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: | $[9, 3, 3]$ |
| Dimension: | $22$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $104$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{22} - 33x^{20} + 462x^{18} - 3592x^{16} + 17053x^{14} - 51199x^{12} + 97422x^{10} - 114830x^{8} + 79961x^{6} - 30261x^{4} + 5211x^{2} - 243\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -17w - 160]$ | $\phantom{-}e$ |
| 2 | $[2, 2, -17w + 177]$ | $\phantom{-}e$ |
| 3 | $[3, 3, -842w + 8767]$ | $\phantom{-}0$ |
| 7 | $[7, 7, -2w + 21]$ | $\phantom{-}\frac{32}{81}e^{20} - \frac{358}{27}e^{18} + \frac{5072}{27}e^{16} - \frac{118562}{81}e^{14} + \frac{554543}{81}e^{12} - \frac{1593620}{81}e^{10} + \frac{921571}{27}e^{8} - \frac{2732233}{81}e^{6} + \frac{1376215}{81}e^{4} - \frac{96284}{27}e^{2} + \frac{538}{3}$ |
| 7 | $[7, 7, 2w + 19]$ | $\phantom{-}\frac{32}{81}e^{20} - \frac{358}{27}e^{18} + \frac{5072}{27}e^{16} - \frac{118562}{81}e^{14} + \frac{554543}{81}e^{12} - \frac{1593620}{81}e^{10} + \frac{921571}{27}e^{8} - \frac{2732233}{81}e^{6} + \frac{1376215}{81}e^{4} - \frac{96284}{27}e^{2} + \frac{538}{3}$ |
| 13 | $[13, 13, -12w - 113]$ | $-\frac{55}{162}e^{20} + \frac{313}{27}e^{18} - \frac{4514}{27}e^{16} + \frac{107420}{81}e^{14} - \frac{1022461}{162}e^{12} + \frac{1493162}{81}e^{10} - \frac{875896}{27}e^{8} + \frac{2626573}{81}e^{6} - \frac{2666903}{162}e^{4} + \frac{93953}{27}e^{2} - \frac{1073}{6}$ |
| 13 | $[13, 13, 12w - 125]$ | $-\frac{55}{162}e^{20} + \frac{313}{27}e^{18} - \frac{4514}{27}e^{16} + \frac{107420}{81}e^{14} - \frac{1022461}{162}e^{12} + \frac{1493162}{81}e^{10} - \frac{875896}{27}e^{8} + \frac{2626573}{81}e^{6} - \frac{2666903}{162}e^{4} + \frac{93953}{27}e^{2} - \frac{1073}{6}$ |
| 17 | $[17, 17, 182w - 1895]$ | $\phantom{-}\frac{425}{162}e^{21} - \frac{2264}{27}e^{19} + \frac{30421}{27}e^{17} - \frac{672526}{81}e^{15} + \frac{5944523}{162}e^{13} - \frac{8084068}{81}e^{11} + \frac{4444412}{27}e^{9} - \frac{12618764}{81}e^{7} + \frac{12272221}{162}e^{5} - \frac{415822}{27}e^{3} + \frac{4447}{6}e$ |
| 17 | $[17, 17, 182w + 1713]$ | $\phantom{-}\frac{425}{162}e^{21} - \frac{2264}{27}e^{19} + \frac{30421}{27}e^{17} - \frac{672526}{81}e^{15} + \frac{5944523}{162}e^{13} - \frac{8084068}{81}e^{11} + \frac{4444412}{27}e^{9} - \frac{12618764}{81}e^{7} + \frac{12272221}{162}e^{5} - \frac{415822}{27}e^{3} + \frac{4447}{6}e$ |
| 23 | $[23, 23, -512w - 4819]$ | $-e^{21} + 30e^{19} - 373e^{17} + 2502e^{15} - 9891e^{13} + 23684e^{11} - 34103e^{9} + 28472e^{7} - 12697e^{5} + 2489e^{3} - 112e$ |
| 23 | $[23, 23, 512w - 5331]$ | $-e^{21} + 30e^{19} - 373e^{17} + 2502e^{15} - 9891e^{13} + 23684e^{11} - 34103e^{9} + 28472e^{7} - 12697e^{5} + 2489e^{3} - 112e$ |
