Base field 4.4.19025.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 13x^{2} + 14x + 44\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16, 2, 2]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $32$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} - x^{8} - 25x^{7} + 25x^{6} + 160x^{5} - 215x^{4} - 174x^{3} + 274x^{2} - 12x - 8\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, w^{2} - 2w - 6]$ | $-1$ |
| 4 | $[4, 2, -w^{2} + 7]$ | $-1$ |
| 5 | $[5, 5, -\frac{1}{2}w^{3} + 2w^{2} + \frac{7}{2}w - 14]$ | $\phantom{-}e$ |
| 5 | $[5, 5, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - 6w - 9]$ | $\phantom{-}e$ |
| 11 | $[11, 11, \frac{1}{2}w^{3} - 2w^{2} - \frac{5}{2}w + 11]$ | $\phantom{-}\frac{7}{20}e^{8} + \frac{3}{20}e^{7} - \frac{173}{20}e^{6} - \frac{67}{20}e^{5} + \frac{531}{10}e^{4} - \frac{57}{20}e^{3} - \frac{349}{5}e^{2} + \frac{53}{10}e + 4$ |
| 11 | $[11, 11, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - 5w - 7]$ | $\phantom{-}\frac{7}{20}e^{8} + \frac{3}{20}e^{7} - \frac{173}{20}e^{6} - \frac{67}{20}e^{5} + \frac{531}{10}e^{4} - \frac{57}{20}e^{3} - \frac{349}{5}e^{2} + \frac{53}{10}e + 4$ |
| 31 | $[31, 31, \frac{1}{2}w^{2} + \frac{1}{2}w - 4]$ | $-\frac{1}{5}e^{8} + 5e^{6} - 32e^{4} + 11e^{3} + \frac{229}{5}e^{2} - 10e - \frac{38}{5}$ |
| 31 | $[31, 31, -\frac{1}{2}w^{2} + \frac{3}{2}w + 3]$ | $-\frac{1}{5}e^{8} + 5e^{6} - 32e^{4} + 11e^{3} + \frac{229}{5}e^{2} - 10e - \frac{38}{5}$ |
| 41 | $[41, 41, \frac{1}{2}w^{2} + \frac{1}{2}w - 6]$ | $-\frac{13}{20}e^{8} - \frac{7}{20}e^{7} + \frac{317}{20}e^{6} + \frac{163}{20}e^{5} - \frac{472}{5}e^{4} - \frac{137}{20}e^{3} + \frac{1167}{10}e^{2} + \frac{83}{10}e - 1$ |
| 41 | $[41, 41, 2w^{3} - \frac{15}{2}w^{2} - \frac{25}{2}w + 50]$ | $-\frac{3}{4}e^{8} - \frac{13}{20}e^{7} + \frac{363}{20}e^{6} + \frac{297}{20}e^{5} - \frac{528}{5}e^{4} - \frac{683}{20}e^{3} + \frac{1349}{10}e^{2} + \frac{327}{10}e - \frac{31}{5}$ |
| 41 | $[41, 41, \frac{5}{2}w^{2} - \frac{1}{2}w - 17]$ | $-\frac{3}{4}e^{8} - \frac{13}{20}e^{7} + \frac{363}{20}e^{6} + \frac{297}{20}e^{5} - \frac{528}{5}e^{4} - \frac{683}{20}e^{3} + \frac{1349}{10}e^{2} + \frac{327}{10}e - \frac{31}{5}$ |
| 41 | $[41, 41, \frac{1}{2}w^{2} - \frac{3}{2}w - 5]$ | $-\frac{13}{20}e^{8} - \frac{7}{20}e^{7} + \frac{317}{20}e^{6} + \frac{163}{20}e^{5} - \frac{472}{5}e^{4} - \frac{137}{20}e^{3} + \frac{1167}{10}e^{2} + \frac{83}{10}e - 1$ |
| 61 | $[61, 61, -\frac{1}{2}w^{3} + w^{2} + \frac{7}{2}w - 1]$ | $\phantom{-}\frac{7}{10}e^{8} + \frac{1}{10}e^{7} - \frac{171}{10}e^{6} - \frac{19}{10}e^{5} + \frac{517}{5}e^{4} - \frac{369}{10}e^{3} - \frac{647}{5}e^{2} + \frac{281}{5}e + \frac{22}{5}$ |
| 61 | $[61, 61, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + 4w - 3]$ | $\phantom{-}\frac{7}{10}e^{8} + \frac{1}{10}e^{7} - \frac{171}{10}e^{6} - \frac{19}{10}e^{5} + \frac{517}{5}e^{4} - \frac{369}{10}e^{3} - \frac{647}{5}e^{2} + \frac{281}{5}e + \frac{22}{5}$ |
| 71 | $[71, 71, \frac{1}{2}w^{3} + w^{2} - \frac{11}{2}w - 13]$ | $-\frac{7}{10}e^{8} - \frac{1}{10}e^{7} + \frac{171}{10}e^{6} + \frac{19}{10}e^{5} - \frac{512}{5}e^{4} + \frac{349}{10}e^{3} + \frac{577}{5}e^{2} - \frac{201}{5}e + \frac{18}{5}$ |
| 71 | $[71, 71, -\frac{1}{2}w^{3} + \frac{5}{2}w^{2} + 2w - 17]$ | $-\frac{7}{10}e^{8} - \frac{1}{10}e^{7} + \frac{171}{10}e^{6} + \frac{19}{10}e^{5} - \frac{512}{5}e^{4} + \frac{349}{10}e^{3} + \frac{577}{5}e^{2} - \frac{201}{5}e + \frac{18}{5}$ |
| 81 | $[81, 3, -3]$ | $-\frac{1}{5}e^{8} + 5e^{6} - 31e^{4} + 10e^{3} + \frac{154}{5}e^{2} - 4e + \frac{82}{5}$ |
| 89 | $[89, 89, -w^{3} + \frac{3}{2}w^{2} + \frac{13}{2}w - 9]$ | $-\frac{1}{4}e^{8} + \frac{1}{4}e^{7} + \frac{25}{4}e^{6} - \frac{25}{4}e^{5} - 40e^{4} + \frac{215}{4}e^{3} + \frac{87}{2}e^{2} - \frac{133}{2}e + 3$ |
| 89 | $[89, 89, w^{3} - \frac{7}{2}w^{2} - \frac{11}{2}w + 20]$ | $\phantom{-}\frac{9}{20}e^{8} - \frac{3}{20}e^{7} - \frac{227}{20}e^{6} + \frac{87}{20}e^{5} + \frac{729}{10}e^{4} - \frac{1163}{20}e^{3} - \frac{452}{5}e^{2} + \frac{807}{10}e + \frac{2}{5}$ |
| 89 | $[89, 89, -w^{3} - \frac{1}{2}w^{2} + \frac{19}{2}w + 12]$ | $\phantom{-}\frac{9}{20}e^{8} - \frac{3}{20}e^{7} - \frac{227}{20}e^{6} + \frac{87}{20}e^{5} + \frac{729}{10}e^{4} - \frac{1163}{20}e^{3} - \frac{452}{5}e^{2} + \frac{807}{10}e + \frac{2}{5}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, w^{2} - 2w - 6]$ | $1$ |
| $4$ | $[4, 2, -w^{2} + 7]$ | $1$ |