Base field 4.4.17069.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 8x^{2} - 4x + 3\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[9, 3, w^{3} - 2w^{2} - 5w + 1]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $19$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} - 18x^{7} - 2x^{6} + 94x^{5} + 26x^{4} - 140x^{3} - 72x^{2} + 20x + 10\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}e$ |
| 3 | $[3, 3, w^{3} - 2w^{2} - 6w + 1]$ | $\phantom{-}e$ |
| 9 | $[9, 3, w^{3} - 2w^{2} - 5w + 1]$ | $-1$ |
| 16 | $[16, 2, 2]$ | $-\frac{425}{747}e^{8} + \frac{268}{747}e^{7} + \frac{7423}{747}e^{6} - \frac{3634}{747}e^{5} - \frac{12155}{249}e^{4} + \frac{9554}{747}e^{3} + \frac{15614}{249}e^{2} + \frac{1964}{249}e - \frac{3211}{747}$ |
| 17 | $[17, 17, w^{3} - w^{2} - 8w - 5]$ | $-\frac{256}{747}e^{8} + \frac{5}{747}e^{7} + \frac{4640}{747}e^{6} + \frac{328}{747}e^{5} - \frac{8218}{249}e^{4} - \frac{4733}{747}e^{3} + \frac{12706}{249}e^{2} + \frac{4147}{249}e - \frac{5576}{747}$ |
| 17 | $[17, 17, -2w^{3} + 4w^{2} + 11w - 2]$ | $\phantom{-}\frac{136}{747}e^{8} - \frac{26}{747}e^{7} - \frac{2465}{747}e^{6} + \frac{386}{747}e^{5} + \frac{4288}{249}e^{4} - \frac{1384}{747}e^{3} - \frac{6112}{249}e^{2} + \frac{248}{249}e + \frac{2402}{747}$ |
| 17 | $[17, 17, -w^{3} + 3w^{2} + 2w - 2]$ | $-\frac{256}{747}e^{8} + \frac{5}{747}e^{7} + \frac{4640}{747}e^{6} + \frac{328}{747}e^{5} - \frac{8218}{249}e^{4} - \frac{4733}{747}e^{3} + \frac{12706}{249}e^{2} + \frac{4147}{249}e - \frac{5576}{747}$ |
| 17 | $[17, 17, w^{3} - 2w^{2} - 4w + 1]$ | $\phantom{-}\frac{136}{747}e^{8} - \frac{26}{747}e^{7} - \frac{2465}{747}e^{6} + \frac{386}{747}e^{5} + \frac{4288}{249}e^{4} - \frac{1384}{747}e^{3} - \frac{6112}{249}e^{2} + \frac{248}{249}e + \frac{2402}{747}$ |
| 25 | $[25, 5, w^{3} - 3w^{2} - 3w + 1]$ | $-\frac{196}{249}e^{8} + \frac{140}{249}e^{7} + \frac{3428}{249}e^{6} - \frac{2021}{249}e^{5} - \frac{5672}{83}e^{4} + \frac{6418}{249}e^{3} + \frac{7666}{83}e^{2} + \frac{82}{83}e - \frac{3242}{249}$ |
| 25 | $[25, 5, w^{2} - 2w - 7]$ | $-\frac{196}{249}e^{8} + \frac{140}{249}e^{7} + \frac{3428}{249}e^{6} - \frac{2021}{249}e^{5} - \frac{5672}{83}e^{4} + \frac{6418}{249}e^{3} + \frac{7666}{83}e^{2} + \frac{82}{83}e - \frac{3242}{249}$ |
| 29 | $[29, 29, w^{3} - 2w^{2} - 6w - 1]$ | $\phantom{-}\frac{65}{249}e^{8} - \frac{82}{249}e^{7} - \frac{1147}{249}e^{6} + \frac{1294}{249}e^{5} + \frac{1942}{83}e^{4} - \frac{5246}{249}e^{3} - \frac{2804}{83}e^{2} + \frac{1344}{83}e + \frac{1906}{249}$ |
