Base field 4.4.17609.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 7x^{2} + 10x - 1\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[7, 7, -w^{3} + 6w - 2]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $16$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 3x^{7} - 9x^{6} + 30x^{5} + 19x^{4} - 89x^{3} + 7x^{2} + 80x - 32\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w + 1]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w^{3} + w^{2} - 4w + 1]$ | $-\frac{1}{4}e^{7} + \frac{1}{4}e^{6} + \frac{13}{4}e^{5} - 3e^{4} - \frac{53}{4}e^{3} + \frac{45}{4}e^{2} + \frac{67}{4}e - 12$ |
| 7 | $[7, 7, -w^{3} + 6w - 2]$ | $-1$ |
| 8 | $[8, 2, w^{3} - 7w + 3]$ | $\phantom{-}\frac{1}{4}e^{7} - \frac{3}{4}e^{6} - \frac{9}{4}e^{5} + \frac{13}{2}e^{4} + \frac{23}{4}e^{3} - \frac{57}{4}e^{2} - \frac{17}{4}e + 9$ |
| 11 | $[11, 11, -w^{3} + 6w - 4]$ | $\phantom{-}\frac{3}{4}e^{7} - \frac{5}{4}e^{6} - \frac{31}{4}e^{5} + \frac{21}{2}e^{4} + \frac{93}{4}e^{3} - \frac{91}{4}e^{2} - \frac{83}{4}e + 13$ |
| 17 | $[17, 17, w^{3} + w^{2} - 6w - 1]$ | $\phantom{-}\frac{5}{4}e^{7} - \frac{9}{4}e^{6} - \frac{53}{4}e^{5} + 19e^{4} + \frac{173}{4}e^{3} - \frac{165}{4}e^{2} - \frac{175}{4}e + 22$ |
| 17 | $[17, 17, -w^{2} - w + 3]$ | $-e^{7} + \frac{3}{2}e^{6} + 11e^{5} - \frac{27}{2}e^{4} - \frac{73}{2}e^{3} + 35e^{2} + \frac{73}{2}e - 27$ |
| 31 | $[31, 31, 2w^{3} + w^{2} - 13w + 3]$ | $-\frac{3}{2}e^{7} + \frac{5}{2}e^{6} + \frac{35}{2}e^{5} - 24e^{4} - \frac{125}{2}e^{3} + \frac{131}{2}e^{2} + \frac{131}{2}e - 49$ |
| 37 | $[37, 37, -3w^{3} - w^{2} + 18w - 3]$ | $\phantom{-}\frac{1}{2}e^{7} - e^{6} - \frac{9}{2}e^{5} + \frac{15}{2}e^{4} + 11e^{3} - \frac{21}{2}e^{2} - 8e + 1$ |
| 41 | $[41, 41, w^{2} + 2w - 4]$ | $-\frac{1}{4}e^{7} + \frac{1}{4}e^{6} + \frac{9}{4}e^{5} - 2e^{4} - \frac{21}{4}e^{3} + \frac{17}{4}e^{2} + \frac{19}{4}e + 1$ |
| 47 | $[47, 47, 2w^{3} + 2w^{2} - 11w - 2]$ | $\phantom{-}e^{5} - e^{4} - 8e^{3} + 7e^{2} + 11e - 10$ |
| 47 | $[47, 47, -3w^{3} - w^{2} + 19w - 6]$ | $\phantom{-}\frac{3}{2}e^{7} - \frac{3}{2}e^{6} - \frac{37}{2}e^{5} + 14e^{4} + \frac{139}{2}e^{3} - \frac{75}{2}e^{2} - \frac{155}{2}e + 32$ |
| 59 | $[59, 59, -2w^{3} + 13w - 6]$ | $\phantom{-}\frac{1}{2}e^{7} - \frac{1}{2}e^{6} - \frac{11}{2}e^{5} + 4e^{4} + \frac{33}{2}e^{3} - \frac{15}{2}e^{2} - \frac{17}{2}e + 3$ |
| 59 | $[59, 59, -w^{3} - w^{2} + 5w + 2]$ | $-\frac{5}{4}e^{7} + \frac{3}{4}e^{6} + \frac{61}{4}e^{5} - \frac{11}{2}e^{4} - \frac{227}{4}e^{3} + \frac{49}{4}e^{2} + \frac{249}{4}e - 16$ |
| 59 | $[59, 59, w^{2} + w - 1]$ | $-\frac{3}{4}e^{7} + \frac{5}{4}e^{6} + \frac{31}{4}e^{5} - \frac{19}{2}e^{4} - \frac{93}{4}e^{3} + \frac{59}{4}e^{2} + \frac{83}{4}e + 2$ |
| 59 | $[59, 59, w^{2} + 2w - 6]$ | $\phantom{-}\frac{1}{4}e^{7} + \frac{5}{4}e^{6} - \frac{21}{4}e^{5} - \frac{23}{2}e^{4} + \frac{107}{4}e^{3} + \frac{99}{4}e^{2} - \frac{133}{4}e - 6$ |
| 61 | $[61, 61, -w^{3} + w^{2} + 8w - 7]$ | $\phantom{-}\frac{1}{4}e^{7} - \frac{1}{4}e^{6} - \frac{17}{4}e^{5} + 4e^{4} + \frac{81}{4}e^{3} - \frac{69}{4}e^{2} - \frac{103}{4}e + 18$ |
| 67 | $[67, 67, w^{3} + 2w^{2} - 4w - 4]$ | $\phantom{-}\frac{1}{2}e^{7} - \frac{1}{2}e^{6} - \frac{15}{2}e^{5} + 6e^{4} + \frac{69}{2}e^{3} - \frac{39}{2}e^{2} - \frac{93}{2}e + 15$ |
| 73 | $[73, 73, -2w^{3} - w^{2} + 12w - 4]$ | $\phantom{-}\frac{1}{4}e^{7} + \frac{1}{4}e^{6} - \frac{13}{4}e^{5} - \frac{5}{2}e^{4} + \frac{47}{4}e^{3} + \frac{11}{4}e^{2} - \frac{49}{4}e + 14$ |
| 81 | $[81, 3, -3]$ | $-\frac{1}{2}e^{7} + 2e^{6} + \frac{7}{2}e^{5} - \frac{33}{2}e^{4} - 4e^{3} + \frac{57}{2}e^{2} - 3e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $7$ | $[7, 7, -w^{3} + 6w - 2]$ | $1$ |