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: | $[13, 13, -w^{2} + 3w + 1]$ |
| Dimension: | $11$ |
| 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:
| \(x^{11} + x^{10} - 14x^{9} - 14x^{8} + 67x^{7} + 67x^{6} - 128x^{5} - 130x^{4} + 75x^{3} + 84x^{2} + 8x - 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w - 2]$ | $\phantom{-}e$ |
| 3 | $[3, 3, -w^{2} + 2w + 3]$ | $-\frac{2}{19}e^{10} - \frac{1}{19}e^{9} + 2e^{8} + \frac{28}{19}e^{7} - \frac{243}{19}e^{6} - \frac{212}{19}e^{5} + \frac{590}{19}e^{4} + \frac{535}{19}e^{3} - \frac{427}{19}e^{2} - \frac{363}{19}e - \frac{15}{19}$ |
| 7 | $[7, 7, -2w^{2} + 5w + 5]$ | $\phantom{-}\frac{4}{19}e^{10} + \frac{2}{19}e^{9} - 3e^{8} - \frac{18}{19}e^{7} + \frac{277}{19}e^{6} + \frac{25}{19}e^{5} - \frac{534}{19}e^{4} + \frac{70}{19}e^{3} + \frac{341}{19}e^{2} - \frac{91}{19}e - \frac{65}{19}$ |
| 9 | $[9, 3, -w^{2} + 5]$ | $\phantom{-}\frac{12}{19}e^{10} + \frac{6}{19}e^{9} - 8e^{8} - \frac{73}{19}e^{7} + \frac{641}{19}e^{6} + \frac{284}{19}e^{5} - \frac{1108}{19}e^{4} - \frac{436}{19}e^{3} + \frac{738}{19}e^{2} + \frac{221}{19}e - \frac{81}{19}$ |
| 13 | $[13, 13, -w^{2} + 3w + 1]$ | $-1$ |
| 13 | $[13, 13, 2w + 5]$ | $-\frac{15}{19}e^{10} + \frac{2}{19}e^{9} + 12e^{8} + \frac{1}{19}e^{7} - \frac{1205}{19}e^{6} - \frac{184}{19}e^{5} + \frac{2563}{19}e^{4} + \frac{754}{19}e^{3} - \frac{1768}{19}e^{2} - \frac{756}{19}e + \frac{30}{19}$ |
| 13 | $[13, 13, w - 1]$ | $\phantom{-}\frac{13}{19}e^{10} + \frac{16}{19}e^{9} - 8e^{8} - \frac{182}{19}e^{7} + \frac{544}{19}e^{6} + \frac{618}{19}e^{5} - \frac{662}{19}e^{4} - \frac{713}{19}e^{3} + \frac{163}{19}e^{2} + \frac{260}{19}e + \frac{31}{19}$ |
| 17 | $[17, 17, w^{2} - w - 5]$ | $-\frac{17}{19}e^{10} - \frac{18}{19}e^{9} + 11e^{8} + \frac{200}{19}e^{7} - \frac{821}{19}e^{6} - \frac{643}{19}e^{5} + \frac{1196}{19}e^{4} + \frac{624}{19}e^{3} - \frac{504}{19}e^{2} - \frac{93}{19}e - \frac{23}{19}$ |
| 19 | $[19, 19, w^{2} - 2w - 5]$ | $-\frac{9}{19}e^{10} - \frac{14}{19}e^{9} + 4e^{8} + \frac{145}{19}e^{7} - \frac{58}{19}e^{6} - \frac{384}{19}e^{5} - \frac{537}{19}e^{4} + \frac{137}{19}e^{3} + \frac{767}{19}e^{2} + \frac{219}{19}e - \frac{58}{19}$ |
| 29 | $[29, 29, w^{2} - 7]$ | $-\frac{9}{19}e^{10} - \frac{33}{19}e^{9} + 4e^{8} + \frac{373}{19}e^{7} - \frac{58}{19}e^{6} - \frac{1258}{19}e^{5} - \frac{518}{19}e^{4} + \frac{1410}{19}e^{3} + \frac{672}{19}e^{2} - \frac{389}{19}e - \frac{20}{19}$ |
