# Properties

 Label 1008.3.cg.q Level $1008$ Weight $3$ Character orbit 1008.cg Analytic conductor $27.466$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1008.cg (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$27.4660106475$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 2 x^{15} + 81 x^{14} - 118 x^{13} + 1960 x^{12} - 366 x^{11} + 37625 x^{10} - 83714 x^{9} + 623931 x^{8} + 289492 x^{7} + 241286 x^{6} - 4777608 x^{5} + \cdots + 1148023744$$ x^16 - 2*x^15 + 81*x^14 - 118*x^13 + 1960*x^12 - 366*x^11 + 37625*x^10 - 83714*x^9 + 623931*x^8 + 289492*x^7 + 241286*x^6 - 4777608*x^5 + 98362220*x^4 - 239083824*x^3 + 523977616*x^2 - 403737152*x + 1148023744 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{18}$$ Twist minimal: no (minimal twist has level 504) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{7} q^{5} + ( - \beta_{5} + \beta_{2} - 1) q^{7}+O(q^{10})$$ q - b7 * q^5 + (-b5 + b2 - 1) * q^7 $$q - \beta_{7} q^{5} + ( - \beta_{5} + \beta_{2} - 1) q^{7} + (\beta_{13} + \beta_{4}) q^{11} + (\beta_{6} + \beta_{5}) q^{13} + (\beta_{14} - \beta_{7} + \beta_{3}) q^{17} + (\beta_{12} - \beta_{9} - \beta_{5} - \beta_{2} + \beta_1 + 1) q^{19} + ( - 2 \beta_{14} - \beta_{10} - \beta_{8} - \beta_{7} + 2 \beta_{3}) q^{23} + ( - \beta_{15} + 2 \beta_{12} - 2 \beta_{9} - \beta_{6} - 4 \beta_{5} + 2 \beta_1) q^{25} + (\beta_{14} - \beta_{10} - 2 \beta_{7} - 2 \beta_{4}) q^{29} + ( - 2 \beta_{15} - \beta_{12} - \beta_{9} + 2 \beta_{6} + 3 \beta_{5} - \beta_{2} + 2 \beta_1 + 6) q^{31} + ( - \beta_{14} + 2 \beta_{13} + \beta_{11} - \beta_{8} + 5 \beta_{7} + \beta_{4} - 4 \beta_{3}) q^{35} + (\beta_{15} - 4 \beta_{12} - 2 \beta_{9} - 2 \beta_{6} - 5 \beta_{5} + 2 \beta_{2} + 4 \beta_1 - 4) q^{37} + (2 \beta_{14} - 4 \beta_{13} + 2 \beta_{10} - 2 \beta_{7} - 2 \beta_{4}) q^{41} + (4 \beta_{15} - \beta_{12} - 5 \beta_{9} - 2 \beta_{6} + 2 \beta_{5} - 4 \beta_{2} + 4 \beta_1 + 1) q^{43} + (2 \beta_{13} - 2 \beta_{11} + \beta_{10} + \beta_{8} + 9 \beta_{7} + 4 \beta_{4} + 2 \beta_{3}) q^{47} + (2 \beta_{15} - 6 \beta_{9} + \beta_{6} - 2 \beta_{2} + 4 \beta_1 - 9) q^{49} + ( - 2 \beta_{13} - 5 \beta_{7} - 2 \beta_{4} - 5 \beta_{3}) q^{53} + ( - 3 \beta_{12} - 4 \beta_{9} - 4 \beta_{6} - 28 \beta_{5} + 7 \beta_{2} + \cdots - 12) q^{55}+ \cdots + (8 \beta_{12} - 12 \beta_{9} + \beta_{6} - 45 \beta_{5} + 4 \beta_{2} + 4 \beta_1 - 23) q^{97}+O(q^{100})$$ q - b7 * q^5 + (-b5 + b2 - 1) * q^7 + (b13 + b4) * q^11 + (b6 + b5) * q^13 + (b14 - b7 + b3) * q^17 + (b12 - b9 - b5 - b2 + b1 + 1) * q^19 + (-2*b14 - b10 - b8 - b7 + 2*b3) * q^23 + (-b15 + 2*b12 - 2*b9 - b6 - 4*b5 + 2*b1) * q^25 + (b14 - b10 - 2*b7 - 2*b4) * q^29 + (-2*b15 - b12 - b9 + 2*b6 + 3*b5 - b2 + 2*b1 + 6) * q^31 + (-b14 + 2*b13 + b11 - b8 + 5*b7 + b4 - 4*b3) * q^35 + (b15 - 4*b12 - 2*b9 - 2*b6 - 5*b5 + 2*b2 + 4*b1 - 4) * q^37 + (2*b14 - 4*b13 + 2*b10 - 2*b7 - 2*b4) * q^41 + (4*b15 - b12 - 5*b9 - 2*b6 + 2*b5 - 4*b2 + 4*b1 + 1) * q^43 + (2*b13 - 2*b11 + b10 + b8 + 9*b7 + 4*b4 + 2*b3) * q^47 + (2*b15 - 6*b9 + b6 - 2*b2 + 4*b1 - 9) * q^49 + (-2*b13 - 5*b7 - 2*b4 - 5*b3) * q^53 + (-3*b12 - 4*b9 - 4*b6 - 28*b5 + 7*b2 + 7*b1 - 12) * q^55 + (-3*b14 - b13 + b11 + b8 + b4 - b3) * q^59 + (4*b12 - 2*b9 - 8*b5 - 2*b2 + 4*b1 + 8) * q^61 + (2*b14 - 2*b13 + b10 + 4*b8 - 7*b7 + 12*b3) * q^65 + (6*b12 - 6*b9 + 9*b5 - b2 + 5*b1) * q^67 + (-b14 + b11 + b10 + 3*b7 - 2*b4 - 3*b3) * q^71 + (-b15 - 8*b12 - 8*b9 + b6 + 10*b5 + 8*b1 + 20) * q^73 + (2*b14 + 2*b13 - 4*b11 + b10 + 4*b8 + 14*b7 - 2*b4 - 14*b3) * q^77 + (-2*b15 - 3*b12 - 4*b9 + 4*b6 - b5 + 4*b2 + 3*b1 - 3) * q^79 + (-b14 - 2*b13 - b11 - b10 + 2*b8 - b4 + 2*b3) * q^83 + (-4*b15 + 6*b12 + 2*b6 - 2*b5 - 6*b2 + 6*b1 + 22) * q^85 + (8*b11 - 2*b10 - 4*b8 + 8*b7) * q^89 + (-6*b15 + b12 - 3*b9 + 4*b6 - 15*b5 - b2 + 9*b1 - 15) * q^91 + (-b14 - b11 - 2*b10 + b8 - 4*b7 - 5*b3) * q^95 + (8*b12 - 12*b9 + b6 - 45*b5 + 4*b2 + 4*b1 - 23) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 4 q^{7}+O(q^{10})$$ 16 * q - 4 * q^7 $$16 q - 4 q^{7} + 24 q^{19} + 36 q^{25} + 84 q^{31} - 68 q^{37} - 80 q^{43} - 184 q^{49} + 216 q^{61} - 56 q^{67} + 156 q^{73} - 28 q^{79} + 448 q^{85} - 48 q^{91}+O(q^{100})$$ 16 * q - 4 * q^7 + 24 * q^19 + 36 * q^25 + 84 * q^31 - 68 * q^37 - 80 * q^43 - 184 * q^49 + 216 * q^61 - 56 * q^67 + 156 * q^73 - 28 * q^79 + 448 * q^85 - 48 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + 81 x^{14} - 118 x^{13} + 1960 x^{12} - 366 x^{11} + 37625 x^{10} - 83714 x^{9} + 623931 x^{8} + 289492 x^{7} + 241286 x^{6} - 4777608 x^{5} + \cdots + 1148023744$$ :

 $$\beta_{1}$$ $$=$$ $$( - 10\!\cdots\!45 \nu^{15} + \cdots + 21\!\cdots\!24 ) / 13\!\cdots\!04$$ (-100556972630329033176752601122320345*v^15 - 4971665538039748120628491175494138700*v^14 - 16642763993085215287405929261356605105*v^13 - 379612817413247452788150010668500673959*v^12 - 947770809803666890747372996856536241814*v^11 - 8347644796175462027251017614250139959645*v^10 - 26176000162481695852882851527051846796295*v^9 - 167585762876128579492743036501190629315174*v^8 - 65894585969511436728304778195514526462537*v^7 - 1109244111007345863639451033650164243718061*v^6 - 6696509818916320807237494143428405645618100*v^5 - 5022494149124071886273731384289935896141069*v^4 + 55663013864112709614887007987978892053722332*v^3 - 133704394673193245733415750569563985542673284*v^2 - 164557731511469534215695726805292681132991568*v + 210855179712442882548541814922251942224880224) / 130760489965855300500766614243316754659410404 $$\beta_{2}$$ $$=$$ $$( - 11\!\cdots\!90 \nu^{15} + \cdots + 62\!\cdots\!96 ) / 12\!\cdots\!92$$ (-119882383833468721065411629686839590*v^15 + 3097652047572484347385565611094609667*v^14 - 15322624620009622764721723350570553644*v^13 + 229378477752771312077901753967594462021*v^12 - 570721650283855855238791095866326197038*v^11 + 4440227284030576997770639284161241813734*v^10 - 6291664184755025267810831640340544257636*v^9 + 93544033795071644508980737928220989502251*v^8 - 353698912780406734218582989009940715021012*v^7 + 1259671069065830250893716144667393679431039*v^6 + 1408212649763204800597832129118469360064420*v^5 - 2497909442228955910890832888821257680072736*v^4 - 43162670153636768086078025088625205229911720*v^3 + 318914566177847980449968236147879275994509336*v^2 - 359969823922360061980079766340359757863156384*v + 623170939865871494574806551139351689779424096) / 126876515016374449990842853424208336204180392 $$\beta_{3}$$ $$=$$ $$( - 11\!\cdots\!90 \nu^{15} + \cdots + 62\!\cdots\!96 ) / 12\!\cdots\!92$$ (-119882383833468721065411629686839590*v^15 + 3097652047572484347385565611094609667*v^14 - 15322624620009622764721723350570553644*v^13 + 229378477752771312077901753967594462021*v^12 - 570721650283855855238791095866326197038*v^11 + 4440227284030576997770639284161241813734*v^10 - 6291664184755025267810831640340544257636*v^9 + 93544033795071644508980737928220989502251*v^8 - 353698912780406734218582989009940715021012*v^7 + 1259671069065830250893716144667393679431039*v^6 + 1408212649763204800597832129118469360064420*v^5 - 2497909442228955910890832888821257680072736*v^4 - 43162670153636768086078025088625205229911720*v^3 + 318914566177847980449968236147879275994509336*v^2 - 613722853955108961961765473188776430271517168*v + 623170939865871494574806551139351689779424096) / 126876515016374449990842853424208336204180392 $$\beta_{4}$$ $$=$$ $$( 32\!