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: | $[27, 3, -w^{4} + w^{3} + 4w^{2} - 2w - 1]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $25$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} - 4x^{8} - 54x^{7} + 258x^{6} + 648x^{5} - 4516x^{4} + 2280x^{3} + 17256x^{2} - 27184x + 9680\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w^{4} - w^{3} - 5w^{2} + 3w + 3]$ | $\phantom{-}1$ |
| 9 | $[9, 3, -w^{4} + 5w^{2} - 3]$ | $-1$ |
| 9 | $[9, 3, -w^{4} + w^{3} + 5w^{2} - 3w - 2]$ | $\phantom{-}e$ |
| 13 | $[13, 13, -w^{4} + w^{3} + 4w^{2} - 3w - 1]$ | $...$ |
| 17 | $[17, 17, w^{4} - w^{3} - 5w^{2} + 3w + 1]$ | $-\frac{3721627}{323898572}e^{8} + \frac{1623847}{80974643}e^{7} + \frac{107168797}{161949286}e^{6} - \frac{240139299}{161949286}e^{5} - \frac{1714788127}{161949286}e^{4} + \frac{2295748533}{80974643}e^{3} + \frac{2788714909}{80974643}e^{2} - \frac{9893512164}{80974643}e + \frac{4545784174}{80974643}$ |
| 19 | $[19, 19, -w^{3} + w^{2} + 4w - 2]$ | $...$ |
| 23 | $[23, 23, -w^{2} + 3]$ | $\phantom{-}\frac{80125}{323898572}e^{8} - \frac{192127}{323898572}e^{7} - \frac{2641657}{323898572}e^{6} + \frac{4407344}{80974643}e^{5} - \frac{3388326}{80974643}e^{4} - \frac{98480951}{80974643}e^{3} + \frac{160813814}{80974643}e^{2} + \frac{536250954}{80974643}e - \frac{365652138}{80974643}$ |
| 31 | $[31, 31, w^{3} - 4w + 2]$ | $-\frac{3137161}{647797144}e^{8} + \frac{3064395}{323898572}e^{7} + \frac{90780243}{323898572}e^{6} - \frac{222840345}{323898572}e^{5} - \frac{361861780}{80974643}e^{4} + \frac{1070038766}{80974643}e^{3} + \frac{1073158857}{80974643}e^{2} - \frac{4802162928}{80974643}e + \frac{2656754536}{80974643}$ |
| 32 | $[32, 2, 2]$ | $...$ |
| 37 | $[37, 37, w^{3} - 3w - 1]$ | $...$ |
| 53 | $[53, 53, -2w^{4} + w^{3} + 9w^{2} - 3w - 2]$ | $...$ |
| 59 | $[59, 59, -w^{4} + 5w^{2} + w - 4]$ | $...$ |
| 61 | $[61, 61, -w^{4} + w^{3} + 5w^{2} - 4w]$ | $\phantom{-}\frac{899159}{161949286}e^{8} - \frac{949801}{323898572}e^{7} - \frac{25438165}{80974643}e^{6} + \frac{28181802}{80974643}e^{5} + \frac{416926355}{80974643}e^{4} - \frac{631429655}{80974643}e^{3} - \frac{1754959296}{80974643}e^{2} + \frac{2842123135}{80974643}e - \frac{22096124}{80974643}$ |
| 67 | $[67, 67, -w^{4} + 6w^{2} + 2w - 4]$ | $...$ |
| 71 | $[71, 71, 2w^{4} - w^{3} - 9w^{2} + 4w + 5]$ | $-\frac{1235107}{161949286}e^{8} + \frac{6306199}{323898572}e^{7} + \frac{144289425}{323898572}e^{6} - \frac{106379731}{80974643}e^{5} - \frac{577669983}{80974643}e^{4} + \frac{1926881968}{80974643}e^{3} + \frac{1707246362}{80974643}e^{2} - \frac{7989117252}{80974643}e + \frac{4447533736}{80974643}$ |
| 79 | $[79, 79, 2w^{4} - w^{3} - 10w^{2} + 2w + 7]$ | $...$ |
| 83 | $[83, 83, -w^{4} + 2w^{3} + 5w^{2} - 7w - 2]$ | $...$ |
| 83 | $[83, 83, -w^{4} + w^{3} + 4w^{2} - 3w + 3]$ | $-\frac{2427677}{161949286}e^{8} + \frac{1721616}{80974643}e^{7} + \frac{280671489}{323898572}e^{6} - \frac{270389261}{161949286}e^{5} - \frac{1144038241}{80974643}e^{4} + \frac{2673026236}{80974643}e^{3} + \frac{4143065467}{80974643}e^{2} - \frac{11748248751}{80974643}e + \frac{3998257342}{80974643}$ |
| 83 | $[83, 83, w^{4} - w^{3} - 5w^{2} + 4w - 1]$ | $...$ |
| 83 | $[83, 83, -w^{4} + w^{3} + 4w^{2} - 4w - 2]$ | $-\frac{2659403}{323898572}e^{8} + \frac{1147772}{80974643}e^{7} + \frac{156377221}{323898572}e^{6} - \frac{168288461}{161949286}e^{5} - \frac{652587593}{80974643}e^{4} + \frac{1607574271}{80974643}e^{3} + \frac{2538919780}{80974643}e^{2} - \frac{6876446018}{80974643}e + \frac{1650166742}{80974643}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, w^{4} - w^{3} - 5w^{2} + 3w + 3]$ | $-1$ |
| $9$ | $[9, 3, -w^{4} + 5w^{2} - 3]$ | $1$ |