Base field 4.4.19025.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 13x^{2} + 14x + 44\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16,4,w^{2} - 2w - 7]$ |
| Dimension: | $14$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $28$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{14} - 3x^{13} - 32x^{12} + 102x^{11} + 330x^{10} - 1156x^{9} - 1253x^{8} + 5179x^{7} + 1948x^{6} - 9758x^{5} - 2037x^{4} + 7016x^{3} + 1988x^{2} - 820x - 85\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, w^{2} - 2w - 6]$ | $\phantom{-}e$ |
| 4 | $[4, 2, -w^{2} + 7]$ | $\phantom{-}0$ |
| 5 | $[5, 5, -\frac{1}{2}w^{3} + 2w^{2} + \frac{7}{2}w - 14]$ | $\phantom{-}\frac{245158343061}{61008426762098}e^{13} + \frac{287776849869}{30504213381049}e^{12} - \frac{5029928412528}{30504213381049}e^{11} - \frac{9014376429462}{30504213381049}e^{10} + \frac{77520863888462}{30504213381049}e^{9} + \frac{93746057200904}{30504213381049}e^{8} - \frac{1099301582384535}{61008426762098}e^{7} - \frac{391715102914054}{30504213381049}e^{6} + \frac{1723561966235973}{30504213381049}e^{5} + \frac{727724377206907}{30504213381049}e^{4} - \frac{4057284523206453}{61008426762098}e^{3} - \frac{1104684579963873}{61008426762098}e^{2} + \frac{877799153546283}{61008426762098}e - \frac{124515961618057}{61008426762098}$ |
| 5 | $[5, 5, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - 6w - 9]$ | $\phantom{-}\frac{724491549097}{61008426762098}e^{13} - \frac{992323120885}{30504213381049}e^{12} - \frac{11130553544166}{30504213381049}e^{11} + \frac{33023251710137}{30504213381049}e^{10} + \frac{104591043581103}{30504213381049}e^{9} - \frac{356537674004926}{30504213381049}e^{8} - \frac{580007510375797}{61008426762098}e^{7} + \frac{1419762368734177}{30504213381049}e^{6} - \frac{34892672298353}{30504213381049}e^{5} - \frac{2086194703075362}{30504213381049}e^{4} + \frac{1448935987914693}{61008426762098}e^{3} + \frac{1797812428714325}{61008426762098}e^{2} - \frac{846605258664891}{61008426762098}e - \frac{92399758532253}{61008426762098}$ |
| 11 | $[11, 11, \frac{1}{2}w^{3} - 2w^{2} - \frac{5}{2}w + 11]$ | $-\frac{1314010975828}{30504213381049}e^{13} + \frac{2660743445829}{30504213381049}e^{12} + \frac{42209057985065}{30504213381049}e^{11} - \frac{89753096627732}{30504213381049}e^{10} - \frac{441853060997170}{30504213381049}e^{9} + \frac{981412998881632}{30504213381049}e^{8} + \frac{1742535189246465}{30504213381049}e^{7} - \frac{3951138441631883}{30504213381049}e^{6} - \frac{2687543714623089}{30504213381049}e^{5} + \frac{5924952825270965}{30504213381049}e^{4} + \frac{1622615432822162}{30504213381049}e^{3} - \frac{2704022001540163}{30504213381049}e^{2} - \frac{514092693686509}{30504213381049}e + \frac{131617113806100}{30504213381049}$ |
| 11 | $[11, 11, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - 5w - 7]$ | $\phantom{-}\frac{590797179729}{30504213381049}e^{13} - \frac{1050851126980}{30504213381049}e^{12} - \frac{18605433243166}{30504213381049}e^{11} + \frac{34751316441797}{30504213381049}e^{10} + \frac{186902696571685}{30504213381049}e^{9} - \frac{360848563925377}{30504213381049}e^{8} - \frac{658242550340450}{30504213381049}e^{7} + \frac{1246489925943409}{30504213381049}e^{6} + \frac{658367582739587}{30504213381049}e^{5} - \frac{1133763942870345}{30504213381049}e^{4} + \frac{319517441259757}{30504213381049}e^{3} - \frac{327474542089357}{30504213381049}e^{2} - \frac{513288610065074}{30504213381049}e + \frac{48920717447669}{30504213381049}$ |
