Base field 4.4.13025.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 12x^{2} + 3x + 29\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[20, 10, \frac{1}{4}w^{3} - \frac{1}{2}w^{2} - \frac{5}{2}w + \frac{5}{4}]$ |
| Dimension: | $7$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $19$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{7} - 3x^{6} - 23x^{5} + 69x^{4} + 136x^{3} - 404x^{2} - 192x + 576\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, \frac{1}{2}w^{3} - 2w^{2} - 2w + \frac{15}{2}]$ | $\phantom{-}1$ |
| 4 | $[4, 2, -w^{2} + w + 8]$ | $\phantom{-}e$ |
| 5 | $[5, 5, -\frac{1}{4}w^{3} + \frac{1}{2}w^{2} + \frac{1}{2}w - \frac{9}{4}]$ | $\phantom{-}1$ |
| 5 | $[5, 5, \frac{1}{2}w^{3} - w^{2} - 4w + \frac{9}{2}]$ | $-\frac{5}{288}e^{6} + \frac{1}{96}e^{5} + \frac{151}{288}e^{4} - \frac{23}{96}e^{3} - \frac{305}{72}e^{2} + \frac{97}{72}e + \frac{43}{6}$ |
| 19 | $[19, 19, \frac{1}{4}w^{3} - \frac{3}{2}w^{2} - \frac{1}{2}w + \frac{21}{4}]$ | $-\frac{35}{288}e^{6} + \frac{7}{96}e^{5} + \frac{769}{288}e^{4} - \frac{161}{96}e^{3} - \frac{1055}{72}e^{2} + \frac{463}{72}e + \frac{103}{6}$ |
| 19 | $[19, 19, \frac{1}{4}w^{3} - \frac{3}{2}w^{2} - \frac{1}{2}w + \frac{41}{4}]$ | $\phantom{-}\frac{1}{36}e^{6} + \frac{1}{12}e^{5} - \frac{23}{36}e^{4} - \frac{17}{12}e^{3} + \frac{34}{9}e^{2} + \frac{71}{18}e - \frac{8}{3}$ |
| 29 | $[29, 29, w]$ | $\phantom{-}\frac{7}{96}e^{6} - \frac{3}{32}e^{5} - \frac{149}{96}e^{4} + \frac{53}{32}e^{3} + \frac{193}{24}e^{2} - \frac{95}{24}e - \frac{13}{2}$ |
| 29 | $[29, 29, \frac{1}{4}w^{3} - \frac{1}{2}w^{2} - \frac{1}{2}w + \frac{1}{4}]$ | $-\frac{17}{144}e^{6} + \frac{1}{48}e^{5} + \frac{391}{144}e^{4} - \frac{23}{48}e^{3} - \frac{307}{18}e^{2} + \frac{13}{36}e + \frac{85}{3}$ |
| 29 | $[29, 29, \frac{1}{4}w^{3} - \frac{1}{2}w^{2} - \frac{5}{2}w + \frac{9}{4}]$ | $-\frac{7}{144}e^{6} - \frac{1}{48}e^{5} + \frac{161}{144}e^{4} - \frac{1}{48}e^{3} - \frac{137}{18}e^{2} + \frac{89}{36}e + \frac{47}{3}$ |
| 29 | $[29, 29, -\frac{1}{2}w^{3} + w^{2} + 4w - \frac{5}{2}]$ | $\phantom{-}\frac{11}{96}e^{6} + \frac{1}{32}e^{5} - \frac{265}{96}e^{4} - \frac{23}{32}e^{3} + \frac{443}{24}e^{2} + \frac{113}{24}e - \frac{63}{2}$ |
