Properties

Label 4.4.5725.1-59.1-c
Base field 4.4.5725.1
Weight $[2, 2, 2, 2]$
Level norm $59$
Level $[59, 59, \frac{2}{3}w^{3} - \frac{13}{3}w + \frac{8}{3}]$
Dimension $13$
CM no
Base change no

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Base field 4.4.5725.1

Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 8x^{2} + 6x + 11\); narrow class number \(1\) and class number \(1\).

Form

Weight: $[2, 2, 2, 2]$
Level: $[59, 59, \frac{2}{3}w^{3} - \frac{13}{3}w + \frac{8}{3}]$
Dimension: $13$
CM: no
Base change: no
Newspace dimension: $18$

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:

\(x^{13} - 13x^{12} + 12x^{11} + 474x^{10} - 1879x^{9} - 1896x^{8} + 18824x^{7} - 13900x^{6} - 42512x^{5} + 37936x^{4} + 44992x^{3} - 23680x^{2} - 23040x - 2304\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
9 $[9, 3, \frac{1}{3}w^{3} - \frac{8}{3}w - \frac{2}{3}]$ $\phantom{-}e$
9 $[9, 3, -w + 1]$ $\phantom{-}\frac{24184495064941}{18612628820003136}e^{12} - \frac{315307409768473}{18612628820003136}e^{11} + \frac{14106139039231}{775526200833464}e^{10} + \frac{1831306505564077}{3102104803333856}e^{9} - \frac{45511898347927507}{18612628820003136}e^{8} - \frac{285073154132073}{193881550208366}e^{7} + \frac{98320522960999241}{4653157205000784}e^{6} - \frac{104236666981410037}{4653157205000784}e^{5} - \frac{13337934174502825}{581644650625098}e^{4} + \frac{32005689872102185}{1163289301250196}e^{3} + \frac{4365514282059497}{581644650625098}e^{2} - \frac{127115021816131}{290822325312549}e + \frac{254291028149069}{96940775104183}$
11 $[11, 11, w]$ $-\frac{12281827278841}{37225257640006272}e^{12} + \frac{205827513311473}{37225257640006272}e^{11} - \frac{28204439840667}{1551052401666928}e^{10} - \frac{1010693541641079}{6204209606667712}e^{9} + \frac{44425686735044215}{37225257640006272}e^{8} - \frac{2576947309384117}{3102104803333856}e^{7} - \frac{49268395107534721}{4653157205000784}e^{6} + \frac{193743349792587703}{9306314410001568}e^{5} + \frac{44748569127445625}{2326578602500392}e^{4} - \frac{118521474991019827}{2326578602500392}e^{3} - \frac{4779938271447437}{290822325312549}e^{2} + \frac{19638093244266475}{581644650625098}e + \frac{867845421206497}{96940775104183}$
11 $[11, 11, -\frac{1}{3}w^{3} + \frac{2}{3}w + \frac{5}{3}]$ $\phantom{-}\frac{9436419644357}{37225257640006272}e^{12} - \frac{284316316599737}{37225257640006272}e^{11} + \frac{44240528670967}{775526200833464}e^{10} + \frac{575695263252589}{6204209606667712}e^{9} - \frac{92287903641339323}{37225257640006272}e^{8} + \frac{10104064752129655}{1551052401666928}e^{7} + \frac{133189444356903253}{9306314410001568}e^{6} - \frac{658209611957589713}{9306314410001568}e^{5} + \frac{28569030544777807}{1163289301250196}e^{4} + \frac{73250512935227177}{581644650625098}e^{3} - \frac{52335983335088735}{1163289301250196}e^{2} - \frac{22858190487619510}{290822325312549}e - \frac{773613097444439}{96940775104183}$
11 $[11, 11, \frac{1}{3}w^{3} - \frac{8}{3}w + \frac{1}{3}]$ $\phantom{-}\frac{6296964233233}{18612628820003136}e^{12} - \frac{65970622664621}{9306314410001568}e^{11} + \frac{234689452958095}{6204209606667712}e^{10} + \frac{436300288709239}{3102104803333856}e^{9} - \frac{35093321195292685}{18612628820003136}e^{8} + \frac{23701938686357777}{6204209606667712}e^{7} + \frac{14212471661129699}{1163289301250196}e^{6} - \frac{27839360253494999}{581644650625098}e^{5} + \frac{41548871497237927}{4653157205000784}e^{4} + \frac{106437118241874415}{1163289301250196}e^{3} - \frac{13336275770368867}{581644650625098}e^{2} - \frac{18012985616823307}{290822325312549}e - \frac{971267000256551}{96940775104183}$
