Base field 6.6.1081856.1
Generator \(w\), with minimal polynomial \(x^{6} - 6x^{4} - 2x^{3} + 7x^{2} + 2x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2, 2]$ |
| Level: | $[41, 41, -w^{5} + w^{4} + 5w^{3} - 2w^{2} - 6w - 1]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $24$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} + 5x^{7} - 16x^{6} - 77x^{5} + 50x^{4} + 130x^{3} - 83x^{2} - 23x + 14\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 7 | $[7, 7, w^{4} - w^{3} - 4w^{2} + w + 1]$ | $\phantom{-}e$ |
| 8 | $[8, 2, w^{5} - w^{4} - 5w^{3} + 2w^{2} + 5w]$ | $\phantom{-}\frac{41838}{211493}e^{7} + \frac{237066}{211493}e^{6} - \frac{511394}{211493}e^{5} - \frac{3579730}{211493}e^{4} - \frac{309869}{211493}e^{3} + \frac{5534872}{211493}e^{2} + \frac{168778}{211493}e - \frac{1436832}{211493}$ |
| 17 | $[17, 17, -w^{2} + w + 2]$ | $-\frac{61733}{211493}e^{7} - \frac{353679}{211493}e^{6} + \frac{744707}{211493}e^{5} + \frac{5360697}{211493}e^{4} + \frac{596683}{211493}e^{3} - \frac{8435910}{211493}e^{2} - \frac{608005}{211493}e + \frac{1601494}{211493}$ |
| 23 | $[23, 23, -w^{4} + 2w^{3} + 3w^{2} - 4w - 1]$ | $-\frac{25940}{211493}e^{7} - \frac{154384}{211493}e^{6} + \frac{263011}{211493}e^{5} + \frac{2206499}{211493}e^{4} + \frac{875453}{211493}e^{3} - \frac{1851694}{211493}e^{2} - \frac{255790}{211493}e - \frac{877586}{211493}$ |
| 25 | $[25, 5, -w^{3} + w^{2} + 4w]$ | $-\frac{23636}{211493}e^{7} - \frac{150064}{211493}e^{6} + \frac{212647}{211493}e^{5} + \frac{2292225}{211493}e^{4} + \frac{1304365}{211493}e^{3} - \frac{4161482}{211493}e^{2} - \frac{1582908}{211493}e + \frac{1366102}{211493}$ |
| 31 | $[31, 31, -w^{3} + 4w + 1]$ | $-\frac{68968}{211493}e^{7} - \frac{340808}{211493}e^{6} + \frac{1111048}{211493}e^{5} + \frac{5222399}{211493}e^{4} - \frac{3404126}{211493}e^{3} - \frac{8426737}{211493}e^{2} + \frac{3797145}{211493}e + \frac{638512}{211493}$ |
| 31 | $[31, 31, w^{5} - 6w^{3} - w^{2} + 5w]$ | $\phantom{-}\frac{78692}{211493}e^{7} + \frac{464787}{211493}e^{6} - \frac{834531}{211493}e^{5} - \frac{6808826}{211493}e^{4} - \frac{2334782}{211493}e^{3} + \frac{8061842}{211493}e^{2} + \frac{1744075}{211493}e - \frac{1153290}{211493}$ |
| 41 | $[41, 41, -w^{5} + w^{4} + 5w^{3} - 2w^{2} - 6w - 1]$ | $\phantom{-}1$ |
| 47 | $[47, 47, -w^{5} + w^{4} + 6w^{3} - 4w^{2} - 8w + 2]$ | $\phantom{-}\frac{149377}{211493}e^{7} + \frac{835251}{211493}e^{6} - \frac{1854233}{211493}e^{5} - \frac{12396017}{211493}e^{4} - \frac{454240}{211493}e^{3} + \frac{16185664}{211493}e^{2} - \frac{2280000}{211493}e - \frac{1175010}{211493}$ |
| 49 | $[49, 7, -w^{5} - w^{4} + 7w^{3} + 6w^{2} - 8w - 3]$ | $-\frac{247543}{211493}e^{7} - \frac{1389425}{211493}e^{6} + \frac{3094626}{211493}e^{5} + \frac{20863600}{211493}e^{4} + \frac{465017}{211493}e^{3} - \frac{30540683}{211493}e^{2} + \frac{3920850}{211493}e + \frac{5337314}{211493}$ |
