Base field 4.4.3981.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 4x^{2} + 2x + 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[53, 53, 2w - 3]$ |
| Dimension: | $7$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $12$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{7} - 3x^{6} - 9x^{5} + 33x^{4} - 10x^{3} - 23x^{2} + 4x + 4\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w + 1]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w^{3} - w^{2} - 3w + 1]$ | $\phantom{-}\frac{1}{4}e^{6} - \frac{5}{4}e^{5} - \frac{3}{4}e^{4} + \frac{51}{4}e^{3} - 18e^{2} - \frac{3}{4}e + \frac{11}{2}$ |
| 9 | $[9, 3, -w^{2} + 2]$ | $\phantom{-}\frac{1}{2}e^{6} - \frac{5}{2}e^{5} - \frac{5}{2}e^{4} + \frac{51}{2}e^{3} - 27e^{2} - \frac{9}{2}e + 7$ |
| 13 | $[13, 13, -w^{3} + w^{2} + 4w]$ | $-\frac{3}{2}e^{6} + \frac{9}{2}e^{5} + \frac{25}{2}e^{4} - \frac{95}{2}e^{3} + 25e^{2} + \frac{25}{2}e - 5$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}\frac{5}{4}e^{6} - \frac{13}{4}e^{5} - \frac{47}{4}e^{4} + \frac{143}{4}e^{3} - 6e^{2} - \frac{83}{4}e + \frac{3}{2}$ |
| 23 | $[23, 23, w^{2} - 2w - 2]$ | $-\frac{1}{2}e^{6} + \frac{5}{2}e^{5} + \frac{7}{2}e^{4} - \frac{53}{2}e^{3} + 17e^{2} + \frac{35}{2}e - 3$ |
| 37 | $[37, 37, w^{3} - 4w + 1]$ | $\phantom{-}2e^{6} - 7e^{5} - 15e^{4} + 73e^{3} - 52e^{2} - 16e + 14$ |
| 37 | $[37, 37, w^{3} - w^{2} - 5w + 1]$ | $\phantom{-}e^{6} - 3e^{5} - 9e^{4} + 31e^{3} - 11e^{2} - 6e + 2$ |
| 41 | $[41, 41, w^{3} - 5w + 1]$ | $\phantom{-}e^{4} + e^{3} - 9e^{2} - 3e + 4$ |
| 43 | $[43, 43, 2w^{3} - w^{2} - 7w]$ | $\phantom{-}\frac{11}{4}e^{6} - \frac{39}{4}e^{5} - \frac{77}{4}e^{4} + \frac{405}{4}e^{3} - 85e^{2} - \frac{73}{4}e + \frac{53}{2}$ |
| 53 | $[53, 53, 2w - 3]$ | $-1$ |
| 59 | $[59, 59, -2w^{3} + w^{2} + 9w - 1]$ | $-\frac{3}{2}e^{6} + \frac{7}{2}e^{5} + \frac{25}{2}e^{4} - \frac{75}{2}e^{3} + 20e^{2} + \frac{7}{2}e + 1$ |
| 67 | $[67, 67, -w - 3]$ | $-\frac{1}{2}e^{6} - \frac{1}{2}e^{5} + \frac{15}{2}e^{4} + \frac{5}{2}e^{3} - 30e^{2} + \frac{19}{2}e + 9$ |
| 67 | $[67, 67, w^{3} + w^{2} - 5w - 4]$ | $\phantom{-}2e^{6} - 7e^{5} - 14e^{4} + 74e^{3} - 61e^{2} - 20e + 19$ |
| 71 | $[71, 71, 2w^{3} - 3w^{2} - 7w + 5]$ | $-\frac{1}{2}e^{6} + \frac{3}{2}e^{5} + \frac{13}{2}e^{4} - \frac{35}{2}e^{3} - 14e^{2} + \frac{53}{2}e + 5$ |
| 73 | $[73, 73, w^{3} - 6w]$ | $\phantom{-}\frac{3}{2}e^{6} - \frac{9}{2}e^{5} - \frac{23}{2}e^{4} + \frac{97}{2}e^{3} - 35e^{2} - \frac{41}{2}e + 19$ |
| 73 | $[73, 73, -w^{3} - w^{2} + 5w + 3]$ | $-2e^{6} + 8e^{5} + 14e^{4} - 83e^{3} + 65e^{2} + 24e - 18$ |
| 79 | $[79, 79, w^{3} - 3w - 4]$ | $-\frac{3}{2}e^{6} + \frac{11}{2}e^{5} + \frac{23}{2}e^{4} - \frac{117}{2}e^{3} + 37e^{2} + \frac{49}{2}e - 9$ |
| 83 | $[83, 83, w^{3} - 2w^{2} - 3w + 1]$ | $-\frac{7}{2}e^{6} + \frac{19}{2}e^{5} + \frac{59}{2}e^{4} - \frac{203}{2}e^{3} + 51e^{2} + \frac{53}{2}e - 13$ |
| 83 | $[83, 83, -2w^{3} + 2w^{2} + 6w - 3]$ | $\phantom{-}\frac{7}{2}e^{6} - \frac{25}{2}e^{5} - \frac{51}{2}e^{4} + \frac{261}{2}e^{3} - 97e^{2} - \frac{67}{2}e + 21$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $53$ | $[53, 53, 2w - 3]$ | $1$ |