Base field 4.4.16225.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 13x^{2} + 6x + 36\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[9, 3, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{7}{2}w - 9]$ |
| Dimension: | $10$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $12$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{10} - 25x^{8} + 182x^{6} - 510x^{4} + 505x^{2} - 121\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, \frac{1}{3}w^{3} - \frac{4}{3}w^{2} - \frac{4}{3}w + 6]$ | $\phantom{-}e$ |
| 4 | $[4, 2, -\frac{1}{3}w^{3} - \frac{2}{3}w^{2} + \frac{10}{3}w + 7]$ | $\phantom{-}\frac{9}{176}e^{9} - \frac{12}{11}e^{7} + \frac{467}{88}e^{5} - \frac{71}{11}e^{3} - \frac{207}{176}e$ |
| 9 | $[9, 3, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{7}{2}w - 9]$ | $-1$ |
| 9 | $[9, 3, \frac{1}{3}w^{3} + \frac{2}{3}w^{2} - \frac{7}{3}w - 5]$ | $-\frac{1}{16}e^{8} + \frac{11}{8}e^{6} - \frac{15}{2}e^{4} + \frac{117}{8}e^{2} - \frac{135}{16}$ |
| 11 | $[11, 11, -\frac{1}{6}w^{3} + \frac{1}{6}w^{2} + \frac{1}{6}w]$ | $\phantom{-}\frac{1}{44}e^{9} - \frac{39}{88}e^{7} + \frac{133}{88}e^{5} + \frac{47}{88}e^{3} - \frac{255}{88}e$ |
| 19 | $[19, 19, w + 1]$ | $\phantom{-}\frac{1}{16}e^{9} - \frac{11}{8}e^{7} + \frac{29}{4}e^{5} - \frac{81}{8}e^{3} - \frac{13}{16}e$ |
| 19 | $[19, 19, \frac{1}{6}w^{3} - \frac{1}{6}w^{2} - \frac{13}{6}w + 2]$ | $-\frac{7}{176}e^{9} + \frac{93}{88}e^{7} - \frac{92}{11}e^{5} + \frac{2027}{88}e^{3} - \frac{2985}{176}e$ |
| 25 | $[25, 5, -\frac{1}{3}w^{3} + \frac{1}{3}w^{2} + \frac{7}{3}w - 1]$ | $-\frac{1}{4}e^{8} + \frac{11}{2}e^{6} - \frac{59}{2}e^{4} + \frac{97}{2}e^{2} - \frac{65}{4}$ |
| 29 | $[29, 29, -\frac{1}{6}w^{3} + \frac{1}{6}w^{2} + \frac{13}{6}w]$ | $\phantom{-}\frac{5}{176}e^{9} - \frac{23}{44}e^{7} + \frac{103}{88}e^{5} + \frac{237}{44}e^{3} - \frac{2491}{176}e$ |
| 29 | $[29, 29, w - 1]$ | $\phantom{-}\frac{7}{88}e^{9} - \frac{41}{22}e^{7} + \frac{505}{44}e^{5} - \frac{469}{22}e^{3} + \frac{367}{88}e$ |
| 31 | $[31, 31, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - \frac{10}{3}w + 3]$ | $-\frac{3}{176}e^{9} + \frac{4}{11}e^{7} - \frac{163}{88}e^{5} + \frac{73}{22}e^{3} + \frac{113}{176}e$ |
| 31 | $[31, 31, \frac{1}{6}w^{3} - \frac{1}{6}w^{2} - \frac{1}{6}w + 2]$ | $-\frac{1}{8}e^{9} + \frac{11}{4}e^{7} - 15e^{5} + \frac{117}{4}e^{3} - \frac{183}{8}e$ |
| 41 | $[41, 41, -\frac{1}{2}w^{3} - \frac{1}{2}w^{2} + \frac{9}{2}w + 5]$ | $\phantom{-}\frac{3}{22}e^{9} - \frac{75}{22}e^{7} + \frac{535}{22}e^{5} - \frac{1321}{22}e^{3} + \frac{389}{11}e$ |
| 41 | $[41, 41, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{5}{2}w - 8]$ | $\phantom{-}\frac{1}{176}e^{9} + \frac{15}{88}e^{7} - \frac{56}{11}e^{5} + \frac{1725}{88}e^{3} - \frac{1321}{176}e$ |
| 59 | $[59, 59, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - \frac{10}{3}w - 1]$ | $-\frac{3}{8}e^{8} + \frac{67}{8}e^{6} - \frac{375}{8}e^{4} + \frac{685}{8}e^{2} - \frac{155}{4}$ |
| 59 | $[59, 59, -\frac{1}{3}w^{3} + \frac{4}{3}w^{2} + \frac{7}{3}w - 7]$ | $-\frac{1}{16}e^{8} + \frac{11}{8}e^{6} - \frac{15}{2}e^{4} + \frac{109}{8}e^{2} - \frac{55}{16}$ |
| 59 | $[59, 59, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{5}{2}w + 2]$ | $-\frac{1}{16}e^{8} + \frac{5}{4}e^{6} - \frac{39}{8}e^{4} + \frac{13}{4}e^{2} - \frac{41}{16}$ |
| 79 | $[79, 79, w^{2} - 11]$ | $\phantom{-}\frac{1}{16}e^{8} - \frac{11}{8}e^{6} + \frac{29}{4}e^{4} - \frac{73}{8}e^{2} + \frac{3}{16}$ |
| 79 | $[79, 79, \frac{1}{6}w^{3} - \frac{7}{6}w^{2} - \frac{7}{6}w + 3]$ | $\phantom{-}\frac{1}{16}e^{8} - \frac{11}{8}e^{6} + 8e^{4} - \frac{181}{8}e^{2} + \frac{383}{16}$ |
| 89 | $[89, 89, -\frac{1}{6}w^{3} + \frac{1}{6}w^{2} + \frac{19}{6}w - 5]$ | $-\frac{47}{176}e^{9} + \frac{549}{88}e^{7} - \frac{1715}{44}e^{5} + \frac{7431}{88}e^{3} - \frac{8973}{176}e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $9$ | $[9, 3, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{7}{2}w - 9]$ | $1$ |