Base field 3.3.1129.1
Generator \(w\), with minimal polynomial \(x^{3} - 7x - 3\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[19, 19, -w^{2} - w + 4]$ |
| Dimension: | $16$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $36$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{16} - 38x^{14} + 570x^{12} - 4330x^{10} + 17802x^{8} - 39556x^{6} + 46044x^{4} - 26272x^{2} + 5776\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $-\frac{2383}{83596}e^{14} + \frac{88451}{83596}e^{12} - \frac{638999}{41798}e^{10} + \frac{2279976}{20899}e^{8} - \frac{8393043}{20899}e^{6} + \frac{30300539}{41798}e^{4} - \frac{11818090}{20899}e^{2} + \frac{3177061}{20899}$ |
| 3 | $[3, 3, w + 1]$ | $\phantom{-}e$ |
| 3 | $[3, 3, w + 2]$ | $\phantom{-}\frac{1469529}{12706592}e^{15} - \frac{717255}{167192}e^{13} + \frac{20739811}{334384}e^{11} - \frac{2822284403}{6353296}e^{9} + \frac{10497720375}{6353296}e^{7} - \frac{4856516049}{1588324}e^{5} + \frac{7978362867}{3176648}e^{3} - \frac{1153443205}{1588324}e$ |
| 8 | $[8, 2, 2]$ | $-\frac{2383}{83596}e^{14} + \frac{88451}{83596}e^{12} - \frac{638999}{41798}e^{10} + \frac{2279976}{20899}e^{8} - \frac{8393043}{20899}e^{6} + \frac{30300539}{41798}e^{4} - \frac{11818090}{20899}e^{2} + \frac{3156162}{20899}$ |
| 11 | $[11, 11, -w^{2} + 5]$ | $\phantom{-}\frac{56599}{12706592}e^{15} - \frac{29191}{167192}e^{13} + \frac{908053}{334384}e^{11} - \frac{136320613}{6353296}e^{9} + \frac{580760993}{6353296}e^{7} - \frac{326035241}{1588324}e^{5} + \frac{697044285}{3176648}e^{3} - \frac{131840835}{1588324}e$ |
| 13 | $[13, 13, w^{2} - w - 7]$ | $\phantom{-}\frac{2077463}{12706592}e^{15} - \frac{1012613}{167192}e^{13} + \frac{29228309}{334384}e^{11} - \frac{3967938869}{6353296}e^{9} + \frac{14709940665}{6353296}e^{7} - \frac{6771366171}{1588324}e^{5} + \frac{11040937269}{3176648}e^{3} - \frac{1580258035}{1588324}e$ |
| 17 | $[17, 17, -w^{2} + w + 4]$ | $-\frac{255633}{3176648}e^{15} + \frac{62364}{20899}e^{13} - \frac{3605457}{83596}e^{11} + \frac{490586225}{1588324}e^{9} - \frac{1825942167}{1588324}e^{7} + \frac{847142287}{397081}e^{5} - \frac{1404114533}{794162}e^{3} + \frac{206647228}{397081}e$ |
| 19 | $[19, 19, -w^{2} - w + 4]$ | $\phantom{-}1$ |
| 29 | $[29, 29, 2w^{2} - 2w - 11]$ | $\phantom{-}\frac{2362103}{12706592}e^{15} - \frac{1149607}{167192}e^{13} + \frac{33111549}{334384}e^{11} - \frac{4481013709}{6353296}e^{9} + \frac{16529895441}{6353296}e^{7} - \frac{7542582645}{1588324}e^{5} + \frac{12088041725}{3176648}e^{3} - \frac{1681021739}{1588324}e$ |
| 31 | $[31, 31, w^{2} - 2w - 4]$ | $\phantom{-}\frac{689689}{6353296}e^{15} - \frac{336335}{83596}e^{13} + \frac{9714711}{167192}e^{11} - \frac{1320222079}{3176648}e^{9} + \frac{4903248391}{3176648}e^{7} - \frac{2265670901}{794162}e^{5} + \frac{3729105827}{1588324}e^{3} - \frac{546340169}{794162}e$ |
