Base field 4.4.19821.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 8x^{2} + 6x + 3\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[27, 3, -w^{3} + w^{2} + 8w - 6]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $43$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} - x^{8} - 24x^{7} + 24x^{6} + 167x^{5} - 186x^{4} - 317x^{3} + 413x^{2} - 32x - 14\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $-1$ |
| 7 | $[7, 7, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - 2w + 2]$ | $\phantom{-}e$ |
| 9 | $[9, 3, w + 1]$ | $\phantom{-}1$ |
| 13 | $[13, 13, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 5w]$ | $-\frac{152986}{1628571}e^{8} - \frac{37084}{1628571}e^{7} + \frac{3582286}{1628571}e^{6} + \frac{954257}{1628571}e^{5} - \frac{23674303}{1628571}e^{4} - \frac{3949898}{1628571}e^{3} + \frac{42074707}{1628571}e^{2} - \frac{637883}{542857}e - \frac{318586}{232653}$ |
| 16 | $[16, 2, 2]$ | $-\frac{119477}{1628571}e^{8} - \frac{167690}{1628571}e^{7} + \frac{2696219}{1628571}e^{6} + \frac{3953449}{1628571}e^{5} - \frac{16249676}{1628571}e^{4} - \frac{23775424}{1628571}e^{3} + \frac{23017130}{1628571}e^{2} + \frac{11949503}{542857}e - \frac{14531}{232653}$ |
| 17 | $[17, 17, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - 3w + 5]$ | $\phantom{-}\frac{6682}{1628571}e^{8} - \frac{102299}{1628571}e^{7} - \frac{223444}{1628571}e^{6} + \frac{2274169}{1628571}e^{5} + \frac{2435791}{1628571}e^{4} - \frac{13677625}{1628571}e^{3} - \frac{6682051}{1628571}e^{2} + \frac{6373560}{542857}e - \frac{425219}{232653}$ |
| 19 | $[19, 19, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 2w - 5]$ | $\phantom{-}\frac{61237}{232653}e^{8} + \frac{1069}{232653}e^{7} - \frac{1444840}{232653}e^{6} + \frac{6301}{232653}e^{5} + \frac{9637648}{232653}e^{4} - \frac{1696759}{232653}e^{3} - \frac{17015884}{232653}e^{2} + \frac{2611890}{77551}e - \frac{253802}{232653}$ |
| 23 | $[23, 23, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 3w - 2]$ | $-\frac{110575}{542857}e^{8} + \frac{29602}{542857}e^{7} + \frac{2622441}{542857}e^{6} - \frac{778328}{542857}e^{5} - \frac{17776765}{542857}e^{4} + \frac{8253991}{542857}e^{3} + \frac{32949324}{542857}e^{2} - \frac{22918723}{542857}e - \frac{54925}{77551}$ |
| 25 | $[25, 5, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 3w]$ | $-\frac{117079}{1628571}e^{8} - \frac{170281}{1628571}e^{7} + \frac{2685736}{1628571}e^{6} + \frac{3840023}{1628571}e^{5} - \frac{16686286}{1628571}e^{4} - \frac{20414384}{1628571}e^{3} + \frac{24328120}{1628571}e^{2} + \frac{5880941}{542857}e + \frac{1125308}{232653}$ |
| 25 | $[25, 5, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + 5w - 3]$ | $\phantom{-}\frac{480154}{1628571}e^{8} + \frac{182071}{1628571}e^{7} - \frac{11183161}{1628571}e^{6} - \frac{3929186}{1628571}e^{5} + \frac{73078984}{1628571}e^{4} + \frac{12041375}{1628571}e^{3} - \frac{128388766}{1628571}e^{2} + \frac{6409840}{542857}e + \frac{521863}{232653}$ |
| 29 | $[29, 29, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + 4w - 3]$ | $-\frac{27745}{1628571}e^{8} + \frac{49673}{1628571}e^{7} + \frac{591931}{1628571}e^{6} - \frac{1120324}{1628571}e^{5} - \frac{3394147}{1628571}e^{4} + \frac{8286799}{1628571}e^{3} + \frac{5131180}{1628571}e^{2} - \frac{6475914}{542857}e - \frac{61543}{232653}$ |
