Base field 4.4.18625.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 14x^{2} + 9x + 41\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[20, 10, w + 1]$ |
| Dimension: | $11$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $32$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{11} + x^{10} - 27x^{9} - 22x^{8} + 259x^{7} + 134x^{6} - 1067x^{5} - 98x^{4} + 1688x^{3} - 683x^{2} - 190x + 52\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, \frac{1}{6}w^{3} - \frac{7}{3}w - \frac{17}{6}]$ | $\phantom{-}e$ |
| 4 | $[4, 2, w - 3]$ | $-1$ |
| 5 | $[5, 5, -\frac{1}{6}w^{3} + w^{2} + \frac{1}{3}w - \frac{25}{6}]$ | $-1$ |
| 9 | $[9, 3, \frac{1}{6}w^{3} - \frac{7}{3}w + \frac{13}{6}]$ | $-\frac{29}{11672}e^{10} - \frac{157}{5836}e^{9} + \frac{917}{11672}e^{8} + \frac{7235}{11672}e^{7} - \frac{1283}{1459}e^{6} - \frac{27223}{5836}e^{5} + \frac{50089}{11672}e^{4} + \frac{149967}{11672}e^{3} - \frac{101353}{11672}e^{2} - \frac{51531}{5836}e + \frac{11137}{2918}$ |
| 9 | $[9, 3, -w - 2]$ | $-\frac{367}{5836}e^{10} + \frac{555}{11672}e^{9} + \frac{17927}{11672}e^{8} - \frac{6545}{5836}e^{7} - \frac{144625}{11672}e^{6} + \frac{104951}{11672}e^{5} + \frac{419789}{11672}e^{4} - \frac{150779}{5836}e^{3} - \frac{252865}{11672}e^{2} + \frac{51043}{5836}e + \frac{1703}{2918}$ |
| 11 | $[11, 11, -\frac{1}{6}w^{3} + \frac{7}{3}w + \frac{11}{6}]$ | $-\frac{51}{5836}e^{10} - \frac{551}{11672}e^{9} + \frac{3175}{11672}e^{8} + \frac{6787}{5836}e^{7} - \frac{36755}{11672}e^{6} - \frac{107733}{11672}e^{5} + \frac{194639}{11672}e^{4} + \frac{70665}{2918}e^{3} - \frac{421333}{11672}e^{2} - \frac{30165}{5836}e + \frac{20607}{2918}$ |
| 11 | $[11, 11, -w + 2]$ | $-\frac{15}{11672}e^{10} + \frac{925}{5836}e^{9} - \frac{1035}{11672}e^{8} - \frac{43147}{11672}e^{7} + \frac{3814}{1459}e^{6} + \frac{79921}{2918}e^{5} - \frac{263477}{11672}e^{4} - \frac{750841}{11672}e^{3} + \frac{756667}{11672}e^{2} - \frac{8341}{5836}e - \frac{8427}{2918}$ |
| 41 | $[41, 41, -w]$ | $\phantom{-}\frac{285}{11672}e^{10} - \frac{67}{5836}e^{9} - \frac{6597}{11672}e^{8} + \frac{2753}{11672}e^{7} + \frac{6320}{1459}e^{6} - \frac{9573}{5836}e^{5} - \frac{147125}{11672}e^{4} + \frac{40729}{11672}e^{3} + \frac{122869}{11672}e^{2} + \frac{21333}{5836}e + \frac{5459}{2918}$ |
| 41 | $[41, 41, \frac{1}{6}w^{3} - \frac{7}{3}w + \frac{1}{6}]$ | $-\frac{559}{11672}e^{10} - \frac{29}{2918}e^{9} + \frac{13953}{11672}e^{8} + \frac{2013}{11672}e^{7} - \frac{58927}{5836}e^{6} + \frac{473}{2918}e^{5} + \frac{388449}{11672}e^{4} - \frac{121639}{11672}e^{3} - \frac{397165}{11672}e^{2} + \frac{145923}{5836}e - \frac{3571}{2918}$ |
| 59 | $[59, 59, -\frac{1}{6}w^{3} + \frac{1}{3}w + \frac{23}{6}]$ | $-\frac{1867}{11672}e^{10} + \frac{357}{5836}e^{9} + \frac{46257}{11672}e^{8} - \frac{17195}{11672}e^{7} - \frac{48287}{1459}e^{6} + \frac{38167}{2918}e^{5} + \frac{1241059}{11672}e^{4} - \frac{548669}{11672}e^{3} - \frac{1187813}{11672}e^{2} + \frac{233877}{5836}e + \frac{25527}{2918}$ |
| 59 | $[59, 59, -\frac{1}{3}w^{3} + \frac{11}{3}w + \frac{11}{3}]$ | $\phantom{-}\frac{174}{1459}e^{10} + \frac{241}{5836}e^{9} - \frac{17631}{5836}e^{8} - \frac{1099}{1459}e^{7} + \frac{151501}{5836}e^{6} + \frac{12571}{5836}e^{5} - \frac{509111}{5836}e^{4} + \frac{37277}{2918}e^{3} + \frac{545985}{5836}e^{2} - \frac{89487}{2918}e - \frac{14881}{1459}$ |