| 29 | $[29, 29, w - 2]$ | $\phantom{-}\frac{65}{249}e^{8} - \frac{82}{249}e^{7} - \frac{1147}{249}e^{6} + \frac{1294}{249}e^{5} + \frac{1942}{83}e^{4} - \frac{5246}{249}e^{3} - \frac{2804}{83}e^{2} + \frac{1344}{83}e + \frac{1906}{249}$ |
| 53 | $[53, 53, w^{3} - 3w^{2} - 4w + 4]$ | $\phantom{-}\frac{146}{747}e^{8} + \frac{38}{747}e^{7} - \frac{2833}{747}e^{6} - \frac{794}{747}e^{5} + \frac{5570}{249}e^{4} + \frac{4666}{747}e^{3} - \frac{10355}{249}e^{2} - \frac{1550}{249}e + \frac{10510}{747}$ |
| 53 | $[53, 53, -w^{3} + 3w^{2} + 4w - 5]$ | $\phantom{-}\frac{146}{747}e^{8} + \frac{38}{747}e^{7} - \frac{2833}{747}e^{6} - \frac{794}{747}e^{5} + \frac{5570}{249}e^{4} + \frac{4666}{747}e^{3} - \frac{10355}{249}e^{2} - \frac{1550}{249}e + \frac{10510}{747}$ |
| 61 | $[61, 61, -w^{3} + 2w^{2} + 7w - 4]$ | $-\frac{278}{249}e^{8} + \frac{163}{249}e^{7} + \frac{4852}{249}e^{6} - \frac{2305}{249}e^{5} - \frac{7984}{83}e^{4} + \frac{7106}{249}e^{3} + \frac{10736}{83}e^{2} - \frac{48}{83}e - \frac{6382}{249}$ |
| 61 | $[61, 61, -4w^{3} + 11w^{2} + 14w - 8]$ | $-\frac{278}{249}e^{8} + \frac{163}{249}e^{7} + \frac{4852}{249}e^{6} - \frac{2305}{249}e^{5} - \frac{7984}{83}e^{4} + \frac{7106}{249}e^{3} + \frac{10736}{83}e^{2} - \frac{48}{83}e - \frac{6382}{249}$ |
| 101 | $[101, 101, -w^{3} + 2w^{2} + 7w - 1]$ | $-\frac{1100}{747}e^{8} + \frac{430}{747}e^{7} + \frac{19564}{747}e^{6} - \frac{6154}{747}e^{5} - \frac{33452}{249}e^{4} + \frac{20246}{747}e^{3} + \frac{48908}{249}e^{2} - \frac{1420}{249}e - \frac{31336}{747}$ |
| 103 | $[103, 103, -w^{3} + 3w^{2} + 2w - 5]$ | $\phantom{-}\frac{70}{249}e^{8} - \frac{50}{249}e^{7} - \frac{1331}{249}e^{6} + \frac{704}{249}e^{5} + \frac{2500}{83}e^{4} - \frac{1972}{249}e^{3} - \frac{4137}{83}e^{2} - \frac{468}{83}e + \frac{1976}{249}$ |
| 103 | $[103, 103, -w^{3} + w^{2} + 8w + 2]$ | $\phantom{-}\frac{70}{249}e^{8} - \frac{50}{249}e^{7} - \frac{1331}{249}e^{6} + \frac{704}{249}e^{5} + \frac{2500}{83}e^{4} - \frac{1972}{249}e^{3} - \frac{4137}{83}e^{2} - \frac{468}{83}e + \frac{1976}{249}$ |
| 113 | $[113, 113, w^{3} - 3w^{2} - 3w + 7]$ | $\phantom{-}\frac{754}{747}e^{8} - \frac{254}{747}e^{7} - \frac{13106}{747}e^{6} + \frac{2909}{747}e^{5} + \frac{21166}{249}e^{4} - \frac{2488}{747}e^{3} - \frac{25654}{249}e^{2} - \frac{8131}{249}e + \frac{5078}{747}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $9$ | $[9, 3, w^{3} - 2w^{2} - 5w + 1]$ | $1$ |