Base field 4.4.13888.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 7x^{2} + 6x + 9\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[28, 14, w^{2} - w - 5]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $28$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 3x^{7} - 26x^{6} + 102x^{5} + 53x^{4} - 507x^{3} + 320x^{2} + 296x - 172\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + \frac{4}{3}w - 1]$ | $\phantom{-}1$ |
| 7 | $[7, 7, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{7}{3}w + 1]$ | $\phantom{-}1$ |
| 7 | $[7, 7, w - 2]$ | $\phantom{-}\frac{11}{312}e^{7} - \frac{9}{104}e^{6} - \frac{103}{104}e^{5} + \frac{931}{312}e^{4} + \frac{49}{12}e^{3} - \frac{29}{2}e^{2} + \frac{35}{78}e + \frac{283}{39}$ |
| 7 | $[7, 7, w - 1]$ | $\phantom{-}e$ |
| 9 | $[9, 3, w]$ | $-\frac{9}{208}e^{7} + \frac{2}{39}e^{6} + \frac{355}{312}e^{5} - \frac{94}{39}e^{4} - \frac{73}{16}e^{3} + \frac{155}{12}e^{2} - \frac{379}{156}e - \frac{1043}{156}$ |
| 9 | $[9, 3, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + \frac{7}{3}w - 2]$ | $\phantom{-}\frac{3}{104}e^{7} - \frac{1}{156}e^{6} - \frac{127}{156}e^{5} + \frac{125}{156}e^{4} + \frac{39}{8}e^{3} - \frac{29}{6}e^{2} - \frac{437}{78}e + \frac{365}{78}$ |
| 23 | $[23, 23, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + \frac{1}{3}w - 2]$ | $-\frac{3}{208}e^{7} + \frac{7}{156}e^{6} + \frac{101}{312}e^{5} - \frac{251}{156}e^{4} + \frac{5}{16}e^{3} + \frac{97}{12}e^{2} - \frac{1097}{156}e - \frac{937}{156}$ |
| 23 | $[23, 23, -\frac{2}{3}w^{3} + \frac{4}{3}w^{2} + \frac{11}{3}w - 4]$ | $\phantom{-}\frac{23}{312}e^{7} - \frac{17}{156}e^{6} - \frac{163}{78}e^{5} + \frac{223}{52}e^{4} + \frac{245}{24}e^{3} - \frac{127}{6}e^{2} - \frac{305}{78}e + \frac{283}{26}$ |
| 31 | $[31, 31, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + \frac{14}{3}w + 2]$ | $\phantom{-}\frac{7}{208}e^{7} - \frac{37}{312}e^{6} - \frac{73}{78}e^{5} + \frac{1193}{312}e^{4} + \frac{49}{16}e^{3} - \frac{227}{12}e^{2} + \frac{445}{156}e + \frac{1883}{156}$ |
| 41 | $[41, 41, -\frac{1}{3}w^{3} + \frac{5}{3}w^{2} + \frac{4}{3}w - 7]$ | $\phantom{-}\frac{53}{624}e^{7} - \frac{2}{39}e^{6} - \frac{745}{312}e^{5} + \frac{253}{78}e^{4} + \frac{653}{48}e^{3} - \frac{227}{12}e^{2} - \frac{801}{52}e + \frac{2317}{156}$ |
| 41 | $[41, 41, -\frac{1}{3}w^{3} + \frac{5}{3}w^{2} + \frac{4}{3}w - 4]$ | $-\frac{5}{39}e^{7} + \frac{19}{312}e^{6} + \frac{1087}{312}e^{5} - \frac{1387}{312}e^{4} - \frac{439}{24}e^{3} + \frac{68}{3}e^{2} + \frac{238}{13}e - \frac{731}{78}$ |
| 47 | $[47, 47, \frac{1}{3}w^{3} - \frac{5}{3}w^{2} - \frac{1}{3}w + 5]$ | $\phantom{-}\frac{55}{624}e^{7} - \frac{3}{104}e^{6} - \frac{119}{52}e^{5} + \frac{865}{312}e^{4} + \frac{493}{48}e^{3} - \frac{53}{4}e^{2} - \frac{605}{156}e + \frac{997}{156}$ |
| 47 | $[47, 47, w^{2} - w - 4]$ | $-\frac{33}{208}e^{7} + \frac{21}{104}e^{6} + \frac{111}{26}e^{5} - \frac{909}{104}e^{4} - \frac{295}{16}e^{3} + \frac{169}{4}e^{2} + \frac{103}{52}e - \frac{775}{52}$ |
| 73 | $[73, 73, -w^{3} + 3w^{2} + 4w - 8]$ | $\phantom{-}\frac{11}{78}e^{7} - \frac{43}{312}e^{6} - \frac{1171}{312}e^{5} + \frac{717}{104}e^{4} + \frac{407}{24}e^{3} - \frac{211}{6}e^{2} - \frac{346}{39}e + \frac{577}{26}$ |
| 73 | $[73, 73, -\frac{2}{3}w^{3} + \frac{7}{3}w^{2} - \frac{1}{3}w - 4]$ | $-\frac{5}{52}e^{7} + \frac{21}{104}e^{6} + \frac{275}{104}e^{5} - \frac{753}{104}e^{4} - \frac{81}{8}e^{3} + \frac{71}{2}e^{2} - \frac{114}{13}e - \frac{381}{26}$ |
| 73 | $[73, 73, -w^{3} + w^{2} + 7w + 1]$ | $-\frac{1}{624}e^{7} + \frac{2}{39}e^{6} + \frac{43}{312}e^{5} - \frac{14}{13}e^{4} - \frac{37}{48}e^{3} + \frac{41}{12}e^{2} - \frac{197}{156}e + \frac{25}{52}$ |
| 73 | $[73, 73, \frac{2}{3}w^{3} - \frac{4}{3}w^{2} - \frac{14}{3}w + 5]$ | $\phantom{-}\frac{11}{312}e^{7} + \frac{25}{312}e^{6} - \frac{257}{312}e^{5} - \frac{421}{312}e^{4} + \frac{43}{12}e^{3} + \frac{29}{3}e^{2} + \frac{3}{26}e - \frac{523}{39}$ |
| 79 | $[79, 79, w^{2} - 5]$ | $\phantom{-}\frac{5}{104}e^{7} - \frac{19}{156}e^{6} - \frac{229}{156}e^{5} + \frac{581}{156}e^{4} + \frac{57}{8}e^{3} - \frac{49}{3}e^{2} - \frac{35}{78}e + \frac{695}{78}$ |
| 79 | $[79, 79, -\frac{2}{3}w^{3} + \frac{7}{3}w^{2} + \frac{8}{3}w - 6]$ | $\phantom{-}\frac{7}{52}e^{7} - \frac{11}{78}e^{6} - \frac{133}{39}e^{5} + \frac{278}{39}e^{4} + \frac{49}{4}e^{3} - \frac{215}{6}e^{2} + \frac{224}{39}e + \frac{661}{39}$ |
| 97 | $[97, 97, -\frac{2}{3}w^{3} + \frac{7}{3}w^{2} + \frac{5}{3}w - 4]$ | $-\frac{73}{624}e^{7} + \frac{11}{312}e^{6} + \frac{121}{39}e^{5} - \frac{1115}{312}e^{4} - \frac{727}{48}e^{3} + \frac{205}{12}e^{2} + \frac{337}{52}e + \frac{769}{156}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + \frac{4}{3}w - 1]$ | $-1$ |
| $7$ | $[7, 7, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{7}{3}w + 1]$ | $-1$ |