Base field 4.4.9225.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 10x^{2} + 7x + 19\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[19, 19, w]$ |
| Dimension: | $12$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $14$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{12} - 60x^{10} + 1261x^{8} - 10888x^{6} + 35376x^{4} - 35520x^{2} + 10816\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -\frac{1}{4}w^{3} + \frac{1}{2}w + \frac{3}{4}]$ | $-\frac{23339}{37431712}e^{10} + \frac{351517}{9357928}e^{8} - \frac{28968439}{37431712}e^{6} + \frac{29111831}{4678964}e^{4} - \frac{77831623}{4678964}e^{2} + \frac{12368623}{1169741}$ |
| 4 | $[4, 2, \frac{1}{2}w^{3} - 4w - \frac{1}{2}]$ | $-\frac{57011}{37431712}e^{10} + \frac{419483}{4678964}e^{8} - \frac{68694095}{37431712}e^{6} + \frac{141169027}{9357928}e^{4} - \frac{199375735}{4678964}e^{2} + \frac{28446106}{1169741}$ |
| 9 | $[9, 3, \frac{1}{2}w^{3} + w^{2} - 3w - \frac{7}{2}]$ | $\phantom{-}e$ |
| 11 | $[11, 11, -\frac{1}{4}w^{3} + \frac{1}{2}w + \frac{7}{4}]$ | $\phantom{-}\frac{701781}{486612256}e^{11} - \frac{5219957}{60826532}e^{9} + \frac{66391149}{37431712}e^{7} - \frac{1787342021}{121653064}e^{5} + \frac{2532465459}{60826532}e^{3} - \frac{320222666}{15206633}e$ |
| 11 | $[11, 11, \frac{1}{2}w^{3} - 4w - \frac{3}{2}]$ | $\phantom{-}\frac{371861}{243306128}e^{11} - \frac{2743555}{30413266}e^{9} + \frac{34357589}{18715856}e^{7} - \frac{896657417}{60826532}e^{5} + \frac{592620349}{15206633}e^{3} - \frac{245214111}{15206633}e$ |
| 19 | $[19, 19, w]$ | $-1$ |
| 19 | $[19, 19, \frac{1}{4}w^{3} - \frac{5}{2}w + \frac{1}{4}]$ | $\phantom{-}\frac{9369}{4678964}e^{10} - \frac{137401}{1169741}e^{8} + \frac{11191789}{4678964}e^{6} - \frac{22906074}{1169741}e^{4} + \frac{65213919}{1169741}e^{2} - \frac{32201656}{1169741}$ |
| 25 | $[25, 5, \frac{1}{2}w^{3} - 3w - \frac{1}{2}]$ | $-\frac{21437}{9357928}e^{10} + \frac{640879}{4678964}e^{8} - \frac{26447689}{9357928}e^{6} + \frac{107768393}{4678964}e^{4} - \frac{72220019}{1169741}e^{2} + \frac{33051428}{1169741}$ |
| 29 | $[29, 29, \frac{3}{4}w^{3} + w^{2} - \frac{9}{2}w - \frac{25}{4}]$ | $\phantom{-}\frac{247123}{486612256}e^{11} - \frac{3611945}{121653064}e^{9} + \frac{22013291}{37431712}e^{7} - \frac{267116085}{60826532}e^{5} + \frac{560205281}{60826532}e^{3} - \frac{19197078}{15206633}e$ |
| 29 | $[29, 29, \frac{1}{2}w^{3} - w^{2} - 3w + \frac{9}{2}]$ | $\phantom{-}\frac{3187355}{486612256}e^{11} - \frac{47318951}{121653064}e^{9} + \frac{299921667}{37431712}e^{7} - \frac{1005109103}{15206633}e^{5} + \frac{11423269407}{60826532}e^{3} - \frac{1584235325}{15206633}e$ |
