Base field \(\Q(\sqrt{305}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 76\); narrow class number \(4\) and class number \(2\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[1, 1, 1]$ |
| Dimension: | $14$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $36$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{14} - 20x^{12} + 151x^{10} - 545x^{8} + 987x^{6} - 872x^{4} + 329x^{2} - 39\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w]$ | $\phantom{-}e$ |
| 2 | $[2, 2, w + 1]$ | $\phantom{-}e$ |
| 5 | $[5, 5, -4w + 37]$ | $\phantom{-}\frac{5}{16}e^{12} - \frac{89}{16}e^{10} + \frac{139}{4}e^{8} - \frac{1441}{16}e^{6} + \frac{169}{2}e^{4} - 13e^{2} - \frac{51}{16}$ |
| 7 | $[7, 7, w + 2]$ | $\phantom{-}\frac{1}{4}e^{13} - \frac{19}{4}e^{11} + 33e^{9} - \frac{413}{4}e^{7} + \frac{287}{2}e^{5} - \frac{149}{2}e^{3} + \frac{27}{4}e$ |
| 7 | $[7, 7, w + 4]$ | $\phantom{-}\frac{1}{4}e^{13} - \frac{19}{4}e^{11} + 33e^{9} - \frac{413}{4}e^{7} + \frac{287}{2}e^{5} - \frac{149}{2}e^{3} + \frac{27}{4}e$ |
| 9 | $[9, 3, 3]$ | $\phantom{-}\frac{17}{32}e^{12} - \frac{317}{32}e^{10} + \frac{535}{8}e^{8} - \frac{6413}{32}e^{6} + \frac{527}{2}e^{4} - \frac{551}{4}e^{2} + \frac{785}{32}$ |
| 17 | $[17, 17, w + 6]$ | $-\frac{9}{32}e^{13} + \frac{181}{32}e^{11} - \frac{343}{8}e^{9} + \frac{4933}{32}e^{7} - \frac{541}{2}e^{5} + \frac{843}{4}e^{3} - \frac{1577}{32}e$ |
| 17 | $[17, 17, w + 10]$ | $-\frac{9}{32}e^{13} + \frac{181}{32}e^{11} - \frac{343}{8}e^{9} + \frac{4933}{32}e^{7} - \frac{541}{2}e^{5} + \frac{843}{4}e^{3} - \frac{1577}{32}e$ |
| 19 | $[19, 19, -2w + 19]$ | $-\frac{9}{16}e^{12} + \frac{165}{16}e^{10} - \frac{271}{4}e^{8} + \frac{3093}{16}e^{6} - 229e^{4} + \frac{189}{2}e^{2} - \frac{185}{16}$ |
| 19 | $[19, 19, -2w - 17]$ | $-\frac{9}{16}e^{12} + \frac{165}{16}e^{10} - \frac{271}{4}e^{8} + \frac{3093}{16}e^{6} - 229e^{4} + \frac{189}{2}e^{2} - \frac{185}{16}$ |
| 23 | $[23, 23, w + 5]$ | $-\frac{1}{16}e^{13} + \frac{21}{16}e^{11} - \frac{43}{4}e^{9} + \frac{701}{16}e^{7} - \frac{183}{2}e^{5} + 86e^{3} - \frac{377}{16}e$ |
| 23 | $[23, 23, w + 17]$ | $-\frac{1}{16}e^{13} + \frac{21}{16}e^{11} - \frac{43}{4}e^{9} + \frac{701}{16}e^{7} - \frac{183}{2}e^{5} + 86e^{3} - \frac{377}{16}e$ |
| 37 | $[37, 37, w + 1]$ | $-\frac{25}{32}e^{13} + \frac{453}{32}e^{11} - \frac{727}{8}e^{9} + \frac{7893}{32}e^{7} - \frac{515}{2}e^{5} + \frac{291}{4}e^{3} - \frac{217}{32}e$ |
| 37 | $[37, 37, w + 35]$ | $-\frac{25}{32}e^{13} + \frac{453}{32}e^{11} - \frac{727}{8}e^{9} + \frac{7893}{32}e^{7} - \frac{515}{2}e^{5} + \frac{291}{4}e^{3} - \frac{217}{32}e$ |
| 41 | $[41, 41, -22w + 203]$ | $-\frac{13}{32}e^{12} + \frac{249}{32}e^{10} - \frac{435}{8}e^{8} + \frac{5433}{32}e^{6} - 231e^{4} + \frac{453}{4}e^{2} - \frac{285}{32}$ |
| 41 | $[41, 41, -6w + 55]$ | $-\frac{13}{32}e^{12} + \frac{249}{32}e^{10} - \frac{435}{8}e^{8} + \frac{5433}{32}e^{6} - 231e^{4} + \frac{453}{4}e^{2} - \frac{285}{32}$ |
| 43 | $[43, 43, w + 20]$ | $-\frac{9}{16}e^{13} + \frac{165}{16}e^{11} - \frac{271}{4}e^{9} + \frac{3093}{16}e^{7} - 228e^{5} + \frac{173}{2}e^{3} + \frac{23}{16}e$ |
| 43 | $[43, 43, w + 22]$ | $-\frac{9}{16}e^{13} + \frac{165}{16}e^{11} - \frac{271}{4}e^{9} + \frac{3093}{16}e^{7} - 228e^{5} + \frac{173}{2}e^{3} + \frac{23}{16}e$ |
| 53 | $[53, 53, w + 13]$ | $-\frac{11}{8}e^{13} + \frac{207}{8}e^{11} - \frac{355}{2}e^{9} + \frac{4383}{8}e^{7} - 763e^{5} + 443e^{3} - \frac{667}{8}e$ |
| 53 | $[53, 53, w + 39]$ | $-\frac{11}{8}e^{13} + \frac{207}{8}e^{11} - \frac{355}{2}e^{9} + \frac{4383}{8}e^{7} - 763e^{5} + 443e^{3} - \frac{667}{8}e$ |
Atkin-Lehner eigenvalues
This form has no Atkin-Lehner eigenvalues since the level is \((1)\).