Base field 4.4.12197.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 5x^{2} + 3x + 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, w^{3} - w^{2} - 4w]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $16$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} - 3x^{8} - 27x^{7} + 72x^{6} + 211x^{5} - 492x^{4} - 389x^{3} + 907x^{2} + 112x - 324\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, w + 1]$ | $\phantom{-}\frac{220337}{4198972}e^{8} - \frac{356171}{4198972}e^{7} - \frac{6350613}{4198972}e^{6} + \frac{1696899}{1049743}e^{5} + \frac{53687529}{4198972}e^{4} - \frac{14514973}{2099486}e^{3} - \frac{109339293}{4198972}e^{2} + \frac{29084347}{4198972}e + \frac{17171347}{2099486}$ |
| 5 | $[5, 5, -w + 2]$ | $\phantom{-}e$ |
| 11 | $[11, 11, w^{3} - w^{2} - 4w + 2]$ | $\phantom{-}\frac{5291}{365128}e^{8} - \frac{7955}{365128}e^{7} - \frac{149655}{365128}e^{6} + \frac{64453}{182564}e^{5} + \frac{1232709}{365128}e^{4} - \frac{104725}{182564}e^{3} - \frac{2374195}{365128}e^{2} - \frac{1003549}{365128}e + \frac{356075}{182564}$ |
| 13 | $[13, 13, w^{3} - w^{2} - 4w]$ | $-1$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}\frac{455217}{8397944}e^{8} - \frac{455893}{8397944}e^{7} - \frac{13848901}{8397944}e^{6} + \frac{3392213}{4198972}e^{5} + \frac{125401011}{8397944}e^{4} - \frac{2829247}{4198972}e^{3} - \frac{293975065}{8397944}e^{2} - \frac{27968267}{8397944}e + \frac{60341341}{4198972}$ |
| 17 | $[17, 17, -w^{2} + w + 3]$ | $-\frac{213783}{4198972}e^{8} + \frac{323337}{4198972}e^{7} + \frac{6178139}{4198972}e^{6} - \frac{1464720}{1049743}e^{5} - \frac{51848055}{4198972}e^{4} + \frac{9997123}{2099486}e^{3} + \frac{99833751}{4198972}e^{2} + \frac{107911}{4198972}e - \frac{13240545}{2099486}$ |
| 19 | $[19, 19, w^{3} - 5w]$ | $\phantom{-}\frac{42791}{2099486}e^{8} - \frac{31781}{2099486}e^{7} - \frac{1190789}{2099486}e^{6} + \frac{42728}{1049743}e^{5} + \frac{9101251}{2099486}e^{4} + \frac{2863405}{1049743}e^{3} - \frac{10591827}{2099486}e^{2} - \frac{17073767}{2099486}e - \frac{1203424}{1049743}$ |
| 19 | $[19, 19, -w + 3]$ | $\phantom{-}\frac{9286}{1049743}e^{8} - \frac{90799}{2099486}e^{7} - \frac{224508}{1049743}e^{6} + \frac{2175213}{2099486}e^{5} + \frac{3058875}{2099486}e^{4} - \frac{6953478}{1049743}e^{3} - \frac{1930634}{1049743}e^{2} + \frac{14232867}{2099486}e - \frac{2353585}{1049743}$ |
| 23 | $[23, 23, 2w^{3} - 2w^{2} - 9w + 4]$ | $\phantom{-}\frac{540575}{4198972}e^{8} - \frac{733599}{4198972}e^{7} - \frac{15838899}{4198972}e^{6} + \frac{6477447}{2099486}e^{5} + \frac{136140869}{4198972}e^{4} - \frac{20955333}{2099486}e^{3} - \frac{287733367}{4198972}e^{2} + \frac{7615515}{4198972}e + \frac{61872907}{2099486}$ |
| 23 | $[23, 23, w^{2} - 2]$ | $\phantom{-}\frac{3277}{2099486}e^{8} - \frac{16417}{2099486}e^{7} - \frac{86237}{2099486}e^{6} + \frac{232179}{1049743}e^{5} + \frac{919737}{2099486}e^{4} - \frac{2258925}{1049743}e^{3} - \frac{4752771}{2099486}e^{2} + \frac{14596129}{2099486}e + \frac{1965401}{1049743}$ |
