Base field 3.3.1345.1
Generator \(w\), with minimal polynomial \(x^{3} - 7x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[7, 7, -w + 1]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $13$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 28x^{6} + 10x^{5} + 241x^{4} - 137x^{3} - 598x^{2} + 228x + 328\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, -w - 1]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w + 2]$ | $-\frac{423}{6416}e^{7} - \frac{529}{3208}e^{6} + \frac{1017}{802}e^{5} + \frac{7309}{3208}e^{4} - \frac{45435}{6416}e^{3} - \frac{40219}{6416}e^{2} + \frac{3406}{401}e + \frac{7561}{1604}$ |
| 7 | $[7, 7, w + 3]$ | $\phantom{-}\frac{913}{19248}e^{7} + \frac{1703}{9624}e^{6} - \frac{635}{802}e^{5} - \frac{27235}{9624}e^{4} + \frac{65213}{19248}e^{3} + \frac{71487}{6416}e^{2} - \frac{1801}{1203}e - \frac{35287}{4812}$ |
| 7 | $[7, 7, -w + 2]$ | $\phantom{-}\frac{457}{9624}e^{7} + \frac{581}{4812}e^{6} - \frac{389}{401}e^{5} - \frac{9163}{4812}e^{4} + \frac{55685}{9624}e^{3} + \frac{24003}{3208}e^{2} - \frac{18457}{2406}e - \frac{16585}{2406}$ |
| 7 | $[7, 7, -w + 1]$ | $\phantom{-}1$ |
| 8 | $[8, 2, 2]$ | $\phantom{-}\frac{155}{9624}e^{7} + \frac{355}{4812}e^{6} - \frac{110}{401}e^{5} - \frac{4961}{4812}e^{4} + \frac{13327}{9624}e^{3} + \frac{6337}{3208}e^{2} - \frac{3575}{2406}e + \frac{8137}{2406}$ |
| 13 | $[13, 13, -w^{2} + w + 4]$ | $-\frac{43}{1203}e^{7} - \frac{361}{4812}e^{6} + \frac{270}{401}e^{5} + \frac{1340}{1203}e^{4} - \frac{8023}{2406}e^{3} - \frac{6155}{1604}e^{2} + \frac{12337}{4812}e + \frac{6785}{2406}$ |
| 19 | $[19, 19, -w^{2} + 5]$ | $\phantom{-}\frac{3445}{19248}e^{7} + \frac{5717}{9624}e^{6} - \frac{2613}{802}e^{5} - \frac{86047}{9624}e^{4} + \frac{339977}{19248}e^{3} + \frac{194967}{6416}e^{2} - \frac{121213}{4812}e - \frac{95761}{4812}$ |
| 23 | $[23, 23, -w^{2} - w + 3]$ | $-\frac{415}{9624}e^{7} - \frac{223}{2406}e^{6} + \frac{398}{401}e^{5} + \frac{8005}{4812}e^{4} - \frac{65951}{9624}e^{3} - \frac{22943}{3208}e^{2} + \frac{59153}{4812}e + \frac{5231}{1203}$ |
| 27 | $[27, 3, 3]$ | $-\frac{155}{9624}e^{7} - \frac{355}{4812}e^{6} + \frac{110}{401}e^{5} + \frac{4961}{4812}e^{4} - \frac{13327}{9624}e^{3} - \frac{9545}{3208}e^{2} + \frac{1169}{2406}e + \frac{15923}{2406}$ |
| 29 | $[29, 29, w^{2} - w - 3]$ | $\phantom{-}\frac{25}{1203}e^{7} + \frac{35}{2406}e^{6} - \frac{129}{401}e^{5} + \frac{340}{1203}e^{4} + \frac{1451}{1203}e^{3} - \frac{3531}{802}e^{2} - \frac{2789}{2406}e + \frac{5987}{1203}$ |
| 31 | $[31, 31, w^{2} - w - 8]$ | $-\frac{1711}{9624}e^{7} - \frac{3065}{4812}e^{6} + \frac{1266}{401}e^{5} + \frac{49237}{4812}e^{4} - \frac{154067}{9624}e^{3} - \frac{129321}{3208}e^{2} + \frac{22778}{1203}e + \frac{73429}{2406}$ |
| 37 | $[37, 37, -w - 4]$ | $-\frac{599}{4812}e^{7} - \frac{751}{2406}e^{6} + \frac{889}{401}e^{5} + \frac{9641}{2406}e^{4} - \frac{51223}{4812}e^{3} - \frac{13189}{1604}e^{2} + \frac{9935}{1203}e + \frac{2285}{1203}$ |
| 43 | $[43, 43, 2w^{2} - 4w - 3]$ | $-\frac{23}{2406}e^{7} - \frac{377}{2406}e^{6} - \frac{77}{401}e^{5} + \frac{3212}{1203}e^{4} + \frac{9059}{2406}e^{3} - \frac{4854}{401}e^{2} - \frac{15535}{2406}e + \frac{12091}{1203}$ |
| 47 | $[47, 47, w^{2} - 3]$ | $\phantom{-}\frac{19}{802}e^{7} + \frac{187}{1604}e^{6} - \frac{123}{401}e^{5} - \frac{753}{401}e^{4} + \frac{399}{401}e^{3} + \frac{12673}{1604}e^{2} - \frac{5071}{1604}e - \frac{5523}{802}$ |
| 53 | $[53, 53, 2w^{2} - w - 11]$ | $-\frac{263}{3208}e^{7} - \frac{473}{1604}e^{6} + \frac{547}{401}e^{5} + \frac{6793}{1604}e^{4} - \frac{22675}{3208}e^{3} - \frac{41395}{3208}e^{2} + \frac{5135}{401}e + \frac{3025}{802}$ |
| 59 | $[59, 59, w^{2} - 2w - 4]$ | $-\frac{3451}{19248}e^{7} - \frac{4877}{9624}e^{6} + \frac{2669}{802}e^{5} + \frac{71089}{9624}e^{4} - \frac{337823}{19248}e^{3} - \frac{151925}{6416}e^{2} + \frac{24643}{1203}e + \frac{71029}{4812}$ |
| 67 | $[67, 67, w^{2} + w - 8]$ | $\phantom{-}\frac{317}{2406}e^{7} + \frac{532}{1203}e^{6} - \frac{874}{401}e^{5} - \frac{7709}{1203}e^{4} + \frac{22537}{2406}e^{3} + \frac{15925}{802}e^{2} - \frac{7987}{1203}e - \frac{12400}{1203}$ |
| 71 | $[71, 71, w^{2} + w - 11]$ | $\phantom{-}\frac{35}{3208}e^{7} - \frac{22}{401}e^{6} - \frac{178}{401}e^{5} + \frac{2243}{1604}e^{4} + \frac{13099}{3208}e^{3} - \frac{34643}{3208}e^{2} - \frac{6931}{1604}e + \frac{6772}{401}$ |
| 73 | $[73, 73, -w^{2} + 4w - 2]$ | $\phantom{-}\frac{2123}{9624}e^{7} + \frac{682}{1203}e^{6} - \frac{1636}{401}e^{5} - \frac{35849}{4812}e^{4} + \frac{206659}{9624}e^{3} + \frac{51079}{3208}e^{2} - \frac{109147}{4812}e - \frac{610}{1203}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $7$ | $[7, 7, -w + 1]$ | $-1$ |