Base field 4.4.8768.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 5x^{2} + 6x + 7\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[23,23,w^{3} - 2w^{2} - 2w + 2]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $12$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 2x^{7} - 16x^{6} + 23x^{5} + 78x^{4} - 48x^{3} - 126x^{2} - 25x - 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -w^{2} + w + 3]$ | $\phantom{-}e$ |
| 7 | $[7, 7, w]$ | $-\frac{247}{397}e^{7} + \frac{1099}{794}e^{6} + \frac{3800}{397}e^{5} - \frac{12917}{794}e^{4} - \frac{35145}{794}e^{3} + \frac{29943}{794}e^{2} + \frac{55843}{794}e + \frac{4019}{794}$ |
| 7 | $[7, 7, -w^{2} + 2w + 1]$ | $\phantom{-}\frac{258}{397}e^{7} - \frac{492}{397}e^{6} - \frac{4183}{397}e^{5} + \frac{5748}{397}e^{4} + \frac{20631}{397}e^{3} - \frac{12893}{397}e^{2} - \frac{33098}{397}e - \frac{3777}{397}$ |
| 7 | $[7, 7, w^{2} - 2]$ | $-\frac{48}{397}e^{7} + \frac{110}{397}e^{6} + \frac{769}{397}e^{5} - \frac{1411}{397}e^{4} - \frac{3746}{397}e^{3} + \frac{3959}{397}e^{2} + \frac{6167}{397}e + \frac{278}{397}$ |
| 7 | $[7, 7, w - 1]$ | $\phantom{-}\frac{263}{794}e^{7} - \frac{260}{397}e^{6} - \frac{4321}{794}e^{5} + \frac{6201}{794}e^{4} + \frac{21931}{794}e^{3} - \frac{14637}{794}e^{2} - \frac{36263}{794}e - \frac{1812}{397}$ |
| 23 | $[23, 23, -w^{3} + w^{2} + 3w - 1]$ | $-\frac{657}{794}e^{7} + \frac{728}{397}e^{6} + \frac{9955}{794}e^{5} - \frac{16807}{794}e^{4} - \frac{44971}{794}e^{3} + \frac{37093}{794}e^{2} + \frac{70491}{794}e + \frac{3565}{397}$ |
| 23 | $[23, 23, w^{3} - 2w^{2} - 2w + 2]$ | $-1$ |
| 31 | $[31, 31, w^{2} - 5]$ | $\phantom{-}\frac{5}{397}e^{7} - \frac{28}{397}e^{6} - \frac{138}{397}e^{5} + \frac{453}{397}e^{4} + \frac{1300}{397}e^{3} - \frac{2141}{397}e^{2} - \frac{3165}{397}e + \frac{1344}{397}$ |
| 31 | $[31, 31, -w^{2} + 2w + 4]$ | $\phantom{-}\frac{271}{794}e^{7} - \frac{203}{397}e^{6} - \frac{4383}{794}e^{5} + \frac{4385}{794}e^{4} + \frac{20835}{794}e^{3} - \frac{7423}{794}e^{2} - \frac{30211}{794}e - \frac{2960}{397}$ |
| 41 | $[41, 41, -w^{3} + 2w^{2} + 2w - 6]$ | $\phantom{-}\frac{618}{397}e^{7} - \frac{1317}{397}e^{6} - \frac{9752}{397}e^{5} + \frac{15338}{397}e^{4} + \frac{46741}{397}e^{3} - \frac{34447}{397}e^{2} - \frac{74785}{397}e - \frac{6656}{397}$ |
| 41 | $[41, 41, w^{3} - w^{2} - 3w - 3]$ | $-\frac{707}{397}e^{7} + \frac{3075}{794}e^{6} + \frac{10938}{397}e^{5} - \frac{35925}{794}e^{4} - \frac{101253}{794}e^{3} + \frac{81673}{794}e^{2} + \frac{153863}{794}e + \frac{12391}{794}$ |
| 47 | $[47, 47, w^{2} - 2w - 5]$ | $\phantom{-}\frac{304}{397}e^{7} - \frac{1261}{794}e^{6} - \frac{4738}{397}e^{5} + \frac{14829}{794}e^{4} + \frac{44141}{794}e^{3} - \frac{34929}{794}e^{2} - \frac{67661}{794}e - \frac{3389}{794}$ |
| 47 | $[47, 47, w^{2} - 6]$ | $\phantom{-}\frac{297}{397}e^{7} - \frac{631}{397}e^{6} - \frac{4783}{397}e^{5} + \frac{7614}{397}e^{4} + \frac{23625}{397}e^{3} - \frac{18715}{397}e^{2} - \frac{38729}{397}e - \frac{3060}{397}$ |
| 71 | $[71, 71, -w^{3} + 2w^{2} + 3w - 1]$ | $\phantom{-}\frac{1}{397}e^{7} - \frac{85}{397}e^{6} + \frac{290}{397}e^{5} + \frac{964}{397}e^{4} - \frac{3313}{397}e^{3} - \frac{1778}{397}e^{2} + \frac{6116}{397}e + \frac{507}{397}$ |
| 71 | $[71, 71, -w^{3} + w^{2} + 4w - 3]$ | $-\frac{37}{397}e^{7} - \frac{31}{397}e^{6} + \frac{783}{397}e^{5} + \frac{459}{397}e^{4} - \frac{4459}{397}e^{3} - \frac{2498}{397}e^{2} + \frac{5159}{397}e + \frac{5458}{397}$ |
| 79 | $[79, 79, -w - 3]$ | $\phantom{-}\frac{143}{397}e^{7} - \frac{887}{794}e^{6} - \frac{1803}{397}e^{5} + \frac{10111}{794}e^{4} + \frac{11237}{794}e^{3} - \frac{20595}{794}e^{2} - \frac{13901}{794}e - \frac{4667}{794}$ |
| 79 | $[79, 79, w - 4]$ | $\phantom{-}\frac{531}{397}e^{7} - \frac{1068}{397}e^{6} - \frac{8780}{397}e^{5} + \frac{12855}{397}e^{4} + \frac{44765}{397}e^{3} - \frac{31018}{397}e^{2} - \frac{74500}{397}e - \frac{5507}{397}$ |
| 81 | $[81, 3, -3]$ | $\phantom{-}\frac{138}{397}e^{7} - \frac{217}{397}e^{6} - \frac{2459}{397}e^{5} + \frac{2816}{397}e^{4} + \frac{13251}{397}e^{3} - \frac{8752}{397}e^{2} - \frac{20658}{397}e + \frac{2079}{397}$ |
| 89 | $[89, 89, w^{2} - 3w - 2]$ | $-\frac{1109}{794}e^{7} + \frac{1279}{397}e^{6} + \frac{17031}{794}e^{5} - \frac{30127}{794}e^{4} - \frac{79121}{794}e^{3} + \frac{68981}{794}e^{2} + \frac{127935}{794}e + \frac{9274}{397}$ |
| 89 | $[89, 89, w^{2} + w - 4]$ | $-\frac{74}{397}e^{7} + \frac{273}{794}e^{6} + \frac{1169}{397}e^{5} - \frac{3325}{794}e^{4} - \frac{11087}{794}e^{3} + \frac{8667}{794}e^{2} + \frac{17857}{794}e - \frac{1591}{794}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $23$ | $[23,23,w^{3} - 2w^{2} - 2w + 2]$ | $1$ |