Base field \(\Q(\sqrt{393}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 98\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[4, 2, 2]$ |
| Dimension: | $11$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $36$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{11} - x^{10} - 33x^{9} + 34x^{8} + 386x^{7} - 400x^{6} - 1866x^{5} + 1848x^{4} + 3272x^{3} - 2624x^{2} - 1920x + 1024\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -17w - 160]$ | $-1$ |
| 2 | $[2, 2, -17w + 177]$ | $-1$ |
| 3 | $[3, 3, -842w + 8767]$ | $\phantom{-}e$ |
| 7 | $[7, 7, -2w + 21]$ | $\phantom{-}\frac{109}{2432}e^{10} - \frac{253}{2432}e^{9} - \frac{3405}{2432}e^{8} + \frac{4045}{1216}e^{7} + \frac{17605}{1216}e^{6} - \frac{5505}{152}e^{5} - \frac{60721}{1216}e^{4} + \frac{44383}{304}e^{3} + \frac{153}{304}e^{2} - \frac{4853}{38}e + \frac{868}{19}$ |
| 7 | $[7, 7, 2w + 19]$ | $\phantom{-}\frac{109}{2432}e^{10} - \frac{253}{2432}e^{9} - \frac{3405}{2432}e^{8} + \frac{4045}{1216}e^{7} + \frac{17605}{1216}e^{6} - \frac{5505}{152}e^{5} - \frac{60721}{1216}e^{4} + \frac{44383}{304}e^{3} + \frac{153}{304}e^{2} - \frac{4853}{38}e + \frac{868}{19}$ |
| 13 | $[13, 13, -12w - 113]$ | $\phantom{-}\frac{135}{2432}e^{10} - \frac{199}{2432}e^{9} - \frac{4167}{2432}e^{8} + \frac{3303}{1216}e^{7} + \frac{21591}{1216}e^{6} - \frac{4619}{152}e^{5} - \frac{78627}{1216}e^{4} + \frac{37761}{304}e^{3} + \frac{8295}{304}e^{2} - \frac{4021}{38}e + \frac{637}{19}$ |
| 13 | $[13, 13, 12w - 125]$ | $\phantom{-}\frac{135}{2432}e^{10} - \frac{199}{2432}e^{9} - \frac{4167}{2432}e^{8} + \frac{3303}{1216}e^{7} + \frac{21591}{1216}e^{6} - \frac{4619}{152}e^{5} - \frac{78627}{1216}e^{4} + \frac{37761}{304}e^{3} + \frac{8295}{304}e^{2} - \frac{4021}{38}e + \frac{637}{19}$ |
| 17 | $[17, 17, 182w - 1895]$ | $-\frac{25}{1216}e^{10} + \frac{65}{1216}e^{9} + \frac{721}{1216}e^{8} - \frac{1093}{608}e^{7} - \frac{3289}{608}e^{6} + \frac{1545}{76}e^{5} + \frac{7805}{608}e^{4} - \frac{12773}{152}e^{3} + \frac{4375}{152}e^{2} + \frac{2827}{38}e - \frac{725}{19}$ |
| 17 | $[17, 17, 182w + 1713]$ | $-\frac{25}{1216}e^{10} + \frac{65}{1216}e^{9} + \frac{721}{1216}e^{8} - \frac{1093}{608}e^{7} - \frac{3289}{608}e^{6} + \frac{1545}{76}e^{5} + \frac{7805}{608}e^{4} - \frac{12773}{152}e^{3} + \frac{4375}{152}e^{2} + \frac{2827}{38}e - \frac{725}{19}$ |
| 23 | $[23, 23, -512w - 4819]$ | $-\frac{203}{2432}e^{10} + \frac{467}{2432}e^{9} + \frac{6195}{2432}e^{8} - \frac{7495}{1216}e^{7} - \frac{30987}{1216}e^{6} + \frac{5127}{76}e^{5} + \frac{99431}{1216}e^{4} - \frac{83367}{304}e^{3} + \frac{7633}{304}e^{2} + \frac{4656}{19}e - \frac{1832}{19}$ |
| 23 | $[23, 23, 512w - 5331]$ | $-\frac{203}{2432}e^{10} + \frac{467}{2432}e^{9} + \frac{6195}{2432}e^{8} - \frac{7495}{1216}e^{7} - \frac{30987}{1216}e^{6} + \frac{5127}{76}e^{5} + \frac{99431}{1216}e^{4} - \frac{83367}{304}e^{3} + \frac{7633}{304}e^{2} + \frac{4656}{19}e - \frac{1832}{19}$ |
