Base field \(\Q(\sqrt{305}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 76\); narrow class number \(4\) and class number \(2\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[4, 2, 2]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $44$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} + x^{7} - 42x^{6} - 36x^{5} + 544x^{4} + 356x^{3} - 2064x^{2} - 528x + 576\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w]$ | $-1$ |
| 2 | $[2, 2, w + 1]$ | $-1$ |
| 5 | $[5, 5, -4w + 37]$ | $\phantom{-}e$ |
| 7 | $[7, 7, w + 2]$ | $\phantom{-}\frac{1}{1884}e^{7} + \frac{41}{3768}e^{6} + \frac{3}{1256}e^{5} - \frac{179}{628}e^{4} - \frac{743}{1884}e^{3} + \frac{1765}{942}e^{2} + \frac{501}{157}e - \frac{558}{157}$ |
| 7 | $[7, 7, w + 4]$ | $\phantom{-}\frac{1}{1884}e^{7} + \frac{41}{3768}e^{6} + \frac{3}{1256}e^{5} - \frac{179}{628}e^{4} - \frac{743}{1884}e^{3} + \frac{1765}{942}e^{2} + \frac{501}{157}e - \frac{558}{157}$ |
| 9 | $[9, 3, 3]$ | $\phantom{-}\frac{3}{2512}e^{7} - \frac{17}{2512}e^{6} - \frac{111}{1256}e^{5} + \frac{29}{628}e^{4} + \frac{895}{628}e^{3} + \frac{999}{628}e^{2} - \frac{1435}{314}e - \frac{392}{157}$ |
| 17 | $[17, 17, w + 6]$ | $\phantom{-}\frac{5}{1256}e^{7} + \frac{3}{157}e^{6} - \frac{213}{1256}e^{5} - \frac{161}{314}e^{4} + \frac{759}{314}e^{3} + \frac{1037}{314}e^{2} - \frac{3475}{314}e - \frac{417}{157}$ |
| 17 | $[17, 17, w + 10]$ | $\phantom{-}\frac{5}{1256}e^{7} + \frac{3}{157}e^{6} - \frac{213}{1256}e^{5} - \frac{161}{314}e^{4} + \frac{759}{314}e^{3} + \frac{1037}{314}e^{2} - \frac{3475}{314}e - \frac{417}{157}$ |
| 19 | $[19, 19, -2w + 19]$ | $\phantom{-}\frac{83}{7536}e^{7} + \frac{53}{7536}e^{6} - \frac{56}{157}e^{5} - \frac{16}{157}e^{4} + \frac{6131}{1884}e^{3} - \frac{935}{1884}e^{2} - \frac{1183}{157}e + \frac{589}{157}$ |
| 19 | $[19, 19, -2w - 17]$ | $\phantom{-}\frac{83}{7536}e^{7} + \frac{53}{7536}e^{6} - \frac{56}{157}e^{5} - \frac{16}{157}e^{4} + \frac{6131}{1884}e^{3} - \frac{935}{1884}e^{2} - \frac{1183}{157}e + \frac{589}{157}$ |
| 23 | $[23, 23, w + 5]$ | $-\frac{1}{628}e^{7} - \frac{41}{1256}e^{6} - \frac{9}{1256}e^{5} + \frac{537}{628}e^{4} + \frac{429}{628}e^{3} - \frac{804}{157}e^{2} - \frac{247}{157}e + \frac{732}{157}$ |
| 23 | $[23, 23, w + 17]$ | $-\frac{1}{628}e^{7} - \frac{41}{1256}e^{6} - \frac{9}{1256}e^{5} + \frac{537}{628}e^{4} + \frac{429}{628}e^{3} - \frac{804}{157}e^{2} - \frac{247}{157}e + \frac{732}{157}$ |
| 37 | $[37, 37, w + 1]$ | $-\frac{17}{7536}e^{7} - \frac{113}{7536}e^{6} + \frac{105}{1256}e^{5} + \frac{329}{628}e^{4} - \frac{2141}{1884}e^{3} - \frac{9115}{1884}e^{2} + \frac{1943}{314}e + \frac{1194}{157}$ |
| 37 | $[37, 37, w + 35]$ | $-\frac{17}{7536}e^{7} - \frac{113}{7536}e^{6} + \frac{105}{1256}e^{5} + \frac{329}{628}e^{4} - \frac{2141}{1884}e^{3} - \frac{9115}{1884}e^{2} + \frac{1943}{314}e + \frac{1194}{157}$ |
| 41 | $[41, 41, -22w + 203]$ | $-\frac{3}{1256}e^{7} + \frac{17}{1256}e^{6} + \frac{111}{628}e^{5} - \frac{215}{628}e^{4} - \frac{1947}{628}e^{3} + \frac{571}{314}e^{2} + \frac{3969}{314}e - \frac{315}{157}$ |
| 41 | $[41, 41, -6w + 55]$ | $-\frac{3}{1256}e^{7} + \frac{17}{1256}e^{6} + \frac{111}{628}e^{5} - \frac{215}{628}e^{4} - \frac{1947}{628}e^{3} + \frac{571}{314}e^{2} + \frac{3969}{314}e - \frac{315}{157}$ |
| 43 | $[43, 43, w + 20]$ | $\phantom{-}\frac{43}{7536}e^{7} + \frac{175}{7536}e^{6} - \frac{321}{1256}e^{5} - \frac{118}{157}e^{4} + \frac{1624}{471}e^{3} + \frac{10865}{1884}e^{2} - \frac{1954}{157}e - \frac{111}{157}$ |
| 43 | $[43, 43, w + 22]$ | $\phantom{-}\frac{43}{7536}e^{7} + \frac{175}{7536}e^{6} - \frac{321}{1256}e^{5} - \frac{118}{157}e^{4} + \frac{1624}{471}e^{3} + \frac{10865}{1884}e^{2} - \frac{1954}{157}e - \frac{111}{157}$ |
| 53 | $[53, 53, w + 13]$ | $\phantom{-}\frac{11}{1256}e^{7} - \frac{5}{628}e^{6} - \frac{343}{1256}e^{5} + \frac{27}{157}e^{4} + \frac{1487}{628}e^{3} + \frac{26}{157}e^{2} - \frac{918}{157}e - \frac{1200}{157}$ |
| 53 | $[53, 53, w + 39]$ | $\phantom{-}\frac{11}{1256}e^{7} - \frac{5}{628}e^{6} - \frac{343}{1256}e^{5} + \frac{27}{157}e^{4} + \frac{1487}{628}e^{3} + \frac{26}{157}e^{2} - \frac{918}{157}e - \frac{1200}{157}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2, 2, w]$ | $1$ |
| $2$ | $[2, 2, w + 1]$ | $1$ |