| 41 | $[41, 41, \frac{3}{4}w^{3} - \frac{5}{2}w^{2} - \frac{7}{2}w + \frac{47}{4}]$ | $-\frac{1}{24}e^{6} + \frac{1}{8}e^{5} + \frac{35}{24}e^{4} - \frac{19}{8}e^{3} - \frac{79}{6}e^{2} + \frac{28}{3}e + 24$ |
| 41 | $[41, 41, \frac{1}{4}w^{3} + \frac{1}{2}w^{2} - \frac{5}{2}w - \frac{15}{4}]$ | $\phantom{-}\frac{1}{48}e^{6} - \frac{1}{16}e^{5} - \frac{47}{48}e^{4} + \frac{15}{16}e^{3} + \frac{34}{3}e^{2} - \frac{35}{12}e - 27$ |
| 61 | $[61, 61, -\frac{3}{2}w^{3} + 3w^{2} + 11w - \frac{21}{2}]$ | $\phantom{-}\frac{17}{288}e^{6} - \frac{13}{96}e^{5} - \frac{283}{288}e^{4} + \frac{251}{96}e^{3} + \frac{173}{72}e^{2} - \frac{625}{72}e + \frac{17}{6}$ |
| 61 | $[61, 61, w^{3} - 2w^{2} - 4w + 6]$ | $-\frac{19}{144}e^{6} - \frac{1}{48}e^{5} + \frac{473}{144}e^{4} + \frac{23}{48}e^{3} - \frac{817}{36}e^{2} - \frac{157}{36}e + \frac{113}{3}$ |
| 79 | $[79, 79, \frac{3}{4}w^{3} - \frac{5}{2}w^{2} - \frac{7}{2}w + \frac{27}{4}]$ | $-\frac{31}{288}e^{6} + \frac{11}{96}e^{5} + \frac{749}{288}e^{4} - \frac{205}{96}e^{3} - \frac{1189}{72}e^{2} + \frac{551}{72}e + \frac{131}{6}$ |
| 79 | $[79, 79, \frac{1}{4}w^{3} + \frac{1}{2}w^{2} - \frac{5}{2}w - \frac{35}{4}]$ | $\phantom{-}\frac{5}{72}e^{6} - \frac{7}{24}e^{5} - \frac{133}{72}e^{4} + \frac{125}{24}e^{3} + \frac{511}{36}e^{2} - \frac{152}{9}e - \frac{80}{3}$ |
| 81 | $[81, 3, -3]$ | $-\frac{11}{96}e^{6} - \frac{1}{32}e^{5} + \frac{265}{96}e^{4} - \frac{9}{32}e^{3} - \frac{443}{24}e^{2} + \frac{151}{24}e + \frac{59}{2}$ |
| 89 | $[89, 89, \frac{7}{4}w^{3} - \frac{11}{2}w^{2} - \frac{21}{2}w + \frac{119}{4}]$ | $-\frac{35}{288}e^{6} + \frac{7}{96}e^{5} + \frac{913}{288}e^{4} - \frac{113}{96}e^{3} - \frac{1667}{72}e^{2} + \frac{67}{72}e + \frac{265}{6}$ |
| 89 | $[89, 89, \frac{1}{4}w^{3} + \frac{3}{2}w^{2} - \frac{3}{2}w - \frac{23}{4}]$ | $\phantom{-}\frac{25}{288}e^{6} - \frac{5}{96}e^{5} - \frac{611}{288}e^{4} + \frac{163}{96}e^{3} + \frac{1057}{72}e^{2} - \frac{593}{72}e - \frac{173}{6}$ |
| 109 | $[109, 109, -\frac{1}{2}w^{3} + 3w^{2} + 2w - \frac{41}{2}]$ | $-\frac{35}{144}e^{6} + \frac{7}{48}e^{5} + \frac{769}{144}e^{4} - \frac{161}{48}e^{3} - \frac{1091}{36}e^{2} + \frac{427}{36}e + \frac{127}{3}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, \frac{1}{2}w^{3} - 2w^{2} - 2w + \frac{15}{2}]$ | $-1$ |
| $5$ | $[5, 5, -\frac{1}{4}w^{3} + \frac{1}{2}w^{2} + \frac{1}{2}w - \frac{9}{4}]$ | $-1$ |