| 29 | $[29, 29, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 2w]$ | $-\frac{74554}{542857}e^{8} - \frac{10450}{542857}e^{7} + \frac{1745314}{542857}e^{6} + \frac{166052}{542857}e^{5} - \frac{11476417}{542857}e^{4} + \frac{1369826}{542857}e^{3} + \frac{19540456}{542857}e^{2} - \frac{8969931}{542857}e + \frac{267473}{77551}$ |
| 37 | $[37, 37, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 2w - 3]$ | $-\frac{172486}{1628571}e^{8} + \frac{1643}{1628571}e^{7} + \frac{4145644}{1628571}e^{6} - \frac{23614}{1628571}e^{5} - \frac{28875250}{1628571}e^{4} + \frac{3824836}{1628571}e^{3} + \frac{57924829}{1628571}e^{2} - \frac{7092133}{542857}e - \frac{1806142}{232653}$ |
| 41 | $[41, 41, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + 5w - 4]$ | $-\frac{19286}{542857}e^{8} + \frac{23102}{542857}e^{7} + \frac{437914}{542857}e^{6} - \frac{584133}{542857}e^{5} - \frac{2663264}{542857}e^{4} + \frac{4654089}{542857}e^{3} + \frac{3002387}{542857}e^{2} - \frac{9879929}{542857}e + \frac{235750}{77551}$ |
| 43 | $[43, 43, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 6w]$ | $\phantom{-}\frac{86647}{1628571}e^{8} + \frac{48319}{1628571}e^{7} - \frac{1984942}{1628571}e^{6} - \frac{1020575}{1628571}e^{5} + \frac{12029194}{1628571}e^{4} + \frac{3515093}{1628571}e^{3} - \frac{13499671}{1628571}e^{2} + \frac{1751947}{542857}e - \frac{1879517}{232653}$ |
| 47 | $[47, 47, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 4w]$ | $-\frac{283562}{1628571}e^{8} - \frac{20960}{1628571}e^{7} + \frac{6782726}{1628571}e^{6} + \frac{158512}{1628571}e^{5} - \frac{46329890}{1628571}e^{4} + \frac{8190113}{1628571}e^{3} + \frac{86739587}{1628571}e^{2} - \frac{14069327}{542857}e - \frac{1377284}{232653}$ |
| 59 | $[59, 59, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 2w - 4]$ | $\phantom{-}\frac{74769}{542857}e^{8} + \frac{48023}{542857}e^{7} - \frac{1703921}{542857}e^{6} - \frac{1008391}{542857}e^{5} + \frac{10593783}{542857}e^{4} + \frac{3731660}{542857}e^{3} - \frac{16154907}{542857}e^{2} + \frac{1270233}{542857}e - \frac{533596}{77551}$ |
| 59 | $[59, 59, \frac{4}{3}w^{3} - \frac{5}{3}w^{2} - 10w + 9]$ | $\phantom{-}\frac{482632}{1628571}e^{8} + \frac{92464}{1628571}e^{7} - \frac{11418109}{1628571}e^{6} - \frac{1957370}{1628571}e^{5} + \frac{77502655}{1628571}e^{4} + \frac{30215}{1628571}e^{3} - \frac{150814708}{1628571}e^{2} + \frac{13702552}{542857}e + \frac{2262442}{232653}$ |
| 67 | $[67, 67, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 3w]$ | $-\frac{25049}{77551}e^{8} + \frac{2131}{77551}e^{7} + \frac{588683}{77551}e^{6} - \frac{54181}{77551}e^{5} - \frac{3909981}{77551}e^{4} + \frac{916994}{77551}e^{3} + \frac{6872279}{77551}e^{2} - \frac{3131569}{77551}e + \frac{63394}{77551}$ |
| 71 | $[71, 71, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 4w + 1]$ | $\phantom{-}\frac{87607}{542857}e^{8} + \frac{89841}{542857}e^{7} - \frac{1964237}{542857}e^{6} - \frac{2054807}{542857}e^{5} + \frac{11724010}{542857}e^{4} + \frac{10677214}{542857}e^{3} - \frac{16057215}{542857}e^{2} - \frac{9909593}{542857}e - \frac{409411}{77551}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, w]$ | $1$ |
| $9$ | $[9, 3, w + 1]$ | $-1$ |