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: | $[49, 7, 2w^{2} - 3w - 3]$ |
Dimension: | $8$ |
CM: | no |
Base change: | no |
Newspace dimension: | $11$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{8} - 20x^{6} + 8x^{5} + 120x^{4} - 96x^{3} - 192x^{2} + 240x - 64\) |
Show full eigenvalues Hide large eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
3 | $[3, 3, w + 1]$ | $\phantom{-}e$ |
3 | $[3, 3, w - 1]$ | $\phantom{-}\frac{1}{16}e^{7} - \frac{5}{4}e^{5} + \frac{1}{4}e^{4} + \frac{15}{2}e^{3} - 3e^{2} - 13e + 8$ |
7 | $[7, 7, w^{2} - 2]$ | $\phantom{-}\frac{1}{8}e^{7} + \frac{1}{8}e^{6} - \frac{5}{2}e^{5} - \frac{5}{4}e^{4} + \frac{31}{2}e^{3} - 28e + 12$ |
8 | $[8, 2, 2]$ | $\phantom{-}\frac{1}{4}e^{7} + \frac{1}{4}e^{6} - \frac{19}{4}e^{5} - \frac{11}{4}e^{4} + 27e^{3} + 3e^{2} - 43e + 17$ |
11 | $[11, 11, -w^{2} + w + 1]$ | $\phantom{-}\frac{1}{4}e^{7} + \frac{1}{4}e^{6} - \frac{19}{4}e^{5} - \frac{5}{2}e^{4} + \frac{55}{2}e^{3} - 47e + 24$ |
23 | $[23, 23, -w - 3]$ | $-\frac{3}{16}e^{7} - \frac{1}{8}e^{6} + \frac{15}{4}e^{5} + e^{4} - 23e^{3} + \frac{7}{2}e^{2} + 42e - 20$ |
29 | $[29, 29, -w^{2} + 2w + 4]$ | $-\frac{5}{16}e^{7} - \frac{3}{8}e^{6} + \frac{23}{4}e^{5} + 4e^{4} - 32e^{3} - \frac{7}{2}e^{2} + 52e - 22$ |
31 | $[31, 31, 2w - 3]$ | $\phantom{-}\frac{5}{16}e^{7} + \frac{1}{2}e^{6} - \frac{11}{2}e^{5} - \frac{25}{4}e^{4} + 29e^{3} + \frac{31}{2}e^{2} - 45e + 8$ |
41 | $[41, 41, -2w^{2} + 3w + 6]$ | $\phantom{-}\frac{5}{16}e^{7} + \frac{1}{4}e^{6} - \frac{25}{4}e^{5} - \frac{9}{4}e^{4} + \frac{77}{2}e^{3} - 3e^{2} - 70e + 34$ |
43 | $[43, 43, w^{2} - 3w + 3]$ | $-\frac{1}{4}e^{7} - \frac{1}{4}e^{6} + \frac{9}{2}e^{5} + 2e^{4} - \frac{49}{2}e^{3} + 4e^{2} + 39e - 24$ |
47 | $[47, 47, w^{2} + w - 4]$ | $-\frac{3}{16}e^{7} + \frac{15}{4}e^{5} - \frac{3}{2}e^{4} - \frac{47}{2}e^{3} + 16e^{2} + 44e - 32$ |
49 | $[49, 7, 2w^{2} - 3w - 3]$ | $\phantom{-}1$ |
53 | $[53, 53, w^{2} - 3w - 2]$ | $-\frac{3}{16}e^{7} - \frac{1}{4}e^{6} + \frac{7}{2}e^{5} + \frac{11}{4}e^{4} - \frac{39}{2}e^{3} - \frac{5}{2}e^{2} + 30e - 14$ |
59 | $[59, 59, 2w^{2} - w - 5]$ | $-\frac{3}{8}e^{7} - \frac{1}{4}e^{6} + \frac{29}{4}e^{5} + \frac{5}{2}e^{4} - 42e^{3} - 2e^{2} + 68e - 20$ |
59 | $[59, 59, w^{2} - w - 7]$ | $-\frac{1}{16}e^{7} - \frac{1}{4}e^{6} + e^{5} + \frac{17}{4}e^{4} - \frac{9}{2}e^{3} - \frac{37}{2}e^{2} + 6e + 16$ |
59 | $[59, 59, -w^{2} - w + 7]$ | $\phantom{-}\frac{1}{4}e^{7} + \frac{1}{8}e^{6} - \frac{19}{4}e^{5} - \frac{3}{4}e^{4} + \frac{55}{2}e^{3} - 6e^{2} - 48e + 32$ |
67 | $[67, 67, 2w^{2} - 3w - 7]$ | $\phantom{-}\frac{7}{16}e^{7} + \frac{1}{2}e^{6} - \frac{31}{4}e^{5} - \frac{21}{4}e^{4} + \frac{81}{2}e^{3} + 5e^{2} - 60e + 28$ |
73 | $[73, 73, -w^{2} + 4w - 5]$ | $\phantom{-}\frac{1}{16}e^{7} + \frac{1}{4}e^{6} - \frac{3}{4}e^{5} - \frac{15}{4}e^{4} + \frac{1}{2}e^{3} + \frac{27}{2}e^{2} + 8e - 10$ |
79 | $[79, 79, w^{2} - 8]$ | $-\frac{7}{16}e^{7} - \frac{3}{8}e^{6} + \frac{33}{4}e^{5} + 3e^{4} - \frac{95}{2}e^{3} + \frac{15}{2}e^{2} + 79e - 48$ |
79 | $[79, 79, w^{2} - 5w + 5]$ | $-\frac{1}{2}e^{7} - \frac{1}{2}e^{6} + \frac{37}{4}e^{5} + 5e^{4} - \frac{105}{2}e^{3} - 2e^{2} + 89e - 36$ |
Atkin-Lehner eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
$49$ | $[49, 7, 2w^{2} - 3w - 3]$ | $-1$ |