Base field 4.4.12400.1
Generator \(w\), with minimal polynomial \(x^{4} - 12x^{2} + 31\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[25, 5, w^{2} - 6]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $30$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} + 2x^{8} - 30x^{7} - 50x^{6} + 293x^{5} + 364x^{4} - 1076x^{3} - 896x^{2} + 1232x + 448\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -\frac{1}{2}w^{3} - \frac{1}{2}w^{2} + \frac{7}{2}w + \frac{11}{2}]$ | $\phantom{-}e$ |
| 5 | $[5, 5, -\frac{1}{2}w^{3} + w^{2} + \frac{5}{2}w - 4]$ | $-1$ |
| 5 | $[5, 5, -\frac{1}{2}w^{2} - w + \frac{3}{2}]$ | $-1$ |
| 9 | $[9, 3, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{7}{2}w - \frac{9}{2}]$ | $\phantom{-}\frac{1}{856}e^{8} + \frac{2}{107}e^{7} - \frac{5}{214}e^{6} - \frac{165}{428}e^{5} - \frac{47}{856}e^{4} + \frac{709}{428}e^{3} + \frac{935}{428}e^{2} - \frac{99}{214}e - \frac{432}{107}$ |
| 9 | $[9, 3, -\frac{1}{2}w^{3} - \frac{1}{2}w^{2} + \frac{7}{2}w + \frac{9}{2}]$ | $\phantom{-}\frac{1}{856}e^{8} + \frac{2}{107}e^{7} - \frac{5}{214}e^{6} - \frac{165}{428}e^{5} - \frac{47}{856}e^{4} + \frac{709}{428}e^{3} + \frac{935}{428}e^{2} - \frac{99}{214}e - \frac{432}{107}$ |
| 19 | $[19, 19, -\frac{1}{2}w^{2} + w + \frac{5}{2}]$ | $\phantom{-}\frac{29}{1712}e^{8} + \frac{9}{428}e^{7} - \frac{397}{856}e^{6} - \frac{291}{856}e^{5} + \frac{6341}{1712}e^{4} + \frac{231}{856}e^{3} - \frac{3295}{428}e^{2} + \frac{865}{214}e - \frac{58}{107}$ |
| 19 | $[19, 19, \frac{1}{2}w^{2} + w - \frac{5}{2}]$ | $\phantom{-}\frac{29}{1712}e^{8} + \frac{9}{428}e^{7} - \frac{397}{856}e^{6} - \frac{291}{856}e^{5} + \frac{6341}{1712}e^{4} + \frac{231}{856}e^{3} - \frac{3295}{428}e^{2} + \frac{865}{214}e - \frac{58}{107}$ |
| 29 | $[29, 29, -\frac{3}{2}w^{2} - w + \frac{17}{2}]$ | $\phantom{-}\frac{5}{856}e^{8} - \frac{27}{856}e^{7} - \frac{25}{214}e^{6} + \frac{88}{107}e^{5} + \frac{407}{856}e^{4} - \frac{5429}{856}e^{3} - \frac{35}{107}e^{2} + \frac{1518}{107}e + \frac{194}{107}$ |
| 29 | $[29, 29, -\frac{3}{2}w^{2} + w + \frac{17}{2}]$ | $\phantom{-}\frac{5}{856}e^{8} - \frac{27}{856}e^{7} - \frac{25}{214}e^{6} + \frac{88}{107}e^{5} + \frac{407}{856}e^{4} - \frac{5429}{856}e^{3} - \frac{35}{107}e^{2} + \frac{1518}{107}e + \frac{194}{107}$ |
| 31 | $[31, 31, \frac{1}{2}w^{3} - \frac{7}{2}w]$ | $\phantom{-}\frac{27}{856}e^{8} + \frac{1}{214}e^{7} - \frac{377}{428}e^{6} + \frac{39}{428}e^{5} + \frac{6435}{856}e^{4} - \frac{1187}{428}e^{3} - \frac{2115}{107}e^{2} + \frac{750}{107}e + \frac{534}{107}$ |
| 59 | $[59, 59, \frac{1}{2}w^{2} + w - \frac{11}{2}]$ | $-\frac{13}{856}e^{8} + \frac{3}{428}e^{7} + \frac{237}{428}e^{6} - \frac{209}{428}e^{5} - \frac{5381}{856}e^{4} + \frac{1437}{214}e^{3} + \frac{4783}{214}e^{2} - \frac{3635}{214}e - \frac{1874}{107}$ |
