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: | $[47, 47, -w^{5} + w^{4} + 6w^{3} - 4w^{2} - 8w + 2]$ |
| Dimension: | $6$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $25$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{6} - 2x^{5} - 21x^{4} + 44x^{3} + 83x^{2} - 210x + 97\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 7 | $[7, 7, w^{4} - w^{3} - 4w^{2} + w + 1]$ | $\phantom{-}e$ |
| 8 | $[8, 2, w^{5} - w^{4} - 5w^{3} + 2w^{2} + 5w]$ | $-\frac{47}{340}e^{5} + \frac{7}{68}e^{4} + \frac{501}{170}e^{3} - \frac{387}{170}e^{2} - \frac{4359}{340}e + \frac{3407}{340}$ |
| 17 | $[17, 17, -w^{2} + w + 2]$ | $-\frac{13}{340}e^{5} + \frac{7}{68}e^{4} + \frac{72}{85}e^{3} - \frac{353}{170}e^{2} - \frac{1231}{340}e + \frac{1503}{340}$ |
| 23 | $[23, 23, -w^{4} + 2w^{3} + 3w^{2} - 4w - 1]$ | $\phantom{-}\frac{3}{68}e^{5} + \frac{5}{68}e^{4} - \frac{14}{17}e^{3} - \frac{31}{34}e^{2} + \frac{101}{68}e - \frac{67}{68}$ |
| 25 | $[25, 5, -w^{3} + w^{2} + 4w]$ | $-\frac{11}{85}e^{5} + \frac{2}{17}e^{4} + \frac{211}{85}e^{3} - \frac{469}{170}e^{2} - \frac{767}{85}e + \frac{2027}{170}$ |
| 31 | $[31, 31, -w^{3} + 4w + 1]$ | $-\frac{21}{340}e^{5} + \frac{5}{34}e^{4} + \frac{213}{170}e^{3} - \frac{223}{85}e^{2} - \frac{1897}{340}e + \frac{1093}{170}$ |
| 31 | $[31, 31, w^{5} - 6w^{3} - w^{2} + 5w]$ | $\phantom{-}\frac{3}{10}e^{5} - \frac{1}{4}e^{4} - \frac{63}{10}e^{3} + \frac{28}{5}e^{2} + \frac{133}{5}e - \frac{571}{20}$ |
| 41 | $[41, 41, -w^{5} + w^{4} + 5w^{3} - 2w^{2} - 6w - 1]$ | $\phantom{-}\frac{1}{20}e^{5} - \frac{1}{4}e^{4} - \frac{13}{10}e^{3} + \frac{23}{5}e^{2} + \frac{157}{20}e - \frac{331}{20}$ |
| 47 | $[47, 47, -w^{5} + w^{4} + 6w^{3} - 4w^{2} - 8w + 2]$ | $\phantom{-}1$ |
| 49 | $[49, 7, -w^{5} - w^{4} + 7w^{3} + 6w^{2} - 8w - 3]$ | $\phantom{-}\frac{39}{170}e^{5} - \frac{25}{68}e^{4} - \frac{779}{170}e^{3} + \frac{634}{85}e^{2} + \frac{1549}{85}e - \frac{9783}{340}$ |
| 71 | $[71, 71, -w^{4} + 5w^{2} + w - 3]$ | $-\frac{93}{340}e^{5} + \frac{3}{68}e^{4} + \frac{502}{85}e^{3} - \frac{433}{170}e^{2} - \frac{9251}{340}e + \frac{7483}{340}$ |
| 71 | $[71, 71, w^{4} - 5w^{2} - 2w + 4]$ | $\phantom{-}\frac{23}{340}e^{5} + \frac{1}{34}e^{4} - \frac{147}{85}e^{3} - \frac{147}{170}e^{2} + \frac{3381}{340}e - \frac{297}{85}$ |
| 73 | $[73, 73, -2w^{5} + w^{4} + 10w^{3} - 9w - 1]$ | $\phantom{-}\frac{8}{85}e^{5} - \frac{29}{68}e^{4} - \frac{223}{85}e^{3} + \frac{1307}{170}e^{2} + \frac{1346}{85}e - \frac{7917}{340}$ |
| 73 | $[73, 73, -w^{5} + 6w^{3} + 2w^{2} - 5w - 1]$ | $-\frac{47}{170}e^{5} - \frac{5}{17}e^{4} + \frac{501}{85}e^{3} + \frac{293}{85}e^{2} - \frac{4699}{170}e + \frac{556}{85}$ |
| 79 | $[79, 79, w^{5} - w^{4} - 6w^{3} + 4w^{2} + 7w - 3]$ | $\phantom{-}\frac{9}{20}e^{5} - \frac{1}{2}e^{4} - \frac{97}{10}e^{3} + \frac{47}{5}e^{2} + \frac{873}{20}e - \frac{377}{10}$ |
| 89 | $[89, 89, w^{5} - 7w^{3} - w^{2} + 9w]$ | $\phantom{-}\frac{73}{340}e^{5} - \frac{1}{17}e^{4} - \frac{352}{85}e^{3} + \frac{79}{85}e^{2} + \frac{5291}{340}e - \frac{562}{85}$ |
| 97 | $[97, 97, 2w^{5} - 2w^{4} - 10w^{3} + 5w^{2} + 10w - 1]$ | $-\frac{9}{170}e^{5} - \frac{3}{34}e^{4} + \frac{67}{85}e^{3} + \frac{246}{85}e^{2} + \frac{207}{170}e - \frac{2961}{170}$ |
| 103 | $[103, 103, w^{5} - w^{4} - 4w^{3} + w^{2} + 3w + 2]$ | $\phantom{-}\frac{59}{340}e^{5} + \frac{7}{34}e^{4} - \frac{281}{85}e^{3} - \frac{451}{170}e^{2} + \frac{4253}{340}e - \frac{141}{85}$ |
| 103 | $[103, 103, -2w^{5} + w^{4} + 11w^{3} - w^{2} - 11w - 1]$ | $-\frac{87}{170}e^{5} + \frac{27}{68}e^{4} + \frac{1947}{170}e^{3} - \frac{767}{85}e^{2} - \frac{4822}{85}e + \frac{17129}{340}$ |
| 103 | $[103, 103, -w^{4} + 2w^{3} + 4w^{2} - 5w - 2]$ | $\phantom{-}\frac{61}{85}e^{5} - \frac{49}{68}e^{4} - \frac{2657}{170}e^{3} + \frac{2709}{170}e^{2} + \frac{12069}{170}e - \frac{23849}{340}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $47$ | $[47, 47, -w^{5} + w^{4} + 6w^{3} - 4w^{2} - 8w + 2]$ | $-1$ |