Base field 6.6.1541581.1
Generator \(w\), with minimal polynomial \(x^{6} - x^{5} - 6x^{4} + 2x^{3} + 9x^{2} + x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2, 2]$ |
| Level: | $[53, 53, -w^{5} + 3w^{4} + 2w^{3} - 8w^{2} + 3]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $56$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} + 7x^{7} + 7x^{6} - 41x^{5} - 77x^{4} + 36x^{3} + 86x^{2} - 28x - 2\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, w^{5} - 2w^{4} - 4w^{3} + 5w^{2} + 5w]$ | $\phantom{-}e$ |
| 11 | $[11, 11, w^{5} - 2w^{4} - 3w^{3} + 4w^{2} + w + 1]$ | $-\frac{12}{167}e^{7} - \frac{149}{167}e^{6} - \frac{404}{167}e^{5} + \frac{586}{167}e^{4} + \frac{2790}{167}e^{3} + \frac{736}{167}e^{2} - \frac{2278}{167}e - \frac{7}{167}$ |
| 11 | $[11, 11, w^{2} - w - 2]$ | $\phantom{-}\frac{113}{167}e^{7} + \frac{582}{167}e^{6} - \frac{148}{167}e^{5} - \frac{3709}{167}e^{4} - \frac{2308}{167}e^{3} + \frac{3980}{167}e^{2} + \frac{1439}{167}e - \frac{282}{167}$ |
| 17 | $[17, 17, w^{5} - 2w^{4} - 4w^{3} + 6w^{2} + 3w - 3]$ | $\phantom{-}\frac{38}{167}e^{7} + \frac{277}{167}e^{6} + \frac{333}{167}e^{5} - \frac{1466}{167}e^{4} - \frac{3157}{167}e^{3} + \frac{564}{167}e^{2} + \frac{2983}{167}e - \frac{451}{167}$ |
| 27 | $[27, 3, w^{5} - w^{4} - 5w^{3} + w^{2} + 6w + 1]$ | $\phantom{-}\frac{59}{167}e^{7} + \frac{162}{167}e^{6} - \frac{797}{167}e^{5} - \frac{1573}{167}e^{4} + \frac{3734}{167}e^{3} + \frac{3451}{167}e^{2} - \frac{6140}{167}e + \frac{1106}{167}$ |
| 27 | $[27, 3, w^{4} - 2w^{3} - 3w^{2} + 5w]$ | $-\frac{48}{167}e^{7} - \frac{95}{167}e^{6} + \frac{889}{167}e^{5} + \frac{1509}{167}e^{4} - \frac{4204}{167}e^{3} - \frac{5072}{167}e^{2} + \frac{4749}{167}e + \frac{139}{167}$ |
| 37 | $[37, 37, -w^{2} + 2w + 2]$ | $\phantom{-}\frac{15}{167}e^{7} + \frac{228}{167}e^{6} + \frac{672}{167}e^{5} - \frac{983}{167}e^{4} - \frac{4740}{167}e^{3} - \frac{920}{167}e^{2} + \frac{4768}{167}e - \frac{33}{167}$ |
| 47 | $[47, 47, -w^{5} + 2w^{4} + 3w^{3} - 3w^{2} - 2w - 1]$ | $\phantom{-}\frac{169}{167}e^{7} + \frac{832}{167}e^{6} - \frac{545}{167}e^{5} - \frac{5887}{167}e^{4} - \frac{1300}{167}e^{3} + \frac{8450}{167}e^{2} - \frac{1012}{167}e - \frac{639}{167}$ |
| 53 | $[53, 53, -w^{5} + 3w^{4} + 2w^{3} - 8w^{2} + 3]$ | $\phantom{-}1$ |
| 59 | $[59, 59, w^{4} - 2w^{3} - 2w^{2} + 3w - 2]$ | $-\frac{25}{167}e^{7} - \frac{46}{167}e^{6} + \frac{383}{167}e^{5} + \frac{525}{167}e^{4} - \frac{1452}{167}e^{3} - \frac{1083}{167}e^{2} + \frac{1127}{167}e - \frac{1448}{167}$ |