| 71 | $[71, 71, -w^{4} + 5w^{2} + w - 3]$ | $-\frac{18303}{211493}e^{7} - \frac{113628}{211493}e^{6} + \frac{119203}{211493}e^{5} + \frac{1433371}{211493}e^{4} + \frac{1558398}{211493}e^{3} + \frac{945970}{211493}e^{2} - \frac{961148}{211493}e - \frac{488064}{211493}$ |
| 71 | $[71, 71, w^{4} - 5w^{2} - 2w + 4]$ | $-\frac{141961}{211493}e^{7} - \frac{821346}{211493}e^{6} + \frac{1612814}{211493}e^{5} + \frac{12182870}{211493}e^{4} + \frac{2786519}{211493}e^{3} - \frac{16297349}{211493}e^{2} - \frac{2665795}{211493}e + \frac{2527950}{211493}$ |
| 73 | $[73, 73, -2w^{5} + w^{4} + 10w^{3} - 9w - 1]$ | $\phantom{-}\frac{1152}{211493}e^{7} + \frac{2160}{211493}e^{6} - \frac{25182}{211493}e^{5} + \frac{42863}{211493}e^{4} + \frac{214456}{211493}e^{3} - \frac{943401}{211493}e^{2} - \frac{240573}{211493}e - \frac{147114}{211493}$ |
| 73 | $[73, 73, -w^{5} + 6w^{3} + 2w^{2} - 5w - 1]$ | $-\frac{33577}{211493}e^{7} - \frac{195140}{211493}e^{6} + \frac{406819}{211493}e^{5} + \frac{2979627}{211493}e^{4} + \frac{192508}{211493}e^{3} - \frac{4860851}{211493}e^{2} + \frac{872554}{211493}e + \frac{1850}{211493}$ |
| 79 | $[79, 79, w^{5} - w^{4} - 6w^{3} + 4w^{2} + 7w - 3]$ | $-\frac{42054}{211493}e^{7} - \frac{237471}{211493}e^{6} + \frac{489679}{211493}e^{5} + \frac{3479165}{211493}e^{4} + \frac{798391}{211493}e^{3} - \frac{3864315}{211493}e^{2} - \frac{2172509}{211493}e - \frac{306838}{211493}$ |
| 89 | $[89, 89, w^{5} - 7w^{3} - w^{2} + 9w]$ | $\phantom{-}\frac{97137}{211493}e^{7} + \frac{525808}{211493}e^{6} - \frac{1293905}{211493}e^{5} - \frac{7798139}{211493}e^{4} + \frac{912409}{211493}e^{3} + \frac{9697222}{211493}e^{2} - \frac{1830774}{211493}e - \frac{55490}{211493}$ |
| 97 | $[97, 97, 2w^{5} - 2w^{4} - 10w^{3} + 5w^{2} + 10w - 1]$ | $\phantom{-}\frac{107491}{211493}e^{7} + \frac{598095}{211493}e^{6} - \frac{1394663}{211493}e^{5} - \frac{9073627}{211493}e^{4} + \frac{551670}{211493}e^{3} + \frac{14100425}{211493}e^{2} - \frac{2306571}{211493}e - \frac{1582612}{211493}$ |
| 103 | $[103, 103, w^{5} - w^{4} - 4w^{3} + w^{2} + 3w + 2]$ | $-\frac{85364}{211493}e^{7} - \frac{477297}{211493}e^{6} + \frac{1033250}{211493}e^{5} + \frac{6992376}{211493}e^{4} + \frac{740236}{211493}e^{3} - \frac{8228979}{211493}e^{2} - \frac{800179}{211493}e + \frac{683494}{211493}$ |
| 103 | $[103, 103, -2w^{5} + w^{4} + 11w^{3} - w^{2} - 11w - 1]$ | $\phantom{-}\frac{56984}{211493}e^{7} + \frac{318338}{211493}e^{6} - \frac{796212}{211493}e^{5} - \frac{5177747}{211493}e^{4} + \frac{867698}{211493}e^{3} + \frac{12362985}{211493}e^{2} + \frac{335741}{211493}e - \frac{4871302}{211493}$ |
| 103 | $[103, 103, -w^{4} + 2w^{3} + 4w^{2} - 5w - 2]$ | $-\frac{62269}{211493}e^{7} - \frac{354684}{211493}e^{6} + \frac{729987}{211493}e^{5} + \frac{5236476}{211493}e^{4} + \frac{931637}{211493}e^{3} - \frac{6538546}{211493}e^{2} - \frac{676722}{211493}e - \frac{312804}{211493}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $41$ | $[41, 41, -w^{5} + w^{4} + 5w^{3} - 2w^{2} - 6w - 1]$ | $-1$ |