Base field 3.3.1937.1
Generator \(w\), with minimal polynomial \(x^{3} - x^{2} - 8x - 1\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[5, 5, w + 1]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $20$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} - 21x^{7} - 6x^{6} + 135x^{5} + 62x^{4} - 239x^{3} - 70x^{2} + 44x - 2\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, -w - 2]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w + 1]$ | $\phantom{-}1$ |
| 7 | $[7, 7, w - 3]$ | $-\frac{3}{16}e^{8} + \frac{1}{16}e^{7} + \frac{15}{4}e^{6} + \frac{3}{8}e^{5} - \frac{367}{16}e^{4} - \frac{157}{16}e^{3} + \frac{147}{4}e^{2} + \frac{119}{8}e - \frac{23}{8}$ |
| 8 | $[8, 2, 2]$ | $\phantom{-}\frac{1}{16}e^{8} - \frac{1}{16}e^{7} - \frac{5}{4}e^{6} + \frac{3}{8}e^{5} + \frac{121}{16}e^{4} + \frac{45}{16}e^{3} - \frac{45}{4}e^{2} - \frac{73}{8}e + \frac{15}{8}$ |
| 9 | $[9, 3, w^{2} - 3w - 2]$ | $-\frac{1}{16}e^{8} + \frac{1}{16}e^{7} + \frac{5}{4}e^{6} - \frac{3}{8}e^{5} - \frac{121}{16}e^{4} - \frac{29}{16}e^{3} + \frac{41}{4}e^{2} + \frac{9}{8}e + \frac{9}{8}$ |
| 13 | $[13, 13, w + 3]$ | $\phantom{-}\frac{1}{8}e^{8} - \frac{5}{2}e^{6} - \frac{3}{4}e^{5} + \frac{123}{8}e^{4} + 7e^{3} - \frac{53}{2}e^{2} - \frac{23}{4}e + 6$ |
| 13 | $[13, 13, -w + 2]$ | $\phantom{-}\frac{1}{4}e^{8} + \frac{1}{8}e^{7} - \frac{11}{2}e^{6} - \frac{7}{2}e^{5} + \frac{73}{2}e^{4} + \frac{199}{8}e^{3} - 65e^{2} - 26e + \frac{29}{4}$ |
| 19 | $[19, 19, -w^{2} + 2w + 4]$ | $\phantom{-}\frac{1}{8}e^{8} - \frac{11}{4}e^{6} - \frac{1}{2}e^{5} + \frac{145}{8}e^{4} + \frac{19}{4}e^{3} - 31e^{2} + \frac{3}{4}e + \frac{7}{2}$ |
| 23 | $[23, 23, w^{2} - 4w + 1]$ | $-\frac{1}{4}e^{8} - \frac{1}{8}e^{7} + \frac{21}{4}e^{6} + \frac{15}{4}e^{5} - \frac{135}{4}e^{4} - \frac{217}{8}e^{3} + \frac{119}{2}e^{2} + \frac{67}{2}e - \frac{31}{4}$ |
| 25 | $[25, 5, w^{2} - 2w - 1]$ | $-\frac{7}{16}e^{8} + \frac{1}{16}e^{7} + 9e^{6} + \frac{13}{8}e^{5} - \frac{903}{16}e^{4} - \frac{361}{16}e^{3} + \frac{377}{4}e^{2} + \frac{215}{8}e - \frac{67}{8}$ |
| 31 | $[31, 31, w^{2} - 2w - 9]$ | $\phantom{-}\frac{5}{16}e^{8} + \frac{1}{16}e^{7} - \frac{27}{4}e^{6} - \frac{25}{8}e^{5} + \frac{705}{16}e^{4} + \frac{443}{16}e^{3} - \frac{301}{4}e^{2} - \frac{289}{8}e + \frac{49}{8}$ |
| 37 | $[37, 37, -w^{2} + 3w + 3]$ | $-\frac{1}{4}e^{6} + \frac{1}{4}e^{5} + \frac{11}{4}e^{4} - \frac{9}{4}e^{3} - \frac{11}{2}e^{2} + \frac{13}{2}e + \frac{7}{2}$ |
| 41 | $[41, 41, w^{2} - w - 1]$ | $-\frac{1}{16}e^{8} - \frac{1}{16}e^{7} + \frac{3}{2}e^{6} + \frac{15}{8}e^{5} - \frac{169}{16}e^{4} - \frac{239}{16}e^{3} + \frac{75}{4}e^{2} + \frac{177}{8}e - \frac{5}{8}$ |
| 41 | $[41, 41, w^{2} - w - 5]$ | $-\frac{1}{4}e^{6} + \frac{1}{4}e^{5} + \frac{15}{4}e^{4} - \frac{9}{4}e^{3} - \frac{31}{2}e^{2} + \frac{5}{2}e + \frac{23}{2}$ |
| 41 | $[41, 41, w^{2} - w - 10]$ | $\phantom{-}\frac{1}{8}e^{8} + \frac{1}{8}e^{7} - \frac{11}{4}e^{6} - 3e^{5} + \frac{147}{8}e^{4} + \frac{161}{8}e^{3} - 34e^{2} - \frac{103}{4}e + \frac{23}{4}$ |
| 43 | $[43, 43, -2w - 3]$ | $\phantom{-}\frac{1}{16}e^{8} + \frac{3}{16}e^{7} - \frac{3}{2}e^{6} - \frac{27}{8}e^{5} + \frac{157}{16}e^{4} + \frac{293}{16}e^{3} - \frac{55}{4}e^{2} - \frac{181}{8}e + \frac{23}{8}$ |
| 47 | $[47, 47, -w^{2} - w + 4]$ | $-\frac{5}{16}e^{8} - \frac{3}{16}e^{7} + \frac{27}{4}e^{6} + \frac{45}{8}e^{5} - \frac{693}{16}e^{4} - \frac{673}{16}e^{3} + \frac{277}{4}e^{2} + \frac{437}{8}e - \frac{67}{8}$ |
| 49 | $[49, 7, -w^{2} + 5w - 5]$ | $\phantom{-}\frac{1}{4}e^{7} - \frac{1}{2}e^{6} - \frac{9}{2}e^{5} + 5e^{4} + \frac{101}{4}e^{3} - 5e^{2} - 32e - \frac{9}{2}$ |
| 59 | $[59, 59, w^{2} - w - 4]$ | $-\frac{1}{8}e^{7} + \frac{1}{4}e^{6} + \frac{9}{4}e^{5} - 3e^{4} - \frac{97}{8}e^{3} + \frac{13}{2}e^{2} + 18e + \frac{25}{4}$ |
| 59 | $[59, 59, w^{2} - 3]$ | $\phantom{-}\frac{3}{8}e^{8} + \frac{1}{8}e^{7} - \frac{15}{2}e^{6} - \frac{19}{4}e^{5} + \frac{355}{8}e^{4} + \frac{307}{8}e^{3} - \frac{127}{2}e^{2} - \frac{211}{4}e + \frac{25}{4}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $5$ | $[5, 5, w + 1]$ | $-1$ |