11 $[11, 11, -\frac{2}{3}w^{3} + \frac{13}{3}w + \frac{4}{3}]$ $\phantom{-}\frac{1242095282095}{37225257640006272}e^{12} - \frac{61984348066201}{37225257640006272}e^{11} + \frac{99827183632235}{6204209606667712}e^{10} + \frac{14440276659073}{6204209606667712}e^{9} - \frac{22908333659749069}{37225257640006272}e^{8} + \frac{12985830861567595}{6204209606667712}e^{7} + \frac{10349047771860505}{4653157205000784}e^{6} - \frac{175918294019322115}{9306314410001568}e^{5} + \frac{82658174664997055}{4653157205000784}e^{4} + \frac{40145864724003745}{2326578602500392}e^{3} - \frac{14535267901604135}{581644650625098}e^{2} + \frac{660028769319871}{290822325312549}e + \frac{666997211243110}{96940775104183}$
16 $[16, 2, 2]$ $\phantom{-}\frac{43874706194765}{12408419213335424}e^{12} - \frac{565574044779307}{12408419213335424}e^{11} + \frac{257276277689915}{6204209606667712}e^{10} + \frac{10124122388105721}{6204209606667712}e^{9} - \frac{80091733889161047}{12408419213335424}e^{8} - \frac{34490542158605421}{6204209606667712}e^{7} + \frac{93091270711949597}{1551052401666928}e^{6} - \frac{164231172332488221}{3102104803333856}e^{5} - \frac{150092650515549915}{1551052401666928}e^{4} + \frac{81384758866050749}{775526200833464}e^{3} + \frac{11027673895626531}{193881550208366}e^{2} - \frac{4034867130972136}{96940775104183}e - \frac{1140911408721468}{96940775104183}$
25 $[25, 5, \frac{2}{3}w^{3} - \frac{10}{3}w - \frac{1}{3}]$ $-\frac{25284102778399}{9306314410001568}e^{12} + \frac{150134350588649}{4653157205000784}e^{11} + \frac{6493927675325}{3102104803333856}e^{10} - \frac{983675623423733}{775526200833464}e^{9} + \frac{34421776033846795}{9306314410001568}e^{8} + \frac{26627872745056569}{3102104803333856}e^{7} - \frac{186415901652925861}{4653157205000784}e^{6} - \frac{549033435245953}{1163289301250196}e^{5} + \frac{223946923152348731}{2326578602500392}e^{4} - \frac{15632059978229237}{1163289301250196}e^{3} - \frac{23143540973479916}{290822325312549}e^{2} - \frac{826976316079063}{290822325312549}e + \frac{820263876436932}{96940775104183}$
29 $[29, 29, -w - 3]$ $-\frac{7896773961793}{6204209606667712}e^{12} + \frac{62828728148287}{3102104803333856}e^{11} - \frac{378153926553337}{6204209606667712}e^{10} - \frac{1811274312593457}{3102104803333856}e^{9} + \frac{25390168131026805}{6204209606667712}e^{8} - \frac{21119267843924271}{6204209606667712}e^{7} - \frac{25008935315991691}{775526200833464}e^{6} + \frac{28687422381971809}{387763100416732}e^{5} + \frac{37384823475506813}{1551052401666928}e^{4} - \frac{56536372299144229}{387763100416732}e^{3} + \frac{1182671358440533}{193881550208366}e^{2} + \frac{8114713401883609}{96940775104183}e + \frac{939930591315101}{96940775104183}$
29 $[29, 29, -\frac{1}{3}w^{3} + \frac{8}{3}w - \frac{10}{3}]$ $-\frac{6381491061953}{18612628820003136}e^{12} + \frac{49138762079399}{18612628820003136}e^{11} + \frac{54796267092387}{3102104803333856}e^{10} - \frac{509903875307243}{3102104803333856}e^{9} - \frac{3619220159952277}{18612628820003136}e^{8} + \frac{9993864931306195}{3102104803333856}e^{7} - \frac{3151883749717673}{2326578602500392}e^{6} - \frac{100804759497101509}{4653157205000784}e^{5} + \frac{44744174648432123}{2326578602500392}e^{4} + \frac{48534347508194959}{1163289301250196}e^{3} - \frac{16073785324631311}{581644650625098}e^{2} - \frac{8719470077523892}{290822325312549}e + \frac{408139961054078}{96940775104183}$
