Base field 5.5.160801.1
Generator \(w\), with minimal polynomial \(x^{5} - x^{4} - 5x^{3} + 4x^{2} + 3x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2]$ |
| Level: | $[39, 39, -w^{4} + 5w^{2} - w - 2]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $39$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} + 9x^{8} - 5x^{7} - 196x^{6} - 208x^{5} + 875x^{4} + 705x^{3} - 96x^{2} - 94x + 8\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w^{4} - w^{3} - 5w^{2} + 3w + 3]$ | $\phantom{-}1$ |
| 9 | $[9, 3, -w^{4} + 5w^{2} - 3]$ | $\phantom{-}e$ |
| 9 | $[9, 3, -w^{4} + w^{3} + 5w^{2} - 3w - 2]$ | $-\frac{9590}{182841}e^{8} - \frac{67604}{182841}e^{7} + \frac{135317}{182841}e^{6} + \frac{1477696}{182841}e^{5} + \frac{106766}{60947}e^{4} - \frac{6747223}{182841}e^{3} - \frac{1653541}{182841}e^{2} + \frac{1854170}{182841}e - \frac{160418}{182841}$ |
| 13 | $[13, 13, -w^{4} + w^{3} + 4w^{2} - 3w - 1]$ | $\phantom{-}1$ |
| 17 | $[17, 17, w^{4} - w^{3} - 5w^{2} + 3w + 1]$ | $\phantom{-}\frac{5533}{182841}e^{8} + \frac{31912}{182841}e^{7} - \frac{109702}{182841}e^{6} - \frac{687302}{182841}e^{5} + \frac{145735}{60947}e^{4} + \frac{2997725}{182841}e^{3} - \frac{1061293}{182841}e^{2} - \frac{205711}{182841}e - \frac{440030}{182841}$ |
| 19 | $[19, 19, -w^{3} + w^{2} + 4w - 2]$ | $\phantom{-}\frac{22919}{182841}e^{8} + \frac{185093}{182841}e^{7} - \frac{284822}{182841}e^{6} - \frac{4270438}{182841}e^{5} - \frac{330837}{60947}e^{4} + \frac{22025248}{182841}e^{3} - \frac{1583390}{182841}e^{2} - \frac{6953468}{182841}e + \frac{715850}{182841}$ |
| 23 | $[23, 23, -w^{2} + 3]$ | $\phantom{-}\frac{81937}{365682}e^{8} + \frac{723691}{365682}e^{7} - \frac{609361}{365682}e^{6} - \frac{8082484}{182841}e^{5} - \frac{2020180}{60947}e^{4} + \frac{76738103}{365682}e^{3} + \frac{29356343}{365682}e^{2} - \frac{5509862}{182841}e - \frac{269185}{182841}$ |
| 31 | $[31, 31, w^{3} - 4w + 2]$ | $\phantom{-}\frac{1469}{121894}e^{8} + \frac{5157}{121894}e^{7} - \frac{72911}{121894}e^{6} - \frac{106021}{60947}e^{5} + \frac{561570}{60947}e^{4} + \frac{2214535}{121894}e^{3} - \frac{5414317}{121894}e^{2} - \frac{1427332}{60947}e + \frac{410679}{60947}$ |
| 32 | $[32, 2, 2]$ | $-\frac{5533}{182841}e^{8} - \frac{31912}{182841}e^{7} + \frac{109702}{182841}e^{6} + \frac{687302}{182841}e^{5} - \frac{145735}{60947}e^{4} - \frac{2997725}{182841}e^{3} + \frac{1061293}{182841}e^{2} + \frac{22870}{182841}e - \frac{108493}{182841}$ |
| 37 | $[37, 37, w^{3} - 3w - 1]$ | $-\frac{61010}{182841}e^{8} - \frac{529037}{182841}e^{7} + \frac{527213}{182841}e^{6} + \frac{11912968}{182841}e^{5} + \frac{2466964}{60947}e^{4} - \frac{57714823}{182841}e^{3} - \frac{14317081}{182841}e^{2} + \frac{9654263}{182841}e - \frac{1405682}{182841}$ |
| 53 | $[53, 53, -2w^{4} + w^{3} + 9w^{2} - 3w - 2]$ | $\phantom{-}\frac{87905}{365682}e^{8} + \frac{696515}{365682}e^{7} - \frac{943409}{365682}e^{6} - \frac{7698161}{182841}e^{5} - \frac{1358440}{60947}e^{4} + \frac{71894779}{365682}e^{3} + \frac{25120003}{365682}e^{2} - \frac{7845430}{182841}e - \frac{1057631}{182841}$ |
