Base field 4.4.18688.1
Generator \(w\), with minimal polynomial \(x^{4} - 10x^{2} - 4x + 14\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[28, 14, \frac{2}{3}w^{3} - \frac{5}{3}w^{2} - 4w + \frac{28}{3}]$ |
| Dimension: | $10$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $20$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{10} - 4x^{9} - 34x^{8} + 139x^{7} + 352x^{6} - 1461x^{5} - 1422x^{4} + 5424x^{3} + 3280x^{2} - 6656x - 4096\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w - 2]$ | $\phantom{-}0$ |
| 7 | $[7, 7, -\frac{2}{3}w^{3} + \frac{2}{3}w^{2} + 5w - \frac{7}{3}]$ | $\phantom{-}1$ |
| 7 | $[7, 7, -\frac{1}{3}w^{3} + \frac{1}{3}w^{2} + w - \frac{5}{3}]$ | $\phantom{-}e$ |
| 9 | $[9, 3, -\frac{1}{3}w^{3} + \frac{1}{3}w^{2} + 3w - \frac{5}{3}]$ | $-\frac{273569}{62329984}e^{9} + \frac{249953}{15582496}e^{8} + \frac{4491585}{31164992}e^{7} - \frac{33141675}{62329984}e^{6} - \frac{2635717}{1947812}e^{5} + \frac{311256629}{62329984}e^{4} + \frac{125346183}{31164992}e^{3} - \frac{53620621}{3895624}e^{2} - \frac{19850997}{3895624}e + \frac{3685569}{486953}$ |
| 9 | $[9, 3, w + 1]$ | $-\frac{343395}{62329984}e^{9} + \frac{122451}{15582496}e^{8} + \frac{6518851}{31164992}e^{7} - \frac{15780353}{62329984}e^{6} - \frac{5146193}{1947812}e^{5} + \frac{130312735}{62329984}e^{4} + \frac{417747829}{31164992}e^{3} - \frac{10822207}{3895624}e^{2} - \frac{88920071}{3895624}e - \frac{4513795}{486953}$ |
| 17 | $[17, 17, w + 3]$ | $-\frac{545195}{62329984}e^{9} + \frac{577711}{15582496}e^{8} + \frac{8116235}{31164992}e^{7} - \frac{73137721}{62329984}e^{6} - \frac{7261635}{3895624}e^{5} + \frac{620365127}{62329984}e^{4} + \frac{33350309}{31164992}e^{3} - \frac{79675841}{3895624}e^{2} + \frac{16445193}{3895624}e + \frac{2131539}{486953}$ |
| 17 | $[17, 17, -\frac{1}{3}w^{3} + \frac{1}{3}w^{2} + 3w - \frac{11}{3}]$ | $-\frac{77309}{62329984}e^{9} + \frac{242505}{15582496}e^{8} + \frac{995261}{31164992}e^{7} - \frac{32428927}{62329984}e^{6} - \frac{490425}{3895624}e^{5} + \frac{319376705}{62329984}e^{4} - \frac{15570093}{31164992}e^{3} - \frac{62317555}{3895624}e^{2} + \frac{2113519}{3895624}e + \frac{5770359}{486953}$ |
| 31 | $[31, 31, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - w - \frac{1}{3}]$ | $-\frac{99175}{7791248}e^{9} + \frac{21609}{486953}e^{8} + \frac{1593891}{3895624}e^{7} - \frac{11167717}{7791248}e^{6} - \frac{7090681}{1947812}e^{5} + \frac{99289755}{7791248}e^{4} + \frac{33730791}{3895624}e^{3} - \frac{15144010}{486953}e^{2} - \frac{1611541}{486953}e + \frac{8196024}{486953}$ |
| 31 | $[31, 31, -\frac{2}{3}w^{3} + \frac{2}{3}w^{2} + 5w - \frac{1}{3}]$ | $-\frac{154241}{15582496}e^{9} + \frac{93101}{3895624}e^{8} + \frac{2752609}{7791248}e^{7} - \frac{12230507}{15582496}e^{6} - \frac{3890955}{973906}e^{5} + \frac{110392341}{15582496}e^{4} + \frac{135773503}{7791248}e^{3} - \frac{16110707}{973906}e^{2} - \frac{28166673}{973906}e + \frac{145680}{486953}$ |
| 41 | $[41, 41, -\frac{2}{3}w^{3} + \frac{5}{3}w^{2} + 4w - \frac{19}{3}]$ | $-\frac{90023}{31164992}e^{9} - \frac{7471}{7791248}e^{8} + \frac{1489247}{15582496}e^{7} + \frac{1134691}{31164992}e^{6} - \frac{4030977}{3895624}e^{5} - \frac{25112797}{31164992}e^{4} + \frac{79626109}{15582496}e^{3} + \frac{13055849}{1947812}e^{2} - \frac{23614923}{1947812}e - \frac{5659905}{486953}$ |
| 41 | $[41, 41, w^{2} - 5]$ | $-\frac{90615}{31164992}e^{9} - \frac{48219}{7791248}e^{8} + \frac{2214527}{15582496}e^{7} + \frac{7401427}{31164992}e^{6} - \frac{9143643}{3895624}e^{5} - \frac{91239277}{31164992}e^{4} + \frac{227658757}{15582496}e^{3} + \frac{26071029}{1947812}e^{2} - \frac{49745335}{1947812}e - \frac{11306905}{486953}$ |
