Base field 5.5.36497.1
Generator \(w\), with minimal polynomial \(x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2]$ |
| Level: | $[47, 47, 2w^{4} - 3w^{3} - 6w^{2} + 6w + 2]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $8$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 20x^{6} + 117x^{4} - 208x^{2} + 64\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w^{4} - 2w^{3} - 3w^{2} + 4w + 1]$ | $\phantom{-}e$ |
| 13 | $[13, 13, w^{3} - 2w^{2} - 2w + 2]$ | $-\frac{1}{64}e^{7} + \frac{7}{16}e^{5} - \frac{213}{64}e^{3} + \frac{47}{8}e$ |
| 23 | $[23, 23, 2w^{4} - 3w^{3} - 6w^{2} + 5w + 1]$ | $\phantom{-}\frac{3}{16}e^{6} - \frac{13}{4}e^{4} + \frac{223}{16}e^{2} - \frac{17}{2}$ |
| 25 | $[25, 5, -w^{2} + 2w + 2]$ | $-\frac{7}{64}e^{7} + \frac{33}{16}e^{5} - \frac{723}{64}e^{3} + \frac{145}{8}e$ |
| 29 | $[29, 29, -w^{4} + w^{3} + 4w^{2} - 3w - 1]$ | $\phantom{-}\frac{7}{64}e^{7} - \frac{33}{16}e^{5} + \frac{659}{64}e^{3} - \frac{89}{8}e$ |
| 31 | $[31, 31, w^{4} - w^{3} - 5w^{2} + 2w + 4]$ | $\phantom{-}\frac{1}{64}e^{7} - \frac{7}{16}e^{5} + \frac{213}{64}e^{3} - \frac{47}{8}e$ |
| 32 | $[32, 2, 2]$ | $-\frac{1}{16}e^{6} + \frac{3}{4}e^{4} - \frac{5}{16}e^{2} - \frac{3}{2}$ |
| 37 | $[37, 37, w^{4} - 2w^{3} - 2w^{2} + 4w + 1]$ | $-2e$ |
| 47 | $[47, 47, w^{4} - w^{3} - 5w^{2} + 2w + 3]$ | $\phantom{-}\frac{3}{32}e^{7} - \frac{13}{8}e^{5} + \frac{223}{32}e^{3} - \frac{17}{4}e$ |
| 47 | $[47, 47, 2w^{4} - 3w^{3} - 6w^{2} + 6w + 2]$ | $\phantom{-}1$ |
| 49 | $[49, 7, -w^{4} + 2w^{3} + 4w^{2} - 6w - 2]$ | $-\frac{3}{16}e^{7} + \frac{13}{4}e^{5} - \frac{223}{16}e^{3} + \frac{21}{2}e$ |
| 53 | $[53, 53, -2w^{4} + 3w^{3} + 7w^{2} - 7w - 2]$ | $\phantom{-}\frac{1}{32}e^{7} - \frac{7}{8}e^{5} + \frac{245}{32}e^{3} - \frac{75}{4}e$ |
| 59 | $[59, 59, -2w^{4} + 3w^{3} + 6w^{2} - 5w - 2]$ | $\phantom{-}\frac{1}{16}e^{6} - \frac{7}{4}e^{4} + \frac{181}{16}e^{2} - \frac{19}{2}$ |
| 67 | $[67, 67, -w^{4} + 3w^{3} + 2w^{2} - 7w]$ | $-\frac{7}{16}e^{6} + \frac{29}{4}e^{4} - \frac{467}{16}e^{2} + \frac{45}{2}$ |
| 67 | $[67, 67, -w^{4} + 3w^{3} + w^{2} - 7w + 1]$ | $\phantom{-}\frac{5}{64}e^{7} - \frac{19}{16}e^{5} + \frac{169}{64}e^{3} + \frac{61}{8}e$ |
| 71 | $[71, 71, w^{4} - 2w^{3} - 4w^{2} + 3w + 3]$ | $-\frac{3}{16}e^{6} + \frac{13}{4}e^{4} - \frac{223}{16}e^{2} + \frac{25}{2}$ |
| 71 | $[71, 71, -w^{2} + 5]$ | $-\frac{1}{8}e^{7} + \frac{5}{2}e^{5} - \frac{109}{8}e^{3} + 16e$ |
| 79 | $[79, 79, 2w^{4} - 3w^{3} - 5w^{2} + 3w + 1]$ | $-e^{2} + 4$ |
| 81 | $[81, 3, -w^{4} + 3w^{3} + 3w^{2} - 8w - 2]$ | $\phantom{-}\frac{9}{32}e^{7} - \frac{39}{8}e^{5} + \frac{669}{32}e^{3} - \frac{67}{4}e$ |
| 83 | $[83, 83, -3w^{4} + 3w^{3} + 10w^{2} - 2w - 2]$ | $-e^{2} + 8$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $47$ | $[47, 47, 2w^{4} - 3w^{3} - 6w^{2} + 6w + 2]$ | $-1$ |