\cdots\!85 \nu^{15} + \cdots - 16\!\cdots\!08 ) / 12\!\cdots\!92$$ (32605338340722171556917720985835970685*v^15 - 257575040992122144425995670235199984806*v^14 + 2475278656635883006012037850356295383969*v^13 - 17902394705350120423170733843962042541864*v^12 + 54916595316657093422763570829489726494816*v^11 - 243301727331372923200794595730798216727540*v^10 + 1214414804045510311844586258271073897507417*v^9 - 5323754099282633061499393998713106113647154*v^8 + 38064221818995907560123029239967073516145943*v^7 + 29067539916397948701981668371191134127191778*v^6 + 40564645261419326185657957867019205260258750*v^5 - 113807875078823383997484103276200359789005506*v^4 + 5025539682225555974768626994315172611613568644*v^3 - 16552796281150155860175308253990011695598144312*v^2 + 37547460010787285637938572343070879255953627008*v - 160285617196229643643080122194961706863321035008) / 12814528016653819449075128195845041956622219592 $$\beta_{5}$$ $$=$$ $$( 40\!\cdots\!73 \nu^{15} + \cdots - 21\!\cdots\!08 ) / 12\!\cdots\!28$$ (4027600444065813751112873*v^15 - 2649016466835195326319299*v^14 + 312725577939578756939380023*v^13 - 106535452460820671971428003*v^12 + 6891013373792712892661231682*v^11 + 3563818366737747545668742542*v^10 + 131033348188392814435619650135*v^9 - 270518005631787299655190336123*v^8 + 1546240617625821656902814905001*v^7 + 1862418993511600406319618071357*v^6 - 1561488823120985975027571944242*v^5 - 42696221425801268092309229505322*v^4 + 253400526336193028451844752924188*v^3 - 608238767983884858530837912378012*v^2 + 982309256143465760138536525431184*v - 2109620953727521097108159077641408) / 1299840607764383914384021345899728 $$\beta_{6}$$ $$=$$ $$( - 16\!\cdots\!11 \nu^{15} + \cdots - 42\!\cdots\!00 ) / 25\!\cdots\!84$$ (-161988701072090189422664175844161458311*v^15 - 917018643774257859693518091698390570185*v^14 - 9723465604646424657177546774635002180805*v^13 - 69535880721556159534372842776901842836173*v^12 - 77440519619126651172689097959626361143418*v^11 - 1387257105738937743685139837414886120974422*v^10 - 1994619944960720151757454598051150821020765*v^9 - 7171928768578933189861350186678370397863249*v^8 + 101440868423805841532589040692227970440492805*v^7 - 347011451289701575601036079450691304515351529*v^6 + 417906241963548922989669434500632185237782854*v^5 + 6191771233823027694720576064872233897067913366*v^4 + 9173174455446882099064803008301588522007405116*v^3 - 46932175698228744406511607339304986566193820796*v^2 + 270838516937007106057239678153532119102549483152*v - 42655905894276241167833912726121238689256446800) / 25629056033307638898150256391690083913244439184 $$\beta_{7}$$ $$=$$ $$( 12\!\cdots\!21 \nu^{15} + \cdots + 91\!\cdots\!12 ) / 18\!\cdots\!56$$ (12208397947434334377888633382820116021*v^15 - 14412118640861282375397566996106702160*v^14 + 859976670121509656837128526714828356969*v^13 - 632558631671731329110938489966443107714*v^12 + 14202070647773741497469263281892005699400*v^11 + 11873705212803841027868320923314928478980*v^10 + 256559959400255815326408759758010971685173*v^9 - 903920451813152858505624234685840756745268*v^8 + 2664006120517227709220934856999949295185615*v^7 + 13745706596456441404841897004211672417469284*v^6 - 39134633789365857470255700557439502897745046*v^5 - 146295238806934626562170104717131820272961742*v^4 + 1087440914458709832003850964918478097145263236*v^3 - 842191705676519327755772713792846510680114512*v^2 - 3217147578470939402821363754018330791056717616*v + 9165238213767562934439990156769594945547098912) / 1830646859521974207010732599406434565231745656 $$\beta_{8}$$ $$=$$ $$( 38\!