Base field 4.4.16448.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 6x^{2} + 2\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, -w^{3} + 3w^{2} + 3w - 1]$ |
| Dimension: | $13$ |
| 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:
| \(x^{13} - 8x^{12} + 11x^{11} + 66x^{10} - 202x^{9} - 64x^{8} + 804x^{7} - 565x^{6} - 985x^{5} + 1282x^{4} + 21x^{3} - 452x^{2} + 83x + 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w]$ | $\phantom{-}e$ |
| 5 | $[5, 5, -w^{3} + 3w^{2} + 2w - 1]$ | $-\frac{336}{97}e^{12} + \frac{2218}{97}e^{11} - \frac{620}{97}e^{10} - \frac{22907}{97}e^{9} + \frac{35964}{97}e^{8} + \frac{70238}{97}e^{7} - \frac{171078}{97}e^{6} - \frac{43455}{97}e^{5} + \frac{264062}{97}e^{4} - \frac{69501}{97}e^{3} - \frac{93499}{97}e^{2} + \frac{22665}{97}e - \frac{38}{97}$ |
| 5 | $[5, 5, w - 1]$ | $-\frac{92}{97}e^{12} + \frac{605}{97}e^{11} - \frac{179}{97}e^{10} - \frac{6174}{97}e^{9} + \frac{9875}{97}e^{8} + \frac{18375}{97}e^{7} - \frac{46531}{97}e^{6} - \frac{9172}{97}e^{5} + \frac{70947}{97}e^{4} - \frac{22283}{97}e^{3} - \frac{24251}{97}e^{2} + \frac{7214}{97}e - \frac{179}{97}$ |
| 11 | $[11, 11, w^{3} - 2w^{2} - 5w - 1]$ | $-\frac{312}{97}e^{12} + \frac{2115}{97}e^{11} - \frac{839}{97}e^{10} - \frac{21638}{97}e^{9} + \frac{36416}{97}e^{8} + \frac{64833}{97}e^{7} - \frac{170609}{97}e^{6} - \frac{34344}{97}e^{5} + \frac{262993}{97}e^{4} - \frac{74105}{97}e^{3} - \frac{94823}{97}e^{2} + \frac{22702}{97}e + \frac{325}{97}$ |
| 13 | $[13, 13, -w^{3} + 3w^{2} + 3w - 1]$ | $-1$ |
| 17 | $[17, 17, 2w^{3} - 5w^{2} - 9w + 5]$ | $...$ |
| 23 | $[23, 23, -w^{3} + 3w^{2} + 4w - 5]$ | $...$ |
| 25 | $[25, 5, -w^{2} + 3w + 1]$ | $...$ |
| 29 | $[29, 29, -2w^{3} + 6w^{2} + 5w - 1]$ | $...$ |
| 31 | $[31, 31, -w^{2} + 2w + 1]$ | $...$ |
| 43 | $[43, 43, -w^{3} + 3w^{2} + 2w - 3]$ | $...$ |
| 43 | $[43, 43, -w^{3} + 2w^{2} + 6w - 3]$ | $\phantom{-}\frac{90}{97}e^{12} - \frac{556}{97}e^{11} - \frac{21}{97}e^{10} + \frac{5947}{97}e^{9} - \frac{7617}{97}e^{8} - \frac{19808}{97}e^{7} + \frac{38837}{97}e^{6} + \frac{18719}{97}e^{5} - \frac{62524}{97}e^{4} + \frac{6791}{97}e^{3} + \frac{24232}{97}e^{2} - \frac{1389}{97}e - \frac{700}{97}$ |
| 59 | $[59, 59, 2w^{3} - 6w^{2} - 7w + 7]$ | $...$ |
| 73 | $[73, 73, 3w^{3} - 6w^{2} - 16w - 5]$ | $...$ |
| 79 | $[79, 79, -5w^{3} + 14w^{2} + 20w - 17]$ | $...$ |
| 81 | $[81, 3, -3]$ | $-\frac{309}{97}e^{12} + \frac{1993}{97}e^{11} - \frac{248}{97}e^{10} - \frac{21152}{97}e^{9} + \frac{29634}{97}e^{8} + \frac{69553}{97}e^{7} - \frac{145876}{97}e^{6} - \frac{63263}{97}e^{5} + \frac{231880}{97}e^{4} - \frac{28751}{97}e^{3} - \frac{89508}{97}e^{2} + \frac{8969}{97}e + \frac{528}{97}$ |
| 83 | $[83, 83, -w - 3]$ | $...$ |
| 83 | $[83, 83, w^{2} - 3w - 7]$ | $...$ |
| 89 | $[89, 89, w^{2} - 2w - 7]$ | $...$ |
| 101 | $[101, 101, -2w^{3} + 5w^{2} + 8w - 3]$ | $-\frac{287}{97}e^{12} + \frac{1842}{97}e^{11} - \frac{279}{97}e^{10} - \frac{19140}{97}e^{9} + \frac{27591}{97}e^{8} + \frac{59708}{97}e^{7} - \frac{132537}{97}e^{6} - \frac{41598}{97}e^{5} + \frac{201889}{97}e^{4} - \frac{49623}{97}e^{3} - \frac{66989}{97}e^{2} + \frac{16577}{97}e - \frac{85}{97}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w^{3} + 3w^{2} + 3w - 1]$ | $1$ |