Base field 4.4.11525.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 11x^{2} + 5x + 25\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[25, 5, \frac{2}{5}w^{3} - \frac{2}{5}w^{2} - \frac{12}{5}w + 1]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $23$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 51x^{6} + 25x^{5} + 711x^{4} - 363x^{3} - 2619x^{2} + 432x + 1620\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, \frac{1}{5}w^{3} + \frac{4}{5}w^{2} - \frac{11}{5}w - 7]$ | $-1$ |
| 5 | $[5, 5, \frac{1}{5}w^{3} - \frac{6}{5}w^{2} - \frac{1}{5}w + 4]$ | $-1$ |
| 11 | $[11, 11, w + 1]$ | $\phantom{-}e$ |
| 11 | $[11, 11, \frac{1}{5}w^{3} - \frac{1}{5}w^{2} - \frac{11}{5}w + 2]$ | $\phantom{-}e$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}\frac{23}{6232}e^{7} - \frac{181}{28044}e^{6} - \frac{10225}{56088}e^{5} + \frac{22093}{56088}e^{4} + \frac{42041}{18696}e^{3} - \frac{26343}{6232}e^{2} - \frac{36387}{6232}e + \frac{12805}{3116}$ |
| 19 | $[19, 19, -\frac{1}{5}w^{3} + \frac{1}{5}w^{2} + \frac{1}{5}w + 1]$ | $\phantom{-}\frac{71}{14022}e^{7} + \frac{32}{7011}e^{6} - \frac{2905}{14022}e^{5} + \frac{421}{4674}e^{4} + \frac{3073}{1558}e^{3} - \frac{15049}{4674}e^{2} - \frac{1981}{1558}e + \frac{5935}{779}$ |
| 19 | $[19, 19, \frac{1}{5}w^{3} - \frac{1}{5}w^{2} - \frac{11}{5}w]$ | $-\frac{13}{28044}e^{7} + \frac{49}{14022}e^{6} + \frac{883}{28044}e^{5} - \frac{5783}{28044}e^{4} - \frac{1989}{3116}e^{3} + \frac{27859}{9348}e^{2} + \frac{12717}{3116}e - \frac{9535}{1558}$ |
| 19 | $[19, 19, -\frac{2}{5}w^{3} + \frac{2}{5}w^{2} + \frac{17}{5}w]$ | $\phantom{-}\frac{71}{14022}e^{7} + \frac{32}{7011}e^{6} - \frac{2905}{14022}e^{5} + \frac{421}{4674}e^{4} + \frac{3073}{1558}e^{3} - \frac{15049}{4674}e^{2} - \frac{1981}{1558}e + \frac{5935}{779}$ |
| 19 | $[19, 19, w - 1]$ | $-\frac{13}{28044}e^{7} + \frac{49}{14022}e^{6} + \frac{883}{28044}e^{5} - \frac{5783}{28044}e^{4} - \frac{1989}{3116}e^{3} + \frac{27859}{9348}e^{2} + \frac{12717}{3116}e - \frac{9535}{1558}$ |
| 29 | $[29, 29, w^{2} - w - 8]$ | $\phantom{-}\frac{65}{28044}e^{7} - \frac{245}{14022}e^{6} - \frac{4415}{28044}e^{5} + \frac{19567}{28044}e^{4} + \frac{6829}{3116}e^{3} - \frac{20465}{3116}e^{2} - \frac{10613}{3116}e + \frac{16515}{1558}$ |
| 29 | $[29, 29, \frac{2}{5}w^{3} - \frac{2}{5}w^{2} - \frac{7}{5}w + 2]$ | $\phantom{-}\frac{65}{28044}e^{7} - \frac{245}{14022}e^{6} - \frac{4415}{28044}e^{5} + \frac{19567}{28044}e^{4} + \frac{6829}{3116}e^{3} - \frac{20465}{3116}e^{2} - \frac{10613}{3116}e + \frac{16515}{1558}$ |
