Base field 3.3.1076.1
Generator \(w\), with minimal polynomial \(x^{3} - 8x - 6\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[13, 13, 2w + 5]$ |
| Dimension: | $16$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $22$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{16} - 32x^{14} + 412x^{12} - 2766x^{10} + 10439x^{8} - 22156x^{6} + 24812x^{4} - 12272x^{2} + 1568\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w - 2]$ | $-\frac{1467}{10232}e^{14} + \frac{10943}{2558}e^{12} - \frac{63620}{1279}e^{10} + \frac{1468997}{5116}e^{8} - \frac{8795461}{10232}e^{6} + \frac{1606563}{1279}e^{4} - \frac{945796}{1279}e^{2} + \frac{128563}{1279}$ |
| 3 | $[3, 3, -w^{2} + 2w + 3]$ | $\phantom{-}e$ |
| 7 | $[7, 7, -2w^{2} + 5w + 5]$ | $\phantom{-}\frac{8817}{71624}e^{15} - \frac{131307}{35812}e^{13} + \frac{761809}{17906}e^{11} - \frac{8777267}{35812}e^{9} + \frac{52492035}{71624}e^{7} - \frac{38413111}{35812}e^{5} + \frac{11402647}{17906}e^{3} - \frac{805753}{8953}e$ |
| 9 | $[9, 3, -w^{2} + 5]$ | $\phantom{-}\frac{4043}{35812}e^{15} - \frac{119723}{35812}e^{13} + \frac{345015}{8953}e^{11} - \frac{3947127}{17906}e^{9} + \frac{23434139}{35812}e^{7} - \frac{34070649}{35812}e^{5} + \frac{5056881}{8953}e^{3} - \frac{755301}{8953}e$ |
| 13 | $[13, 13, -w^{2} + 3w + 1]$ | $-\frac{1969}{71624}e^{15} + \frac{6931}{8953}e^{13} - \frac{151357}{17906}e^{11} + \frac{1645463}{35812}e^{9} - \frac{9360159}{71624}e^{7} + \frac{3268185}{17906}e^{5} - \frac{1742757}{17906}e^{3} + \frac{43892}{8953}e$ |
| 13 | $[13, 13, 2w + 5]$ | $\phantom{-}1$ |
| 13 | $[13, 13, w - 1]$ | $-\frac{3531}{17906}e^{15} + \frac{209723}{35812}e^{13} - \frac{606027}{8953}e^{11} + \frac{3471977}{8953}e^{9} - \frac{10287028}{8953}e^{7} + \frac{59158049}{35812}e^{5} - \frac{8448145}{8953}e^{3} + \frac{1089555}{8953}e$ |
| 17 | $[17, 17, w^{2} - w - 5]$ | $\phantom{-}\frac{1411}{35812}e^{15} - \frac{44095}{35812}e^{13} + \frac{269813}{17906}e^{11} - \frac{1645289}{17906}e^{9} + \frac{10473303}{35812}e^{7} - \frac{16543013}{35812}e^{5} + \frac{5522245}{17906}e^{3} - \frac{503959}{8953}e$ |
| 19 | $[19, 19, w^{2} - 2w - 5]$ | $-\frac{214}{1279}e^{14} + \frac{6394}{1279}e^{12} - \frac{74388}{1279}e^{10} + \frac{428791}{1279}e^{8} - \frac{1276407}{1279}e^{6} + \frac{1837937}{1279}e^{4} - \frac{1046652}{1279}e^{2} + \frac{135788}{1279}$ |
| 29 | $[29, 29, w^{2} - 7]$ | $-\frac{2293}{35812}e^{15} + \frac{68105}{35812}e^{13} - \frac{393153}{17906}e^{11} + \frac{2243971}{17906}e^{9} - \frac{13175877}{35812}e^{7} + \frac{18516747}{35812}e^{5} - \frac{4944031}{17906}e^{3} + \frac{249019}{8953}e$ |
