Base field 4.4.16609.1
Generator \(w\), with minimal polynomial \(x^{4} - 7x^{2} - x + 9\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16, 2, 2]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $23$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - x^{7} - 16x^{6} + 17x^{5} + 70x^{4} - 76x^{3} - 62x^{2} + 58x + 8\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w^{2} + w + 4]$ | $\phantom{-}1$ |
| 3 | $[3, 3, -w^{2} + w + 3]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w^{2} - 4]$ | $-\frac{1}{14}e^{7} + \frac{1}{14}e^{6} + \frac{9}{7}e^{5} - \frac{17}{14}e^{4} - \frac{46}{7}e^{3} + \frac{37}{7}e^{2} + \frac{52}{7}e - \frac{19}{7}$ |
| 8 | $[8, 2, w^{3} - w^{2} - 4w + 5]$ | $\phantom{-}1$ |
| 17 | $[17, 17, -w^{2} + w + 5]$ | $\phantom{-}\frac{5}{14}e^{7} + \frac{1}{14}e^{6} - \frac{38}{7}e^{5} - \frac{9}{14}e^{4} + \frac{157}{7}e^{3} + \frac{11}{7}e^{2} - \frac{132}{7}e - \frac{19}{7}$ |
| 27 | $[27, 3, -w^{3} + 4w - 2]$ | $-\frac{2}{7}e^{6} + \frac{29}{7}e^{4} - \frac{5}{7}e^{3} - 16e^{2} + \frac{36}{7}e + \frac{76}{7}$ |
| 29 | $[29, 29, -w^{2} + w + 1]$ | $-\frac{1}{14}e^{7} - \frac{3}{14}e^{6} + \frac{9}{7}e^{5} + \frac{41}{14}e^{4} - \frac{51}{7}e^{3} - \frac{75}{7}e^{2} + \frac{81}{7}e + \frac{57}{7}$ |
| 29 | $[29, 29, -w^{3} + 4w + 2]$ | $-\frac{1}{2}e^{7} + \frac{1}{14}e^{6} + 8e^{5} - \frac{25}{14}e^{4} - \frac{242}{7}e^{3} + 9e^{2} + \frac{187}{7}e - \frac{19}{7}$ |
| 37 | $[37, 37, -2w^{3} + 3w^{2} + 9w - 13]$ | $-\frac{1}{2}e^{7} + \frac{1}{14}e^{6} + 8e^{5} - \frac{25}{14}e^{4} - \frac{242}{7}e^{3} + 9e^{2} + \frac{201}{7}e - \frac{19}{7}$ |
| 41 | $[41, 41, w^{3} - w^{2} - 5w + 2]$ | $-\frac{1}{14}e^{7} + \frac{3}{14}e^{6} + \frac{9}{7}e^{5} - \frac{39}{14}e^{4} - \frac{47}{7}e^{3} + \frac{65}{7}e^{2} + \frac{62}{7}e - \frac{43}{7}$ |
| 43 | $[43, 43, -w^{3} + w^{2} + 3w - 4]$ | $\phantom{-}\frac{4}{7}e^{7} + \frac{1}{7}e^{6} - \frac{65}{7}e^{5} - \frac{15}{7}e^{4} + \frac{293}{7}e^{3} + \frac{68}{7}e^{2} - \frac{268}{7}e - \frac{80}{7}$ |
| 61 | $[61, 61, w^{2} - 2w - 2]$ | $\phantom{-}\frac{1}{14}e^{7} - \frac{1}{14}e^{6} - \frac{9}{7}e^{5} + \frac{3}{14}e^{4} + \frac{39}{7}e^{3} + \frac{19}{7}e^{2} - \frac{3}{7}e - \frac{23}{7}$ |
| 61 | $[61, 61, 2w^{2} - w - 8]$ | $-\frac{1}{2}e^{7} + \frac{3}{14}e^{6} + 8e^{5} - \frac{47}{14}e^{4} - \frac{243}{7}e^{3} + 12e^{2} + \frac{211}{7}e - \frac{15}{7}$ |
| 67 | $[67, 67, -4w^{3} + 5w^{2} + 18w - 22]$ | $-\frac{2}{7}e^{6} + \frac{29}{7}e^{4} - \frac{5}{7}e^{3} - 13e^{2} + \frac{43}{7}e - \frac{8}{7}$ |
| 73 | $[73, 73, -w^{2} - 2w + 2]$ | $-\frac{5}{14}e^{7} + \frac{3}{14}e^{6} + \frac{38}{7}e^{5} - \frac{9}{2}e^{4} - \frac{152}{7}e^{3} + \frac{164}{7}e^{2} + \frac{89}{7}e - \frac{71}{7}$ |
| 79 | $[79, 79, w^{2} - 2w - 4]$ | $\phantom{-}\frac{3}{7}e^{7} + \frac{1}{7}e^{6} - \frac{47}{7}e^{5} - e^{4} + \frac{209}{7}e^{3} - \frac{5}{7}e^{2} - \frac{223}{7}e + \frac{60}{7}$ |
| 89 | $[89, 89, w^{2} + w - 5]$ | $-\frac{1}{2}e^{7} - \frac{3}{14}e^{6} + 8e^{5} + \frac{33}{14}e^{4} - \frac{247}{7}e^{3} - 4e^{2} + \frac{216}{7}e - \frac{27}{7}$ |
| 89 | $[89, 89, 2w^{3} - 2w^{2} - 9w + 8]$ | $-\frac{1}{14}e^{7} + \frac{5}{14}e^{6} + \frac{9}{7}e^{5} - \frac{61}{14}e^{4} - \frac{41}{7}e^{3} + \frac{79}{7}e^{2} + \frac{23}{7}e - \frac{11}{7}$ |
| 89 | $[89, 89, w^{3} - 5w - 5]$ | $\phantom{-}\frac{11}{14}e^{7} - \frac{1}{14}e^{6} - \frac{85}{7}e^{5} + \frac{5}{2}e^{4} + \frac{354}{7}e^{3} - \frac{99}{7}e^{2} - \frac{277}{7}e + \frac{33}{7}$ |
| 89 | $[89, 89, -w^{3} + 2w^{2} + 4w - 4]$ | $\phantom{-}\frac{5}{14}e^{7} - \frac{1}{14}e^{6} - \frac{38}{7}e^{5} + \frac{27}{14}e^{4} + \frac{151}{7}e^{3} - \frac{66}{7}e^{2} - \frac{93}{7}e + \frac{5}{7}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2, 2, -w^{2} + w + 4]$ | $-1$ |
| $8$ | $[8, 2, w^{3} - w^{2} - 4w + 5]$ | $-1$ |