Base field 3.3.169.1
Generator \(w\), with minimal polynomial \(x^{3} - x^{2} - 4x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[47,47,2w^{2} - 4w - 7]$ |
| Dimension: | $4$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $5$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{4} + 2x^{3} - 11x^{2} - 19x + 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, -w^{2} + 2w + 3]$ | $-\frac{2}{7}e^{3} - \frac{1}{7}e^{2} + \frac{20}{7}e + \frac{22}{7}$ |
| 5 | $[5, 5, -w^{2} + w + 2]$ | $\phantom{-}e$ |
| 5 | $[5, 5, -w + 1]$ | $\phantom{-}\frac{1}{7}e^{3} - \frac{3}{7}e^{2} - \frac{10}{7}e + \frac{17}{7}$ |
| 8 | $[8, 2, 2]$ | $\phantom{-}\frac{3}{7}e^{3} + \frac{5}{7}e^{2} - \frac{30}{7}e - \frac{33}{7}$ |
| 13 | $[13, 13, -w^{2} + 3]$ | $-\frac{3}{7}e^{3} - \frac{5}{7}e^{2} + \frac{30}{7}e + \frac{33}{7}$ |
| 27 | $[27, 3, 3]$ | $\phantom{-}\frac{5}{7}e^{3} + \frac{6}{7}e^{2} - \frac{50}{7}e - \frac{41}{7}$ |
| 31 | $[31, 31, -2w^{2} + 3w + 3]$ | $-\frac{2}{7}e^{3} + \frac{6}{7}e^{2} + \frac{20}{7}e - \frac{48}{7}$ |
| 31 | $[31, 31, -w^{2} + 5]$ | $\phantom{-}\frac{5}{7}e^{3} + \frac{6}{7}e^{2} - \frac{71}{7}e - \frac{48}{7}$ |
| 31 | $[31, 31, -w^{2} + 3w + 4]$ | $-\frac{1}{7}e^{3} - \frac{4}{7}e^{2} + \frac{3}{7}e + \frac{60}{7}$ |
| 47 | $[47, 47, 2w - 3]$ | $\phantom{-}e^{3} + e^{2} - 13e - 11$ |
| 47 | $[47, 47, 2w^{2} - 4w - 7]$ | $\phantom{-}1$ |
| 47 | $[47, 47, 2w^{2} - 2w - 3]$ | $\phantom{-}2$ |
| 53 | $[53, 53, 3w^{2} - 4w - 8]$ | $\phantom{-}e^{3} + e^{2} - 12e - 11$ |
| 53 | $[53, 53, 4w^{2} - 6w - 11]$ | $\phantom{-}\frac{2}{7}e^{3} - \frac{6}{7}e^{2} - \frac{27}{7}e + \frac{48}{7}$ |
| 53 | $[53, 53, 3w^{2} - 5w - 6]$ | $-\frac{3}{7}e^{3} - \frac{5}{7}e^{2} + \frac{30}{7}e - \frac{9}{7}$ |
| 73 | $[73, 73, w^{2} - 4w - 4]$ | $\phantom{-}\frac{2}{7}e^{3} + \frac{1}{7}e^{2} - \frac{13}{7}e + \frac{13}{7}$ |
| 73 | $[73, 73, 2w^{2} - w - 8]$ | $-e^{3} - e^{2} + 10e + 13$ |
| 73 | $[73, 73, 3w^{2} - 5w - 5]$ | $\phantom{-}\frac{2}{7}e^{3} + \frac{1}{7}e^{2} - \frac{34}{7}e - \frac{22}{7}$ |
| 79 | $[79, 79, -3w^{2} + 5w + 4]$ | $\phantom{-}\frac{5}{7}e^{3} + \frac{6}{7}e^{2} - \frac{71}{7}e - \frac{34}{7}$ |
| 79 | $[79, 79, 2w^{2} - w - 9]$ | $-\frac{2}{7}e^{3} - \frac{8}{7}e^{2} + \frac{34}{7}e + \frac{64}{7}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $47$ | $[47,47,2w^{2} - 4w - 7]$ | $-1$ |