 + b13 - b12 + b9 - 2b6 + b4 - b3 + 2b2 + b1 + 3) q^73 + (-b4 - b1) * q^75 + (-b17 + 2b15 + b14 - b11 - b8 - 2b7 + 7b5 - 2b1 + 7) * q^77 + (-2b15 - 2b14 - 2b8 - b7 - 3b5 - b2 + b1 - 3) * q^79 + (-b17 - 2b15 - b14 - 3b11 - b8 + b7 - b5 - b2 - 3b1 - 1) q^81 + (-b17 - b13 + b10 + b8 - 3b5) q^83 + (-b12 - b6 + b2) * q^85 + (-b14 - b8 - b7 + b5 - b2 - 3b1 + 1) q^87 + (3b16 - 3b15 - 2b11 + 2b9 - 3b6 + 3b5 + 2b4) q^89 + (-b14 - 2b13 - b12 + 2b8 + 2b7 + b6 - b5 - b4 - 2b3) * q^91 + (-b15 - b14 - 2b8 + b5 - 2b2 - b1 + 1) * q^93 + (b16 - b15 - b14 - b12 + b11 - b9 + b7 - b5 - b3) * q^95 + (b16 + b13 - b12 + 2b10 - 2b9 + 2b6 - 2b4 - 2b2 - 2b1 + 3) * q^97 + (2b17 + b16 - b15 + b14 - b13 + b12 - 2b10 + b8 + 2b7 - b6 - 5b5 - 4b4 - 2b3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 2 q^{3} + 9 q^{5} - 5 q^{9} - 6 q^{11} + 2 q^{13} - 2 q^{15} + q^{17} - 5 q^{21} + 12 q^{23} - 9 q^{25} - 10 q^{27} - 20 q^{29} + 28 q^{31} + 11 q^{33} - 5 q^{39} - 5 q^{41} + 2 q^{43} - 10 q^{45}+ \cdots + 30 q^{99}+O(q^{100}) \) 18 * q + 2 * q^3 + 9 * q^5 - 5 * q^9 - 6 * q^11 + 2 * q^13 - 2 * q^15 + q^17 - 5 * q^21 + 12 * q^23 - 9 * q^25 - 10 * q^27 - 20 * q^29 + 28 * q^31 + 11 * q^33 - 5 * q^39 - 5 * q^41 + 2 * q^43 - 10 * q^45 - 36 * q^47 + 7 * q^49 - 2 * q^51 - 10 * q^53 - 3 * q^55 + 22 * q^57 - 10 * q^59 + 16 * q^61 + 42 * q^63 - 2 * q^65 - 4 * q^67 - 13 * q^69 + 17 * q^71 + 56 * q^73 - 4 * q^75 + 43 * q^77 - 23 * q^79 - 17 * q^81 + 29 * q^83 + 2 * q^85 - q^87 - 16 * q^89 - 4 * q^91 + 5 * q^93 + 26 * q^97 + 30 * 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	1480.2.q.b

Level	$1480$
Weight	$2$
Character orbit	1480.q
Analytic conductor	$11.818$
Analytic rank	$0$
Dimension	$18$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(11.8178594991\)
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:	\( x^{18} - 2 x^{17} + 18 x^{16} - 22 x^{15} + 194 x^{14} - 224 x^{13} + 1053 x^{12} - 770 x^{11} + \cdots + 441 \) x^18 - 2x^17 + 18x^16 - 22x^15 + 194x^14 - 224x^13 + 1053x^12 - 770x^11 + 3419x^10 - 2128x^9 + 7324x^8 - 2915x^7 + 9758x^6 - 3440x^5 + 7942x^4 - 786x^3 + 2419x^2 - 483*x + 441
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 12\!\cdots\!47 \nu^{17} + \cdots + 24\!\cdots\!26 ) / 42\!\cdots\!35 \) (12680671535375582589547v^17 - 94863536676533277513218v^16 + 479153006301364052562732v^15 - 1567463638010797924720558v^14 + 5406913685417593127671094v^13 - 15029889728807433230083886v^12 + 42584818285871773142484993v^11 - 68413640558360347586559986v^10 + 132652809600937454298172715v^9 - 125430771716727569133289801v^8 + 279047983588669187941201105v^7 - 139492745111501398810253975v^6 + 272694499340730051325877381v^5 + 60208349106461323659405313v^4 + 129712561413030203549322418v^3 + 73305388858880269592688312v^2 - 44288992522951642305319856*v + 24709257899243144154824526) / 42826887877547270447500935