| 64 | $[64, 2, -2]$ | $-\frac{38}{167}e^{7} - \frac{444}{167}e^{6} - \frac{1168}{167}e^{5} + \frac{1967}{167}e^{4} + \frac{9002}{167}e^{3} + \frac{772}{167}e^{2} - \frac{11166}{167}e + \frac{1453}{167}$ |
| 67 | $[67, 67, w^{5} - 2w^{4} - 3w^{3} + 4w^{2} + w - 2]$ | $\phantom{-}\frac{94}{167}e^{7} + \frac{360}{167}e^{6} - \frac{732}{167}e^{5} - \frac{2809}{167}e^{4} + \frac{2026}{167}e^{3} + \frac{5034}{167}e^{2} - \frac{3643}{167}e - \frac{1142}{167}$ |
| 67 | $[67, 67, -w^{5} + 3w^{4} + w^{3} - 7w^{2} + 3w + 2]$ | $\phantom{-}\frac{72}{167}e^{7} + \frac{393}{167}e^{6} - \frac{81}{167}e^{5} - \frac{2681}{167}e^{4} - \frac{1376}{167}e^{3} + \frac{3600}{167}e^{2} - \frac{527}{167}e - \frac{960}{167}$ |
| 71 | $[71, 71, w^{4} - w^{3} - 4w^{2} + 2w + 1]$ | $\phantom{-}\frac{114}{167}e^{7} + \frac{664}{167}e^{6} + \frac{164}{167}e^{5} - \frac{3897}{167}e^{4} - \frac{3626}{167}e^{3} + \frac{2527}{167}e^{2} + \frac{98}{167}e + \frac{818}{167}$ |
| 71 | $[71, 71, w^{4} - w^{3} - 5w^{2} + 2w + 4]$ | $\phantom{-}\frac{3}{167}e^{7} + \frac{79}{167}e^{6} + \frac{268}{167}e^{5} - \frac{564}{167}e^{4} - \frac{2284}{167}e^{3} + \frac{1152}{167}e^{2} + \frac{3659}{167}e - \frac{1209}{167}$ |
| 73 | $[73, 73, 2w^{5} - 4w^{4} - 7w^{3} + 10w^{2} + 5w - 3]$ | $\phantom{-}\frac{33}{167}e^{7} + \frac{368}{167}e^{6} + \frac{944}{167}e^{5} - \frac{1528}{167}e^{4} - \frac{7088}{167}e^{3} - \frac{1189}{167}e^{2} + \frac{7851}{167}e - \frac{1108}{167}$ |
| 83 | $[83, 83, w^{4} - 2w^{3} - 4w^{2} + 5w + 2]$ | $-\frac{6}{167}e^{7} - \frac{158}{167}e^{6} - \frac{536}{167}e^{5} + \frac{627}{167}e^{4} + \frac{3232}{167}e^{3} + \frac{702}{167}e^{2} - \frac{2308}{167}e - \frac{87}{167}$ |
| 83 | $[83, 83, w^{5} - 3w^{4} - 2w^{3} + 9w^{2} - 3]$ | $\phantom{-}\frac{113}{167}e^{7} + \frac{248}{167}e^{6} - \frac{1651}{167}e^{5} - \frac{2373}{167}e^{4} + \frac{7712}{167}e^{3} + \frac{4982}{167}e^{2} - \frac{10919}{167}e + \frac{1555}{167}$ |
| 89 | $[89, 89, -w^{5} + w^{4} + 6w^{3} - 2w^{2} - 8w + 1]$ | $-\frac{410}{167}e^{7} - \frac{2224}{167}e^{6} + \frac{2}{167}e^{5} + \frac{13787}{167}e^{4} + \frac{11658}{167}e^{3} - \frac{13486}{167}e^{2} - \frac{7636}{167}e + \frac{2572}{167}$ |
| 97 | $[97, 97, 2w^{5} - 4w^{4} - 7w^{3} + 8w^{2} + 7w]$ | $-\frac{137}{167}e^{7} - \frac{1047}{167}e^{6} - \frac{1495}{167}e^{5} + \frac{5382}{167}e^{4} + \frac{13566}{167}e^{3} - \frac{1339}{167}e^{2} - \frac{13510}{167}e + \frac{1771}{167}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $53$ | $[53, 53, -w^{5} + 3w^{4} + 2w^{3} - 8w^{2} + 3]$ | $-1$ |