Base field \(\Q(\sqrt{229}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 57\); narrow class number \(3\) and class number \(3\).
Form
Weight: | $[2, 2]$ |
Level: | $[4, 2, 2]$ |
Dimension: | $8$ |
CM: | no |
Base change: | yes |
Newspace dimension: | $60$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{8} + 3x^{7} - 15x^{6} - 44x^{5} + 66x^{4} + 191x^{3} - 75x^{2} - 234x - 4\) |
Show full eigenvalues Hide large eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
3 | $[3, 3, w]$ | $\phantom{-}e$ |
3 | $[3, 3, w + 2]$ | $\phantom{-}e$ |
4 | $[4, 2, 2]$ | $-1$ |
5 | $[5, 5, w + 1]$ | $-\frac{221}{722}e^{7} - \frac{275}{722}e^{6} + \frac{3739}{722}e^{5} + \frac{1570}{361}e^{4} - \frac{9543}{361}e^{3} - \frac{7827}{722}e^{2} + \frac{1413}{38}e - \frac{59}{361}$ |
5 | $[5, 5, w + 3]$ | $-\frac{221}{722}e^{7} - \frac{275}{722}e^{6} + \frac{3739}{722}e^{5} + \frac{1570}{361}e^{4} - \frac{9543}{361}e^{3} - \frac{7827}{722}e^{2} + \frac{1413}{38}e - \frac{59}{361}$ |
11 | $[11, 11, w + 1]$ | $-\frac{15}{361}e^{7} - \frac{35}{361}e^{6} + \frac{128}{361}e^{5} + \frac{334}{361}e^{4} + \frac{111}{361}e^{3} - \frac{773}{361}e^{2} - \frac{72}{19}e + \frac{812}{361}$ |
11 | $[11, 11, w + 9]$ | $-\frac{15}{361}e^{7} - \frac{35}{361}e^{6} + \frac{128}{361}e^{5} + \frac{334}{361}e^{4} + \frac{111}{361}e^{3} - \frac{773}{361}e^{2} - \frac{72}{19}e + \frac{812}{361}$ |
17 | $[17, 17, w + 2]$ | $\phantom{-}\frac{27}{722}e^{7} + \frac{63}{722}e^{6} - \frac{447}{722}e^{5} - \frac{445}{361}e^{4} + \frac{1308}{361}e^{3} + \frac{3413}{722}e^{2} - \frac{315}{38}e - \frac{1164}{361}$ |
17 | $[17, 17, w + 14]$ | $\phantom{-}\frac{27}{722}e^{7} + \frac{63}{722}e^{6} - \frac{447}{722}e^{5} - \frac{445}{361}e^{4} + \frac{1308}{361}e^{3} + \frac{3413}{722}e^{2} - \frac{315}{38}e - \frac{1164}{361}$ |
19 | $[19, 19, w]$ | $\phantom{-}\frac{91}{361}e^{7} + \frac{92}{361}e^{6} - \frac{1667}{361}e^{5} - \frac{1208}{361}e^{4} + \frac{9218}{361}e^{3} + \frac{4136}{361}e^{2} - \frac{745}{19}e - \frac{2712}{361}$ |
19 | $[19, 19, w + 18]$ | $\phantom{-}\frac{91}{361}e^{7} + \frac{92}{361}e^{6} - \frac{1667}{361}e^{5} - \frac{1208}{361}e^{4} + \frac{9218}{361}e^{3} + \frac{4136}{361}e^{2} - \frac{745}{19}e - \frac{2712}{361}$ |
37 | $[37, 37, -w - 4]$ | $\phantom{-}\frac{541}{722}e^{7} + \frac{781}{722}e^{6} - \frac{9117}{722}e^{5} - \frac{4531}{361}e^{4} + \frac{23521}{361}e^{3} + \frac{23355}{722}e^{2} - \frac{3639}{38}e - \frac{1262}{361}$ |
37 | $[37, 37, w - 5]$ | $\phantom{-}\frac{541}{722}e^{7} + \frac{781}{722}e^{6} - \frac{9117}{722}e^{5} - \frac{4531}{361}e^{4} + \frac{23521}{361}e^{3} + \frac{23355}{722}e^{2} - \frac{3639}{38}e - \frac{1262}{361}$ |
43 | $[43, 43, w + 16]$ | $-\frac{45}{361}e^{7} - \frac{105}{361}e^{6} + \frac{745}{361}e^{5} + \frac{1363}{361}e^{4} - \frac{3638}{361}e^{3} - \frac{4485}{361}e^{2} + \frac{221}{19}e + \frac{1714}{361}$ |
43 | $[43, 43, w + 26]$ | $-\frac{45}{361}e^{7} - \frac{105}{361}e^{6} + \frac{745}{361}e^{5} + \frac{1363}{361}e^{4} - \frac{3638}{361}e^{3} - \frac{4485}{361}e^{2} + \frac{221}{19}e + \frac{1714}{361}$ |
49 | $[49, 7, -7]$ | $-\frac{125}{722}e^{7} - \frac{51}{722}e^{6} + \frac{1909}{722}e^{5} + \frac{68}{361}e^{4} - \frac{4050}{361}e^{3} + \frac{1741}{722}e^{2} + \frac{483}{38}e + \frac{1097}{361}$ |
53 | $[53, 53, -w - 10]$ | $-\frac{203}{722}e^{7} - \frac{233}{722}e^{6} + \frac{3441}{722}e^{5} + \frac{1514}{361}e^{4} - \frac{8671}{361}e^{3} - \frac{10365}{722}e^{2} + \frac{1203}{38}e + \frac{2775}{361}$ |
53 | $[53, 53, w - 11]$ | $-\frac{203}{722}e^{7} - \frac{233}{722}e^{6} + \frac{3441}{722}e^{5} + \frac{1514}{361}e^{4} - \frac{8671}{361}e^{3} - \frac{10365}{722}e^{2} + \frac{1203}{38}e + \frac{2775}{361}$ |
61 | $[61, 61, w + 15]$ | $\phantom{-}\frac{559}{722}e^{7} + \frac{823}{722}e^{6} - \frac{9415}{722}e^{5} - \frac{4587}{361}e^{4} + \frac{24393}{361}e^{3} + \frac{20817}{722}e^{2} - \frac{3773}{38}e + \frac{850}{361}$ |
61 | $[61, 61, w + 45]$ | $\phantom{-}\frac{559}{722}e^{7} + \frac{823}{722}e^{6} - \frac{9415}{722}e^{5} - \frac{4587}{361}e^{4} + \frac{24393}{361}e^{3} + \frac{20817}{722}e^{2} - \frac{3773}{38}e + \frac{850}{361}$ |
Atkin-Lehner eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
$4$ | $[4, 2, 2]$ | $1$ |