\(\beta_{3}\)	\(=\)	\( ( 18\!\cdots\!50 \nu^{17} + \cdots - 48\!\cdots\!90 ) / 40\!\cdots\!47 \) (181600926925644441750v^17 + 624378093456778121973v^16 - 1282726640238060100053v^15 + 20794987748689678594360v^14 - 33065051984732062831298v^13 + 226683741635134835604622v^12 - 489094609494010091823804v^11 + 1628338065639933100804399v^10 - 2396227562291923143014694v^9 + 4824837344160572947122401v^8 - 5521272071706087517647735v^7 + 9303104961415135621827056v^6 - 7949106569914252995309973v^5 + 8314991050190498362530339v^4 - 3489373132247660923358185v^3 + 2879764816893023346900334v^2 - 841356661601286485972898*v - 489102531790320097642490) / 407875122643307337595247
\(\beta_{4}\)	\(=\)	\( ( - 10\!\cdots\!96 \nu^{17} + \cdots + 72\!\cdots\!37 ) / 20\!\cdots\!35 \) (-1037642949860136960596v^17 + 2622848317066160087574v^16 - 19552196099529772852926v^15 + 31781012042576880686364v^14 - 208690086099848986514302v^13 + 326520109527191144869628v^12 - 1167554956562423020389704v^11 + 1250407520868564542978838v^10 - 3686984357611538745344185v^9 + 3565660003496783107829358v^8 - 7987457317178262302808140v^7 + 5558485385378892190449045v^6 - 10057908869013073652171208v^5 + 6491969360534991905893876v^4 - 9164155469633511424817094v^3 + 2444534945700792213723374v^2 - 2935660261964297358201517*v + 722416291408921669368837) / 2039375613216536687976235
\(\beta_{5}\)	\(=\)	\( ( 34\!\cdots\!97 \nu^{17} + \cdots + 22\!\cdots\!71 ) / 42\!\cdots\!35 \) (34400775781377222350897v^17 - 47011049615691568529278v^16 + 564134149406400640477092v^15 - 346220949100173661808288v^14 + 6006349248693066641660374v^13 - 3323281966931669089800586v^12 + 29367094597719201093232353v^11 - 1969943263849577782006906v^10 + 91357694458288867815161245v^9 + 4221820647071584489519069v^8 + 177072421749374331233553110v^7 + 67458342258028905206106185v^6 + 218954576981722199700622981v^5 + 92877417561336901808509688v^4 + 136879604684463069887052578v^3 + 165408255098141243153353932v^2 + 31880242755434864378628989*v + 2206402921297775679247671) / 42826887877547270447500935
\(\beta_{6}\)	\(=\)	\( ( 41\!\cdots\!03 \nu^{17} + \cdots + 46\!\cdots\!44 ) / 42\!\cdots\!35 \) (41692589723432743719103v^17 - 40152045874674507025862v^16 + 666171508309303633978833v^15 + 3074153466271978631138v^14 + 6652890800949860698703996v^13 + 1737009147905958071659066v^12 + 28437225146174223838686477v^11 + 39102572296942621851893656v^10 + 49904176936452349208653625v^9 + 186068773884626904714302951v^8 + 27226324125028280862570745v^7 + 427323492599703794648713315v^6 - 118486054058919070625480701v^5 + 572641243915351886848386487v^4 - 185237072847574121642396693v^3 + 245319333294410890935427608v^2 - 89779861948274806574392124*v + 46181966509252861481906544) / 42826887877547270447500935
