Base field 3.3.1772.1
Generator \(w\), with minimal polynomial \(x^{3} - x^{2} - 12x + 8\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[8, 4, w]$ |
| Dimension: | $12$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $17$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{12} - 20x^{10} + x^{9} + 145x^{8} - 11x^{7} - 456x^{6} + 33x^{5} + 570x^{4} - 34x^{3} - 157x^{2} + 37x - 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -\frac{1}{2}w^{2} - \frac{1}{2}w + 4]$ | $\phantom{-}0$ |
| 2 | $[2, 2, -w^{2} + 11]$ | $\phantom{-}e$ |
| 3 | $[3, 3, -\frac{1}{2}w^{2} + \frac{1}{2}w + 7]$ | $-\frac{19}{54}e^{11} - \frac{2}{27}e^{10} + \frac{191}{27}e^{9} + \frac{10}{9}e^{8} - \frac{2785}{54}e^{7} - \frac{178}{27}e^{6} + \frac{4394}{27}e^{5} + \frac{1135}{54}e^{4} - \frac{5422}{27}e^{3} - \frac{773}{27}e^{2} + \frac{419}{9}e - \frac{205}{54}$ |
| 5 | $[5, 5, -\frac{3}{2}w^{2} - \frac{1}{2}w + 15]$ | $\phantom{-}\frac{13}{27}e^{11} + \frac{5}{54}e^{10} - \frac{259}{27}e^{9} - \frac{14}{9}e^{8} + \frac{3731}{54}e^{7} + \frac{290}{27}e^{6} - \frac{5785}{27}e^{5} - \frac{1021}{27}e^{4} + \frac{13879}{54}e^{3} + \frac{1405}{27}e^{2} - \frac{493}{9}e + \frac{173}{54}$ |
| 9 | $[9, 3, -\frac{1}{2}w^{2} + \frac{5}{2}w - 1]$ | $-\frac{8}{27}e^{11} - \frac{1}{54}e^{10} + \frac{325}{54}e^{9} + \frac{1}{9}e^{8} - \frac{1202}{27}e^{7} - \frac{4}{27}e^{6} + \frac{3884}{27}e^{5} + \frac{80}{27}e^{4} - \frac{10109}{54}e^{3} - \frac{427}{54}e^{2} + \frac{509}{9}e - \frac{182}{27}$ |
| 25 | $[25, 5, \frac{7}{2}w^{2} - \frac{31}{2}w + 9]$ | $\phantom{-}\frac{4}{27}e^{11} - \frac{2}{27}e^{10} - \frac{79}{27}e^{9} + \frac{29}{18}e^{8} + \frac{1103}{54}e^{7} - \frac{313}{27}e^{6} - \frac{1600}{27}e^{5} + \frac{851}{27}e^{4} + \frac{1706}{27}e^{3} - \frac{773}{27}e^{2} - \frac{179}{18}e + \frac{443}{54}$ |
| 29 | $[29, 29, -\frac{5}{2}w^{2} + \frac{1}{2}w + 29]$ | $-\frac{1}{18}e^{11} - \frac{1}{18}e^{10} + \frac{19}{18}e^{9} + e^{8} - \frac{127}{18}e^{7} - \frac{67}{9}e^{6} + \frac{161}{9}e^{5} + \frac{505}{18}e^{4} - \frac{101}{18}e^{3} - \frac{787}{18}e^{2} - 23e + \frac{125}{18}$ |
| 41 | $[41, 41, -w^{2} + 3w - 1]$ | $-\frac{7}{18}e^{11} + \frac{1}{9}e^{10} + \frac{71}{9}e^{9} - \frac{5}{2}e^{8} - \frac{521}{9}e^{7} + \frac{161}{9}e^{6} + \frac{1658}{9}e^{5} - \frac{839}{18}e^{4} - \frac{2131}{9}e^{3} + \frac{355}{9}e^{2} + \frac{153}{2}e - \frac{107}{9}$ |
| 41 | $[41, 41, -\frac{1}{2}w^{2} - \frac{3}{2}w + 3]$ | $\phantom{-}\frac{4}{9}e^{11} + \frac{1}{9}e^{10} - \frac{155}{18}e^{9} - \frac{11}{6}e^{8} + \frac{541}{9}e^{7} + \frac{116}{9}e^{6} - \frac{1615}{9}e^{5} - \frac{436}{9}e^{4} + \frac{1829}{9}e^{3} + \frac{1349}{18}e^{2} - \frac{187}{6}e - \frac{56}{9}$ |
| 41 | $[41, 41, 2w + 7]$ | $-\frac{26}{27}e^{11} - \frac{5}{27}e^{10} + \frac{518}{27}e^{9} + \frac{47}{18}e^{8} - \frac{7489}{54}e^{7} - \frac{391}{27}e^{6} + \frac{11732}{27}e^{5} + \frac{1205}{27}e^{4} - \frac{14419}{27}e^{3} - \frac{1622}{27}e^{2} + \frac{2251}{18}e - \frac{589}{54}$ |
