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: | $16$ |
| 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:
| \(x^{16} - 28x^{14} + 323x^{12} - 1979x^{10} + 6912x^{8} - 13683x^{6} + 14188x^{4} - 6015x^{2} + 121\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w + 2]$ | $\phantom{-}e$ |
| 4 | $[4, 2, -w^{2} - w + 9]$ | $-\frac{1}{275}e^{15} - \frac{16}{275}e^{13} + \frac{623}{275}e^{11} - \frac{5809}{275}e^{9} + \frac{23767}{275}e^{7} - \frac{45244}{275}e^{5} + \frac{35851}{275}e^{3} - \frac{7416}{275}e$ |
| 5 | $[5, 5, -2w^{2} - w + 21]$ | $\phantom{-}\frac{1}{25}e^{14} - \frac{24}{25}e^{12} + \frac{227}{25}e^{10} - \frac{1071}{25}e^{8} + \frac{2628}{25}e^{6} - \frac{3171}{25}e^{4} + \frac{1529}{25}e^{2} - \frac{49}{25}$ |
| 5 | $[5, 5, w - 1]$ | $\phantom{-}\frac{1}{25}e^{14} - \frac{24}{25}e^{12} + \frac{227}{25}e^{10} - \frac{1071}{25}e^{8} + \frac{2628}{25}e^{6} - \frac{3171}{25}e^{4} + \frac{1529}{25}e^{2} - \frac{49}{25}$ |
| 13 | $[13, 13, -2w^{2} - w + 19]$ | $\phantom{-}\frac{1}{55}e^{15} - \frac{28}{55}e^{13} + \frac{312}{55}e^{11} - \frac{1759}{55}e^{9} + \frac{1048}{11}e^{7} - \frac{1531}{11}e^{5} + \frac{3892}{55}e^{3} + \frac{607}{55}e$ |
| 13 | $[13, 13, w^{2} - 9]$ | $\phantom{-}1$ |
| 13 | $[13, 13, 3w^{2} + 2w - 29]$ | $\phantom{-}\frac{1}{55}e^{15} - \frac{28}{55}e^{13} + \frac{312}{55}e^{11} - \frac{1759}{55}e^{9} + \frac{1048}{11}e^{7} - \frac{1531}{11}e^{5} + \frac{3892}{55}e^{3} + \frac{607}{55}e$ |
| 17 | $[17, 17, -w^{2} - 2w + 7]$ | $\phantom{-}\frac{2}{55}e^{15} - \frac{56}{55}e^{13} + \frac{624}{55}e^{11} - \frac{3518}{55}e^{9} + \frac{2107}{11}e^{7} - \frac{3216}{11}e^{5} + \frac{11029}{55}e^{3} - \frac{2526}{55}e$ |
| 23 | $[23, 23, -w^{2} + 3]$ | $\phantom{-}\frac{16}{275}e^{15} - \frac{349}{275}e^{13} + \frac{2957}{275}e^{11} - \frac{12491}{275}e^{9} + \frac{29093}{275}e^{7} - \frac{42026}{275}e^{5} + \frac{40294}{275}e^{3} - \frac{18074}{275}e$ |
| 27 | $[27, 3, -3]$ | $\phantom{-}\frac{4}{25}e^{14} - \frac{86}{25}e^{12} + \frac{708}{25}e^{10} - \frac{2789}{25}e^{8} + \frac{5357}{25}e^{6} - \frac{4549}{25}e^{4} + \frac{1221}{25}e^{2} - \frac{86}{25}$ |
| 31 | $[31, 31, -w^{2} + 7]$ | $-\frac{68}{275}e^{15} + \frac{1607}{275}e^{13} - \frac{15111}{275}e^{11} + \frac{72103}{275}e^{9} - \frac{183829}{275}e^{7} + \frac{239028}{275}e^{5} - \frac{129422}{275}e^{3} + \frac{9632}{275}e$ |
| 43 | $[43, 43, w^{2} - 13]$ | $\phantom{-}\frac{3}{25}e^{14} - \frac{72}{25}e^{12} + \frac{681}{25}e^{10} - \frac{3213}{25}e^{8} + \frac{7859}{25}e^{6} - \frac{9288}{25}e^{4} + \frac{4137}{25}e^{2} - \frac{47}{25}$ |
| 47 | $[47, 47, 2w^{2} + 3w - 11]$ | $-\frac{4}{25}e^{15} + \frac{91}{25}e^{13} - \frac{808}{25}e^{11} + \frac{3524}{25}e^{9} - \frac{7722}{25}e^{7} + \frac{7404}{25}e^{5} - \frac{1081}{25}e^{3} - \frac{1384}{25}e$ |
| 59 | $[59, 59, w^{2} - 5]$ | $\phantom{-}\frac{9}{275}e^{15} - \frac{241}{275}e^{13} + \frac{2643}{275}e^{11} - \frac{15314}{275}e^{9} + \frac{50152}{275}e^{7} - \frac{90389}{275}e^{5} + \frac{76861}{275}e^{3} - \frac{19441}{275}e$ |
| 61 | $[61, 61, 5w^{2} + 4w - 47]$ | $\phantom{-}\frac{52}{275}e^{15} - \frac{1258}{275}e^{13} + \frac{12154}{275}e^{11} - \frac{59887}{275}e^{9} + \frac{159136}{275}e^{7} - \frac{220652}{275}e^{5} + \frac{137528}{275}e^{3} - \frac{22083}{275}e$ |
| 61 | $[61, 61, 4w - 7]$ | $\phantom{-}\frac{52}{275}e^{15} - \frac{1258}{275}e^{13} + \frac{12154}{275}e^{11} - \frac{59887}{275}e^{9} + \frac{158861}{275}e^{7} - \frac{217352}{275}e^{5} + \frac{126253}{275}e^{3} - \frac{12733}{275}e$ |
| 61 | $[61, 61, w - 5]$ | $-\frac{14}{25}e^{14} + \frac{326}{25}e^{12} - \frac{3003}{25}e^{10} + \frac{13924}{25}e^{8} - \frac{34112}{25}e^{6} + \frac{41934}{25}e^{4} - \frac{20686}{25}e^{2} + \frac{701}{25}$ |
| 71 | $[71, 71, -2w^{2} - 2w + 21]$ | $-\frac{1}{55}e^{15} + \frac{28}{55}e^{13} - \frac{312}{55}e^{11} + \frac{1759}{55}e^{9} - \frac{1048}{11}e^{7} + \frac{1531}{11}e^{5} - \frac{3892}{55}e^{3} - \frac{717}{55}e$ |
| 73 | $[73, 73, -2w^{2} + 4w - 1]$ | $-\frac{49}{275}e^{15} + \frac{1141}{275}e^{13} - \frac{10448}{275}e^{11} + \frac{47559}{275}e^{9} - \frac{111417}{275}e^{7} + \frac{123569}{275}e^{5} - \frac{47026}{275}e^{3} - \frac{1759}{275}e$ |
| 79 | $[79, 79, w + 5]$ | $\phantom{-}\frac{9}{25}e^{14} - \frac{206}{25}e^{12} + \frac{1868}{25}e^{10} - \frac{8569}{25}e^{8} + \frac{20997}{25}e^{6} - \frac{26304}{25}e^{4} + \frac{13541}{25}e^{2} - \frac{656}{25}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, w^{2} - 9]$ | $-1$ |