| 59 | $[59, 59, \frac{1}{2}w^{2} - w - \frac{11}{2}]$ | $-\frac{13}{856}e^{8} + \frac{3}{428}e^{7} + \frac{237}{428}e^{6} - \frac{209}{428}e^{5} - \frac{5381}{856}e^{4} + \frac{1437}{214}e^{3} + \frac{4783}{214}e^{2} - \frac{3635}{214}e - \frac{1874}{107}$ |
| 61 | $[61, 61, -\frac{1}{2}w^{3} + w^{2} + \frac{7}{2}w - 5]$ | $-\frac{3}{428}e^{8} + \frac{11}{856}e^{7} + \frac{15}{107}e^{6} - \frac{187}{428}e^{5} - \frac{45}{107}e^{4} + \frac{3583}{856}e^{3} - \frac{795}{428}e^{2} - \frac{987}{107}e + \frac{24}{107}$ |
| 61 | $[61, 61, \frac{1}{2}w^{3} + w^{2} - \frac{7}{2}w - 5]$ | $-\frac{3}{428}e^{8} + \frac{11}{856}e^{7} + \frac{15}{107}e^{6} - \frac{187}{428}e^{5} - \frac{45}{107}e^{4} + \frac{3583}{856}e^{3} - \frac{795}{428}e^{2} - \frac{987}{107}e + \frac{24}{107}$ |
| 71 | $[71, 71, -\frac{1}{2}w^{3} - 2w^{2} + \frac{11}{2}w + 13]$ | $-\frac{19}{856}e^{8} + \frac{17}{856}e^{7} + \frac{297}{428}e^{6} - \frac{91}{214}e^{5} - \frac{5741}{856}e^{4} + \frac{1627}{856}e^{3} + \frac{8771}{428}e^{2} + \frac{245}{107}e - \frac{1208}{107}$ |
| 71 | $[71, 71, \frac{3}{2}w^{3} + 2w^{2} - \frac{23}{2}w - 18]$ | $-\frac{19}{856}e^{8} + \frac{17}{856}e^{7} + \frac{297}{428}e^{6} - \frac{91}{214}e^{5} - \frac{5741}{856}e^{4} + \frac{1627}{856}e^{3} + \frac{8771}{428}e^{2} + \frac{245}{107}e - \frac{1208}{107}$ |
| 79 | $[79, 79, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{7}{2}w - \frac{1}{2}]$ | $\phantom{-}\frac{3}{214}e^{8} - \frac{11}{428}e^{7} - \frac{227}{428}e^{6} + \frac{187}{214}e^{5} + \frac{1357}{214}e^{4} - \frac{3797}{428}e^{3} - \frac{10501}{428}e^{2} + \frac{5125}{214}e + \frac{2306}{107}$ |
| 79 | $[79, 79, -2w^{2} + w + 10]$ | $-\frac{19}{428}e^{8} - \frac{73}{856}e^{7} + \frac{487}{428}e^{6} + \frac{813}{428}e^{5} - \frac{1747}{214}e^{4} - \frac{9265}{856}e^{3} + \frac{1550}{107}e^{2} + \frac{3441}{214}e + \frac{366}{107}$ |
| 79 | $[79, 79, 2w^{2} + w - 10]$ | $-\frac{19}{428}e^{8} - \frac{73}{856}e^{7} + \frac{487}{428}e^{6} + \frac{813}{428}e^{5} - \frac{1747}{214}e^{4} - \frac{9265}{856}e^{3} + \frac{1550}{107}e^{2} + \frac{3441}{214}e + \frac{366}{107}$ |
| 79 | $[79, 79, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w + \frac{1}{2}]$ | $\phantom{-}\frac{3}{214}e^{8} - \frac{11}{428}e^{7} - \frac{227}{428}e^{6} + \frac{187}{214}e^{5} + \frac{1357}{214}e^{4} - \frac{3797}{428}e^{3} - \frac{10501}{428}e^{2} + \frac{5125}{214}e + \frac{2306}{107}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $5$ | $[5,5,-\frac{1}{2}w^{3}+w^{2}+\frac{5}{2}w-4]$ | $1$ |
| $5$ | $[5,5,-\frac{1}{2}w^{2}-w+\frac{3}{2}]$ | $1$ |