Base field 3.3.321.1
Generator \(w\), with minimal polynomial \(x^{3} - x^{2} - 4x + 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[29, 29, -w^{2} + 2w + 4]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $8$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 2x^{7} - 17x^{6} + 36x^{5} + 65x^{4} - 162x^{3} + 44x^{2} + 38x - 9\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w + 1]$ | $\phantom{-}e$ |
| 3 | $[3, 3, w - 1]$ | $\phantom{-}\frac{124}{793}e^{7} - \frac{154}{793}e^{6} - \frac{2148}{793}e^{5} + \frac{226}{61}e^{4} + \frac{689}{61}e^{3} - \frac{13848}{793}e^{2} + \frac{407}{793}e + \frac{2552}{793}$ |
| 7 | $[7, 7, w^{2} - 2]$ | $\phantom{-}\frac{62}{793}e^{7} - \frac{77}{793}e^{6} - \frac{1074}{793}e^{5} + \frac{113}{61}e^{4} + \frac{314}{61}e^{3} - \frac{6924}{793}e^{2} + \frac{3772}{793}e + \frac{483}{793}$ |
| 8 | $[8, 2, 2]$ | $\phantom{-}\frac{37}{793}e^{7} + \frac{18}{793}e^{6} - \frac{820}{793}e^{5} - \frac{28}{61}e^{4} + \frac{394}{61}e^{3} + \frac{1598}{793}e^{2} - \frac{6958}{793}e + \frac{608}{793}$ |
| 11 | $[11, 11, -w^{2} + w + 1]$ | $\phantom{-}\frac{30}{793}e^{7} - \frac{114}{793}e^{6} - \frac{622}{793}e^{5} + \frac{157}{61}e^{4} + \frac{270}{61}e^{3} - \frac{9592}{793}e^{2} - \frac{2984}{793}e + \frac{4608}{793}$ |
| 23 | $[23, 23, -w - 3]$ | $-\frac{332}{793}e^{7} + \frac{310}{793}e^{6} + \frac{5879}{793}e^{5} - \frac{428}{61}e^{4} - \frac{2073}{61}e^{3} + \frac{25054}{793}e^{2} + \frac{14361}{793}e - \frac{6270}{793}$ |
| 29 | $[29, 29, -w^{2} + 2w + 4]$ | $\phantom{-}1$ |
| 31 | $[31, 31, 2w - 3]$ | $-\frac{231}{793}e^{7} + \frac{402}{793}e^{6} + \frac{4155}{793}e^{5} - \frac{544}{61}e^{4} - \frac{1408}{61}e^{3} + \frac{30402}{793}e^{2} + \frac{297}{793}e - \frac{2810}{793}$ |
| 41 | $[41, 41, -2w^{2} + 3w + 6]$ | $-\frac{526}{793}e^{7} + \frac{730}{793}e^{6} + \frac{9214}{793}e^{5} - \frac{1000}{61}e^{4} - \frac{3026}{61}e^{3} + \frac{57054}{793}e^{2} + \frac{2942}{793}e - \frac{8472}{793}$ |
| 43 | $[43, 43, w^{2} - 3w + 3]$ | $-\frac{168}{793}e^{7} + \frac{4}{793}e^{6} + \frac{3166}{793}e^{5} - \frac{13}{61}e^{4} - \frac{1268}{61}e^{3} + \frac{1060}{793}e^{2} + \frac{17662}{793}e - \frac{746}{793}$ |
| 47 | $[47, 47, w^{2} + w - 4]$ | $-\frac{124}{793}e^{7} + \frac{154}{793}e^{6} + \frac{2148}{793}e^{5} - \frac{226}{61}e^{4} - \frac{750}{61}e^{3} + \frac{13848}{793}e^{2} + \frac{6730}{793}e - \frac{5724}{793}$ |
| 49 | $[49, 7, 2w^{2} - 3w - 3]$ | $\phantom{-}\frac{140}{793}e^{7} - \frac{532}{793}e^{6} - \frac{2374}{793}e^{5} + \frac{692}{61}e^{4} + \frac{650}{61}e^{3} - \frac{37890}{793}e^{2} + \frac{10922}{793}e + \frac{8816}{793}$ |
| 53 | $[53, 53, w^{2} - 3w - 2]$ | $\phantom{-}\frac{124}{793}e^{7} - \frac{154}{793}e^{6} - \frac{2148}{793}e^{5} + \frac{226}{61}e^{4} + \frac{750}{61}e^{3} - \frac{15434}{793}e^{2} - \frac{6730}{793}e + \frac{12068}{793}$ |
| 59 | $[59, 59, 2w^{2} - w - 5]$ | $\phantom{-}\frac{124}{793}e^{7} - \frac{154}{793}e^{6} - \frac{2148}{793}e^{5} + \frac{226}{61}e^{4} + \frac{750}{61}e^{3} - \frac{14641}{793}e^{2} - \frac{6730}{793}e + \frac{4138}{793}$ |
| 59 | $[59, 59, w^{2} - w - 7]$ | $-\frac{618}{793}e^{7} + \frac{921}{793}e^{6} + \frac{10910}{793}e^{5} - \frac{1270}{61}e^{4} - \frac{3610}{61}e^{3} + \frac{72777}{793}e^{2} - \frac{1018}{793}e - \frac{8012}{793}$ |
| 59 | $[59, 59, -w^{2} - w + 7]$ | $-\frac{264}{793}e^{7} + \frac{686}{793}e^{6} + \frac{4522}{793}e^{5} - \frac{918}{61}e^{4} - \frac{1278}{61}e^{3} + \frac{51738}{793}e^{2} - \frac{18466}{793}e - \frac{9782}{793}$ |
| 67 | $[67, 67, 2w^{2} - 3w - 7]$ | $-\frac{446}{793}e^{7} + \frac{426}{793}e^{6} + \frac{8084}{793}e^{5} - \frac{622}{61}e^{4} - \frac{2916}{61}e^{3} + \frac{38348}{793}e^{2} + \frac{19832}{793}e - \frac{8872}{793}$ |
| 73 | $[73, 73, -w^{2} + 4w - 5]$ | $\phantom{-}\frac{446}{793}e^{7} - \frac{426}{793}e^{6} - \frac{8084}{793}e^{5} + \frac{622}{61}e^{4} + \frac{2916}{61}e^{3} - \frac{38348}{793}e^{2} - \frac{16660}{793}e + \frac{8872}{793}$ |
| 79 | $[79, 79, w^{2} - 8]$ | $\phantom{-}\frac{248}{793}e^{7} - \frac{308}{793}e^{6} - \frac{4296}{793}e^{5} + \frac{452}{61}e^{4} + \frac{1378}{61}e^{3} - \frac{29282}{793}e^{2} + \frac{2400}{793}e + \frac{14620}{793}$ |
| 79 | $[79, 79, w^{2} - 5w + 5]$ | $\phantom{-}\frac{204}{793}e^{7} - \frac{458}{793}e^{6} - \frac{3278}{793}e^{5} + \frac{604}{61}e^{4} + \frac{860}{61}e^{3} - \frac{34140}{793}e^{2} + \frac{10160}{793}e + \frac{11668}{793}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $29$ | $[29, 29, -w^{2} + 2w + 4]$ | $-1$ |