Base field 4.4.16225.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 13x^{2} + 6x + 36\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16, 2, 2]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $22$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} + x^{8} - 45x^{7} - 21x^{6} + 674x^{5} - 70x^{4} - 3666x^{3} + 2082x^{2} + 3232x - 160\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, \frac{1}{3}w^{3} - \frac{4}{3}w^{2} - \frac{4}{3}w + 6]$ | $-1$ |
| 4 | $[4, 2, -\frac{1}{3}w^{3} - \frac{2}{3}w^{2} + \frac{10}{3}w + 7]$ | $-1$ |
| 9 | $[9, 3, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{7}{2}w - 9]$ | $\phantom{-}e$ |
| 9 | $[9, 3, \frac{1}{3}w^{3} + \frac{2}{3}w^{2} - \frac{7}{3}w - 5]$ | $\phantom{-}e$ |
| 11 | $[11, 11, -\frac{1}{6}w^{3} + \frac{1}{6}w^{2} + \frac{1}{6}w]$ | $\phantom{-}\frac{20877}{224432}e^{8} + \frac{63945}{224432}e^{7} - \frac{712669}{224432}e^{6} - \frac{1885261}{224432}e^{5} + \frac{3466431}{112216}e^{4} + \frac{6700957}{112216}e^{3} - \frac{10423657}{112216}e^{2} - \frac{8178415}{112216}e + \frac{174323}{28054}$ |
| 19 | $[19, 19, w + 1]$ | $\phantom{-}\frac{2651}{112216}e^{8} + \frac{22423}{112216}e^{7} - \frac{89163}{112216}e^{6} - \frac{731979}{112216}e^{5} + \frac{522945}{56108}e^{4} + \frac{2982203}{56108}e^{3} - \frac{2939739}{56108}e^{2} - \frac{3329857}{56108}e + \frac{52825}{14027}$ |
| 19 | $[19, 19, \frac{1}{6}w^{3} - \frac{1}{6}w^{2} - \frac{13}{6}w + 2]$ | $\phantom{-}\frac{2651}{112216}e^{8} + \frac{22423}{112216}e^{7} - \frac{89163}{112216}e^{6} - \frac{731979}{112216}e^{5} + \frac{522945}{56108}e^{4} + \frac{2982203}{56108}e^{3} - \frac{2939739}{56108}e^{2} - \frac{3329857}{56108}e + \frac{52825}{14027}$ |
| 25 | $[25, 5, -\frac{1}{3}w^{3} + \frac{1}{3}w^{2} + \frac{7}{3}w - 1]$ | $\phantom{-}\frac{28715}{224432}e^{8} + \frac{110431}{224432}e^{7} - \frac{973835}{224432}e^{6} - \frac{3351259}{224432}e^{5} + \frac{4827473}{112216}e^{4} + \frac{12404723}{112216}e^{3} - \frac{16296911}{112216}e^{2} - \frac{14501665}{112216}e + \frac{308749}{28054}$ |
| 29 | $[29, 29, -\frac{1}{6}w^{3} + \frac{1}{6}w^{2} + \frac{13}{6}w]$ | $\phantom{-}\frac{12483}{224432}e^{8} + \frac{41879}{224432}e^{7} - \frac{428739}{224432}e^{6} - \frac{1246099}{224432}e^{5} + \frac{2157537}{112216}e^{4} + \frac{4447187}{112216}e^{3} - \frac{7332919}{112216}e^{2} - \frac{4641409}{112216}e + \frac{317845}{28054}$ |
| 29 | $[29, 29, w - 1]$ | $\phantom{-}\frac{12483}{224432}e^{8} + \frac{41879}{224432}e^{7} - \frac{428739}{224432}e^{6} - \frac{1246099}{224432}e^{5} + \frac{2157537}{112216}e^{4} + \frac{4447187}{112216}e^{3} - \frac{7332919}{112216}e^{2} - \frac{4641409}{112216}e + \frac{317845}{28054}$ |
| 31 | $[31, 31, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - \frac{10}{3}w + 3]$ | $-\frac{11057}{112216}e^{8} - \frac{31045}{112216}e^{7} + \frac{372777}{112216}e^{6} + \frac{902817}{112216}e^{5} - \frac{1735451}{56108}e^{4} - \frac{3197329}{56108}e^{3} + \frac{4490221}{56108}e^{2} + \frac{4783611}{56108}e + \frac{25249}{14027}$ |
