Base field 4.4.19773.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 11x^{2} - 9x + 3\); narrow class number \(4\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, w^{3} - 2w^{2} - 6w + 1]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $40$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 80x^{6} + 1840x^{4} - 12800x^{2} + 6400\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w^{3} - 2w^{2} - 8w - 3]$ | $\phantom{-}\frac{1}{1520}e^{6} - \frac{13}{304}e^{4} + \frac{12}{19}e^{2} - \frac{37}{19}$ |
| 3 | $[3, 3, w^{3} - 3w^{2} - 5w + 2]$ | $-\frac{1}{1520}e^{6} + \frac{13}{304}e^{4} - \frac{12}{19}e^{2} + \frac{18}{19}$ |
| 13 | $[13, 13, w^{3} - 2w^{2} - 6w + 1]$ | $\phantom{-}1$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}4$ |
| 17 | $[17, 17, -2w^{3} + 5w^{2} + 14w - 1]$ | $\phantom{-}e$ |
| 17 | $[17, 17, -w^{3} + 3w^{2} + 5w - 4]$ | $-\frac{1}{1520}e^{7} + \frac{21}{380}e^{5} - \frac{105}{76}e^{3} + \frac{189}{19}e$ |
| 17 | $[17, 17, -w^{3} + 2w^{2} + 8w + 1]$ | $\phantom{-}\frac{7}{6080}e^{7} - \frac{8}{95}e^{5} + \frac{61}{38}e^{3} - \frac{136}{19}e$ |
| 17 | $[17, 17, w - 1]$ | $-\frac{3}{6080}e^{7} + \frac{63}{1520}e^{5} - \frac{37}{38}e^{3} + \frac{99}{19}e$ |
| 23 | $[23, 23, 2w^{3} - 6w^{2} - 11w + 7]$ | $-\frac{3}{1900}e^{7} + \frac{35}{304}e^{5} - \frac{421}{190}e^{3} + \frac{218}{19}e$ |
| 23 | $[23, 23, w^{3} - 3w^{2} - 7w + 4]$ | $\phantom{-}\frac{27}{30400}e^{7} - \frac{51}{760}e^{5} + \frac{533}{380}e^{3} - \frac{182}{19}e$ |
| 23 | $[23, 23, -w^{2} + 2w + 5]$ | $\phantom{-}\frac{23}{30400}e^{7} - \frac{89}{1520}e^{5} + \frac{447}{380}e^{3} - \frac{91}{19}e$ |
| 23 | $[23, 23, 3w^{3} - 8w^{2} - 20w + 5]$ | $-\frac{1}{15200}e^{7} - \frac{3}{1520}e^{5} + \frac{147}{380}e^{3} - \frac{116}{19}e$ |
| 61 | $[61, 61, -w^{3} + 2w^{2} + 9w + 1]$ | $-\frac{3}{80}e^{4} + 2e^{2} - 15$ |
| 61 | $[61, 61, -2w^{3} + 5w^{2} + 13w - 2]$ | $\phantom{-}\frac{1}{380}e^{6} - \frac{279}{1520}e^{4} + \frac{48}{19}e^{2} + \frac{42}{19}$ |
| 61 | $[61, 61, w^{3} - 2w^{2} - 9w - 2]$ | $\phantom{-}\frac{1}{760}e^{6} - \frac{73}{1520}e^{4} - \frac{14}{19}e^{2} + \frac{211}{19}$ |
| 61 | $[61, 61, 2w^{3} - 5w^{2} - 13w + 1]$ | $-\frac{3}{760}e^{6} + \frac{409}{1520}e^{4} - \frac{72}{19}e^{2} - \frac{6}{19}$ |
| 79 | $[79, 79, 4w^{3} - 10w^{2} - 28w + 5]$ | $-\frac{3}{760}e^{6} + \frac{333}{1520}e^{4} - \frac{34}{19}e^{2} - \frac{253}{19}$ |
| 79 | $[79, 79, w^{3} - 3w^{2} - 4w + 4]$ | $-\frac{1}{380}e^{6} + \frac{317}{1520}e^{4} - \frac{86}{19}e^{2} + \frac{243}{19}$ |
| 79 | $[79, 79, w^{3} - 4w^{2} - 2w + 10]$ | $\phantom{-}\frac{1}{152}e^{6} - \frac{669}{1520}e^{4} + \frac{120}{19}e^{2} - \frac{256}{19}$ |
| 79 | $[79, 79, 2w^{3} - 6w^{2} - 10w + 5]$ | $\phantom{-}\frac{1}{80}e^{4} - 16$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, w^{3} - 2w^{2} - 6w + 1]$ | $-1$ |