| 31 | $[31, 31, \frac{1}{5}w^{3} - \frac{1}{5}w^{2} - \frac{1}{5}w + 2]$ | $-\frac{16}{7011}e^{7} - \frac{179}{7011}e^{6} + \frac{128}{7011}e^{5} + \frac{634}{779}e^{4} + \frac{668}{779}e^{3} - \frac{14789}{2337}e^{2} - \frac{4842}{779}e + \frac{7090}{779}$ |
| 31 | $[31, 31, \frac{2}{5}w^{3} - \frac{2}{5}w^{2} - \frac{17}{5}w + 3]$ | $-\frac{16}{7011}e^{7} - \frac{179}{7011}e^{6} + \frac{128}{7011}e^{5} + \frac{634}{779}e^{4} + \frac{668}{779}e^{3} - \frac{14789}{2337}e^{2} - \frac{4842}{779}e + \frac{7090}{779}$ |
| 61 | $[61, 61, -\frac{2}{5}w^{3} - \frac{3}{5}w^{2} + \frac{12}{5}w + 5]$ | $\phantom{-}\frac{1}{76}e^{7} + \frac{7}{342}e^{6} - \frac{395}{684}e^{5} - \frac{229}{684}e^{4} + \frac{1547}{228}e^{3} + \frac{17}{228}e^{2} - \frac{1321}{76}e - \frac{5}{38}$ |
| 61 | $[61, 61, -\frac{4}{5}w^{3} - \frac{6}{5}w^{2} + \frac{39}{5}w + 15]$ | $\phantom{-}\frac{1}{76}e^{7} + \frac{7}{342}e^{6} - \frac{395}{684}e^{5} - \frac{229}{684}e^{4} + \frac{1547}{228}e^{3} + \frac{17}{228}e^{2} - \frac{1321}{76}e - \frac{5}{38}$ |
| 61 | $[61, 61, \frac{3}{5}w^{3} + \frac{2}{5}w^{2} - \frac{18}{5}w - 3]$ | $-\frac{49}{14022}e^{7} - \frac{55}{7011}e^{6} + \frac{2729}{14022}e^{5} + \frac{481}{1558}e^{4} - \frac{14701}{4674}e^{3} - \frac{14959}{4674}e^{2} + \frac{20129}{1558}e + \frac{5647}{779}$ |
| 61 | $[61, 61, \frac{7}{5}w^{3} + \frac{3}{5}w^{2} - \frac{62}{5}w - 13]$ | $-\frac{49}{14022}e^{7} - \frac{55}{7011}e^{6} + \frac{2729}{14022}e^{5} + \frac{481}{1558}e^{4} - \frac{14701}{4674}e^{3} - \frac{14959}{4674}e^{2} + \frac{20129}{1558}e + \frac{5647}{779}$ |
| 71 | $[71, 71, \frac{1}{5}w^{3} - \frac{6}{5}w^{2} - \frac{6}{5}w + 4]$ | $\phantom{-}\frac{71}{14022}e^{7} + \frac{32}{7011}e^{6} - \frac{2905}{14022}e^{5} + \frac{421}{4674}e^{4} + \frac{10777}{4674}e^{3} - \frac{4497}{1558}e^{2} - \frac{12887}{1558}e + \frac{4377}{779}$ |
| 71 | $[71, 71, w^{2} - 8]$ | $\phantom{-}\frac{71}{14022}e^{7} + \frac{32}{7011}e^{6} - \frac{2905}{14022}e^{5} + \frac{421}{4674}e^{4} + \frac{10777}{4674}e^{3} - \frac{4497}{1558}e^{2} - \frac{12887}{1558}e + \frac{4377}{779}$ |
| 79 | $[79, 79, \frac{3}{5}w^{3} + \frac{12}{5}w^{2} - \frac{33}{5}w - 22]$ | $-\frac{421}{28044}e^{7} - \frac{91}{14022}e^{6} + \frac{19727}{28044}e^{5} - \frac{1465}{9348}e^{4} - \frac{77947}{9348}e^{3} + \frac{32839}{9348}e^{2} + \frac{61405}{3116}e - \frac{6775}{1558}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $5$ | $[5, 5, \frac{1}{5}w^{3} + \frac{4}{5}w^{2} - \frac{11}{5}w - 7]$ | $1$ |
| $5$ | $[5, 5, \frac{1}{5}w^{3} - \frac{6}{5}w^{2} - \frac{1}{5}w + 4]$ | $1$ |