Base field 4.4.8000.1
Generator \(w\), with minimal polynomial \(x^{4} - 10x^{2} + 20\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16, 2, 2]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $12$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 4x^{7} - 92x^{6} + 528x^{5} + 352x^{4} - 3776x^{3} - 1728x^{2} + 6912x + 4864\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, \frac{1}{2}w^{3} - 3w + 2]$ | $\phantom{-}0$ |
| 5 | $[5, 5, w^{2} - w - 5]$ | $-\frac{591}{123008}e^{7} + \frac{781}{30752}e^{6} + \frac{6261}{15376}e^{5} - \frac{189}{62}e^{4} + \frac{4701}{1922}e^{3} + \frac{13185}{961}e^{2} - \frac{18861}{1922}e - \frac{15494}{961}$ |
| 11 | $[11, 11, -\frac{1}{2}w^{3} - \frac{1}{2}w^{2} + 3w + 3]$ | $\phantom{-}\frac{591}{123008}e^{7} - \frac{781}{30752}e^{6} - \frac{6261}{15376}e^{5} + \frac{189}{62}e^{4} - \frac{4701}{1922}e^{3} - \frac{13185}{961}e^{2} + \frac{20783}{1922}e + \frac{15494}{961}$ |
| 11 | $[11, 11, -\frac{1}{2}w^{2} + w + 2]$ | $-\frac{243}{123008}e^{7} + \frac{163}{15376}e^{6} + \frac{2467}{15376}e^{5} - \frac{311}{248}e^{4} + \frac{1525}{961}e^{3} + \frac{6057}{1922}e^{2} - \frac{6555}{1922}e + \frac{1132}{961}$ |
| 11 | $[11, 11, -\frac{1}{2}w^{2} - w + 2]$ | $-\frac{243}{123008}e^{7} + \frac{163}{15376}e^{6} + \frac{2467}{15376}e^{5} - \frac{311}{248}e^{4} + \frac{1525}{961}e^{3} + \frac{6057}{1922}e^{2} - \frac{6555}{1922}e + \frac{1132}{961}$ |
| 11 | $[11, 11, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - 3w + 3]$ | $\phantom{-}\frac{591}{123008}e^{7} - \frac{781}{30752}e^{6} - \frac{6261}{15376}e^{5} + \frac{189}{62}e^{4} - \frac{4701}{1922}e^{3} - \frac{13185}{961}e^{2} + \frac{20783}{1922}e + \frac{15494}{961}$ |
| 29 | $[29, 29, -\frac{1}{2}w^{2} - w + 4]$ | $\phantom{-}\frac{79}{123008}e^{7} + \frac{53}{61504}e^{6} - \frac{577}{7688}e^{5} + \frac{9}{248}e^{4} + \frac{4201}{1922}e^{3} - \frac{2782}{961}e^{2} - \frac{18133}{1922}e + \frac{4785}{961}$ |
| 29 | $[29, 29, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + 3w - 1]$ | $\phantom{-}\frac{11}{123008}e^{7} + \frac{275}{61504}e^{6} - \frac{129}{7688}e^{5} - \frac{89}{248}e^{4} + \frac{1588}{961}e^{3} + \frac{379}{961}e^{2} - \frac{5493}{1922}e - \frac{1645}{961}$ |
| 29 | $[29, 29, -\frac{1}{2}w^{3} - \frac{1}{2}w^{2} + 3w + 1]$ | $\phantom{-}\frac{11}{123008}e^{7} + \frac{275}{61504}e^{6} - \frac{129}{7688}e^{5} - \frac{89}{248}e^{4} + \frac{1588}{961}e^{3} + \frac{379}{961}e^{2} - \frac{5493}{1922}e - \frac{1645}{961}$ |
