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: | $[19, 19, w^{3} - 4w^{2} - 5w + 20]$ |
| Dimension: | $8$ |
| 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:
| \(x^{8} - 77x^{6} + 1744x^{4} - 12772x^{2} + 20480\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 9 | $[9, 3, -2w^{3} + 6w^{2} + 13w - 26]$ | $\phantom{-}\frac{61}{53632}e^{6} - \frac{4411}{53632}e^{4} + \frac{38895}{26816}e^{2} - \frac{895}{419}$ |
| 11 | $[11, 11, w^{2} - w - 8]$ | $\phantom{-}e$ |
| 11 | $[11, 11, -4w^{3} + 13w^{2} + 23w - 55]$ | $\phantom{-}\frac{15}{107264}e^{7} - \frac{1277}{107264}e^{5} + \frac{17491}{53632}e^{3} - \frac{43395}{13408}e$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}3$ |
| 19 | $[19, 19, -w - 1]$ | $-\frac{79}{53632}e^{6} + \frac{5273}{53632}e^{4} - \frac{36085}{26816}e^{2} + \frac{850}{419}$ |
| 19 | $[19, 19, -w^{3} + 3w^{2} + 7w - 14]$ | $\phantom{-}\frac{165}{53632}e^{6} - \frac{12371}{53632}e^{4} + \frac{119495}{26816}e^{2} - \frac{6920}{419}$ |
| 19 | $[19, 19, w^{3} - 4w^{2} - 5w + 20]$ | $-1$ |
| 19 | $[19, 19, -3w^{3} + 10w^{2} + 17w - 43]$ | $\phantom{-}\frac{79}{53632}e^{6} - \frac{5273}{53632}e^{4} + \frac{36085}{26816}e^{2} + \frac{1245}{419}$ |
| 25 | $[25, 5, -w^{3} + 3w^{2} + 6w - 10]$ | $-\frac{61}{53632}e^{6} + \frac{4411}{53632}e^{4} - \frac{38895}{26816}e^{2} + \frac{1314}{419}$ |
| 29 | $[29, 29, 4w^{3} - 13w^{2} - 23w + 57]$ | $-\frac{149}{107264}e^{7} + \frac{10115}{107264}e^{5} - \frac{86983}{53632}e^{3} + \frac{11825}{1676}e$ |
| 29 | $[29, 29, w^{2} - w - 6]$ | $\phantom{-}\frac{1}{53632}e^{7} - \frac{141}{53632}e^{5} + \frac{673}{3352}e^{3} - \frac{54809}{13408}e$ |
| 31 | $[31, 31, -2w^{3} + 7w^{2} + 11w - 31]$ | $-\frac{97}{26816}e^{6} + \frac{6135}{26816}e^{4} - \frac{46683}{13408}e^{2} + \frac{4962}{419}$ |
| 31 | $[31, 31, -2w^{3} + 7w^{2} + 11w - 32]$ | $\phantom{-}\frac{151}{53632}e^{6} - \frac{8721}{53632}e^{4} + \frac{51661}{26816}e^{2} + \frac{587}{419}$ |
| 41 | $[41, 41, 3w^{3} - 10w^{2} - 18w + 43]$ | $-\frac{3}{6704}e^{7} + \frac{1273}{26816}e^{5} - \frac{39131}{26816}e^{3} + \frac{151015}{13408}e$ |
| 41 | $[41, 41, 3w^{3} - 9w^{2} - 19w + 37]$ | $\phantom{-}\frac{17}{107264}e^{7} - \frac{1559}{107264}e^{5} + \frac{28259}{53632}e^{3} - \frac{24551}{3352}e$ |
| 41 | $[41, 41, 2w^{3} - 6w^{2} - 11w + 24]$ | $\phantom{-}\frac{67}{53632}e^{7} - \frac{4419}{53632}e^{5} + \frac{17373}{13408}e^{3} - \frac{51205}{13408}e$ |
| 41 | $[41, 41, -2w^{3} + 7w^{2} + 12w - 33]$ | $\phantom{-}\frac{67}{53632}e^{7} - \frac{4419}{53632}e^{5} + \frac{17373}{13408}e^{3} - \frac{51205}{13408}e$ |
| 59 | $[59, 59, 3w^{3} - 10w^{2} - 17w + 46]$ | $-\frac{83}{53632}e^{7} + \frac{5837}{53632}e^{5} - \frac{57621}{26816}e^{3} + \frac{44849}{3352}e$ |
| 59 | $[59, 59, -w^{3} + 4w^{2} + 5w - 17]$ | $\phantom{-}\frac{9}{3352}e^{7} - \frac{1281}{6704}e^{5} + \frac{24129}{6704}e^{3} - \frac{52009}{3352}e$ |
| 61 | $[61, 61, 4w^{3} - 12w^{2} - 25w + 50]$ | $-\frac{97}{53632}e^{6} + \frac{6135}{53632}e^{4} - \frac{33275}{26816}e^{2} - \frac{33}{419}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $19$ | $[19, 19, w^{3} - 4w^{2} - 5w + 20]$ | $1$ |