Base field 5.5.176281.1
Generator \(w\), with minimal polynomial \(x^{5} - x^{4} - 5x^{3} + 3x^{2} + 4x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2]$ |
| Level: | $[13, 13, -w^{4} + w^{3} + 4w^{2} - 2w - 2]$ |
| Dimension: | $5$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $20$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{5} + 3x^{4} - 10x^{3} - 31x^{2} + 5x + 25\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, w^{4} - w^{3} - 4w^{2} + 2w + 1]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w^{2} - w - 2]$ | $-\frac{6}{25}e^{4} - \frac{8}{25}e^{3} + \frac{13}{5}e^{2} + \frac{61}{25}e - \frac{23}{5}$ |
| 7 | $[7, 7, -w^{2} + 2]$ | $\phantom{-}\frac{2}{5}e^{4} + \frac{1}{5}e^{3} - 4e^{2} - \frac{12}{5}e + 3$ |
| 11 | $[11, 11, -w^{2} + 3]$ | $-\frac{2}{25}e^{4} - \frac{11}{25}e^{3} + \frac{6}{5}e^{2} + \frac{87}{25}e - \frac{21}{5}$ |
| 13 | $[13, 13, -w^{4} + w^{3} + 4w^{2} - 2w - 2]$ | $-1$ |
| 23 | $[23, 23, -w^{4} + 5w^{2} + w - 4]$ | $-\frac{4}{5}e^{4} - \frac{2}{5}e^{3} + 7e^{2} + \frac{19}{5}e - 1$ |
| 31 | $[31, 31, -w^{4} + w^{3} + 5w^{2} - 2w - 2]$ | $\phantom{-}\frac{7}{25}e^{4} + \frac{1}{25}e^{3} - \frac{16}{5}e^{2} - \frac{42}{25}e + \frac{36}{5}$ |
| 32 | $[32, 2, 2]$ | $-\frac{1}{25}e^{4} + \frac{7}{25}e^{3} + \frac{8}{5}e^{2} - \frac{44}{25}e - \frac{43}{5}$ |
| 37 | $[37, 37, w^{3} - 3w - 1]$ | $-\frac{1}{25}e^{4} + \frac{7}{25}e^{3} + \frac{3}{5}e^{2} - \frac{19}{25}e - \frac{13}{5}$ |
| 41 | $[41, 41, w^{4} - w^{3} - 3w^{2} + w + 1]$ | $-\frac{19}{25}e^{4} + \frac{8}{25}e^{3} + \frac{42}{5}e^{2} - \frac{61}{25}e - \frac{82}{5}$ |
| 47 | $[47, 47, -w^{3} + w^{2} + 4w]$ | $-\frac{31}{25}e^{4} - \frac{8}{25}e^{3} + \frac{58}{5}e^{2} + \frac{36}{25}e - \frac{28}{5}$ |
| 61 | $[61, 61, -w^{4} + 6w^{2} + w - 4]$ | $\phantom{-}\frac{2}{5}e^{4} - \frac{4}{5}e^{3} - 4e^{2} + \frac{38}{5}e + 9$ |
| 67 | $[67, 67, -w^{4} + w^{3} + 5w^{2} - 4w - 3]$ | $\phantom{-}\frac{8}{25}e^{4} - \frac{31}{25}e^{3} - \frac{19}{5}e^{2} + \frac{252}{25}e + \frac{29}{5}$ |
| 71 | $[71, 71, w^{4} - 6w^{2} + 7]$ | $-\frac{46}{25}e^{4} - \frac{28}{25}e^{3} + \frac{98}{5}e^{2} + \frac{276}{25}e - \frac{108}{5}$ |
| 73 | $[73, 73, -w^{4} + w^{3} + 4w^{2} - 4w - 2]$ | $\phantom{-}\frac{8}{5}e^{4} + \frac{4}{5}e^{3} - 16e^{2} - \frac{38}{5}e + 15$ |
| 83 | $[83, 83, -w^{4} + 2w^{3} + 4w^{2} - 6w - 2]$ | $\phantom{-}\frac{36}{25}e^{4} + \frac{23}{25}e^{3} - \frac{73}{5}e^{2} - \frac{241}{25}e + \frac{58}{5}$ |
| 83 | $[83, 83, -w^{4} + 5w^{2} + 2w - 4]$ | $-\frac{2}{25}e^{4} - \frac{36}{25}e^{3} + \frac{1}{5}e^{2} + \frac{312}{25}e + \frac{24}{5}$ |
| 83 | $[83, 83, w^{4} - 5w^{2} + 2]$ | $\phantom{-}\frac{44}{25}e^{4} + \frac{17}{25}e^{3} - \frac{92}{5}e^{2} - \frac{189}{25}e + \frac{102}{5}$ |
| 89 | $[89, 89, -2w^{4} + 2w^{3} + 9w^{2} - 6w - 4]$ | $-\frac{61}{25}e^{4} - \frac{23}{25}e^{3} + \frac{133}{5}e^{2} + \frac{266}{25}e - \frac{158}{5}$ |
| 101 | $[101, 101, -2w^{4} + 2w^{3} + 8w^{2} - 3w - 3]$ | $-\frac{11}{5}e^{4} + \frac{2}{5}e^{3} + 23e^{2} + \frac{1}{5}e - 20$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w^{4} + w^{3} + 4w^{2} - 2w - 2]$ | $1$ |