Base field \(\Q(\zeta_{28})^+\)
Generator \(w\), with minimal polynomial \(x^{6} - 7x^{4} + 14x^{2} - 7\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2, 2]$ |
| Level: | $[29,29,w^{4} + w^{3} - 5w^{2} - 3w + 4]$ |
| Dimension: | $12$ |
| 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:
| \(x^{12} - 58x^{10} + 1261x^{8} - 12484x^{6} + 53386x^{4} - 71194x^{2} + 4802\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 7 | $[7, 7, w]$ | $\phantom{-}e$ |
| 8 | $[8, 2, w^{3} + w^{2} - 2w - 1]$ | $\phantom{-}\frac{11294}{7291721}e^{10} - \frac{460410}{7291721}e^{8} + \frac{6029359}{7291721}e^{6} - \frac{26291365}{7291721}e^{4} + \frac{21512178}{7291721}e^{2} - \frac{5023969}{7291721}$ |
| 27 | $[27, 3, -w^{5} - w^{4} + 5w^{3} + 5w^{2} - 5w - 6]$ | $\phantom{-}\frac{31588}{51042047}e^{11} - \frac{1504644}{51042047}e^{9} + \frac{26076523}{51042047}e^{7} - \frac{201342370}{51042047}e^{5} + \frac{658899082}{51042047}e^{3} - \frac{431431786}{51042047}e$ |
| 27 | $[27, 3, w^{4} - 4w^{2} - w + 1]$ | $-\frac{48904}{51042047}e^{11} + \frac{2219585}{51042047}e^{9} - \frac{35400815}{51042047}e^{7} + \frac{238470728}{51042047}e^{5} - \frac{676116687}{51042047}e^{3} + \frac{732395511}{51042047}e$ |
| 29 | $[29, 29, w^{2} + w - 3]$ | $\phantom{-}\frac{50051}{7291721}e^{10} - \frac{2157878}{7291721}e^{8} + \frac{31406592}{7291721}e^{6} - \frac{173590730}{7291721}e^{4} + \frac{291434761}{7291721}e^{2} - \frac{32024404}{7291721}$ |
| 29 | $[29, 29, -w^{4} - w^{3} + 5w^{2} + 3w - 4]$ | $-1$ |
| 29 | $[29, 29, w^{5} + w^{4} - 5w^{3} - 4w^{2} + 5w + 3]$ | $-\frac{9460}{7291721}e^{10} + \frac{419218}{7291721}e^{8} - \frac{6389301}{7291721}e^{6} + \frac{39371298}{7291721}e^{4} - \frac{91756035}{7291721}e^{2} + \frac{30955212}{7291721}$ |
| 29 | $[29, 29, -w^{5} + w^{4} + 5w^{3} - 4w^{2} - 5w + 3]$ | $\phantom{-}\frac{7106}{7291721}e^{10} - \frac{247071}{7291721}e^{8} + \frac{2374484}{7291721}e^{6} - \frac{2374439}{7291721}e^{4} - \frac{29853664}{7291721}e^{2} + \frac{39761632}{7291721}$ |
| 29 | $[29, 29, w^{4} - w^{3} - 5w^{2} + 3w + 4]$ | $-\frac{9460}{7291721}e^{10} + \frac{419218}{7291721}e^{8} - \frac{6389301}{7291721}e^{6} + \frac{39371298}{7291721}e^{4} - \frac{91756035}{7291721}e^{2} + \frac{30955212}{7291721}$ |
| 29 | $[29, 29, -w^{2} + w + 3]$ | $\phantom{-}\frac{17316}{7291721}e^{10} - \frac{714941}{7291721}e^{8} + \frac{9324292}{7291721}e^{6} - \frac{37128358}{7291721}e^{4} - \frac{4657558}{7291721}e^{2} + \frac{56330604}{7291721}$ |
