Base field 4.4.19025.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 13x^{2} + 14x + 44\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[20, 10, w + 1]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $35$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} + x^{8} - 22x^{7} - 20x^{6} + 132x^{5} + 110x^{4} - 206x^{3} - 168x^{2} + 55x + 45\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, w^{2} - 2w - 6]$ | $\phantom{-}e$ |
| 4 | $[4, 2, -w^{2} + 7]$ | $-1$ |
| 5 | $[5, 5, -\frac{1}{2}w^{3} + 2w^{2} + \frac{7}{2}w - 14]$ | $-1$ |
| 5 | $[5, 5, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - 6w - 9]$ | $-\frac{7}{44}e^{8} - \frac{1}{22}e^{7} + \frac{157}{44}e^{6} + \frac{31}{44}e^{5} - \frac{244}{11}e^{4} - \frac{30}{11}e^{3} + \frac{1655}{44}e^{2} + \frac{107}{44}e - \frac{529}{44}$ |
| 11 | $[11, 11, \frac{1}{2}w^{3} - 2w^{2} - \frac{5}{2}w + 11]$ | $\phantom{-}\frac{1}{8}e^{8} - \frac{11}{4}e^{6} + \frac{1}{4}e^{5} + \frac{65}{4}e^{4} - \frac{5}{2}e^{3} - \frac{95}{4}e^{2} + \frac{13}{4}e + \frac{65}{8}$ |
| 11 | $[11, 11, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - 5w - 7]$ | $-\frac{9}{88}e^{8} - \frac{1}{88}e^{7} + \frac{205}{88}e^{6} - \frac{1}{88}e^{5} - \frac{1313}{88}e^{4} + \frac{149}{88}e^{3} + \frac{2395}{88}e^{2} - \frac{359}{88}e - \frac{465}{44}$ |
| 31 | $[31, 31, \frac{1}{2}w^{2} + \frac{1}{2}w - 4]$ | $-\frac{1}{4}e^{8} + \frac{11}{2}e^{6} - \frac{1}{2}e^{5} - \frac{65}{2}e^{4} + 5e^{3} + \frac{93}{2}e^{2} - \frac{9}{2}e - \frac{45}{4}$ |
| 31 | $[31, 31, -\frac{1}{2}w^{2} + \frac{3}{2}w + 3]$ | $\phantom{-}\frac{39}{88}e^{8} + \frac{1}{11}e^{7} - \frac{435}{44}e^{6} - \frac{51}{44}e^{5} + \frac{2667}{44}e^{4} + \frac{38}{11}e^{3} - \frac{4333}{44}e^{2} - \frac{357}{44}e + \frac{2875}{88}$ |
| 41 | $[41, 41, \frac{1}{2}w^{2} + \frac{1}{2}w - 6]$ | $\phantom{-}\frac{2}{11}e^{8} + \frac{3}{88}e^{7} - \frac{351}{88}e^{6} - \frac{41}{88}e^{5} + \frac{2069}{88}e^{4} + \frac{213}{88}e^{3} - \frac{2961}{88}e^{2} - \frac{683}{88}e + \frac{535}{88}$ |
| 41 | $[41, 41, 2w^{3} - \frac{15}{2}w^{2} - \frac{25}{2}w + 50]$ | $-\frac{1}{88}e^{8} + \frac{3}{44}e^{7} + \frac{3}{11}e^{6} - \frac{63}{44}e^{5} - \frac{65}{44}e^{4} + \frac{323}{44}e^{3} - \frac{6}{11}e^{2} - \frac{199}{44}e - \frac{85}{88}$ |
| 41 | $[41, 41, \frac{5}{2}w^{2} - \frac{1}{2}w - 17]$ | $\phantom{-}\frac{1}{2}e^{8} + \frac{1}{4}e^{7} - 11e^{6} - \frac{17}{4}e^{5} + \frac{131}{2}e^{4} + \frac{71}{4}e^{3} - 98e^{2} - \frac{67}{4}e + 25$ |
