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: | $[9, 3, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - 2w + 4]$ |
| Dimension: | $6$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $18$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{6} + 6x^{5} - 27x^{4} - 192x^{3} + 63x^{2} + 1260x + 716\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}0$ |
| 7 | $[7, 7, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - 2w + 2]$ | $\phantom{-}e$ |
| 9 | $[9, 3, w + 1]$ | $-\frac{45}{704}e^{5} - \frac{27}{176}e^{4} + \frac{133}{64}e^{3} + \frac{1405}{352}e^{2} - \frac{10135}{704}e - \frac{5179}{352}$ |
| 13 | $[13, 13, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 5w]$ | $-\frac{41}{704}e^{5} - \frac{7}{176}e^{4} + \frac{129}{64}e^{3} + \frac{361}{352}e^{2} - \frac{10955}{704}e - \frac{1535}{352}$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}\frac{5}{352}e^{5} + \frac{3}{88}e^{4} - \frac{13}{32}e^{3} - \frac{205}{176}e^{2} + \frac{559}{352}e + \frac{1123}{176}$ |
| 17 | $[17, 17, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - 3w + 5]$ | $\phantom{-}\frac{15}{704}e^{5} + \frac{9}{176}e^{4} - \frac{55}{64}e^{3} - \frac{527}{352}e^{2} + \frac{6077}{704}e + \frac{3017}{352}$ |
| 19 | $[19, 19, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 2w - 5]$ | $\phantom{-}\frac{5}{352}e^{5} + \frac{3}{88}e^{4} - \frac{13}{32}e^{3} - \frac{205}{176}e^{2} + \frac{559}{352}e + \frac{595}{176}$ |
| 23 | $[23, 23, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 3w - 2]$ | $-\frac{5}{88}e^{5} - \frac{3}{22}e^{4} + \frac{17}{8}e^{3} + \frac{183}{44}e^{2} - \frac{1571}{88}e - \frac{903}{44}$ |
| 25 | $[25, 5, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 3w]$ | $-\frac{15}{704}e^{5} - \frac{9}{176}e^{4} + \frac{55}{64}e^{3} + \frac{527}{352}e^{2} - \frac{5373}{704}e - \frac{3017}{352}$ |
| 25 | $[25, 5, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + 5w - 3]$ | $\phantom{-}\frac{35}{704}e^{5} + \frac{21}{176}e^{4} - \frac{107}{64}e^{3} - \frac{995}{352}e^{2} + \frac{9017}{704}e + \frac{2581}{352}$ |
| 29 | $[29, 29, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + 4w - 3]$ | $\phantom{-}\frac{91}{704}e^{5} + \frac{37}{176}e^{4} - \frac{291}{64}e^{3} - \frac{1883}{352}e^{2} + \frac{24641}{704}e + \frac{5373}{352}$ |
| 29 | $[29, 29, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 2w]$ | $\phantom{-}\frac{111}{704}e^{5} + \frac{49}{176}e^{4} - \frac{343}{64}e^{3} - \frac{2703}{352}e^{2} + \frac{27581}{704}e + \frac{10921}{352}$ |
| 37 | $[37, 37, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 2w - 3]$ | $-\frac{95}{704}e^{5} - \frac{57}{176}e^{4} + \frac{295}{64}e^{3} + \frac{2927}{352}e^{2} - \frac{24525}{704}e - \frac{10425}{352}$ |
| 41 | $[41, 41, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + 5w - 4]$ | $-\frac{101}{704}e^{5} - \frac{43}{176}e^{4} + \frac{317}{64}e^{3} + \frac{2293}{352}e^{2} - \frac{27167}{704}e - \frac{9731}{352}$ |
| 43 | $[43, 43, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 6w]$ | $-\frac{23}{352}e^{5} - \frac{5}{88}e^{4} + \frac{79}{32}e^{3} + \frac{239}{176}e^{2} - \frac{7605}{352}e - \frac{801}{176}$ |
| 47 | $[47, 47, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 4w]$ | $\phantom{-}\frac{19}{176}e^{5} + \frac{7}{44}e^{4} - \frac{59}{16}e^{3} - \frac{339}{88}e^{2} + \frac{4641}{176}e + \frac{589}{88}$ |
| 59 | $[59, 59, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 2w - 4]$ | $\phantom{-}\frac{17}{88}e^{5} + \frac{4}{11}e^{4} - \frac{53}{8}e^{3} - \frac{455}{44}e^{2} + \frac{4259}{88}e + \frac{2023}{44}$ |
| 59 | $[59, 59, \frac{4}{3}w^{3} - \frac{5}{3}w^{2} - 10w + 9]$ | $\phantom{-}\frac{5}{176}e^{5} + \frac{3}{44}e^{4} - \frac{21}{16}e^{3} - \frac{161}{88}e^{2} + \frac{2407}{176}e + \frac{155}{88}$ |
| 67 | $[67, 67, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 3w]$ | $\phantom{-}\frac{5}{88}e^{5} + \frac{3}{22}e^{4} - \frac{17}{8}e^{3} - \frac{139}{44}e^{2} + \frac{1571}{88}e + \frac{23}{44}$ |
| 71 | $[71, 71, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 4w + 1]$ | $-\frac{19}{176}e^{5} - \frac{7}{44}e^{4} + \frac{59}{16}e^{3} + \frac{339}{88}e^{2} - \frac{5169}{176}e - \frac{589}{88}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, w]$ | $-1$ |