Base field 4.4.16400.1
Generator \(w\), with minimal polynomial \(x^{4} - 13x^{2} + 41\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[20, 10, w^{2} + w - 7]$ |
| Dimension: | $12$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $28$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{12} - 80x^{10} + 2148x^{8} - 22800x^{6} + 87232x^{4} - 43776x^{2} + 4096\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -w^{3} + 2w^{2} + 6w - 11]$ | $\phantom{-}1$ |
| 5 | $[5, 5, -w - 2]$ | $\phantom{-}1$ |
| 5 | $[5, 5, -w + 2]$ | $\phantom{-}\frac{29}{704512}e^{10} - \frac{629}{176128}e^{8} + \frac{19065}{176128}e^{6} - \frac{29557}{22016}e^{4} + \frac{58937}{11008}e^{2} + \frac{1337}{688}$ |
| 11 | $[11, 11, -2w^{2} - w + 12]$ | $\phantom{-}e$ |
| 11 | $[11, 11, 2w^{2} - w - 12]$ | $\phantom{-}\frac{715}{1409024}e^{11} - \frac{14227}{352256}e^{9} + \frac{378191}{352256}e^{7} - \frac{490851}{44032}e^{5} + \frac{887471}{22016}e^{3} - \frac{16145}{1376}e$ |
| 19 | $[19, 19, -w^{2} - w + 4]$ | $-\frac{355}{352256}e^{11} + \frac{7083}{88064}e^{9} - \frac{189255}{88064}e^{7} + \frac{248251}{11008}e^{5} - \frac{458583}{5504}e^{3} + \frac{8597}{344}e$ |
| 19 | $[19, 19, w^{2} - w - 4]$ | $\phantom{-}\frac{985}{1409024}e^{11} - \frac{19585}{352256}e^{9} + \frac{520581}{352256}e^{7} - \frac{677961}{44032}e^{5} + \frac{1246805}{22016}e^{3} - \frac{25547}{1376}e$ |
| 29 | $[29, 29, -w - 1]$ | $\phantom{-}\frac{299}{704512}e^{10} - \frac{5987}{176128}e^{8} + \frac{161455}{176128}e^{6} - \frac{216667}{22016}e^{4} + \frac{418271}{11008}e^{2} - \frac{6689}{688}$ |
| 29 | $[29, 29, w - 1]$ | $-\frac{531}{704512}e^{10} + \frac{10331}{176128}e^{8} - \frac{264439}{176128}e^{6} + \frac{322059}{22016}e^{4} - \frac{535447}{11008}e^{2} + \frac{9065}{688}$ |
| 31 | $[31, 31, -2w^{2} - w + 14]$ | $\phantom{-}\frac{493}{1409024}e^{11} - \frac{10005}{352256}e^{9} + \frac{274569}{352256}e^{7} - \frac{374157}{44032}e^{5} + \frac{705401}{22016}e^{3} - \frac{6167}{1376}e$ |
| 31 | $[31, 31, w^{3} - 3w^{2} - 6w + 19]$ | $-\frac{39}{1409024}e^{11} + \frac{751}{352256}e^{9} - \frac{18427}{352256}e^{7} + \frac{18255}{44032}e^{5} + \frac{5957}{22016}e^{3} - \frac{11691}{1376}e$ |
| 31 | $[31, 31, -w^{3} - 3w^{2} + 6w + 19]$ | $\phantom{-}\frac{425}{704512}e^{11} - \frac{8625}{176128}e^{9} + \frac{237077}{176128}e^{7} - \frac{326345}{22016}e^{5} + \frac{652421}{11008}e^{3} - \frac{20571}{688}e$ |
| 31 | $[31, 31, -2w^{2} + w + 14]$ | $-\frac{613}{704512}e^{11} + \frac{12157}{176128}e^{9} - \frac{321953}{176128}e^{7} + \frac{417757}{22016}e^{5} - \frac{776353}{11008}e^{3} + \frac{22959}{688}e$ |
| 41 | $[41, 41, -w]$ | $-\frac{155}{704512}e^{10} + \frac{3267}{176128}e^{8} - \frac{94687}{176128}e^{6} + \frac{139235}{22016}e^{4} - \frac{287583}{11008}e^{2} + \frac{5665}{688}$ |
| 71 | $[71, 71, w^{3} + 3w^{2} - 7w - 19]$ | $\phantom{-}\frac{1285}{1409024}e^{11} - \frac{25309}{352256}e^{9} + \frac{661057}{352256}e^{7} - \frac{831165}{44032}e^{5} + \frac{1416961}{22016}e^{3} - \frac{10767}{1376}e$ |
| 71 | $[71, 71, w^{3} - 3w^{2} - 7w + 19]$ | $\phantom{-}\frac{63}{704512}e^{11} - \frac{1319}{176128}e^{9} + \frac{37811}{176128}e^{7} - \frac{54839}{22016}e^{5} + \frac{117075}{11008}e^{3} - \frac{7629}{688}e$ |
| 79 | $[79, 79, w^{3} + 3w^{2} - 9w - 25]$ | $\phantom{-}\frac{1583}{1409024}e^{11} - \frac{31559}{352256}e^{9} + \frac{843491}{352256}e^{7} - \frac{1114047}{44032}e^{5} + \frac{2152147}{22016}e^{3} - \frac{82269}{1376}e$ |
| 79 | $[79, 79, w^{3} - 3w^{2} - 9w + 25]$ | $-\frac{2569}{1409024}e^{11} + \frac{51569}{352256}e^{9} - \frac{1392629}{352256}e^{7} + \frac{1862361}{44032}e^{5} - \frac{3562949}{22016}e^{3} + \frac{91851}{1376}e$ |
| 81 | $[81, 3, -3]$ | $\phantom{-}\frac{63}{352256}e^{10} - \frac{1319}{88064}e^{8} + \frac{37811}{88064}e^{6} - \frac{54839}{11008}e^{4} + \frac{114323}{5504}e^{2} - \frac{1437}{344}$ |
| 89 | $[89, 89, -w^{3} - 4w^{2} + 6w + 24]$ | $\phantom{-}\frac{23}{88064}e^{10} - \frac{487}{22016}e^{8} + \frac{14219}{22016}e^{6} - \frac{21099}{2752}e^{4} + \frac{43315}{1376}e^{2} - \frac{369}{86}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, -w^{3} + 2w^{2} + 6w - 11]$ | $-1$ |
| $5$ | $[5, 5, -w - 2]$ | $-1$ |