| 31 | $[31, 31, \frac{1}{6}w^{3} - \frac{1}{6}w^{2} - \frac{1}{6}w + 2]$ | $-\frac{11057}{112216}e^{8} - \frac{31045}{112216}e^{7} + \frac{372777}{112216}e^{6} + \frac{902817}{112216}e^{5} - \frac{1735451}{56108}e^{4} - \frac{3197329}{56108}e^{3} + \frac{4490221}{56108}e^{2} + \frac{4783611}{56108}e + \frac{25249}{14027}$ |
| 41 | $[41, 41, -\frac{1}{2}w^{3} - \frac{1}{2}w^{2} + \frac{9}{2}w + 5]$ | $-\frac{17639}{224432}e^{8} - \frac{63267}{224432}e^{7} + \frac{592207}{224432}e^{6} + \frac{1899471}{224432}e^{5} - \frac{2842529}{112216}e^{4} - \frac{6936663}{112216}e^{3} + \frac{8938547}{112216}e^{2} + \frac{8574653}{112216}e - \frac{300507}{28054}$ |
| 41 | $[41, 41, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{5}{2}w - 8]$ | $-\frac{17639}{224432}e^{8} - \frac{63267}{224432}e^{7} + \frac{592207}{224432}e^{6} + \frac{1899471}{224432}e^{5} - \frac{2842529}{112216}e^{4} - \frac{6936663}{112216}e^{3} + \frac{8938547}{112216}e^{2} + \frac{8574653}{112216}e - \frac{300507}{28054}$ |
| 59 | $[59, 59, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - \frac{10}{3}w - 1]$ | $\phantom{-}\frac{17059}{224432}e^{8} + \frac{114575}{224432}e^{7} - \frac{570075}{224432}e^{6} - \frac{3662955}{224432}e^{5} + \frac{3087453}{112216}e^{4} + \frac{14463947}{112216}e^{3} - \frac{14801575}{112216}e^{2} - \frac{15302969}{112216}e + \frac{279235}{28054}$ |
| 59 | $[59, 59, -\frac{1}{3}w^{3} + \frac{4}{3}w^{2} + \frac{7}{3}w - 7]$ | $\phantom{-}\frac{17059}{224432}e^{8} + \frac{114575}{224432}e^{7} - \frac{570075}{224432}e^{6} - \frac{3662955}{224432}e^{5} + \frac{3087453}{112216}e^{4} + \frac{14463947}{112216}e^{3} - \frac{14801575}{112216}e^{2} - \frac{15302969}{112216}e + \frac{279235}{28054}$ |
| 59 | $[59, 59, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{5}{2}w + 2]$ | $\phantom{-}\frac{1619}{56108}e^{8} + \frac{339}{56108}e^{7} - \frac{60231}{56108}e^{6} + \frac{7105}{56108}e^{5} + \frac{311951}{28054}e^{4} - \frac{117853}{28054}e^{3} - \frac{798663}{28054}e^{2} + \frac{142011}{28054}e + \frac{182410}{14027}$ |
| 79 | $[79, 79, w^{2} - 11]$ | $\phantom{-}\frac{1003}{17264}e^{8} + \frac{1343}{17264}e^{7} - \frac{34731}{17264}e^{6} - \frac{32251}{17264}e^{5} + \frac{161953}{8632}e^{4} + \frac{85339}{8632}e^{3} - \frac{350031}{8632}e^{2} - \frac{272185}{8632}e + \frac{28965}{2158}$ |
| 79 | $[79, 79, \frac{1}{6}w^{3} - \frac{7}{6}w^{2} - \frac{7}{6}w + 3]$ | $\phantom{-}\frac{1003}{17264}e^{8} + \frac{1343}{17264}e^{7} - \frac{34731}{17264}e^{6} - \frac{32251}{17264}e^{5} + \frac{161953}{8632}e^{4} + \frac{85339}{8632}e^{3} - \frac{350031}{8632}e^{2} - \frac{272185}{8632}e + \frac{28965}{2158}$ |
| 89 | $[89, 89, -\frac{1}{6}w^{3} + \frac{1}{6}w^{2} + \frac{19}{6}w - 5]$ | $\phantom{-}\frac{14667}{112216}e^{8} + \frac{63823}{112216}e^{7} - \frac{483443}{112216}e^{6} - \frac{1968587}{112216}e^{5} + \frac{2288941}{56108}e^{4} + \frac{7542123}{56108}e^{3} - \frac{7434111}{56108}e^{2} - \frac{9997669}{56108}e - \frac{86325}{14027}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, \frac{1}{3}w^{3} - \frac{4}{3}w^{2} - \frac{4}{3}w + 6]$ | $1$ |
| $4$ | $[4, 2, -\frac{1}{3}w^{3} - \frac{2}{3}w^{2} + \frac{10}{3}w + 7]$ | $1$ |