Base field 3.3.1129.1
Generator \(w\), with minimal polynomial \(x^{3} - 7x - 3\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[13, 13, w^{2} - w - 7]$ |
| Dimension: | $10$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $20$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{10} - 2x^{9} - 15x^{8} + 28x^{7} + 74x^{6} - 126x^{5} - 139x^{4} + 192x^{3} + 109x^{2} - 84x - 29\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}e$ |
| 3 | $[3, 3, w + 1]$ | $-\frac{431}{2127}e^{9} - \frac{49}{2127}e^{8} + \frac{6470}{2127}e^{7} + \frac{373}{709}e^{6} - \frac{29968}{2127}e^{5} - \frac{4102}{2127}e^{4} + \frac{46654}{2127}e^{3} + \frac{2950}{2127}e^{2} - \frac{18457}{2127}e - \frac{25}{2127}$ |
| 3 | $[3, 3, w + 2]$ | $-\frac{20}{2127}e^{9} - \frac{175}{2127}e^{8} + \frac{14}{2127}e^{7} + \frac{927}{709}e^{6} + \frac{2360}{2127}e^{5} - \frac{12523}{2127}e^{4} - \frac{11135}{2127}e^{3} + \frac{16309}{2127}e^{2} + \frac{10958}{2127}e - \frac{4951}{2127}$ |
| 8 | $[8, 2, 2]$ | $-\frac{289}{2127}e^{9} + \frac{130}{2127}e^{8} + \frac{4669}{2127}e^{7} - \frac{324}{709}e^{6} - \frac{25454}{2127}e^{5} - \frac{56}{2127}e^{4} + \frac{54458}{2127}e^{3} + \frac{9884}{2127}e^{2} - \frac{37979}{2127}e - \frac{5924}{2127}$ |
| 11 | $[11, 11, -w^{2} + 5]$ | $\phantom{-}\frac{338}{709}e^{9} - \frac{233}{709}e^{8} - \frac{4916}{709}e^{7} + \frac{2702}{709}e^{6} + \frac{21799}{709}e^{5} - \frac{9782}{709}e^{4} - \frac{30545}{709}e^{3} + \frac{7907}{709}e^{2} + \frac{10352}{709}e - \frac{770}{709}$ |
| 13 | $[13, 13, w^{2} - w - 7]$ | $-1$ |
| 17 | $[17, 17, -w^{2} + w + 4]$ | $\phantom{-}\frac{1342}{2127}e^{9} + \frac{44}{2127}e^{8} - \frac{19657}{2127}e^{7} - \frac{1015}{709}e^{6} + \frac{88376}{2127}e^{5} + \frac{17357}{2127}e^{4} - \frac{134483}{2127}e^{3} - \frac{31472}{2127}e^{2} + \frac{63194}{2127}e + \frac{10397}{2127}$ |
| 19 | $[19, 19, -w^{2} - w + 4]$ | $\phantom{-}\frac{878}{2127}e^{9} + \frac{238}{2127}e^{8} - \frac{13802}{2127}e^{7} - \frac{1913}{709}e^{6} + \frac{70810}{2127}e^{5} + \frac{32686}{2127}e^{4} - \frac{135448}{2127}e^{3} - \frac{60211}{2127}e^{2} + \frac{73240}{2127}e + \frac{30811}{2127}$ |
| 29 | $[29, 29, 2w^{2} - 2w - 11]$ | $-\frac{611}{2127}e^{9} + \frac{503}{2127}e^{8} + \frac{8723}{2127}e^{7} - \frac{1919}{709}e^{6} - \frac{38506}{2127}e^{5} + \frac{21446}{2127}e^{4} + \frac{57043}{2127}e^{3} - \frac{22556}{2127}e^{2} - \frac{30439}{2127}e - \frac{2044}{2127}$ |
| 31 | $[31, 31, w^{2} - 2w - 4]$ | $-\frac{1927}{2127}e^{9} - \frac{377}{2127}e^{8} + \frac{29638}{2127}e^{7} + \frac{2783}{709}e^{6} - \frac{144839}{2127}e^{5} - \frac{40676}{2127}e^{4} + \frac{255986}{2127}e^{3} + \frac{64499}{2127}e^{2} - \frac{135104}{2127}e - \frac{31316}{2127}$ |
