Base field \(\Q(\zeta_{16})^+\)
Generator \(w\), with minimal polynomial \(x^{4} - 4x^{2} + 2\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[49,7,-w^{2} + 5]$ |
| Dimension: | $6$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $6$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{6} - 2x^{5} - 10x^{4} + 20x^{3} + 16x^{2} - 32x + 4\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w]$ | $\phantom{-}e$ |
| 17 | $[17, 17, -w^{2} - w + 3]$ | $\phantom{-}\frac{1}{5}e^{5} - \frac{3}{5}e^{4} - \frac{12}{5}e^{3} + \frac{22}{5}e^{2} + \frac{24}{5}e - \frac{6}{5}$ |
| 17 | $[17, 17, -w^{3} - w^{2} + 3w + 1]$ | $-\frac{7}{10}e^{5} + \frac{3}{5}e^{4} + \frac{37}{5}e^{3} - \frac{32}{5}e^{2} - \frac{74}{5}e + \frac{46}{5}$ |
| 17 | $[17, 17, w^{3} - w^{2} - 3w + 1]$ | $-\frac{7}{10}e^{5} + \frac{3}{5}e^{4} + \frac{37}{5}e^{3} - \frac{32}{5}e^{2} - \frac{74}{5}e + \frac{46}{5}$ |
| 17 | $[17, 17, w^{2} - w - 3]$ | $\phantom{-}\frac{1}{5}e^{5} - \frac{3}{5}e^{4} - \frac{12}{5}e^{3} + \frac{22}{5}e^{2} + \frac{24}{5}e - \frac{6}{5}$ |
| 31 | $[31, 31, w^{3} + w^{2} - 2w - 3]$ | $\phantom{-}\frac{9}{10}e^{5} - \frac{1}{5}e^{4} - \frac{44}{5}e^{3} + \frac{14}{5}e^{2} + \frac{78}{5}e - \frac{12}{5}$ |
| 31 | $[31, 31, -w^{3} + w^{2} + 4w - 1]$ | $-\frac{1}{5}e^{5} - \frac{2}{5}e^{4} + \frac{12}{5}e^{3} + \frac{18}{5}e^{2} - \frac{44}{5}e - \frac{14}{5}$ |
| 31 | $[31, 31, w^{3} + w^{2} - 4w - 1]$ | $-\frac{1}{5}e^{5} - \frac{2}{5}e^{4} + \frac{12}{5}e^{3} + \frac{18}{5}e^{2} - \frac{44}{5}e - \frac{14}{5}$ |
| 31 | $[31, 31, -w^{3} + w^{2} + 2w - 3]$ | $\phantom{-}\frac{9}{10}e^{5} - \frac{1}{5}e^{4} - \frac{44}{5}e^{3} + \frac{14}{5}e^{2} + \frac{78}{5}e - \frac{12}{5}$ |
| 47 | $[47, 47, -2w^{3} + w^{2} + 5w - 1]$ | $\phantom{-}\frac{4}{5}e^{5} - \frac{7}{5}e^{4} - \frac{38}{5}e^{3} + \frac{58}{5}e^{2} + \frac{56}{5}e - \frac{54}{5}$ |
| 47 | $[47, 47, 2w^{3} + w^{2} - 6w - 1]$ | $\phantom{-}\frac{3}{10}e^{5} + \frac{3}{5}e^{4} - \frac{18}{5}e^{3} - \frac{22}{5}e^{2} + \frac{46}{5}e + \frac{16}{5}$ |
| 47 | $[47, 47, -2w^{3} + w^{2} + 6w - 1]$ | $\phantom{-}\frac{3}{10}e^{5} + \frac{3}{5}e^{4} - \frac{18}{5}e^{3} - \frac{22}{5}e^{2} + \frac{46}{5}e + \frac{16}{5}$ |
| 47 | $[47, 47, 2w^{3} + w^{2} - 5w - 1]$ | $\phantom{-}\frac{4}{5}e^{5} - \frac{7}{5}e^{4} - \frac{38}{5}e^{3} + \frac{58}{5}e^{2} + \frac{56}{5}e - \frac{54}{5}$ |
| 49 | $[49, 7, w^{2} + 1]$ | $-\frac{2}{5}e^{5} + \frac{1}{5}e^{4} + \frac{14}{5}e^{3} - \frac{14}{5}e^{2} + \frac{12}{5}e + \frac{42}{5}$ |
| 49 | $[49, 7, -2w^{2} + 3]$ | $-1$ |
| 79 | $[79, 79, -w^{3} - w^{2} + 4w - 1]$ | $-\frac{1}{5}e^{5} - \frac{2}{5}e^{4} + \frac{2}{5}e^{3} + \frac{28}{5}e^{2} + \frac{36}{5}e - \frac{84}{5}$ |
| 79 | $[79, 79, -w^{3} + w^{2} + 2w - 5]$ | $-\frac{11}{10}e^{5} + \frac{9}{5}e^{4} + \frac{56}{5}e^{3} - \frac{86}{5}e^{2} - \frac{102}{5}e + \frac{88}{5}$ |
| 79 | $[79, 79, w^{3} + w^{2} - 2w - 5]$ | $-\frac{11}{10}e^{5} + \frac{9}{5}e^{4} + \frac{56}{5}e^{3} - \frac{86}{5}e^{2} - \frac{102}{5}e + \frac{88}{5}$ |
| 79 | $[79, 79, w^{3} - w^{2} - 4w - 1]$ | $-\frac{1}{5}e^{5} - \frac{2}{5}e^{4} + \frac{2}{5}e^{3} + \frac{28}{5}e^{2} + \frac{36}{5}e - \frac{84}{5}$ |
| 81 | $[81, 3, -3]$ | $-\frac{6}{5}e^{5} + \frac{3}{5}e^{4} + \frac{62}{5}e^{3} - \frac{22}{5}e^{2} - \frac{124}{5}e + \frac{46}{5}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $49$ | $[49,7,-w^{2} + 5]$ | $1$ |