Base field \(\Q(\zeta_{13})^+\)
Generator \(w\), with minimal polynomial \(x^{6} - x^{5} - 5x^{4} + 4x^{3} + 6x^{2} - 3x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2, 2]$ |
| Level: | $[79, 79, -2w^{5} + w^{4} + 9w^{3} - 3w^{2} - 9w + 2]$ |
| Dimension: | $5$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $9$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{5} - 7x^{4} - 8x^{3} + 106x^{2} - 19x - 361\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 13 | $[13, 13, w^{5} - 5w^{3} + 4w]$ | $\phantom{-}e$ |
| 25 | $[25, 5, w^{5} - 5w^{3} + 6w - 1]$ | $\phantom{-}\frac{17}{38}e^{4} - \frac{31}{19}e^{3} - \frac{163}{19}e^{2} + \frac{312}{19}e + \frac{91}{2}$ |
| 25 | $[25, 5, -w^{3} + w^{2} + 3w - 1]$ | $\phantom{-}\frac{5}{19}e^{4} - \frac{16}{19}e^{3} - \frac{116}{19}e^{2} + \frac{188}{19}e + 30$ |
| 25 | $[25, 5, w^{5} - 4w^{3} - w^{2} + 3w + 2]$ | $-\frac{7}{38}e^{4} + \frac{15}{19}e^{3} + \frac{47}{19}e^{2} - \frac{124}{19}e - \frac{17}{2}$ |
| 27 | $[27, 3, w^{4} - w^{3} - 4w^{2} + 2w + 2]$ | $-\frac{7}{38}e^{4} + \frac{15}{19}e^{3} + \frac{66}{19}e^{2} - \frac{162}{19}e - \frac{37}{2}$ |
| 27 | $[27, 3, w^{4} - w^{3} - 4w^{2} + 2w + 3]$ | $\phantom{-}\frac{4}{19}e^{4} - \frac{9}{19}e^{3} - \frac{108}{19}e^{2} + \frac{101}{19}e + 32$ |
| 53 | $[53, 53, -w^{4} + w^{3} + 3w^{2} - 2w + 1]$ | $-\frac{17}{38}e^{4} + \frac{31}{19}e^{3} + \frac{163}{19}e^{2} - \frac{312}{19}e - \frac{83}{2}$ |
| 53 | $[53, 53, -w^{4} + w^{3} + 4w^{2} - 3w - 4]$ | $-\frac{17}{38}e^{4} + \frac{31}{19}e^{3} + \frac{163}{19}e^{2} - \frac{312}{19}e - \frac{83}{2}$ |
| 53 | $[53, 53, -w^{5} + w^{4} + 4w^{3} - 4w^{2} - 3w + 1]$ | $\phantom{-}\frac{7}{38}e^{4} - \frac{15}{19}e^{3} - \frac{47}{19}e^{2} + \frac{124}{19}e + \frac{25}{2}$ |
| 53 | $[53, 53, w^{3} - 2w - 2]$ | $-\frac{17}{38}e^{4} + \frac{31}{19}e^{3} + \frac{163}{19}e^{2} - \frac{312}{19}e - \frac{83}{2}$ |
| 53 | $[53, 53, w^{5} - 5w^{3} - w^{2} + 5w]$ | $\phantom{-}\frac{3}{19}e^{4} - \frac{21}{19}e^{3} - \frac{24}{19}e^{2} + \frac{204}{19}e + 9$ |
| 53 | $[53, 53, -w^{4} + 4w^{2} + w - 4]$ | $\phantom{-}\frac{3}{38}e^{4} - \frac{1}{19}e^{3} - \frac{31}{19}e^{2} - \frac{50}{19}e + \frac{17}{2}$ |
| 64 | $[64, 2, -2]$ | $\phantom{-}\frac{12}{19}e^{4} - \frac{46}{19}e^{3} - \frac{229}{19}e^{2} + \frac{436}{19}e + 68$ |
| 79 | $[79, 79, -2w^{5} + w^{4} + 9w^{3} - 3w^{2} - 9w + 2]$ | $-1$ |
| 79 | $[79, 79, -w^{5} - w^{4} + 5w^{3} + 4w^{2} - 6w - 1]$ | $-\frac{5}{38}e^{4} + \frac{8}{19}e^{3} + \frac{39}{19}e^{2} - \frac{94}{19}e - \frac{7}{2}$ |
| 79 | $[79, 79, w^{3} - w^{2} - 4w + 1]$ | $-\frac{2}{19}e^{4} + \frac{14}{19}e^{3} + \frac{16}{19}e^{2} - \frac{174}{19}e$ |
| 79 | $[79, 79, -2w^{5} + 2w^{4} + 9w^{3} - 7w^{2} - 9w + 3]$ | $\phantom{-}\frac{5}{19}e^{4} - \frac{16}{19}e^{3} - \frac{116}{19}e^{2} + \frac{226}{19}e + 37$ |
| 79 | $[79, 79, -2w^{4} + w^{3} + 7w^{2} - 3w - 3]$ | $-\frac{21}{38}e^{4} + \frac{45}{19}e^{3} + \frac{160}{19}e^{2} - \frac{410}{19}e - \frac{71}{2}$ |
| 79 | $[79, 79, -w^{5} + 6w^{3} - w^{2} - 8w + 1]$ | $-\frac{10}{19}e^{4} + \frac{32}{19}e^{3} + \frac{232}{19}e^{2} - \frac{338}{19}e - 74$ |
| 103 | $[103, 103, 2w^{4} - 7w^{2} - w + 3]$ | $-\frac{5}{19}e^{4} + \frac{16}{19}e^{3} + \frac{116}{19}e^{2} - \frac{188}{19}e - 33$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $79$ | $[79, 79, -2w^{5} + w^{4} + 9w^{3} - 3w^{2} - 9w + 2]$ | $1$ |