\cdots\!09 \nu^{15} + \cdots - 20\!\cdots\!64 ) / 46\!\cdots\!64$$ (382173777066895284751616052978160009*v^15 + 2413426775320279471441937332715044560*v^14 + 27680748103882019359592606064656192829*v^13 + 190312441811758249764519812867243436068*v^12 + 571729543412827173263898905476747607204*v^11 + 4367711488300094829258117045725087899162*v^10 + 14445272679891265714757691588168943061189*v^9 + 46527587621690380007855592346406105154352*v^8 - 68483846612223662686047190582561570596281*v^7 + 869577711588503846387377157035113864480882*v^6 + 916810815118530634202953155928857479304694*v^5 - 10027588649275129217494072956547091777153620*v^4 - 9639502998123387422873522967269098176609124*v^3 + 109003744142760262325150529890756981619831192*v^2 - 208117077576461894525896998608364512869648112*v - 20023042728826187750917655509390801915772864) / 46095424520337480032644346028219575383533164 $$\beta_{9}$$ $$=$$ $$( - 10\!\cdots\!46 \nu^{15} + \cdots - 10\!\cdots\!40 ) / 12\!\cdots\!92$$ (-106411824056666340363940752850425056246*v^15 - 706887335904565042747610716087266294725*v^14 - 8879866379061558448310466634256994628988*v^13 - 57415968439715604607321262036581223999801*v^12 - 253527363864044310288681175223706378745850*v^11 - 1454162027478123096915702287206764764662640*v^10 - 6454174491208485454478206700156734053032928*v^9 - 21174429920990458139598330462605040927696801*v^8 - 36384199867767148342704435054176449355652320*v^7 - 323399373343067510801646317498870128229804367*v^6 - 960460945378497061610807935263169129219144320*v^5 + 401263201105853047607334698703545283842128174*v^4 - 749177910630449269918122020005710038834454128*v^3 - 30606761909823435485283000268618874110153954016*v^2 + 7038670773012701768459786370554403498797742272*v - 10303631696872657655143522725911770442174554240) / 12814528016653819449075128195845041956622219592 $$\beta_{10}$$ $$=$$ $$( - 83\!\cdots\!77 \nu^{15} + \cdots + 12\!\cdots\!16 ) / 64\!\cdots\!96$$ (-83764037978781081056492420292835892177*v^15 + 542759512937705019428649565682114448714*v^14 - 5260285697907763663757580086779095205101*v^13 + 42429661363701114000649089965737158801231*v^12 - 31167890270826108553858264208759096442354*v^11 + 995286117233125596376913939682619983991339*v^10 + 790305156821866023656724290902747788301457*v^9 + 29930345596738093030970181102966528559766072*v^8 + 1587900756790349743655517093587957521097227*v^7 + 187225482563172553152926313721511265177656101*v^6 + 831205643705129903437971193468608794161133028*v^5 + 2727882636304736043725953650013178696867918527*v^4 - 8121225481341340063707511128371781479311835752*v^3 + 31173205185364336684180298854940366153343487148*v^2 - 20960066095405742038379796100099845851921789856*v + 121618201337609848389599716116499051484818025616) / 6407264008326909724537564097922520978311109796 $$\beta_{11}$$ $$=$$ $$( - 25\!\cdots\!43 \nu^{15} + \cdots + 12\!\cdots\!92 ) / 18\!\cdots\!56$$ (-25780989016044576578524279111177470743*v^15 + 84894735650046731469632777268069329004*v^14 - 1980913769641666124463090146171443455027*v^13 + 5490228633457430647275659463698907405978*v^12 - 42980147144985491061572014736399417449240*v^11 + 57385956089526048353959853443621292848508*v^10 - 845347405597570223625333523989884423671919*v^9 + 3162365940233509977358930980207473009248472*v^8 - 16868033152672019036631840607613070755923461*v^7 - 7589986596522774914499895339762227094335248*v^6 + 20365040991482873026887046356999504254489402*v^5 + 234708161723534139083215119519497269356349386*v^4 - 2947584161289706152618769777991780538998089820*v^3 + 6788997641181016585691925411803829070180443936*v^2 - 7553809172292589653090502213801318887630754768*v + 12382657975262530473209332095271563297919565792) / 1830646859521974207010732599406434565231745656 $$\beta_{12}$$ $$=$$ $$( 96\!