Base field 4.4.11324.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 5x^{2} + 4x + 2\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, -w^{2} + 3]$ |
| Dimension: | $14$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $19$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{14} - 5x^{13} - 12x^{12} + 89x^{11} + 15x^{10} - 575x^{9} + 271x^{8} + 1694x^{7} - 1131x^{6} - 2383x^{5} + 1514x^{4} + 1486x^{3} - 644x^{2} - 284x + 24\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w]$ | $\phantom{-}e$ |
| 4 | $[4, 2, -w^{3} + 4w + 1]$ | $-\frac{343}{6332}e^{13} + \frac{969}{6332}e^{12} + \frac{2761}{3166}e^{11} - \frac{16431}{6332}e^{10} - \frac{29731}{6332}e^{9} + \frac{97967}{6332}e^{8} + \frac{60121}{6332}e^{7} - \frac{62584}{1583}e^{6} - \frac{25459}{6332}e^{5} + \frac{269523}{6332}e^{4} - \frac{12782}{1583}e^{3} - \frac{50643}{3166}e^{2} + \frac{12150}{1583}e + \frac{2873}{1583}$ |
| 5 | $[5, 5, w + 1]$ | $\phantom{-}\frac{2699}{3166}e^{13} - \frac{3284}{1583}e^{12} - \frac{24901}{1583}e^{11} + \frac{56955}{1583}e^{10} + \frac{171278}{1583}e^{9} - \frac{698739}{3166}e^{8} - \frac{1122535}{3166}e^{7} + \frac{923264}{1583}e^{6} + \frac{927025}{1583}e^{5} - \frac{1030268}{1583}e^{4} - \frac{704964}{1583}e^{3} + \frac{764929}{3166}e^{2} + \frac{164340}{1583}e - \frac{10342}{1583}$ |
| 13 | $[13, 13, -w^{2} + 3]$ | $-1$ |
| 17 | $[17, 17, -w^{3} + w^{2} + 3w - 1]$ | $\phantom{-}\frac{599}{1583}e^{13} - \frac{1443}{1583}e^{12} - \frac{11051}{1583}e^{11} + \frac{25233}{1583}e^{10} + \frac{75209}{1583}e^{9} - \frac{156769}{1583}e^{8} - \frac{237097}{1583}e^{7} + \frac{423021}{1583}e^{6} + \frac{353307}{1583}e^{5} - \frac{490159}{1583}e^{4} - \frac{211870}{1583}e^{3} + \frac{198988}{1583}e^{2} + \frac{23528}{1583}e - \frac{8282}{1583}$ |
| 19 | $[19, 19, -w^{3} + 3w - 1]$ | $\phantom{-}\frac{2286}{1583}e^{13} - \frac{5692}{1583}e^{12} - \frac{84077}{3166}e^{11} + \frac{197911}{3166}e^{10} + \frac{287503}{1583}e^{9} - \frac{1218707}{3166}e^{8} - \frac{1865391}{3166}e^{7} + \frac{3241431}{3166}e^{6} + \frac{3026933}{3166}e^{5} - \frac{1827790}{1583}e^{4} - \frac{2231309}{3166}e^{3} + \frac{1376305}{3166}e^{2} + \frac{241207}{1583}e - \frac{17960}{1583}$ |
| 23 | $[23, 23, -w + 3]$ | $\phantom{-}\frac{417}{6332}e^{13} - \frac{735}{6332}e^{12} - \frac{2216}{1583}e^{11} + \frac{12767}{6332}e^{10} + \frac{73325}{6332}e^{9} - \frac{75711}{6332}e^{8} - \frac{299853}{6332}e^{7} + \frac{84537}{3166}e^{6} + \frac{626095}{6332}e^{5} - \frac{67505}{6332}e^{4} - \frac{299057}{3166}e^{3} - \frac{31241}{1583}e^{2} + \frac{41146}{1583}e + \frac{7902}{1583}$ |
