Base field 4.4.13968.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 7x^{2} + 8x + 4\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, \frac{1}{2}w^{3} - w^{2} - \frac{5}{2}w + 2]$ |
| Dimension: | $22$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $30$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{22} - 38x^{20} + 621x^{18} - 5710x^{16} + 32467x^{14} - 118294x^{12} + 277439x^{10} - 411354x^{8} + 370104x^{6} - 187524x^{4} + 46720x^{2} - 4400\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, \frac{1}{2}w^{2} + \frac{1}{2}w - 1]$ | $\phantom{-}e$ |
| 2 | $[2, 2, -\frac{1}{2}w^{2} + \frac{3}{2}w]$ | $...$ |
| 9 | $[9, 3, -\frac{1}{2}w^{2} + \frac{1}{2}w + 2]$ | $-\frac{15}{1612}e^{20} + \frac{639}{1612}e^{18} - \frac{11529}{1612}e^{16} + \frac{57695}{806}e^{14} - \frac{704265}{1612}e^{12} + \frac{2712899}{1612}e^{10} - \frac{507743}{124}e^{8} + \frac{2466728}{403}e^{6} - \frac{2122801}{403}e^{4} + \frac{914098}{403}e^{2} - \frac{135010}{403}$ |
| 13 | $[13, 13, \frac{1}{2}w^{3} - w^{2} - \frac{5}{2}w + 2]$ | $-1$ |
| 13 | $[13, 13, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + 3w - 1]$ | $-\frac{287}{8060}e^{20} + \frac{2714}{2015}e^{18} - \frac{176097}{8060}e^{16} + \frac{80017}{403}e^{14} - \frac{8934739}{8060}e^{12} + \frac{7908767}{2015}e^{10} - \frac{5452841}{620}e^{8} + \frac{24377472}{2015}e^{6} - \frac{19260107}{2015}e^{4} + \frac{7617397}{2015}e^{2} - \frac{212024}{403}$ |
| 23 | $[23, 23, \frac{1}{2}w^{3} - w^{2} - \frac{5}{2}w]$ | $...$ |
| 23 | $[23, 23, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + 3w - 3]$ | $...$ |
| 37 | $[37, 37, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - 3w - 3]$ | $-\frac{165}{1612}e^{20} + \frac{6223}{1612}e^{18} - \frac{25156}{403}e^{16} + \frac{455713}{806}e^{14} - \frac{5071801}{1612}e^{12} + \frac{17904223}{1612}e^{10} - \frac{1539861}{62}e^{8} + \frac{13771334}{403}e^{6} - \frac{10938411}{403}e^{4} + \frac{4383659}{403}e^{2} - \frac{619466}{403}$ |
| 37 | $[37, 37, \frac{1}{2}w^{3} - w^{2} - \frac{5}{2}w + 6]$ | $-\frac{287}{8060}e^{20} + \frac{2714}{2015}e^{18} - \frac{176097}{8060}e^{16} + \frac{80017}{403}e^{14} - \frac{8934739}{8060}e^{12} + \frac{7908767}{2015}e^{10} - \frac{5452841}{620}e^{8} + \frac{24377472}{2015}e^{6} - \frac{19258092}{2015}e^{4} + \frac{7603292}{2015}e^{2} - \frac{209606}{403}$ |
| 59 | $[59, 59, -\frac{1}{2}w^{3} + \frac{7}{2}w]$ | $...$ |
| 59 | $[59, 59, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + 2w - 3]$ | $...$ |
| 61 | $[61, 61, -w^{3} + 2w^{2} + 7w - 7]$ | $\phantom{-}\frac{57}{806}e^{20} - \frac{4131}{1612}e^{18} + \frac{64327}{1612}e^{16} - \frac{281715}{806}e^{14} + \frac{762821}{403}e^{12} - \frac{10577771}{1612}e^{10} + \frac{1807583}{124}e^{8} - \frac{8135095}{403}e^{6} + \frac{6579286}{403}e^{4} - \frac{2702104}{403}e^{2} + \frac{389336}{403}$ |
| 61 | $[61, 61, w^{3} - w^{2} - 6w + 3]$ | $...$ |
| 61 | $[61, 61, w^{3} - 2w^{2} - 5w + 3]$ | $-\frac{51}{1612}e^{20} + \frac{523}{403}e^{18} - \frac{36297}{1612}e^{16} + \frac{86999}{403}e^{14} - \frac{2020517}{1612}e^{12} + \frac{1831613}{403}e^{10} - \frac{1272139}{124}e^{8} + \frac{5636239}{403}e^{6} - \frac{4359528}{403}e^{4} + \frac{1683570}{403}e^{2} - \frac{230936}{403}$ |
| 61 | $[61, 61, w^{3} - w^{2} - 8w + 1]$ | $-\frac{227}{2015}e^{20} + \frac{34409}{8060}e^{18} - \frac{279709}{4030}e^{16} + \frac{509917}{806}e^{14} - \frac{7147344}{2015}e^{12} + \frac{101858477}{8060}e^{10} - \frac{8862117}{310}e^{8} + \frac{80405113}{2015}e^{6} - \frac{64941183}{2015}e^{4} + \frac{26453398}{2015}e^{2} - \frac{755390}{403}$ |
| 71 | $[71, 71, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + 5w - 5]$ | $...$ |
| 71 | $[71, 71, -w^{3} + \frac{3}{2}w^{2} + \frac{15}{2}w - 6]$ | $...$ |
| 83 | $[83, 83, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + 2w - 7]$ | $...$ |
| 83 | $[83, 83, \frac{1}{2}w^{3} - \frac{7}{2}w - 4]$ | $...$ |
| 97 | $[97, 97, 2w - 1]$ | $...$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, \frac{1}{2}w^{3} - w^{2} - \frac{5}{2}w + 2]$ | $1$ |