Base field 4.4.16225.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 13x^{2} + 6x + 36\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16,4,-\frac{1}{6}w^{3} + \frac{1}{6}w^{2} + \frac{1}{6}w + 1]$ |
| Dimension: | $10$ |
| 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:
| \(x^{10} - 28x^{8} + 283x^{6} - 1275x^{4} + 2478x^{2} - 1587\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, \frac{1}{3}w^{3} - \frac{4}{3}w^{2} - \frac{4}{3}w + 6]$ | $\phantom{-}e$ |
| 4 | $[4, 2, -\frac{1}{3}w^{3} - \frac{2}{3}w^{2} + \frac{10}{3}w + 7]$ | $\phantom{-}0$ |
| 9 | $[9, 3, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{7}{2}w - 9]$ | $-\frac{3}{2}e^{8} + \frac{65}{2}e^{6} - 218e^{4} + \frac{1037}{2}e^{2} - \frac{721}{2}$ |
| 9 | $[9, 3, \frac{1}{3}w^{3} + \frac{2}{3}w^{2} - \frac{7}{3}w - 5]$ | $\phantom{-}\frac{5}{6}e^{8} - \frac{109}{6}e^{6} + 123e^{4} - \frac{591}{2}e^{2} + \frac{411}{2}$ |
| 11 | $[11, 11, -\frac{1}{6}w^{3} + \frac{1}{6}w^{2} + \frac{1}{6}w]$ | $\phantom{-}\frac{37}{69}e^{9} - \frac{806}{69}e^{7} + \frac{1819}{23}e^{5} - \frac{4386}{23}e^{3} + \frac{3054}{23}e$ |
| 19 | $[19, 19, w + 1]$ | $-\frac{61}{69}e^{9} + \frac{1340}{69}e^{7} - \frac{3071}{23}e^{5} + \frac{7617}{23}e^{3} - \frac{5605}{23}e$ |
| 19 | $[19, 19, \frac{1}{6}w^{3} - \frac{1}{6}w^{2} - \frac{13}{6}w + 2]$ | $\phantom{-}\frac{25}{69}e^{9} - \frac{539}{69}e^{7} + \frac{1193}{23}e^{5} - \frac{2782}{23}e^{3} + \frac{1905}{23}e$ |
| 25 | $[25, 5, -\frac{1}{3}w^{3} + \frac{1}{3}w^{2} + \frac{7}{3}w - 1]$ | $-\frac{7}{3}e^{8} + \frac{152}{3}e^{6} - 341e^{4} + 815e^{2} - 571$ |
| 29 | $[29, 29, -\frac{1}{6}w^{3} + \frac{1}{6}w^{2} + \frac{13}{6}w]$ | $\phantom{-}\frac{93}{46}e^{9} - \frac{2029}{46}e^{7} + \frac{6892}{23}e^{5} - \frac{33567}{46}e^{3} + \frac{24305}{46}e$ |
| 29 | $[29, 29, w - 1]$ | $-\frac{107}{138}e^{9} + \frac{2329}{138}e^{7} - \frac{2628}{23}e^{5} + \frac{12769}{46}e^{3} - \frac{9377}{46}e$ |
| 31 | $[31, 31, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - \frac{10}{3}w + 3]$ | $\phantom{-}\frac{197}{69}e^{9} - \frac{4297}{69}e^{7} + \frac{9721}{23}e^{5} - \frac{23580}{23}e^{3} + \frac{16833}{23}e$ |
| 31 | $[31, 31, \frac{1}{6}w^{3} - \frac{1}{6}w^{2} - \frac{1}{6}w + 2]$ | $-\frac{74}{69}e^{9} + \frac{1612}{69}e^{7} - \frac{3638}{23}e^{5} + \frac{8795}{23}e^{3} - \frac{6269}{23}e$ |
| 41 | $[41, 41, -\frac{1}{2}w^{3} - \frac{1}{2}w^{2} + \frac{9}{2}w + 5]$ | $\phantom{-}\frac{3}{46}e^{9} - \frac{61}{46}e^{7} + \frac{183}{23}e^{5} - \frac{743}{46}e^{3} + \frac{649}{46}e$ |
| 41 | $[41, 41, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{5}{2}w - 8]$ | $-\frac{427}{138}e^{9} + \frac{9311}{138}e^{7} - \frac{10530}{23}e^{5} + \frac{51111}{46}e^{3} - \frac{36613}{46}e$ |
| 59 | $[59, 59, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - \frac{10}{3}w - 1]$ | $\phantom{-}e^{8} - 22e^{6} + 151e^{4} - 370e^{2} + 264$ |
| 59 | $[59, 59, -\frac{1}{3}w^{3} + \frac{4}{3}w^{2} + \frac{7}{3}w - 7]$ | $-\frac{23}{3}e^{8} + \frac{502}{3}e^{6} - 1137e^{4} + 2764e^{2} - 1972$ |
| 59 | $[59, 59, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{5}{2}w + 2]$ | $-\frac{7}{3}e^{8} + \frac{152}{3}e^{6} - 341e^{4} + 816e^{2} - 578$ |
| 79 | $[79, 79, w^{2} - 11]$ | $\phantom{-}\frac{13}{3}e^{8} - \frac{284}{3}e^{6} + 644e^{4} - 1567e^{2} + 1128$ |
| 79 | $[79, 79, \frac{1}{6}w^{3} - \frac{7}{6}w^{2} - \frac{7}{6}w + 3]$ | $\phantom{-}\frac{35}{3}e^{8} - \frac{763}{3}e^{6} + 1725e^{4} - 4184e^{2} + 2999$ |
| 89 | $[89, 89, -\frac{1}{6}w^{3} + \frac{1}{6}w^{2} + \frac{19}{6}w - 5]$ | $-\frac{157}{138}e^{9} + \frac{3407}{138}e^{7} - \frac{3821}{23}e^{5} + \frac{18333}{46}e^{3} - \frac{13095}{46}e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4,2,\frac{1}{3}w^{3} + \frac{2}{3}w^{2} - \frac{10}{3}w - 7]$ | $1$ |