| 37 | $[37, 37, -2w^{2} + 3w + 7]$ | $-\frac{1408}{2127}e^{9} + \frac{442}{2127}e^{8} + \frac{20554}{2127}e^{7} - \frac{818}{709}e^{6} - \frac{93350}{2127}e^{5} - \frac{4019}{2127}e^{4} + \frac{143468}{2127}e^{3} + \frac{42539}{2127}e^{2} - \frac{59363}{2127}e - \frac{37583}{2127}$ |
| 41 | $[41, 41, w^{2} - 2]$ | $-\frac{408}{709}e^{9} + \frac{684}{709}e^{8} + \frac{5674}{709}e^{7} - \frac{8921}{709}e^{6} - \frac{23465}{709}e^{5} + \frac{35788}{709}e^{4} + \frac{26668}{709}e^{3} - \frac{40514}{709}e^{2} - \frac{5322}{709}e + \frac{10029}{709}$ |
| 59 | $[59, 59, 2w^{2} - 13]$ | $-\frac{268}{709}e^{9} - \frac{218}{709}e^{8} + \frac{4158}{709}e^{7} + \frac{3517}{709}e^{6} - \frac{20842}{709}e^{5} - \frac{15515}{709}e^{4} + \frac{40094}{709}e^{3} + \frac{22573}{709}e^{2} - \frac{23890}{709}e - \frac{12743}{709}$ |
| 61 | $[61, 61, w^{2} + w - 10]$ | $-\frac{3680}{2127}e^{9} + \frac{1832}{2127}e^{8} + \frac{53624}{2127}e^{7} - \frac{5973}{709}e^{6} - \frac{242146}{2127}e^{5} + \frac{50357}{2127}e^{4} + \frac{367432}{2127}e^{3} + \frac{6040}{2127}e^{2} - \frac{161776}{2127}e - \frac{45295}{2127}$ |
| 67 | $[67, 67, 2w^{2} - w - 11]$ | $\phantom{-}\frac{2984}{2127}e^{9} + \frac{586}{2127}e^{8} - \frac{45905}{2127}e^{7} - \frac{4591}{709}e^{6} + \frac{224305}{2127}e^{5} + \frac{73669}{2127}e^{4} - \frac{393340}{2127}e^{3} - \frac{137419}{2127}e^{2} + \frac{198115}{2127}e + \frac{69535}{2127}$ |
| 73 | $[73, 73, -w^{2} - 1]$ | $-\frac{328}{2127}e^{9} - \frac{2870}{2127}e^{8} + \frac{7036}{2127}e^{7} + \frac{13643}{709}e^{6} - \frac{46376}{2127}e^{5} - \frac{174323}{2127}e^{4} + \frac{119420}{2127}e^{3} + \frac{225353}{2127}e^{2} - \frac{83186}{2127}e - \frac{65882}{2127}$ |
| 83 | $[83, 83, w^{2} - w - 10]$ | $-\frac{1124}{2127}e^{9} + \frac{800}{2127}e^{8} + \frac{16952}{2127}e^{7} - \frac{2921}{709}e^{6} - \frac{80068}{2127}e^{5} + \frac{23216}{2127}e^{4} + \frac{129298}{2127}e^{3} + \frac{15994}{2127}e^{2} - \frac{66502}{2127}e - \frac{23857}{2127}$ |
| 89 | $[89, 89, -w^{2} + 4w - 2]$ | $-\frac{128}{709}e^{9} - \frac{411}{709}e^{8} + \frac{2642}{709}e^{7} + \frac{5320}{709}e^{6} - \frac{17510}{709}e^{5} - \frac{18606}{709}e^{4} + \frac{45012}{709}e^{3} + \frac{12633}{709}e^{2} - \frac{35368}{709}e - \frac{774}{709}$ |
| 97 | $[97, 97, w^{2} - 2w - 7]$ | $\phantom{-}\frac{3437}{2127}e^{9} - \frac{236}{2127}e^{8} - \frac{51965}{2127}e^{7} - \frac{808}{709}e^{6} + \frac{247423}{2127}e^{5} + \frac{23695}{2127}e^{4} - \frac{416047}{2127}e^{3} - \frac{71740}{2127}e^{2} + \frac{209623}{2127}e + \frac{48844}{2127}$ |
| 97 | $[97, 97, -w^{2} - 4w - 5]$ | $\phantom{-}\frac{2729}{2127}e^{9} - \frac{50}{2127}e^{8} - \frac{40409}{2127}e^{7} - \frac{2166}{709}e^{6} + \frac{188458}{2127}e^{5} + \frac{51724}{2127}e^{4} - \frac{310381}{2127}e^{3} - \frac{123568}{2127}e^{2} + \frac{147463}{2127}e + \frac{69688}{2127}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, w^{2} - w - 7]$ | $1$ |