| 41 | $[41, 41, -w^{3} + 7w + 3]$ | $-\frac{346185}{121653064}e^{11} + \frac{10222491}{60826532}e^{9} - \frac{32228209}{9357928}e^{7} + \frac{1719375205}{60826532}e^{5} - \frac{1208875619}{15206633}e^{3} + \frac{591826161}{15206633}e$ |
| 41 | $[41, 41, -\frac{3}{4}w^{3} + \frac{11}{2}w - \frac{3}{4}]$ | $-\frac{400953}{486612256}e^{11} + \frac{5937855}{121653064}e^{9} - \frac{37443793}{37431712}e^{7} + \frac{494275949}{60826532}e^{5} - \frac{1299251201}{60826532}e^{3} + \frac{50051933}{15206633}e$ |
| 41 | $[41, 41, \frac{1}{4}w^{3} + w^{2} - \frac{5}{2}w - \frac{11}{4}]$ | $-\frac{1841977}{486612256}e^{11} + \frac{27340613}{121653064}e^{9} - \frac{173311777}{37431712}e^{7} + \frac{581247218}{15206633}e^{5} - \frac{6647186027}{60826532}e^{3} + \frac{1041985876}{15206633}e$ |
| 71 | $[71, 71, \frac{1}{2}w^{3} - 4w + \frac{5}{2}]$ | $-\frac{195771}{486612256}e^{11} + \frac{357527}{15206633}e^{9} - \frac{17754531}{37431712}e^{7} + \frac{460292541}{121653064}e^{5} - \frac{577061857}{60826532}e^{3} - \frac{44153700}{15206633}e$ |
| 71 | $[71, 71, \frac{3}{4}w^{3} + w^{2} - \frac{11}{2}w - \frac{21}{4}]$ | $-\frac{104921}{30413266}e^{11} + \frac{3099660}{15206633}e^{9} - \frac{9746659}{2339482}e^{7} + \frac{515961400}{15206633}e^{5} - \frac{1440753155}{15206633}e^{3} + \frac{872771688}{15206633}e$ |
| 79 | $[79, 79, \frac{1}{4}w^{3} - 2w^{2} + \frac{1}{2}w + \frac{25}{4}]$ | $\phantom{-}\frac{16347}{9357928}e^{10} - \frac{115561}{1169741}e^{8} + \frac{17753495}{9357928}e^{6} - \frac{32274917}{2339482}e^{4} + \frac{32956647}{1169741}e^{2} - \frac{4058108}{1169741}$ |
| 79 | $[79, 79, -\frac{3}{4}w^{3} + \frac{13}{2}w + \frac{9}{4}]$ | $\phantom{-}\frac{4601}{9357928}e^{10} - \frac{76715}{2339482}e^{8} + \frac{6584861}{9357928}e^{6} - \frac{6205757}{1169741}e^{4} + \frac{11447963}{1169741}e^{2} + \frac{1443020}{1169741}$ |
| 89 | $[89, 89, -\frac{7}{4}w^{3} - w^{2} + \frac{27}{2}w + \frac{41}{4}]$ | $\phantom{-}\frac{2625061}{243306128}e^{11} - \frac{38781477}{60826532}e^{9} + \frac{244329901}{18715856}e^{7} - \frac{1622786977}{15206633}e^{5} + \frac{9046673253}{30413266}e^{3} - \frac{2415040637}{15206633}e$ |
| 89 | $[89, 89, -\frac{7}{4}w^{3} - w^{2} + \frac{27}{2}w + \frac{33}{4}]$ | $\phantom{-}\frac{236225}{30413266}e^{11} - \frac{28052823}{60826532}e^{9} + \frac{22242603}{2339482}e^{7} - \frac{4785243127}{60826532}e^{5} + \frac{3436142609}{15206633}e^{3} - \frac{2031837357}{15206633}e$ |
| 101 | $[101, 101, \frac{1}{4}w^{3} + w^{2} - \frac{1}{2}w - \frac{27}{4}]$ | $\phantom{-}\frac{1445503}{243306128}e^{11} - \frac{10707067}{30413266}e^{9} + \frac{135106327}{18715856}e^{7} - \frac{3580656855}{60826532}e^{5} + \frac{4902946855}{30413266}e^{3} - \frac{1176493453}{15206633}e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $19$ | $[19, 19, w]$ | $1$ |