Base field 4.4.18097.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 7x^{2} + 6x + 4\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[1, 1, 1]$ |
| 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:
| \(x^{8} - 21x^{6} + 127x^{4} - 180x^{2} + 64\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, -w + 1]$ | $\phantom{-}e$ |
| 4 | $[4, 2, w]$ | $-\frac{5}{48}e^{7} + \frac{97}{48}e^{5} - \frac{499}{48}e^{3} + \frac{83}{12}e$ |
| 4 | $[4, 2, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{7}{2}w - 3]$ | $-\frac{1}{48}e^{7} + \frac{29}{48}e^{5} - \frac{215}{48}e^{3} + \frac{67}{12}e$ |
| 7 | $[7, 7, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{5}{2}w - 2]$ | $-\frac{1}{6}e^{6} + \frac{17}{6}e^{4} - \frac{77}{6}e^{2} + \frac{32}{3}$ |
| 13 | $[13, 13, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{7}{2}w - 1]$ | $\phantom{-}\frac{1}{6}e^{6} - \frac{17}{6}e^{4} + \frac{71}{6}e^{2} - \frac{14}{3}$ |
| 17 | $[17, 17, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{5}{2}w]$ | $\phantom{-}\frac{1}{8}e^{7} - \frac{21}{8}e^{5} + \frac{119}{8}e^{3} - \frac{27}{2}e$ |
| 27 | $[27, 3, -w^{3} + 7w + 1]$ | $\phantom{-}e^{3} - 9e$ |
| 31 | $[31, 31, w + 3]$ | $\phantom{-}\frac{1}{6}e^{6} - \frac{23}{6}e^{4} + \frac{143}{6}e^{2} - \frac{56}{3}$ |
| 31 | $[31, 31, -w^{2} + 5]$ | $-\frac{1}{12}e^{7} + \frac{17}{12}e^{5} - \frac{83}{12}e^{3} + \frac{28}{3}e$ |
| 37 | $[37, 37, w^{2} - 3]$ | $-\frac{1}{24}e^{7} + \frac{29}{24}e^{5} - \frac{215}{24}e^{3} + \frac{61}{6}e$ |
| 41 | $[41, 41, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{3}{2}w - 2]$ | $-\frac{1}{8}e^{7} + \frac{21}{8}e^{5} - \frac{127}{8}e^{3} + \frac{41}{2}e$ |
| 47 | $[47, 47, w^{3} - 5w - 3]$ | $\phantom{-}\frac{1}{4}e^{7} - \frac{21}{4}e^{5} + \frac{119}{4}e^{3} - 24e$ |
| 53 | $[53, 53, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w]$ | $\phantom{-}\frac{1}{8}e^{7} - \frac{21}{8}e^{5} + \frac{127}{8}e^{3} - \frac{41}{2}e$ |
| 61 | $[61, 61, w^{3} - w^{2} - 5w + 3]$ | $\phantom{-}\frac{1}{3}e^{6} - \frac{17}{3}e^{4} + \frac{71}{3}e^{2} - \frac{22}{3}$ |
| 83 | $[83, 83, w^{3} + w^{2} - 6w - 7]$ | $\phantom{-}\frac{1}{2}e^{7} - \frac{19}{2}e^{5} + \frac{97}{2}e^{3} - 35e$ |
| 83 | $[83, 83, -w^{3} + 5w + 1]$ | $\phantom{-}\frac{1}{2}e^{6} - \frac{19}{2}e^{4} + \frac{99}{2}e^{2} - 36$ |
| 83 | $[83, 83, 2w - 1]$ | $\phantom{-}\frac{1}{2}e^{7} - \frac{19}{2}e^{5} + \frac{97}{2}e^{3} - 35e$ |
| 83 | $[83, 83, -\frac{3}{2}w^{3} - \frac{1}{2}w^{2} + \frac{19}{2}w + 2]$ | $-e^{4} + 10e^{2} - 12$ |
| 89 | $[89, 89, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w - 2]$ | $\phantom{-}2e^{4} - 22e^{2} + 26$ |
| 89 | $[89, 89, w^{3} - 5w + 5]$ | $-6$ |
Atkin-Lehner eigenvalues
This form has no Atkin-Lehner eigenvalues since the level is \((1)\).