Base field 4.4.18097.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 7x^{2} + 6x + 4\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{7}{2}w - 1]$ |
| Dimension: | $18$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $28$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{18} - 36x^{16} + 533x^{14} - 4195x^{12} + 18957x^{10} - 49667x^{8} + 72711x^{6} - 53660x^{4} + 14928x^{2} - 64\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, -w + 1]$ | $\phantom{-}e$ |
| 4 | $[4, 2, w]$ | $\phantom{-}\frac{28809}{2094208}e^{17} - \frac{104645}{261776}e^{15} + \frac{9406877}{2094208}e^{13} - \frac{51669935}{2094208}e^{11} + \frac{145556401}{2094208}e^{9} - \frac{205121247}{2094208}e^{7} + \frac{127482939}{2094208}e^{5} - \frac{1047981}{261776}e^{3} - \frac{1378597}{130888}e$ |
| 4 | $[4, 2, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{7}{2}w - 3]$ | $-\frac{5733}{523552}e^{17} + \frac{11805}{32722}e^{15} - \frac{2492841}{523552}e^{13} + \frac{16805675}{523552}e^{11} - \frac{60867317}{523552}e^{9} + \frac{112681611}{523552}e^{7} - \frac{86803703}{523552}e^{5} + \frac{217223}{16361}e^{3} + \frac{259597}{16361}e$ |
| 7 | $[7, 7, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{5}{2}w - 2]$ | $\phantom{-}\frac{41121}{1047104}e^{16} - \frac{161051}{130888}e^{14} + \frac{16063957}{1047104}e^{12} - \frac{102060295}{1047104}e^{10} + \frac{352029849}{1047104}e^{8} - \frac{649056887}{1047104}e^{6} + \frac{577893587}{1047104}e^{4} - \frac{22998255}{130888}e^{2} + \frac{33339}{65444}$ |
| 13 | $[13, 13, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{7}{2}w - 1]$ | $-1$ |
| 17 | $[17, 17, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{5}{2}w]$ | $\phantom{-}\frac{2429}{130888}e^{17} - \frac{8505}{16361}e^{15} + \frac{725849}{130888}e^{13} - \frac{3715667}{130888}e^{11} + \frac{9785701}{130888}e^{9} - \frac{15331899}{130888}e^{7} + \frac{20821607}{130888}e^{5} - \frac{2835733}{16361}e^{3} + \frac{1239258}{16361}e$ |
| 27 | $[27, 3, -w^{3} + 7w + 1]$ | $\phantom{-}\frac{6281}{261776}e^{17} - \frac{104897}{130888}e^{15} + \frac{2841533}{261776}e^{13} - \frac{20116185}{261776}e^{11} + \frac{80083679}{261776}e^{9} - \frac{179959801}{261776}e^{7} + \frac{218049145}{261776}e^{5} - \frac{63783483}{130888}e^{3} + \frac{3337961}{32722}e$ |
| 31 | $[31, 31, w + 3]$ | $-\frac{25245}{523552}e^{16} + \frac{24232}{16361}e^{14} - \frac{9422481}{523552}e^{12} + \frac{57887555}{523552}e^{10} - \frac{191035501}{523552}e^{8} + \frac{331528275}{523552}e^{6} - \frac{266743967}{523552}e^{4} + \frac{2014577}{16361}e^{2} + \frac{263159}{32722}$ |
| 31 | $[31, 31, -w^{2} + 5]$ | $-\frac{515}{523552}e^{17} - \frac{395}{32722}e^{15} + \frac{477697}{523552}e^{13} - \frac{6783011}{523552}e^{11} + \frac{43486909}{523552}e^{9} - \frac{142731891}{523552}e^{7} + \frac{240120207}{523552}e^{5} - \frac{5879031}{16361}e^{3} + \frac{3075799}{32722}e$ |
| 37 | $[37, 37, w^{2} - 3]$ | $-\frac{69275}{2094208}e^{17} + \frac{249069}{261776}e^{15} - \frac{21823895}{2094208}e^{13} + \frac{111909373}{2094208}e^{11} - \frac{253030435}{2094208}e^{9} + \frac{87879757}{2094208}e^{7} + \frac{553658431}{2094208}e^{5} - \frac{97158439}{261776}e^{3} + \frac{17377855}{130888}e$ |
