Base field 4.4.13824.1
Generator \(w\), with minimal polynomial \(x^{4} - 6x^{2} + 6\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[22, 22, -w^{3} + w^{2} + 3w - 2]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $26$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} + x^{8} - 19x^{7} - 14x^{6} + 119x^{5} + 48x^{4} - 276x^{3} - 13x^{2} + 174x - 36\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w^{2} + w + 2]$ | $-1$ |
| 3 | $[3, 3, w^{2} - w - 3]$ | $\phantom{-}e$ |
| 11 | $[11, 11, -w^{2} + w + 1]$ | $\phantom{-}\frac{1}{8}e^{7} - 2e^{5} - \frac{1}{4}e^{4} + \frac{73}{8}e^{3} + \frac{17}{8}e^{2} - \frac{41}{4}e - \frac{3}{2}$ |
| 11 | $[11, 11, -w^{2} - w + 1]$ | $-1$ |
| 13 | $[13, 13, w^{3} - 4w + 1]$ | $-\frac{1}{24}e^{8} + \frac{1}{12}e^{7} + \frac{2}{3}e^{6} - \frac{11}{12}e^{5} - \frac{77}{24}e^{4} + \frac{11}{8}e^{3} + \frac{9}{2}e^{2} + \frac{11}{3}e + 1$ |
| 13 | $[13, 13, -w^{3} + 4w + 1]$ | $\phantom{-}\frac{1}{12}e^{8} + \frac{5}{24}e^{7} - \frac{4}{3}e^{6} - \frac{19}{6}e^{5} + \frac{20}{3}e^{4} + \frac{109}{8}e^{3} - \frac{101}{8}e^{2} - \frac{193}{12}e + \frac{17}{2}$ |
| 25 | $[25, 5, -w^{2} - 2w + 1]$ | $\phantom{-}\frac{1}{12}e^{8} + \frac{1}{12}e^{7} - \frac{4}{3}e^{6} - \frac{7}{6}e^{5} + \frac{71}{12}e^{4} + \frac{9}{2}e^{3} - \frac{23}{4}e^{2} - \frac{29}{6}e + 1$ |
| 25 | $[25, 5, w^{2} - 2w - 1]$ | $\phantom{-}\frac{1}{12}e^{8} + \frac{1}{12}e^{7} - \frac{11}{6}e^{6} - \frac{7}{6}e^{5} + \frac{155}{12}e^{4} + \frac{9}{2}e^{3} - \frac{117}{4}e^{2} - \frac{13}{3}e + 10$ |
| 37 | $[37, 37, w^{3} - 3w - 1]$ | $-\frac{1}{6}e^{8} - \frac{1}{6}e^{7} + \frac{8}{3}e^{6} + \frac{7}{3}e^{5} - \frac{77}{6}e^{4} - 8e^{3} + \frac{39}{2}e^{2} + \frac{5}{3}e - 2$ |
| 37 | $[37, 37, w^{3} - 3w + 1]$ | $\phantom{-}\frac{1}{12}e^{8} - \frac{1}{24}e^{7} - \frac{4}{3}e^{6} + \frac{5}{6}e^{5} + \frac{37}{6}e^{4} - \frac{37}{8}e^{3} - \frac{63}{8}e^{2} + \frac{65}{12}e + \frac{5}{2}$ |
| 59 | $[59, 59, w^{2} - w - 5]$ | $\phantom{-}\frac{1}{4}e^{7} - 4e^{5} - \frac{1}{2}e^{4} + \frac{77}{4}e^{3} + \frac{17}{4}e^{2} - \frac{55}{2}e - 3$ |
| 59 | $[59, 59, -w^{2} - w + 5]$ | $\phantom{-}\frac{1}{4}e^{8} + \frac{1}{2}e^{7} - \frac{9}{2}e^{6} - \frac{15}{2}e^{5} + \frac{101}{4}e^{4} + \frac{123}{4}e^{3} - \frac{95}{2}e^{2} - \frac{59}{2}e + 21$ |
| 61 | $[61, 61, -w^{3} + w^{2} + 4w - 7]$ | $\phantom{-}\frac{5}{24}e^{8} + \frac{5}{24}e^{7} - \frac{10}{3}e^{6} - \frac{41}{12}e^{5} + \frac{379}{24}e^{4} + \frac{63}{4}e^{3} - \frac{183}{8}e^{2} - \frac{211}{12}e + \frac{17}{2}$ |
| 61 | $[61, 61, w^{3} - 3w^{2} - 6w + 11]$ | $-\frac{1}{24}e^{8} - \frac{1}{24}e^{7} + \frac{7}{6}e^{6} + \frac{13}{12}e^{5} - \frac{239}{24}e^{4} - \frac{31}{4}e^{3} + \frac{231}{8}e^{2} + \frac{161}{12}e - \frac{37}{2}$ |
| 73 | $[73, 73, 2w^{2} - w - 5]$ | $-\frac{1}{8}e^{8} - \frac{1}{2}e^{7} + 2e^{6} + \frac{29}{4}e^{5} - \frac{73}{8}e^{4} - \frac{229}{8}e^{3} + \frac{43}{4}e^{2} + \frac{57}{2}e + 2$ |
| 73 | $[73, 73, 2w - 1]$ | $-\frac{5}{12}e^{8} - \frac{13}{24}e^{7} + \frac{43}{6}e^{6} + \frac{47}{6}e^{5} - \frac{115}{3}e^{4} - \frac{237}{8}e^{3} + \frac{553}{8}e^{2} + \frac{287}{12}e - \frac{49}{2}$ |
| 73 | $[73, 73, -2w - 1]$ | $\phantom{-}\frac{1}{12}e^{8} + \frac{1}{3}e^{7} - \frac{4}{3}e^{6} - \frac{31}{6}e^{5} + \frac{77}{12}e^{4} + \frac{95}{4}e^{3} - \frac{23}{2}e^{2} - \frac{97}{3}e + 10$ |
| 73 | $[73, 73, 2w^{2} + w - 5]$ | $\phantom{-}\frac{1}{8}e^{8} + \frac{1}{2}e^{7} - \frac{5}{2}e^{6} - \frac{29}{4}e^{5} + \frac{129}{8}e^{4} + \frac{221}{8}e^{3} - \frac{141}{4}e^{2} - 20e + 14$ |
| 83 | $[83, 83, 2w^{3} + w^{2} - 9w - 7]$ | $\phantom{-}\frac{1}{8}e^{7} - 2e^{5} - \frac{1}{4}e^{4} + \frac{65}{8}e^{3} + \frac{25}{8}e^{2} - \frac{17}{4}e - \frac{9}{2}$ |
| 83 | $[83, 83, -2w^{3} + w^{2} + 7w + 1]$ | $-\frac{1}{8}e^{8} - \frac{3}{8}e^{7} + 2e^{6} + \frac{25}{4}e^{5} - \frac{83}{8}e^{4} - \frac{61}{2}e^{3} + \frac{183}{8}e^{2} + \frac{161}{4}e - \frac{33}{2}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2,2,-w^{2}+w+2]$ | $1$ |
| $11$ | $[11,11,-w^{2}-w+1]$ | $1$ |