\cdots\!19 \nu^{15} + \cdots - 13\!\cdots\!76 ) / 64\!\cdots\!96$$ (96026467331219647573425582473263442619*v^15 - 183631835618669873834521242376201107845*v^14 + 6644972275602167675853469583543056385283*v^13 - 12102891972126511820667385803931959014846*v^12 + 99373774267912101414070116852543792366490*v^11 - 160592521734061592873462938934189272749555*v^10 + 1555017001395713319272764977545952761851279*v^9 - 12694011221927095674594178427970628262207831*v^8 + 16190107524783059281592479900025951260729039*v^7 + 23135413479196825058836402963208670867289114*v^6 - 368192632151755886561643813400018328040870218*v^5 - 1919469713957408598251862330800538959154896905*v^4 + 7302820762161942868655758840509014466851216336*v^3 - 14068725109885822602520472306817517191461790168*v^2 - 3212377732760199764005748216638726909659897632*v - 13505386331862466836084144822849764499094718576) / 6407264008326909724537564097922520978311109796 $$\beta_{13}$$ $$=$$ $$( 26\!\cdots\!95 \nu^{15} + \cdots - 11\!\cdots\!92 ) / 12\!\cdots\!92$$ (264841838352519635916978466645964866295*v^15 - 261824036015188831766934294062389515515*v^14 + 21442500687824482314774340901398267911973*v^13 - 12554126486151821324913354622073390733177*v^12 + 515822946943157739669348016213540421958118*v^11 + 174732521037476101684185084336783146857844*v^10 + 9622623946869196462354915968997164976962657*v^9 - 19470532711823085768822495185101317472282091*v^8 + 123277890711049397144499751487702929376419847*v^7 + 10633903262662884932830320114123152851961019*v^6 - 236346215942697945882853814658079948795079138*v^5 - 2769617725170987503021376351417085108672319996*v^4 + 15936557070820784050019384410784385656576887244*v^3 - 56780129650804073326837214061375680341715529552*v^2 + 92908999167460291722388590859726761973464792480*v - 111262370002407036239602502719457972313533839392) / 12814528016653819449075128195845041956622219592 $$\beta_{14}$$ $$=$$ $$( - 27\!\cdots\!00 \nu^{15} + \cdots - 18\!\cdots\!16 ) / 12\!\cdots\!92$$ (-272216252730060430680657253305170552400*v^15 - 1137037760527704083170157950951255699957*v^14 - 23022936158529012947931125391081822153122*v^13 - 98075930279525382423437270372616853223465*v^12 - 659338562234787391290404931506973716114934*v^11 - 2885095771880104672079573258676120166314004*v^10 - 16245458585421800320702895161535969526239714*v^9 - 42255684991304972828822524769468912358722929*v^8 - 158994156951299776241861120322145270389977434*v^7 - 776622597643179949600996046084478541576249723*v^6 - 2209944829517735126931556489233370530731533592*v^5 - 1760208146689851672146370401764117443510639874*v^4 - 12546799703955332986848369444007983147137653024*v^3 - 37899973621857333632651231955682462166541747096*v^2 - 79230723815768542684872896230102845229686712736*v - 181170626688351247286126149251690449630106094816) / 12814528016653819449075128195845041956622219592 $$\beta_{15}$$ $$=$$ $$( - 30\!\cdots\!35 \nu^{15} + \cdots - 18\!\cdots\!96 ) / 12\!\cdots\!92$$ (-304840072705013275753754440353389826135*v^15 - 820118425177494782182805033107321305067*v^14 - 24919678845879232618742807987341295546573*v^13 - 77317469824385909949318929260719746559837*v^12 - 652494144192560406236722182558075705949430*v^11 - 2535631663727677707980935570998922735477588*v^10 - 14976993986183529664056342977532024432647165*v^9 - 27284677786224150651391050793369424224661599*v^8 - 142357540658302033692667244156115696133530015*v^7 - 680323353251134485172428148423507005624289661*v^6 - 1129991073691319377527499700594556804135110554*v^5 + 102317824755046920263112685022588189044954568*v^4 - 17749601477404559149757847202591453818636252548*v^3 - 10661433694317295636984604732507094152624376188*v^2 + 28012423836115247125812966760756814902792366416*v - 187682962067908185956793621422283192686344422296) / 12814528016653819449075128195845041956622219592