| 31 | $[31, 31, -w^{2} - 2w + 1]$ | $\phantom{-}\frac{769}{1583}e^{13} - \frac{3565}{3166}e^{12} - \frac{13894}{1583}e^{11} + \frac{60029}{3166}e^{10} + \frac{185441}{3166}e^{9} - \frac{175164}{1583}e^{8} - \frac{293081}{1583}e^{7} + \frac{422148}{1583}e^{6} + \frac{949109}{3166}e^{5} - \frac{776585}{3166}e^{4} - \frac{376005}{1583}e^{3} + \frac{163597}{3166}e^{2} + \frac{102294}{1583}e + \frac{7132}{1583}$ |
| 41 | $[41, 41, w^{3} + w^{2} - 5w - 3]$ | $-\frac{3126}{1583}e^{13} + \frac{7742}{1583}e^{12} + \frac{57064}{1583}e^{11} - \frac{134268}{1583}e^{10} - \frac{385577}{1583}e^{9} + \frac{823767}{1583}e^{8} + \frac{1228458}{1583}e^{7} - \frac{2177296}{1583}e^{6} - \frac{1952456}{1583}e^{5} + \frac{2423331}{1583}e^{4} + \frac{1422489}{1583}e^{3} - \frac{879228}{1583}e^{2} - \frac{308496}{1583}e + \frac{17222}{1583}$ |
| 43 | $[43, 43, 2w - 1]$ | $\phantom{-}\frac{6691}{3166}e^{13} - \frac{16235}{3166}e^{12} - \frac{61950}{1583}e^{11} + \frac{283123}{3166}e^{10} + \frac{856327}{3166}e^{9} - \frac{1752909}{3166}e^{8} - \frac{2824019}{3166}e^{7} + \frac{2354531}{1583}e^{6} + \frac{4703927}{3166}e^{5} - \frac{5415003}{3166}e^{4} - \frac{1810525}{1583}e^{3} + \frac{1070279}{1583}e^{2} + \frac{424360}{1583}e - \frac{47892}{1583}$ |
| 53 | $[53, 53, -w - 3]$ | $\phantom{-}\frac{3041}{1583}e^{13} - \frac{7968}{1583}e^{12} - \frac{54851}{1583}e^{11} + \frac{138605}{1583}e^{10} + \frac{364553}{1583}e^{9} - \frac{854936}{1583}e^{8} - \frac{1138729}{1583}e^{7} + \frac{2284585}{1583}e^{6} + \frac{1783001}{1583}e^{5} - \frac{2612381}{1583}e^{4} - \frac{1306387}{1583}e^{3} + \frac{1032407}{1583}e^{2} + \frac{302356}{1583}e - \frac{36010}{1583}$ |
| 53 | $[53, 53, w^{3} - w^{2} - 4w + 1]$ | $-\frac{959}{6332}e^{13} + \frac{1417}{6332}e^{12} + \frac{5362}{1583}e^{11} - \frac{26233}{6332}e^{10} - \frac{184179}{6332}e^{9} + \frac{171885}{6332}e^{8} + \frac{755871}{6332}e^{7} - \frac{233819}{3166}e^{6} - \frac{1489097}{6332}e^{5} + \frac{444847}{6332}e^{4} + \frac{615321}{3166}e^{3} - \frac{2744}{1583}e^{2} - \frac{68440}{1583}e - \frac{9768}{1583}$ |
| 61 | $[61, 61, w^{3} - 3w - 5]$ | $-\frac{730}{1583}e^{13} + \frac{1970}{1583}e^{12} + \frac{12860}{1583}e^{11} - \frac{33336}{1583}e^{10} - \frac{83158}{1583}e^{9} + \frac{196691}{1583}e^{8} + \frac{254742}{1583}e^{7} - \frac{488378}{1583}e^{6} - \frac{406256}{1583}e^{5} + \frac{495938}{1583}e^{4} + \frac{321729}{1583}e^{3} - \frac{165592}{1583}e^{2} - \frac{81412}{1583}e + \frac{15754}{1583}$ |
| 67 | $[67, 67, w^{3} + w^{2} - 5w - 1]$ | $\phantom{-}\frac{6625}{6332}e^{13} - \frac{16187}{6332}e^{12} - \frac{30294}{1583}e^{11} + \frac{277663}{6332}e^{10} + \frac{823949}{6332}e^{9} - \frac{1673415}{6332}e^{8} - \frac{2666593}{6332}e^{7} + \frac{2149255}{3166}e^{6} + \frac{4375127}{6332}e^{5} - \frac{4605849}{6332}e^{4} - \frac{1681309}{3166}e^{3} + \frac{412869}{1583}e^{2} + \frac{204563}{1583}e - \frac{12806}{1583}$ |
