Base field \(\Q(\sqrt{67}) \)
Generator \(w\), with minimal polynomial \(x^{2} - 67\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[8, 4, -54w + 442]$ |
| Dimension: | $16$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $32$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{16} - 5x^{15} - 19x^{14} + 117x^{13} + 113x^{12} - 1025x^{11} - 180x^{10} + 4236x^{9} - 211x^{8} - 8619x^{7} + 104x^{6} + 7972x^{5} + 1364x^{4} - 2444x^{3} - 1081x^{2} - 135x - 4\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -27w + 221]$ | $\phantom{-}0$ |
| 3 | $[3, 3, -w + 8]$ | $\phantom{-}\frac{1484033845}{47213134919}e^{15} - \frac{11165805075}{47213134919}e^{14} - \frac{10347914513}{47213134919}e^{13} + \frac{241366429943}{47213134919}e^{12} - \frac{237219831576}{47213134919}e^{11} - \frac{1857112344771}{47213134919}e^{10} + \frac{3117267529700}{47213134919}e^{9} + \frac{6112280111180}{47213134919}e^{8} - \frac{13317840707743}{47213134919}e^{7} - \frac{8019570866461}{47213134919}e^{6} + \frac{23843050091360}{47213134919}e^{5} + \frac{2066505033608}{47213134919}e^{4} - \frac{16602940078783}{47213134919}e^{3} + \frac{1270238510476}{47213134919}e^{2} + \frac{3111662316011}{47213134919}e + \frac{277623991539}{47213134919}$ |
| 3 | $[3, 3, -w - 8]$ | $\phantom{-}e$ |
| 7 | $[7, 7, -11w + 90]$ | $\phantom{-}\frac{22057621824}{47213134919}e^{15} - \frac{113719299385}{47213134919}e^{14} - \frac{383035857809}{47213134919}e^{13} + \frac{2597700412127}{47213134919}e^{12} + \frac{1650582618073}{47213134919}e^{11} - \frac{21948811948557}{47213134919}e^{10} + \frac{3374021350772}{47213134919}e^{9} + \frac{85875891091546}{47213134919}e^{8} - \frac{34784696388137}{47213134919}e^{7} - \frac{161441567680319}{47213134919}e^{6} + \frac{62978919039576}{47213134919}e^{5} + \frac{134410061488696}{47213134919}e^{4} - \frac{25272502372776}{47213134919}e^{3} - \frac{37523442006042}{47213134919}e^{2} - \frac{6367496616542}{47213134919}e - \frac{307884619041}{47213134919}$ |
| 7 | $[7, 7, -11w - 90]$ | $-\frac{16914611506}{47213134919}e^{15} + \frac{87442601346}{47213134919}e^{14} + \frac{296834930812}{47213134919}e^{13} - \frac{2007843893488}{47213134919}e^{12} - \frac{1337791963144}{47213134919}e^{11} + \frac{17103269931012}{47213134919}e^{10} - \frac{1964379478914}{47213134919}e^{9} - \frac{67798271763620}{47213134919}e^{8} + \frac{24143065877239}{47213134919}e^{7} + \frac{130141792639476}{47213134919}e^{6} - \frac{43217165630477}{47213134919}e^{5} - \frac{111953786273082}{47213134919}e^{4} + \frac{14709683253770}{47213134919}e^{3} + \frac{32741564957768}{47213134919}e^{2} + \frac{6443745455229}{47213134919}e + \frac{408558954554}{47213134919}$ |
| 11 | $[11, 11, 6w - 49]$ | $-\frac{32443001154}{47213134919}e^{15} + \frac{175866060905}{47213134919}e^{14} + \frac{545772717188}{47213134919}e^{13} - \frac{4031565055465}{47213134919}e^{12} - \frac{2052816476509}{47213134919}e^{11} + \frac{34246538423005}{47213134919}e^{10} - \frac{7789178332691}{47213134919}e^{9} - \frac{135112057869458}{47213134919}e^{8} + \frac{60188551160868}{47213134919}e^{7} + \frac{257317846835757}{47213134919}e^{6} - \frac{103797504867856}{47213134919}e^{5} - \frac{218462464249374}{47213134919}e^{4} + \frac{39571282416445}{47213134919}e^{3} + \frac{62778585120644}{47213134919}e^{2} + \frac{11599313865139}{47213134919}e + \frac{652628169103}{47213134919}$ |
