Base field 4.4.7053.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 4x^{2} + 3x + 3\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[19, 19, -w^{3} + 2w^{2} + 2w - 1]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $10$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - x^{7} - 15x^{6} + 11x^{5} + 70x^{4} - 22x^{3} - 111x^{2} - 15x + 18\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}e$ |
| 7 | $[7, 7, w^{2} - w - 2]$ | $-\frac{1}{6}e^{7} + \frac{1}{6}e^{6} + 2e^{5} - \frac{7}{3}e^{4} - \frac{37}{6}e^{3} + \frac{23}{3}e^{2} + \frac{11}{2}e - 3$ |
| 9 | $[9, 3, w^{2} - 2w - 1]$ | $\phantom{-}\frac{1}{2}e^{6} + \frac{1}{2}e^{5} - \frac{11}{2}e^{4} - 5e^{3} + 14e^{2} + \frac{23}{2}e - 2$ |
| 13 | $[13, 13, w^{3} - 3w^{2} - w + 5]$ | $\phantom{-}\frac{1}{3}e^{7} - \frac{1}{3}e^{6} - \frac{9}{2}e^{5} + \frac{11}{3}e^{4} + \frac{101}{6}e^{3} - \frac{47}{6}e^{2} - \frac{35}{2}e + 2$ |
| 13 | $[13, 13, w^{3} - 3w^{2} - w + 2]$ | $-\frac{1}{3}e^{7} + \frac{1}{3}e^{6} + 4e^{5} - \frac{11}{3}e^{4} - \frac{37}{3}e^{3} + \frac{25}{3}e^{2} + 10e - 3$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}\frac{1}{6}e^{7} + \frac{1}{3}e^{6} - 2e^{5} - \frac{19}{6}e^{4} + \frac{17}{3}e^{3} + \frac{29}{6}e^{2} + \frac{1}{2}e + 6$ |
| 17 | $[17, 17, -w^{3} + 2w^{2} + 2w - 2]$ | $\phantom{-}e^{3} + e^{2} - 6e - 3$ |
| 19 | $[19, 19, -w^{3} + 2w^{2} + 2w - 1]$ | $-1$ |
| 29 | $[29, 29, -2w^{3} + 5w^{2} + 4w - 7]$ | $-e^{6} + 11e^{4} - 28e^{2} - 4e + 9$ |
| 31 | $[31, 31, w^{3} - 2w^{2} - 4w + 2]$ | $-\frac{1}{3}e^{7} + \frac{1}{3}e^{6} + 4e^{5} - \frac{11}{3}e^{4} - \frac{37}{3}e^{3} + \frac{22}{3}e^{2} + 9e$ |
| 47 | $[47, 47, -w^{3} + 4w^{2} - w - 5]$ | $\phantom{-}\frac{1}{3}e^{7} + \frac{2}{3}e^{6} - 4e^{5} - \frac{22}{3}e^{4} + \frac{37}{3}e^{3} + \frac{53}{3}e^{2} - 5e + 4$ |
| 53 | $[53, 53, -w^{3} + 4w^{2} - w - 7]$ | $\phantom{-}e^{3} - e^{2} - 8e + 3$ |
| 67 | $[67, 67, 2w^{3} - 3w^{2} - 8w + 1]$ | $-\frac{2}{3}e^{7} - \frac{1}{3}e^{6} + 7e^{5} + \frac{11}{3}e^{4} - \frac{44}{3}e^{3} - \frac{34}{3}e^{2} - 5e - 4$ |
| 67 | $[67, 67, 2w^{3} - 5w^{2} - 4w + 4]$ | $-\frac{1}{3}e^{7} - \frac{1}{6}e^{6} + \frac{9}{2}e^{5} + \frac{5}{6}e^{4} - \frac{58}{3}e^{3} + \frac{4}{3}e^{2} + \frac{57}{2}e + 1$ |
| 71 | $[71, 71, w^{3} - 3w^{2} - 2w + 2]$ | $-\frac{1}{3}e^{7} - \frac{2}{3}e^{6} + 3e^{5} + \frac{22}{3}e^{4} - \frac{4}{3}e^{3} - \frac{56}{3}e^{2} - 21e - 4$ |
| 79 | $[79, 79, w^{2} - 3w - 4]$ | $-\frac{1}{3}e^{7} - \frac{2}{3}e^{6} + 4e^{5} + \frac{19}{3}e^{4} - \frac{40}{3}e^{3} - \frac{35}{3}e^{2} + 11e + 4$ |
| 83 | $[83, 83, 2w^{2} - 3w - 4]$ | $\phantom{-}\frac{1}{3}e^{7} - \frac{1}{3}e^{6} - 3e^{5} + \frac{14}{3}e^{4} + \frac{10}{3}e^{3} - \frac{49}{3}e^{2} + 6e + 16$ |
| 89 | $[89, 89, -3w^{3} + 7w^{2} + 8w - 11]$ | $-e^{6} + e^{5} + 11e^{4} - 10e^{3} - 24e^{2} + 15e$ |
| 101 | $[101, 101, w^{3} - 2w^{2} - 3w - 2]$ | $\phantom{-}\frac{1}{3}e^{7} - \frac{1}{3}e^{6} - 3e^{5} + \frac{11}{3}e^{4} + \frac{7}{3}e^{3} - \frac{16}{3}e^{2} + 8e - 5$ |
| 103 | $[103, 103, -2w^{3} + 4w^{2} + 5w - 1]$ | $\phantom{-}\frac{1}{3}e^{7} - \frac{1}{3}e^{6} - 4e^{5} + \frac{11}{3}e^{4} + \frac{34}{3}e^{3} - \frac{22}{3}e^{2} - 2e + 2$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $19$ | $[19, 19, -w^{3} + 2w^{2} + 2w - 1]$ | $1$ |