Base field 4.4.13025.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 12x^{2} + 3x + 29\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16, 4, \frac{1}{4}w^{3} - \frac{1}{2}w^{2} - \frac{5}{2}w + \frac{13}{4}]$ |
| Dimension: | $8$ |
| 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:
| \(x^{8} - 5x^{7} - 6x^{6} + 62x^{5} - 62x^{4} - 91x^{3} + 140x^{2} - 34x - 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, \frac{1}{2}w^{3} - 2w^{2} - 2w + \frac{15}{2}]$ | $\phantom{-}0$ |
| 4 | $[4, 2, -w^{2} + w + 8]$ | $\phantom{-}e$ |
| 5 | $[5, 5, -\frac{1}{4}w^{3} + \frac{1}{2}w^{2} + \frac{1}{2}w - \frac{9}{4}]$ | $-\frac{1}{82}e^{7} - \frac{27}{82}e^{6} + \frac{85}{82}e^{5} + \frac{321}{82}e^{4} - \frac{941}{82}e^{3} - \frac{189}{41}e^{2} + \frac{1499}{82}e - \frac{148}{41}$ |
| 5 | $[5, 5, \frac{1}{2}w^{3} - w^{2} - 4w + \frac{9}{2}]$ | $\phantom{-}\frac{11}{82}e^{7} - \frac{31}{82}e^{6} - \frac{115}{82}e^{5} + \frac{405}{82}e^{4} - \frac{63}{82}e^{3} - \frac{381}{41}e^{2} + \frac{649}{82}e + \frac{70}{41}$ |
| 19 | $[19, 19, \frac{1}{4}w^{3} - \frac{3}{2}w^{2} - \frac{1}{2}w + \frac{21}{4}]$ | $-\frac{17}{41}e^{7} + \frac{74}{41}e^{6} + \frac{133}{41}e^{5} - \frac{939}{41}e^{4} + \frac{649}{41}e^{3} + \frac{1569}{41}e^{2} - \frac{1577}{41}e + \frac{216}{41}$ |
| 19 | $[19, 19, \frac{1}{4}w^{3} - \frac{3}{2}w^{2} - \frac{1}{2}w + \frac{41}{4}]$ | $\phantom{-}\frac{2}{41}e^{7} - \frac{28}{41}e^{6} + \frac{35}{41}e^{5} + \frac{342}{41}e^{4} - \frac{701}{41}e^{3} - \frac{474}{41}e^{2} + \frac{1102}{41}e - \frac{146}{41}$ |
| 29 | $[29, 29, w]$ | $\phantom{-}\frac{35}{82}e^{7} - \frac{81}{41}e^{6} - \frac{155}{41}e^{5} + \frac{1045}{41}e^{4} - \frac{445}{41}e^{3} - \frac{3949}{82}e^{2} + \frac{2393}{82}e + \frac{397}{82}$ |
| 29 | $[29, 29, \frac{1}{4}w^{3} - \frac{1}{2}w^{2} - \frac{1}{2}w + \frac{1}{4}]$ | $-\frac{1}{41}e^{7} - \frac{13}{82}e^{6} + \frac{47}{82}e^{5} + \frac{191}{82}e^{4} - \frac{283}{82}e^{3} - \frac{633}{82}e^{2} + \frac{23}{41}e + \frac{433}{82}$ |
| 29 | $[29, 29, \frac{1}{4}w^{3} - \frac{1}{2}w^{2} - \frac{5}{2}w + \frac{9}{4}]$ | $-\frac{9}{82}e^{7} + \frac{22}{41}e^{6} + \frac{34}{41}e^{5} - \frac{298}{41}e^{4} + \frac{173}{41}e^{3} + \frac{1313}{82}e^{2} - \frac{777}{82}e + \frac{247}{82}$ |
| 29 | $[29, 29, -\frac{1}{2}w^{3} + w^{2} + 4w - \frac{5}{2}]$ | $\phantom{-}\frac{3}{41}e^{7} - \frac{43}{82}e^{6} + \frac{23}{82}e^{5} + \frac{493}{82}e^{4} - \frac{1119}{82}e^{3} - \frac{315}{82}e^{2} + \frac{1161}{41}e - \frac{725}{82}$ |