| 59 | $[59, 59, -w^{4} + 5w^{2} + w - 4]$ | $-\frac{18188}{182841}e^{8} - \frac{139235}{182841}e^{7} + \frac{306284}{182841}e^{6} + \frac{3358153}{182841}e^{5} - \frac{367287}{60947}e^{4} - \frac{18918076}{182841}e^{3} + \frac{11197400}{182841}e^{2} + \frac{6777938}{182841}e - \frac{3037820}{182841}$ |
| 61 | $[61, 61, -w^{4} + w^{3} + 5w^{2} - 4w]$ | $\phantom{-}\frac{19756}{182841}e^{8} + \frac{209545}{182841}e^{7} + \frac{115748}{182841}e^{6} - \frac{4319915}{182841}e^{5} - \frac{2882496}{60947}e^{4} + \frac{15983822}{182841}e^{3} + \frac{32738060}{182841}e^{2} + \frac{7080401}{182841}e - \frac{3474452}{182841}$ |
| 67 | $[67, 67, -w^{4} + 6w^{2} + 2w - 4]$ | $-\frac{13984}{182841}e^{8} - \frac{108379}{182841}e^{7} + \frac{228814}{182841}e^{6} + \frac{2603297}{182841}e^{5} - \frac{240706}{60947}e^{4} - \frac{14380901}{182841}e^{3} + \frac{8454886}{182841}e^{2} + \frac{2857585}{182841}e - \frac{2407954}{182841}$ |
| 71 | $[71, 71, 2w^{4} - w^{3} - 9w^{2} + 4w + 5]$ | $\phantom{-}\frac{64151}{365682}e^{8} + \frac{463019}{365682}e^{7} - \frac{997139}{365682}e^{6} - \frac{5255903}{182841}e^{5} + \frac{157227}{60947}e^{4} + \frac{52182817}{365682}e^{3} - \frac{14826803}{365682}e^{2} - \frac{5981113}{182841}e + \frac{3416455}{182841}$ |
| 79 | $[79, 79, 2w^{4} - w^{3} - 10w^{2} + 2w + 7]$ | $-\frac{113023}{182841}e^{8} - \frac{947329}{182841}e^{7} + \frac{977791}{182841}e^{6} + \frac{20892653}{182841}e^{5} + \frac{4939363}{60947}e^{4} - \frac{96445307}{182841}e^{3} - \frac{43551275}{182841}e^{2} + \frac{13233256}{182841}e + \frac{2183258}{182841}$ |
| 83 | $[83, 83, -w^{4} + 2w^{3} + 5w^{2} - 7w - 2]$ | $\phantom{-}\frac{8801}{182841}e^{8} + \frac{56246}{182841}e^{7} - \frac{312230}{182841}e^{6} - \frac{1761727}{182841}e^{5} + \frac{1470612}{60947}e^{4} + \frac{14460124}{182841}e^{3} - \frac{25802855}{182841}e^{2} - \frac{11944340}{182841}e + \frac{3248882}{182841}$ |
| 83 | $[83, 83, -w^{4} + w^{3} + 4w^{2} - 3w + 3]$ | $-\frac{62147}{365682}e^{8} - \frac{566261}{365682}e^{7} + \frac{566345}{365682}e^{6} + \frac{6639965}{182841}e^{5} + \frac{1002550}{60947}e^{4} - \frac{70826119}{365682}e^{3} + \frac{1430399}{365682}e^{2} + \frac{12630097}{182841}e - \frac{1409911}{182841}$ |
| 83 | $[83, 83, w^{4} - w^{3} - 5w^{2} + 4w - 1]$ | $-\frac{2866}{60947}e^{8} - \frac{23877}{60947}e^{7} + \frac{56989}{60947}e^{6} + \frac{626819}{60947}e^{5} - \frac{474053}{60947}e^{4} - \frac{4056951}{60947}e^{3} + \frac{4283647}{60947}e^{2} + \frac{1702203}{60947}e - \frac{1081028}{60947}$ |
| 83 | $[83, 83, -w^{4} + w^{3} + 4w^{2} - 4w - 2]$ | $-\frac{40894}{182841}e^{8} - \frac{387307}{182841}e^{7} + \frac{198796}{182841}e^{6} + \frac{8728622}{182841}e^{5} + \frac{2700705}{60947}e^{4} - \frac{42552074}{182841}e^{3} - \frac{21373808}{182841}e^{2} + \frac{9924538}{182841}e + \frac{1707020}{182841}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, w^{4} - w^{3} - 5w^{2} + 3w + 3]$ | $-1$ |
| $13$ | $[13, 13, -w^{4} + w^{3} + 4w^{2} - 3w - 1]$ | $-1$ |