Base field \(\Q(\sqrt{5}, \sqrt{17})\)
Generator \(w\), with minimal polynomial \(x^{4} - 11x^{2} + 9\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[25, 5, \frac{1}{3}w^{3} - \frac{8}{3}w]$ |
| Dimension: | $6$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $15$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{6} - 88x^{4} + 172x^{3} + 1936x^{2} - 7400x + 6052\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, \frac{1}{6}w^{3} - \frac{7}{3}w - \frac{3}{2}]$ | $-\frac{13}{298}e^{5} - \frac{101}{596}e^{4} + \frac{1867}{596}e^{3} + \frac{1439}{298}e^{2} - \frac{18919}{298}e + \frac{19641}{298}$ |
| 4 | $[4, 2, \frac{1}{6}w^{3} - \frac{7}{3}w + \frac{3}{2}]$ | $-\frac{13}{298}e^{5} - \frac{101}{596}e^{4} + \frac{1867}{596}e^{3} + \frac{1439}{298}e^{2} - \frac{18919}{298}e + \frac{19641}{298}$ |
| 9 | $[9, 3, -\frac{1}{3}w^{3} + \frac{11}{3}w]$ | $-\frac{13}{298}e^{5} - \frac{101}{596}e^{4} + \frac{1867}{596}e^{3} + \frac{1439}{298}e^{2} - \frac{18919}{298}e + \frac{19045}{298}$ |
| 9 | $[9, 3, w]$ | $-\frac{13}{298}e^{5} - \frac{101}{596}e^{4} + \frac{1867}{596}e^{3} + \frac{1439}{298}e^{2} - \frac{18919}{298}e + \frac{19045}{298}$ |
| 19 | $[19, 19, w + 2]$ | $\phantom{-}e$ |
| 19 | $[19, 19, -\frac{1}{3}w^{3} + \frac{11}{3}w - 2]$ | $\phantom{-}e$ |
| 19 | $[19, 19, -\frac{1}{3}w^{3} + \frac{11}{3}w + 2]$ | $\phantom{-}\frac{1}{149}e^{5} + \frac{27}{596}e^{4} - \frac{207}{596}e^{3} - \frac{485}{298}e^{2} + \frac{1455}{298}e + \frac{795}{298}$ |
| 19 | $[19, 19, -w + 2]$ | $\phantom{-}\frac{1}{149}e^{5} + \frac{27}{596}e^{4} - \frac{207}{596}e^{3} - \frac{485}{298}e^{2} + \frac{1455}{298}e + \frac{795}{298}$ |
| 25 | $[25, 5, \frac{1}{3}w^{3} - \frac{8}{3}w]$ | $-1$ |
| 49 | $[49, 7, \frac{2}{3}w^{3} - \frac{19}{3}w]$ | $\phantom{-}\frac{1}{149}e^{5} + \frac{27}{596}e^{4} - \frac{207}{596}e^{3} - \frac{485}{298}e^{2} + \frac{1753}{298}e + \frac{795}{298}$ |
| 49 | $[49, 7, -\frac{1}{3}w^{3} + \frac{5}{3}w]$ | $\phantom{-}\frac{1}{149}e^{5} + \frac{27}{596}e^{4} - \frac{207}{596}e^{3} - \frac{485}{298}e^{2} + \frac{1753}{298}e + \frac{795}{298}$ |
| 59 | $[59, 59, \frac{1}{6}w^{3} + \frac{1}{2}w^{2} - \frac{11}{6}w]$ | $-\frac{93}{596}e^{5} - \frac{367}{596}e^{4} + \frac{6787}{596}e^{3} + \frac{2714}{149}e^{2} - \frac{69775}{298}e + \frac{69389}{298}$ |
| 59 | $[59, 59, \frac{1}{2}w^{2} + \frac{1}{2}w - \frac{11}{2}]$ | $\phantom{-}\frac{19}{596}e^{5} + \frac{91}{596}e^{4} - \frac{1393}{596}e^{3} - \frac{798}{149}e^{2} + \frac{14771}{298}e - \frac{10863}{298}$ |
| 59 | $[59, 59, \frac{1}{2}w^{2} - \frac{1}{2}w - \frac{11}{2}]$ | $-\frac{93}{596}e^{5} - \frac{367}{596}e^{4} + \frac{6787}{596}e^{3} + \frac{2714}{149}e^{2} - \frac{69775}{298}e + \frac{69389}{298}$ |
| 59 | $[59, 59, \frac{1}{6}w^{3} - \frac{1}{2}w^{2} - \frac{11}{6}w]$ | $\phantom{-}\frac{19}{596}e^{5} + \frac{91}{596}e^{4} - \frac{1393}{596}e^{3} - \frac{798}{149}e^{2} + \frac{14771}{298}e - \frac{10863}{298}$ |
| 89 | $[89, 89, \frac{2}{3}w^{3} + \frac{1}{2}w^{2} - \frac{35}{6}w - \frac{7}{2}]$ | $\phantom{-}\frac{19}{596}e^{5} + \frac{91}{596}e^{4} - \frac{1393}{596}e^{3} - \frac{798}{149}e^{2} + \frac{15069}{298}e - \frac{9671}{298}$ |
| 89 | $[89, 89, -\frac{5}{6}w^{3} + \frac{26}{3}w - \frac{3}{2}]$ | $-\frac{89}{596}e^{5} - \frac{85}{149}e^{4} + \frac{1645}{149}e^{3} + \frac{4943}{298}e^{2} - \frac{34160}{149}e + \frac{35688}{149}$ |
| 89 | $[89, 89, \frac{1}{6}w^{3} - \frac{1}{2}w^{2} + \frac{1}{6}w - 2]$ | $\phantom{-}\frac{19}{596}e^{5} + \frac{91}{596}e^{4} - \frac{1393}{596}e^{3} - \frac{798}{149}e^{2} + \frac{15069}{298}e - \frac{9671}{298}$ |
| 89 | $[89, 89, -\frac{1}{6}w^{3} - w^{2} + \frac{7}{3}w + \frac{5}{2}]$ | $-\frac{89}{596}e^{5} - \frac{85}{149}e^{4} + \frac{1645}{149}e^{3} + \frac{4943}{298}e^{2} - \frac{34160}{149}e + \frac{35688}{149}$ |
| 101 | $[101, 101, \frac{1}{6}w^{3} + \frac{1}{2}w^{2} - \frac{17}{6}w + 1]$ | $\phantom{-}\frac{15}{1192}e^{5} + \frac{8}{149}e^{4} - \frac{593}{596}e^{3} - \frac{1111}{596}e^{2} + \frac{6509}{298}e - \frac{7021}{298}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $25$ | $[25,5,\frac{1}{3}w^{3}-\frac{8}{3}w]$ | $1$ |