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: | $[9, 3, 3]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $99$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} - 11x^{8} + 31x^{7} + 56x^{6} - 403x^{5} + 492x^{4} + 336x^{3} - 908x^{2} + 265x + 157\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}1$ |
| 3 | $[3, 3, w + 2]$ | $\phantom{-}1$ |
| 4 | $[4, 2, 2]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w + 1]$ | $\phantom{-}\frac{7}{12}e^{8} - \frac{11}{2}e^{7} + \frac{37}{4}e^{6} + \frac{193}{4}e^{5} - \frac{949}{6}e^{4} + \frac{53}{2}e^{3} + \frac{1469}{6}e^{2} - \frac{731}{6}e - \frac{619}{12}$ |
| 5 | $[5, 5, w + 3]$ | $\phantom{-}\frac{7}{12}e^{8} - \frac{11}{2}e^{7} + \frac{37}{4}e^{6} + \frac{193}{4}e^{5} - \frac{949}{6}e^{4} + \frac{53}{2}e^{3} + \frac{1469}{6}e^{2} - \frac{731}{6}e - \frac{619}{12}$ |
| 11 | $[11, 11, w + 1]$ | $\phantom{-}\frac{23}{6}e^{8} - \frac{109}{3}e^{7} + \frac{383}{6}e^{6} + \frac{1861}{6}e^{5} - \frac{3223}{3}e^{4} + \frac{808}{3}e^{3} + 1686e^{2} - \frac{2839}{3}e - \frac{807}{2}$ |
| 11 | $[11, 11, w + 9]$ | $\phantom{-}\frac{23}{6}e^{8} - \frac{109}{3}e^{7} + \frac{383}{6}e^{6} + \frac{1861}{6}e^{5} - \frac{3223}{3}e^{4} + \frac{808}{3}e^{3} + 1686e^{2} - \frac{2839}{3}e - \frac{807}{2}$ |
| 17 | $[17, 17, w + 2]$ | $\phantom{-}\frac{25}{12}e^{8} - \frac{119}{6}e^{7} + \frac{421}{12}e^{6} + \frac{2033}{12}e^{5} - \frac{3527}{6}e^{4} + \frac{869}{6}e^{3} + \frac{1837}{2}e^{2} - \frac{3047}{6}e - \frac{863}{4}$ |
| 17 | $[17, 17, w + 14]$ | $\phantom{-}\frac{25}{12}e^{8} - \frac{119}{6}e^{7} + \frac{421}{12}e^{6} + \frac{2033}{12}e^{5} - \frac{3527}{6}e^{4} + \frac{869}{6}e^{3} + \frac{1837}{2}e^{2} - \frac{3047}{6}e - \frac{863}{4}$ |
| 19 | $[19, 19, w]$ | $\phantom{-}\frac{7}{3}e^{8} - \frac{67}{3}e^{7} + \frac{121}{3}e^{6} + \frac{566}{3}e^{5} - \frac{2008}{3}e^{4} + \frac{559}{3}e^{3} + 1044e^{2} - \frac{1828}{3}e - 248$ |
| 19 | $[19, 19, w + 18]$ | $\phantom{-}\frac{7}{3}e^{8} - \frac{67}{3}e^{7} + \frac{121}{3}e^{6} + \frac{566}{3}e^{5} - \frac{2008}{3}e^{4} + \frac{559}{3}e^{3} + 1044e^{2} - \frac{1828}{3}e - 248$ |
| 37 | $[37, 37, -w - 4]$ | $\phantom{-}\frac{5}{4}e^{8} - \frac{23}{2}e^{7} + \frac{73}{4}e^{6} + \frac{409}{4}e^{5} - \frac{647}{2}e^{4} + \frac{85}{2}e^{3} + \frac{1031}{2}e^{2} - \frac{493}{2}e - \frac{481}{4}$ |
| 37 | $[37, 37, w - 5]$ | $\phantom{-}\frac{5}{4}e^{8} - \frac{23}{2}e^{7} + \frac{73}{4}e^{6} + \frac{409}{4}e^{5} - \frac{647}{2}e^{4} + \frac{85}{2}e^{3} + \frac{1031}{2}e^{2} - \frac{493}{2}e - \frac{481}{4}$ |
| 43 | $[43, 43, w + 16]$ | $-\frac{11}{6}e^{8} + \frac{53}{3}e^{7} - \frac{193}{6}e^{6} - \frac{899}{6}e^{5} + 532e^{4} - \frac{422}{3}e^{3} - \frac{2510}{3}e^{2} + \frac{1411}{3}e + \frac{1219}{6}$ |
| 43 | $[43, 43, w + 26]$ | $-\frac{11}{6}e^{8} + \frac{53}{3}e^{7} - \frac{193}{6}e^{6} - \frac{899}{6}e^{5} + 532e^{4} - \frac{422}{3}e^{3} - \frac{2510}{3}e^{2} + \frac{1411}{3}e + \frac{1219}{6}$ |
| 49 | $[49, 7, -7]$ | $\phantom{-}\frac{5}{4}e^{8} - \frac{25}{2}e^{7} + \frac{101}{4}e^{6} + \frac{405}{4}e^{5} - \frac{799}{2}e^{4} + \frac{303}{2}e^{3} + \frac{1269}{2}e^{2} - \frac{819}{2}e - \frac{653}{4}$ |
| 53 | $[53, 53, -w - 10]$ | $\phantom{-}\frac{31}{12}e^{8} - \frac{149}{6}e^{7} + \frac{547}{12}e^{6} + \frac{2495}{12}e^{5} - \frac{4505}{6}e^{4} + \frac{1361}{6}e^{3} + \frac{2341}{2}e^{2} - \frac{4199}{6}e - \frac{1137}{4}$ |
| 53 | $[53, 53, w - 11]$ | $\phantom{-}\frac{31}{12}e^{8} - \frac{149}{6}e^{7} + \frac{547}{12}e^{6} + \frac{2495}{12}e^{5} - \frac{4505}{6}e^{4} + \frac{1361}{6}e^{3} + \frac{2341}{2}e^{2} - \frac{4199}{6}e - \frac{1137}{4}$ |
| 61 | $[61, 61, w + 15]$ | $\phantom{-}\frac{19}{12}e^{8} - \frac{31}{2}e^{7} + \frac{117}{4}e^{6} + \frac{521}{4}e^{5} - \frac{2857}{6}e^{4} + \frac{273}{2}e^{3} + \frac{4535}{6}e^{2} - \frac{2615}{6}e - \frac{2335}{12}$ |
| 61 | $[61, 61, w + 45]$ | $\phantom{-}\frac{19}{12}e^{8} - \frac{31}{2}e^{7} + \frac{117}{4}e^{6} + \frac{521}{4}e^{5} - \frac{2857}{6}e^{4} + \frac{273}{2}e^{3} + \frac{4535}{6}e^{2} - \frac{2615}{6}e - \frac{2335}{12}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, w]$ | $-1$ |
| $3$ | $[3, 3, w + 2]$ | $-1$ |