| 83 | $[83, 83, w^{4} - w^{3} - 4w^{2} + 2w + 1]$ | $\phantom{-}\frac{18926}{51042047}e^{11} - \frac{138820}{51042047}e^{9} - \frac{17375472}{51042047}e^{7} + \frac{345730839}{51042047}e^{5} - \frac{1893760884}{51042047}e^{3} + \frac{2086377441}{51042047}e$ |
| 83 | $[83, 83, w^{5} - w^{4} - 4w^{3} + 6w^{2} + 3w - 5]$ | $\phantom{-}\frac{81926}{51042047}e^{11} - \frac{4225588}{51042047}e^{9} + \frac{80749163}{51042047}e^{7} - \frac{695586641}{51042047}e^{5} + \frac{2570488339}{51042047}e^{3} - \frac{2904991375}{51042047}e$ |
| 83 | $[83, 83, -w^{5} - w^{4} + 5w^{3} + 5w^{2} - 4w - 6]$ | $-\frac{7626}{7291721}e^{11} + \frac{378026}{7291721}e^{9} - \frac{6749243}{7291721}e^{7} + \frac{52451231}{7291721}e^{5} - \frac{169291613}{7291721}e^{3} + \frac{144387107}{7291721}e$ |
| 83 | $[83, 83, w^{5} - w^{4} - 5w^{3} + 5w^{2} + 4w - 6]$ | $-\frac{11064}{7291721}e^{11} + \frac{542713}{7291721}e^{9} - \frac{9843611}{7291721}e^{7} + \frac{80985536}{7291721}e^{5} - \frac{294509105}{7291721}e^{3} + \frac{360656149}{7291721}e$ |
| 83 | $[83, 83, -w^{5} + w^{4} + 6w^{3} - 3w^{2} - 9w + 2]$ | $\phantom{-}\frac{23228}{51042047}e^{11} - \frac{1642897}{51042047}e^{9} + \frac{40440003}{51042047}e^{7} - \frac{421589790}{51042047}e^{5} + \frac{1710572732}{51042047}e^{3} - \frac{1656895430}{51042047}e$ |
| 83 | $[83, 83, -w^{4} - w^{3} + 4w^{2} + 2w - 1]$ | $\phantom{-}\frac{23228}{51042047}e^{11} - \frac{1642897}{51042047}e^{9} + \frac{40440003}{51042047}e^{7} - \frac{421589790}{51042047}e^{5} + \frac{1710572732}{51042047}e^{3} - \frac{1656895430}{51042047}e$ |
| 113 | $[113, 113, -w^{5} - w^{4} + 5w^{3} + 3w^{2} - 5w - 1]$ | $-\frac{41508}{7291721}e^{10} + \frac{1759256}{7291721}e^{8} - \frac{24837320}{7291721}e^{6} + \frac{131325326}{7291721}e^{4} - \frac{226536426}{7291721}e^{2} + \frac{86541804}{7291721}$ |
| 113 | $[113, 113, 2w^{4} - w^{3} - 9w^{2} + 3w + 6]$ | $\phantom{-}\frac{12044}{7291721}e^{10} - \frac{509062}{7291721}e^{8} + \frac{6589866}{7291721}e^{6} - \frac{21673986}{7291721}e^{4} - \frac{52339472}{7291721}e^{2} + \frac{137292588}{7291721}$ |
| 113 | $[113, 113, w^{4} - 6w^{2} - w + 8]$ | $-\frac{8376}{7291721}e^{10} + \frac{426678}{7291721}e^{8} - \frac{7309750}{7291721}e^{6} + \frac{47833852}{7291721}e^{4} - \frac{102731684}{7291721}e^{2} + \frac{104154644}{7291721}$ |
| 113 | $[113, 113, w^{4} - 6w^{2} + w + 8]$ | $\phantom{-}\frac{44779}{7291721}e^{10} - \frac{1951999}{7291721}e^{8} + \frac{28672166}{7291721}e^{6} - \frac{158136358}{7291721}e^{4} + \frac{243752847}{7291721}e^{2} + \frac{48937580}{7291721}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $29$ | $[29,29,w^{4} + w^{3} - 5w^{2} - 3w + 4]$ | $1$ |