Base field \(\Q(\sqrt{2}, \sqrt{13})\)
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 9x^{2} + 10x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16, 2, 2]$ |
| Dimension: | $4$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $12$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{4} + 8x^{3} - 6x^{2} - 88x - 79\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -\frac{2}{5}w^{3} + \frac{3}{5}w^{2} + \frac{17}{5}w - \frac{9}{5}]$ | $\phantom{-}0$ |
| 9 | $[9, 3, -\frac{2}{5}w^{3} + \frac{3}{5}w^{2} + \frac{22}{5}w - \frac{9}{5}]$ | $\phantom{-}\frac{1}{10}e^{3} + \frac{3}{5}e^{2} - \frac{13}{10}e - \frac{21}{5}$ |
| 9 | $[9, 3, \frac{2}{5}w^{3} - \frac{3}{5}w^{2} - \frac{22}{5}w + \frac{14}{5}]$ | $-\frac{1}{10}e^{3} - \frac{3}{5}e^{2} + \frac{13}{10}e + \frac{21}{5}$ |
| 17 | $[17, 17, w + 1]$ | $-\frac{1}{5}e^{3} - \frac{6}{5}e^{2} + \frac{18}{5}e + \frac{42}{5}$ |
| 17 | $[17, 17, -\frac{4}{5}w^{3} + \frac{6}{5}w^{2} + \frac{39}{5}w - \frac{13}{5}]$ | $-e - 4$ |
| 17 | $[17, 17, -\frac{4}{5}w^{3} + \frac{6}{5}w^{2} + \frac{39}{5}w - \frac{28}{5}]$ | $\phantom{-}\frac{1}{5}e^{3} + \frac{6}{5}e^{2} - \frac{18}{5}e - \frac{62}{5}$ |
| 17 | $[17, 17, -w + 2]$ | $\phantom{-}e$ |
| 23 | $[23, 23, -\frac{1}{5}w^{3} + \frac{4}{5}w^{2} + \frac{6}{5}w - \frac{22}{5}]$ | $\phantom{-}\frac{1}{10}e^{3} + \frac{3}{5}e^{2} - \frac{13}{10}e - \frac{26}{5}$ |
| 23 | $[23, 23, -\frac{1}{5}w^{3} - \frac{1}{5}w^{2} + \frac{11}{5}w + \frac{3}{5}]$ | $-\frac{1}{10}e^{3} - \frac{3}{5}e^{2} + \frac{13}{10}e + \frac{16}{5}$ |
| 23 | $[23, 23, -\frac{1}{5}w^{3} + \frac{4}{5}w^{2} + \frac{6}{5}w - \frac{12}{5}]$ | $\phantom{-}\frac{1}{10}e^{3} + \frac{3}{5}e^{2} - \frac{13}{10}e - \frac{26}{5}$ |
| 23 | $[23, 23, -\frac{1}{5}w^{3} - \frac{1}{5}w^{2} + \frac{11}{5}w + \frac{13}{5}]$ | $-\frac{1}{10}e^{3} - \frac{3}{5}e^{2} + \frac{13}{10}e + \frac{16}{5}$ |
| 25 | $[25, 5, -\frac{4}{5}w^{3} + \frac{6}{5}w^{2} + \frac{39}{5}w - \frac{33}{5}]$ | $\phantom{-}\frac{1}{5}e^{3} + \frac{6}{5}e^{2} - \frac{23}{5}e - \frac{57}{5}$ |
| 25 | $[25, 5, -w^{3} + w^{2} + 10w - 2]$ | $-\frac{1}{5}e^{3} - \frac{6}{5}e^{2} + \frac{23}{5}e + \frac{67}{5}$ |
| 49 | $[49, 7, -\frac{4}{5}w^{3} + \frac{6}{5}w^{2} + \frac{34}{5}w - \frac{23}{5}]$ | $-\frac{1}{2}e^{2} - 2e + \frac{1}{2}$ |
| 49 | $[49, 7, \frac{4}{5}w^{3} - \frac{6}{5}w^{2} - \frac{34}{5}w + \frac{13}{5}]$ | $\phantom{-}\frac{1}{2}e^{2} + 2e - \frac{21}{2}$ |
| 79 | $[79, 79, \frac{3}{5}w^{3} - \frac{7}{5}w^{2} - \frac{28}{5}w + \frac{21}{5}]$ | $\phantom{-}\frac{1}{10}e^{3} + \frac{11}{10}e^{2} + \frac{17}{10}e - \frac{77}{10}$ |
| 79 | $[79, 79, \frac{1}{5}w^{3} + \frac{1}{5}w^{2} - \frac{16}{5}w - \frac{13}{5}]$ | $\phantom{-}\frac{3}{10}e^{3} + \frac{13}{10}e^{2} - \frac{69}{10}e - \frac{91}{10}$ |
| 79 | $[79, 79, -\frac{1}{5}w^{3} + \frac{4}{5}w^{2} + \frac{11}{5}w - \frac{27}{5}]$ | $-\frac{1}{10}e^{3} - \frac{1}{10}e^{2} + \frac{23}{10}e - \frac{33}{10}$ |
| 79 | $[79, 79, \frac{3}{5}w^{3} - \frac{2}{5}w^{2} - \frac{33}{5}w + \frac{11}{5}]$ | $-\frac{3}{10}e^{3} - \frac{23}{10}e^{2} + \frac{29}{10}e + \frac{201}{10}$ |
| 103 | $[103, 103, -\frac{1}{5}w^{3} + \frac{4}{5}w^{2} + \frac{1}{5}w - \frac{17}{5}]$ | $\phantom{-}\frac{1}{2}e^{3} + \frac{7}{2}e^{2} - \frac{15}{2}e - \frac{57}{2}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, -\frac{2}{5}w^{3} + \frac{3}{5}w^{2} + \frac{17}{5}w - \frac{9}{5}]$ | $-1$ |