| 31 | $[31, 31, \frac{1}{2}w^{2} + \frac{1}{2}w - 4]$ | $-\frac{764070604366}{30504213381049}e^{13} + \frac{3125582619587}{30504213381049}e^{12} + \frac{23336531350144}{30504213381049}e^{11} - \frac{102846790370100}{30504213381049}e^{10} - \frac{211855652891187}{30504213381049}e^{9} + \frac{1110423460817399}{30504213381049}e^{8} + \frac{492282521267854}{30504213381049}e^{7} - \frac{4572353150632197}{30504213381049}e^{6} + \frac{415757217873006}{30504213381049}e^{5} + \frac{7376243116014792}{30504213381049}e^{4} - \frac{1281267507625666}{30504213381049}e^{3} - \frac{3998251799698584}{30504213381049}e^{2} - \frac{198583845232793}{30504213381049}e + \frac{172607255806355}{30504213381049}$ |
| 31 | $[31, 31, -\frac{1}{2}w^{2} + \frac{3}{2}w + 3]$ | $-\frac{227768363395}{30504213381049}e^{13} + \frac{1297032621777}{30504213381049}e^{12} + \frac{5656055753364}{30504213381049}e^{11} - \frac{43225064346478}{30504213381049}e^{10} - \frac{20908663523537}{30504213381049}e^{9} + \frac{476222755037624}{30504213381049}e^{8} - \frac{303802796503668}{30504213381049}e^{7} - \frac{2039041364723278}{30504213381049}e^{6} + \frac{1962591022430955}{30504213381049}e^{5} + \frac{3607009215549850}{30504213381049}e^{4} - \frac{3244520784453246}{30504213381049}e^{3} - \frac{2408442354363671}{30504213381049}e^{2} + \frac{1148246136991231}{30504213381049}e + \frac{234341334253012}{30504213381049}$ |
| 41 | $[41, 41, \frac{1}{2}w^{2} + \frac{1}{2}w - 6]$ | $-\frac{2137344189783}{61008426762098}e^{13} + \frac{3134267100175}{30504213381049}e^{12} + \frac{31995992386315}{30504213381049}e^{11} - \frac{105300605393879}{30504213381049}e^{10} - \frac{282748773515184}{30504213381049}e^{9} + \frac{1156419499291804}{30504213381049}e^{8} + \frac{1208937659516985}{61008426762098}e^{7} - \frac{4784384823193038}{30504213381049}e^{6} + \frac{946402614057614}{30504213381049}e^{5} + \frac{7733133977216373}{30504213381049}e^{4} - \frac{5519502558374195}{61008426762098}e^{3} - \frac{8389295945398331}{61008426762098}e^{2} + \frac{796157092098945}{61008426762098}e + \frac{26312681118727}{61008426762098}$ |
| 41 | $[41, 41, 2w^{3} - \frac{15}{2}w^{2} - \frac{25}{2}w + 50]$ | $-\frac{483157872941}{30504213381049}e^{13} - \frac{465607159774}{30504213381049}e^{12} + \frac{19005557236340}{30504213381049}e^{11} + \frac{15175442171468}{30504213381049}e^{10} - \frac{275786740354973}{30504213381049}e^{9} - \frac{173522729172359}{30504213381049}e^{8} + \frac{1825175526057643}{30504213381049}e^{7} + \frac{873407350571504}{30504213381049}e^{6} - \frac{5448592868424743}{30504213381049}e^{5} - \frac{2019417790590720}{30504213381049}e^{4} + \frac{6264264715706768}{30504213381049}e^{3} + \frac{1718010651111784}{30504213381049}e^{2} - \frac{1513840842049543}{30504213381049}e - \frac{22277612733750}{30504213381049}$ |
| 41 | $[41, 41, \frac{5}{2}w^{2} - \frac{1}{2}w - 17]$ | $\phantom{-}\frac{267261158158}{30504213381049}e^{13} - \frac{867171600838}{30504213381049}e^{12} - \frac{7849672349207}{30504213381049}e^{11} + \frac{29120915341375}{30504213381049}e^{10} + \frac{67181874933320}{30504213381049}e^{9} - \frac{318825029436526}{30504213381049}e^{8} - \frac{139473597722790}{30504213381049}e^{7} + \frac{1301776904182471}{30504213381049}e^{6} - \frac{93205942568927}{30504213381049}e^{5} - \frac{2009970119756937}{30504213381049}e^{4} + \frac{69806880985185}{30504213381049}e^{3} + \frac{888466908781409}{30504213381049}e^{2} + \frac{226925508473570}{30504213381049}e + \frac{156470450385501}{30504213381049}$ |
