Base field 6.6.1397493.1
Generator \(w\), with minimal polynomial \(x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2, 2]$ |
| Level: | $[51, 51, -w^{5} + 3w^{4} + 2w^{3} - 7w^{2} - 3w]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $30$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} + 12x^{7} + 14x^{6} - 295x^{5} - 943x^{4} + 1065x^{3} + 5559x^{2} + 967x - 5366\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w - 1]$ | $\phantom{-}1$ |
| 17 | $[17, 17, -w^{2} + 2w + 1]$ | $-1$ |
| 17 | $[17, 17, -w^{3} + w^{2} + 4w]$ | $\phantom{-}e$ |
| 19 | $[19, 19, w^{5} - 3w^{4} - 2w^{3} + 8w^{2} + w - 4]$ | $\phantom{-}\frac{573723}{228952021}e^{7} + \frac{5247625}{228952021}e^{6} - \frac{6547881}{228952021}e^{5} - \frac{135015560}{228952021}e^{4} - \frac{114348577}{228952021}e^{3} + \frac{563180557}{228952021}e^{2} + \frac{286781322}{228952021}e - \frac{563382596}{228952021}$ |
| 19 | $[19, 19, w^{2} - w - 1]$ | $-\frac{4687666}{228952021}e^{7} - \frac{36290142}{228952021}e^{6} + \frac{76919624}{228952021}e^{5} + \frac{986093543}{228952021}e^{4} + \frac{529790127}{228952021}e^{3} - \frac{5390906146}{228952021}e^{2} - \frac{4102939853}{228952021}e + \frac{5141128442}{228952021}$ |
| 37 | $[37, 37, w^{4} - 2w^{3} - 3w^{2} + 3w + 2]$ | $\phantom{-}\frac{11660334}{228952021}e^{7} + \frac{75274240}{228952021}e^{6} - \frac{275675227}{228952021}e^{5} - \frac{2047786176}{228952021}e^{4} + \frac{811610989}{228952021}e^{3} + \frac{11109184830}{228952021}e^{2} + \frac{2311532442}{228952021}e - \frac{11673502270}{228952021}$ |
| 37 | $[37, 37, w^{4} - 4w^{3} + w^{2} + 7w - 3]$ | $-\frac{3058449}{228952021}e^{7} - \frac{23331757}{228952021}e^{6} + \frac{52580106}{228952021}e^{5} + \frac{634483136}{228952021}e^{4} + \frac{270025326}{228952021}e^{3} - \frac{3578007616}{228952021}e^{2} - \frac{2538866005}{228952021}e + \frac{4155569284}{228952021}$ |
| 53 | $[53, 53, 2w^{5} - 6w^{4} - 6w^{3} + 18w^{2} + 9w - 6]$ | $-\frac{8528343}{228952021}e^{7} - \frac{62613208}{228952021}e^{6} + \frac{168741347}{228952021}e^{5} + \frac{1718629652}{228952021}e^{4} + \frac{153032415}{228952021}e^{3} - \frac{9789577632}{228952021}e^{2} - \frac{3740150749}{228952021}e + \frac{11181170544}{228952021}$ |
| 53 | $[53, 53, -w^{5} + 3w^{4} + 3w^{3} - 9w^{2} - 3w + 2]$ | $\phantom{-}\frac{6759947}{228952021}e^{7} + \frac{51590278}{228952021}e^{6} - \frac{110762184}{228952021}e^{5} - \frac{1374537043}{228952021}e^{4} - \frac{732017682}{228952021}e^{3} + \frac{7080728362}{228952021}e^{2} + \frac{5592660259}{228952021}e - \frac{6765786624}{228952021}$ |
| 64 | $[64, 2, -2]$ | $-\frac{5593754}{228952021}e^{7} - \frac{37155996}{228952021}e^{6} + \frac{127912596}{228952021}e^{5} + \frac{1017437681}{228952021}e^{4} - \frac{275017721}{228952021}e^{3} - \frac{5738122920}{228952021}e^{2} - \frac{1685650254}{228952021}e + \frac{7353743953}{228952021}$ |
| 71 | $[71, 71, -w^{5} + 4w^{4} - w^{3} - 8w^{2} + 3w - 1]$ | $-\frac{5468085}{228952021}e^{7} - \frac{33295710}{228952021}e^{6} + \frac{138864479}{228952021}e^{5} + \frac{908674702}{228952021}e^{4} - \frac{619336883}{228952021}e^{3} - \frac{4856780368}{228952021}e^{2} - \frac{13275743}{228952021}e + \frac{2683358290}{228952021}$ |
