Base field 3.3.321.1
Generator \(w\), with minimal polynomial \(x^{3} - x^{2} - 4x + 1\); narrow class number \(1\) and class number \(1\).
Form
Weight: | $[2, 2, 2]$ |
Level: | $[47, 47, w^{2} + w - 4]$ |
Dimension: | $9$ |
CM: | no |
Base change: | no |
Newspace dimension: | $13$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{9} - 2x^{8} - 15x^{7} + 24x^{6} + 81x^{5} - 86x^{4} - 181x^{3} + 84x^{2} + 122x - 16\) |
Show full eigenvalues Hide large eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
3 | $[3, 3, w + 1]$ | $\phantom{-}e$ |
3 | $[3, 3, w - 1]$ | $-\frac{1}{2}e^{7} + \frac{1}{2}e^{6} + 6e^{5} - 4e^{4} - \frac{43}{2}e^{3} + \frac{15}{2}e^{2} + 20e - 4$ |
7 | $[7, 7, w^{2} - 2]$ | $\phantom{-}\frac{1}{2}e^{7} - \frac{13}{2}e^{5} - \frac{1}{2}e^{4} + 24e^{3} + \frac{5}{2}e^{2} - 20e + 2$ |
8 | $[8, 2, 2]$ | $\phantom{-}e^{2} - e - 3$ |
11 | $[11, 11, -w^{2} + w + 1]$ | $\phantom{-}\frac{1}{2}e^{8} - \frac{1}{2}e^{7} - \frac{13}{2}e^{6} + 5e^{5} + \frac{51}{2}e^{4} - \frac{29}{2}e^{3} - \frac{55}{2}e^{2} + 14e + 4$ |
23 | $[23, 23, -w - 3]$ | $\phantom{-}\frac{1}{2}e^{8} - \frac{1}{2}e^{7} - 6e^{6} + 4e^{5} + \frac{41}{2}e^{4} - \frac{13}{2}e^{3} - 13e^{2} + e - 6$ |
29 | $[29, 29, -w^{2} + 2w + 4]$ | $-\frac{1}{2}e^{8} + \frac{13}{2}e^{6} + e^{5} - \frac{51}{2}e^{4} - 6e^{3} + \frac{57}{2}e^{2} + 5e - 6$ |
31 | $[31, 31, 2w - 3]$ | $\phantom{-}e^{3} - e^{2} - 5e + 4$ |
41 | $[41, 41, -2w^{2} + 3w + 6]$ | $-\frac{1}{2}e^{6} + \frac{1}{2}e^{5} + \frac{9}{2}e^{4} - \frac{9}{2}e^{3} - 10e^{2} + 10e + 6$ |
43 | $[43, 43, w^{2} - 3w + 3]$ | $\phantom{-}\frac{1}{2}e^{8} - \frac{1}{2}e^{7} - 6e^{6} + \frac{9}{2}e^{5} + 22e^{4} - 11e^{3} - \frac{53}{2}e^{2} + 7e + 12$ |
47 | $[47, 47, w^{2} + w - 4]$ | $\phantom{-}1$ |
49 | $[49, 7, 2w^{2} - 3w - 3]$ | $-\frac{1}{2}e^{7} + \frac{1}{2}e^{6} + \frac{15}{2}e^{5} - \frac{11}{2}e^{4} - 33e^{3} + 15e^{2} + 36e - 6$ |
53 | $[53, 53, w^{2} - 3w - 2]$ | $-\frac{1}{2}e^{8} + e^{7} + \frac{11}{2}e^{6} - 10e^{5} - \frac{35}{2}e^{4} + 30e^{3} + \frac{25}{2}e^{2} - 29e + 2$ |
59 | $[59, 59, 2w^{2} - w - 5]$ | $\phantom{-}e^{8} - e^{7} - 13e^{6} + 10e^{5} + 51e^{4} - 28e^{3} - 55e^{2} + 21e + 8$ |
59 | $[59, 59, w^{2} - w - 7]$ | $-\frac{1}{2}e^{5} - \frac{1}{2}e^{4} + \frac{7}{2}e^{3} + \frac{9}{2}e^{2} - e - 6$ |
59 | $[59, 59, -w^{2} - w + 7]$ | $\phantom{-}\frac{1}{2}e^{8} + e^{7} - \frac{15}{2}e^{6} - 14e^{5} + \frac{67}{2}e^{4} + 55e^{3} - \frac{81}{2}e^{2} - 44e + 10$ |
67 | $[67, 67, 2w^{2} - 3w - 7]$ | $-\frac{1}{2}e^{7} + 6e^{5} - \frac{41}{2}e^{3} + 4e^{2} + 15e - 8$ |
73 | $[73, 73, -w^{2} + 4w - 5]$ | $-e^{7} + 12e^{5} + 3e^{4} - 42e^{3} - 17e^{2} + 35e + 12$ |
79 | $[79, 79, w^{2} - 8]$ | $\phantom{-}e^{8} + \frac{1}{2}e^{7} - \frac{27}{2}e^{6} - 9e^{5} + 53e^{4} + \frac{83}{2}e^{3} - \frac{101}{2}e^{2} - 33e + 6$ |
79 | $[79, 79, w^{2} - 5w + 5]$ | $\phantom{-}2e^{3} - 12e$ |
Atkin-Lehner eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
$47$ | $[47, 47, w^{2} + w - 4]$ | $-1$ |