Base field 4.4.19225.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 15x^{2} + 2x + 44\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16, 2, 2]$ |
| Dimension: | $7$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $29$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{7} - 5x^{6} - 39x^{5} + 171x^{4} + 315x^{3} - 663x^{2} - 1105x - 250\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, w + 2]$ | $-1$ |
| 4 | $[4, 2, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{9}{2}w - 10]$ | $-1$ |
| 9 | $[9, 3, \frac{1}{2}w^{3} - \frac{5}{2}w^{2} - \frac{5}{2}w + 17]$ | $\phantom{-}e$ |
| 9 | $[9, 3, \frac{3}{2}w^{3} - \frac{11}{2}w^{2} - \frac{19}{2}w + 28]$ | $\phantom{-}e$ |
| 11 | $[11, 11, \frac{1}{2}w^{3} - \frac{3}{2}w^{2} - \frac{9}{2}w + 11]$ | $\phantom{-}\frac{1}{135}e^{6} - \frac{1}{9}e^{5} - \frac{8}{45}e^{4} + \frac{167}{45}e^{3} - \frac{7}{9}e^{2} - \frac{172}{15}e - \frac{122}{27}$ |
| 11 | $[11, 11, -w - 3]$ | $\phantom{-}\frac{1}{135}e^{6} - \frac{1}{9}e^{5} - \frac{8}{45}e^{4} + \frac{167}{45}e^{3} - \frac{7}{9}e^{2} - \frac{172}{15}e - \frac{122}{27}$ |
| 25 | $[25, 5, w^{3} - 3w^{2} - 7w + 15]$ | $-\frac{2}{135}e^{6} + \frac{1}{45}e^{5} + \frac{22}{45}e^{4} - \frac{22}{45}e^{3} - \frac{47}{45}e^{2} - 4e - \frac{8}{27}$ |
| 29 | $[29, 29, w + 1]$ | $\phantom{-}\frac{4}{135}e^{6} - \frac{7}{45}e^{5} - \frac{13}{15}e^{4} + \frac{26}{5}e^{3} - \frac{7}{15}e^{2} - \frac{154}{9}e - \frac{80}{27}$ |
| 29 | $[29, 29, \frac{1}{2}w^{3} - \frac{3}{2}w^{2} - \frac{9}{2}w + 9]$ | $\phantom{-}\frac{4}{135}e^{6} - \frac{7}{45}e^{5} - \frac{13}{15}e^{4} + \frac{26}{5}e^{3} - \frac{7}{15}e^{2} - \frac{154}{9}e - \frac{80}{27}$ |
| 31 | $[31, 31, -\frac{1}{2}w^{3} + \frac{5}{2}w^{2} + \frac{5}{2}w - 16]$ | $-\frac{2}{135}e^{6} + \frac{1}{5}e^{5} + \frac{4}{9}e^{4} - \frac{311}{45}e^{3} - \frac{43}{45}e^{2} + \frac{1199}{45}e + \frac{358}{27}$ |
| 31 | $[31, 31, \frac{1}{2}w^{3} - \frac{3}{2}w^{2} - \frac{9}{2}w + 5]$ | $\phantom{-}\frac{4}{135}e^{6} - \frac{4}{45}e^{5} - \frac{17}{15}e^{4} + 3e^{3} + \frac{116}{15}e^{2} - \frac{521}{45}e - \frac{332}{27}$ |
| 31 | $[31, 31, -w + 3]$ | $\phantom{-}\frac{4}{135}e^{6} - \frac{4}{45}e^{5} - \frac{17}{15}e^{4} + 3e^{3} + \frac{116}{15}e^{2} - \frac{521}{45}e - \frac{332}{27}$ |
| 31 | $[31, 31, \frac{3}{2}w^{3} - \frac{11}{2}w^{2} - \frac{19}{2}w + 29]$ | $-\frac{2}{135}e^{6} + \frac{1}{5}e^{5} + \frac{4}{9}e^{4} - \frac{311}{45}e^{3} - \frac{43}{45}e^{2} + \frac{1199}{45}e + \frac{358}{27}$ |
| 59 | $[59, 59, 2w^{2} - w - 13]$ | $\phantom{-}\frac{1}{135}e^{6} - \frac{1}{15}e^{5} - \frac{16}{45}e^{4} + \frac{106}{45}e^{3} + \frac{206}{45}e^{2} - \frac{95}{9}e - \frac{440}{27}$ |
| 59 | $[59, 59, \frac{9}{2}w^{3} - \frac{31}{2}w^{2} - \frac{61}{2}w + 85]$ | $\phantom{-}\frac{1}{135}e^{6} - \frac{1}{15}e^{5} - \frac{16}{45}e^{4} + \frac{106}{45}e^{3} + \frac{206}{45}e^{2} - \frac{95}{9}e - \frac{440}{27}$ |
| 61 | $[61, 61, 2w^{3} - 6w^{2} - 15w + 31]$ | $-\frac{8}{135}e^{6} + \frac{4}{15}e^{5} + \frac{86}{45}e^{4} - \frac{392}{45}e^{3} - \frac{214}{45}e^{2} + \frac{1214}{45}e + \frac{298}{27}$ |
| 61 | $[61, 61, -\frac{3}{2}w^{3} + \frac{11}{2}w^{2} + \frac{21}{2}w - 34]$ | $-\frac{8}{135}e^{6} + \frac{4}{15}e^{5} + \frac{86}{45}e^{4} - \frac{392}{45}e^{3} - \frac{214}{45}e^{2} + \frac{1214}{45}e + \frac{298}{27}$ |
| 71 | $[71, 71, \frac{3}{2}w^{3} - \frac{11}{2}w^{2} - \frac{19}{2}w + 32]$ | $\phantom{-}\frac{1}{135}e^{6} + \frac{1}{9}e^{5} - \frac{2}{5}e^{4} - \frac{61}{15}e^{3} + \frac{14}{3}e^{2} + \frac{904}{45}e + \frac{88}{27}$ |
| 71 | $[71, 71, -\frac{1}{2}w^{3} + \frac{5}{2}w^{2} + \frac{5}{2}w - 13]$ | $\phantom{-}\frac{1}{135}e^{6} + \frac{1}{9}e^{5} - \frac{2}{5}e^{4} - \frac{61}{15}e^{3} + \frac{14}{3}e^{2} + \frac{904}{45}e + \frac{88}{27}$ |
| 79 | $[79, 79, -3w^{3} + 10w^{2} + 19w - 51]$ | $\phantom{-}\frac{1}{45}e^{6} - \frac{4}{45}e^{5} - \frac{38}{45}e^{4} + \frac{128}{45}e^{3} + \frac{253}{45}e^{2} - \frac{59}{9}e - 10$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, w + 2]$ | $1$ |
| $4$ | $[4, 2, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{9}{2}w - 10]$ | $1$ |