| 41 | $[41, 41, \frac{1}{2}w^{2} - \frac{3}{2}w - 5]$ | $\phantom{-}\frac{1410930516553}{61008426762098}e^{13} + \frac{480892813445}{30504213381049}e^{12} - \frac{24916846805825}{30504213381049}e^{11} - \frac{13912734581239}{30504213381049}e^{10} + \frac{315662047408520}{30504213381049}e^{9} + \frac{146207199892961}{30504213381049}e^{8} - \frac{3598063160361637}{61008426762098}e^{7} - \frac{784008460537157}{30504213381049}e^{6} + \frac{4712563728411289}{30504213381049}e^{5} + \frac{2079598824328833}{30504213381049}e^{4} - \frac{9616365892402197}{61008426762098}e^{3} - \frac{3976195267586919}{61008426762098}e^{2} + \frac{1456437363366759}{61008426762098}e - \frac{97534546436321}{61008426762098}$ |
| 61 | $[61, 61, -\frac{1}{2}w^{3} + w^{2} + \frac{7}{2}w - 1]$ | $\phantom{-}\frac{676494251443}{30504213381049}e^{13} - \frac{3340322215900}{30504213381049}e^{12} - \frac{19352934995622}{30504213381049}e^{11} + \frac{108780719167799}{30504213381049}e^{10} + \frac{144066473788664}{30504213381049}e^{9} - \frac{1152787695250510}{30504213381049}e^{8} + \frac{29826222671964}{30504213381049}e^{7} + \frac{4550782830449811}{30504213381049}e^{6} - \frac{2180147230647128}{30504213381049}e^{5} - \frac{6596593895931865}{30504213381049}e^{4} + \frac{3762691647183042}{30504213381049}e^{3} + \frac{2481267662461831}{30504213381049}e^{2} - \frac{1015744274474262}{30504213381049}e + \frac{177844439555493}{30504213381049}$ |
| 61 | $[61, 61, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + 4w - 3]$ | $-\frac{213776903799}{30504213381049}e^{13} - \frac{1875288180656}{30504213381049}e^{12} + \frac{11039245480301}{30504213381049}e^{11} + \frac{62611938784430}{30504213381049}e^{10} - \frac{209903362529794}{30504213381049}e^{9} - \frac{710398808485156}{30504213381049}e^{8} + \frac{1744054527384398}{30504213381049}e^{7} + \frac{3336458059467325}{30504213381049}e^{6} - \frac{5900735110158981}{30504213381049}e^{5} - \frac{6985895014488076}{30504213381049}e^{4} + \frac{6802213681232461}{30504213381049}e^{3} + \frac{5791776521910486}{30504213381049}e^{2} - \frac{744026821499671}{30504213381049}e - \frac{332914713611244}{30504213381049}$ |
| 71 | $[71, 71, \frac{1}{2}w^{3} + w^{2} - \frac{11}{2}w - 13]$ | $-\frac{25912598770}{30504213381049}e^{13} + \frac{1027634347021}{30504213381049}e^{12} + \frac{561412170941}{30504213381049}e^{11} - \frac{35117202015954}{30504213381049}e^{10} + \frac{3022267601835}{30504213381049}e^{9} + \frac{417642812264756}{30504213381049}e^{8} - \frac{109383614934439}{30504213381049}e^{7} - \frac{2141376829823771}{30504213381049}e^{6} + \frac{434082208578110}{30504213381049}e^{5} + \frac{4941776844698285}{30504213381049}e^{4} - \frac{102667337394628}{30504213381049}e^{3} - \frac{4242121819508996}{30504213381049}e^{2} - \frac{845917633423487}{30504213381049}e + \frac{220736601302529}{30504213381049}$ |
