Base field 3.3.1345.1
Generator \(w\), with minimal polynomial \(x^{3} - 7x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[13, 13, -w^{2} + w + 4]$ |
| 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:
| \(x^{11} - 12x^{10} + 42x^{9} + 20x^{8} - 408x^{7} + 720x^{6} + 16x^{5} - 984x^{4} + 552x^{3} + 240x^{2} - 224x + 32\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, -w - 1]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w + 2]$ | $-\frac{11}{752}e^{10} + \frac{3}{47}e^{9} + \frac{183}{376}e^{8} - \frac{120}{47}e^{7} - \frac{343}{188}e^{6} + \frac{931}{47}e^{5} - \frac{967}{94}e^{4} - \frac{2767}{94}e^{3} + \frac{1201}{94}e^{2} + \frac{682}{47}e - \frac{137}{47}$ |
| 7 | $[7, 7, w + 3]$ | $\phantom{-}\frac{99}{376}e^{10} - \frac{545}{188}e^{9} + \frac{1549}{188}e^{8} + \frac{609}{47}e^{7} - \frac{17655}{188}e^{6} + \frac{9489}{94}e^{5} + \frac{4332}{47}e^{4} - \frac{16229}{94}e^{3} + \frac{95}{47}e^{2} + \frac{2858}{47}e - \frac{542}{47}$ |
| 7 | $[7, 7, -w + 2]$ | $-\frac{7}{47}e^{10} + \frac{599}{376}e^{9} - \frac{393}{94}e^{8} - \frac{1593}{188}e^{7} + \frac{4727}{94}e^{6} - \frac{4033}{94}e^{5} - \frac{2919}{47}e^{4} + \frac{7517}{94}e^{3} + \frac{769}{47}e^{2} - \frac{1328}{47}e + \frac{139}{47}$ |
| 7 | $[7, 7, -w + 1]$ | $\phantom{-}\frac{15}{188}e^{10} - \frac{42}{47}e^{9} + \frac{505}{188}e^{8} + \frac{305}{94}e^{7} - \frac{1361}{47}e^{6} + \frac{3589}{94}e^{5} + \frac{1647}{94}e^{4} - \frac{5865}{94}e^{3} + \frac{946}{47}e^{2} + \frac{1027}{47}e - \frac{432}{47}$ |
| 8 | $[8, 2, 2]$ | $\phantom{-}\frac{125}{752}e^{10} - \frac{175}{94}e^{9} + \frac{2065}{376}e^{8} + \frac{1451}{188}e^{7} - \frac{2902}{47}e^{6} + \frac{3341}{47}e^{5} + \frac{5511}{94}e^{4} - \frac{11549}{94}e^{3} + \frac{777}{94}e^{2} + \frac{2073}{47}e - \frac{571}{47}$ |
| 13 | $[13, 13, -w^{2} + w + 4]$ | $\phantom{-}1$ |
| 19 | $[19, 19, -w^{2} + 5]$ | $-\frac{309}{752}e^{10} + \frac{221}{47}e^{9} - \frac{5507}{376}e^{8} - \frac{3071}{188}e^{7} + \frac{29785}{188}e^{6} - \frac{9734}{47}e^{5} - \frac{5718}{47}e^{4} + \frac{30945}{94}e^{3} - \frac{3239}{94}e^{2} - \frac{5329}{47}e + \frac{1219}{47}$ |
| 23 | $[23, 23, -w^{2} - w + 3]$ | $-\frac{9}{188}e^{10} + \frac{129}{188}e^{9} - \frac{585}{188}e^{8} + \frac{245}{188}e^{7} + \frac{5137}{188}e^{6} - \frac{3046}{47}e^{5} + \frac{873}{94}e^{4} + \frac{8501}{94}e^{3} - \frac{2175}{47}e^{2} - \frac{1152}{47}e + \frac{560}{47}$ |
| 27 | $[27, 3, 3]$ | $-\frac{67}{376}e^{10} + \frac{741}{376}e^{9} - \frac{1073}{188}e^{8} - \frac{1519}{188}e^{7} + \frac{2944}{47}e^{6} - \frac{3421}{47}e^{5} - \frac{4247}{94}e^{4} + \frac{9597}{94}e^{3} - \frac{662}{47}e^{2} - \frac{1233}{47}e + \frac{476}{47}$ |