 $$\nu$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{15} + \beta_{13} + \beta_{10} - 3\beta_{9} - \beta_{8} + 4\beta_{7} - \beta_{4} + \beta _1 - 18 ) / 2$$ (2*b15 + b13 + b10 - 3*b9 - b8 + 4*b7 - b4 + b1 - 18) / 2 $$\nu^{3}$$ $$=$$ $$( - \beta_{15} + 2 \beta_{14} - 19 \beta_{13} + 4 \beta_{12} + \beta_{11} + 5 \beta_{10} + 15 \beta_{9} + 25 \beta_{8} - 6 \beta_{7} + 22 \beta_{6} + 176 \beta_{5} - 8 \beta_{4} + 41 \beta_{3} - 45 \beta_{2} + 2 \beta _1 + 47 ) / 4$$ (-b15 + 2*b14 - 19*b13 + 4*b12 + b11 + 5*b10 + 15*b9 + 25*b8 - 6*b7 + 22*b6 + 176*b5 - 8*b4 + 41*b3 - 45*b2 + 2*b1 + 47) / 4 $$\nu^{4}$$ $$=$$ $$( - 109 \beta_{15} - 4 \beta_{14} + 16 \beta_{13} + 59 \beta_{12} + 32 \beta_{11} - 50 \beta_{10} + 105 \beta_{9} - 305 \beta_{7} + 27 \beta_{6} - 317 \beta_{5} + 41 \beta_{4} + 135 \beta_{3} - 9 \beta_{2} + 29 \beta _1 + 332 ) / 2$$ (-109*b15 - 4*b14 + 16*b13 + 59*b12 + 32*b11 - 50*b10 + 105*b9 - 305*b7 + 27*b6 - 317*b5 + 41*b4 + 135*b3 - 9*b2 + 29*b1 + 332) / 2 $$\nu^{5}$$ $$=$$ $$( 279 \beta_{15} - 468 \beta_{14} + 871 \beta_{13} - 442 \beta_{12} + 198 \beta_{11} - 563 \beta_{10} - 1557 \beta_{9} - 941 \beta_{8} + 1315 \beta_{7} - 1338 \beta_{6} - 14216 \beta_{5} + 428 \beta_{4} - 3351 \beta_{3} + \cdots - 6081 ) / 4$$ (279*b15 - 468*b14 + 871*b13 - 442*b12 + 198*b11 - 563*b10 - 1557*b9 - 941*b8 + 1315*b7 - 1338*b6 - 14216*b5 + 428*b4 - 3351*b3 + 1363*b2 + 684*b1 - 6081) / 4 $$\nu^{6}$$ $$=$$ $$( 5209 \beta_{15} - 24 \beta_{14} - 1485 \beta_{13} - 5865 \beta_{12} - 3017 \beta_{11} + 1797 \beta_{10} - 2202 \beta_{9} + 707 \beta_{8} + 16955 \beta_{7} - 1410 \beta_{6} + 25140 \beta_{5} - 2985 \beta_{4} + \cdots - 22583 ) / 2$$ (5209*b15 - 24*b14 - 1485*b13 - 5865*b12 - 3017*b11 + 1797*b10 - 2202*b9 + 707*b8 + 16955*b7 - 1410*b6 + 25140*b5 - 2985*b4 - 6292*b3 + 2237*b2 - 3558*b1 - 22583) / 2 $$\nu^{7}$$ $$=$$ $$( - 24675 \beta_{15} + 32956 \beta_{14} - 45339 \beta_{13} + 43890 \beta_{12} - 8589 \beta_{11} + 27685 \beta_{10} + 79415 \beta_{9} + 39298 \beta_{8} - 94080 \beta_{7} + 82082 \beta_{6} + \cdots + 456701 ) / 4$$ (-24675*b15 + 32956*b14 - 45339*b13 + 43890*b12 - 8589*b11 + 27685*b10 + 79415*b9 + 39298*b8 - 94080*b7 + 82082*b6 + 719768*b5 - 14350*b4 + 217355*b3 - 85069*b2 - 57806*b1 + 456701) / 4 $$\nu^{8}$$ $$=$$ $$( - 281111 \beta_{15} + 7840 \beta_{14} + 70358 \beta_{13} + 314552 \beta_{12} + 189210 \beta_{11} - 87940 \beta_{10} + 59028 \beta_{9} - 58808 \beta_{8} - 929365 \beta_{7} + 47320 \beta_{6} + \cdots + 1501223 ) / 2$$ (-281111*b15 + 7840*b14 + 70358*b13 + 314552*b12 + 189210*b11 - 87940*b10 + 59028*b9 - 58808*b8 - 929365*b7 + 47320*b6 - 1230306*b5 + 181481*b4 + 229957*b3 - 76153*b2 + 264867*b1 + 1501223) / 2 $$\nu^{9}$$ $$=$$ $$( 1823989 \beta_{15} - 1925334 \beta_{14} + 2500899 \beta_{13} - 2474216 \beta_{12} + 189113 \beta_{11} - 1228677 \beta_{10} - 4490331 \beta_{9} - 2225317 \beta_{8} + 6002452 \beta_{7} + \cdots - 31098363 ) / 4$$ (1823989*b15 - 1925334*b14 + 2500899*b13 - 2474216*b12 + 189113*b11 - 1228677*b10 - 4490331*b9 - 2225317*b8 + 6002452*b7 - 4727246*b6 - 38669728*b5 + 303120*b4 - 12550189*b3 + 4942921*b2 + 3121022*b1 - 31098363) / 4 $$\nu^{10}$$ $$=$$ $$( 15761336 \beta_{15} - 406340 \beta_{14} - 4194951 \beta_{13} - 16548277 \beta_{12} - 10315623 \beta_{11} + 5308169 \beta_{10} - 1948744 \beta_{9} + 4285624 \beta_{8} + \cdots - 80877241 ) / 2$$ (15761336*b15 - 406340*b14 - 4194951*b13 - 16548277*b12 - 10315623*b11 + 5308169*b10 - 1948744*b9 + 4285624*b8 + 51539477*b7 - 1190869*b6 + 73768133*b5 - 10002357*b4 - 8223094*b3 + 904804*b2 - 15796997*b1 - 80877241) / 2 $$\nu^{11}$$ $$=$$ $$( - 127051415 \beta_{15} + 107202502 \beta_{14} - 138044043 \beta_{13} + 143357780 \beta_{12} - 3359816 \beta_{11} + 55725989 \beta_{10} + 271490321 \beta_{9} + \cdots + 1912446727 ) / 4$$ (-127051415*b15 + 107202502*b14 - 138044043*b13 + 143357780*b12 - 3359816*b11 + 55725989*b10 + 271490321*b9 + 131145571*b8 - 404302711*b7 + 268756368*b6 + 2100648250*b5 - 3042542*b4 + 720984693*b3 - 276347269*b2 - 158111438*b1 + 1912446727) / 4 $$\nu^{12}$$ $$=$$ $$( - 869585982 \beta_{15} - 516652 \beta_{14} + 290868584 \beta_{13} + 894758565 \beta_{12} + 576135224 \beta_{11} - 329410426 \beta_{10} + 28956455 \beta_{9} + \cdots + 3998642990 ) / 2$$ (-869585982*b15 - 516652*b14 + 290868584*b13 + 894758565*b12 + 576135224*b11 - 329410426*b10 + 28956455*b9 - 298604894*b8 - 2803955965*b7 - 13193596*b6 - 4982968400*b5 + 573466513*b4 + 243712463*b3 + 72579490*b2 + 909783964*b1 + 3998642990) / 2 $$\nu^{13}$$ $$=$$ $$( 8475803371 \beta_{15} - 5956688682 \beta_{14} + 7417316179 \beta_{13} - 9265432204 \beta_{12} - 419542935 \beta_{11} - 2598095535 \beta_{10} - 16031100855 \beta_{9} + \cdots - 112654415615 ) / 4$$ (8475803371*b15 - 5956688682*b14 + 7417316179*b13 - 9265432204*b12 - 419542935*b11 - 2598095535*b10 - 16031100855*b9 - 7268635024*b8 + 26972704206*b7 - 15303684172*b6 - 111579242054*b5 - 536255720*b4 - 41658215433*b3 + 15674790735*b2 + 7977907196*b1 - 112654415615) / 4 $$\nu^{14}$$ $$=$$ $$( 46926047268 \beta_{15} + 2026278086 \beta_{14} - 19502362417 \beta_{13} - 47573182678 \beta_{12} - 32663193668 \beta_{11} + 20102185643 \beta_{10} + \cdots - 192474230932 ) / 2$$ (46926047268*b15 + 2026278086*b14 - 19502362417*b13 - 47573182678*b12 - 32663193668*b11 + 20102185643*b10 + 3661721165*b9 + 19560181311*b8 + 150280975646*b7 + 5862040688*b6 + 327477745084*b5 - 33069615321*b4 - 325222898*b3 - 9506939442*b2 - 53876336289*b1 - 192474230932) / 2 $$\nu^{15}$$ $$=$$ $$( - 546882905353 \beta_{15} + 333119813700 \beta_{14} - 393537231435 \beta_{13} + 600336340426 \beta_{12} + 70136917089 \beta_{11} + 120588039003 \beta_{10} + \cdots + 6633855205101 ) / 4$$ (-546882905353*b15 + 333119813700*b14 - 393537231435*b13 + 600336340426*b12 + 70136917089*b11 + 120588039003*b10 + 915374510583*b9 + 388861144781*b8 - 1748493282024*b7 + 866170242316*b6 + 5877370016834*b5 + 75405101106*b4 + 2389070904947*b3 - 892301525643*b2 - 383133526900*b1 + 6633855205101) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$577$$ $$757$$ $$785$$ $$\chi(n)$$ $$1$$ $$1 + \beta_{5}$$ $$1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 −0.338813 − 1.51822i −3.20903 − 3.57433i 1.75486 − 4.85286i 2.29298 + 1.67052i 2.29298 + 3.14719i 1.75486 − 2.33310i −3.20903 + 1.64043i −0.338813 + 7.55243i −0.338813 + 1.51822i −3.20903 + 3.57433i 1.75486 + 4.85286i 2.29298 − 1.67052i 2.29298 − 3.14719i 1.75486 + 2.33310i −3.20903 − 1.64043i −0.338813 − 7.55243i
0 0 0 −7.85541 4.53532i 0 −1.17763 + 6.90023i 0 0 0
145.2 0 0 0 −4.51611 2.60738i 0 −6.91807 1.06788i 0 0 0
145.3 0 0 0 −2.18217 1.25988i 0 3.00972 6.31993i 0 0 0
145.4 0 0 0 −1.27883 0.738332i 0 4.08597 + 5.68374i 0 0 0
145.5 0 0 0 1.27883 + 0.738332i 0 4.08597 + 5.68374i 0 0 0
145.6 0 0 0 2.18217 + 1.25988i 0 3.00972 6.31993i 0 0 0
145.7 0 0 0 4.51611 + 2.60738i 0 −6.91807 1.06788i 0 0 0
