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: | $[16, 2, 2]$ |
| Dimension: | $11$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $30$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{11} + x^{10} - 53x^{9} - 46x^{8} + 1006x^{7} + 720x^{6} - 8252x^{5} - 4144x^{4} + 28752x^{3} + 3520x^{2} - 35840x + 15360\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, \frac{1}{6}w^{3} - \frac{7}{3}w - \frac{17}{6}]$ | $-1$ |
| 4 | $[4, 2, w - 3]$ | $-1$ |
| 5 | $[5, 5, -\frac{1}{6}w^{3} + w^{2} + \frac{1}{3}w - \frac{25}{6}]$ | $\phantom{-}e$ |
| 9 | $[9, 3, \frac{1}{6}w^{3} - \frac{7}{3}w + \frac{13}{6}]$ | $-\frac{781}{692704}e^{10} + \frac{3883}{1385408}e^{9} + \frac{84647}{1385408}e^{8} - \frac{171237}{1385408}e^{7} - \frac{775117}{692704}e^{6} + \frac{1278823}{692704}e^{5} + \frac{1361663}{173176}e^{4} - \frac{3711511}{346352}e^{3} - \frac{1459625}{86588}e^{2} + \frac{1835407}{86588}e - \frac{93110}{21647}$ |
| 9 | $[9, 3, -w - 2]$ | $-\frac{781}{692704}e^{10} + \frac{3883}{1385408}e^{9} + \frac{84647}{1385408}e^{8} - \frac{171237}{1385408}e^{7} - \frac{775117}{692704}e^{6} + \frac{1278823}{692704}e^{5} + \frac{1361663}{173176}e^{4} - \frac{3711511}{346352}e^{3} - \frac{1459625}{86588}e^{2} + \frac{1835407}{86588}e - \frac{93110}{21647}$ |
| 11 | $[11, 11, -\frac{1}{6}w^{3} + \frac{7}{3}w + \frac{11}{6}]$ | $-\frac{3019}{5541632}e^{10} + \frac{3389}{5541632}e^{9} + \frac{150175}{5541632}e^{8} - \frac{108239}{2770816}e^{7} - \frac{1226461}{2770816}e^{6} + \frac{137875}{173176}e^{5} + \frac{3355213}{1385408}e^{4} - \frac{2103165}{346352}e^{3} - \frac{277443}{346352}e^{2} + \frac{1383593}{86588}e - \frac{233667}{21647}$ |
| 11 | $[11, 11, -w + 2]$ | $-\frac{3019}{5541632}e^{10} + \frac{3389}{5541632}e^{9} + \frac{150175}{5541632}e^{8} - \frac{108239}{2770816}e^{7} - \frac{1226461}{2770816}e^{6} + \frac{137875}{173176}e^{5} + \frac{3355213}{1385408}e^{4} - \frac{2103165}{346352}e^{3} - \frac{277443}{346352}e^{2} + \frac{1383593}{86588}e - \frac{233667}{21647}$ |
| 41 | $[41, 41, -w]$ | $\phantom{-}\frac{10663}{5541632}e^{10} + \frac{41635}{5541632}e^{9} - \frac{554935}{5541632}e^{8} - \frac{959703}{2770816}e^{7} + \frac{5245565}{2770816}e^{6} + \frac{3798443}{692704}e^{5} - \frac{21539113}{1385408}e^{4} - \frac{5856529}{173176}e^{3} + \frac{17935679}{346352}e^{2} + \frac{2696077}{43294}e - \frac{1226802}{21647}$ |
| 41 | $[41, 41, \frac{1}{6}w^{3} - \frac{7}{3}w + \frac{1}{6}]$ | $\phantom{-}\frac{10663}{5541632}e^{10} + \frac{41635}{5541632}e^{9} - \frac{554935}{5541632}e^{8} - \frac{959703}{2770816}e^{7} + \frac{5245565}{2770816}e^{6} + \frac{3798443}{692704}e^{5} - \frac{21539113}{1385408}e^{4} - \frac{5856529}{173176}e^{3} + \frac{17935679}{346352}e^{2} + \frac{2696077}{43294}e - \frac{1226802}{21647}$ |
| 59 | $[59, 59, -\frac{1}{6}w^{3} + \frac{1}{3}w + \frac{23}{6}]$ | $\phantom{-}\frac{1737}{5541632}e^{10} + \frac{10469}{5541632}e^{9} - \frac{94865}{5541632}e^{8} - \frac{238717}{2770816}e^{7} + \frac{961171}{2770816}e^{6} + \frac{1007575}{692704}e^{5} - \frac{4535023}{1385408}e^{4} - \frac{950311}{86588}e^{3} + \frac{5335893}{346352}e^{2} + \frac{1249283}{43294}e - \frac{631320}{21647}$ |
| 59 | $[59, 59, -\frac{1}{3}w^{3} + \frac{11}{3}w + \frac{11}{3}]$ | $\phantom{-}\frac{1737}{5541632}e^{10} + \frac{10469}{5541632}e^{9} - \frac{94865}{5541632}e^{8} - \frac{238717}{2770816}e^{7} + \frac{961171}{2770816}e^{6} + \frac{1007575}{692704}e^{5} - \frac{4535023}{1385408}e^{4} - \frac{950311}{86588}e^{3} + \frac{5335893}{346352}e^{2} + \frac{1249283}{43294}e - \frac{631320}{21647}$ |
