Base field 4.4.13824.1
Generator \(w\), with minimal polynomial \(x^{4} - 6x^{2} + 6\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13,13,-w^{3} + 4w + 1]$ |
| Dimension: | $14$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $20$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{14} - 23x^{12} + 209x^{10} - 953x^{8} + 2293x^{6} - 2811x^{4} + 1529x^{2} - 261\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w^{2} + w + 2]$ | $\phantom{-}e$ |
| 3 | $[3, 3, w^{2} - w - 3]$ | $-\frac{1}{24}e^{13} + \frac{5}{6}e^{11} - \frac{149}{24}e^{9} + \frac{253}{12}e^{7} - \frac{775}{24}e^{5} + \frac{85}{4}e^{3} - \frac{191}{24}e$ |
| 11 | $[11, 11, -w^{2} + w + 1]$ | $-\frac{1}{12}e^{13} + \frac{5}{3}e^{11} - \frac{155}{12}e^{9} + \frac{295}{6}e^{7} - \frac{1147}{12}e^{5} + \frac{173}{2}e^{3} - \frac{305}{12}e$ |
| 11 | $[11, 11, -w^{2} - w + 1]$ | $\phantom{-}\frac{1}{6}e^{13} - \frac{43}{12}e^{11} + \frac{355}{12}e^{9} - \frac{701}{6}e^{7} + \frac{1345}{6}e^{5} - \frac{759}{4}e^{3} + \frac{619}{12}e$ |
| 13 | $[13, 13, w^{3} - 4w + 1]$ | $-\frac{1}{4}e^{12} + 5e^{10} - \frac{151}{4}e^{8} + \frac{265}{2}e^{6} - \frac{855}{4}e^{4} + \frac{271}{2}e^{2} - \frac{85}{4}$ |
| 13 | $[13, 13, -w^{3} + 4w + 1]$ | $-1$ |
| 25 | $[25, 5, -w^{2} - 2w + 1]$ | $-\frac{3}{8}e^{12} + \frac{15}{2}e^{10} - \frac{451}{8}e^{8} + \frac{783}{4}e^{6} - \frac{2477}{8}e^{4} + \frac{765}{4}e^{2} - \frac{257}{8}$ |
| 25 | $[25, 5, w^{2} - 2w - 1]$ | $\phantom{-}\frac{1}{4}e^{12} - 5e^{10} + \frac{149}{4}e^{8} - \frac{251}{2}e^{6} + \frac{727}{4}e^{4} - \frac{169}{2}e^{2} + \frac{31}{4}$ |
| 37 | $[37, 37, w^{3} - 3w - 1]$ | $-\frac{3}{8}e^{12} + \frac{33}{4}e^{10} - \frac{549}{8}e^{8} + \frac{1065}{4}e^{6} - \frac{3845}{8}e^{4} + \frac{717}{2}e^{2} - \frac{627}{8}$ |
| 37 | $[37, 37, w^{3} - 3w + 1]$ | $-\frac{1}{4}e^{10} + \frac{15}{4}e^{8} - \frac{37}{2}e^{6} + 32e^{4} - \frac{43}{4}e^{2} - \frac{9}{4}$ |
| 59 | $[59, 59, w^{2} - w - 5]$ | $\phantom{-}\frac{1}{12}e^{13} - \frac{5}{3}e^{11} + \frac{143}{12}e^{9} - \frac{205}{6}e^{7} + \frac{271}{12}e^{5} + \frac{73}{2}e^{3} - \frac{271}{12}e$ |
| 59 | $[59, 59, -w^{2} - w + 5]$ | $-\frac{7}{24}e^{13} + \frac{73}{12}e^{11} - \frac{1145}{24}e^{9} + \frac{2065}{12}e^{7} - \frac{6649}{24}e^{5} + \frac{317}{2}e^{3} - \frac{239}{24}e$ |
| 61 | $[61, 61, -w^{3} + w^{2} + 4w - 7]$ | $-\frac{1}{8}e^{12} + \frac{5}{2}e^{10} - \frac{149}{8}e^{8} + \frac{249}{4}e^{6} - \frac{703}{8}e^{4} + \frac{167}{4}e^{2} - \frac{47}{8}$ |
| 61 | $[61, 61, w^{3} - 3w^{2} - 6w + 11]$ | $-\frac{1}{4}e^{10} + \frac{15}{4}e^{8} - \frac{39}{2}e^{6} + 44e^{4} - \frac{203}{4}e^{2} + \frac{107}{4}$ |
| 73 | $[73, 73, 2w^{2} - w - 5]$ | $-\frac{1}{4}e^{12} + 5e^{10} - \frac{149}{4}e^{8} + \frac{253}{2}e^{6} - \frac{775}{4}e^{4} + \frac{247}{2}e^{2} - \frac{119}{4}$ |
| 73 | $[73, 73, 2w - 1]$ | $\phantom{-}\frac{1}{2}e^{12} - 10e^{10} + \frac{151}{2}e^{8} - 266e^{6} + \frac{873}{2}e^{4} - 291e^{2} + \frac{117}{2}$ |
| 73 | $[73, 73, -2w - 1]$ | $-\frac{3}{8}e^{12} + \frac{33}{4}e^{10} - \frac{553}{8}e^{8} + \frac{1089}{4}e^{6} - \frac{4021}{8}e^{4} + \frac{767}{2}e^{2} - \frac{663}{8}$ |
| 73 | $[73, 73, 2w^{2} + w - 5]$ | $-e^{6} + 10e^{4} - 28e^{2} + 21$ |
| 83 | $[83, 83, 2w^{3} + w^{2} - 9w - 7]$ | $\phantom{-}\frac{1}{8}e^{13} - \frac{9}{4}e^{11} + \frac{115}{8}e^{9} - \frac{147}{4}e^{7} + \frac{175}{8}e^{5} + 33e^{3} - \frac{155}{8}e$ |
| 83 | $[83, 83, -2w^{3} + w^{2} + 7w + 1]$ | $-\frac{1}{24}e^{13} + \frac{5}{6}e^{11} - \frac{149}{24}e^{9} + \frac{253}{12}e^{7} - \frac{823}{24}e^{5} + \frac{145}{4}e^{3} - \frac{695}{24}e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13,13,-w^{3} + 4w + 1]$ | $1$ |