| 31 | $[31, 31, 2w^{2} - 3w - 11]$ | $\phantom{-}\frac{4}{19}e^{10} - \frac{17}{19}e^{9} - 5e^{8} + \frac{172}{19}e^{7} + \frac{676}{19}e^{6} - \frac{431}{19}e^{5} - \frac{1655}{19}e^{4} + \frac{70}{19}e^{3} + \frac{1044}{19}e^{2} + \frac{308}{19}e + \frac{68}{19}$ |
| 49 | $[49, 7, -2w^{2} - 4w + 1]$ | $-\frac{52}{19}e^{10} - \frac{64}{19}e^{9} + 33e^{8} + \frac{728}{19}e^{7} - \frac{2404}{19}e^{6} - \frac{2491}{19}e^{5} + \frac{3465}{19}e^{4} + \frac{2985}{19}e^{3} - \frac{1583}{19}e^{2} - \frac{1230}{19}e - \frac{67}{19}$ |
| 59 | $[59, 59, w^{2} - 2w - 11]$ | $-\frac{15}{19}e^{10} + \frac{40}{19}e^{9} + 14e^{8} - \frac{417}{19}e^{7} - \frac{1623}{19}e^{6} + \frac{1146}{19}e^{5} + \frac{3874}{19}e^{4} - \frac{557}{19}e^{3} - \frac{2984}{19}e^{2} - \frac{414}{19}e + \frac{220}{19}$ |
| 71 | $[71, 71, -2w + 7]$ | $\phantom{-}\frac{13}{19}e^{10} + \frac{54}{19}e^{9} - 5e^{8} - \frac{600}{19}e^{7} - \frac{64}{19}e^{6} + \frac{1948}{19}e^{5} + \frac{1124}{19}e^{4} - \frac{1986}{19}e^{3} - \frac{1129}{19}e^{2} + \frac{431}{19}e - \frac{64}{19}$ |
| 73 | $[73, 73, w^{2} - 3w - 13]$ | $\phantom{-}\frac{21}{19}e^{10} + \frac{1}{19}e^{9} - 17e^{8} - \frac{85}{19}e^{7} + \frac{1725}{19}e^{6} + \frac{839}{19}e^{5} - \frac{3687}{19}e^{4} - \frac{2549}{19}e^{3} + \frac{2498}{19}e^{2} + \frac{2187}{19}e + \frac{91}{19}$ |
| 73 | $[73, 73, 2w^{2} - 2w - 17]$ | $\phantom{-}\frac{41}{19}e^{10} + \frac{68}{19}e^{9} - 27e^{8} - \frac{802}{19}e^{7} + \frac{2103}{19}e^{6} + \frac{2940}{19}e^{5} - \frac{3431}{19}e^{4} - \frac{3928}{19}e^{3} + \frac{1999}{19}e^{2} + \frac{1637}{19}e - \frac{44}{19}$ |
| 73 | $[73, 73, -w^{2} - 4w - 5]$ | $\phantom{-}\frac{4}{19}e^{10} + \frac{2}{19}e^{9} - e^{8} + \frac{58}{19}e^{7} - \frac{122}{19}e^{6} - \frac{773}{19}e^{5} + \frac{625}{19}e^{4} + \frac{2331}{19}e^{3} - \frac{552}{19}e^{2} - \frac{1630}{19}e - \frac{46}{19}$ |
| 79 | $[79, 79, 2w - 1]$ | $\phantom{-}\frac{2}{19}e^{10} + \frac{20}{19}e^{9} - e^{8} - \frac{237}{19}e^{7} + \frac{34}{19}e^{6} + \frac{877}{19}e^{5} + \frac{75}{19}e^{4} - \frac{1143}{19}e^{3} - \frac{219}{19}e^{2} + \frac{382}{19}e - \frac{42}{19}$ |
| 79 | $[79, 79, w^{2} + w - 5]$ | $\phantom{-}\frac{4}{19}e^{10} - \frac{17}{19}e^{9} - 4e^{8} + \frac{191}{19}e^{7} + \frac{486}{19}e^{6} - \frac{640}{19}e^{5} - \frac{1180}{19}e^{4} + \frac{697}{19}e^{3} + \frac{854}{19}e^{2} - \frac{129}{19}e - \frac{8}{19}$ |
| 79 | $[79, 79, w - 5]$ | $\phantom{-}e^{10} - 12e^{8} + e^{7} + 44e^{6} - 12e^{5} - 50e^{4} + 40e^{3} - 34e + 2$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w^{2} + 3w + 1]$ | $1$ |