| 41 | $[41, 41, \frac{3}{4}w^{3} - \frac{5}{2}w^{2} - \frac{7}{2}w + \frac{47}{4}]$ | $-\frac{3}{82}e^{7} + \frac{21}{41}e^{6} - \frac{16}{41}e^{5} - \frac{277}{41}e^{4} + \frac{413}{41}e^{3} + \frac{1203}{82}e^{2} - \frac{1407}{82}e - \frac{355}{82}$ |
| 41 | $[41, 41, \frac{1}{4}w^{3} + \frac{1}{2}w^{2} - \frac{5}{2}w - \frac{15}{4}]$ | $\phantom{-}\frac{73}{82}e^{7} - \frac{142}{41}e^{6} - \frac{335}{41}e^{5} + \frac{1793}{41}e^{4} - \frac{770}{41}e^{3} - \frac{5821}{82}e^{2} + \frac{4881}{82}e - \frac{737}{82}$ |
| 61 | $[61, 61, -\frac{3}{2}w^{3} + 3w^{2} + 11w - \frac{21}{2}]$ | $-\frac{34}{41}e^{7} + \frac{337}{82}e^{6} + \frac{491}{82}e^{5} - \frac{4289}{82}e^{4} + \frac{3129}{82}e^{3} + \frac{7465}{82}e^{2} - \frac{3646}{41}e + \frac{741}{82}$ |
| 61 | $[61, 61, w^{3} - 2w^{2} - 4w + 6]$ | $\phantom{-}\frac{16}{41}e^{7} - \frac{161}{82}e^{6} - \frac{219}{82}e^{5} + \frac{1987}{82}e^{4} - \frac{1663}{82}e^{3} - \frac{2951}{82}e^{2} + \frac{1969}{41}e - \frac{901}{82}$ |
| 79 | $[79, 79, \frac{3}{4}w^{3} - \frac{5}{2}w^{2} - \frac{7}{2}w + \frac{27}{4}]$ | $\phantom{-}\frac{16}{41}e^{7} - \frac{60}{41}e^{6} - \frac{130}{41}e^{5} + \frac{768}{41}e^{4} - \frac{606}{41}e^{3} - \frac{1332}{41}e^{2} + \frac{2010}{41}e - \frac{20}{41}$ |
| 79 | $[79, 79, \frac{1}{4}w^{3} + \frac{1}{2}w^{2} - \frac{5}{2}w - \frac{35}{4}]$ | $-\frac{9}{41}e^{7} + \frac{44}{41}e^{6} + \frac{68}{41}e^{5} - \frac{555}{41}e^{4} + \frac{428}{41}e^{3} + \frac{903}{41}e^{2} - \frac{1351}{41}e + \frac{370}{41}$ |
| 81 | $[81, 3, -3]$ | $-\frac{31}{41}e^{7} + \frac{147}{41}e^{6} + \frac{216}{41}e^{5} - \frac{1857}{41}e^{4} + \frac{1538}{41}e^{3} + \frac{3042}{41}e^{2} - \frac{3633}{41}e + \frac{500}{41}$ |
| 89 | $[89, 89, \frac{7}{4}w^{3} - \frac{11}{2}w^{2} - \frac{21}{2}w + \frac{119}{4}]$ | $\phantom{-}\frac{12}{41}e^{7} - \frac{213}{82}e^{6} - \frac{31}{82}e^{5} + \frac{2751}{82}e^{4} - \frac{2959}{82}e^{3} - \frac{5401}{82}e^{2} + \frac{2717}{41}e - \frac{71}{82}$ |
| 89 | $[89, 89, \frac{1}{4}w^{3} + \frac{3}{2}w^{2} - \frac{3}{2}w - \frac{23}{4}]$ | $\phantom{-}\frac{11}{41}e^{7} - \frac{21}{82}e^{6} - \frac{271}{82}e^{5} + \frac{195}{82}e^{4} + \frac{407}{82}e^{3} + \frac{567}{82}e^{2} + \frac{403}{41}e - \frac{1155}{82}$ |
| 109 | $[109, 109, -\frac{1}{2}w^{3} + 3w^{2} + 2w - \frac{41}{2}]$ | $-\frac{21}{82}e^{7} + \frac{24}{41}e^{6} + \frac{134}{41}e^{5} - \frac{299}{41}e^{4} - \frac{225}{41}e^{3} + \frac{959}{82}e^{2} - \frac{337}{82}e - \frac{189}{82}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, \frac{1}{2}w^{3} - 2w^{2} - 2w + \frac{15}{2}]$ | $1$ |