Base field 4.4.13725.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 13x^{2} + x + 31\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[19, 19, -w - 1]$ |
| Dimension: | $14$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $22$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{14} - 109x^{12} + 4711x^{10} - 101035x^{8} + 1081928x^{6} - 4802880x^{4} + 2626048x^{2} - 299008\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 9 | $[9, 3, -2w^{3} + 6w^{2} + 13w - 26]$ | $\phantom{-}\frac{3506309}{327248403712}e^{12} - \frac{261079129}{327248403712}e^{10} + \frac{6680328795}{327248403712}e^{8} - \frac{58572765215}{327248403712}e^{6} - \frac{7534165987}{20453025232}e^{4} + \frac{46223698429}{5113256308}e^{2} + \frac{583589068}{1278314077}$ |
| 11 | $[11, 11, w^{2} - w - 8]$ | $\phantom{-}e$ |
| 11 | $[11, 11, -4w^{3} + 13w^{2} + 23w - 55]$ | $-\frac{18174691}{654496807424}e^{13} + \frac{1736324615}{654496807424}e^{11} - \frac{65000023541}{654496807424}e^{9} + \frac{1192975040897}{654496807424}e^{7} - \frac{1349873928903}{81812100928}e^{5} + \frac{310993695757}{5113256308}e^{3} - \frac{26659357863}{1278314077}e$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}\frac{6636649}{163624201856}e^{12} - \frac{540241633}{163624201856}e^{10} + \frac{16066658619}{163624201856}e^{8} - \frac{203932282871}{163624201856}e^{6} + \frac{215349395547}{40906050464}e^{4} + \frac{16630414871}{2556628154}e^{2} - \frac{3348438101}{1278314077}$ |
| 19 | $[19, 19, -w - 1]$ | $-1$ |
| 19 | $[19, 19, -w^{3} + 3w^{2} + 7w - 14]$ | $\phantom{-}\frac{27643929}{327248403712}e^{12} - \frac{2436890365}{327248403712}e^{10} + \frac{80101223239}{327248403712}e^{8} - \frac{1179430705819}{327248403712}e^{6} + \frac{440263422265}{20453025232}e^{4} - \frac{34495033036}{1278314077}e^{2} + \frac{14401980434}{1278314077}$ |
| 19 | $[19, 19, w^{3} - 4w^{2} - 5w + 20]$ | $-\frac{218739}{10226512616}e^{12} + \frac{42413329}{20453025232}e^{10} - \frac{1595477893}{20453025232}e^{8} + \frac{28845453959}{20453025232}e^{6} - \frac{248746293565}{20453025232}e^{4} + \frac{104081239881}{2556628154}e^{2} - \frac{12210490920}{1278314077}$ |
| 19 | $[19, 19, -3w^{3} + 10w^{2} + 17w - 43]$ | $\phantom{-}\frac{14107243}{81812100928}e^{12} - \frac{1217684841}{81812100928}e^{10} + \frac{39069383151}{81812100928}e^{8} - \frac{557887515983}{81812100928}e^{6} + \frac{1584247750627}{40906050464}e^{4} - \frac{198187969809}{5113256308}e^{2} + \frac{13426948042}{1278314077}$ |
| 25 | $[25, 5, -w^{3} + 3w^{2} + 6w - 10]$ | $\phantom{-}\frac{618705}{163624201856}e^{12} - \frac{50822725}{163624201856}e^{10} + \frac{1975924511}{163624201856}e^{8} - \frac{49123475283}{163624201856}e^{6} + \frac{11156237253}{2556628154}e^{4} - \frac{33503639909}{1278314077}e^{2} + \frac{12157897978}{1278314077}$ |
| 29 | $[29, 29, 4w^{3} - 13w^{2} - 23w + 57]$ | $-\frac{2634001}{163624201856}e^{13} + \frac{166387359}{163624201856}e^{11} - \frac{2246377233}{163624201856}e^{9} - \frac{49801602743}{163624201856}e^{7} + \frac{734311106529}{81812100928}e^{5} - \frac{621953439471}{10226512616}e^{3} + \frac{82650844697}{2556628154}e$ |