| 81 | $[81, 3, -3]$ | $-\frac{2979}{1583}e^{13} + \frac{7779}{1583}e^{12} + \frac{106455}{3166}e^{11} - \frac{268247}{3166}e^{10} - \frac{348864}{1583}e^{9} + \frac{1631179}{3166}e^{8} + \frac{2138309}{3166}e^{7} - \frac{4255125}{3166}e^{6} - \frac{3276801}{3166}e^{5} + \frac{2329757}{1583}e^{4} + \frac{2348999}{3166}e^{3} - \frac{1677293}{3166}e^{2} - \frac{263517}{1583}e + \frac{26770}{1583}$ |
| 83 | $[83, 83, -w^{3} + 5w - 3]$ | $-\frac{20061}{6332}e^{13} + \frac{50711}{6332}e^{12} + \frac{91506}{1583}e^{11} - \frac{882051}{6332}e^{10} - \frac{2476173}{6332}e^{9} + \frac{5438263}{6332}e^{8} + \frac{7938601}{6332}e^{7} - \frac{7252783}{3166}e^{6} - \frac{12842875}{6332}e^{5} + \frac{16466873}{6332}e^{4} + \frac{4854319}{3166}e^{3} - \frac{1588728}{1583}e^{2} - \frac{568813}{1583}e + \frac{67510}{1583}$ |
| 89 | $[89, 89, w^{2} + 1]$ | $\phantom{-}\frac{7173}{1583}e^{13} - \frac{17449}{1583}e^{12} - \frac{265281}{3166}e^{11} + \frac{608147}{3166}e^{10} + \frac{914721}{1583}e^{9} - \frac{3760141}{3166}e^{8} - \frac{6014089}{3166}e^{7} + \frac{10071087}{3166}e^{6} + \frac{9989617}{3166}e^{5} - \frac{5744492}{1583}e^{4} - \frac{7731185}{3166}e^{3} + \frac{4415681}{3166}e^{2} + \frac{947323}{1583}e - \frac{61914}{1583}$ |
| 97 | $[97, 97, w^{3} - w^{2} - 5w + 1]$ | $-\frac{12867}{3166}e^{13} + \frac{32815}{3166}e^{12} + \frac{116870}{1583}e^{11} - \frac{568685}{3166}e^{10} - \frac{1571585}{3166}e^{9} + \frac{3485337}{3166}e^{8} + \frac{4997673}{3166}e^{7} - \frac{4601879}{1583}e^{6} - \frac{8008313}{3166}e^{5} + \frac{10265321}{3166}e^{4} + \frac{2996867}{1583}e^{3} - \frac{1890062}{1583}e^{2} - \frac{696030}{1583}e + \frac{39842}{1583}$ |
| 97 | $[97, 97, 3w^{3} - 5w^{2} - 14w + 21]$ | $-\frac{117}{1583}e^{13} + \frac{229}{1583}e^{12} + \frac{3102}{1583}e^{11} - \frac{5074}{1583}e^{10} - \frac{31062}{1583}e^{9} + \frac{41480}{1583}e^{8} + \frac{145094}{1583}e^{7} - \frac{152736}{1583}e^{6} - \frac{314914}{1583}e^{5} + \frac{246152}{1583}e^{4} + \frac{272055}{1583}e^{3} - \frac{135663}{1583}e^{2} - \frac{49982}{1583}e + \frac{7326}{1583}$ |
| 97 | $[97, 97, w^{3} - 3w - 3]$ | $-\frac{6265}{3166}e^{13} + \frac{16213}{3166}e^{12} + \frac{56181}{1583}e^{11} - \frac{279829}{3166}e^{10} - \frac{740307}{3166}e^{9} + \frac{1705789}{3166}e^{8} + \frac{2280791}{3166}e^{7} - \frac{2238548}{1583}e^{6} - \frac{3486285}{3166}e^{5} + \frac{4983591}{3166}e^{4} + \frac{1205166}{1583}e^{3} - \frac{941906}{1583}e^{2} - \frac{217524}{1583}e + \frac{35042}{1583}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w^{2} + 3]$ | $1$ |