Base field 4.4.18097.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 7x^{2} + 6x + 4\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16, 2, 2]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $24$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} - 2x^{8} - 17x^{7} + 30x^{6} + 86x^{5} - 104x^{4} - 163x^{3} + 66x^{2} + 44x - 16\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, -w + 1]$ | $\phantom{-}e$ |
| 4 | $[4, 2, w]$ | $\phantom{-}1$ |
| 4 | $[4, 2, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{7}{2}w - 3]$ | $\phantom{-}1$ |
| 7 | $[7, 7, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{5}{2}w - 2]$ | $\phantom{-}e^{8} - \frac{3}{2}e^{7} - 18e^{6} + \frac{43}{2}e^{5} + 100e^{4} - 61e^{3} - 202e^{2} - \frac{33}{2}e + 42$ |
| 13 | $[13, 13, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{7}{2}w - 1]$ | $-\frac{3}{4}e^{8} + e^{7} + \frac{51}{4}e^{6} - 14e^{5} - \frac{129}{2}e^{4} + 36e^{3} + \frac{469}{4}e^{2} + 12e - 22$ |
| 17 | $[17, 17, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{5}{2}w]$ | $\phantom{-}e^{8} - \frac{3}{2}e^{7} - 17e^{6} + \frac{43}{2}e^{5} + 86e^{4} - 61e^{3} - 158e^{2} - \frac{7}{2}e + 32$ |
| 27 | $[27, 3, -w^{3} + 7w + 1]$ | $\phantom{-}\frac{1}{2}e^{8} - e^{7} - \frac{17}{2}e^{6} + 14e^{5} + 43e^{4} - 39e^{3} - \frac{165}{2}e^{2} - 3e + 16$ |
| 31 | $[31, 31, w + 3]$ | $\phantom{-}\frac{7}{4}e^{8} - 3e^{7} - \frac{123}{4}e^{6} + 43e^{5} + \frac{331}{2}e^{4} - 124e^{3} - \frac{1337}{4}e^{2} - 17e + 72$ |
| 31 | $[31, 31, -w^{2} + 5]$ | $\phantom{-}\frac{1}{2}e^{8} - e^{7} - \frac{17}{2}e^{6} + 14e^{5} + 43e^{4} - 39e^{3} - \frac{165}{2}e^{2} - 2e + 16$ |
| 37 | $[37, 37, w^{2} - 3]$ | $\phantom{-}2e^{8} - 3e^{7} - 35e^{6} + 43e^{5} + 186e^{4} - 122e^{3} - 359e^{2} - 20e + 70$ |
| 41 | $[41, 41, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{3}{2}w - 2]$ | $-e^{8} + 2e^{7} + 18e^{6} - 29e^{5} - 100e^{4} + 90e^{3} + 207e^{2} - 7e - 42$ |
| 47 | $[47, 47, w^{3} - 5w - 3]$ | $-e^{8} + \frac{3}{2}e^{7} + 17e^{6} - \frac{43}{2}e^{5} - 85e^{4} + 62e^{3} + 149e^{2} - \frac{3}{2}e - 26$ |
| 53 | $[53, 53, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w]$ | $\phantom{-}\frac{5}{2}e^{8} - 4e^{7} - \frac{87}{2}e^{6} + 57e^{5} + 229e^{4} - 161e^{3} - \frac{883}{2}e^{2} - 26e + 86$ |
| 61 | $[61, 61, w^{3} - w^{2} - 5w + 3]$ | $\phantom{-}\frac{3}{4}e^{8} - e^{7} - \frac{55}{4}e^{6} + 14e^{5} + \frac{157}{2}e^{4} - 36e^{3} - \frac{649}{4}e^{2} - 25e + 38$ |
| 83 | $[83, 83, w^{3} + w^{2} - 6w - 7]$ | $\phantom{-}\frac{1}{2}e^{7} + e^{6} - \frac{15}{2}e^{5} - 14e^{4} + 27e^{3} + 48e^{2} + \frac{11}{2}e - 2$ |
| 83 | $[83, 83, -w^{3} + 5w + 1]$ | $-\frac{5}{2}e^{8} + 4e^{7} + \frac{87}{2}e^{6} - 57e^{5} - 230e^{4} + 160e^{3} + \frac{905}{2}e^{2} + 27e - 96$ |
| 83 | $[83, 83, 2w - 1]$ | $\phantom{-}\frac{11}{4}e^{8} - 4e^{7} - \frac{195}{4}e^{6} + 57e^{5} + \frac{531}{2}e^{4} - 156e^{3} - \frac{2125}{4}e^{2} - 59e + 116$ |
| 83 | $[83, 83, -\frac{3}{2}w^{3} - \frac{1}{2}w^{2} + \frac{19}{2}w + 2]$ | $-\frac{1}{2}e^{8} + e^{7} + \frac{19}{2}e^{6} - 14e^{5} - 57e^{4} + 39e^{3} + \frac{255}{2}e^{2} + 14e - 28$ |
| 89 | $[89, 89, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w - 2]$ | $-\frac{5}{2}e^{8} + 4e^{7} + \frac{87}{2}e^{6} - 58e^{5} - 229e^{4} + 172e^{3} + \frac{879}{2}e^{2} + 5e - 82$ |
| 89 | $[89, 89, w^{3} - 5w + 5]$ | $-\frac{5}{2}e^{8} + 4e^{7} + \frac{87}{2}e^{6} - 57e^{5} - 229e^{4} + 160e^{3} + \frac{883}{2}e^{2} + 29e - 82$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, w]$ | $-1$ |
| $4$ | $[4, 2, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{7}{2}w - 3]$ | $-1$ |