| 43 | $[43, 43, -\frac{7}{2}w^{2} - \frac{1}{2}w + 37]$ | $-\frac{17}{54}e^{11} + \frac{2}{27}e^{10} + \frac{347}{54}e^{9} - \frac{13}{9}e^{8} - \frac{1294}{27}e^{7} + \frac{259}{27}e^{6} + \frac{4228}{27}e^{5} - \frac{1369}{54}e^{4} - \frac{5540}{27}e^{3} + \frac{1357}{54}e^{2} + \frac{556}{9}e - \frac{325}{27}$ |
| 43 | $[43, 43, \frac{1}{2}w^{2} - \frac{9}{2}w + 3]$ | $\phantom{-}\frac{7}{27}e^{11} + \frac{11}{54}e^{10} - \frac{281}{54}e^{9} - \frac{67}{18}e^{8} + \frac{2063}{54}e^{7} + \frac{692}{27}e^{6} - \frac{3304}{27}e^{5} - \frac{2176}{27}e^{4} + \frac{8167}{54}e^{3} + \frac{5399}{54}e^{2} - \frac{425}{18}e - \frac{505}{54}$ |
| 43 | $[43, 43, \frac{1}{2}w^{2} - \frac{1}{2}w + 1]$ | $\phantom{-}\frac{7}{54}e^{11} + \frac{5}{27}e^{10} - \frac{145}{54}e^{9} - \frac{34}{9}e^{8} + \frac{545}{27}e^{7} + \frac{769}{27}e^{6} - \frac{1751}{27}e^{5} - \frac{5065}{54}e^{4} + \frac{1999}{27}e^{3} + \frac{6187}{54}e^{2} + \frac{37}{9}e - \frac{358}{27}$ |
| 47 | $[47, 47, -w^{2} + w + 15]$ | $-\frac{19}{18}e^{11} - \frac{7}{18}e^{10} + \frac{188}{9}e^{9} + \frac{37}{6}e^{8} - \frac{2689}{18}e^{7} - \frac{325}{9}e^{6} + \frac{4136}{9}e^{5} + \frac{1765}{18}e^{4} - \frac{9833}{18}e^{3} - \frac{941}{9}e^{2} + \frac{701}{6}e - \frac{175}{18}$ |
| 53 | $[53, 53, -2w^{2} + 2w + 29]$ | $\phantom{-}\frac{26}{27}e^{11} + \frac{19}{54}e^{10} - \frac{509}{27}e^{9} - \frac{49}{9}e^{8} + \frac{7201}{54}e^{7} + \frac{832}{27}e^{6} - \frac{10985}{27}e^{5} - \frac{2177}{27}e^{4} + \frac{26399}{54}e^{3} + \frac{2261}{27}e^{2} - \frac{1154}{9}e + \frac{499}{54}$ |
| 59 | $[59, 59, w^{2} - w - 13]$ | $-\frac{16}{27}e^{11} - \frac{11}{54}e^{10} + \frac{316}{27}e^{9} + \frac{55}{18}e^{8} - \frac{2260}{27}e^{7} - \frac{449}{27}e^{6} + \frac{6967}{27}e^{5} + \frac{1132}{27}e^{4} - \frac{16591}{54}e^{3} - \frac{1120}{27}e^{2} + \frac{1121}{18}e - \frac{319}{27}$ |
| 67 | $[67, 67, -\frac{1}{2}w^{2} + \frac{1}{2}w + 9]$ | $-\frac{2}{3}e^{11} - \frac{1}{6}e^{10} + \frac{79}{6}e^{9} + \frac{5}{2}e^{8} - \frac{565}{6}e^{7} - \frac{43}{3}e^{6} + \frac{878}{3}e^{5} + \frac{125}{3}e^{4} - \frac{2201}{6}e^{3} - \frac{301}{6}e^{2} + \frac{221}{2}e - \frac{61}{6}$ |
| 71 | $[71, 71, 2w - 3]$ | $-\frac{5}{54}e^{11} - \frac{11}{54}e^{10} + \frac{46}{27}e^{9} + \frac{73}{18}e^{8} - \frac{605}{54}e^{7} - \frac{773}{27}e^{6} + \frac{838}{27}e^{5} + \frac{4505}{54}e^{4} - \frac{1579}{54}e^{3} - \frac{2281}{27}e^{2} - \frac{139}{18}e + \frac{469}{54}$ |
| 73 | $[73, 73, \frac{3}{2}w^{2} - \frac{11}{2}w + 1]$ | $\phantom{-}\frac{3}{2}e^{11} + \frac{1}{6}e^{10} - \frac{89}{3}e^{9} - \frac{11}{6}e^{8} + \frac{1273}{6}e^{7} + \frac{25}{3}e^{6} - \frac{1966}{3}e^{5} - \frac{67}{2}e^{4} + \frac{4781}{6}e^{3} + \frac{188}{3}e^{2} - \frac{1207}{6}e + \frac{119}{6}$ |
| 79 | $[79, 79, -\frac{3}{2}w^{2} + \frac{15}{2}w - 5]$ | $\phantom{-}\frac{19}{27}e^{11} + \frac{17}{54}e^{10} - \frac{373}{27}e^{9} - \frac{91}{18}e^{8} + \frac{2668}{27}e^{7} + \frac{797}{27}e^{6} - \frac{8338}{27}e^{5} - \frac{2053}{27}e^{4} + \frac{20869}{54}e^{3} + \frac{1888}{27}e^{2} - \frac{1949}{18}e + \frac{403}{27}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2, 2, -\frac{1}{2}w^{2} - \frac{1}{2}w + 4]$ | $-1$ |