| 25 | $[25, 5, w^{2} - 3]$ | $\phantom{-}\frac{276325}{2099486}e^{8} - \frac{184268}{1049743}e^{7} - \frac{8173151}{2099486}e^{6} + \frac{6627913}{2099486}e^{5} + \frac{35962968}{1049743}e^{4} - \frac{12109781}{1049743}e^{3} - \frac{166222373}{2099486}e^{2} + \frac{11100308}{1049743}e + \frac{39369898}{1049743}$ |
| 37 | $[37, 37, -w^{3} + 2w^{2} + 4w - 6]$ | $-\frac{174269}{2099486}e^{8} + \frac{307973}{2099486}e^{7} + \frac{5073587}{2099486}e^{6} - \frac{3118891}{1049743}e^{5} - \frac{43666541}{2099486}e^{4} + \frac{15119453}{1049743}e^{3} + \frac{93994695}{2099486}e^{2} - \frac{29462499}{2099486}e - \frac{16409370}{1049743}$ |
| 37 | $[37, 37, -w^{3} + w^{2} + 6w - 4]$ | $-\frac{167571}{8397944}e^{8} + \frac{412803}{8397944}e^{7} + \frac{4649383}{8397944}e^{6} - \frac{4732925}{4198972}e^{5} - \frac{37029653}{8397944}e^{4} + \frac{29834653}{4198972}e^{3} + \frac{58160699}{8397944}e^{2} - \frac{97284739}{8397944}e + \frac{6284381}{4198972}$ |
| 41 | $[41, 41, w^{2} - w - 5]$ | $\phantom{-}\frac{166593}{2099486}e^{8} - \frac{149014}{1049743}e^{7} - \frac{4776767}{2099486}e^{6} + \frac{5855137}{2099486}e^{5} + \frac{20200457}{1049743}e^{4} - \frac{13407619}{1049743}e^{3} - \frac{85008765}{2099486}e^{2} + \frac{13432961}{1049743}e + \frac{20639884}{1049743}$ |
| 47 | $[47, 47, w^{3} - 6w + 1]$ | $\phantom{-}\frac{88268}{1049743}e^{8} - \frac{240207}{2099486}e^{7} - \frac{2615634}{1049743}e^{6} + \frac{4248433}{2099486}e^{5} + \frac{46891447}{2099486}e^{4} - \frac{7208203}{1049743}e^{3} - \frac{58589363}{1049743}e^{2} + \frac{8495333}{2099486}e + \frac{30865338}{1049743}$ |
| 61 | $[61, 61, -w^{3} + w^{2} + 6w - 2]$ | $-\frac{141451}{1049743}e^{8} + \frac{189691}{1049743}e^{7} + \frac{4127307}{1049743}e^{6} - \frac{3329711}{1049743}e^{5} - \frac{34969519}{1049743}e^{4} + \frac{10129441}{1049743}e^{3} + \frac{68483295}{1049743}e^{2} + \frac{2280671}{1049743}e - \frac{17393178}{1049743}$ |
| 67 | $[67, 67, 2w^{3} - w^{2} - 9w + 2]$ | $\phantom{-}\frac{20839}{2099486}e^{8} - \frac{23033}{2099486}e^{7} - \frac{606697}{2099486}e^{6} + \frac{92932}{1049743}e^{5} + \frac{6283781}{2099486}e^{4} + \frac{957772}{1049743}e^{3} - \frac{27045299}{2099486}e^{2} - \frac{10933271}{2099486}e + \frac{14201869}{1049743}$ |
| 67 | $[67, 67, w^{3} - 7w + 3]$ | $\phantom{-}\frac{134755}{4198972}e^{8} - \frac{292609}{4198972}e^{7} - \frac{3969035}{4198972}e^{6} + \frac{1654171}{1049743}e^{5} + \frac{35485027}{4198972}e^{4} - \frac{20241783}{2099486}e^{3} - \frac{92354611}{4198972}e^{2} + \frac{67430853}{4198972}e + \frac{34274597}{2099486}$ |
| 73 | $[73, 73, 2w^{3} - w^{2} - 9w]$ | $-\frac{59297}{2099486}e^{8} - \frac{905}{1049743}e^{7} + \frac{1964715}{2099486}e^{6} + \frac{852331}{2099486}e^{5} - \frac{10112118}{1049743}e^{4} - \frac{6403126}{1049743}e^{3} + \frac{61654713}{2099486}e^{2} + \frac{12521819}{1049743}e - \frac{14453561}{1049743}$ |
| 81 | $[81, 3, -3]$ | $\phantom{-}\frac{395719}{4198972}e^{8} - \frac{649859}{4198972}e^{7} - \frac{11401595}{4198972}e^{6} + \frac{6369553}{2099486}e^{5} + \frac{95987245}{4198972}e^{4} - \frac{28686565}{2099486}e^{3} - \frac{195526479}{4198972}e^{2} + \frac{52842191}{4198972}e + \frac{35465845}{2099486}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, w^{3} - w^{2} - 4w]$ | $1$ |