Base field 3.3.1765.1
Generator \(w\), with minimal polynomial \(x^{3} - x^{2} - 11x + 16\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[13, 13, w^{2} - 9]$ |
| Dimension: | $20$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $48$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{20} - 27x^{18} + 302x^{16} - 1831x^{14} + 6585x^{12} - 14410x^{10} + 18912x^{8} - 14151x^{6} + 5498x^{4} - 916x^{2} + 36\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w + 2]$ | $\phantom{-}e$ |
| 4 | $[4, 2, -w^{2} - w + 9]$ | $-\frac{1059}{81203}e^{19} + \frac{51337}{162406}e^{17} - \frac{505725}{162406}e^{15} + \frac{1336715}{81203}e^{13} - \frac{8457529}{162406}e^{11} + \frac{16971311}{162406}e^{9} - \frac{10971725}{81203}e^{7} + \frac{8321155}{81203}e^{5} - \frac{4807191}{162406}e^{3} - \frac{253558}{81203}e$ |
| 5 | $[5, 5, -2w^{2} - w + 21]$ | $-\frac{2957}{81203}e^{18} + \frac{73475}{81203}e^{16} - \frac{736997}{81203}e^{14} + \frac{3886883}{81203}e^{12} - \frac{11756000}{81203}e^{10} + \frac{20976973}{81203}e^{8} - \frac{22147592}{81203}e^{6} + \frac{13494506}{81203}e^{4} - \frac{4000587}{81203}e^{2} + \frac{212685}{81203}$ |
| 5 | $[5, 5, w - 1]$ | $-\frac{389}{81203}e^{18} + \frac{13071}{81203}e^{16} - \frac{177882}{81203}e^{14} + \frac{1271374}{81203}e^{12} - \frac{5177074}{81203}e^{10} + \frac{12184416}{81203}e^{8} - \frac{16149325}{81203}e^{6} + \frac{11503006}{81203}e^{4} - \frac{4211425}{81203}e^{2} + \frac{470298}{81203}$ |
| 13 | $[13, 13, -2w^{2} - w + 19]$ | $-\frac{14374}{81203}e^{19} + \frac{382372}{81203}e^{17} - \frac{4182989}{81203}e^{15} + \frac{24517030}{81203}e^{13} - \frac{83591548}{81203}e^{11} + \frac{167482969}{81203}e^{9} - \frac{188121492}{81203}e^{7} + \frac{104056007}{81203}e^{5} - \frac{20233046}{81203}e^{3} - \frac{551943}{81203}e$ |
| 13 | $[13, 13, w^{2} - 9]$ | $-1$ |
| 13 | $[13, 13, 3w^{2} + 2w - 29]$ | $...$ |
| 17 | $[17, 17, -w^{2} - 2w + 7]$ | $-\frac{8140}{81203}e^{19} + \frac{423663}{162406}e^{17} - \frac{4521005}{162406}e^{15} + \frac{12924603}{81203}e^{13} - \frac{86543973}{162406}e^{11} + \frac{173985699}{162406}e^{9} - \frac{103714884}{81203}e^{7} + \frac{69900388}{81203}e^{5} - \frac{47214931}{162406}e^{3} + \frac{2638345}{81203}e$ |
| 23 | $[23, 23, -w^{2} + 3]$ | $...$ |
| 27 | $[27, 3, -3]$ | $...$ |
| 31 | $[31, 31, -w^{2} + 7]$ | $...$ |
| 43 | $[43, 43, w^{2} - 13]$ | $\phantom{-}\frac{8366}{81203}e^{18} - \frac{200325}{81203}e^{16} + \frac{1898444}{81203}e^{14} - \frac{9129860}{81203}e^{12} + \frac{23444482}{81203}e^{10} - \frac{30129152}{81203}e^{8} + \frac{13860731}{81203}e^{6} + \frac{3499072}{81203}e^{4} - \frac{2571104}{81203}e^{2} - \frac{449185}{81203}$ |
| 47 | $[47, 47, 2w^{2} + 3w - 11]$ | $-\frac{13089}{81203}e^{19} + \frac{368210}{81203}e^{17} - \frac{4292185}{81203}e^{15} + \frac{26975058}{81203}e^{13} - \frac{99190288}{81203}e^{11} + \frac{215861011}{81203}e^{9} - \frac{267003920}{81203}e^{7} + \frac{169215515}{81203}e^{5} - \frac{44657644}{81203}e^{3} + \frac{2946414}{81203}e$ |
| 59 | $[59, 59, w^{2} - 5]$ | $...$ |
| 61 | $[61, 61, 5w^{2} + 4w - 47]$ | $...$ |
| 61 | $[61, 61, 4w - 7]$ | $...$ |
| 61 | $[61, 61, w - 5]$ | $...$ |
| 71 | $[71, 71, -2w^{2} - 2w + 21]$ | $...$ |
| 73 | $[73, 73, -2w^{2} + 4w - 1]$ | $...$ |
| 79 | $[79, 79, w + 5]$ | $-\frac{21313}{162406}e^{18} + \frac{572531}{162406}e^{16} - \frac{3168996}{81203}e^{14} + \frac{37667135}{162406}e^{12} - \frac{130501197}{162406}e^{10} + \frac{133253386}{81203}e^{8} - \frac{153449293}{81203}e^{6} + \frac{175944399}{162406}e^{4} - \frac{17999876}{81203}e^{2} + \frac{6584}{81203}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, w^{2} - 9]$ | $1$ |