| 29 | $[29, 29, w^{2} - w - 6]$ | $-\frac{15242809}{327248403712}e^{13} + \frac{1304828093}{327248403712}e^{11} - \frac{41108671143}{327248403712}e^{9} + \frac{562772985595}{327248403712}e^{7} - \frac{175402517161}{20453025232}e^{5} - \frac{2158849617}{10226512616}e^{3} + \frac{2784664847}{2556628154}e$ |
| 31 | $[31, 31, -2w^{3} + 7w^{2} + 11w - 31]$ | $-\frac{10386631}{81812100928}e^{12} + \frac{882183977}{81812100928}e^{10} - \frac{27309620471}{81812100928}e^{8} + \frac{358421226671}{81812100928}e^{6} - \frac{770061165649}{40906050464}e^{4} - \frac{124179715421}{5113256308}e^{2} + \frac{14449843338}{1278314077}$ |
| 31 | $[31, 31, -2w^{3} + 7w^{2} + 11w - 32]$ | $\phantom{-}\frac{59550571}{327248403712}e^{12} - \frac{5128398759}{327248403712}e^{10} + \frac{164170965637}{327248403712}e^{8} - \frac{2342750921121}{327248403712}e^{6} + \frac{419117819087}{10226512616}e^{4} - \frac{57740048166}{1278314077}e^{2} + \frac{14934716930}{1278314077}$ |
| 41 | $[41, 41, 3w^{3} - 10w^{2} - 18w + 43]$ | $\phantom{-}\frac{11635339}{1308993614848}e^{13} - \frac{1660066327}{1308993614848}e^{11} + \frac{88800157941}{1308993614848}e^{9} - \frac{2264706949233}{1308993614848}e^{7} + \frac{1739746492157}{81812100928}e^{5} - \frac{2108034456227}{20453025232}e^{3} + \frac{43229621997}{1278314077}e$ |
| 41 | $[41, 41, 3w^{3} - 9w^{2} - 19w + 37]$ | $-\frac{39858919}{1308993614848}e^{13} + \frac{3196473715}{1308993614848}e^{11} - \frac{91030460265}{1308993614848}e^{9} + \frac{1015876560693}{1308993614848}e^{7} - \frac{30567844003}{20453025232}e^{5} - \frac{357309648451}{20453025232}e^{3} - \frac{151110949}{1278314077}e$ |
| 41 | $[41, 41, 2w^{3} - 6w^{2} - 11w + 24]$ | $\phantom{-}\frac{17711167}{1308993614848}e^{13} - \frac{1671224283}{1308993614848}e^{11} + \frac{61322158465}{1308993614848}e^{9} - \frac{1088362155533}{1308993614848}e^{7} + \frac{581449530085}{81812100928}e^{5} - \frac{487951836397}{20453025232}e^{3} + \frac{10866199548}{1278314077}e$ |
| 41 | $[41, 41, -2w^{3} + 7w^{2} + 12w - 33]$ | $-\frac{20063751}{654496807424}e^{13} + \frac{2082738379}{654496807424}e^{11} - \frac{85683071009}{654496807424}e^{9} + \frac{1748061224221}{654496807424}e^{7} - \frac{2219851419941}{81812100928}e^{5} + \frac{575263555551}{5113256308}e^{3} - \frac{96846788991}{2556628154}e$ |
| 59 | $[59, 59, 3w^{3} - 10w^{2} - 17w + 46]$ | $-\frac{2067803}{654496807424}e^{13} + \frac{136220703}{654496807424}e^{11} - \frac{1842135101}{654496807424}e^{9} - \frac{47518799831}{654496807424}e^{7} + \frac{170441703201}{81812100928}e^{5} - \frac{16236434208}{1278314077}e^{3} - \frac{15181811639}{1278314077}e$ |
| 59 | $[59, 59, -w^{3} + 4w^{2} + 5w - 17]$ | $\phantom{-}\frac{54041065}{1308993614848}e^{13} - \frac{4382274237}{1308993614848}e^{11} + \frac{124356351719}{1308993614848}e^{9} - \frac{1305783130523}{1308993614848}e^{7} + \frac{9407968129}{40906050464}e^{5} + \frac{776975093435}{20453025232}e^{3} - \frac{11573950579}{2556628154}e$ |
| 61 | $[61, 61, 4w^{3} - 12w^{2} - 25w + 50]$ | $\phantom{-}\frac{9348359}{81812100928}e^{12} - \frac{803322071}{81812100928}e^{10} + \frac{25642104525}{81812100928}e^{8} - \frac{365101296353}{81812100928}e^{6} + \frac{524952781999}{20453025232}e^{4} - \frac{157300879425}{5113256308}e^{2} + \frac{12887850970}{1278314077}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $19$ | $[19, 19, -w - 1]$ | $1$ |