| 31 | $[31, 31, 2w^{2} - 3w - 11]$ | $-\frac{753}{5116}e^{14} + \frac{11315}{2558}e^{12} - \frac{66232}{1279}e^{10} + \frac{768625}{2558}e^{8} - \frac{4612423}{5116}e^{6} + \frac{3357561}{2558}e^{4} - \frac{969415}{1279}e^{2} + \frac{128408}{1279}$ |
| 49 | $[49, 7, -2w^{2} - 4w + 1]$ | $\phantom{-}\frac{16995}{71624}e^{15} - \frac{125999}{17906}e^{13} + \frac{1454455}{17906}e^{11} - \frac{16652681}{35812}e^{9} + \frac{98737413}{71624}e^{7} - \frac{17799081}{8953}e^{5} + \frac{20392525}{17906}e^{3} - \frac{1189516}{8953}e$ |
| 59 | $[59, 59, w^{2} - 2w - 11]$ | $-\frac{1317}{5116}e^{14} + \frac{9709}{1279}e^{12} - \frac{111575}{1279}e^{10} + \frac{1275945}{2558}e^{8} - \frac{7602059}{5116}e^{6} + \frac{2784031}{1279}e^{4} - \frac{1658649}{1279}e^{2} + \frac{232148}{1279}$ |
| 71 | $[71, 71, -2w + 7]$ | $\phantom{-}\frac{337}{2558}e^{14} - \frac{4658}{1279}e^{12} + \frac{49583}{1279}e^{10} - \frac{260581}{1279}e^{8} + \frac{1419605}{2558}e^{6} - \frac{943919}{1279}e^{4} + \frac{496226}{1279}e^{2} - \frac{46852}{1279}$ |
| 73 | $[73, 73, w^{2} - 3w - 13]$ | $\phantom{-}\frac{1363}{5116}e^{14} - \frac{9951}{1279}e^{12} + \frac{112883}{1279}e^{10} - \frac{1269083}{2558}e^{8} + \frac{7392929}{5116}e^{6} - \frac{2627163}{1279}e^{4} + \frac{1498313}{1279}e^{2} - \frac{182918}{1279}$ |
| 73 | $[73, 73, 2w^{2} - 2w - 17]$ | $\phantom{-}\frac{301}{2558}e^{14} - \frac{4168}{1279}e^{12} + \frac{44366}{1279}e^{10} - \frac{232308}{1279}e^{8} + \frac{1254847}{2558}e^{6} - \frac{822879}{1279}e^{4} + \frac{422988}{1279}e^{2} - \frac{46834}{1279}$ |
| 73 | $[73, 73, -w^{2} - 4w - 5]$ | $-\frac{2957}{5116}e^{14} + \frac{22007}{1279}e^{12} - \frac{255134}{1279}e^{10} + \frac{2934563}{2558}e^{8} - \frac{17483799}{5116}e^{6} + \frac{6338865}{1279}e^{4} - \frac{3676157}{1279}e^{2} + \frac{488358}{1279}$ |
| 79 | $[79, 79, 2w - 1]$ | $\phantom{-}\frac{32581}{71624}e^{15} - \frac{484889}{35812}e^{13} + \frac{2810067}{17906}e^{11} - \frac{32306615}{35812}e^{9} + \frac{192281151}{71624}e^{7} - \frac{139039297}{35812}e^{5} + \frac{39866585}{17906}e^{3} - \frac{2371022}{8953}e$ |
| 79 | $[79, 79, w^{2} + w - 5]$ | $\phantom{-}\frac{8291}{71624}e^{15} - \frac{123437}{35812}e^{13} + \frac{718047}{17906}e^{11} - \frac{8342353}{35812}e^{9} + \frac{50823289}{71624}e^{7} - \frac{38633285}{35812}e^{5} + \frac{12470545}{17906}e^{3} - \frac{1161569}{8953}e$ |
| 79 | $[79, 79, w - 5]$ | $-\frac{862}{1279}e^{14} + \frac{25313}{1279}e^{12} - \frac{289059}{1279}e^{10} + \frac{1637190}{1279}e^{8} - \frac{4808648}{1279}e^{6} + \frac{6896269}{1279}e^{4} - \frac{3977390}{1279}e^{2} + \frac{525252}{1279}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, 2w + 5]$ | $-1$ |