| 29 | $[29, 29, w^{2} - w - 3]$ | $\phantom{-}\frac{43}{376}e^{10} - \frac{491}{376}e^{9} + \frac{763}{188}e^{8} + \frac{843}{188}e^{7} - \frac{8201}{188}e^{6} + \frac{2728}{47}e^{5} + \frac{1413}{47}e^{4} - \frac{8759}{94}e^{3} + \frac{911}{47}e^{2} + \frac{1671}{47}e - \frac{450}{47}$ |
| 31 | $[31, 31, w^{2} - w - 8]$ | $-\frac{19}{188}e^{10} + \frac{341}{376}e^{9} - \frac{201}{188}e^{8} - \frac{919}{94}e^{7} + \frac{1091}{47}e^{6} + \frac{959}{47}e^{5} - \frac{3379}{47}e^{4} + \frac{119}{47}e^{3} + \frac{2427}{47}e^{2} - \frac{505}{47}e - \frac{186}{47}$ |
| 37 | $[37, 37, -w - 4]$ | $-\frac{115}{376}e^{10} + \frac{691}{188}e^{9} - \frac{2351}{188}e^{8} - \frac{1837}{188}e^{7} + \frac{6212}{47}e^{6} - \frac{9084}{47}e^{5} - \frac{4790}{47}e^{4} + \frac{15013}{47}e^{3} - \frac{1480}{47}e^{2} - \frac{5386}{47}e + \frac{1092}{47}$ |
| 43 | $[43, 43, 2w^{2} - 4w - 3]$ | $-\frac{73}{376}e^{10} + \frac{437}{188}e^{9} - \frac{1503}{188}e^{8} - \frac{234}{47}e^{7} + \frac{7551}{94}e^{6} - \frac{12147}{94}e^{5} - \frac{1367}{47}e^{4} + \frac{16833}{94}e^{3} - \frac{2233}{47}e^{2} - \frac{2369}{47}e + \frac{600}{47}$ |
| 47 | $[47, 47, w^{2} - 3]$ | $\phantom{-}\frac{127}{376}e^{10} - \frac{365}{94}e^{9} + \frac{2271}{188}e^{8} + \frac{673}{47}e^{7} - \frac{6277}{47}e^{6} + \frac{7762}{47}e^{5} + \frac{12053}{94}e^{4} - \frac{13037}{47}e^{3} - \frac{360}{47}e^{2} + \frac{4509}{47}e - \frac{353}{47}$ |
| 53 | $[53, 53, 2w^{2} - w - 11]$ | $\phantom{-}\frac{103}{188}e^{10} - \frac{1163}{188}e^{9} + \frac{1773}{94}e^{8} + \frac{2063}{94}e^{7} - \frac{9521}{47}e^{6} + \frac{24939}{94}e^{5} + \frac{12287}{94}e^{4} - \frac{18938}{47}e^{3} + \frac{3820}{47}e^{2} + \frac{5993}{47}e - \frac{1688}{47}$ |
| 59 | $[59, 59, w^{2} - 2w - 4]$ | $-\frac{17}{188}e^{10} + \frac{409}{376}e^{9} - \frac{341}{94}e^{8} - \frac{165}{47}e^{7} + \frac{7395}{188}e^{6} - \frac{4619}{94}e^{5} - \frac{1925}{47}e^{4} + \frac{6271}{94}e^{3} + \frac{1234}{47}e^{2} - \frac{672}{47}e - \frac{300}{47}$ |
| 67 | $[67, 67, w^{2} + w - 8]$ | $-\frac{26}{47}e^{10} + \frac{573}{94}e^{9} - \frac{3235}{188}e^{8} - \frac{5373}{188}e^{7} + \frac{9430}{47}e^{6} - \frac{19203}{94}e^{5} - \frac{10842}{47}e^{4} + \frac{36293}{94}e^{3} + \frac{1312}{47}e^{2} - \frac{6967}{47}e + \frac{1040}{47}$ |
| 71 | $[71, 71, w^{2} + w - 11]$ | $\phantom{-}\frac{71}{188}e^{10} - \frac{1581}{376}e^{9} + \frac{578}{47}e^{8} + \frac{3279}{188}e^{7} - \frac{6464}{47}e^{6} + \frac{14945}{94}e^{5} + \frac{11443}{94}e^{4} - \frac{24847}{94}e^{3} + \frac{1241}{47}e^{2} + \frac{4200}{47}e - \frac{1462}{47}$ |
| 73 | $[73, 73, -w^{2} + 4w - 2]$ | $-\frac{99}{188}e^{10} + \frac{2227}{376}e^{9} - \frac{845}{47}e^{8} - \frac{983}{47}e^{7} + \frac{9039}{47}e^{6} - \frac{23913}{94}e^{5} - \frac{5374}{47}e^{4} + \frac{18015}{47}e^{3} - \frac{4326}{47}e^{2} - \frac{5810}{47}e + \frac{1742}{47}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w^{2} + w + 4]$ | $-1$ |