Base field 4.4.19429.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 7x^{2} - x + 5\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[21, 21, w^{2} - w - 6]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $37$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} - 2x^{8} - 18x^{7} + 36x^{6} + 85x^{5} - 182x^{4} - 81x^{3} + 225x^{2} - 16x - 24\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, -w^{3} + 2w^{2} + 5w - 3]$ | $\phantom{-}1$ |
| 5 | $[5, 5, w]$ | $\phantom{-}e$ |
| 7 | $[7, 7, -w^{3} + 2w^{2} + 4w - 3]$ | $\phantom{-}1$ |
| 13 | $[13, 13, -w^{2} + w + 4]$ | $\phantom{-}\frac{229}{1687}e^{8} - \frac{121}{1687}e^{7} - \frac{4359}{1687}e^{6} + \frac{2146}{1687}e^{5} + \frac{23264}{1687}e^{4} - \frac{12577}{1687}e^{3} - \frac{5215}{241}e^{2} + \frac{17974}{1687}e + \frac{9917}{1687}$ |
| 13 | $[13, 13, -w^{3} + 2w^{2} + 5w - 2]$ | $\phantom{-}\frac{167}{1687}e^{8} + \frac{258}{1687}e^{7} - \frac{3142}{1687}e^{6} - \frac{4520}{1687}e^{5} + \frac{16133}{1687}e^{4} + \frac{18159}{1687}e^{3} - \frac{3681}{241}e^{2} - \frac{17774}{1687}e + \frac{6488}{1687}$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}\frac{841}{3374}e^{8} - \frac{108}{1687}e^{7} - \frac{7669}{1687}e^{6} + \frac{1539}{1687}e^{5} + \frac{74335}{3374}e^{4} - \frac{8033}{1687}e^{3} - \frac{12785}{482}e^{2} + \frac{953}{3374}e + \frac{305}{1687}$ |
| 17 | $[17, 17, -w + 2]$ | $-\frac{276}{1687}e^{8} + \frac{109}{1687}e^{7} + \frac{4819}{1687}e^{6} - \frac{1975}{1687}e^{5} - \frac{20915}{1687}e^{4} + \frac{13184}{1687}e^{3} + \frac{2483}{241}e^{2} - \frac{17265}{1687}e + \frac{6612}{1687}$ |
| 19 | $[19, 19, -w^{3} + 2w^{2} + 3w - 2]$ | $\phantom{-}\frac{228}{1687}e^{8} + \frac{130}{1687}e^{7} - \frac{4421}{1687}e^{6} - \frac{2696}{1687}e^{5} + \frac{24319}{1687}e^{4} + \frac{13387}{1687}e^{3} - \frac{5750}{241}e^{2} - \frac{19771}{1687}e + \frac{8474}{1687}$ |
| 27 | $[27, 3, w^{3} - 3w^{2} - 4w + 7]$ | $-\frac{61}{1687}e^{8} + \frac{128}{1687}e^{7} + \frac{1279}{1687}e^{6} - \frac{1824}{1687}e^{5} - \frac{8186}{1687}e^{4} + \frac{4772}{1687}e^{3} + \frac{2310}{241}e^{2} + \frac{1997}{1687}e - \frac{5360}{1687}$ |
| 31 | $[31, 31, w^{3} - 3w^{2} - 3w + 7]$ | $-\frac{41}{1687}e^{8} + \frac{169}{1687}e^{7} + \frac{832}{1687}e^{6} - \frac{2830}{1687}e^{5} - \frac{5668}{1687}e^{4} + \frac{11836}{1687}e^{3} + \frac{2165}{241}e^{2} - \frac{9001}{1687}e - \frac{10240}{1687}$ |
| 31 | $[31, 31, -w^{3} + w^{2} + 6w + 1]$ | $-\frac{291}{1687}e^{8} + \frac{500}{1687}e^{7} + \frac{5576}{1687}e^{6} - \frac{8812}{1687}e^{5} - \frac{30395}{1687}e^{4} + \frac{43313}{1687}e^{3} + \frac{6508}{241}e^{2} - \frac{50348}{1687}e - \frac{6598}{1687}$ |
