Base field \(\Q(\sqrt{229}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 57\); narrow class number \(3\) and class number \(3\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[3,3,-w + 1]$ |
| Dimension: | $6$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $30$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{6} + x^{5} + 3x^{4} + 5x^{2} + 2x + 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}e$ |
| 3 | $[3, 3, w + 2]$ | $-\frac{6}{13}e^{5} - \frac{5}{13}e^{4} - \frac{15}{13}e^{3} + \frac{9}{13}e^{2} - \frac{25}{13}e - \frac{10}{13}$ |
| 4 | $[4, 2, 2]$ | $-\frac{3}{13}e^{5} - \frac{9}{13}e^{4} - \frac{14}{13}e^{3} - \frac{15}{13}e^{2} - \frac{6}{13}e - \frac{31}{13}$ |
| 5 | $[5, 5, w + 1]$ | $-e^{5} - e^{4} - 3e^{3} - e^{2} - 5e - 2$ |
| 5 | $[5, 5, w + 3]$ | $-\frac{4}{13}e^{5} + \frac{1}{13}e^{4} - \frac{10}{13}e^{3} + \frac{6}{13}e^{2} - \frac{21}{13}e + \frac{2}{13}$ |
| 11 | $[11, 11, w + 1]$ | $-2e$ |
| 11 | $[11, 11, w + 9]$ | $-\frac{2}{13}e^{5} - \frac{6}{13}e^{4} - \frac{18}{13}e^{3} - \frac{23}{13}e^{2} - \frac{30}{13}e - \frac{12}{13}$ |
| 17 | $[17, 17, w + 2]$ | $\phantom{-}\frac{24}{13}e^{5} + \frac{7}{13}e^{4} + \frac{60}{13}e^{3} - \frac{36}{13}e^{2} + \frac{113}{13}e - \frac{12}{13}$ |
| 17 | $[17, 17, w + 14]$ | $\phantom{-}\frac{31}{13}e^{5} + \frac{28}{13}e^{4} + \frac{84}{13}e^{3} - \frac{1}{13}e^{2} + \frac{140}{13}e + \frac{56}{13}$ |
| 19 | $[19, 19, w]$ | $-\frac{41}{13}e^{5} - \frac{32}{13}e^{4} - \frac{96}{13}e^{3} + \frac{29}{13}e^{2} - \frac{160}{13}e - \frac{64}{13}$ |
| 19 | $[19, 19, w + 18]$ | $-\frac{22}{13}e^{5} - \frac{14}{13}e^{4} - \frac{55}{13}e^{3} + \frac{33}{13}e^{2} - \frac{148}{13}e + \frac{11}{13}$ |
| 37 | $[37, 37, -w - 4]$ | $\phantom{-}\frac{9}{13}e^{5} + \frac{27}{13}e^{4} + \frac{42}{13}e^{3} + \frac{45}{13}e^{2} + \frac{18}{13}e + \frac{106}{13}$ |
| 37 | $[37, 37, w - 5]$ | $-\frac{16}{13}e^{5} - \frac{48}{13}e^{4} - \frac{53}{13}e^{3} - \frac{80}{13}e^{2} - \frac{32}{13}e - \frac{135}{13}$ |
| 43 | $[43, 43, w + 16]$ | $\phantom{-}\frac{14}{13}e^{5} + \frac{3}{13}e^{4} + \frac{35}{13}e^{3} - \frac{21}{13}e^{2} + \frac{41}{13}e - \frac{7}{13}$ |
| 43 | $[43, 43, w + 26]$ | $\phantom{-}\frac{45}{13}e^{5} + \frac{44}{13}e^{4} + \frac{132}{13}e^{3} + \frac{4}{13}e^{2} + \frac{220}{13}e + \frac{88}{13}$ |
| 49 | $[49, 7, -7]$ | $\phantom{-}\frac{1}{13}e^{5} + \frac{3}{13}e^{4} + \frac{9}{13}e^{3} + \frac{5}{13}e^{2} + \frac{2}{13}e - \frac{85}{13}$ |
| 53 | $[53, 53, -w - 10]$ | $-\frac{19}{13}e^{5} - \frac{57}{13}e^{4} - \frac{67}{13}e^{3} - \frac{95}{13}e^{2} - \frac{38}{13}e - \frac{192}{13}$ |
| 53 | $[53, 53, w - 11]$ | $-\frac{1}{13}e^{5} - \frac{3}{13}e^{4} + \frac{30}{13}e^{3} - \frac{5}{13}e^{2} - \frac{2}{13}e - \frac{58}{13}$ |
| 61 | $[61, 61, w + 15]$ | $\phantom{-}\frac{17}{13}e^{5} + \frac{12}{13}e^{4} + \frac{36}{13}e^{3} - \frac{84}{13}e^{2} + \frac{60}{13}e + \frac{24}{13}$ |
| 61 | $[61, 61, w + 45]$ | $\phantom{-}\frac{18}{13}e^{5} + \frac{15}{13}e^{4} + \frac{45}{13}e^{3} - \frac{27}{13}e^{2} + \frac{127}{13}e - \frac{9}{13}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3,3,-w + 1]$ | $\frac{6}{13}e^{5} + \frac{5}{13}e^{4} + \frac{15}{13}e^{3} - \frac{9}{13}e^{2} + \frac{25}{13}e + \frac{10}{13}$ |