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 | $[5, 5, -w^{3} + 3w^{2} + 2w - 1]$ | $...$ |
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 | $[11, 11, w^{3} - 2w^{2} - 5w - 1]$ | $...$ |
13 | $[13, 13, -w^{3} + 3w^{2} + 3w - 1]$ | $\phantom{-}1$ |
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 | $[31, 31, -w^{2} + 2w + 1]$ | $-\frac{461}{97}e^{12} - \frac{3001}{97}e^{11} - \frac{607}{97}e^{10} + \frac{31323}{97}e^{9} + \frac{46624}{97}e^{8} - \frac{98870}{97}e^{7} - \frac{224959}{97}e^{6} + \frac{73727}{97}e^{5} + \frac{351041}{97}e^{4} + \frac{72795}{97}e^{3} - \frac{128976}{97}e^{2} - \frac{24093}{97}e + \frac{945}{97}$ |
43 | $[43, 43, -w^{3} + 3w^{2} + 2w - 3]$ | $...$ |
43 | $[43, 43, -w^{3} + 2w^{2} + 6w - 3]$ | $-\frac{90}{97}e^{12} - \frac{556}{97}e^{11} + \frac{21}{97}e^{10} + \frac{5947}{97}e^{9} + \frac{7617}{97}e^{8} - \frac{19808}{97}e^{7} - \frac{38837}{97}e^{6} + \frac{18719}{97}e^{5} + \frac{62524}{97}e^{4} + \frac{6791}{97}e^{3} - \frac{24232}{97}e^{2} - \frac{1389}{97}e + \frac{700}{97}$ |
59 | $[59, 59, 2w^{3} - 6w^{2} - 7w + 7]$ | $...$ |
73 | $[73, 73, 3w^{3} - 6w^{2} - 16w - 5]$ | $-\frac{193}{97}e^{12} - \frac{1285}{97}e^{11} - \frac{385}{97}e^{10} + \frac{13321}{97}e^{9} + \frac{21084}{97}e^{8} - \frac{41117}{97}e^{7} - \frac{100331}{97}e^{6} + \frac{26065}{97}e^{5} + \frac{155208}{97}e^{4} + \frac{39978}{97}e^{3} - \frac{55135}{97}e^{2} - \frac{12559}{97}e + \frac{197}{97}$ |
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 | $[101, 101, -2w^{3} + 5w^{2} + 8w - 3]$ | $...$ |
Atkin-Lehner eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
$13$ | $[13, 13, -w^{3} + 3w^{2} + 3w - 1]$ | $-1$ |