\(\beta_{7}\)	\(=\)	\( ( - 18\!\cdots\!87 \nu^{17} + \cdots - 92\!\cdots\!86 ) / 42\!\cdots\!35 \) (-182718739511839510543787v^17 + 505714578242012274681523v^16 - 3628211086669761563100552v^15 + 6487106118795336197105858v^14 - 38709330869712992026136734v^13 + 64800341116838836564381606v^12 - 221694525902415718087443183v^11 + 251368962682677702123649171v^10 - 660096434092972657890480655v^9 + 573149111295100952074286411v^8 - 1231369926399541216603384550v^7 + 680270543424319487647422895v^6 - 1261465516806643112211460801v^5 + 280012005882288794247191272v^4 - 564081183583974677497404608v^3 - 254561952946687195471342662v^2 - 93137916281150697186676274*v - 92371972217345299958107986) / 42826887877547270447500935
\(\beta_{8}\)	\(=\)	\( ( - 42\!\cdots\!77 \nu^{17} + \cdots - 13\!\cdots\!76 ) / 85\!\cdots\!70 \) (-425463218375596624509677v^17 + 1861856308294243261445443v^16 - 10684489322273948480883132v^15 + 28144790750337667474537013v^14 - 116319165706687798414500859v^13 + 278239489095834361082623426v^12 - 770613425146213735786787658v^11 + 1237351115384426929409164516v^10 - 2302832102437776811797937675v^9 + 2652134795986650333764042351v^8 - 4262703683913953791139410250v^7 + 3530943995475257300062720930v^6 - 4205528264644983060656542846v^5 + 1158435220616371524749730532v^4 - 1653252852048032036530335008v^3 - 105345172810787042975921382v^2 - 821827613828959945905333149*v - 132394609685758432922749476) / 85653775755094540895001870
\(\beta_{9}\)	\(=\)	\( ( 19\!\cdots\!32 \nu^{17} + \cdots + 25\!\cdots\!11 ) / 37\!\cdots\!70 \) (1972106707796190539632v^17 + 646296427317819391507v^16 + 14608993794857171640927v^15 + 70630860460756355462792v^14 + 67977275504098680427269v^13 + 787549245594823332446839v^12 - 1065594228267594378570262v^11 + 6535196672273923767763964v^10 - 7314361826772622281607080v^9 + 19428631770087858053737209v^8 - 17668682586436318396300095v^7 + 37273820781117138990395870v^6 - 27031594007422639477533874v^5 + 32940105845701404408673578v^4 - 10358521757576683369977292v^3 + 11177271259594608218339642v^2 - 3053869242921870348141516*v + 2540228724080437888043311) / 370795566039370306904770
\(\beta_{10}\)	\(=\)	\( ( - 27\!\cdots\!44 \nu^{17} + \cdots - 51\!\cdots\!77 ) / 40\!\cdots\!70 \) (-27622570560546571509344v^17 - 45319511682205423435529v^16 - 54869107870670027204469v^15 - 1849227475000633974002444v^14 + 1253329940432800659161387v^13 - 20289411663626711616053713v^12 + 36938575250809509419271194v^11 - 150783446291688691518319508v^10 + 199196475881432557368859410v^9 - 446589872067154385312156963v^8 + 467411472317104765460681115v^7 - 842088135020241268464825820v^6 + 687435646828946607577302888v^5 - 763326698583266223094550256v^4 + 310589780986326106963671474v^3 - 262141971037551650740297124v^2 + 74572018705902866115283302*v - 51663813636032526974250877) / 4078751226433073375952470
\(\beta_{11}\)	\(=\)	\( ( - 58\!\cdots\!73 \nu^{17} + \cdots + 27\!\cdots\!36 ) / 77\!\cdots\!70 \) (-58068945589908680473573v^17 + 295977268906634048472617v^16 - 1522024126916467486747968v^15 + 4553534709211293401296927v^14 - 16412614467286694855560421v^13 + 45550046693936614893636314v^12 - 110251923114307297969437882v^11 + 207236240501182099931730944v^10 - 321470454830978287479635675v^9 + 478407951241102319474111389v^8 - 584239730875203568262497810v^7 + 709951254088596610561375820v^6 - 518378896091010064954164104v^5 + 430736903112305844489209438v^4 - 143871374350096329576312142v^3 + 171267552995993129798366052v^2 - 17090066096111894064634771*v + 27890874812767028887765836) / 7786706886826776445000170
