Base field \(\Q(\sqrt{353}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 88\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[8,4,2w + 18]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $16$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 3x^{7} - 30x^{6} + 81x^{5} + 202x^{4} - 416x^{3} - 424x^{2} + 476x + 392\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w + 10]$ | $\phantom{-}1$ |
| 2 | $[2, 2, w + 9]$ | $\phantom{-}0$ |
| 9 | $[9, 3, 3]$ | $\phantom{-}e$ |
| 11 | $[11, 11, -10w + 99]$ | $-\frac{1478}{29463}e^{7} + \frac{14153}{58926}e^{6} + \frac{71579}{58926}e^{5} - \frac{195143}{29463}e^{4} - \frac{154145}{58926}e^{3} + \frac{342721}{9821}e^{2} - \frac{13697}{1281}e - \frac{147332}{4209}$ |
| 11 | $[11, 11, 10w + 89]$ | $\phantom{-}\frac{5113}{117852}e^{7} - \frac{24929}{117852}e^{6} - \frac{30980}{29463}e^{5} + \frac{681637}{117852}e^{4} + \frac{70526}{29463}e^{3} - \frac{292753}{9821}e^{2} + \frac{10894}{1281}e + \frac{132139}{4209}$ |
| 17 | $[17, 17, -66w - 587]$ | $-\frac{7507}{58926}e^{7} + \frac{11089}{29463}e^{6} + \frac{215081}{58926}e^{5} - \frac{572521}{58926}e^{4} - \frac{1198397}{58926}e^{3} + \frac{408651}{9821}e^{2} + \frac{20317}{1281}e - \frac{106796}{4209}$ |
| 17 | $[17, 17, -66w + 653]$ | $\phantom{-}\frac{7471}{117852}e^{7} - \frac{14909}{117852}e^{6} - \frac{116713}{58926}e^{5} + \frac{356623}{117852}e^{4} + \frac{837871}{58926}e^{3} - \frac{89317}{9821}e^{2} - \frac{28715}{1281}e - \frac{11591}{4209}$ |
| 19 | $[19, 19, -28w + 277]$ | $-\frac{1419}{19642}e^{7} + \frac{2083}{9821}e^{6} + \frac{20242}{9821}e^{5} - \frac{54057}{9821}e^{4} - \frac{225115}{19642}e^{3} + \frac{466999}{19642}e^{2} + \frac{4835}{427}e - \frac{17081}{1403}$ |
| 19 | $[19, 19, 28w + 249]$ | $-\frac{53}{9821}e^{7} + \frac{1699}{9821}e^{6} - \frac{1637}{9821}e^{5} - \frac{51046}{9821}e^{4} + \frac{68485}{9821}e^{3} + \frac{331147}{9821}e^{2} - \frac{10996}{427}e - \frac{61426}{1403}$ |
| 23 | $[23, 23, -8w - 71]$ | $\phantom{-}\frac{7543}{117852}e^{7} - \frac{29447}{117852}e^{6} - \frac{49184}{29463}e^{5} + \frac{788419}{117852}e^{4} + \frac{180263}{29463}e^{3} - \frac{319334}{9821}e^{2} + \frac{8398}{1281}e + \frac{118387}{4209}$ |
| 23 | $[23, 23, 8w - 79]$ | $-\frac{7507}{58926}e^{7} + \frac{11089}{29463}e^{6} + \frac{215081}{58926}e^{5} - \frac{572521}{58926}e^{4} - \frac{1198397}{58926}e^{3} + \frac{408651}{9821}e^{2} + \frac{20317}{1281}e - \frac{106796}{4209}$ |
| 25 | $[25, 5, -5]$ | $\phantom{-}\frac{5113}{117852}e^{7} - \frac{24929}{117852}e^{6} - \frac{30980}{29463}e^{5} + \frac{681637}{117852}e^{4} + \frac{70526}{29463}e^{3} - \frac{292753}{9821}e^{2} + \frac{8332}{1281}e + \frac{123721}{4209}$ |
| 29 | $[29, 29, -2w + 19]$ | $-\frac{53}{9821}e^{7} + \frac{1699}{9821}e^{6} - \frac{1637}{9821}e^{5} - \frac{51046}{9821}e^{4} + \frac{68485}{9821}e^{3} + \frac{331147}{9821}e^{2} - \frac{10569}{427}e - \frac{61426}{1403}$ |
| 29 | $[29, 29, 2w + 17]$ | $-\frac{601}{39284}e^{7} - \frac{2229}{39284}e^{6} + \frac{6709}{9821}e^{5} + \frac{76941}{39284}e^{4} - \frac{169369}{19642}e^{3} - \frac{318379}{19642}e^{2} + \frac{10762}{427}e + \frac{37094}{1403}$ |
| 41 | $[41, 41, -6w - 53]$ | $\phantom{-}\frac{4135}{58926}e^{7} - \frac{9143}{58926}e^{6} - \frac{63166}{29463}e^{5} + \frac{227779}{58926}e^{4} + \frac{425482}{29463}e^{3} - \frac{139285}{9821}e^{2} - \frac{24631}{1281}e + \frac{12020}{4209}$ |
| 41 | $[41, 41, 6w - 59]$ | $-\frac{10463}{58926}e^{7} + \frac{36331}{58926}e^{6} + \frac{143330}{29463}e^{5} - \frac{962807}{58926}e^{4} - \frac{676271}{29463}e^{3} + \frac{751372}{9821}e^{2} + \frac{7901}{1281}e - \frac{245710}{4209}$ |
| 43 | $[43, 43, -4w + 39]$ | $-\frac{1637}{29463}e^{7} + \frac{24347}{58926}e^{6} + \frac{61757}{58926}e^{5} - \frac{348281}{29463}e^{4} + \frac{256765}{58926}e^{3} + \frac{673868}{9821}e^{2} - \frac{49247}{1281}e - \frac{323192}{4209}$ |
| 43 | $[43, 43, 4w + 35]$ | $\phantom{-}\frac{1888}{29463}e^{7} - \frac{19315}{58926}e^{6} - \frac{46196}{29463}e^{5} + \frac{531683}{58926}e^{4} + \frac{238249}{58926}e^{3} - \frac{929421}{19642}e^{2} + \frac{10315}{1281}e + \frac{188299}{4209}$ |
| 47 | $[47, 47, 2w - 21]$ | $\phantom{-}\frac{31}{366}e^{7} - \frac{67}{183}e^{6} - \frac{400}{183}e^{5} + \frac{1817}{183}e^{4} + \frac{2843}{366}e^{3} - \frac{6103}{122}e^{2} + \frac{1483}{183}e + \frac{9053}{183}$ |
| 47 | $[47, 47, -2w - 19]$ | $\phantom{-}\frac{4465}{29463}e^{7} - \frac{11939}{29463}e^{6} - \frac{130037}{29463}e^{5} + \frac{301570}{29463}e^{4} + \frac{771989}{29463}e^{3} - \frac{393644}{9821}e^{2} - \frac{37112}{1281}e + \frac{75184}{4209}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2,2,w + 9]$ | $-1$ |
| $2$ | $[2,2,-w + 10]$ | $-1$ |