| 11 | $[11, 11, 6w + 49]$ | $-\frac{6091745255}{47213134919}e^{15} + \frac{34173417791}{47213134919}e^{14} + \frac{106430787344}{47213134919}e^{13} - \frac{801055198765}{47213134919}e^{12} - \frac{480366200342}{47213134919}e^{11} + \frac{7032626950877}{47213134919}e^{10} - \frac{605431577715}{47213134919}e^{9} - \frac{29140423289714}{47213134919}e^{8} + \frac{7627655203095}{47213134919}e^{7} + \frac{59535562520805}{47213134919}e^{6} - \frac{11656726634263}{47213134919}e^{5} - \frac{55631358700566}{47213134919}e^{4} - \frac{124862057392}{47213134919}e^{3} + \frac{17988097556014}{47213134919}e^{2} + \frac{4684944286838}{47213134919}e + \frac{286199274763}{47213134919}$ |
| 17 | $[17, 17, 4w + 33]$ | $-\frac{17528972974}{47213134919}e^{15} + \frac{98500505534}{47213134919}e^{14} + \frac{274541491180}{47213134919}e^{13} - \frac{2233683713851}{47213134919}e^{12} - \frac{639632168074}{47213134919}e^{11} + \frac{18651567838689}{47213134919}e^{10} - \frac{8235531037228}{47213134919}e^{9} - \frac{71544595549966}{47213134919}e^{8} + \frac{48628820628054}{47213134919}e^{7} + \frac{130091541896563}{47213134919}e^{6} - \frac{87392794638668}{47213134919}e^{5} - \frac{102319752489335}{47213134919}e^{4} + \frac{48892091463092}{47213134919}e^{3} + \frac{26455824170898}{47213134919}e^{2} - \frac{2139192767472}{47213134919}e - \frac{476663271890}{47213134919}$ |
| 17 | $[17, 17, -4w + 33]$ | $\phantom{-}\frac{1582350418}{47213134919}e^{15} - \frac{20395531748}{47213134919}e^{14} + \frac{9991899502}{47213134919}e^{13} + \frac{457630874073}{47213134919}e^{12} - \frac{710351874318}{47213134919}e^{11} - \frac{3751420525521}{47213134919}e^{10} + \frac{6898266304904}{47213134919}e^{9} + \frac{13914055709842}{47213134919}e^{8} - \frac{26429414263422}{47213134919}e^{7} - \frac{23738541516555}{47213134919}e^{6} + \frac{43406942219222}{47213134919}e^{5} + \frac{16172824436931}{47213134919}e^{4} - \frac{27134036382674}{47213134919}e^{3} - \frac{3053551394476}{47213134919}e^{2} + \frac{4257373581220}{47213134919}e + \frac{450728575528}{47213134919}$ |
| 25 | $[25, 5, -5]$ | $\phantom{-}\frac{24029044949}{47213134919}e^{15} - \frac{126401245220}{47213134919}e^{14} - \frac{406379124189}{47213134919}e^{13} + \frac{2879263079134}{47213134919}e^{12} + \frac{1548263755597}{47213134919}e^{11} - \frac{24219858520869}{47213134919}e^{10} + \frac{5801126529594}{47213134919}e^{9} + \frac{94085423756763}{47213134919}e^{8} - \frac{46297510507199}{47213134919}e^{7} - \frac{174919296425743}{47213134919}e^{6} + \frac{84606480861704}{47213134919}e^{5} + \frac{143405781808277}{47213134919}e^{4} - \frac{40936803070726}{47213134919}e^{3} - \frac{39878919820349}{47213134919}e^{2} - \frac{3334484921151}{47213134919}e + \frac{213849934094}{47213134919}$ |
| 29 | $[29, 29, -70w + 573]$ | $\phantom{-}\frac{40322607358}{47213134919}e^{15} - \frac{216597915261}{47213134919}e^{14} - \frac{696147417706}{47213134919}e^{13} + \frac{4996448965777}{47213134919}e^{12} + \frac{2959545630120}{47213134919}e^{11} - \frac{42847190250697}{47213134919}e^{10} + \frac{6214126218982}{47213134919}e^{9} + \frac{171513079472453}{47213134919}e^{8} - \frac{61097657252452}{47213134919}e^{7} - \frac{333639265455492}{47213134919}e^{6} + \frac{102737039082920}{47213134919}e^{5} + \frac{291556481384650}{47213134919}e^{4} - \frac{26940552184272}{47213134919}e^{3} - \frac{86636726315391}{47213134919}e^{2} - \frac{20615741005118}{47213134919}e - \frac{1103970089301}{47213134919}$ |