| 41 | $[41, 41, 2w + 3]$ | $-\frac{54747}{31164992}e^{9} + \frac{12171}{7791248}e^{8} + \frac{754739}{15582496}e^{7} - \frac{1781129}{31164992}e^{6} - \frac{145512}{486953}e^{5} + \frac{18017863}{31164992}e^{4} - \frac{723539}{15582496}e^{3} - \frac{2777525}{973906}e^{2} + \frac{498625}{1947812}e + \frac{3389625}{486953}$ |
| 41 | $[41, 41, \frac{2}{3}w^{3} + \frac{4}{3}w^{2} - 5w - \frac{29}{3}]$ | $\phantom{-}\frac{253735}{31164992}e^{9} - \frac{174031}{7791248}e^{8} - \frac{4750479}{15582496}e^{7} + \frac{22679885}{31164992}e^{6} + \frac{3599931}{973906}e^{5} - \frac{202766819}{31164992}e^{4} - \frac{272270545}{15582496}e^{3} + \frac{7153544}{486953}e^{2} + \frac{56831971}{1947812}e - \frac{651679}{486953}$ |
| 47 | $[47, 47, -\frac{2}{3}w^{3} + \frac{5}{3}w^{2} + 3w - \frac{19}{3}]$ | $-\frac{8129}{3895624}e^{9} + \frac{72005}{1947812}e^{8} - \frac{82637}{1947812}e^{7} - \frac{4732655}{3895624}e^{6} + \frac{5505149}{1947812}e^{5} + \frac{44285417}{3895624}e^{4} - \frac{13143158}{486953}e^{3} - \frac{16026271}{486953}e^{2} + \frac{26218601}{486953}e + \frac{17446240}{486953}$ |
| 47 | $[47, 47, \frac{1}{3}w^{3} + \frac{2}{3}w^{2} - 3w - \frac{13}{3}]$ | $-\frac{177857}{15582496}e^{9} + \frac{73215}{3895624}e^{8} + \frac{3363161}{7791248}e^{7} - \frac{9242171}{15582496}e^{6} - \frac{10545275}{1947812}e^{5} + \frac{75811653}{15582496}e^{4} + \frac{214012043}{7791248}e^{3} - \frac{8026951}{973906}e^{2} - \frac{47298899}{973906}e - \frac{4712904}{486953}$ |
| 49 | $[49, 7, -\frac{1}{3}w^{3} + \frac{1}{3}w^{2} + 2w - \frac{11}{3}]$ | $\phantom{-}\frac{38821}{7791248}e^{9} - \frac{48923}{1947812}e^{8} - \frac{458139}{3895624}e^{7} + \frac{6173743}{7791248}e^{6} + \frac{35927}{486953}e^{5} - \frac{53925693}{7791248}e^{4} + \frac{31767781}{3895624}e^{3} + \frac{37193559}{1947812}e^{2} - \frac{10690067}{486953}e - \frac{11859031}{486953}$ |
| 73 | $[73, 73, -\frac{1}{3}w^{3} + \frac{1}{3}w^{2} + 5w + \frac{19}{3}]$ | $-\frac{716091}{62329984}e^{9} + \frac{448983}{15582496}e^{8} + \frac{11755867}{31164992}e^{7} - \frac{56912809}{62329984}e^{6} - \frac{13690937}{3895624}e^{5} + \frac{474941335}{62329984}e^{4} + \frac{285416421}{31164992}e^{3} - \frac{58834525}{3895624}e^{2} - \frac{11110223}{3895624}e + \frac{1451759}{486953}$ |
| 73 | $[73, 73, -\frac{2}{3}w^{3} - \frac{1}{3}w^{2} + 4w + \frac{11}{3}]$ | $\phantom{-}\frac{590383}{62329984}e^{9} - \frac{402251}{15582496}e^{8} - \frac{9728559}{31164992}e^{7} + \frac{50368421}{62329984}e^{6} + \frac{11776809}{3895624}e^{5} - \frac{396800091}{62329984}e^{4} - \frac{305199441}{31164992}e^{3} + \frac{30309661}{3895624}e^{2} + \frac{41268427}{3895624}e + \frac{4539075}{486953}$ |
| 73 | $[73, 73, -\frac{1}{3}w^{3} + \frac{4}{3}w^{2} + 2w - \frac{17}{3}]$ | $\phantom{-}\frac{11463}{1947812}e^{9} - \frac{21027}{973906}e^{8} - \frac{86072}{486953}e^{7} + \frac{1236637}{1947812}e^{6} + \frac{1340715}{973906}e^{5} - \frac{8295885}{1947812}e^{4} - \frac{1394624}{486953}e^{3} - \frac{175797}{973906}e^{2} + \frac{2423842}{486953}e + \frac{8236686}{486953}$ |
| 103 | $[103, 103, -\frac{1}{3}w^{3} + \frac{4}{3}w^{2} + w - \frac{23}{3}]$ | $-\frac{147089}{15582496}e^{9} + \frac{85267}{3895624}e^{8} + \frac{2939849}{7791248}e^{7} - \frac{10237547}{15582496}e^{6} - \frac{9990473}{1947812}e^{5} + \frac{70554389}{15582496}e^{4} + \frac{227122947}{7791248}e^{3} + \frac{3797453}{973906}e^{2} - \frac{57482667}{973906}e - \frac{22920600}{486953}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2, 2, -w - 2]$ | $-1$ |
| $7$ | $[7, 7, -\frac{2}{3}w^{3} + \frac{2}{3}w^{2} + 5w - \frac{7}{3}]$ | $-1$ |