Base field 3.3.1425.1
Generator \(w\), with minimal polynomial \(x^{3} - x^{2} - 8x - 3\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[13, 13, w^{2} - 2w - 8]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $28$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} + 2x^{7} - 9x^{6} - 14x^{5} + 22x^{4} + 28x^{3} - 14x^{2} - 13x + 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}e$ |
| 3 | $[3, 3, w + 1]$ | $-\frac{7}{17}e^{7} - \frac{12}{17}e^{6} + \frac{64}{17}e^{5} + \frac{70}{17}e^{4} - \frac{157}{17}e^{3} - \frac{88}{17}e^{2} + \frac{77}{17}e - \frac{16}{17}$ |
| 5 | $[5, 5, w^{2} - w - 7]$ | $-\frac{4}{17}e^{7} - \frac{2}{17}e^{6} + \frac{39}{17}e^{5} - \frac{11}{17}e^{4} - \frac{97}{17}e^{3} + \frac{76}{17}e^{2} + \frac{78}{17}e - \frac{65}{17}$ |
| 8 | $[8, 2, 2]$ | $\phantom{-}\frac{10}{17}e^{7} + \frac{5}{17}e^{6} - \frac{106}{17}e^{5} + \frac{2}{17}e^{4} + \frac{285}{17}e^{3} - \frac{37}{17}e^{2} - \frac{195}{17}e + \frac{1}{17}$ |
| 11 | $[11, 11, w - 1]$ | $-\frac{2}{17}e^{7} - \frac{1}{17}e^{6} + \frac{28}{17}e^{5} + \frac{3}{17}e^{4} - \frac{125}{17}e^{3} + \frac{4}{17}e^{2} + \frac{158}{17}e - \frac{7}{17}$ |
| 13 | $[13, 13, w^{2} - 2w - 8]$ | $-1$ |
| 17 | $[17, 17, w^{2} - 2w - 7]$ | $-\frac{10}{17}e^{7} - \frac{5}{17}e^{6} + \frac{106}{17}e^{5} + \frac{15}{17}e^{4} - \frac{251}{17}e^{3} - \frac{48}{17}e^{2} + \frac{93}{17}e - \frac{1}{17}$ |
| 19 | $[19, 19, -w^{2} + 2w + 4]$ | $\phantom{-}\frac{5}{17}e^{7} + \frac{11}{17}e^{6} - \frac{53}{17}e^{5} - \frac{101}{17}e^{4} + \frac{151}{17}e^{3} + \frac{245}{17}e^{2} - \frac{89}{17}e - \frac{59}{17}$ |
| 19 | $[19, 19, -2w^{2} + 3w + 16]$ | $\phantom{-}\frac{8}{17}e^{7} + \frac{4}{17}e^{6} - \frac{78}{17}e^{5} + \frac{22}{17}e^{4} + \frac{194}{17}e^{3} - \frac{135}{17}e^{2} - \frac{139}{17}e + \frac{113}{17}$ |
| 23 | $[23, 23, -w^{2} + 2w + 2]$ | $\phantom{-}\frac{11}{17}e^{7} + \frac{14}{17}e^{6} - \frac{103}{17}e^{5} - \frac{59}{17}e^{4} + \frac{254}{17}e^{3} + \frac{12}{17}e^{2} - \frac{155}{17}e + \frac{30}{17}$ |
| 31 | $[31, 31, 2w^{2} - 3w - 13]$ | $-\frac{4}{17}e^{7} + \frac{15}{17}e^{6} + \frac{56}{17}e^{5} - \frac{164}{17}e^{4} - \frac{148}{17}e^{3} + \frac{365}{17}e^{2} + \frac{78}{17}e - \frac{133}{17}$ |
| 37 | $[37, 37, w^{2} - w - 10]$ | $-\frac{5}{17}e^{7} + \frac{6}{17}e^{6} + \frac{36}{17}e^{5} - \frac{103}{17}e^{4} + \frac{36}{17}e^{3} + \frac{231}{17}e^{2} - \frac{183}{17}e - \frac{60}{17}$ |
| 43 | $[43, 43, 3w^{2} - 5w - 19]$ | $-\frac{22}{17}e^{7} - \frac{11}{17}e^{6} + \frac{240}{17}e^{5} + \frac{16}{17}e^{4} - \frac{661}{17}e^{3} - \frac{7}{17}e^{2} + \frac{480}{17}e + \frac{25}{17}$ |
| 43 | $[43, 43, w^{2} - w - 4]$ | $-\frac{40}{17}e^{7} - \frac{37}{17}e^{6} + \frac{390}{17}e^{5} + \frac{162}{17}e^{4} - \frac{885}{17}e^{3} - \frac{277}{17}e^{2} + \frac{338}{17}e + \frac{81}{17}$ |
| 43 | $[43, 43, 2w^{2} - 2w - 17]$ | $-\frac{16}{17}e^{7} - \frac{25}{17}e^{6} + \frac{156}{17}e^{5} + \frac{160}{17}e^{4} - \frac{439}{17}e^{3} - \frac{308}{17}e^{2} + \frac{346}{17}e + \frac{97}{17}$ |
| 47 | $[47, 47, w^{2} - 2]$ | $-\frac{2}{17}e^{7} - \frac{1}{17}e^{6} + \frac{45}{17}e^{5} + \frac{37}{17}e^{4} - \frac{227}{17}e^{3} - \frac{149}{17}e^{2} + \frac{243}{17}e + \frac{61}{17}$ |
| 67 | $[67, 67, w^{2} - 3w - 5]$ | $-\frac{5}{17}e^{7} + \frac{6}{17}e^{6} + \frac{87}{17}e^{5} - \frac{35}{17}e^{4} - \frac{338}{17}e^{3} + \frac{27}{17}e^{2} + \frac{344}{17}e - \frac{26}{17}$ |
| 79 | $[79, 79, -w^{2} + 3w - 1]$ | $-\frac{36}{17}e^{7} - \frac{52}{17}e^{6} + \frac{351}{17}e^{5} + \frac{309}{17}e^{4} - \frac{941}{17}e^{3} - \frac{523}{17}e^{2} + \frac{600}{17}e + \frac{231}{17}$ |
| 83 | $[83, 83, w^{2} + w - 4]$ | $-\frac{1}{17}e^{7} + \frac{8}{17}e^{6} - \frac{3}{17}e^{5} - \frac{126}{17}e^{4} + \frac{65}{17}e^{3} + \frac{359}{17}e^{2} - \frac{40}{17}e - \frac{182}{17}$ |
| 97 | $[97, 97, w^{2} - 11]$ | $\phantom{-}\frac{18}{17}e^{7} - \frac{25}{17}e^{6} - \frac{235}{17}e^{5} + \frac{262}{17}e^{4} + \frac{632}{17}e^{3} - \frac{478}{17}e^{2} - \frac{334}{17}e + \frac{148}{17}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, w^{2} - 2w - 8]$ | $1$ |