Base field 6.6.1397493.1
Generator \(w\), with minimal polynomial \(x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2, 2]$ |
| Level: | $[53,53,w^{5} - 3w^{4} - 3w^{3} + 9w^{2} + 3w - 2]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $46$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} + 2x^{7} - 12x^{6} - 18x^{5} + 44x^{4} + 43x^{3} - 63x^{2} - 27x + 27\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w - 1]$ | $\phantom{-}e$ |
| 17 | $[17, 17, -w^{2} + 2w + 1]$ | $\phantom{-}\frac{4}{9}e^{7} + \frac{8}{9}e^{6} - \frac{13}{3}e^{5} - 7e^{4} + \frac{77}{9}e^{3} + \frac{127}{9}e^{2} - e - 10$ |
| 17 | $[17, 17, -w^{3} + w^{2} + 4w]$ | $\phantom{-}\frac{2}{9}e^{7} + \frac{10}{9}e^{6} - \frac{4}{3}e^{5} - 10e^{4} + \frac{7}{9}e^{3} + \frac{170}{9}e^{2} - \frac{7}{3}e - 4$ |
| 19 | $[19, 19, w^{5} - 3w^{4} - 2w^{3} + 8w^{2} + w - 4]$ | $\phantom{-}\frac{1}{9}e^{7} - \frac{1}{9}e^{6} - 2e^{5} + e^{4} + \frac{80}{9}e^{3} + \frac{1}{9}e^{2} - \frac{22}{3}e - 4$ |
| 19 | $[19, 19, w^{2} - w - 1]$ | $\phantom{-}\frac{1}{27}e^{7} - \frac{10}{27}e^{6} - e^{5} + \frac{13}{3}e^{4} + \frac{125}{27}e^{3} - \frac{350}{27}e^{2} - \frac{43}{9}e + \frac{28}{3}$ |
| 37 | $[37, 37, w^{4} - 2w^{3} - 3w^{2} + 3w + 2]$ | $\phantom{-}\frac{5}{27}e^{7} + \frac{49}{27}e^{6} + \frac{1}{3}e^{5} - \frac{55}{3}e^{4} - \frac{266}{27}e^{3} + \frac{1202}{27}e^{2} + \frac{115}{9}e - \frac{73}{3}$ |
| 37 | $[37, 37, w^{4} - 4w^{3} + w^{2} + 7w - 3]$ | $-\frac{17}{27}e^{7} - \frac{55}{27}e^{6} + \frac{13}{3}e^{5} + \frac{49}{3}e^{4} - \frac{19}{27}e^{3} - \frac{764}{27}e^{2} - \frac{64}{9}e + \frac{25}{3}$ |
| 53 | $[53, 53, 2w^{5} - 6w^{4} - 6w^{3} + 18w^{2} + 9w - 6]$ | $\phantom{-}\frac{26}{27}e^{7} + \frac{64}{27}e^{6} - 9e^{5} - \frac{58}{3}e^{4} + \frac{496}{27}e^{3} + \frac{998}{27}e^{2} - \frac{110}{9}e - \frac{55}{3}$ |
| 53 | $[53, 53, -w^{5} + 3w^{4} + 3w^{3} - 9w^{2} - 3w + 2]$ | $\phantom{-}1$ |
| 64 | $[64, 2, -2]$ | $\phantom{-}\frac{16}{27}e^{7} + \frac{11}{27}e^{6} - \frac{19}{3}e^{5} + \frac{4}{3}e^{4} + \frac{407}{27}e^{3} - \frac{560}{27}e^{2} - \frac{82}{9}e + \frac{61}{3}$ |
| 71 | $[71, 71, -w^{5} + 4w^{4} - w^{3} - 8w^{2} + 3w - 1]$ | $\phantom{-}\frac{8}{27}e^{7} + \frac{37}{27}e^{6} - \frac{7}{3}e^{5} - \frac{40}{3}e^{4} + \frac{136}{27}e^{3} + \frac{809}{27}e^{2} - \frac{62}{9}e - \frac{49}{3}$ |
| 71 | $[71, 71, 2w^{5} - 5w^{4} - 8w^{3} + 15w^{2} + 12w - 6]$ | $\phantom{-}e^{7} - 13e^{5} + 6e^{4} + 41e^{3} - 29e^{2} - 29e + 29$ |
| 71 | $[71, 71, 2w^{5} - 6w^{4} - 6w^{3} + 18w^{2} + 9w - 7]$ | $-\frac{2}{9}e^{7} - \frac{1}{9}e^{6} + \frac{10}{3}e^{5} - \frac{142}{9}e^{3} + \frac{28}{9}e^{2} + \frac{67}{3}e - 4$ |
| 73 | $[73, 73, -2w^{5} + 6w^{4} + 5w^{3} - 16w^{2} - 7w + 3]$ | $-\frac{32}{27}e^{7} - \frac{40}{27}e^{6} + \frac{40}{3}e^{5} + \frac{28}{3}e^{4} - \frac{976}{27}e^{3} - \frac{320}{27}e^{2} + \frac{212}{9}e + \frac{10}{3}$ |
| 73 | $[73, 73, -2w^{5} + 6w^{4} + 5w^{3} - 17w^{2} - 6w + 6]$ | $-\frac{29}{27}e^{7} - \frac{43}{27}e^{6} + \frac{34}{3}e^{5} + \frac{31}{3}e^{4} - \frac{736}{27}e^{3} - \frac{344}{27}e^{2} + \frac{110}{9}e - \frac{2}{3}$ |
| 89 | $[89, 89, w^{5} - 2w^{4} - 5w^{3} + 6w^{2} + 8w - 4]$ | $-\frac{32}{27}e^{7} - \frac{49}{27}e^{6} + \frac{41}{3}e^{5} + \frac{40}{3}e^{4} - \frac{1111}{27}e^{3} - \frac{554}{27}e^{2} + \frac{263}{9}e - \frac{8}{3}$ |
| 89 | $[89, 89, w^{5} - w^{4} - 9w^{3} + 7w^{2} + 16w - 6]$ | $-\frac{2}{9}e^{7} - \frac{16}{9}e^{6} + 16e^{4} + \frac{74}{9}e^{3} - \frac{272}{9}e^{2} - \frac{28}{3}e + 10$ |
| 89 | $[89, 89, 2w^{5} - 6w^{4} - 4w^{3} + 15w^{2} + 3w - 3]$ | $\phantom{-}\frac{5}{27}e^{7} + \frac{4}{27}e^{6} - 2e^{5} + \frac{2}{3}e^{4} + \frac{166}{27}e^{3} - \frac{211}{27}e^{2} - \frac{44}{9}e + \frac{17}{3}$ |
| 89 | $[89, 89, w^{5} - 2w^{4} - 6w^{3} + 8w^{2} + 10w - 6]$ | $-\frac{2}{9}e^{7} + \frac{2}{9}e^{6} + 4e^{5} - 2e^{4} - \frac{169}{9}e^{3} + \frac{7}{9}e^{2} + \frac{68}{3}e + 1$ |
| 107 | $[107, 107, w^{5} - 2w^{4} - 7w^{3} + 10w^{2} + 12w - 6]$ | $-\frac{8}{27}e^{7} - \frac{55}{27}e^{6} + 2e^{5} + \frac{67}{3}e^{4} - \frac{55}{27}e^{3} - \frac{1655}{27}e^{2} - \frac{34}{9}e + \frac{112}{3}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $53$ | $[53,53,w^{5} - 3w^{4} - 3w^{3} + 9w^{2} + 3w - 2]$ | $-1$ |