| 25 | $[25, 5, -5]$ | $-\frac{137}{81}e^{20} + \frac{1225}{27}e^{18} - \frac{12845}{27}e^{16} + \frac{194564}{81}e^{14} - \frac{433292}{81}e^{12} - \frac{14359}{81}e^{10} + \frac{635813}{27}e^{8} - \frac{3329021}{81}e^{6} + \frac{2153699}{81}e^{4} - \frac{168157}{27}e^{2} + \frac{986}{3}$ |
| 29 | $[29, 29, 22w - 229]$ | $\phantom{-}\frac{671}{162}e^{21} - \frac{3482}{27}e^{19} + \frac{45367}{27}e^{17} - \frac{967975}{81}e^{15} + \frac{8224397}{162}e^{13} - \frac{10725286}{81}e^{11} + \frac{5658956}{27}e^{9} - \frac{15515201}{81}e^{7} + \frac{14761555}{162}e^{5} - \frac{499681}{27}e^{3} + \frac{5707}{6}e$ |
| 29 | $[29, 29, 22w + 207]$ | $\phantom{-}\frac{671}{162}e^{21} - \frac{3482}{27}e^{19} + \frac{45367}{27}e^{17} - \frac{967975}{81}e^{15} + \frac{8224397}{162}e^{13} - \frac{10725286}{81}e^{11} + \frac{5658956}{27}e^{9} - \frac{15515201}{81}e^{7} + \frac{14761555}{162}e^{5} - \frac{499681}{27}e^{3} + \frac{5707}{6}e$ |
| 43 | $[43, 43, 114w + 1073]$ | $\phantom{-}\frac{434}{81}e^{20} - \frac{4528}{27}e^{18} + \frac{59366}{27}e^{16} - \frac{1275968}{81}e^{14} + \frac{5465579}{81}e^{12} - \frac{14381285}{81}e^{10} + \frac{7652764}{27}e^{8} - \frac{21119722}{81}e^{6} + \frac{10066852}{81}e^{4} - \frac{676556}{27}e^{2} + \frac{3700}{3}$ |
| 43 | $[43, 43, 114w - 1187]$ | $\phantom{-}\frac{434}{81}e^{20} - \frac{4528}{27}e^{18} + \frac{59366}{27}e^{16} - \frac{1275968}{81}e^{14} + \frac{5465579}{81}e^{12} - \frac{14381285}{81}e^{10} + \frac{7652764}{27}e^{8} - \frac{21119722}{81}e^{6} + \frac{10066852}{81}e^{4} - \frac{676556}{27}e^{2} + \frac{3700}{3}$ |
| 47 | $[47, 47, 8w - 83]$ | $\phantom{-}\frac{2}{9}e^{21} - 13e^{19} + \frac{797}{3}e^{17} - \frac{24383}{9}e^{15} + \frac{140132}{9}e^{13} - \frac{472133}{9}e^{11} + 102664e^{9} - \frac{995290}{9}e^{7} + \frac{530401}{9}e^{5} - 12861e^{3} + 684e$ |
| 47 | $[47, 47, -8w - 75]$ | $\phantom{-}\frac{2}{9}e^{21} - 13e^{19} + \frac{797}{3}e^{17} - \frac{24383}{9}e^{15} + \frac{140132}{9}e^{13} - \frac{472133}{9}e^{11} + 102664e^{9} - \frac{995290}{9}e^{7} + \frac{530401}{9}e^{5} - 12861e^{3} + 684e$ |
| 61 | $[61, 61, -1172w - 11031]$ | $\phantom{-}\frac{601}{162}e^{20} - \frac{3037}{27}e^{18} + \frac{38267}{27}e^{16} - \frac{782990}{81}e^{14} + \frac{6317851}{162}e^{12} - \frac{7744925}{81}e^{10} + \frac{3809389}{27}e^{8} - \frac{9715744}{81}e^{6} + \frac{8660339}{162}e^{4} - \frac{276746}{27}e^{2} + \frac{2915}{6}$ |
| 61 | $[61, 61, 1172w - 12203]$ | $\phantom{-}\frac{601}{162}e^{20} - \frac{3037}{27}e^{18} + \frac{38267}{27}e^{16} - \frac{782990}{81}e^{14} + \frac{6317851}{162}e^{12} - \frac{7744925}{81}e^{10} + \frac{3809389}{27}e^{8} - \frac{9715744}{81}e^{6} + \frac{8660339}{162}e^{4} - \frac{276746}{27}e^{2} + \frac{2915}{6}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, -842w + 8767]$ | $1$ |