| 41 | $[41, 41, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{3}{2}w - 2]$ | $-\frac{97149}{2094208}e^{17} + \frac{367697}{261776}e^{15} - \frac{35041121}{2094208}e^{13} + \frac{209364571}{2094208}e^{11} - \frac{664315973}{2094208}e^{9} + \frac{1088170251}{2094208}e^{7} - \frac{778697239}{2094208}e^{5} + \frac{9923649}{261776}e^{3} + \frac{4224197}{130888}e$ |
| 47 | $[47, 47, w^{3} - 5w - 3]$ | $\phantom{-}\frac{41813}{2094208}e^{17} - \frac{129809}{261776}e^{15} + \frac{8643353}{2094208}e^{13} - \frac{20755331}{2094208}e^{11} - \frac{68648675}{2094208}e^{9} + \frac{405454157}{2094208}e^{7} - \frac{601274785}{2094208}e^{5} + \frac{33998799}{261776}e^{3} - \frac{741349}{130888}e$ |
| 53 | $[53, 53, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w]$ | $\phantom{-}\frac{2429}{130888}e^{17} - \frac{8505}{16361}e^{15} + \frac{725849}{130888}e^{13} - \frac{3715667}{130888}e^{11} + \frac{9785701}{130888}e^{9} - \frac{15331899}{130888}e^{7} + \frac{20821607}{130888}e^{5} - \frac{2819372}{16361}e^{3} + \frac{1124731}{16361}e$ |
| 61 | $[61, 61, w^{3} - w^{2} - 5w + 3]$ | $\phantom{-}\frac{18865}{1047104}e^{16} - \frac{60491}{130888}e^{14} + \frac{4363557}{1047104}e^{12} - \frac{14820535}{1047104}e^{10} + \frac{589481}{1047104}e^{8} + \frac{90247961}{1047104}e^{6} - \frac{142641725}{1047104}e^{4} + \frac{6554833}{130888}e^{2} - \frac{70541}{65444}$ |
| 83 | $[83, 83, w^{3} + w^{2} - 6w - 7]$ | $-\frac{49681}{1047104}e^{17} + \frac{202245}{130888}e^{15} - \frac{21289029}{1047104}e^{13} + \frac{146024679}{1047104}e^{11} - \frac{562973305}{1047104}e^{9} + \frac{1226775255}{1047104}e^{7} - \frac{1436225683}{1047104}e^{5} + \frac{100027813}{130888}e^{3} - \frac{10353035}{65444}e$ |
| 83 | $[83, 83, -w^{3} + 5w + 1]$ | $\phantom{-}\frac{159047}{1047104}e^{16} - \frac{613945}{130888}e^{14} + \frac{60132275}{1047104}e^{12} - \frac{373560369}{1047104}e^{10} + \frac{1255007407}{1047104}e^{8} - \frac{2249175681}{1047104}e^{6} + \frac{1940701253}{1047104}e^{4} - \frac{73030765}{130888}e^{2} - \frac{54555}{65444}$ |
| 83 | $[83, 83, 2w - 1]$ | $\phantom{-}\frac{131923}{1047104}e^{17} - \frac{524347}{130888}e^{15} + \frac{53416943}{1047104}e^{13} - \frac{350145269}{1047104}e^{11} + \frac{1267033003}{1047104}e^{9} - \frac{2524889029}{1047104}e^{7} + \frac{2590965753}{1047104}e^{5} - \frac{144715443}{130888}e^{3} + \frac{9241721}{65444}e$ |
| 83 | $[83, 83, -\frac{3}{2}w^{3} - \frac{1}{2}w^{2} + \frac{19}{2}w + 2]$ | $-\frac{39701}{523552}e^{16} + \frac{39338}{16361}e^{14} - \frac{15962937}{523552}e^{12} + \frac{103947035}{523552}e^{10} - \frac{371806181}{523552}e^{8} + \frac{723174827}{523552}e^{6} - \frac{695459463}{523552}e^{4} + \frac{15352553}{32722}e^{2} - \frac{22665}{32722}$ |
| 89 | $[89, 89, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w - 2]$ | $\phantom{-}\frac{1597}{130888}e^{16} - \frac{7515}{65444}e^{14} - \frac{362931}{130888}e^{12} + \frac{6612667}{130888}e^{10} - \frac{40003649}{130888}e^{8} + \frac{105642571}{130888}e^{6} - \frac{115627487}{130888}e^{4} + \frac{18565589}{65444}e^{2} + \frac{255183}{16361}$ |
| 89 | $[89, 89, w^{3} - 5w + 5]$ | $-\frac{2189}{32722}e^{16} + \frac{128151}{65444}e^{14} - \frac{727671}{32722}e^{12} + \frac{8122797}{65444}e^{10} - \frac{23470611}{65444}e^{8} + \frac{34641325}{65444}e^{6} - \frac{24804919}{65444}e^{4} + \frac{7816233}{65444}e^{2} - \frac{96287}{16361}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{7}{2}w - 1]$ | $1$ |