| 71 | $[71, 71, 2w^{5} - 5w^{4} - 8w^{3} + 15w^{2} + 12w - 6]$ | $-\frac{1294618}{228952021}e^{7} - \frac{5012199}{228952021}e^{6} + \frac{47798750}{228952021}e^{5} + \frac{105441277}{228952021}e^{4} - \frac{626701831}{228952021}e^{3} - \frac{59120129}{228952021}e^{2} + \frac{3205765666}{228952021}e - \frac{958874824}{228952021}$ |
| 71 | $[71, 71, 2w^{5} - 6w^{4} - 6w^{3} + 18w^{2} + 9w - 7]$ | $\phantom{-}\frac{2413591}{228952021}e^{7} + \frac{12419252}{228952021}e^{6} - \frac{69681372}{228952021}e^{5} - \frac{353440551}{228952021}e^{4} + \frac{424285676}{228952021}e^{3} + \frac{2298504838}{228952021}e^{2} - \frac{79879329}{228952021}e - \frac{5021910594}{228952021}$ |
| 73 | $[73, 73, -2w^{5} + 6w^{4} + 5w^{3} - 16w^{2} - 7w + 3]$ | $\phantom{-}\frac{8959535}{228952021}e^{7} + \frac{69336236}{228952021}e^{6} - \frac{150684838}{228952021}e^{5} - \frac{1916721970}{228952021}e^{4} - \frac{884819858}{228952021}e^{3} + \frac{11330826803}{228952021}e^{2} + \frac{6920154303}{228952021}e - \frac{14868256844}{228952021}$ |
| 73 | $[73, 73, -2w^{5} + 6w^{4} + 5w^{3} - 17w^{2} - 6w + 6]$ | $\phantom{-}\frac{2806425}{228952021}e^{7} + \frac{22074132}{228952021}e^{6} - \frac{47782273}{228952021}e^{5} - \frac{597719248}{228952021}e^{4} - \frac{284423654}{228952021}e^{3} + \frac{3153178990}{228952021}e^{2} + \frac{2983838036}{228952021}e - \frac{3018280110}{228952021}$ |
| 89 | $[89, 89, w^{5} - 2w^{4} - 5w^{3} + 6w^{2} + 8w - 4]$ | $-\frac{6352753}{228952021}e^{7} - \frac{35046707}{228952021}e^{6} + \frac{178471925}{228952021}e^{5} + \frac{897801647}{228952021}e^{4} - \frac{1171228567}{228952021}e^{3} - \frac{3607828824}{228952021}e^{2} + \frac{1290669769}{228952021}e - \frac{898955928}{228952021}$ |
| 89 | $[89, 89, w^{5} - w^{4} - 9w^{3} + 7w^{2} + 16w - 6]$ | $-\frac{4555198}{228952021}e^{7} - \frac{27398527}{228952021}e^{6} + \frac{102578023}{228952021}e^{5} + \frac{712439262}{228952021}e^{4} - \frac{144439668}{228952021}e^{3} - \frac{3578811648}{228952021}e^{2} - \frac{2095892046}{228952021}e + \frac{4079930194}{228952021}$ |
| 89 | $[89, 89, 2w^{5} - 6w^{4} - 4w^{3} + 15w^{2} + 3w - 3]$ | $\phantom{-}\frac{4432373}{228952021}e^{7} + \frac{26114271}{228952021}e^{6} - \frac{115426391}{228952021}e^{5} - \frac{689318600}{228952021}e^{4} + \frac{623924730}{228952021}e^{3} + \frac{3524297040}{228952021}e^{2} - \frac{1000664039}{228952021}e - \frac{5790781040}{228952021}$ |
| 89 | $[89, 89, w^{5} - 2w^{4} - 6w^{3} + 8w^{2} + 10w - 6]$ | $-\frac{3311508}{228952021}e^{7} - \frac{21179671}{228952021}e^{6} + \frac{81977662}{228952021}e^{5} + \frac{581417527}{228952021}e^{4} - \frac{381882874}{228952021}e^{3} - \frac{3245922517}{228952021}e^{2} + \frac{747248622}{228952021}e + \frac{2981763598}{228952021}$ |
| 107 | $[107, 107, w^{5} - 2w^{4} - 7w^{3} + 10w^{2} + 12w - 6]$ | $-\frac{1326614}{228952021}e^{7} - \frac{15960653}{228952021}e^{6} - \frac{9758580}{228952021}e^{5} + \frac{431359550}{228952021}e^{4} + \frac{974882925}{228952021}e^{3} - \frac{2471733680}{228952021}e^{2} - \frac{5114346313}{228952021}e + \frac{2760754234}{228952021}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, w - 1]$ | $-1$ |
| $17$ | $[17, 17, -w^{2} + 2w + 1]$ | $1$ |