\(\beta_{12}\)	\(=\)	\( ( - 68\!\cdots\!12 \nu^{17} + \cdots - 11\!\cdots\!51 ) / 81\!\cdots\!94 \) (-6880022405497196588112v^17 - 828035755066547820007v^16 - 53321141454116422392129v^15 - 225021026579411701792696v^14 - 254952189580423516690329v^13 - 2531463856320105018919753v^12 + 3518484950648902478812670v^11 - 21906820457878676638523836v^10 + 25164935484208258669007562v^9 - 65332495179557461793660341v^8 + 59486752195992901765871035v^7 - 125647587976092066097856524v^6 + 91167867825542570455609890v^5 - 111063282647937597113419418v^4 + 28613525433197437099264044v^3 - 37695765406057135012145066v^2 + 10302608408675428387262532*v - 11432375237386254833618351) / 815750245286614675190494
\(\beta_{13}\)	\(=\)	\( ( 38\!\cdots\!68 \nu^{17} + \cdots - 54\!\cdots\!11 ) / 40\!\cdots\!70 \) (38464327690197992233568v^17 - 47719271646193873989507v^16 + 546356808679758718386493v^15 - 94598095685383722038392v^14 + 5226176975455428037844331v^13 - 533869794541566586232449v^12 + 18074806858936204994812262v^11 + 24077778044530247474383436v^10 + 29472804734868725728668020v^9 + 73903595624729446511712391v^8 + 43063882659263886740145095v^7 + 167330821566401465734628980v^6 + 16740704612782488283983874v^5 + 113243610760940364423385622v^4 + 74125469922776885904705892v^3 + 32774657246218626592072158v^2 - 3647832301872245690208684*v - 5408331602787815043021811) / 4078751226433073375952470
\(\beta_{14}\)	\(=\)	\( ( - 85\!\cdots\!31 \nu^{17} + \cdots + 88\!\cdots\!12 ) / 85\!\cdots\!70 \) (-853200470796681402282631v^17 + 5112581917057836613131569v^16 - 24191402264332952998668096v^15 + 80753106288984068440491349v^14 - 260703541997839112881971797v^13 + 810306955507781537862523838v^12 - 1807545004528587108068931534v^11 + 3766973851423442295040490708v^10 - 5212237472677083674740864025v^9 + 8844729779341457740747716463v^8 - 9321797322727245376146249820v^7 + 13638017404082309944647184340v^6 - 7794485167905590492211698888v^5 + 9140793858590157607757125946v^4 - 1535642827460068055625880474v^3 + 4522546121822366296104048684v^2 - 330187156344821857541870467*v + 885544067277092208962851812) / 85653775755094540895001870
\(\beta_{15}\)	\(=\)	\( ( 61\!\cdots\!82 \nu^{17} + \cdots - 33\!\cdots\!64 ) / 61\!\cdots\!05 \) (61103803116721773530182v^17 - 334467647236142053289603v^16 + 1675690302102244380311382v^15 - 5237977745840894181941203v^14 + 18139835836550408234257019v^13 - 52316931610359970967907176v^12 + 124355151294538438721834058v^11 - 241010907921662896202131061v^10 + 363040904534792510580373010v^9 - 550384112492047158799480711v^8 + 655373427550153255904018350v^7 - 818810077876761475133025185v^6 + 583665980058641253305534861v^5 - 472356896802152901929926922v^4 + 158825072687493090505856773v^3 - 203562837519947224009888203v^2 + 59339496643050164985072874*v - 33914575080637570937761164) / 6118126839649610063928705
