Base field 6.6.300125.1
Generator \(w\), with minimal polynomial \(x^{6} - x^{5} - 7x^{4} + 2x^{3} + 7x^{2} - 2x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2, 2]$ |
| Level: | $[169,13,w^{5} - w^{4} - 6w^{3} + w^{2} + 2w - 2]$ |
| Dimension: | $6$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $15$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{6} - 9x^{5} - 48x^{4} + 304x^{3} + 816x^{2} - 2112x - 4800\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 29 | $[29, 29, -9w^{5} + 3w^{4} + 64w^{3} + 26w^{2} - 40w - 10]$ | $\phantom{-}\frac{31}{4176}e^{5} - \frac{295}{4176}e^{4} - \frac{275}{1044}e^{3} + \frac{213}{116}e^{2} + \frac{499}{174}e - \frac{634}{87}$ |
| 29 | $[29, 29, w^{5} - 7w^{3} - 5w^{2} + 2w + 2]$ | $-e + 2$ |
| 29 | $[29, 29, w^{4} - w^{3} - 6w^{2} + 2]$ | $-e + 2$ |
| 29 | $[29, 29, 5w^{5} - w^{4} - 36w^{3} - 19w^{2} + 21w + 9]$ | $\phantom{-}e$ |
| 29 | $[29, 29, -w^{5} + w^{4} + 7w^{3} - 2w^{2} - 6w + 1]$ | $\phantom{-}\frac{31}{4176}e^{5} - \frac{295}{4176}e^{4} - \frac{275}{1044}e^{3} + \frac{213}{116}e^{2} + \frac{499}{174}e - \frac{634}{87}$ |
| 29 | $[29, 29, 2w^{5} - 15w^{3} - 10w^{2} + 11w + 5]$ | $\phantom{-}e$ |
| 41 | $[41, 41, 5w^{5} - w^{4} - 36w^{3} - 18w^{2} + 21w + 5]$ | $\phantom{-}\frac{1}{174}e^{5} - \frac{17}{174}e^{4} + \frac{5}{29}e^{3} + \frac{88}{29}e^{2} - \frac{162}{29}e - \frac{622}{29}$ |
| 41 | $[41, 41, -5w^{5} + 2w^{4} + 36w^{3} + 11w^{2} - 25w - 2]$ | $-\frac{1}{1044}e^{5} + \frac{5}{2088}e^{4} + \frac{259}{2088}e^{3} - \frac{49}{116}e^{2} - \frac{331}{174}e + \frac{514}{87}$ |
| 41 | $[41, 41, 6w^{5} - w^{4} - 44w^{3} - 23w^{2} + 30w + 8]$ | $-\frac{43}{4176}e^{5} + \frac{499}{4176}e^{4} + \frac{185}{1044}e^{3} - \frac{389}{116}e^{2} - \frac{50}{87}e + \frac{1306}{87}$ |
| 41 | $[41, 41, 13w^{5} - 4w^{4} - 93w^{3} - 39w^{2} + 59w + 16]$ | $-\frac{1}{1044}e^{5} + \frac{5}{2088}e^{4} + \frac{259}{2088}e^{3} - \frac{49}{116}e^{2} - \frac{331}{174}e + \frac{514}{87}$ |
| 41 | $[41, 41, -4w^{5} + 30w^{3} + 19w^{2} - 19w - 8]$ | $\phantom{-}\frac{1}{174}e^{5} - \frac{17}{174}e^{4} + \frac{5}{29}e^{3} + \frac{88}{29}e^{2} - \frac{162}{29}e - \frac{622}{29}$ |
| 41 | $[41, 41, w^{5} - 7w^{3} - 6w^{2} + 2w + 3]$ | $-\frac{43}{4176}e^{5} + \frac{499}{4176}e^{4} + \frac{185}{1044}e^{3} - \frac{389}{116}e^{2} - \frac{50}{87}e + \frac{1306}{87}$ |
| 49 | $[49, 7, -5w^{5} + w^{4} + 36w^{3} + 19w^{2} - 22w - 6]$ | $\phantom{-}\frac{7}{174}e^{5} - \frac{151}{348}e^{4} - \frac{121}{116}e^{3} + \frac{384}{29}e^{2} + \frac{84}{29}e - \frac{1918}{29}$ |
| 64 | $[64, 2, -2]$ | $\phantom{-}\frac{61}{4176}e^{5} - \frac{805}{4176}e^{4} - \frac{25}{522}e^{3} + \frac{341}{58}e^{2} - \frac{445}{87}e - \frac{2575}{87}$ |
| 71 | $[71, 71, -8w^{5} + w^{4} + 58w^{3} + 34w^{2} - 34w - 16]$ | $\phantom{-}\frac{37}{1392}e^{5} - \frac{397}{1392}e^{4} - \frac{115}{174}e^{3} + \frac{495}{58}e^{2} + \frac{12}{29}e - \frac{1144}{29}$ |
| 71 | $[71, 71, -6w^{5} + 2w^{4} + 42w^{3} + 18w^{2} - 23w - 6]$ | $-\frac{23}{2088}e^{5} + \frac{47}{522}e^{4} + \frac{1021}{2088}e^{3} - \frac{259}{116}e^{2} - \frac{805}{174}e + \frac{604}{87}$ |
| 71 | $[71, 71, -8w^{5} + 2w^{4} + 58w^{3} + 27w^{2} - 38w - 10]$ | $\phantom{-}\frac{97}{4176}e^{5} - \frac{1069}{4176}e^{4} - \frac{119}{261}e^{3} + \frac{833}{116}e^{2} - \frac{130}{87}e - \frac{3460}{87}$ |
| 71 | $[71, 71, 4w^{5} - 30w^{3} - 19w^{2} + 20w + 8]$ | $\phantom{-}\frac{97}{4176}e^{5} - \frac{1069}{4176}e^{4} - \frac{119}{261}e^{3} + \frac{833}{116}e^{2} - \frac{130}{87}e - \frac{3460}{87}$ |
| 71 | $[71, 71, -10w^{5} + 3w^{4} + 72w^{3} + 30w^{2} - 48w - 10]$ | $\phantom{-}\frac{37}{1392}e^{5} - \frac{397}{1392}e^{4} - \frac{115}{174}e^{3} + \frac{495}{58}e^{2} + \frac{12}{29}e - \frac{1144}{29}$ |
| 71 | $[71, 71, -8w^{5} + 2w^{4} + 58w^{3} + 26w^{2} - 37w - 8]$ | $-\frac{23}{2088}e^{5} + \frac{47}{522}e^{4} + \frac{1021}{2088}e^{3} - \frac{259}{116}e^{2} - \frac{805}{174}e + \frac{604}{87}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $169$ | $[169,13,w^{5} - w^{4} - 6w^{3} + w^{2} + 2w - 2]$ | $1$ |