Base field \(\Q(\sqrt{493}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 123\); narrow class number \(2\) and class number \(2\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[4, 2, 2]$ |
| Dimension: | $20$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $118$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{20} + 44x^{18} + 801x^{16} + 7851x^{14} + 45332x^{12} + 159192x^{10} + 339461x^{8} + 425066x^{6} + 288672x^{4} + 89596x^{2} + 8712\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}e$ |
| 3 | $[3, 3, w + 2]$ | $-e$ |
| 4 | $[4, 2, 2]$ | $-1$ |
| 11 | $[11, 11, w + 1]$ | $\phantom{-}\frac{4406306959}{1065175357092}e^{19} + \frac{1231282805}{6916723098}e^{17} + \frac{1114447340949}{355058452364}e^{15} + \frac{10418144780077}{355058452364}e^{13} + \frac{83957319567775}{532587678546}e^{11} + \frac{3992888498453}{8069510281}e^{9} + \frac{940226012183699}{1065175357092}e^{7} + \frac{220267039288460}{266293839273}e^{5} + \frac{30821949987819}{88764613091}e^{3} + \frac{12465656820889}{266293839273}e$ |
| 11 | $[11, 11, w + 9]$ | $-\frac{4406306959}{1065175357092}e^{19} - \frac{1231282805}{6916723098}e^{17} - \frac{1114447340949}{355058452364}e^{15} - \frac{10418144780077}{355058452364}e^{13} - \frac{83957319567775}{532587678546}e^{11} - \frac{3992888498453}{8069510281}e^{9} - \frac{940226012183699}{1065175357092}e^{7} - \frac{220267039288460}{266293839273}e^{5} - \frac{30821949987819}{88764613091}e^{3} - \frac{12465656820889}{266293839273}e$ |
| 13 | $[13, 13, w - 11]$ | $-\frac{396921807}{32278041124}e^{18} - \frac{605207129}{1152787183}e^{16} - \frac{295679580027}{32278041124}e^{14} - \frac{2725787356253}{32278041124}e^{12} - \frac{3592748719443}{8069510281}e^{10} - \frac{10987428762704}{8069510281}e^{8} - \frac{75443410208275}{32278041124}e^{6} - \frac{33107432672637}{16139020562}e^{4} - \frac{6080518169140}{8069510281}e^{2} - \frac{631058450044}{8069510281}$ |
| 13 | $[13, 13, w + 10]$ | $-\frac{396921807}{32278041124}e^{18} - \frac{605207129}{1152787183}e^{16} - \frac{295679580027}{32278041124}e^{14} - \frac{2725787356253}{32278041124}e^{12} - \frac{3592748719443}{8069510281}e^{10} - \frac{10987428762704}{8069510281}e^{8} - \frac{75443410208275}{32278041124}e^{6} - \frac{33107432672637}{16139020562}e^{4} - \frac{6080518169140}{8069510281}e^{2} - \frac{631058450044}{8069510281}$ |
| 17 | $[17, 17, w + 8]$ | $\phantom{-}0$ |
| 25 | $[25, 5, -5]$ | $\phantom{-}\frac{682631449}{16139020562}e^{18} + \frac{2069066338}{1152787183}e^{16} + \frac{501440901369}{16139020562}e^{14} + \frac{4574638023299}{16139020562}e^{12} + \frac{11895353376339}{8069510281}e^{10} + \frac{35750127871618}{8069510281}e^{8} + \frac{120189941070275}{16139020562}e^{6} + \frac{51543497725991}{8069510281}e^{4} + \frac{18562807951785}{8069510281}e^{2} + \frac{1996628973272}{8069510281}$ |
| 29 | $[29, 29, w + 14]$ | $\phantom{-}0$ |
| 31 | $[31, 31, w + 5]$ | $\phantom{-}\frac{6605066247}{355058452364}e^{19} + \frac{1825636069}{2305574366}e^{17} + \frac{4886642477103}{355058452364}e^{15} + \frac{44819165744543}{355058452364}e^{13} + \frac{117357825335393}{177529226182}e^{11} + \frac{16174868709138}{8069510281}e^{9} + \frac{1209265156118095}{355058452364}e^{7} + \frac{262285688257438}{88764613091}e^{5} + \frac{94474997940326}{88764613091}e^{3} + \frac{8557492264879}{88764613091}e$ |
| 31 | $[31, 31, w + 25]$ | $-\frac{6605066247}{355058452364}e^{19} - \frac{1825636069}{2305574366}e^{17} - \frac{4886642477103}{355058452364}e^{15} - \frac{44819165744543}{355058452364}e^{13} - \frac{117357825335393}{177529226182}e^{11} - \frac{16174868709138}{8069510281}e^{9} - \frac{1209265156118095}{355058452364}e^{7} - \frac{262285688257438}{88764613091}e^{5} - \frac{94474997940326}{88764613091}e^{3} - \frac{8557492264879}{88764613091}e$ |