| 41 | $[41, 41, \frac{1}{2}w^{2} - \frac{3}{2}w - 5]$ | $-\frac{61}{88}e^{8} - \frac{19}{88}e^{7} + \frac{1343}{88}e^{6} + \frac{289}{88}e^{5} - \frac{8007}{88}e^{4} - \frac{1085}{88}e^{3} + \frac{12109}{88}e^{2} + \frac{1363}{88}e - \frac{815}{22}$ |
| 61 | $[61, 61, -\frac{1}{2}w^{3} + w^{2} + \frac{7}{2}w - 1]$ | $\phantom{-}\frac{15}{44}e^{8} + \frac{7}{88}e^{7} - \frac{665}{88}e^{6} - \frac{81}{88}e^{5} + \frac{3999}{88}e^{4} + \frac{57}{88}e^{3} - \frac{6007}{88}e^{2} + \frac{445}{88}e + \frac{1439}{88}$ |
| 61 | $[61, 61, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + 4w - 3]$ | $-\frac{6}{11}e^{8} - \frac{5}{22}e^{7} + \frac{133}{11}e^{6} + \frac{83}{22}e^{5} - \frac{1615}{22}e^{4} - \frac{355}{22}e^{3} + \frac{1296}{11}e^{2} + \frac{515}{22}e - \frac{789}{22}$ |
| 71 | $[71, 71, \frac{1}{2}w^{3} + w^{2} - \frac{11}{2}w - 13]$ | $\phantom{-}\frac{57}{88}e^{8} + \frac{5}{44}e^{7} - \frac{629}{44}e^{6} - \frac{61}{44}e^{5} + \frac{940}{11}e^{4} + \frac{201}{44}e^{3} - \frac{5683}{44}e^{2} - \frac{625}{44}e + \frac{2799}{88}$ |
| 71 | $[71, 71, -\frac{1}{2}w^{3} + \frac{5}{2}w^{2} + 2w - 17]$ | $-\frac{5}{8}e^{8} + 14e^{6} - \frac{5}{4}e^{5} - \frac{173}{2}e^{4} + \frac{25}{2}e^{3} + \frac{287}{2}e^{2} - \frac{45}{4}e - \frac{371}{8}$ |
| 81 | $[81, 3, -3]$ | $-\frac{6}{11}e^{8} + \frac{1}{44}e^{7} + \frac{133}{11}e^{6} - \frac{65}{44}e^{5} - \frac{802}{11}e^{4} + \frac{555}{44}e^{3} + \frac{1252}{11}e^{2} - \frac{675}{44}e - \frac{400}{11}$ |
| 89 | $[89, 89, -w^{3} + \frac{3}{2}w^{2} + \frac{13}{2}w - 9]$ | $-\frac{17}{88}e^{8} - \frac{19}{88}e^{7} + \frac{375}{88}e^{6} + \frac{377}{88}e^{5} - \frac{2287}{88}e^{4} - \frac{1965}{88}e^{3} + \frac{4013}{88}e^{2} + \frac{2155}{88}e - \frac{215}{11}$ |
| 89 | $[89, 89, w^{3} - \frac{7}{2}w^{2} - \frac{11}{2}w + 20]$ | $\phantom{-}\frac{53}{88}e^{8} + \frac{3}{22}e^{7} - \frac{148}{11}e^{6} - \frac{41}{22}e^{5} + \frac{3643}{44}e^{4} + \frac{68}{11}e^{3} - \frac{1497}{11}e^{2} - \frac{116}{11}e + \frac{3405}{88}$ |
| 89 | $[89, 89, -w^{3} - \frac{1}{2}w^{2} + \frac{19}{2}w + 12]$ | $\phantom{-}\frac{3}{44}e^{8} + \frac{1}{11}e^{7} - \frac{18}{11}e^{6} - \frac{21}{11}e^{5} + \frac{261}{22}e^{4} + \frac{115}{11}e^{3} - \frac{316}{11}e^{2} - \frac{125}{11}e + \frac{915}{44}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, -w^{2} + 7]$ | $1$ |
| $5$ | $[5, 5, -\frac{1}{2}w^{3} + 2w^{2} + \frac{7}{2}w - 14]$ | $1$ |