| 37 | $[37, 37, -2w^{2} + 3w + 7]$ | $\phantom{-}\frac{35345}{3176648}e^{15} - \frac{8829}{20899}e^{13} + \frac{528511}{83596}e^{11} - \frac{75786853}{1588324}e^{9} + \frac{305806907}{1588324}e^{7} - \frac{160640415}{397081}e^{5} + \frac{315649835}{794162}e^{3} - \frac{54309623}{397081}e$ |
| 41 | $[41, 41, w^{2} - 2]$ | $\phantom{-}\frac{307687}{12706592}e^{15} - \frac{150791}{167192}e^{13} + \frac{4388957}{334384}e^{11} - \frac{603875821}{6353296}e^{9} + \frac{2290253169}{6353296}e^{7} - \frac{1098112709}{1588324}e^{5} + \frac{1917865165}{3176648}e^{3} - \frac{290975915}{1588324}e$ |
| 59 | $[59, 59, 2w^{2} - 13]$ | $-\frac{11533}{83596}e^{14} + \frac{213411}{41798}e^{12} - \frac{3075671}{41798}e^{10} + \frac{21928215}{41798}e^{8} - \frac{80984315}{41798}e^{6} + \frac{73984492}{20899}e^{4} - \frac{59278825}{20899}e^{2} + \frac{16394892}{20899}$ |
| 61 | $[61, 61, w^{2} + w - 10]$ | $-\frac{127283}{334384}e^{14} + \frac{1179425}{83596}e^{12} - \frac{34059207}{167192}e^{10} + \frac{243387753}{167192}e^{8} - \frac{901513477}{167192}e^{6} + \frac{413535145}{41798}e^{4} - \frac{667435953}{83596}e^{2} + \frac{93835003}{41798}$ |
| 67 | $[67, 67, 2w^{2} - w - 11]$ | $-\frac{3427159}{6353296}e^{15} + \frac{417776}{20899}e^{13} - \frac{48259645}{167192}e^{11} + \frac{6556211577}{3176648}e^{9} - \frac{24328731965}{3176648}e^{7} + \frac{5605936520}{397081}e^{5} - \frac{18286799729}{1588324}e^{3} + \frac{2615630243}{794162}e$ |
| 73 | $[73, 73, -w^{2} - 1]$ | $-\frac{6767}{83596}e^{14} + \frac{62480}{20899}e^{12} - \frac{1797673}{41798}e^{10} + \frac{12808311}{41798}e^{8} - \frac{47412143}{41798}e^{6} + \frac{43683953}{20899}e^{4} - \frac{35684443}{20899}e^{2} + \frac{10416952}{20899}$ |
| 83 | $[83, 83, w^{2} - w - 10]$ | $-\frac{132927}{334384}e^{14} + \frac{1231561}{83596}e^{12} - \frac{35563975}{167192}e^{10} + \frac{254218365}{167192}e^{8} - \frac{942734297}{167192}e^{6} + \frac{433906537}{41798}e^{4} - \frac{706290329}{83596}e^{2} + \frac{100889251}{41798}$ |
| 89 | $[89, 89, -w^{2} + 4w - 2]$ | $-\frac{9841}{41798}e^{14} + \frac{363607}{41798}e^{12} - \frac{2615096}{20899}e^{10} + \frac{18606314}{20899}e^{8} - \frac{68607519}{20899}e^{6} + \frac{125453971}{20899}e^{4} - \frac{101405260}{20899}e^{2} + \frac{28870464}{20899}$ |
| 97 | $[97, 97, w^{2} - 2w - 7]$ | $\phantom{-}\frac{302079}{1588324}e^{15} - \frac{587859}{83596}e^{13} + \frac{4229323}{41798}e^{11} - \frac{285640662}{397081}e^{9} + \frac{1049445459}{397081}e^{7} - \frac{3797945209}{794162}e^{5} + \frac{1495815780}{397081}e^{3} - \frac{406684372}{397081}e$ |
| 97 | $[97, 97, -w^{2} - 4w - 5]$ | $\phantom{-}\frac{104013}{334384}e^{14} - \frac{962367}{83596}e^{12} + \frac{27748729}{167192}e^{10} - \frac{198058143}{167192}e^{8} + \frac{733647515}{167192}e^{6} - \frac{337718525}{41798}e^{4} + \frac{551954875}{83596}e^{2} - \frac{79566617}{41798}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $19$ | $[19, 19, -w^{2} - w + 4]$ | $-1$ |