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: | $[13, 13, w^{3} - 3w^{2} - w + 2]$ |
| Dimension: | $7$ |
| 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:
| \(x^{7} - 5x^{6} - 5x^{5} + 47x^{4} - 20x^{3} - 74x^{2} + 21x + 33\) |
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| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}e$ |
| 7 | $[7, 7, w^{2} - w - 2]$ | $\phantom{-}\frac{1}{2}e^{5} - e^{4} - \frac{9}{2}e^{3} + \frac{13}{2}e^{2} + \frac{7}{2}e - 1$ |
| 9 | $[9, 3, w^{2} - 2w - 1]$ | $\phantom{-}e^{5} - 3e^{4} - 8e^{3} + 24e^{2} + e - 17$ |
| 13 | $[13, 13, w^{3} - 3w^{2} - w + 5]$ | $-\frac{1}{2}e^{5} + \frac{3}{2}e^{4} + \frac{9}{2}e^{3} - 13e^{2} - 4e + \frac{23}{2}$ |
| 13 | $[13, 13, w^{3} - 3w^{2} - w + 2]$ | $-1$ |
| 16 | $[16, 2, 2]$ | $-\frac{1}{2}e^{6} + \frac{5}{2}e^{5} + e^{4} - 20e^{3} + \frac{45}{2}e^{2} + 12e - \frac{31}{2}$ |
| 17 | $[17, 17, -w^{3} + 2w^{2} + 2w - 2]$ | $-\frac{1}{2}e^{5} + \frac{3}{2}e^{4} + \frac{7}{2}e^{3} - 12e^{2} + 4e + \frac{21}{2}$ |
| 19 | $[19, 19, -w^{3} + 2w^{2} + 2w - 1]$ | $\phantom{-}\frac{1}{2}e^{6} - \frac{5}{2}e^{5} - \frac{5}{2}e^{4} + 22e^{3} - 9e^{2} - \frac{41}{2}e + 4$ |
| 29 | $[29, 29, -2w^{3} + 5w^{2} + 4w - 7]$ | $-e^{3} + e^{2} + 8e - 3$ |
| 31 | $[31, 31, w^{3} - 2w^{2} - 4w + 2]$ | $-e^{5} + 3e^{4} + 8e^{3} - 25e^{2} + 22$ |
| 47 | $[47, 47, -w^{3} + 4w^{2} - w - 5]$ | $-e^{6} + 4e^{5} + 6e^{4} - 33e^{3} + 12e^{2} + 23e - 3$ |
| 53 | $[53, 53, -w^{3} + 4w^{2} - w - 7]$ | $-e^{5} + 2e^{4} + 9e^{3} - 13e^{2} - 8e + 6$ |
| 67 | $[67, 67, 2w^{3} - 3w^{2} - 8w + 1]$ | $-2e^{4} + 3e^{3} + 19e^{2} - 15e - 17$ |
| 67 | $[67, 67, 2w^{3} - 5w^{2} - 4w + 4]$ | $\phantom{-}e^{4} - e^{3} - 10e^{2} + 7e + 8$ |
| 71 | $[71, 71, w^{3} - 3w^{2} - 2w + 2]$ | $\phantom{-}\frac{1}{2}e^{6} - \frac{5}{2}e^{5} - \frac{1}{2}e^{4} + 19e^{3} - 26e^{2} - \frac{13}{2}e + 15$ |
| 79 | $[79, 79, w^{2} - 3w - 4]$ | $\phantom{-}\frac{1}{2}e^{5} - \frac{11}{2}e^{3} - \frac{7}{2}e^{2} + \frac{21}{2}e + 11$ |
| 83 | $[83, 83, 2w^{2} - 3w - 4]$ | $-e^{5} + 5e^{4} + 5e^{3} - 42e^{2} + 14e + 33$ |
| 89 | $[89, 89, -3w^{3} + 7w^{2} + 8w - 11]$ | $\phantom{-}e^{6} - 3e^{5} - 9e^{4} + 26e^{3} + 10e^{2} - 33e$ |
| 101 | $[101, 101, w^{3} - 2w^{2} - 3w - 2]$ | $\phantom{-}\frac{3}{2}e^{5} - e^{4} - \frac{33}{2}e^{3} + \frac{5}{2}e^{2} + \frac{49}{2}e + 6$ |
| 103 | $[103, 103, -2w^{3} + 4w^{2} + 5w - 1]$ | $-2e^{5} + 4e^{4} + 19e^{3} - 27e^{2} - 24e + 13$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, w^{3} - 3w^{2} - w + 2]$ | $1$ |