\(\beta_{16}\)	\(=\)	\( ( - 27\!\cdots\!46 \nu^{17} + \cdots - 16\!\cdots\!53 ) / 20\!\cdots\!35 \) (-27112025457055013284946v^17 + 19412843239538261286329v^16 - 316117164013651135125686v^15 - 305551536681166919283026v^14 - 2624853084545112662822057v^13 - 3667599237353663966724422v^12 - 2231352679371820326026664v^11 - 44252341692903667601917217v^10 + 26596716764359814605076895v^9 - 132929381208444750869900727v^8 + 74175518805975953201810660v^7 - 267171250014182451768998770v^6 + 134656155816479175876565972v^5 - 220805985634736222849579734v^4 + 15704636130570202751380881v^3 - 72541317251617332887683501v^2 + 17571484504984304791838823*v - 16627524837408725657017153) / 2039375613216536687976235
\(\beta_{17}\)	\(=\)	\( ( 18\!\cdots\!17 \nu^{17} + \cdots - 53\!\cdots\!84 ) / 11\!\cdots\!10 \) (18658018221063834078317v^17 - 85675416180508022280943v^16 + 462475243370006272398612v^15 - 1287521317988838157152893v^14 + 4974381597658228193462149v^13 - 12883342214507701590236146v^12 + 32535127214087944080165558v^11 - 57585978583491122770099696v^10 + 95084646581380100414729395v^9 - 133276487102885346982313891v^8 + 173617133034043663767083480v^7 - 194510284704348049548340690v^6 + 157066574332528127585847766v^5 - 118163555603860300323810712v^4 + 47744064868065167602743308v^3 - 38811425817065760062730798v^2 + 1639987278188805714686879*v - 5331727741067987069277684) / 1112386698118110920714310

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( -\beta_{16} + \beta_{15} - \beta_{13} + \beta_{11} - \beta_{9} + \beta_{8} + \beta_{6} + 3\beta_{5} - \beta_{4} \) -b16 + b15 - b13 + b11 - b9 + b8 + b6 + 3*b5 - b4
\(\nu^{3}\)	\(=\)	\( -\beta_{16} - \beta_{13} + \beta_{12} - \beta_{10} - \beta_{9} - 7\beta_{4} - \beta_{3} - 7\beta_1 \) -b16 - b13 + b12 - b10 - b9 - 7b4 - b3 - 7b1
\(\nu^{4}\)	\(=\)	\( - \beta_{17} - 11 \beta_{15} - \beta_{14} - 12 \beta_{11} - 10 \beta_{8} + \beta_{7} - 19 \beta_{5} + \cdots - 19 \) -b17 - 11b15 - b14 - 12b11 - 10b8 + b7 - 19b5 - 10b2 - 12b1 - 19
\(\nu^{5}\)	\(=\)	\( - 12 \beta_{17} + 14 \beta_{16} - 14 \beta_{15} - 11 \beta_{14} + 12 \beta_{13} - 11 \beta_{12} + \cdots + 11 \beta_{3} \) -12b17 + 14b16 - 14b15 - 11b14 + 12b13 - 11b12 - 17b11 + 12b10 + 17b9 - 12b8 - 11b7 - b6 - 2b5 + 65b4 + 11b3
\(\nu^{6}\)	\(=\)	\( 111 \beta_{16} + 98 \beta_{13} - 14 \beta_{12} + 17 \beta_{10} + 129 \beta_{9} - 95 \beta_{6} + \cdots + 159 \) 111b16 + 98b13 - 14b12 + 17b10 + 129b9 - 95b6 + 136b4 - 9b3 + 95b2 + 136b1 + 159
\(\nu^{7}\)	\(=\)	\( 129 \beta_{17} + 171 \beta_{15} + 111 \beta_{14} + 222 \beta_{11} + 142 \beta_{8} + 107 \beta_{7} + \cdots + 52 \) 129b17 + 171b15 + 111b14 + 222b11 + 142b8 + 107b7 + 52b5 + 33b2 + 657*b1 + 52