| 25 | $[25, 5, -5]$ | $\phantom{-}\frac{69}{2432}e^{10} - \frac{149}{2432}e^{9} - \frac{2069}{2432}e^{8} + \frac{2509}{1216}e^{7} + \frac{10245}{1216}e^{6} - \frac{3565}{152}e^{5} - \frac{33337}{1216}e^{4} + \frac{29935}{304}e^{3} - \frac{979}{304}e^{2} - \frac{1784}{19}e + \frac{554}{19}$ |
| 29 | $[29, 29, 22w - 229]$ | $\phantom{-}\frac{277}{2432}e^{10} - \frac{629}{2432}e^{9} - \frac{8469}{2432}e^{8} + \frac{10101}{1216}e^{7} + \frac{42741}{1216}e^{6} - \frac{13805}{152}e^{5} - \frac{141929}{1216}e^{4} + \frac{112011}{304}e^{3} - \frac{3939}{304}e^{2} - \frac{12549}{38}e + \frac{2183}{19}$ |
| 29 | $[29, 29, 22w + 207]$ | $\phantom{-}\frac{277}{2432}e^{10} - \frac{629}{2432}e^{9} - \frac{8469}{2432}e^{8} + \frac{10101}{1216}e^{7} + \frac{42741}{1216}e^{6} - \frac{13805}{152}e^{5} - \frac{141929}{1216}e^{4} + \frac{112011}{304}e^{3} - \frac{3939}{304}e^{2} - \frac{12549}{38}e + \frac{2183}{19}$ |
| 43 | $[43, 43, 114w + 1073]$ | $-\frac{117}{1216}e^{10} + \frac{213}{1216}e^{9} + \frac{3581}{1216}e^{8} - \frac{3425}{608}e^{7} - \frac{18241}{608}e^{6} + \frac{4677}{76}e^{5} + \frac{63249}{608}e^{4} - \frac{37727}{152}e^{3} - \frac{2591}{152}e^{2} + \frac{8149}{38}e - \frac{1512}{19}$ |
| 43 | $[43, 43, 114w - 1187]$ | $-\frac{117}{1216}e^{10} + \frac{213}{1216}e^{9} + \frac{3581}{1216}e^{8} - \frac{3425}{608}e^{7} - \frac{18241}{608}e^{6} + \frac{4677}{76}e^{5} + \frac{63249}{608}e^{4} - \frac{37727}{152}e^{3} - \frac{2591}{152}e^{2} + \frac{8149}{38}e - \frac{1512}{19}$ |
| 47 | $[47, 47, 8w - 83]$ | $-\frac{213}{2432}e^{10} + \frac{493}{2432}e^{9} + \frac{6605}{2432}e^{8} - \frac{7689}{1216}e^{7} - \frac{34005}{1216}e^{6} + \frac{5113}{76}e^{5} + \frac{118969}{1216}e^{4} - \frac{80785}{304}e^{3} - \frac{5361}{304}e^{2} + \frac{4344}{19}e - \frac{1312}{19}$ |
| 47 | $[47, 47, -8w - 75]$ | $-\frac{213}{2432}e^{10} + \frac{493}{2432}e^{9} + \frac{6605}{2432}e^{8} - \frac{7689}{1216}e^{7} - \frac{34005}{1216}e^{6} + \frac{5113}{76}e^{5} + \frac{118969}{1216}e^{4} - \frac{80785}{304}e^{3} - \frac{5361}{304}e^{2} + \frac{4344}{19}e - \frac{1312}{19}$ |
| 61 | $[61, 61, -1172w - 11031]$ | $\phantom{-}\frac{35}{2432}e^{10} - \frac{91}{2432}e^{9} - \frac{1131}{2432}e^{8} + \frac{1439}{1216}e^{7} + \frac{5851}{1216}e^{6} - \frac{977}{76}e^{5} - \frac{18223}{1216}e^{4} + \frac{15739}{304}e^{3} - \frac{3541}{304}e^{2} - \frac{808}{19}e + \frac{517}{19}$ |
| 61 | $[61, 61, 1172w - 12203]$ | $\phantom{-}\frac{35}{2432}e^{10} - \frac{91}{2432}e^{9} - \frac{1131}{2432}e^{8} + \frac{1439}{1216}e^{7} + \frac{5851}{1216}e^{6} - \frac{977}{76}e^{5} - \frac{18223}{1216}e^{4} + \frac{15739}{304}e^{3} - \frac{3541}{304}e^{2} - \frac{808}{19}e + \frac{517}{19}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2, 2, -17w - 160]$ | $1$ |
| $2$ | $[2, 2, -17w + 177]$ | $1$ |