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: | $[21, 21, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 5w - 1]$ |
| Dimension: | $6$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $37$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{6} + 2x^{5} - 34x^{4} - 88x^{3} + 100x^{2} + 184x + 56\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}1$ |
| 7 | $[7, 7, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - 2w + 2]$ | $-1$ |
| 9 | $[9, 3, w + 1]$ | $\phantom{-}e$ |
| 13 | $[13, 13, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 5w]$ | $-\frac{1}{40}e^{5} - \frac{1}{40}e^{4} + e^{3} + \frac{6}{5}e^{2} - \frac{67}{10}e - \frac{29}{10}$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}\frac{1}{8}e^{5} + \frac{1}{8}e^{4} - \frac{17}{4}e^{3} - 7e^{2} + 17e + \frac{17}{2}$ |
| 17 | $[17, 17, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - 3w + 5]$ | $-\frac{7}{40}e^{5} - \frac{3}{10}e^{4} + \frac{25}{4}e^{3} + \frac{129}{10}e^{2} - \frac{132}{5}e - \frac{84}{5}$ |
| 19 | $[19, 19, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 2w - 5]$ | $\phantom{-}\frac{1}{10}e^{5} + \frac{1}{10}e^{4} - \frac{7}{2}e^{3} - \frac{29}{5}e^{2} + \frac{79}{5}e + \frac{48}{5}$ |
| 23 | $[23, 23, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 3w - 2]$ | $-\frac{1}{8}e^{5} - \frac{1}{8}e^{4} + \frac{17}{4}e^{3} + \frac{15}{2}e^{2} - 17e - \frac{29}{2}$ |
| 25 | $[25, 5, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 3w]$ | $\phantom{-}\frac{1}{40}e^{5} + \frac{1}{40}e^{4} - \frac{3}{4}e^{3} - \frac{17}{10}e^{2} + \frac{6}{5}e - \frac{11}{10}$ |
| 25 | $[25, 5, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + 5w - 3]$ | $\phantom{-}\frac{1}{8}e^{5} + \frac{1}{4}e^{4} - \frac{9}{2}e^{3} - \frac{21}{2}e^{2} + \frac{37}{2}e + 18$ |
| 29 | $[29, 29, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + 4w - 3]$ | $\phantom{-}\frac{1}{40}e^{5} + \frac{1}{40}e^{4} - e^{3} - \frac{6}{5}e^{2} + \frac{67}{10}e - \frac{11}{10}$ |
| 29 | $[29, 29, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 2w]$ | $-\frac{1}{10}e^{5} - \frac{1}{10}e^{4} + \frac{7}{2}e^{3} + \frac{53}{10}e^{2} - \frac{79}{5}e - \frac{33}{5}$ |
| 37 | $[37, 37, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 2w - 3]$ | $\phantom{-}\frac{1}{10}e^{5} + \frac{1}{10}e^{4} - \frac{7}{2}e^{3} - \frac{53}{10}e^{2} + \frac{69}{5}e + \frac{3}{5}$ |
| 41 | $[41, 41, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + 5w - 4]$ | $-\frac{1}{40}e^{5} - \frac{1}{40}e^{4} + \frac{3}{4}e^{3} + \frac{17}{10}e^{2} - \frac{1}{5}e - \frac{9}{10}$ |
| 43 | $[43, 43, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 6w]$ | $-\frac{13}{40}e^{5} - \frac{9}{20}e^{4} + 11e^{3} + \frac{221}{10}e^{2} - \frac{391}{10}e - \frac{156}{5}$ |
| 47 | $[47, 47, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 4w]$ | $\phantom{-}\frac{1}{4}e^{5} + \frac{3}{8}e^{4} - \frac{35}{4}e^{3} - \frac{35}{2}e^{2} + \frac{67}{2}e + \frac{51}{2}$ |
| 59 | $[59, 59, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 2w - 4]$ | $-\frac{1}{40}e^{5} - \frac{1}{40}e^{4} + e^{3} + \frac{6}{5}e^{2} - \frac{67}{10}e - \frac{69}{10}$ |
| 59 | $[59, 59, \frac{4}{3}w^{3} - \frac{5}{3}w^{2} - 10w + 9]$ | $\phantom{-}\frac{1}{8}e^{5} + \frac{1}{4}e^{4} - \frac{9}{2}e^{3} - \frac{21}{2}e^{2} + \frac{33}{2}e + 18$ |
| 67 | $[67, 67, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 3w]$ | $-\frac{3}{20}e^{5} - \frac{2}{5}e^{4} + \frac{11}{2}e^{3} + \frac{76}{5}e^{2} - \frac{106}{5}e - \frac{132}{5}$ |
| 71 | $[71, 71, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 4w + 1]$ | $\phantom{-}\frac{13}{40}e^{5} + \frac{9}{20}e^{4} - \frac{45}{4}e^{3} - \frac{108}{5}e^{2} + \frac{213}{5}e + \frac{166}{5}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, w]$ | $-1$ |
| $7$ | $[7, 7, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - 2w + 2]$ | $1$ |