\(\nu^{8}\)	\(=\)	\( 222 \beta_{17} - 1124 \beta_{16} + 1124 \beta_{15} + 171 \beta_{14} - 980 \beta_{13} + \cdots + 56 \beta_{3} \) 222b17 - 1124b16 + 1124b15 + 171b14 - 980b13 + 171b12 + 1351b11 - 222b10 - 1351b9 + 980b8 - 56b7 + 918b6 + 1476b5 - 1527b4 + 56*b3
\(\nu^{9}\)	\(=\)	\( - 2019 \beta_{16} - 1672 \beta_{13} + 1124 \beta_{12} - 1351 \beta_{10} - 2663 \beta_{9} + 601 \beta_{6} + \cdots - 910 \) -2019b16 - 1672b13 + 1124b12 - 1351b10 - 2663b9 + 601b6 - 6819b4 - 1025b3 - 601b2 - 6819b1 - 910
\(\nu^{10}\)	\(=\)	\( - 2663 \beta_{17} - 11510 \beta_{15} - 2019 \beta_{14} - 14079 \beta_{11} - 9969 \beta_{8} + \cdots - 14303 \) -2663b17 - 11510b15 - 2019b14 - 14079b11 - 9969b8 + 197b7 - 14303b5 - 9028b2 - 17033*b1 - 14303
\(\nu^{11}\)	\(=\)	\( - 14079 \beta_{17} + 23356 \beta_{16} - 23356 \beta_{15} - 11510 \beta_{14} + 19412 \beta_{13} + \cdots + 9869 \beta_{3} \) -14079b17 + 23356b16 - 23356b15 - 11510b14 + 19412b13 - 11510b12 - 30773b11 + 14079b10 + 30773b9 - 19412b8 - 9869b7 - 8783b6 - 13269b5 + 71545b4 + 9869*b3
\(\nu^{12}\)	\(=\)	\( 119007 \beta_{16} + 102649 \beta_{13} - 23356 \beta_{12} + 30773 \beta_{10} + 147016 \beta_{9} + \cdots + 141737 \) 119007b16 + 102649b13 - 23356b12 + 30773b10 + 147016b9 - 90043b6 + 188816b4 + 1483b3 + 90043b2 + 188816b1 + 141737
\(\nu^{13}\)	\(=\)	\( 147016 \beta_{17} + 265975 \beta_{15} + 119007 \beta_{14} + 348532 \beta_{11} + 222027 \beta_{8} + \cdots + 174956 \) 147016b17 + 265975b15 + 119007b14 + 348532b11 + 222027b8 + 95977b7 + 174956b5 + 115253b2 + 755256*b1 + 174956
\(\nu^{14}\)	\(=\)	\( 348532 \beta_{17} - 1239969 \beta_{16} + 1239969 \beta_{15} + 265975 \beta_{14} - 1066414 \beta_{13} + \cdots - 47285 \beta_{3} \) 348532b17 - 1239969b16 + 1239969b15 + 265975b14 - 1066414b13 + 265975b12 + 1541499b11 - 348532b10 - 1541499b9 + 1066414b8 + 47285b7 + 908649b6 + 1425442b5 - 2081696b4 - 47285*b3
\(\nu^{15}\)	\(=\)	\( - 2992251 \beta_{16} - 2507303 \beta_{13} + 1239969 \beta_{12} - 1541499 \beta_{10} - 3898425 \beta_{9} + \cdots - 2170884 \) -2992251b16 - 2507303b13 + 1239969b12 - 1541499b10 - 3898425b9 + 1422829b6 - 8006949b4 - 943921b3 - 1422829b2 - 8006949b1 - 2170884
\(\nu^{16}\)	\(=\)	\( - 3898425 \beta_{17} - 13000230 \beta_{15} - 2992251 \beta_{14} - 16234381 \beta_{11} - 11155617 \beta_{8} + \cdots - 14499638 \) -3898425b17 - 13000230b15 - 2992251b14 - 16234381b11 - 11155617b8 - 780438b7 - 14499638b5 - 9262782b2 - 22847710*b1 - 14499638
\(\nu^{17}\)	\(=\)	\( - 16234381 \beta_{17} + 33350098 \beta_{16} - 33350098 \beta_{15} - 13000230 \beta_{14} + \cdots + 9388518 \beta_{3} \) -16234381b17 + 33350098b16 - 33350098b15 - 13000230b14 + 28029870b13 - 13000230b12 - 43233461b11 + 16234381b10 + 43233461b9 - 28029870b8 - 9388518b7 - 16909494b6 - 25905412b5 + 85167226b4 + 9388518*b3

\(n\)	\(297\)	\(741\)	\(1001\)	\(1111\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1 - \beta_{5}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 121.9