| 61 | $[61, 61, -\frac{5}{3}w^{3} - 3w^{2} + \frac{46}{3}w + \frac{79}{3}]$ | $\phantom{-}\frac{553}{5836}e^{10} - \frac{603}{5836}e^{9} - \frac{3227}{1459}e^{8} + \frac{14577}{5836}e^{7} + \frac{94751}{5836}e^{6} - \frac{118255}{5836}e^{5} - \frac{102333}{2918}e^{4} + \frac{331369}{5836}e^{3} - \frac{57291}{2918}e^{2} - \frac{17291}{1459}e + \frac{18000}{1459}$ |
| 61 | $[61, 61, \frac{5}{6}w^{3} + w^{2} - \frac{23}{3}w - \frac{55}{6}]$ | $\phantom{-}\frac{341}{11672}e^{10} - \frac{29}{1459}e^{9} - \frac{8569}{11672}e^{8} + \frac{5485}{11672}e^{7} + \frac{38259}{5836}e^{6} - \frac{25829}{5836}e^{5} - \frac{301303}{11672}e^{4} + \frac{245487}{11672}e^{3} + \frac{502721}{11672}e^{2} - \frac{234853}{5836}e - \frac{23191}{2918}$ |
| 61 | $[61, 61, w^{3} + 2w^{2} - 9w - 17]$ | $\phantom{-}\frac{1233}{11672}e^{10} - \frac{167}{5836}e^{9} - \frac{28725}{11672}e^{8} + \frac{7733}{11672}e^{7} + \frac{26728}{1459}e^{6} - \frac{33399}{5836}e^{5} - \frac{527161}{11672}e^{4} + \frac{197769}{11672}e^{3} + \frac{129869}{11672}e^{2} + \frac{57093}{5836}e + \frac{4635}{2918}$ |
| 61 | $[61, 61, -\frac{1}{6}w^{3} + \frac{10}{3}w + \frac{35}{6}]$ | $\phantom{-}\frac{29}{5836}e^{10} + \frac{2087}{11672}e^{9} - \frac{3293}{11672}e^{8} - \frac{24743}{5836}e^{7} + \frac{57003}{11672}e^{6} + \frac{372971}{11672}e^{5} - \frac{408027}{11672}e^{4} - \frac{435931}{5836}e^{3} + \frac{1076647}{11672}e^{2} - \frac{76395}{5836}e - \frac{14979}{2918}$ |
| 71 | $[71, 71, -\frac{1}{2}w^{3} + 2w^{2} + 4w - \frac{33}{2}]$ | $-\frac{115}{2918}e^{10} - \frac{327}{5836}e^{9} + \frac{6015}{5836}e^{8} + \frac{2145}{1459}e^{7} - \frac{56903}{5836}e^{6} - \frac{70937}{5836}e^{5} + \frac{243157}{5836}e^{4} + \frac{45369}{1459}e^{3} - \frac{432091}{5836}e^{2} + \frac{5598}{1459}e + \frac{12309}{1459}$ |
| 71 | $[71, 71, \frac{1}{6}w^{3} - w^{2} - \frac{4}{3}w + \frac{67}{6}]$ | $-\frac{99}{11672}e^{10} - \frac{921}{11672}e^{9} + \frac{1691}{5836}e^{8} + \frac{19869}{11672}e^{7} - \frac{40523}{11672}e^{6} - \frac{131985}{11672}e^{5} + \frac{26248}{1459}e^{4} + \frac{274089}{11672}e^{3} - \frac{104649}{2918}e^{2} - \frac{4498}{1459}e + \frac{7061}{1459}$ |
| 79 | $[79, 79, \frac{1}{2}w^{3} - 3w + \frac{1}{2}]$ | $-\frac{101}{5836}e^{10} - \frac{47}{1459}e^{9} + \frac{1785}{5836}e^{8} + \frac{4973}{5836}e^{7} - \frac{2227}{2918}e^{6} - \frac{22163}{2918}e^{5} - \frac{49735}{5836}e^{4} + \frac{160165}{5836}e^{3} + \frac{191873}{5836}e^{2} - \frac{52933}{1459}e - \frac{2467}{1459}$ |
| 79 | $[79, 79, -\frac{2}{3}w^{3} + \frac{19}{3}w - \frac{2}{3}]$ | $\phantom{-}\frac{703}{5836}e^{10} - \frac{1731}{11672}e^{9} - \frac{32837}{11672}e^{8} + \frac{20019}{5836}e^{7} + \frac{246025}{11672}e^{6} - \frac{311261}{11672}e^{5} - \frac{612501}{11672}e^{4} + \frac{220595}{2918}e^{3} + \frac{139371}{11672}e^{2} - \frac{213111}{5836}e + \frac{13411}{2918}$ |
| 89 | $[89, 89, \frac{1}{2}w^{3} + w^{2} - 5w - \frac{15}{2}]$ | $-\frac{973}{11672}e^{10} - \frac{121}{11672}e^{9} + \frac{10931}{5836}e^{8} + \frac{2867}{11672}e^{7} - \frac{150831}{11672}e^{6} - \frac{14327}{11672}e^{5} + \frac{145777}{5836}e^{4} + \frac{25269}{11672}e^{3} + \frac{45923}{2918}e^{2} - \frac{30045}{1459}e - \frac{8799}{1459}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, w - 3]$ | $1$ |
| $5$ | $[5, 5, -\frac{1}{6}w^{3} + w^{2} + \frac{1}{3}w - \frac{25}{6}]$ | $1$ |