Base field 5.5.81589.1
Generator \(w\), with minimal polynomial \(x^{5} - 6x^{3} + 8x - 1\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2]$ |
| Level: | $[31, 31, w^{4} - 4w^{2} - w + 3]$ |
| Dimension: | $9$ |
| 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:
| \(x^{9} + 4x^{8} - 7x^{7} - 40x^{6} + 2x^{5} + 115x^{4} + 34x^{3} - 98x^{2} - 20x + 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w^{2} - 3]$ | $\phantom{-}e$ |
| 11 | $[11, 11, -w^{4} + 5w^{2} + w - 4]$ | $-\frac{1}{5}e^{8} - \frac{1}{5}e^{7} + 2e^{6} + 2e^{5} - \frac{27}{5}e^{4} - \frac{34}{5}e^{3} + \frac{18}{5}e^{2} + \frac{44}{5}e - \frac{7}{5}$ |
| 13 | $[13, 13, -w^{3} + w^{2} + 3w - 4]$ | $-e^{8} - 2e^{7} + 10e^{6} + 18e^{5} - 29e^{4} - 41e^{3} + 26e^{2} + 18e - 1$ |
| 16 | $[16, 2, w^{4} - w^{3} - 3w^{2} + 3w - 1]$ | $-e^{7} - e^{6} + 10e^{5} + 8e^{4} - 27e^{3} - 15e^{2} + 17e + 4$ |
| 17 | $[17, 17, w^{4} - 4w^{2} + 2]$ | $\phantom{-}\frac{2}{5}e^{8} + \frac{7}{5}e^{7} - 3e^{6} - 14e^{5} + \frac{14}{5}e^{4} + \frac{198}{5}e^{3} + \frac{34}{5}e^{2} - \frac{153}{5}e - \frac{11}{5}$ |
| 31 | $[31, 31, w^{4} - 4w^{2} - w + 3]$ | $-1$ |
| 43 | $[43, 43, -w^{3} + 4w - 2]$ | $\phantom{-}\frac{9}{5}e^{8} + \frac{19}{5}e^{7} - 18e^{6} - 35e^{5} + \frac{258}{5}e^{4} + \frac{421}{5}e^{3} - \frac{217}{5}e^{2} - \frac{231}{5}e - \frac{22}{5}$ |
| 53 | $[53, 53, w^{4} - 3w^{2} - 1]$ | $\phantom{-}e^{8} + 3e^{7} - 9e^{6} - 28e^{5} + 21e^{4} + 69e^{3} - 12e^{2} - 40e - 7$ |
| 53 | $[53, 53, w^{4} - w^{3} - 4w^{2} + 2w + 1]$ | $\phantom{-}\frac{2}{5}e^{8} + \frac{7}{5}e^{7} - 4e^{6} - 13e^{5} + \frac{64}{5}e^{4} + \frac{158}{5}e^{3} - \frac{81}{5}e^{2} - \frac{93}{5}e + \frac{14}{5}$ |
| 53 | $[53, 53, -w^{3} - w^{2} + 3w + 4]$ | $\phantom{-}\frac{2}{5}e^{8} + \frac{7}{5}e^{7} - 3e^{6} - 15e^{5} + \frac{14}{5}e^{4} + \frac{243}{5}e^{3} + \frac{34}{5}e^{2} - \frac{248}{5}e - \frac{16}{5}$ |
| 61 | $[61, 61, -w^{3} + w^{2} + 5w - 2]$ | $\phantom{-}\frac{3}{5}e^{8} + \frac{13}{5}e^{7} - 5e^{6} - 26e^{5} + \frac{51}{5}e^{4} + \frac{372}{5}e^{3} - \frac{34}{5}e^{2} - \frac{317}{5}e + \frac{1}{5}$ |
| 61 | $[61, 61, w^{4} - 4w^{2} + w + 1]$ | $-\frac{16}{5}e^{8} - \frac{26}{5}e^{7} + 35e^{6} + 46e^{5} - \frac{597}{5}e^{4} - \frac{494}{5}e^{3} + \frac{698}{5}e^{2} + \frac{139}{5}e - \frac{47}{5}$ |
| 61 | $[61, 61, w^{3} + w^{2} - 4w - 3]$ | $\phantom{-}2e^{8} + 3e^{7} - 22e^{6} - 26e^{5} + 74e^{4} + 52e^{3} - 78e^{2} - 6e - 5$ |
| 67 | $[67, 67, -2w^{3} + w^{2} + 6w - 2]$ | $-\frac{1}{5}e^{8} + \frac{4}{5}e^{7} + 3e^{6} - 10e^{5} - \frac{77}{5}e^{4} + \frac{191}{5}e^{3} + \frac{138}{5}e^{2} - \frac{236}{5}e - \frac{12}{5}$ |
| 71 | $[71, 71, -2w^{3} + w^{2} + 7w - 3]$ | $\phantom{-}\frac{6}{5}e^{8} + \frac{16}{5}e^{7} - 10e^{6} - 30e^{5} + \frac{87}{5}e^{4} + \frac{389}{5}e^{3} + \frac{7}{5}e^{2} - \frac{299}{5}e + \frac{2}{5}$ |
| 71 | $[71, 71, 2w^{4} - 4w^{3} - 7w^{2} + 13w - 1]$ | $\phantom{-}2e^{8} + 2e^{7} - 23e^{6} - 17e^{5} + 81e^{4} + 32e^{3} - 87e^{2} + 3e - 9$ |
| 73 | $[73, 73, -w^{4} + 2w^{3} + 3w^{2} - 7w]$ | $-\frac{7}{5}e^{8} - \frac{12}{5}e^{7} + 16e^{6} + 23e^{5} - \frac{284}{5}e^{4} - \frac{288}{5}e^{3} + \frac{326}{5}e^{2} + \frac{133}{5}e - \frac{9}{5}$ |
| 73 | $[73, 73, w^{4} + w^{3} - 3w^{2} - 4w - 2]$ | $\phantom{-}\frac{3}{5}e^{8} - \frac{7}{5}e^{7} - 10e^{6} + 17e^{5} + \frac{261}{5}e^{4} - \frac{333}{5}e^{3} - \frac{439}{5}e^{2} + \frac{443}{5}e + \frac{51}{5}$ |
| 79 | $[79, 79, w^{4} - w^{3} - 5w^{2} + 4w + 6]$ | $\phantom{-}\frac{4}{5}e^{8} + \frac{4}{5}e^{7} - 9e^{6} - 6e^{5} + \frac{163}{5}e^{4} + \frac{31}{5}e^{3} - \frac{227}{5}e^{2} + \frac{54}{5}e + \frac{38}{5}$ |
| 83 | $[83, 83, -2w^{4} + w^{3} + 9w^{2} - 3w - 6]$ | $-\frac{2}{5}e^{8} - \frac{12}{5}e^{7} + 3e^{6} + 26e^{5} - \frac{14}{5}e^{4} - \frac{413}{5}e^{3} - \frac{49}{5}e^{2} + \frac{363}{5}e + \frac{26}{5}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $31$ | $[31, 31, w^{4} - 4w^{2} - w + 3]$ | $1$ |