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{18} \) T^18
$3$	\( T^{18} - 2 T^{17} + \cdots + 441 \) T^18 - 2T^17 + 18T^16 - 22T^15 + 194T^14 - 224T^13 + 1053T^12 - 770T^11 + 3419T^10 - 2128T^9 + 7324T^8 - 2915T^7 + 9758T^6 - 3440T^5 + 7942T^4 - 786T^3 + 2419T^2 - 483*T + 441
$5$	\( (T^{2} - T + 1)^{9} \) (T^2 - T + 1)^9
$7$	\( T^{18} + 28 T^{16} + \cdots + 35344 \) T^18 + 28T^16 + 48T^15 + 559T^14 + 1000T^13 + 5984T^12 + 12037T^11 + 46203T^10 + 78084T^9 + 196454T^8 + 272054T^7 + 538910T^6 + 584790T^5 + 718573T^4 + 348310T^3 + 212664T^2 - 36472T + 35344
$11$	\( (T^{9} + 3 T^{8} + \cdots - 7164)^{2} \) (T^9 + 3T^8 - 49T^7 - 110T^6 + 788T^5 + 1117T^4 - 5240T^3 - 2225T^2 + 13222T - 7164)^2
$13$	\( T^{18} - 2 T^{17} + \cdots + 790321 \) T^18 - 2T^17 + 67T^16 - 110T^15 + 3016T^14 - 5049T^13 + 70021T^12 - 129872T^11 + 1189502T^10 - 2076747T^9 + 11914103T^8 - 17421889T^7 + 80280765T^6 - 89615619T^5 + 228227723T^4 + 28670404T^3 + 20953977T^2 - 2523871*T + 790321
$17$	\( T^{18} - T^{17} + \cdots + 26584336 \) T^18 - T^17 + 88T^16 - 199T^15 + 5779T^14 - 11640T^13 + 165215T^12 - 238774T^11 + 3254150T^10 - 3737676T^9 + 24450270T^8 + 2852226T^7 + 90697721T^6 + 51439058T^5 + 281849449T^4 + 263415690T^3 + 357027768T^2 + 102862200T + 26584336
$19$	\( T^{18} + \cdots + 44522688016 \) T^18 + 124T^16 - 106T^15 + 10402T^14 - 10386T^13 + 480073T^12 - 608757T^11 + 16100070T^10 - 20267824T^9 + 309752819T^8 - 428665272T^7 + 4163833518T^6 - 4703877092T^5 + 18059638817T^4 - 15253147076T^3 + 52757860988T^2 - 40724616016T + 44522688016
$23$	\( (T^{9} - 6 T^{8} + \cdots + 6572)^{2} \) (T^9 - 6T^8 - 89T^7 + 420T^6 + 2484T^5 - 7657T^4 - 24571T^3 + 33225T^2 + 35178T + 6572)^2
$29$	\( (T^{9} + 10 T^{8} + \cdots + 3548)^{2} \) (T^9 + 10T^8 - 8T^7 - 350T^6 - 698T^5 + 2844T^4 + 8243T^3 - 5969T^2 - 21472T + 3548)^2
$31$	\( (T^{9} - 14 T^{8} + \cdots - 4123)^{2} \) (T^9 - 14T^8 - 30T^7 + 818T^6 + 336T^5 - 13652T^4 - 13034T^3 + 14767T^2 + 10359T - 4123)^2
$37$	\( T^{18} + \cdots + 129961739795077 \) T^18 + 90T^16 - 470T^15 + 1449T^14 - 41670T^13 + 30338T^12 - 670437T^11 + 8329023T^10 + 16966273T^9 + 308173851T^8 - 917828253T^7 + 1536710714T^6 - 78096288870T^5 + 100479393693T^4 - 1205891412230T^3 + 8543868941970*T^2 + 129961739795077
$41$	\( T^{18} + 5 T^{17} + \cdots + 23843689 \) T^18 + 5T^17 + 192T^16 + 359T^15 + 23229T^14 + 38029T^13 + 1308964T^12 - 320528T^11 + 45197524T^10 - 22666670T^9 + 967136620T^8 - 1120481528T^7 + 12808153293T^6 - 6379987935T^5 + 67291962879T^4 - 32587293713T^3 + 252454285680T^2 + 2451856843*T + 23843689
$43$	\( (T^{9} - T^{8} - 214 T^{7} + \cdots + 25)^{2} \) (T^9 - T^8 - 214T^7 + 192T^6 + 12697T^5 - 9465T^4 - 228054T^3 + 144583T^2 + 5160*T + 25)^2
$47$	\( (T^{9} + 18 T^{8} + \cdots - 107144572)^{2} \) (T^9 + 18T^8 - 223T^7 - 5569T^6 + 283T^5 + 490161T^4 + 1847571T^3 - 11529243T^2 - 77088590T - 107144572)^2
