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: | $[8, 4, -34w - 320]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $18$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} - 2x^{8} - 15x^{7} + 24x^{6} + 75x^{5} - 81x^{4} - 140x^{3} + 54x^{2} + 79x + 14\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -17w - 160]$ | $\phantom{-}0$ |
| 2 | $[2, 2, -17w + 177]$ | $\phantom{-}1$ |
| 3 | $[3, 3, -842w + 8767]$ | $\phantom{-}e$ |
| 7 | $[7, 7, -2w + 21]$ | $-\frac{514}{1863}e^{8} + \frac{1177}{1863}e^{7} + \frac{6865}{1863}e^{6} - \frac{13757}{1863}e^{5} - \frac{28607}{1863}e^{4} + \frac{2030}{81}e^{3} + \frac{12893}{621}e^{2} - \frac{13039}{621}e - \frac{14446}{1863}$ |
| 7 | $[7, 7, 2w + 19]$ | $-\frac{50}{1863}e^{8} + \frac{71}{1863}e^{7} + \frac{1052}{1863}e^{6} - \frac{1186}{1863}e^{5} - \frac{6748}{1863}e^{4} + \frac{223}{81}e^{3} + \frac{4480}{621}e^{2} - \frac{1034}{621}e - \frac{3290}{1863}$ |
| 13 | $[13, 13, -12w - 113]$ | $\phantom{-}\frac{329}{1863}e^{8} - \frac{728}{1863}e^{7} - \frac{4463}{1863}e^{6} + \frac{8251}{1863}e^{5} + \frac{18916}{1863}e^{4} - \frac{1132}{81}e^{3} - \frac{8116}{621}e^{2} + \frac{5363}{621}e + \frac{5999}{1863}$ |
| 13 | $[13, 13, 12w - 125]$ | $-\frac{44}{1863}e^{8} + \frac{137}{1863}e^{7} - \frac{43}{1863}e^{6} - \frac{373}{1863}e^{5} + \frac{4271}{1863}e^{4} - \frac{212}{81}e^{3} - \frac{5621}{621}e^{2} + \frac{4381}{621}e + \frac{14617}{1863}$ |
| 17 | $[17, 17, 182w - 1895]$ | $-\frac{19}{81}e^{8} + \frac{61}{81}e^{7} + \frac{241}{81}e^{6} - \frac{752}{81}e^{5} - \frac{941}{81}e^{4} + \frac{2644}{81}e^{3} + \frac{401}{27}e^{2} - \frac{718}{27}e - \frac{667}{81}$ |
| 17 | $[17, 17, 182w + 1713]$ | $-\frac{482}{1863}e^{8} + \frac{908}{1863}e^{7} + \frac{7235}{1863}e^{6} - \frac{11284}{1863}e^{5} - \frac{34423}{1863}e^{4} + \frac{1735}{81}e^{3} + \frac{18223}{621}e^{2} - \frac{9620}{621}e - \frac{20165}{1863}$ |
| 23 | $[23, 23, -512w - 4819]$ | $-\frac{53}{621}e^{8} + \frac{38}{621}e^{7} + \frac{668}{621}e^{6} - \frac{40}{621}e^{5} - \frac{2632}{621}e^{4} - \frac{32}{27}e^{3} + \frac{1147}{207}e^{2} - \frac{326}{207}e - \frac{134}{621}$ |
| 23 | $[23, 23, 512w - 5331]$ | $\phantom{-}\frac{59}{621}e^{8} + \frac{28}{621}e^{7} - \frac{1142}{621}e^{6} - \frac{389}{621}e^{5} + \frac{6820}{621}e^{4} + \frac{56}{27}e^{3} - \frac{4624}{207}e^{2} + \frac{152}{207}e + \frac{6242}{621}$ |
| 25 | $[25, 5, -5]$ | $-\frac{4}{27}e^{8} + \frac{10}{27}e^{7} + \frac{55}{27}e^{6} - \frac{110}{27}e^{5} - \frac{272}{27}e^{4} + \frac{325}{27}e^{3} + \frac{182}{9}e^{2} - \frac{55}{9}e - \frac{220}{27}$ |
| 29 | $[29, 29, 22w - 229]$ | $\phantom{-}\frac{197}{1863}e^{8} - \frac{317}{1863}e^{7} - \frac{2729}{1863}e^{6} + \frac{3406}{1863}e^{5} + \frac{11236}{1863}e^{4} - \frac{391}{81}e^{3} - \frac{4486}{621}e^{2} + \frac{1118}{621}e + \frac{3275}{1863}$ |
| 29 | $[29, 29, 22w + 207]$ | $-\frac{17}{81}e^{8} + \frac{29}{81}e^{7} + \frac{254}{81}e^{6} - \frac{319}{81}e^{5} - \frac{1264}{81}e^{4} + \frac{929}{81}e^{3} + \frac{778}{27}e^{2} - \frac{155}{27}e - \frac{827}{81}$ |
| 43 | $[43, 43, 114w + 1073]$ | $\phantom{-}\frac{382}{1863}e^{8} - \frac{766}{1863}e^{7} - \frac{5131}{1863}e^{6} + \frac{8912}{1863}e^{5} + \frac{19064}{1863}e^{4} - \frac{1208}{81}e^{3} - \frac{4916}{621}e^{2} + \frac{4447}{621}e + \frac{4270}{1863}$ |
| 43 | $[43, 43, 114w - 1187]$ | $-\frac{2}{81}e^{8} + \frac{32}{81}e^{7} - \frac{13}{81}e^{6} - \frac{433}{81}e^{5} + \frac{323}{81}e^{4} + \frac{1634}{81}e^{3} - \frac{377}{27}e^{2} - \frac{374}{27}e + \frac{484}{81}$ |
| 47 | $[47, 47, 8w - 83]$ | $\phantom{-}\frac{17}{81}e^{8} - \frac{29}{81}e^{7} - \frac{254}{81}e^{6} + \frac{319}{81}e^{5} + \frac{1264}{81}e^{4} - \frac{929}{81}e^{3} - \frac{751}{27}e^{2} + \frac{182}{27}e + \frac{908}{81}$ |
| 47 | $[47, 47, -8w - 75]$ | $-\frac{494}{1863}e^{8} + \frac{776}{1863}e^{7} + \frac{7562}{1863}e^{6} - \frac{9184}{1863}e^{5} - \frac{35968}{1863}e^{4} + \frac{1309}{81}e^{3} + \frac{17311}{621}e^{2} - \frac{6167}{621}e - \frac{9404}{1863}$ |
| 61 | $[61, 61, -1172w - 11031]$ | $\phantom{-}\frac{223}{1863}e^{8} - \frac{652}{1863}e^{7} - \frac{3127}{1863}e^{6} + \frac{8792}{1863}e^{5} + \frac{14894}{1863}e^{4} - \frac{1547}{81}e^{3} - \frac{10169}{621}e^{2} + \frac{12163}{621}e + \frac{24361}{1863}$ |
| 61 | $[61, 61, 1172w - 12203]$ | $-\frac{355}{1863}e^{8} + \frac{1063}{1863}e^{7} + \frac{4861}{1863}e^{6} - \frac{13637}{1863}e^{5} - \frac{22574}{1863}e^{4} + \frac{2207}{81}e^{3} + \frac{14420}{621}e^{2} - \frac{15166}{621}e - \frac{23359}{1863}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2, 2, -17w - 160]$ | $-1$ |
| $2$ | $[2, 2, -17w + 177]$ | $-1$ |