31 $[31, 31, w^{3} - 6w + 1]$ $\phantom{-}\frac{38638117885359}{6204209606667712}e^{12} - \frac{502494307642465}{6204209606667712}e^{11} + \frac{256630389911417}{3102104803333856}e^{10} + \frac{8866243628345933}{3102104803333856}e^{9} - \frac{72455996383863897}{6204209606667712}e^{8} - \frac{25594536882201315}{3102104803333856}e^{7} + \frac{81746191727900695}{775526200833464}e^{6} - \frac{40712527412175839}{387763100416732}e^{5} - \frac{114762400454339723}{775526200833464}e^{4} + \frac{18146998932603883}{96940775104183}e^{3} + \frac{25356089959749451}{387763100416732}e^{2} - \frac{6476328263381545}{96940775104183}e - \frac{408552490193811}{96940775104183}$
31 $[31, 31, w^{3} - 6w - 2]$ $\phantom{-}\frac{73269786512251}{18612628820003136}e^{12} - \frac{116605280181113}{2326578602500392}e^{11} + \frac{244402384223235}{6204209606667712}e^{10} + \frac{5604134238111411}{3102104803333856}e^{9} - \frac{127973545279981015}{18612628820003136}e^{8} - \frac{41658538535457763}{6204209606667712}e^{7} + \frac{18392531044173503}{290822325312549}e^{6} - \frac{233297318214328351}{4653157205000784}e^{5} - \frac{438775603115093285}{4653157205000784}e^{4} + \frac{92186892099062929}{1163289301250196}e^{3} + \frac{60876583782693337}{1163289301250196}e^{2} - \frac{3784754514969514}{290822325312549}e - \frac{105312841944910}{96940775104183}$
41 $[41, 41, \frac{2}{3}w^{3} + w^{2} - \frac{13}{3}w - \frac{10}{3}]$ $\phantom{-}\frac{4264331381115}{6204209606667712}e^{12} - \frac{63465670838881}{6204209606667712}e^{11} + \frac{75264321509033}{3102104803333856}e^{10} + \frac{981410207062777}{3102104803333856}e^{9} - \frac{11568487541864321}{6204209606667712}e^{8} + \frac{2379058703426671}{3102104803333856}e^{7} + \frac{23192882510491911}{1551052401666928}e^{6} - \frac{42364824155571321}{1551052401666928}e^{5} - \frac{10307503022750443}{775526200833464}e^{4} + \frac{7924507497703289}{193881550208366}e^{3} + \frac{2510531922636049}{193881550208366}e^{2} - \frac{1731276951942800}{96940775104183}e - \frac{942489068488680}{96940775104183}$
41 $[41, 41, \frac{2}{3}w^{3} - \frac{13}{3}w + \frac{5}{3}]$ $-\frac{7524167981673}{3102104803333856}e^{12} + \frac{93507038586049}{3102104803333856}e^{11} - \frac{24205301498297}{1551052401666928}e^{10} - \frac{215600903124175}{193881550208366}e^{9} + \frac{12090292600842223}{3102104803333856}e^{8} + \frac{1980349540385123}{387763100416732}e^{7} - \frac{56949202451565825}{1551052401666928}e^{6} + \frac{16083040926375309}{775526200833464}e^{5} + \frac{22447758238243963}{387763100416732}e^{4} - \frac{10805970665079817}{387763100416732}e^{3} - \frac{3053445838973114}{96940775104183}e^{2} - \frac{432784757020450}{96940775104183}e + \frac{60230438897106}{96940775104183}$
59 $[59, 59, \frac{2}{3}w^{3} - \frac{13}{3}w + \frac{8}{3}]$ $-1$
