Base field 4.4.18432.1
Generator \(w\), with minimal polynomial \(x^{4} - 12x^{2} + 18\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[9, 3, w - 3]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $18$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} + x^{7} - 33x^{6} + 8x^{5} + 305x^{4} - 353x^{3} - 373x^{2} + 354x + 211\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -\frac{1}{3}w^{3} - \frac{1}{3}w^{2} + 3w + 4]$ | $\phantom{-}\frac{47}{4705}e^{7} + \frac{42}{941}e^{6} - \frac{1103}{4705}e^{5} - \frac{822}{941}e^{4} + \frac{1678}{941}e^{3} + \frac{19814}{4705}e^{2} - \frac{3998}{941}e - \frac{16911}{4705}$ |
| 7 | $[7, 7, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 3w - 3]$ | $\phantom{-}e$ |
| 7 | $[7, 7, -\frac{1}{3}w^{2} + w + 1]$ | $\phantom{-}\frac{103}{10351}e^{7} + \frac{359}{51755}e^{6} - \frac{13908}{51755}e^{5} + \frac{4187}{10351}e^{4} + \frac{17826}{10351}e^{3} - \frac{75304}{10351}e^{2} + \frac{225453}{51755}e + \frac{294969}{51755}$ |
| 7 | $[7, 7, \frac{1}{3}w^{2} + w - 1]$ | $-\frac{8}{10351}e^{7} + \frac{2384}{51755}e^{6} + \frac{5904}{51755}e^{5} - \frac{14897}{10351}e^{4} - \frac{12640}{10351}e^{3} + \frac{127046}{10351}e^{2} - \frac{222722}{51755}e - \frac{544882}{51755}$ |
| 7 | $[7, 7, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - 3w + 3]$ | $\phantom{-}\frac{42}{51755}e^{7} - \frac{433}{51755}e^{6} - \frac{4129}{51755}e^{5} + \frac{1668}{10351}e^{4} + \frac{13272}{10351}e^{3} - \frac{40756}{51755}e^{2} - \frac{274376}{51755}e + \frac{12137}{51755}$ |
| 9 | $[9, 3, w - 3]$ | $-1$ |
| 41 | $[41, 41, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - 3w + 1]$ | $-\frac{3183}{51755}e^{7} - \frac{4892}{51755}e^{6} + \frac{19849}{10351}e^{5} + \frac{5195}{10351}e^{4} - \frac{167397}{10351}e^{3} + \frac{700599}{51755}e^{2} + \frac{459981}{51755}e - \frac{64630}{10351}$ |
| 41 | $[41, 41, -\frac{1}{3}w^{2} + w + 3]$ | $\phantom{-}\frac{7321}{51755}e^{7} + \frac{14972}{51755}e^{6} - \frac{215236}{51755}e^{5} - \frac{28654}{10351}e^{4} + \frac{357097}{10351}e^{3} - \frac{950243}{51755}e^{2} - \frac{1391271}{51755}e + \frac{175768}{51755}$ |
| 41 | $[41, 41, \frac{1}{3}w^{2} + w - 3]$ | $\phantom{-}\frac{953}{51755}e^{7} + \frac{3237}{51755}e^{6} - \frac{28872}{51755}e^{5} - \frac{15386}{10351}e^{4} + \frac{52724}{10351}e^{3} + \frac{477541}{51755}e^{2} - \frac{698781}{51755}e - \frac{476039}{51755}$ |
| 41 | $[41, 41, -\frac{1}{3}w^{3} - \frac{1}{3}w^{2} + 3w + 1]$ | $-\frac{1472}{51755}e^{7} + \frac{2853}{51755}e^{6} + \frac{59932}{51755}e^{5} - \frac{24449}{10351}e^{4} - \frac{113218}{10351}e^{3} + \frac{1297781}{51755}e^{2} + \frac{90841}{51755}e - \frac{575216}{51755}$ |
