Base field 4.4.18097.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 7x^{2} + 6x + 4\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16, 4, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{3}{2}w - 1]$ |
| Dimension: | $14$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $22$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{14} - 33x^{12} + 418x^{10} - 2551x^{8} + 7661x^{6} - 9788x^{4} + 2560x^{2} - 64\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, -w + 1]$ | $\phantom{-}e$ |
| 4 | $[4, 2, w]$ | $-\frac{713}{165408}e^{13} + \frac{20117}{165408}e^{11} - \frac{99143}{82704}e^{9} + \frac{258637}{55136}e^{7} - \frac{856777}{165408}e^{5} - \frac{65123}{20676}e^{3} - \frac{42457}{10338}e$ |
| 4 | $[4, 2, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{7}{2}w - 3]$ | $\phantom{-}0$ |
| 7 | $[7, 7, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{5}{2}w - 2]$ | $-\frac{134}{5169}e^{12} + \frac{3875}{5169}e^{10} - \frac{81005}{10338}e^{8} + \frac{61604}{1723}e^{6} - \frac{351310}{5169}e^{4} + \frac{384487}{10338}e^{2} - \frac{8470}{5169}$ |
| 13 | $[13, 13, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{7}{2}w - 1]$ | $-\frac{46}{5169}e^{12} + \frac{2429}{10338}e^{10} - \frac{10625}{5169}e^{8} + \frac{9961}{1723}e^{6} + \frac{52355}{10338}e^{4} - \frac{170507}{5169}e^{2} + \frac{26872}{5169}$ |
| 17 | $[17, 17, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{5}{2}w]$ | $\phantom{-}\frac{127}{10338}e^{13} - \frac{1817}{5169}e^{11} + \frac{36863}{10338}e^{9} - \frac{51211}{3446}e^{7} + \frac{101095}{5169}e^{5} + \frac{128807}{10338}e^{3} - \frac{74177}{5169}e$ |
| 27 | $[27, 3, -w^{3} + 7w + 1]$ | $\phantom{-}\frac{1991}{82704}e^{13} - \frac{61163}{82704}e^{11} + \frac{349781}{41352}e^{9} - \frac{1227651}{27568}e^{7} + \frac{8919607}{82704}e^{5} - \frac{504209}{5169}e^{3} + \frac{47149}{5169}e$ |
| 31 | $[31, 31, w + 3]$ | $\phantom{-}\frac{355}{6892}e^{12} - \frac{10253}{6892}e^{10} + \frac{26483}{1723}e^{8} - \frac{465209}{6892}e^{6} + \frac{782017}{6892}e^{4} - \frac{110205}{3446}e^{2} + \frac{13254}{1723}$ |
| 31 | $[31, 31, -w^{2} + 5]$ | $\phantom{-}\frac{647}{27568}e^{13} - \frac{21783}{27568}e^{11} + \frac{141675}{13784}e^{9} - \frac{1790433}{27568}e^{7} + \frac{5616667}{27568}e^{5} - \frac{1887699}{6892}e^{3} + \frac{124312}{1723}e$ |
| 37 | $[37, 37, w^{2} - 3]$ | $-\frac{271}{82704}e^{13} + \frac{10807}{82704}e^{11} - \frac{82819}{41352}e^{9} + \frac{404819}{27568}e^{7} - \frac{4282187}{82704}e^{5} + \frac{1559287}{20676}e^{3} - \frac{113993}{5169}e$ |
| 41 | $[41, 41, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{3}{2}w - 2]$ | $\phantom{-}\frac{2827}{82704}e^{13} - \frac{92899}{82704}e^{11} + \frac{584095}{41352}e^{9} - \frac{2342847}{27568}e^{7} + \frac{20490551}{82704}e^{5} - \frac{6122239}{20676}e^{3} + \frac{288785}{5169}e$ |
| 47 | $[47, 47, w^{3} - 5w - 3]$ | $\phantom{-}\frac{51}{27568}e^{13} - \frac{279}{27568}e^{11} - \frac{8587}{13784}e^{9} + \frac{241987}{27568}e^{7} - \frac{1099549}{27568}e^{5} + \frac{207977}{3446}e^{3} - \frac{6001}{1723}e$ |
| 53 | $[53, 53, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w]$ | $\phantom{-}\frac{423}{27568}e^{13} - \frac{14071}{27568}e^{11} + \frac{90335}{13784}e^{9} - \frac{1123521}{27568}e^{7} + \frac{3436219}{27568}e^{5} - \frac{1093985}{6892}e^{3} + \frac{57053}{1723}e$ |
| 61 | $[61, 61, w^{3} - w^{2} - 5w + 3]$ | $\phantom{-}\frac{73}{3446}e^{12} - \frac{2021}{3446}e^{10} + \frac{19371}{3446}e^{8} - \frac{71133}{3446}e^{6} + \frac{49945}{3446}e^{4} + \frac{153161}{3446}e^{2} - \frac{21622}{1723}$ |
| 83 | $[83, 83, w^{3} + w^{2} - 6w - 7]$ | $\phantom{-}\frac{3629}{82704}e^{13} - \frac{117557}{82704}e^{11} + \frac{726497}{41352}e^{9} - \frac{2867033}{27568}e^{7} + \frac{24926161}{82704}e^{5} - \frac{7538513}{20676}e^{3} + \frac{366142}{5169}e$ |
| 83 | $[83, 83, -w^{3} + 5w + 1]$ | $-\frac{881}{20676}e^{12} + \frac{25901}{20676}e^{10} - \frac{68824}{5169}e^{8} + \frac{425365}{6892}e^{6} - \frac{2450761}{20676}e^{4} + \frac{650953}{10338}e^{2} + \frac{16070}{5169}$ |
| 83 | $[83, 83, 2w - 1]$ | $-\frac{1029}{27568}e^{13} + \frac{31981}{27568}e^{11} - \frac{186953}{13784}e^{9} + \frac{2057395}{27568}e^{7} - \frac{5453929}{27568}e^{5} + \frac{1473313}{6892}e^{3} - \frac{42606}{1723}e$ |
| 83 | $[83, 83, -\frac{3}{2}w^{3} - \frac{1}{2}w^{2} + \frac{19}{2}w + 2]$ | $-\frac{127}{5169}e^{12} + \frac{3634}{5169}e^{10} - \frac{36863}{5169}e^{8} + \frac{51211}{1723}e^{6} - \frac{207359}{5169}e^{4} - \frac{66779}{5169}e^{2} + \frac{34636}{5169}$ |
| 89 | $[89, 89, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w - 2]$ | $-\frac{847}{10338}e^{12} + \frac{11996}{5169}e^{10} - \frac{241373}{10338}e^{8} + \frac{337471}{3446}e^{6} - \frac{751360}{5169}e^{4} - \frac{31523}{10338}e^{2} + \frac{54122}{5169}$ |
| 89 | $[89, 89, w^{3} - 5w + 5]$ | $-\frac{923}{20676}e^{12} + \frac{27347}{20676}e^{10} - \frac{72991}{5169}e^{8} + \frac{446371}{6892}e^{6} - \frac{2384047}{20676}e^{4} + \frac{365791}{10338}e^{2} + \frac{28946}{5169}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{7}{2}w - 3]$ | $1$ |