| 41 | $[41, 41, w^{2} - w - 1]$ | $-\frac{150}{241}e^{8} + \frac{54}{241}e^{7} + \frac{2750}{241}e^{6} - \frac{890}{241}e^{5} - \frac{13583}{241}e^{4} + \frac{5101}{241}e^{3} + \frac{18337}{241}e^{2} - \frac{4757}{241}e - \frac{3888}{241}$ |
| 43 | $[43, 43, 2w^{3} - 5w^{2} - 6w + 4]$ | $-\frac{228}{1687}e^{8} - \frac{130}{1687}e^{7} + \frac{4421}{1687}e^{6} + \frac{2696}{1687}e^{5} - \frac{24319}{1687}e^{4} - \frac{13387}{1687}e^{3} + \frac{5509}{241}e^{2} + \frac{18084}{1687}e + \frac{3335}{1687}$ |
| 47 | $[47, 47, -w^{3} + 2w^{2} + 5w - 1]$ | $\phantom{-}\frac{229}{1687}e^{8} - \frac{121}{1687}e^{7} - \frac{4359}{1687}e^{6} + \frac{2146}{1687}e^{5} + \frac{23264}{1687}e^{4} - \frac{12577}{1687}e^{3} - \frac{4974}{241}e^{2} + \frac{17974}{1687}e + \frac{6543}{1687}$ |
| 53 | $[53, 53, -w - 3]$ | $-\frac{1301}{3374}e^{8} + \frac{480}{1687}e^{7} + \frac{11966}{1687}e^{6} - \frac{8527}{1687}e^{5} - \frac{118753}{3374}e^{4} + \frac{46574}{1687}e^{3} + \frac{21663}{482}e^{2} - \frac{105643}{3374}e - \frac{4917}{1687}$ |
| 53 | $[53, 53, -w^{3} + 3w^{2} + 3w - 6]$ | $\phantom{-}\frac{251}{482}e^{8} - \frac{50}{241}e^{7} - \frac{2341}{241}e^{6} + \frac{833}{241}e^{5} + \frac{23913}{482}e^{4} - \frac{4500}{241}e^{3} - \frac{33645}{482}e^{2} + \frac{7033}{482}e + \frac{2877}{241}$ |
| 59 | $[59, 59, 2w^{2} - 3w - 6]$ | $-\frac{625}{3374}e^{8} - \frac{8}{1687}e^{7} + \frac{5930}{1687}e^{6} + \frac{114}{1687}e^{5} - \frac{62661}{3374}e^{4} + \frac{2654}{1687}e^{3} + \frac{13629}{482}e^{2} - \frac{6365}{3374}e - \frac{3039}{1687}$ |
| 59 | $[59, 59, w^{3} - w^{2} - 7w - 3]$ | $-\frac{337}{1687}e^{8} + \frac{237}{1687}e^{7} + \frac{6098}{1687}e^{6} - \frac{3799}{1687}e^{5} - \frac{29101}{1687}e^{4} + \frac{17956}{1687}e^{3} + \frac{4793}{241}e^{2} - \frac{20329}{1687}e - \frac{435}{1687}$ |
| 79 | $[79, 79, 2w^{3} - 4w^{2} - 8w + 3]$ | $-\frac{333}{3374}e^{8} + \frac{460}{1687}e^{7} + \frac{3173}{1687}e^{6} - \frac{8242}{1687}e^{5} - \frac{33321}{3374}e^{4} + \frac{39713}{1687}e^{3} + \frac{5005}{482}e^{2} - \frac{92033}{3374}e + \frac{8573}{1687}$ |
| 79 | $[79, 79, w^{2} - 2w - 1]$ | $\phantom{-}\frac{1629}{3374}e^{8} - \frac{1156}{1687}e^{7} - \frac{15294}{1687}e^{6} + \frac{19847}{1687}e^{5} + \frac{157349}{3374}e^{4} - \frac{92231}{1687}e^{3} - \frac{27897}{482}e^{2} + \frac{170903}{3374}e + \frac{2015}{1687}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, -w^{3} + 2w^{2} + 5w - 3]$ | $-1$ |
| $7$ | $[7, 7, -w^{3} + 2w^{2} + 4w - 3]$ | $-1$ |