Base field 3.3.1257.1
Generator \(w\), with minimal polynomial \(x^{3} - x^{2} - 8x + 9\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[13, 13, -2w + 5]$ |
| Dimension: | $10$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $24$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{10} + 3x^{9} - 12x^{8} - 32x^{7} + 52x^{6} + 102x^{5} - 94x^{4} - 103x^{3} + 54x^{2} + 29x - 4\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, -w + 3]$ | $\phantom{-}e$ |
| 3 | $[3, 3, w - 2]$ | $\phantom{-}\frac{133}{4705}e^{9} - \frac{17}{4705}e^{8} - \frac{3347}{4705}e^{7} - \frac{827}{4705}e^{6} + \frac{5049}{941}e^{5} + \frac{7231}{4705}e^{4} - \frac{67559}{4705}e^{3} - \frac{13581}{4705}e^{2} + \frac{48069}{4705}e + \frac{5304}{4705}$ |
| 5 | $[5, 5, w^{2} + w - 5]$ | $\phantom{-}\frac{4}{4705}e^{9} + \frac{424}{4705}e^{8} + \frac{1279}{4705}e^{7} - \frac{4836}{4705}e^{6} - \frac{2657}{941}e^{5} + \frac{20028}{4705}e^{4} + \frac{39358}{4705}e^{3} - \frac{30478}{4705}e^{2} - \frac{33718}{4705}e + \frac{8862}{4705}$ |
| 8 | $[8, 2, 2]$ | $\phantom{-}\frac{7}{941}e^{9} - \frac{199}{941}e^{8} - \frac{820}{941}e^{7} + \frac{1888}{941}e^{6} + \frac{7569}{941}e^{5} - \frac{5414}{941}e^{4} - \frac{20048}{941}e^{3} + \frac{5476}{941}e^{2} + \frac{12039}{941}e - \frac{959}{941}$ |
| 13 | $[13, 13, w + 2]$ | $\phantom{-}\frac{644}{4705}e^{9} + \frac{2394}{4705}e^{8} - \frac{5806}{4705}e^{7} - \frac{25796}{4705}e^{6} + \frac{2260}{941}e^{5} + \frac{86273}{4705}e^{4} + \frac{13118}{4705}e^{3} - \frac{103153}{4705}e^{2} - \frac{27258}{4705}e + \frac{34102}{4705}$ |
| 13 | $[13, 13, -2w + 5]$ | $\phantom{-}1$ |
| 13 | $[13, 13, -w^{2} - w + 4]$ | $\phantom{-}\frac{230}{941}e^{9} + \frac{855}{941}e^{8} - \frac{2208}{941}e^{7} - \frac{8944}{941}e^{6} + \frac{6321}{941}e^{5} + \frac{27115}{941}e^{4} - \frac{5666}{941}e^{3} - \frac{21986}{941}e^{2} + \frac{616}{941}e + \frac{2366}{941}$ |
| 19 | $[19, 19, -w^{2} + w + 4]$ | $-\frac{2026}{4705}e^{9} - \frac{7736}{4705}e^{8} + \frac{17944}{4705}e^{7} + \frac{78114}{4705}e^{6} - \frac{8799}{941}e^{5} - \frac{230747}{4705}e^{4} + \frac{23783}{4705}e^{3} + \frac{202317}{4705}e^{2} + \frac{8427}{4705}e - \frac{32968}{4705}$ |
| 23 | $[23, 23, -w^{2} - w + 7]$ | $-\frac{271}{4705}e^{9} - \frac{496}{4705}e^{8} + \frac{3919}{4705}e^{7} + \frac{2994}{4705}e^{6} - \frac{4189}{941}e^{5} + \frac{7553}{4705}e^{4} + \frac{36518}{4705}e^{3} - \frac{50013}{4705}e^{2} + \frac{4822}{4705}e + \frac{41832}{4705}$ |
| 25 | $[25, 5, w^{2} - 2w - 1]$ | $-\frac{1928}{4705}e^{9} - \frac{6758}{4705}e^{8} + \frac{18697}{4705}e^{7} + \frac{67847}{4705}e^{6} - \frac{11319}{941}e^{5} - \frac{191741}{4705}e^{4} + \frac{61169}{4705}e^{3} + \frac{128421}{4705}e^{2} - \frac{17814}{4705}e + \frac{10066}{4705}$ |