| 61 | $[61, 61, -\frac{5}{3}w^{3} - 3w^{2} + \frac{46}{3}w + \frac{79}{3}]$ | $\phantom{-}\frac{12121}{2770816}e^{10} - \frac{9247}{2770816}e^{9} - \frac{602709}{2770816}e^{8} + \frac{271317}{1385408}e^{7} + \frac{5125007}{1385408}e^{6} - \frac{159275}{43294}e^{5} - \frac{16914823}{692704}e^{4} + \frac{4393159}{173176}e^{3} + \frac{8692781}{173176}e^{2} - \frac{2491217}{43294}e + \frac{249328}{21647}$ |
| 61 | $[61, 61, \frac{5}{6}w^{3} + w^{2} - \frac{23}{3}w - \frac{55}{6}]$ | $\phantom{-}\frac{12121}{2770816}e^{10} - \frac{9247}{2770816}e^{9} - \frac{602709}{2770816}e^{8} + \frac{271317}{1385408}e^{7} + \frac{5125007}{1385408}e^{6} - \frac{159275}{43294}e^{5} - \frac{16914823}{692704}e^{4} + \frac{4393159}{173176}e^{3} + \frac{8692781}{173176}e^{2} - \frac{2491217}{43294}e + \frac{249328}{21647}$ |
| 61 | $[61, 61, w^{3} + 2w^{2} - 9w - 17]$ | $\phantom{-}\frac{5995}{2770816}e^{10} + \frac{16043}{2770816}e^{9} - \frac{306127}{2770816}e^{8} - \frac{387193}{1385408}e^{7} + \frac{2691833}{1385408}e^{6} + \frac{814439}{173176}e^{5} - \frac{9588761}{692704}e^{4} - \frac{5520191}{173176}e^{3} + \frac{6983069}{173176}e^{2} + \frac{1495463}{21647}e - \frac{1150172}{21647}$ |
| 61 | $[61, 61, -\frac{1}{6}w^{3} + \frac{10}{3}w + \frac{35}{6}]$ | $\phantom{-}\frac{5995}{2770816}e^{10} + \frac{16043}{2770816}e^{9} - \frac{306127}{2770816}e^{8} - \frac{387193}{1385408}e^{7} + \frac{2691833}{1385408}e^{6} + \frac{814439}{173176}e^{5} - \frac{9588761}{692704}e^{4} - \frac{5520191}{173176}e^{3} + \frac{6983069}{173176}e^{2} + \frac{1495463}{21647}e - \frac{1150172}{21647}$ |
| 71 | $[71, 71, -\frac{1}{2}w^{3} + 2w^{2} + 4w - \frac{33}{2}]$ | $-\frac{12391}{2770816}e^{10} - \frac{16607}{2770816}e^{9} + \frac{614763}{2770816}e^{8} + \frac{352165}{1385408}e^{7} - \frac{5400929}{1385408}e^{6} - \frac{321505}{86588}e^{5} + \frac{19868345}{692704}e^{4} + \frac{3791655}{173176}e^{3} - \frac{14014003}{173176}e^{2} - \frac{1870339}{43294}e + \frac{1331448}{21647}$ |
| 71 | $[71, 71, \frac{1}{6}w^{3} - w^{2} - \frac{4}{3}w + \frac{67}{6}]$ | $-\frac{12391}{2770816}e^{10} - \frac{16607}{2770816}e^{9} + \frac{614763}{2770816}e^{8} + \frac{352165}{1385408}e^{7} - \frac{5400929}{1385408}e^{6} - \frac{321505}{86588}e^{5} + \frac{19868345}{692704}e^{4} + \frac{3791655}{173176}e^{3} - \frac{14014003}{173176}e^{2} - \frac{1870339}{43294}e + \frac{1331448}{21647}$ |
| 79 | $[79, 79, \frac{1}{2}w^{3} - 3w + \frac{1}{2}]$ | $-\frac{3965}{2770816}e^{10} - \frac{14077}{2770816}e^{9} + \frac{196257}{2770816}e^{8} + \frac{378491}{1385408}e^{7} - \frac{1685227}{1385408}e^{6} - \frac{866919}{173176}e^{5} + \frac{6011899}{692704}e^{4} + \frac{6141475}{173176}e^{3} - \frac{4758197}{173176}e^{2} - \frac{3378527}{43294}e + \frac{1116410}{21647}$ |
| 79 | $[79, 79, -\frac{2}{3}w^{3} + \frac{19}{3}w - \frac{2}{3}]$ | $-\frac{3965}{2770816}e^{10} - \frac{14077}{2770816}e^{9} + \frac{196257}{2770816}e^{8} + \frac{378491}{1385408}e^{7} - \frac{1685227}{1385408}e^{6} - \frac{866919}{173176}e^{5} + \frac{6011899}{692704}e^{4} + \frac{6141475}{173176}e^{3} - \frac{4758197}{173176}e^{2} - \frac{3378527}{43294}e + \frac{1116410}{21647}$ |
| 89 | $[89, 89, \frac{1}{2}w^{3} + w^{2} - 5w - \frac{15}{2}]$ | $\phantom{-}\frac{3715}{5541632}e^{10} - \frac{241}{5541632}e^{9} - \frac{175475}{5541632}e^{8} - \frac{4835}{2770816}e^{7} + \frac{1400881}{2770816}e^{6} + \frac{114495}{692704}e^{5} - \frac{4091725}{1385408}e^{4} - \frac{418741}{173176}e^{3} + \frac{1094683}{346352}e^{2} + \frac{253291}{43294}e + \frac{129540}{21647}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, \frac{1}{6}w^{3} - \frac{7}{3}w - \frac{17}{6}]$ | $1$ |
| $4$ | $[4, 2, w - 3]$ | $1$ |