Base field 5.5.170701.1
Generator \(w\), with minimal polynomial \(x^{5} - x^{4} - 6x^{3} + 4x + 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2]$ |
| Level: | $[8, 2, -2w^{4} + 3w^{3} + 10w^{2} - 5w - 3]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $15$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 4x^{7} - 13x^{6} + 55x^{5} + 56x^{4} - 231x^{3} - 88x^{2} + 260x + 8\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -w^{4} + w^{3} + 6w^{2} - w - 3]$ | $\phantom{-}e$ |
| 7 | $[7, 7, w^{4} - w^{3} - 6w^{2} + w + 2]$ | $\phantom{-}\frac{1}{2}e^{7} - e^{6} - \frac{13}{2}e^{5} + \frac{15}{2}e^{4} + 27e^{3} - \frac{11}{2}e^{2} - 26e - 6$ |
| 8 | $[8, 2, -2w^{4} + 3w^{3} + 10w^{2} - 5w - 3]$ | $\phantom{-}1$ |
| 11 | $[11, 11, w^{4} - w^{3} - 6w^{2} + 2]$ | $\phantom{-}\frac{1}{2}e^{7} - \frac{1}{2}e^{6} - \frac{17}{2}e^{5} + 4e^{4} + \frac{85}{2}e^{3} - \frac{1}{2}e^{2} - \frac{95}{2}e - 2$ |
| 13 | $[13, 13, -w^{4} + 2w^{3} + 5w^{2} - 5w - 3]$ | $\phantom{-}\frac{1}{4}e^{7} - \frac{1}{2}e^{6} - \frac{13}{4}e^{5} + \frac{17}{4}e^{4} + \frac{25}{2}e^{3} - \frac{27}{4}e^{2} - \frac{15}{2}e + 4$ |
| 23 | $[23, 23, -w^{2} + 2]$ | $-\frac{1}{2}e^{7} + e^{6} + \frac{13}{2}e^{5} - \frac{15}{2}e^{4} - 28e^{3} + \frac{15}{2}e^{2} + 32e - 1$ |
| 23 | $[23, 23, -w + 2]$ | $-\frac{3}{4}e^{7} + \frac{59}{4}e^{5} + \frac{7}{4}e^{4} - 80e^{3} - \frac{83}{4}e^{2} + 94e + 10$ |
| 31 | $[31, 31, -w^{4} + 2w^{3} + 5w^{2} - 6w - 4]$ | $-\frac{5}{4}e^{7} + e^{6} + \frac{89}{4}e^{5} - \frac{35}{4}e^{4} - 115e^{3} + \frac{23}{4}e^{2} + 135e - 6$ |
| 43 | $[43, 43, -w^{3} + 2w^{2} + 3w - 2]$ | $\phantom{-}\frac{1}{2}e^{6} - 2e^{5} - \frac{7}{2}e^{4} + \frac{31}{2}e^{3} + 7e^{2} - \frac{43}{2}e - 6$ |
| 47 | $[47, 47, -w^{4} + 2w^{3} + 5w^{2} - 4w - 4]$ | $\phantom{-}\frac{1}{2}e^{7} - 2e^{6} - \frac{7}{2}e^{5} + \frac{33}{2}e^{4} + 4e^{3} - \frac{57}{2}e^{2} + 5e + 6$ |
| 53 | $[53, 53, w^{2} - w - 4]$ | $\phantom{-}\frac{3}{2}e^{7} - 2e^{6} - \frac{47}{2}e^{5} + \frac{31}{2}e^{4} + 112e^{3} - \frac{17}{2}e^{2} - 119e - 4$ |
| 53 | $[53, 53, -w^{4} + 2w^{3} + 4w^{2} - 4w - 3]$ | $-\frac{7}{4}e^{7} + 3e^{6} + \frac{99}{4}e^{5} - \frac{89}{4}e^{4} - 113e^{3} + \frac{61}{4}e^{2} + 129e + 2$ |
| 53 | $[53, 53, -w^{3} + w^{2} + 4w - 1]$ | $-e^{3} + 8e - 1$ |
| 71 | $[71, 71, w^{4} - w^{3} - 7w^{2} + 3w + 6]$ | $-\frac{3}{2}e^{7} + 2e^{6} + \frac{47}{2}e^{5} - \frac{33}{2}e^{4} - 112e^{3} + \frac{33}{2}e^{2} + 124e - 2$ |
| 71 | $[71, 71, -2w^{4} + 2w^{3} + 11w^{2} - 2]$ | $-e^{4} + e^{3} + 8e^{2} - 3e - 5$ |
| 71 | $[71, 71, -w^{4} + 3w^{3} + 3w^{2} - 9w - 1]$ | $\phantom{-}\frac{1}{4}e^{7} - \frac{3}{2}e^{6} - \frac{1}{4}e^{5} + \frac{53}{4}e^{4} - \frac{17}{2}e^{3} - \frac{127}{4}e^{2} + \frac{19}{2}e + 16$ |
| 73 | $[73, 73, 2w^{4} - 4w^{3} - 9w^{2} + 10w + 5]$ | $-\frac{1}{2}e^{7} + \frac{21}{2}e^{5} + \frac{1}{2}e^{4} - 60e^{3} - \frac{25}{2}e^{2} + 76e + 18$ |
| 79 | $[79, 79, 2w^{4} - 3w^{3} - 11w^{2} + 4w + 8]$ | $\phantom{-}e^{6} - 3e^{5} - 10e^{4} + 25e^{3} + 27e^{2} - 38e - 2$ |
| 83 | $[83, 83, -3w^{4} + 5w^{3} + 15w^{2} - 9w - 10]$ | $\phantom{-}e^{7} - e^{6} - 17e^{5} + 8e^{4} + 87e^{3} - 5e^{2} - 103e + 2$ |
| 89 | $[89, 89, w^{3} - w^{2} - 5w + 1]$ | $\phantom{-}\frac{1}{2}e^{7} - \frac{19}{2}e^{5} - \frac{3}{2}e^{4} + 50e^{3} + \frac{35}{2}e^{2} - 61e - 16$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $8$ | $[8, 2, -2w^{4} + 3w^{3} + 10w^{2} - 5w - 3]$ | $-1$ |