Base field 4.4.8725.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 10x^{2} + 2x + 19\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[31,31,\frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{8}{3}w + \frac{16}{3}]$ |
| Dimension: | $6$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $21$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{6} + 11x^{5} + 33x^{4} - 30x^{3} - 289x^{2} - 363x - 29\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 9 | $[9, 3, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{8}{3}w + \frac{10}{3}]$ | $-\frac{10}{7}e^{5} - \frac{72}{7}e^{4} - \frac{55}{7}e^{3} + \frac{516}{7}e^{2} + \frac{918}{7}e + \frac{66}{7}$ |
| 9 | $[9, 3, w + 1]$ | $\phantom{-}e$ |
| 11 | $[11, 11, w^{2} - w - 6]$ | $\phantom{-}\frac{6}{7}e^{5} + \frac{46}{7}e^{4} + \frac{47}{7}e^{3} - \frac{325}{7}e^{2} - \frac{660}{7}e - \frac{62}{7}$ |
| 11 | $[11, 11, \frac{2}{3}w^{3} - \frac{7}{3}w^{2} - \frac{7}{3}w + \frac{23}{3}]$ | $\phantom{-}\frac{8}{7}e^{5} + \frac{59}{7}e^{4} + \frac{51}{7}e^{3} - \frac{417}{7}e^{2} - \frac{789}{7}e - \frac{106}{7}$ |
| 16 | $[16, 2, 2]$ | $-\frac{15}{7}e^{5} - \frac{115}{7}e^{4} - \frac{114}{7}e^{3} + \frac{816}{7}e^{2} + \frac{1615}{7}e + \frac{134}{7}$ |
| 19 | $[19, 19, w]$ | $\phantom{-}\frac{10}{7}e^{5} + \frac{72}{7}e^{4} + \frac{55}{7}e^{3} - \frac{523}{7}e^{2} - \frac{925}{7}e - \frac{17}{7}$ |
| 19 | $[19, 19, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{8}{3}w + \frac{7}{3}]$ | $-\frac{6}{7}e^{5} - \frac{46}{7}e^{4} - \frac{47}{7}e^{3} + \frac{325}{7}e^{2} + \frac{667}{7}e + \frac{83}{7}$ |
| 19 | $[19, 19, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{2}{3}w + \frac{10}{3}]$ | $\phantom{-}\frac{8}{7}e^{5} + \frac{66}{7}e^{4} + \frac{79}{7}e^{3} - \frac{466}{7}e^{2} - \frac{992}{7}e - \frac{113}{7}$ |
| 19 | $[19, 19, -\frac{2}{3}w^{3} + \frac{4}{3}w^{2} + \frac{13}{3}w - \frac{17}{3}]$ | $\phantom{-}\frac{12}{7}e^{5} + \frac{85}{7}e^{4} + \frac{59}{7}e^{3} - \frac{615}{7}e^{2} - \frac{1054}{7}e - \frac{61}{7}$ |
| 25 | $[25, 5, \frac{2}{3}w^{3} - \frac{4}{3}w^{2} - \frac{10}{3}w + \frac{11}{3}]$ | $\phantom{-}2e^{5} + 15e^{4} + 14e^{3} - 107e^{2} - 208e - 18$ |
| 31 | $[31, 31, w + 3]$ | $-\frac{4}{7}e^{5} - \frac{26}{7}e^{4} - \frac{8}{7}e^{3} + \frac{191}{7}e^{2} + \frac{265}{7}e + \frac{4}{7}$ |
| 31 | $[31, 31, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + \frac{8}{3}w - \frac{16}{3}]$ | $\phantom{-}1$ |
| 31 | $[31, 31, \frac{2}{3}w^{3} - \frac{7}{3}w^{2} - \frac{10}{3}w + \frac{23}{3}]$ | $-\frac{31}{7}e^{5} - \frac{233}{7}e^{4} - \frac{216}{7}e^{3} + \frac{1664}{7}e^{2} + \frac{3200}{7}e + \frac{234}{7}$ |
| 31 | $[31, 31, -\frac{1}{3}w^{3} + \frac{5}{3}w^{2} + \frac{5}{3}w - \frac{25}{3}]$ | $\phantom{-}\frac{10}{7}e^{5} + \frac{72}{7}e^{4} + \frac{55}{7}e^{3} - \frac{516}{7}e^{2} - \frac{932}{7}e - \frac{101}{7}$ |
| 59 | $[59, 59, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + \frac{10}{3}w + \frac{1}{3}]$ | $-\frac{38}{7}e^{5} - \frac{282}{7}e^{4} - \frac{244}{7}e^{3} + \frac{2028}{7}e^{2} + \frac{3767}{7}e + \frac{241}{7}$ |
| 59 | $[59, 59, \frac{5}{3}w^{3} - \frac{13}{3}w^{2} - \frac{25}{3}w + \frac{47}{3}]$ | $\phantom{-}\frac{10}{7}e^{5} + \frac{79}{7}e^{4} + \frac{83}{7}e^{3} - \frac{551}{7}e^{2} - \frac{1093}{7}e - \frac{178}{7}$ |
| 61 | $[61, 61, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - \frac{11}{3}w - \frac{8}{3}]$ | $-e^{3} - 3e^{2} + 6e + 11$ |
| 61 | $[61, 61, -\frac{2}{3}w^{3} + \frac{7}{3}w^{2} + \frac{4}{3}w - \frac{26}{3}]$ | $\phantom{-}\frac{2}{7}e^{5} + \frac{13}{7}e^{4} + \frac{11}{7}e^{3} - \frac{85}{7}e^{2} - \frac{206}{7}e - \frac{58}{7}$ |
| 71 | $[71, 71, \frac{2}{3}w^{3} - \frac{4}{3}w^{2} - \frac{13}{3}w + \frac{5}{3}]$ | $-\frac{12}{7}e^{5} - \frac{92}{7}e^{4} - \frac{94}{7}e^{3} + \frac{650}{7}e^{2} + \frac{1313}{7}e + \frac{131}{7}$ |
| 71 | $[71, 71, -w^{3} + 3w^{2} + 5w - 10]$ | $\phantom{-}\frac{4}{7}e^{5} + \frac{33}{7}e^{4} + \frac{43}{7}e^{3} - \frac{233}{7}e^{2} - \frac{545}{7}e - \frac{46}{7}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $31$ | $[31,31,\frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{8}{3}w + \frac{16}{3}]$ | $-1$ |