Base field 4.4.13824.1
Generator \(w\), with minimal polynomial \(x^4 - 6 x^2 + 6\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, w^3 - 4 w + 1]$ |
| 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:
| \(x^{14} - 23 x^{12} + 209 x^{10} - 953 x^8 + 2293 x^6 - 2811 x^4 + 1529 x^2 - 261\) |
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 | $[11, 11, -w^2 + w + 1]$ | $\phantom{-}\frac{1}{6} e^{13} - \frac{43}{12} e^{11} + \frac{355}{12} e^9 - \frac{701}{6} e^7 + \frac{1345}{6} e^5 - \frac{759}{4} e^3 + \frac{619}{12} e$ |
| 11 | $[11, 11, -w^2 - w + 1]$ | $-\frac{1}{12} e^{13} + \frac{5}{3} e^{11} - \frac{155}{12} e^9 + \frac{295}{6} e^7 - \frac{1147}{12} e^5 + \frac{173}{2} e^3 - \frac{305}{12} e$ |
| 13 | $[13, 13, w^3 - 4 w + 1]$ | $-1$ |
| 13 | $[13, 13, -w^3 + 4 w + 1]$ | $-\frac{1}{4} e^{12} + 5 e^{10} - \frac{151}{4} e^8 + \frac{265}{2} e^6 - \frac{855}{4} e^4 + \frac{271}{2} e^2 - \frac{85}{4}$ |
| 25 | $[25, 5, -w^2 - 2 w + 1]$ | $\phantom{-}\frac{1}{4} e^{12} - 5 e^{10} + \frac{149}{4} e^8 - \frac{251}{2} e^6 + \frac{727}{4} e^4 - \frac{169}{2} e^2 + \frac{31}{4}$ |
| 25 | $[25, 5, w^2 - 2 w - 1]$ | $-\frac{3}{8} e^{12} + \frac{15}{2} e^{10} - \frac{451}{8} e^8 + \frac{783}{4} e^6 - \frac{2477}{8} e^4 + \frac{765}{4} e^2 - \frac{257}{8}$ |
| 37 | $[37, 37, w^3 - 3 w - 1]$ | $-\frac{1}{4} e^{10} + \frac{15}{4} e^8 - \frac{37}{2} e^6 + 32 e^4 - \frac{43}{4} e^2 - \frac{9}{4}$ |
| 37 | $[37, 37, w^3 - 3 w + 1]$ | $-\frac{3}{8} e^{12} + \frac{33}{4} e^{10} - \frac{549}{8} e^8 + \frac{1065}{4} e^6 - \frac{3845}{8} e^4 + \frac{717}{2} e^2 - \frac{627}{8}$ |
| 59 | $[59, 59, w^2 - w - 5]$ | $-\frac{7}{24} e^{13} + \frac{73}{12} e^{11} - \frac{1145}{24} e^9 + \frac{2065}{12} e^7 - \frac{6649}{24} e^5 + \frac{317}{2} e^3 - \frac{239}{24} e$ |
| 59 | $[59, 59, -w^2 - w + 5]$ | $\phantom{-}\frac{1}{12} e^{13} - \frac{5}{3} e^{11} + \frac{143}{12} e^9 - \frac{205}{6} e^7 + \frac{271}{12} e^5 + \frac{73}{2} e^3 - \frac{271}{12} e$ |
| 61 | $[61, 61, -w^3 + w^2 + 4 w - 7]$ | $-\frac{1}{4} e^{10} + \frac{15}{4} e^8 - \frac{39}{2} e^6 + 44 e^4 - \frac{203}{4} e^2 + \frac{107}{4}$ |
| 61 | $[61, 61, w^3 - 3 w^2 - 6 w + 11]$ | $-\frac{1}{8} e^{12} + \frac{5}{2} e^{10} - \frac{149}{8} e^8 + \frac{249}{4} e^6 - \frac{703}{8} e^4 + \frac{167}{4} e^2 - \frac{47}{8}$ |
| 73 | $[73, 73, 2 w^2 - w - 5]$ | $-e^6 + 10 e^4 - 28 e^2 + 21$ |
| 73 | $[73, 73, 2 w - 1]$ | $-\frac{3}{8} e^{12} + \frac{33}{4} e^{10} - \frac{553}{8} e^8 + \frac{1089}{4} e^6 - \frac{4021}{8} e^4 + \frac{767}{2} e^2 - \frac{663}{8}$ |
| 73 | $[73, 73, -2 w - 1]$ | $\phantom{-}\frac{1}{2} e^{12} - 10 e^{10} + \frac{151}{2} e^8 - 266 e^6 + \frac{873}{2} e^4 - 291 e^2 + \frac{117}{2}$ |
| 73 | $[73, 73, 2 w^2 + w - 5]$ | $-\frac{1}{4} e^{12} + 5 e^{10} - \frac{149}{4} e^8 + \frac{253}{2} e^6 - \frac{775}{4} e^4 + \frac{247}{2} e^2 - \frac{119}{4}$ |
| 83 | $[83, 83, 2 w^3 + w^2 - 9 w - 7]$ | $-\frac{1}{24} e^{13} + \frac{5}{6} e^{11} - \frac{149}{24} e^9 + \frac{253}{12} e^7 - \frac{823}{24} e^5 + \frac{145}{4} e^3 - \frac{695}{24} e$ |
| 83 | $[83, 83, -2 w^3 + w^2 + 7 w + 1]$ | $\phantom{-}\frac{1}{8} e^{13} - \frac{9}{4} e^{11} + \frac{115}{8} e^9 - \frac{147}{4} e^7 + \frac{175}{8} e^5 + 33 e^3 - \frac{155}{8} e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13,13,w^3-4 w+1]$ | $1$ |