| 29 | $[29, 29, -\frac{1}{2}w^{2} + w + 4]$ | $\phantom{-}\frac{79}{123008}e^{7} + \frac{53}{61504}e^{6} - \frac{577}{7688}e^{5} + \frac{9}{248}e^{4} + \frac{4201}{1922}e^{3} - \frac{2782}{961}e^{2} - \frac{18133}{1922}e + \frac{4785}{961}$ |
| 41 | $[41, 41, w^{3} - \frac{1}{2}w^{2} - 6w + 6]$ | $\phantom{-}\frac{517}{123008}e^{7} - \frac{745}{30752}e^{6} - \frac{5399}{15376}e^{5} + \frac{715}{248}e^{4} - \frac{5449}{1922}e^{3} - \frac{14861}{961}e^{2} + \frac{26285}{1922}e + \frac{27434}{961}$ |
| 41 | $[41, 41, -\frac{1}{2}w^{3} + \frac{5}{2}w^{2} + 5w - 14]$ | $-\frac{517}{123008}e^{7} + \frac{745}{30752}e^{6} + \frac{5399}{15376}e^{5} - \frac{715}{248}e^{4} + \frac{5449}{1922}e^{3} + \frac{14861}{961}e^{2} - \frac{26285}{1922}e - \frac{23590}{961}$ |
| 41 | $[41, 41, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - 3w + 6]$ | $-\frac{517}{123008}e^{7} + \frac{745}{30752}e^{6} + \frac{5399}{15376}e^{5} - \frac{715}{248}e^{4} + \frac{5449}{1922}e^{3} + \frac{14861}{961}e^{2} - \frac{26285}{1922}e - \frac{23590}{961}$ |
| 41 | $[41, 41, -\frac{3}{2}w^{2} - 2w + 6]$ | $\phantom{-}\frac{517}{123008}e^{7} - \frac{745}{30752}e^{6} - \frac{5399}{15376}e^{5} + \frac{715}{248}e^{4} - \frac{5449}{1922}e^{3} - \frac{14861}{961}e^{2} + \frac{26285}{1922}e + \frac{27434}{961}$ |
| 79 | $[79, 79, -w^{3} - w^{2} + 4w - 1]$ | $-\frac{159}{15376}e^{7} + \frac{699}{15376}e^{6} + \frac{6967}{7688}e^{5} - \frac{179}{31}e^{4} + \frac{2025}{1922}e^{3} + \frac{28773}{961}e^{2} - \frac{7048}{961}e - \frac{34128}{961}$ |
| 79 | $[79, 79, -\frac{3}{2}w^{2} - w + 9]$ | $-\frac{159}{15376}e^{7} + \frac{699}{15376}e^{6} + \frac{6967}{7688}e^{5} - \frac{179}{31}e^{4} + \frac{2025}{1922}e^{3} + \frac{28773}{961}e^{2} - \frac{7048}{961}e - \frac{34128}{961}$ |
| 79 | $[79, 79, -w^{3} + \frac{7}{2}w^{2} + 9w - 21]$ | $-\frac{159}{15376}e^{7} + \frac{699}{15376}e^{6} + \frac{6967}{7688}e^{5} - \frac{179}{31}e^{4} + \frac{2025}{1922}e^{3} + \frac{28773}{961}e^{2} - \frac{7048}{961}e - \frac{34128}{961}$ |
| 79 | $[79, 79, w^{3} - w^{2} - 6w + 9]$ | $-\frac{159}{15376}e^{7} + \frac{699}{15376}e^{6} + \frac{6967}{7688}e^{5} - \frac{179}{31}e^{4} + \frac{2025}{1922}e^{3} + \frac{28773}{961}e^{2} - \frac{7048}{961}e - \frac{34128}{961}$ |
| 81 | $[81, 3, -3]$ | $\phantom{-}10$ |
| 109 | $[109, 109, w^{2} - w - 7]$ | $\phantom{-}\frac{249}{123008}e^{7} - \frac{251}{30752}e^{6} - \frac{2433}{15376}e^{5} + \frac{127}{124}e^{4} - \frac{3327}{1922}e^{3} + \frac{1328}{961}e^{2} + \frac{7927}{1922}e - \frac{4126}{961}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, \frac{1}{2}w^{3} - 3w + 2]$ | $1$ |