| 37 | $[37, 37, w + 3]$ | $\phantom{-}\frac{6059420075}{355058452364}e^{19} + \frac{1663198577}{2305574366}e^{17} + \frac{4411770277333}{355058452364}e^{15} + \frac{39991946299147}{355058452364}e^{13} + \frac{51584595079824}{88764613091}e^{11} + \frac{27933879386463}{16139020562}e^{9} + \frac{1024381232343683}{355058452364}e^{7} + \frac{219084021820955}{88764613091}e^{5} + \frac{161466584585331}{177529226182}e^{3} + \frac{9895078533047}{88764613091}e$ |
| 37 | $[37, 37, w + 33]$ | $-\frac{6059420075}{355058452364}e^{19} - \frac{1663198577}{2305574366}e^{17} - \frac{4411770277333}{355058452364}e^{15} - \frac{39991946299147}{355058452364}e^{13} - \frac{51584595079824}{88764613091}e^{11} - \frac{27933879386463}{16139020562}e^{9} - \frac{1024381232343683}{355058452364}e^{7} - \frac{219084021820955}{88764613091}e^{5} - \frac{161466584585331}{177529226182}e^{3} - \frac{9895078533047}{88764613091}e$ |
| 41 | $[41, 41, w]$ | $-\frac{11115981868}{266293839273}e^{19} - \frac{12261224585}{6916723098}e^{17} - \frac{5453977881551}{177529226182}e^{15} - \frac{49843136612925}{177529226182}e^{13} - \frac{389879269404638}{266293839273}e^{11} - \frac{71321851729297}{16139020562}e^{9} - \frac{1991843629070165}{266293839273}e^{7} - \frac{3454417780325473}{532587678546}e^{5} - \frac{421171457079027}{177529226182}e^{3} - \frac{65449201215037}{266293839273}e$ |
| 41 | $[41, 41, w + 40]$ | $\phantom{-}\frac{11115981868}{266293839273}e^{19} + \frac{12261224585}{6916723098}e^{17} + \frac{5453977881551}{177529226182}e^{15} + \frac{49843136612925}{177529226182}e^{13} + \frac{389879269404638}{266293839273}e^{11} + \frac{71321851729297}{16139020562}e^{9} + \frac{1991843629070165}{266293839273}e^{7} + \frac{3454417780325473}{532587678546}e^{5} + \frac{421171457079027}{177529226182}e^{3} + \frac{65449201215037}{266293839273}e$ |
| 49 | $[49, 7, -7]$ | $-\frac{385236631}{32278041124}e^{18} - \frac{1159912769}{2305574366}e^{16} - \frac{278421625127}{32278041124}e^{14} - \frac{2504254157119}{32278041124}e^{12} - \frac{6372428382599}{16139020562}e^{10} - \frac{9252382172617}{8069510281}e^{8} - \frac{58680584690027}{32278041124}e^{6} - \frac{11251698688348}{8069510281}e^{4} - \frac{3149521420132}{8069510281}e^{2} - \frac{172683922066}{8069510281}$ |
| 53 | $[53, 53, -4w - 43]$ | $\phantom{-}\frac{508799227}{32278041124}e^{18} + \frac{775606943}{1152787183}e^{16} + \frac{378636361219}{32278041124}e^{14} + \frac{3484102492705}{32278041124}e^{12} + \frac{4574311216040}{8069510281}e^{10} + \frac{13882476421589}{8069510281}e^{8} + \frac{94033174008415}{32278041124}e^{6} + \frac{40458044894277}{16139020562}e^{4} + \frac{7375368534603}{8069510281}e^{2} + \frac{864833744568}{8069510281}$ |
| 53 | $[53, 53, 4w - 47]$ | $\phantom{-}\frac{508799227}{32278041124}e^{18} + \frac{775606943}{1152787183}e^{16} + \frac{378636361219}{32278041124}e^{14} + \frac{3484102492705}{32278041124}e^{12} + \frac{4574311216040}{8069510281}e^{10} + \frac{13882476421589}{8069510281}e^{8} + \frac{94033174008415}{32278041124}e^{6} + \frac{40458044894277}{16139020562}e^{4} + \frac{7375368534603}{8069510281}e^{2} + \frac{864833744568}{8069510281}$ |
| 59 | $[59, 59, -w - 13]$ | $\phantom{-}\frac{647220799}{16139020562}e^{18} + \frac{1963633277}{1152787183}e^{16} + \frac{476476336045}{16139020562}e^{14} + \frac{4353342541371}{16139020562}e^{12} + \frac{11337428212000}{8069510281}e^{10} + \frac{34105176557895}{8069510281}e^{8} + \frac{114475176197769}{16139020562}e^{6} + \frac{48611911950592}{8069510281}e^{4} + \frac{16877286742654}{8069510281}e^{2} + \frac{1620261988332}{8069510281}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, 2]$ | $1$ |