| 47 | $[47, 47, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 4w - 1]$ | $\phantom{-}\frac{2072}{51755}e^{7} + \frac{2791}{51755}e^{6} - \frac{65684}{51755}e^{5} - \frac{520}{10351}e^{4} + \frac{116500}{10351}e^{3} - \frac{526986}{51755}e^{2} - \frac{548828}{51755}e + \frac{63957}{51755}$ |
| 47 | $[47, 47, -\frac{1}{3}w^{3} + \frac{1}{3}w^{2} + 2w - 3]$ | $-\frac{4338}{51755}e^{7} - \frac{8511}{51755}e^{6} + \frac{124809}{51755}e^{5} + \frac{11080}{10351}e^{4} - \frac{201145}{10351}e^{3} + \frac{941554}{51755}e^{2} + \frac{469793}{51755}e - \frac{599987}{51755}$ |
| 47 | $[47, 47, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 2w - 3]$ | $\phantom{-}\frac{4338}{51755}e^{7} + \frac{8511}{51755}e^{6} - \frac{124809}{51755}e^{5} - \frac{11080}{10351}e^{4} + \frac{201145}{10351}e^{3} - \frac{941554}{51755}e^{2} - \frac{469793}{51755}e + \frac{651742}{51755}$ |
| 47 | $[47, 47, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - 4w + 1]$ | $-\frac{2072}{51755}e^{7} - \frac{2791}{51755}e^{6} + \frac{65684}{51755}e^{5} + \frac{520}{10351}e^{4} - \frac{116500}{10351}e^{3} + \frac{526986}{51755}e^{2} + \frac{548828}{51755}e - \frac{12202}{51755}$ |
| 89 | $[89, 89, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 3w - 3]$ | $-\frac{6416}{51755}e^{7} - \frac{3431}{10351}e^{6} + \frac{185168}{51755}e^{5} + \frac{51287}{10351}e^{4} - \frac{319541}{10351}e^{3} - \frac{263127}{51755}e^{2} + \frac{481281}{10351}e + \frac{675516}{51755}$ |
| 89 | $[89, 89, \frac{2}{3}w^{2} + w - 5]$ | $-\frac{1050}{10351}e^{7} - \frac{18332}{51755}e^{6} + \frac{133138}{51755}e^{5} + \frac{60626}{10351}e^{4} - \frac{199509}{10351}e^{3} - \frac{171465}{10351}e^{2} + \frac{1028886}{51755}e + \frac{1546771}{51755}$ |
| 89 | $[89, 89, \frac{2}{3}w^{2} - w - 5]$ | $\phantom{-}\frac{4068}{51755}e^{7} + \frac{3901}{51755}e^{6} - \frac{136712}{51755}e^{5} + \frac{4814}{10351}e^{4} + \frac{250388}{10351}e^{3} - \frac{1234069}{51755}e^{2} - \frac{1005348}{51755}e + \frac{570761}{51755}$ |
| 89 | $[89, 89, \frac{1}{3}w^{3} + \frac{2}{3}w^{2} - 3w - 3]$ | $\phantom{-}\frac{589}{10351}e^{7} + \frac{10796}{51755}e^{6} - \frac{72397}{51755}e^{5} - \frac{35349}{10351}e^{4} + \frac{102540}{10351}e^{3} + \frac{78587}{10351}e^{2} - \frac{450933}{51755}e + \frac{19751}{51755}$ |
| 97 | $[97, 97, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 6w + 5]$ | $\phantom{-}\frac{502}{4705}e^{7} + \frac{1322}{4705}e^{6} - \frac{13683}{4705}e^{5} - \frac{3454}{941}e^{4} + \frac{21246}{941}e^{3} - \frac{6101}{4705}e^{2} - \frac{77806}{4705}e - \frac{55591}{4705}$ |
| 97 | $[97, 97, -w^{3} - \frac{5}{3}w^{2} + 10w + 15]$ | $\phantom{-}\frac{1636}{51755}e^{7} + \frac{2443}{10351}e^{6} - \frac{28243}{51755}e^{5} - \frac{56282}{10351}e^{4} + \frac{40830}{10351}e^{3} + \frac{1801177}{51755}e^{2} - \frac{247920}{10351}e - \frac{1571311}{51755}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $9$ | $[9, 3, w - 3]$ | $1$ |