145.8 0 0 0 7.85541 + 4.53532i 0 −1.17763 + 6.90023i 0 0 0
577.1 0 0 0 −7.85541 + 4.53532i 0 −1.17763 6.90023i 0 0 0
577.2 0 0 0 −4.51611 + 2.60738i 0 −6.91807 + 1.06788i 0 0 0
577.3 0 0 0 −2.18217 + 1.25988i 0 3.00972 + 6.31993i 0 0 0
577.4 0 0 0 −1.27883 + 0.738332i 0 4.08597 5.68374i 0 0 0
577.5 0 0 0 1.27883 0.738332i 0 4.08597 5.68374i 0 0 0
577.6 0 0 0 2.18217 1.25988i 0 3.00972 + 6.31993i 0 0 0
577.7 0 0 0 4.51611 2.60738i 0 −6.91807 + 1.06788i 0 0 0
577.8 0 0 0 7.85541 4.53532i 0 −1.17763 6.90023i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 577.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.cg.q 16
3.b odd 2 1 inner 1008.3.cg.q 16
4.b odd 2 1 504.3.by.d 16
7.d odd 6 1 inner 1008.3.cg.q 16
12.b even 2 1 504.3.by.d 16
21.g even 6 1 inner 1008.3.cg.q 16
28.f even 6 1 504.3.by.d 16
28.f even 6 1 3528.3.f.i 16
28.g odd 6 1 3528.3.f.i 16
84.j odd 6 1 504.3.by.d 16
84.j odd 6 1 3528.3.f.i 16
84.n even 6 1 3528.3.f.i 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.3.by.d 16 4.b odd 2 1
504.3.by.d 16 12.b even 2 1
504.3.by.d 16 28.f even 6 1
504.3.by.d 16 84.j odd 6 1
1008.3.cg.q 16 1.a even 1 1 trivial
1008.3.cg.q 16 3.b odd 2 1 inner
1008.3.cg.q 16 7.d odd 6 1 inner
1008.3.cg.q 16 21.g even 6 1 inner
3528.3.f.i 16 28.f even 6 1
3528.3.f.i 16 28.g odd 6 1
3528.3.f.i 16 84.j odd 6 1
3528.3.f.i 16 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1008, [\chi])$$:

 $$T_{5}^{16} - 118 T_{5}^{14} + 10739 T_{5}^{12} - 334630 T_{5}^{10} + 7682449 T_{5}^{8} - 58300664 T_{5}^{6} + 325701440 T_{5}^{4} - 638105600 T_{5}^{2} + 959512576$$ T5^16 - 118*T5^14 + 10739*T5^12 - 334630*T5^10 + 7682449*T5^8 - 58300664*T5^6 + 325701440*T5^4 - 638105600*T5^2 + 959512576 $$T_{11}^{16} + 666 T_{11}^{14} + 303067 T_{11}^{12} + 73620906 T_{11}^{10} + 12902674833 T_{11}^{8} + 1144098699168 T_{11}^{6} + 72351409853440 T_{11}^{4} + \cdots + 37\!\cdots\!36$$ T11^16 + 666*T11^14 + 303067*T11^12 + 73620906*T11^10 + 12902674833*T11^8 + 1144098699168*T11^6 + 72351409853440*T11^4 + 1923438961360896*T11^2 + 37201361225383936 $$T_{13}^{8} + 766T_{13}^{6} + 165641T_{13}^{4} + 7200248T_{13}^{2} + 63043600$$ T13^8 + 766*T13^6 + 165641*T13^4 + 7200248*T13^2 + 63043600

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$T^{16} - 118 T^{14} + \cdots + 959512576$$
$7$ $$(T^{8} + 2 T^{7} + 48 T^{6} + \cdots + 5764801)^{2}$$
$11$ $$T^{16} + 666 T^{14} + \cdots + 37\!\cdots\!36$$
$13$ $$(T^{8} + 766 T^{6} + 165641 T^{4} + \cdots + 63043600)^{2}$$
$17$ $$T^{16} - 920 T^{14} + \cdots + 44\!\cdots\!96$$
$19$ $$(T^{8} - 12 T^{7} - 95 T^{6} + \cdots + 2458624)^{2}$$
$23$ $$T^{16} + 3152 T^{14} + \cdots + 36\!\cdots\!56$$
$29$ $$(T^{8} - 4554 T^{6} + \cdots + 142968684544)^{2}$$
$31$ $$(T^{8} - 42 T^{7} - 1544 T^{6} + \cdots + 74287318249)^{2}$$
$37$ $$(T^{8} + 34 T^{7} + \cdots + 212012360704)^{2}$$
$41$ $$(T^{8} + 10520 T^{6} + \cdots + 12478867111936)^{2}$$
$43$ $$(T^{4} + 20 T^{3} - 5901 T^{2} + \cdots + 3352240)^{4}$$
$47$ $$T^{16} - 12976 T^{14} + \cdots + 47\!\cdots\!36$$
$53$ $$T^{16} + 7314 T^{14} + \cdots + 74\!\cdots\!76$$
$59$ $$T^{16} - 10038 T^{14} + \cdots + 13\!\cdots\!76$$
$61$ $$(T^{8} - 108 T^{7} + \cdots + 2040555110400)^{2}$$
$67$ $$(T^{8} + 28 T^{7} + 6317 T^{6} + \cdots + 2997781504)^{2}$$
$71$ $$(T^{8} - 6544 T^{6} + \cdots + 21724401664)^{2}$$
$73$ $$(T^{8} - 78 T^{7} + \cdots + 999648030976)^{2}$$
$79$ $$(T^{8} + 14 T^{7} + \cdots + 3096660231289)^{2}$$
$83$ $$(T^{8} + 5366 T^{6} + \cdots + 1728025927936)^{2}$$
$89$ $$T^{16} - 46136 T^{14} + \cdots + 46\!\cdots\!76$$
$97$ $$(T^{8} + 48470 T^{6} + \cdots + 14\!\cdots\!00)^{2}$$