| 29 | $[29, 29, 151w - 1236]$ | $-\frac{48083611098}{47213134919}e^{15} + \frac{260078087454}{47213134919}e^{14} + \frac{808995012948}{47213134919}e^{13} - \frac{5964049779906}{47213134919}e^{12} - \frac{3031988537544}{47213134919}e^{11} + \frac{50692926838206}{47213134919}e^{10} - \frac{11783223243564}{47213134919}e^{9} - \frac{200216292579905}{47213134919}e^{8} + \frac{90947668785518}{47213134919}e^{7} + \frac{382046229901293}{47213134919}e^{6} - \frac{159145337409430}{47213134919}e^{5} - \frac{325660912934912}{47213134919}e^{4} + \frac{65435687626744}{47213134919}e^{3} + \frac{95009291053093}{47213134919}e^{2} + \frac{14246473008056}{47213134919}e + \frac{135548429850}{47213134919}$ |
| 31 | $[31, 31, -w - 6]$ | $\phantom{-}\frac{76065586561}{47213134919}e^{15} - \frac{401632221934}{47213134919}e^{14} - \frac{1314197447615}{47213134919}e^{13} + \frac{9222941110296}{47213134919}e^{12} + \frac{5573407846333}{47213134919}e^{11} - \frac{78558978455970}{47213134919}e^{10} + \frac{12205937674215}{47213134919}e^{9} + \frac{311294104339802}{47213134919}e^{8} - \frac{119600138547569}{47213134919}e^{7} - \frac{596929317969770}{47213134919}e^{6} + \frac{209086394273641}{47213134919}e^{5} + \frac{512516794293966}{47213134919}e^{4} - \frac{71780045629763}{47213134919}e^{3} - \frac{150370327101456}{47213134919}e^{2} - \frac{29765412528988}{47213134919}e - \frac{1337843693710}{47213134919}$ |
| 31 | $[31, 31, w - 6]$ | $\phantom{-}\frac{37635139276}{47213134919}e^{15} - \frac{200362037351}{47213134919}e^{14} - \frac{652147858449}{47213134919}e^{13} + \frac{4619007684173}{47213134919}e^{12} + \frac{2804760268596}{47213134919}e^{11} - \frac{39582066075595}{47213134919}e^{10} + \frac{5599756014037}{47213134919}e^{9} + \frac{158358895957090}{47213134919}e^{8} - \frac{57216188345116}{47213134919}e^{7} - \frac{308273820275083}{47213134919}e^{6} + \frac{99074978008850}{47213134919}e^{5} + \frac{270925389688948}{47213134919}e^{4} - \frac{31102414147146}{47213134919}e^{3} - \frac{82485578287894}{47213134919}e^{2} - \frac{16496446651129}{47213134919}e - \frac{317896683133}{47213134919}$ |
| 37 | $[37, 37, -21w - 172]$ | $-\frac{6054986534}{47213134919}e^{15} + \frac{28088289539}{47213134919}e^{14} + \frac{103446013134}{47213134919}e^{13} - \frac{615237989881}{47213134919}e^{12} - \frac{401868334142}{47213134919}e^{11} + \frac{4849089781352}{47213134919}e^{10} - \frac{1508607208390}{47213134919}e^{9} - \frac{16756292698457}{47213134919}e^{8} + \frac{12675872415600}{47213134919}e^{7} + \frac{24717091651121}{47213134919}e^{6} - \frac{25402556266026}{47213134919}e^{5} - \frac{11271248935935}{47213134919}e^{4} + \frac{15834517570244}{47213134919}e^{3} - \frac{1447593625462}{47213134919}e^{2} - \frac{1253134778054}{47213134919}e + \frac{431798934157}{47213134919}$ |