| 71 | $[71, 71, -\frac{1}{2}w^{3} + \frac{5}{2}w^{2} + 2w - 17]$ | $\phantom{-}\frac{1936337610085}{30504213381049}e^{13} - \frac{625841642267}{30504213381049}e^{12} - \frac{67977632746991}{30504213381049}e^{11} + \frac{23139109101353}{30504213381049}e^{10} + \frac{842751100349072}{30504213381049}e^{9} - \frac{236648134418597}{30504213381049}e^{8} - \frac{4595654674773186}{30504213381049}e^{7} + \frac{445449732591414}{30504213381049}e^{6} + \frac{11490365044624685}{30504213381049}e^{5} + \frac{1585312663388440}{30504213381049}e^{4} - \frac{11263430031781572}{30504213381049}e^{3} - \frac{3763790297906291}{30504213381049}e^{2} + \frac{1593410766228381}{30504213381049}e + \frac{309795674721854}{30504213381049}$ |
| 81 | $[81, 3, -3]$ | $\phantom{-}\frac{1512166116837}{30504213381049}e^{13} - \frac{2788276268507}{30504213381049}e^{12} - \frac{48913945106271}{30504213381049}e^{11} + \frac{91265496189455}{30504213381049}e^{10} + \frac{517157676435289}{30504213381049}e^{9} - \frac{940373384210493}{30504213381049}e^{8} - \frac{2060988203069857}{30504213381049}e^{7} + \frac{3256167429131063}{30504213381049}e^{6} + \frac{3062537821284350}{30504213381049}e^{5} - \frac{3056682781179917}{30504213381049}e^{4} - \frac{871444881804857}{30504213381049}e^{3} - \frac{628611971161895}{30504213381049}e^{2} - \frac{1159740369573271}{30504213381049}e - \frac{79406531359396}{30504213381049}$ |
| 89 | $[89, 89, -w^{3} + \frac{3}{2}w^{2} + \frac{13}{2}w - 9]$ | $\phantom{-}\frac{388020439123}{61008426762098}e^{13} + \frac{2431726866770}{30504213381049}e^{12} - \frac{10552114688568}{30504213381049}e^{11} - \frac{79311058851962}{30504213381049}e^{10} + \frac{211168396110634}{30504213381049}e^{9} + \frac{866204043579734}{30504213381049}e^{8} - \frac{3657084793879871}{61008426762098}e^{7} - \frac{3809957990082287}{30504213381049}e^{6} + \frac{6306664669267442}{30504213381049}e^{5} + \frac{7220791982336219}{30504213381049}e^{4} - \frac{14562414619827029}{61008426762098}e^{3} - \frac{10816937327127815}{61008426762098}e^{2} + \frac{1648778005510735}{61008426762098}e + \frac{433167903031513}{61008426762098}$ |
| 89 | $[89, 89, w^{3} - \frac{7}{2}w^{2} - \frac{11}{2}w + 20]$ | $-\frac{599575491973}{30504213381049}e^{13} - \frac{3755527029899}{30504213381049}e^{12} + \frac{26438439737344}{30504213381049}e^{11} + \frac{122834419047131}{30504213381049}e^{10} - \frac{443729257802598}{30504213381049}e^{9} - \frac{1361309989750908}{30504213381049}e^{8} + \frac{3388250116545605}{30504213381049}e^{7} + \frac{6238818367868256}{30504213381049}e^{6} - \frac{10836391060873808}{30504213381049}e^{5} - \frac{12659486383120271}{30504213381049}e^{4} + \frac{11843637435618456}{30504213381049}e^{3} + \frac{10071535243889807}{30504213381049}e^{2} - \frac{1157556732270445}{30504213381049}e - \frac{622191908293953}{30504213381049}$ |
| 89 | $[89, 89, -w^{3} - \frac{1}{2}w^{2} + \frac{19}{2}w + 12]$ | $-\frac{4014785610868}{30504213381049}e^{13} + \frac{9335321019941}{30504213381049}e^{12} + \frac{126520631936517}{30504213381049}e^{11} - \frac{313410955225064}{30504213381049}e^{10} - \frac{1268742343379982}{30504213381049}e^{9} + \frac{3415327754466642}{30504213381049}e^{8} + \frac{4454918585685918}{30504213381049}e^{7} - \frac{13742078420759026}{30504213381049}e^{6} - \frac{4876081903252656}{30504213381049}e^{5} + \frac{20570608120678448}{30504213381049}e^{4} + \frac{583099771515386}{30504213381049}e^{3} - \frac{9237645400475250}{30504213381049}e^{2} - \frac{138170671783418}{30504213381049}e + \frac{583960171616276}{30504213381049}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4,2,w^{2} - 7]$ | $1$ |