$53$	\( T^{18} + \cdots + 7650341105625 \) T^18 + 10T^17 + 369T^16 + 1928T^15 + 76648T^14 + 340151T^13 + 8368942T^12 + 1727177T^11 + 411602712T^10 - 453833160T^9 + 15268608045T^8 - 34980080529T^7 + 281223309319T^6 - 514599120597T^5 + 3011659264646T^4 - 5831372123350T^3 + 16712584180450T^2 - 11915051715000*T + 7650341105625
$59$	\( T^{18} + \cdots + 17956541850256 \) T^18 + 10T^17 + 304T^16 + 1878T^15 + 50258T^14 + 276894T^13 + 4720061T^12 + 14368439T^11 + 237860776T^10 + 597467856T^9 + 8050337137T^8 + 11480710402T^7 + 155577268834T^6 + 167386850200T^5 + 2005759145137T^4 + 324346155854T^3 + 7125683148864T^2 + 3745396316856*T + 17956541850256
$61$	\( T^{18} + \cdots + 111805161849616 \) T^18 - 16T^17 + 414T^16 - 3178T^15 + 62284T^14 - 329976T^13 + 6330369T^12 - 13575009T^11 + 336663738T^10 + 493588334T^9 + 14729419001T^8 + 46340086040T^7 + 475159674870T^6 + 2155664346324T^5 + 11174523000393T^4 + 32654085617640T^3 + 81255334314852T^2 + 108942158549472*T + 111805161849616
$67$	\( T^{18} + \cdots + 44472039837696 \) T^18 + 4T^17 + 391T^16 - 888T^15 + 97080T^14 - 335980T^13 + 14359273T^12 - 83064743T^11 + 1482461658T^10 - 7220437123T^9 + 76753719079T^8 - 387098549916T^7 + 2734319029663T^6 - 9163115612591T^5 + 30557404291033T^4 - 31824846627712T^3 + 43622397509440T^2 + 14275576430592*T + 44472039837696
$71$	\( T^{18} + \cdots + 1911870351616 \) T^18 - 17T^17 + 339T^16 - 4342T^15 + 63225T^14 - 682664T^13 + 6531114T^12 - 46180186T^11 + 276750881T^10 - 1273231372T^9 + 5246422541T^8 - 17440578264T^7 + 57216889839T^6 - 144702999437T^5 + 359986870685T^4 - 499761018852T^3 + 833403576000T^2 - 222775737664*T + 1911870351616
$73$	\( (T^{9} - 28 T^{8} + \cdots - 1989644)^{2} \) (T^9 - 28T^8 + 152T^7 + 1686T^6 - 14739T^5 - 22418T^4 + 307580T^3 + 181447T^2 - 2018062T - 1989644)^2
$79$	\( T^{18} + \cdots + 362504540360704 \) T^18 + 23T^17 + 670T^16 + 10183T^15 + 205024T^14 + 2754321T^13 + 39879945T^12 + 422926123T^11 + 4681110596T^10 + 41437926723T^9 + 356553072525T^8 + 2301434798952T^7 + 12682948813400T^6 + 45960562726514T^5 + 130053280908985T^4 + 165800132200544T^3 + 214519223749184T^2 - 16327710529536*T + 362504540360704
$83$	\( T^{18} + \cdots + 6249851136 \) T^18 - 29T^17 + 564T^16 - 6913T^15 + 66259T^14 - 462053T^13 + 2732425T^12 - 12561315T^11 + 57503976T^10 - 211184053T^9 + 840505909T^8 - 2159579790T^7 + 6916505697T^6 - 14620434822T^5 + 37518690105T^4 - 45689431536T^3 + 43880166288T^2 - 19227684096*T + 6249851136
$89$	\( T^{18} + \cdots + 149023837531024 \) T^18 + 16T^17 + 586T^16 + 5128T^15 + 165190T^14 + 1234430T^13 + 27848590T^12 + 87791497T^11 + 2150074186T^10 + 1888533482T^9 + 130195895226T^8 - 114229331582T^7 + 3018830458915T^6 - 2842659176951T^5 + 49707564230077T^4 - 57281834115372T^3 + 242763325201092T^2 + 156522973268752*T + 149023837531024
$97$	\( (T^{9} - 13 T^{8} + \cdots - 4302148)^{2} \) (T^9 - 13T^8 - 459T^7 + 4494T^6 + 70948T^5 - 392235T^4 - 4038204T^3 + 8486767T^2 + 68621228T - 4302148)^2
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