Base field 4.4.13625.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 11x^{2} + 12x + 31\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16, 2, 2]$ |
| Dimension: | $6$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $20$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{6} + x^{5} - 40x^{4} - 37x^{3} + 474x^{2} + 340x - 1416\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -\frac{1}{2}w^{2} - \frac{1}{2}w + \frac{3}{2}]$ | $\phantom{-}1$ |
| 4 | $[4, 2, -\frac{1}{2}w^{2} + \frac{3}{2}w + \frac{1}{2}]$ | $\phantom{-}1$ |
| 5 | $[5, 5, w - 3]$ | $\phantom{-}e$ |
| 11 | $[11, 11, \frac{1}{2}w^{3} - 4w - \frac{7}{2}]$ | $-\frac{3}{64}e^{5} - \frac{5}{64}e^{4} + \frac{37}{32}e^{3} + \frac{75}{64}e^{2} - \frac{87}{16}e + \frac{39}{16}$ |
| 11 | $[11, 11, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{5}{2}w - 7]$ | $-\frac{3}{64}e^{5} - \frac{5}{64}e^{4} + \frac{37}{32}e^{3} + \frac{75}{64}e^{2} - \frac{87}{16}e + \frac{39}{16}$ |
| 19 | $[19, 19, -w^{2} + 2w + 4]$ | $\phantom{-}\frac{7}{64}e^{5} - \frac{31}{64}e^{4} - \frac{97}{32}e^{3} + \frac{721}{64}e^{2} + \frac{315}{16}e - \frac{763}{16}$ |
| 19 | $[19, 19, -w^{2} + 5]$ | $\phantom{-}\frac{7}{64}e^{5} - \frac{31}{64}e^{4} - \frac{97}{32}e^{3} + \frac{721}{64}e^{2} + \frac{315}{16}e - \frac{763}{16}$ |
| 31 | $[31, 31, w]$ | $\phantom{-}\frac{9}{64}e^{5} - \frac{17}{64}e^{4} - \frac{127}{32}e^{3} + \frac{415}{64}e^{2} + \frac{413}{16}e - \frac{517}{16}$ |
| 31 | $[31, 31, w + 3]$ | $\phantom{-}\frac{1}{16}e^{5} + \frac{3}{16}e^{4} - \frac{9}{8}e^{3} - \frac{73}{16}e^{2} + \frac{1}{2}e + \frac{77}{4}$ |
| 31 | $[31, 31, -w + 4]$ | $\phantom{-}\frac{1}{16}e^{5} + \frac{3}{16}e^{4} - \frac{9}{8}e^{3} - \frac{73}{16}e^{2} + \frac{1}{2}e + \frac{77}{4}$ |
| 31 | $[31, 31, w - 1]$ | $\phantom{-}\frac{9}{64}e^{5} - \frac{17}{64}e^{4} - \frac{127}{32}e^{3} + \frac{415}{64}e^{2} + \frac{413}{16}e - \frac{517}{16}$ |
| 41 | $[41, 41, -w^{2} + 2]$ | $-\frac{1}{16}e^{5} + \frac{5}{16}e^{4} + \frac{13}{8}e^{3} - \frac{119}{16}e^{2} - 11e + \frac{135}{4}$ |
| 41 | $[41, 41, \frac{1}{2}w^{3} - \frac{5}{2}w^{2} - \frac{3}{2}w + 12]$ | $-\frac{1}{16}e^{5} + \frac{5}{16}e^{4} + \frac{13}{8}e^{3} - \frac{119}{16}e^{2} - 11e + \frac{135}{4}$ |
| 59 | $[59, 59, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{9}{2}w - 5]$ | $\phantom{-}\frac{7}{32}e^{5} - \frac{7}{32}e^{4} - \frac{85}{16}e^{3} + \frac{209}{32}e^{2} + \frac{205}{8}e - \frac{375}{8}$ |
| 59 | $[59, 59, -\frac{1}{2}w^{3} + 2w^{2} + 2w - \frac{17}{2}]$ | $\phantom{-}\frac{7}{32}e^{5} - \frac{7}{32}e^{4} - \frac{85}{16}e^{3} + \frac{209}{32}e^{2} + \frac{205}{8}e - \frac{375}{8}$ |
| 79 | $[79, 79, -w^{2} + 8]$ | $-\frac{1}{32}e^{5} + \frac{1}{32}e^{4} + \frac{19}{16}e^{3} - \frac{39}{32}e^{2} - \frac{83}{8}e + \frac{97}{8}$ |
| 79 | $[79, 79, w^{2} - 2w - 7]$ | $-\frac{1}{32}e^{5} + \frac{1}{32}e^{4} + \frac{19}{16}e^{3} - \frac{39}{32}e^{2} - \frac{83}{8}e + \frac{97}{8}$ |
| 81 | $[81, 3, -3]$ | $-\frac{7}{64}e^{5} + \frac{47}{64}e^{4} + \frac{105}{32}e^{3} - \frac{1105}{64}e^{2} - \frac{383}{16}e + \frac{1363}{16}$ |
| 101 | $[101, 101, -w^{3} + \frac{1}{2}w^{2} + \frac{15}{2}w + \frac{3}{2}]$ | $\phantom{-}\frac{5}{32}e^{5} - \frac{21}{32}e^{4} - \frac{71}{16}e^{3} + \frac{515}{32}e^{2} + \frac{227}{8}e - \frac{621}{8}$ |
| 101 | $[101, 101, w^{3} - \frac{5}{2}w^{2} - \frac{11}{2}w + \frac{17}{2}]$ | $\phantom{-}\frac{5}{32}e^{5} - \frac{21}{32}e^{4} - \frac{71}{16}e^{3} + \frac{515}{32}e^{2} + \frac{227}{8}e - \frac{621}{8}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, -\frac{1}{2}w^{2} - \frac{1}{2}w + \frac{3}{2}]$ | $-1$ |
| $4$ | $[4, 2, -\frac{1}{2}w^{2} + \frac{3}{2}w + \frac{1}{2}]$ | $-1$ |