Base field 4.4.7053.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 4x^{2} + 3x + 3\); narrow class number \(1\) and class number \(1\).
Form
Weight: | $[2, 2, 2, 2]$ |
Level: | $[13, 13, w^{3} - 3w^{2} - w + 5]$ |
Dimension: | $6$ |
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:
\(x^{6} - 14x^{4} + 2x^{3} + 42x^{2} - 8x - 27\) |
Show full eigenvalues Hide large eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
3 | $[3, 3, w]$ | $\phantom{-}e$ |
7 | $[7, 7, w^{2} - w - 2]$ | $-\frac{5}{31}e^{5} - \frac{6}{31}e^{4} + \frac{69}{31}e^{3} + \frac{48}{31}e^{2} - \frac{171}{31}e - \frac{4}{31}$ |
9 | $[9, 3, w^{2} - 2w - 1]$ | $\phantom{-}\frac{5}{62}e^{5} - \frac{25}{62}e^{4} - \frac{69}{62}e^{3} + \frac{293}{62}e^{2} + \frac{171}{62}e - \frac{337}{62}$ |
13 | $[13, 13, w^{3} - 3w^{2} - w + 5]$ | $-1$ |
13 | $[13, 13, w^{3} - 3w^{2} - w + 2]$ | $-\frac{17}{62}e^{5} - \frac{39}{62}e^{4} + \frac{185}{62}e^{3} + \frac{405}{62}e^{2} - \frac{259}{62}e - \frac{553}{62}$ |
16 | $[16, 2, 2]$ | $-\frac{3}{62}e^{5} + \frac{15}{62}e^{4} + \frac{29}{62}e^{3} - \frac{213}{62}e^{2} + \frac{71}{62}e + \frac{413}{62}$ |
17 | $[17, 17, -w^{3} + 2w^{2} + 2w - 2]$ | $\phantom{-}\frac{16}{31}e^{5} + \frac{13}{31}e^{4} - \frac{196}{31}e^{3} - \frac{135}{31}e^{2} + \frac{324}{31}e + \frac{267}{31}$ |
19 | $[19, 19, -w^{3} + 2w^{2} + 2w - 1]$ | $-\frac{5}{31}e^{5} - \frac{6}{31}e^{4} + \frac{69}{31}e^{3} + \frac{79}{31}e^{2} - \frac{171}{31}e - \frac{128}{31}$ |
29 | $[29, 29, -2w^{3} + 5w^{2} + 4w - 7]$ | $\phantom{-}\frac{6}{31}e^{5} + \frac{1}{31}e^{4} - \frac{58}{31}e^{3} - \frac{8}{31}e^{2} + \frac{44}{31}e + \frac{42}{31}$ |
31 | $[31, 31, w^{3} - 2w^{2} - 4w + 2]$ | $-\frac{5}{62}e^{5} + \frac{25}{62}e^{4} + \frac{69}{62}e^{3} - \frac{355}{62}e^{2} - \frac{109}{62}e + \frac{647}{62}$ |
47 | $[47, 47, -w^{3} + 4w^{2} - w - 5]$ | $\phantom{-}\frac{38}{31}e^{5} + \frac{27}{31}e^{4} - \frac{450}{31}e^{3} - \frac{216}{31}e^{2} + \frac{754}{31}e + \frac{297}{31}$ |
53 | $[53, 53, -w^{3} + 4w^{2} - w - 7]$ | $\phantom{-}e^{4} - 11e^{2} + 2e + 15$ |
67 | $[67, 67, 2w^{3} - 3w^{2} - 8w + 1]$ | $\phantom{-}\frac{1}{31}e^{5} - \frac{5}{31}e^{4} + \frac{11}{31}e^{3} + \frac{9}{31}e^{2} - \frac{189}{31}e + \frac{193}{31}$ |
67 | $[67, 67, 2w^{3} - 5w^{2} - 4w + 4]$ | $\phantom{-}\frac{9}{62}e^{5} + \frac{17}{62}e^{4} - \frac{149}{62}e^{3} - \frac{291}{62}e^{2} + \frac{531}{62}e + \frac{931}{62}$ |
71 | $[71, 71, w^{3} - 3w^{2} - 2w + 2]$ | $\phantom{-}\frac{21}{62}e^{5} + \frac{19}{62}e^{4} - \frac{265}{62}e^{3} - \frac{183}{62}e^{2} + \frac{743}{62}e - \frac{39}{62}$ |
79 | $[79, 79, w^{2} - 3w - 4]$ | $-\frac{47}{62}e^{5} - \frac{13}{62}e^{4} + \frac{537}{62}e^{3} + \frac{135}{62}e^{2} - \frac{727}{62}e - \frac{143}{62}$ |
83 | $[83, 83, 2w^{2} - 3w - 4]$ | $\phantom{-}\frac{25}{62}e^{5} - \frac{1}{62}e^{4} - \frac{283}{62}e^{3} + \frac{163}{62}e^{2} + \frac{297}{62}e - \frac{507}{62}$ |
89 | $[89, 89, -3w^{3} + 7w^{2} + 8w - 11]$ | $\phantom{-}\frac{28}{31}e^{5} + \frac{46}{31}e^{4} - \frac{343}{31}e^{3} - \frac{492}{31}e^{2} + \frac{691}{31}e + \frac{630}{31}$ |
101 | $[101, 101, w^{3} - 2w^{2} - 3w - 2]$ | $\phantom{-}\frac{19}{62}e^{5} + \frac{29}{62}e^{4} - \frac{287}{62}e^{3} - \frac{263}{62}e^{2} + \frac{873}{62}e - \frac{177}{62}$ |
103 | $[103, 103, -2w^{3} + 4w^{2} + 5w - 1]$ | $\phantom{-}\frac{5}{31}e^{5} + \frac{6}{31}e^{4} - \frac{69}{31}e^{3} - \frac{79}{31}e^{2} + \frac{171}{31}e + \frac{283}{31}$ |
Atkin-Lehner eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
$13$ | $[13, 13, w^{3} - 3w^{2} - w + 5]$ | $1$ |