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:
| \(x^{9} + x^{8} - 28x^{7} - 24x^{6} + 267x^{5} + 166x^{4} - 1063x^{3} - 323x^{2} + 1512x - 162\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}1$ |
| 7 | $[7, 7, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - 2w + 2]$ | $\phantom{-}e$ |
| 9 | $[9, 3, w + 1]$ | $-1$ |
| 13 | $[13, 13, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 5w]$ | $\phantom{-}\frac{12}{17}e^{8} - \frac{20}{17}e^{7} - \frac{290}{17}e^{6} + \frac{481}{17}e^{5} + \frac{2087}{17}e^{4} - \frac{3490}{17}e^{3} - \frac{4497}{17}e^{2} + 461e - \frac{882}{17}$ |
| 16 | $[16, 2, 2]$ | $-\frac{5}{17}e^{8} + \frac{14}{17}e^{7} + \frac{125}{17}e^{6} - \frac{327}{17}e^{5} - \frac{948}{17}e^{4} + \frac{2246}{17}e^{3} + \frac{2154}{17}e^{2} - \frac{4717}{17}e + \frac{443}{17}$ |
| 17 | $[17, 17, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - 3w + 5]$ | $-\frac{8}{17}e^{8} + \frac{13}{17}e^{7} + \frac{192}{17}e^{6} - \frac{313}{17}e^{5} - \frac{1365}{17}e^{4} + \frac{2281}{17}e^{3} + \frac{2883}{17}e^{2} - \frac{5176}{17}e + \frac{613}{17}$ |
| 19 | $[19, 19, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 2w - 5]$ | $\phantom{-}\frac{3}{17}e^{8} - \frac{5}{17}e^{7} - \frac{72}{17}e^{6} + 7e^{5} + \frac{512}{17}e^{4} - \frac{855}{17}e^{3} - \frac{1076}{17}e^{2} + \frac{1918}{17}e - \frac{274}{17}$ |
| 23 | $[23, 23, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 3w - 2]$ | $\phantom{-}\frac{15}{17}e^{8} - \frac{20}{17}e^{7} - \frac{361}{17}e^{6} + \frac{490}{17}e^{5} + \frac{2583}{17}e^{4} - \frac{3679}{17}e^{3} - \frac{5588}{17}e^{2} + \frac{8629}{17}e - \frac{773}{17}$ |
| 25 | $[25, 5, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 3w]$ | $\phantom{-}\frac{1}{17}e^{8} - \frac{1}{17}e^{7} - \frac{24}{17}e^{6} + \frac{31}{17}e^{5} + 10e^{4} - \frac{304}{17}e^{3} - \frac{352}{17}e^{2} + \frac{881}{17}e - \frac{196}{17}$ |
| 25 | $[25, 5, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + 5w - 3]$ | $-\frac{8}{17}e^{8} + \frac{19}{17}e^{7} + \frac{197}{17}e^{6} - \frac{446}{17}e^{5} - \frac{1460}{17}e^{4} + \frac{3101}{17}e^{3} + 190e^{2} - \frac{6652}{17}e + \frac{751}{17}$ |
| 29 | $[29, 29, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + 4w - 3]$ | $\phantom{-}\frac{3}{17}e^{8} - \frac{11}{17}e^{7} - \frac{77}{17}e^{6} + \frac{252}{17}e^{5} + \frac{607}{17}e^{4} - \frac{1675}{17}e^{3} - \frac{1440}{17}e^{2} + \frac{3394}{17}e - \frac{293}{17}$ |
| 29 | $[29, 29, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 2w]$ | $-\frac{6}{17}e^{8} + \frac{8}{17}e^{7} + \frac{146}{17}e^{6} - \frac{200}{17}e^{5} - \frac{1061}{17}e^{4} + \frac{1548}{17}e^{3} + \frac{2308}{17}e^{2} - 221e + \frac{495}{17}$ |
| 37 | $[37, 37, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 2w - 3]$ | $-\frac{12}{17}e^{8} + \frac{9}{17}e^{7} + \frac{284}{17}e^{6} - 14e^{5} - \frac{1976}{17}e^{4} + \frac{2004}{17}e^{3} + 243e^{2} - \frac{5235}{17}e + \frac{582}{17}$ |
| 41 | $[41, 41, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + 5w - 4]$ | $\phantom{-}\frac{10}{17}e^{7} + \frac{10}{17}e^{6} - \frac{223}{17}e^{5} - \frac{188}{17}e^{4} + \frac{1391}{17}e^{3} + \frac{691}{17}e^{2} - \frac{2589}{17}e + \frac{182}{17}$ |
| 43 | $[43, 43, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 6w]$ | $-\frac{3}{17}e^{8} + \frac{5}{17}e^{7} + \frac{72}{17}e^{6} - 7e^{5} - \frac{512}{17}e^{4} + \frac{855}{17}e^{3} + \frac{1093}{17}e^{2} - \frac{1935}{17}e + \frac{155}{17}$ |
| 47 | $[47, 47, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 4w]$ | $\phantom{-}\frac{8}{17}e^{8} - \frac{14}{17}e^{7} - \frac{196}{17}e^{6} + \frac{336}{17}e^{5} + \frac{1444}{17}e^{4} - \frac{2435}{17}e^{3} - \frac{3211}{17}e^{2} + \frac{5509}{17}e - \frac{640}{17}$ |
| 59 | $[59, 59, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 2w - 4]$ | $\phantom{-}\frac{7}{17}e^{8} - \frac{7}{17}e^{7} - \frac{163}{17}e^{6} + \frac{179}{17}e^{5} + \frac{1101}{17}e^{4} - \frac{1426}{17}e^{3} - \frac{2177}{17}e^{2} + 205e - \frac{564}{17}$ |
| 59 | $[59, 59, \frac{4}{3}w^{3} - \frac{5}{3}w^{2} - 10w + 9]$ | $-\frac{20}{17}e^{8} + \frac{30}{17}e^{7} + \frac{483}{17}e^{6} - \frac{732}{17}e^{5} - \frac{3477}{17}e^{4} + \frac{5441}{17}e^{3} + \frac{7574}{17}e^{2} - \frac{12568}{17}e + \frac{1162}{17}$ |
| 67 | $[67, 67, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 3w]$ | $\phantom{-}\frac{19}{17}e^{8} - \frac{43}{17}e^{7} - \frac{471}{17}e^{6} + \frac{1015}{17}e^{5} + \frac{3521}{17}e^{4} - \frac{7130}{17}e^{3} - \frac{7857}{17}e^{2} + \frac{15507}{17}e - 106$ |
| 71 | $[71, 71, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 4w + 1]$ | $-\frac{7}{17}e^{8} - \frac{1}{17}e^{7} + \frac{159}{17}e^{6} + \frac{3}{17}e^{5} - \frac{1032}{17}e^{4} + \frac{256}{17}e^{3} + \frac{2017}{17}e^{2} - \frac{1295}{17}e + \frac{55}{17}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, w]$ | $-1$ |
| $9$ | $[9, 3, w + 1]$ | $1$ |