| 37 | $[37, 37, -21w + 172]$ | $-\frac{13618942526}{47213134919}e^{15} + \frac{75990617036}{47213134919}e^{14} + \frac{232570644896}{47213134919}e^{13} - \frac{1764491358306}{47213134919}e^{12} - \frac{954776842594}{47213134919}e^{11} + \frac{15284396089765}{47213134919}e^{10} - \frac{2312679265764}{47213134919}e^{9} - \frac{62158551060135}{47213134919}e^{8} + \frac{20545008979170}{47213134919}e^{7} + \frac{123961864698784}{47213134919}e^{6} - \frac{32217647532596}{47213134919}e^{5} - \frac{112672087047673}{47213134919}e^{4} + \frac{5113383823376}{47213134919}e^{3} + \frac{35876676859334}{47213134919}e^{2} + \frac{8385221146764}{47213134919}e + \frac{142632513882}{47213134919}$ |
| 43 | $[43, 43, 2w - 15]$ | $\phantom{-}\frac{115201477774}{47213134919}e^{15} - \frac{620122055082}{47213134919}e^{14} - \frac{1962365456445}{47213134919}e^{13} + \frac{14251343507156}{47213134919}e^{12} + \frac{7833351889139}{47213134919}e^{11} - \frac{121532139385036}{47213134919}e^{10} + \frac{23217465095969}{47213134919}e^{9} + \frac{482462724684008}{47213134919}e^{8} - \frac{197210457682236}{47213134919}e^{7} - \frac{927849412493338}{47213134919}e^{6} + \frac{339800197219504}{47213134919}e^{5} + \frac{800151974903944}{47213134919}e^{4} - \frac{119553350882566}{47213134919}e^{3} - \frac{236526951676606}{47213134919}e^{2} - \frac{45117746354169}{47213134919}e - \frac{1532333826698}{47213134919}$ |
| 43 | $[43, 43, 2w + 15]$ | $-\frac{19622317121}{47213134919}e^{15} + \frac{100177857472}{47213134919}e^{14} + \frac{346058639034}{47213134919}e^{13} - \frac{2292571211218}{47213134919}e^{12} - \frac{1592398134683}{47213134919}e^{11} + \frac{19422559494642}{47213134919}e^{10} - \frac{1924142952499}{47213134919}e^{9} - \frac{76293323332806}{47213134919}e^{8} + \frac{26599486249595}{47213134919}e^{7} + \frac{144259907936414}{47213134919}e^{6} - \frac{47700330582838}{47213134919}e^{5} - \frac{121179984121832}{47213134919}e^{4} + \frac{15785885147515}{47213134919}e^{3} + \frac{34395673132904}{47213134919}e^{2} + \frac{7215570050973}{47213134919}e + \frac{305232401058}{47213134919}$ |
| 67 | $[67, 67, -w]$ | $-\frac{136890760531}{47213134919}e^{15} + \frac{724304525595}{47213134919}e^{14} + \frac{2337795682297}{47213134919}e^{13} - \frac{16579119164515}{47213134919}e^{12} - \frac{9388982987140}{47213134919}e^{11} + \frac{140523407171709}{47213134919}e^{10} - \frac{27603716328408}{47213134919}e^{9} - \frac{552589071388122}{47213134919}e^{8} + \frac{238591229958445}{47213134919}e^{7} + \frac{1047509427813183}{47213134919}e^{6} - \frac{423854930836298}{47213134919}e^{5} - \frac{884915441180018}{47213134919}e^{4} + \frac{173066052576399}{47213134919}e^{3} + \frac{255574101271874}{47213134919}e^{2} + \frac{39663844666994}{47213134919}e + \frac{668574520367}{47213134919}$ |
| 73 | $[73, 73, -3w - 26]$ | $-\frac{48065318465}{47213134919}e^{15} + \frac{260996004445}{47213134919}e^{14} + \frac{806052189207}{47213134919}e^{13} - \frac{5983171971314}{47213134919}e^{12} - \frac{2975349395551}{47213134919}e^{11} + \frac{50827825234353}{47213134919}e^{10} - \frac{12198838581288}{47213134919}e^{9} - \frac{200571598533435}{47213134919}e^{8} + \frac{92323982222427}{47213134919}e^{7} + \frac{382252624196107}{47213134919}e^{6} - \frac{161151823304792}{47213134919}e^{5} - \frac{325402943779590}{47213134919}e^{4} + \frac{66255881641666}{47213134919}e^{3} + \frac{94558170719435}{47213134919}e^{2} + \frac{14310480249351}{47213134919}e + \frac{587289658066}{47213134919}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2, 2, -27w + 221]$ | $1$ |