Base field 3.3.1940.1
Generator \(w\), with minimal polynomial \(x^{3} - 8x - 2\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[4, 2, -w^{2} + 8]$ |
| 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:
| \(x^{8} - 22x^{6} + 155x^{4} - 360x^{2} + 64\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w]$ | $\phantom{-}0$ |
| 3 | $[3, 3, w^{2} - 7]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w + 1]$ | $-\frac{1}{112}e^{7} + \frac{3}{56}e^{5} + \frac{53}{112}e^{3} - \frac{17}{14}e$ |
| 5 | $[5, 5, -w - 3]$ | $\phantom{-}\frac{3}{28}e^{6} - \frac{23}{14}e^{4} + \frac{177}{28}e^{2} - \frac{24}{7}$ |
| 9 | $[9, 3, w^{2} - 2w - 1]$ | $-\frac{3}{112}e^{7} + \frac{9}{56}e^{5} + \frac{271}{112}e^{3} - \frac{191}{14}e$ |
| 17 | $[17, 17, -2w^{2} + 15]$ | $-\frac{5}{28}e^{6} + \frac{43}{14}e^{4} - \frac{351}{28}e^{2} + \frac{40}{7}$ |
| 17 | $[17, 17, 3w + 1]$ | $\phantom{-}\frac{19}{112}e^{7} - \frac{169}{56}e^{5} + \frac{1681}{112}e^{3} - \frac{251}{14}e$ |
| 17 | $[17, 17, -w^{2} + w + 5]$ | $\phantom{-}\frac{1}{112}e^{7} - \frac{3}{56}e^{5} - \frac{165}{112}e^{3} + \frac{129}{14}e$ |
| 19 | $[19, 19, -2w^{2} + w + 15]$ | $-\frac{1}{14}e^{7} + \frac{10}{7}e^{5} - \frac{129}{14}e^{3} + \frac{135}{7}e$ |
| 29 | $[29, 29, w^{2} - w - 1]$ | $\phantom{-}\frac{9}{112}e^{7} - \frac{83}{56}e^{5} + \frac{979}{112}e^{3} - \frac{253}{14}e$ |
| 41 | $[41, 41, w^{2} - 5]$ | $\phantom{-}\frac{9}{112}e^{7} - \frac{83}{56}e^{5} + \frac{867}{112}e^{3} - \frac{155}{14}e$ |
| 43 | $[43, 43, w^{2} + w - 3]$ | $\phantom{-}\frac{1}{56}e^{7} - \frac{3}{28}e^{5} - \frac{53}{56}e^{3} + \frac{17}{7}e$ |
| 47 | $[47, 47, 2w - 1]$ | $\phantom{-}\frac{3}{14}e^{6} - \frac{23}{7}e^{4} + \frac{135}{14}e^{2} + \frac{64}{7}$ |
| 53 | $[53, 53, -2w - 3]$ | $-\frac{1}{16}e^{7} + \frac{11}{8}e^{5} - \frac{155}{16}e^{3} + \frac{45}{2}e$ |
| 59 | $[59, 59, w^{2} - w - 11]$ | $-\frac{1}{14}e^{6} + \frac{3}{7}e^{4} + \frac{53}{14}e^{2} - \frac{68}{7}$ |
| 71 | $[71, 71, w^{2} - 3]$ | $\phantom{-}\frac{9}{56}e^{7} - \frac{83}{28}e^{5} + \frac{867}{56}e^{3} - \frac{148}{7}e$ |
| 73 | $[73, 73, 6w^{2} - 2w - 47]$ | $\phantom{-}\frac{9}{112}e^{7} - \frac{83}{56}e^{5} + \frac{755}{112}e^{3} - \frac{43}{14}e$ |
| 83 | $[83, 83, w^{2} - 4w + 1]$ | $-\frac{1}{14}e^{6} + \frac{10}{7}e^{4} - \frac{115}{14}e^{2} + \frac{16}{7}$ |
| 83 | $[83, 83, 3w^{2} - 25]$ | $\phantom{-}\frac{1}{14}e^{6} - \frac{10}{7}e^{4} + \frac{87}{14}e^{2} + \frac{40}{7}$ |
| 83 | $[83, 83, w - 5]$ | $\phantom{-}\frac{1}{7}e^{6} - \frac{20}{7}e^{4} + \frac{101}{7}e^{2} - \frac{60}{7}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2, 2, -w]$ | $-1$ |