Base field 4.4.17600.1
Generator \(w\), with minimal polynomial \(x^{4} - 14x^{2} + 44\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[19,19,-\frac{1}{2}w^{3} + w^{2} + 4w - 9]$ |
| Dimension: | $12$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $34$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{12} - 20x^{10} + 138x^{8} - 428x^{6} + 604x^{4} - 317x^{2} + 20\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -\frac{1}{2}w^{3} + w^{2} + 4w - 8]$ | $\phantom{-}e$ |
| 11 | $[11, 11, \frac{1}{2}w^{3} - \frac{3}{2}w^{2} - 4w + 11]$ | $\phantom{-}\frac{81}{181}e^{10} - \frac{1384}{181}e^{8} + \frac{7197}{181}e^{6} - \frac{14490}{181}e^{4} + \frac{9924}{181}e^{2} - \frac{872}{181}$ |
| 11 | $[11, 11, -\frac{1}{2}w^{2} - w + 2]$ | $-\frac{47}{181}e^{11} + \frac{812}{181}e^{9} - \frac{4290}{181}e^{7} + \frac{8629}{181}e^{5} - \frac{5269}{181}e^{3} - \frac{475}{181}e$ |
| 11 | $[11, 11, -\frac{1}{2}w^{2} + w + 2]$ | $\phantom{-}\frac{28}{181}e^{11} - \frac{503}{181}e^{9} + \frac{2937}{181}e^{7} - \frac{7563}{181}e^{5} + \frac{8646}{181}e^{3} - \frac{3079}{181}e$ |
| 19 | $[19, 19, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - 4w + 5]$ | $\phantom{-}\frac{4}{181}e^{11} - \frac{46}{181}e^{9} - \frac{20}{181}e^{7} + \frac{1195}{181}e^{5} - \frac{3264}{181}e^{3} + \frac{1939}{181}e$ |
| 19 | $[19, 19, \frac{1}{2}w^{3} + w^{2} - 4w - 9]$ | $-\frac{18}{181}e^{10} + \frac{388}{181}e^{8} - \frac{2806}{181}e^{6} + \frac{7926}{181}e^{4} - \frac{7937}{181}e^{2} + \frac{1320}{181}$ |
| 19 | $[19, 19, \frac{1}{2}w^{3} - w^{2} - 4w + 9]$ | $\phantom{-}1$ |
| 19 | $[19, 19, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - 4w - 5]$ | $-\frac{81}{181}e^{11} + \frac{1384}{181}e^{9} - \frac{7197}{181}e^{7} + \frac{14490}{181}e^{5} - \frac{9924}{181}e^{3} + \frac{691}{181}e$ |
| 25 | $[25, 5, \frac{1}{2}w^{2} - 1]$ | $-\frac{8}{181}e^{10} + \frac{92}{181}e^{8} - \frac{141}{181}e^{6} + \frac{144}{181}e^{4} - \frac{1436}{181}e^{2} + \frac{466}{181}$ |
| 29 | $[29, 29, \frac{1}{2}w^{3} - 4w - 1]$ | $-\frac{3}{181}e^{11} + \frac{125}{181}e^{9} - \frac{1433}{181}e^{7} + \frac{5665}{181}e^{5} - \frac{7869}{181}e^{3} + \frac{2211}{181}e$ |
| 29 | $[29, 29, -\frac{1}{2}w^{3} + 4w - 1]$ | $\phantom{-}\frac{85}{181}e^{11} - \frac{1430}{181}e^{9} + \frac{7177}{181}e^{7} - \frac{13295}{181}e^{5} + \frac{6660}{181}e^{3} + \frac{1791}{181}e$ |
| 31 | $[31, 31, -w - 1]$ | $-\frac{8}{181}e^{11} + \frac{92}{181}e^{9} - \frac{141}{181}e^{7} + \frac{325}{181}e^{5} - \frac{3608}{181}e^{3} + \frac{5172}{181}e$ |
| 31 | $[31, 31, w - 1]$ | $\phantom{-}\frac{76}{181}e^{11} - \frac{1417}{181}e^{9} + \frac{8670}{181}e^{7} - \frac{22545}{181}e^{5} + \frac{24502}{181}e^{3} - \frac{8228}{181}e$ |
| 41 | $[41, 41, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - 4w - 2]$ | $-\frac{67}{181}e^{11} + \frac{1223}{181}e^{9} - \frac{7267}{181}e^{7} + \frac{18582}{181}e^{5} - \frac{20805}{181}e^{3} + \frac{8473}{181}e$ |
| 41 | $[41, 41, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + 4w - 2]$ | $-\frac{67}{181}e^{11} + \frac{1223}{181}e^{9} - \frac{7267}{181}e^{7} + \frac{18582}{181}e^{5} - \frac{20805}{181}e^{3} + \frac{8473}{181}e$ |
| 49 | $[49, 7, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + 5w - 7]$ | $-\frac{32}{181}e^{11} + \frac{549}{181}e^{9} - \frac{2917}{181}e^{7} + \frac{6368}{181}e^{5} - \frac{5382}{181}e^{3} + \frac{959}{181}e$ |
| 49 | $[49, 7, \frac{3}{2}w^{3} - \frac{7}{2}w^{2} - 13w + 29]$ | $\phantom{-}\frac{147}{181}e^{11} - \frac{2505}{181}e^{9} + \frac{13021}{181}e^{7} - \frac{26900}{181}e^{5} + \frac{22314}{181}e^{3} - \frac{8065}{181}e$ |
| 59 | $[59, 59, -\frac{1}{2}w^{3} - \frac{3}{2}w^{2} + 3w + 10]$ | $-\frac{85}{181}e^{10} + \frac{1430}{181}e^{8} - \frac{7177}{181}e^{6} + \frac{13476}{181}e^{4} - \frac{8289}{181}e^{2} + \frac{200}{181}$ |
| 59 | $[59, 59, \frac{1}{2}w^{3} - \frac{3}{2}w^{2} - 3w + 10]$ | $-\frac{30}{181}e^{10} + \frac{345}{181}e^{8} - \frac{212}{181}e^{6} - \frac{3623}{181}e^{4} + \frac{5656}{181}e^{2} - \frac{1420}{181}$ |
| 61 | $[61, 61, \frac{1}{2}w^{2} + w - 6]$ | $-\frac{120}{181}e^{10} + \frac{1923}{181}e^{8} - \frac{8631}{181}e^{6} + \frac{12477}{181}e^{4} - \frac{2535}{181}e^{2} - \frac{612}{181}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $19$ | $[19,19,-\frac{1}{2}w^{3} + w^{2} + 4w - 9]$ | $-1$ |