59 $[59, 59, w^{3} + w^{2} - 6w - 4]$ $-\frac{26014154114051}{6204209606667712}e^{12} + \frac{331689396190987}{6204209606667712}e^{11} - \frac{67347513161527}{1551052401666928}e^{10} - \frac{5958960710922597}{3102104803333856}e^{9} + \frac{45838123018796333}{6204209606667712}e^{8} + \frac{10636572802011289}{1551052401666928}e^{7} - \frac{105536936529116111}{1551052401666928}e^{6} + \frac{89568206298778945}{1551052401666928}e^{5} + \frac{39785681446636833}{387763100416732}e^{4} - \frac{20591252444285555}{193881550208366}e^{3} - \frac{5606771302416906}{96940775104183}e^{2} + \frac{4229813003996291}{96940775104183}e + \frac{1289099583851632}{96940775104183}$
79 $[79, 79, \frac{2}{3}w^{3} + w^{2} - \frac{10}{3}w - \frac{19}{3}]$ $\phantom{-}\frac{14462953454785}{2326578602500392}e^{12} - \frac{202359804515047}{2326578602500392}e^{11} + \frac{243051794142165}{1551052401666928}e^{10} + \frac{4403704661662673}{1551052401666928}e^{9} - \frac{16850931452383853}{1163289301250196}e^{8} + \frac{552877006541953}{775526200833464}e^{7} + \frac{565429760340692551}{4653157205000784}e^{6} - \frac{453541441946668705}{2326578602500392}e^{5} - \frac{144031175783956099}{1163289301250196}e^{4} + \frac{421350786871702043}{1163289301250196}e^{3} + \frac{8936685580770661}{581644650625098}e^{2} - \frac{52501841983723202}{290822325312549}e - \frac{1651879740335654}{96940775104183}$
79 $[79, 79, \frac{1}{3}w^{3} + w^{2} - \frac{2}{3}w - \frac{17}{3}]$ $\phantom{-}\frac{2916422008801}{9306314410001568}e^{12} - \frac{3789543435431}{1163289301250196}e^{11} - \frac{15415944220229}{3102104803333856}e^{10} + \frac{107879162520187}{775526200833464}e^{9} - \frac{2177865886366153}{9306314410001568}e^{8} - \frac{4061046535565805}{3102104803333856}e^{7} + \frac{14381324508360931}{4653157205000784}e^{6} + \frac{3796784844106459}{1163289301250196}e^{5} - \frac{21016264810584947}{2326578602500392}e^{4} + \frac{291093353864329}{581644650625098}e^{3} + \frac{3234937087751033}{290822325312549}e^{2} - \frac{2986808465494790}{290822325312549}e - \frac{516701957694861}{96940775104183}$
89 $[89, 89, \frac{4}{3}w^{3} - \frac{23}{3}w - \frac{5}{3}]$ $\phantom{-}\frac{40671869664077}{9306314410001568}e^{12} - \frac{265803700900663}{4653157205000784}e^{11} + \frac{47035229444673}{775526200833464}e^{10} + \frac{6253017369681211}{3102104803333856}e^{9} - \frac{77591050304453927}{9306314410001568}e^{8} - \frac{17776397966994981}{3102104803333856}e^{7} + \frac{709584193813797283}{9306314410001568}e^{6} - \frac{360362444664128359}{4653157205000784}e^{5} - \frac{267874448797414165}{2326578602500392}e^{4} + \frac{362251218140488787}{2326578602500392}e^{3} + \frac{59657872975878853}{1163289301250196}e^{2} - \frac{34284981325663667}{581644650625098}e - \frac{443377696836892}{96940775104183}$
89 $[89, 89, \frac{1}{3}w^{3} + 2w^{2} - \frac{5}{3}w - \frac{32}{3}]$ $-\frac{8674899994465}{12408419213335424}e^{12} + \frac{163674633845965}{12408419213335424}e^{11} - \frac{97424322601657}{1551052401666928}e^{10} - \frac{1656207784670051}{6204209606667712}e^{9} + \frac{40011222497179887}{12408419213335424}e^{8} - \frac{2639940682164183}{387763100416732}e^{7} - \frac{56816534769865493}{3102104803333856}e^{6} + \frac{260141797486041181}{3102104803333856}e^{5} - \frac{8469515795205799}{193881550208366}e^{4} - \frac{13648723665060761}{96940775104183}e^{3} + \frac{36315625144330831}{387763100416732}e^{2} + \frac{7513913262236424}{96940775104183}e - \frac{726944313972025}{96940775104183}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$59$ $[59, 59, \frac{2}{3}w^{3} - \frac{13}{3}w + \frac{8}{3}]$ $1$