| 29 | $[29, 29, w^{2} - 5]$ | $\phantom{-}\frac{743}{4705}e^{9} + \frac{3478}{4705}e^{8} - \frac{3557}{4705}e^{7} - \frac{32567}{4705}e^{6} - \frac{2571}{941}e^{5} + \frac{83236}{4705}e^{4} + \frac{67401}{4705}e^{3} - \frac{45871}{4705}e^{2} - \frac{40756}{4705}e - \frac{2986}{4705}$ |
| 31 | $[31, 31, w^{2} + w - 8]$ | $\phantom{-}\frac{8}{941}e^{9} - \frac{93}{941}e^{8} - \frac{265}{941}e^{7} + \frac{1620}{941}e^{6} + \frac{2601}{941}e^{5} - \frac{8876}{941}e^{4} - \frac{8797}{941}e^{3} + \frac{15265}{941}e^{2} + \frac{5962}{941}e - \frac{1096}{941}$ |
| 41 | $[41, 41, w^{2} + w - 11]$ | $-\frac{3122}{4705}e^{9} - \frac{10992}{4705}e^{8} + \frac{29783}{4705}e^{7} + \frac{109303}{4705}e^{6} - \frac{17584}{941}e^{5} - \frac{312374}{4705}e^{4} + \frac{89421}{4705}e^{3} + \frac{258374}{4705}e^{2} - \frac{31101}{4705}e - \frac{42786}{4705}$ |
| 47 | $[47, 47, w^{2} + 2w - 1]$ | $-\frac{1468}{4705}e^{9} - \frac{5048}{4705}e^{8} + \frac{15222}{4705}e^{7} + \frac{52782}{4705}e^{6} - \frac{10108}{941}e^{5} - \frac{161036}{4705}e^{4} + \frac{51719}{4705}e^{3} + \frac{133381}{4705}e^{2} + \frac{356}{4705}e - \frac{5904}{4705}$ |
| 59 | $[59, 59, w^{2} + w - 1]$ | $\phantom{-}\frac{243}{4705}e^{9} + \frac{2233}{4705}e^{8} + \frac{1243}{4705}e^{7} - \frac{25602}{4705}e^{6} - \frac{6383}{941}e^{5} + \frac{92206}{4705}e^{4} + \frac{106721}{4705}e^{3} - \frac{113041}{4705}e^{2} - \frac{60506}{4705}e + \frac{37284}{4705}$ |
| 61 | $[61, 61, -w^{2} + 5w - 7]$ | $\phantom{-}\frac{504}{941}e^{9} + \frac{1669}{941}e^{8} - \frac{5403}{941}e^{7} - \frac{17447}{941}e^{6} + \frac{19890}{941}e^{5} + \frac{54344}{941}e^{4} - \frac{30074}{941}e^{3} - \frac{50821}{941}e^{2} + \frac{17085}{941}e + \frac{8114}{941}$ |
| 67 | $[67, 67, 2w^{2} - 13]$ | $-\frac{3176}{4705}e^{9} - \frac{7306}{4705}e^{8} + \frac{43099}{4705}e^{7} + \frac{71079}{4705}e^{6} - \frac{43350}{941}e^{5} - \frac{178122}{4705}e^{4} + \frac{442628}{4705}e^{3} + \frac{86407}{4705}e^{2} - \frac{239313}{4705}e - \frac{16568}{4705}$ |
| 89 | $[89, 89, -2w^{2} + w + 20]$ | $\phantom{-}\frac{1647}{4705}e^{9} + \frac{5202}{4705}e^{8} - \frac{20328}{4705}e^{7} - \frac{57468}{4705}e^{6} + \frac{18948}{941}e^{5} + \frac{191569}{4705}e^{4} - \frac{198326}{4705}e^{3} - \frac{196339}{4705}e^{2} + \frac{125751}{4705}e + \frac{23726}{4705}$ |
| 89 | $[89, 89, 5w^{2} + 8w - 19]$ | $\phantom{-}\frac{737}{4705}e^{9} + \frac{2842}{4705}e^{8} - \frac{7828}{4705}e^{7} - \frac{34723}{4705}e^{6} + \frac{4708}{941}e^{5} + \frac{133179}{4705}e^{4} - \frac{19866}{4705}e^{3} - \frac{174239}{4705}e^{2} + \frac{23936}{4705}e + \frac{63706}{4705}$ |
| 89 | $[89, 89, w^{2} + 2w - 7]$ | $-\frac{2473}{4705}e^{9} - \frac{8068}{4705}e^{8} + \frac{26752}{4705}e^{7} + \frac{82167}{4705}e^{6} - \frac{20292}{941}e^{5} - \frac{238706}{4705}e^{4} + \frac{163499}{4705}e^{3} + \frac{185346}{4705}e^{2} - \frac{79334}{4705}e + \frac{4746}{4705}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -2w + 5]$ | $-1$ |