Base field 4.4.17417.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 5x^{2} + 3x + 4\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16, 2, 2]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $24$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} - 6x^{8} - 14x^{7} + 126x^{6} - 26x^{5} - 722x^{4} + 737x^{3} + 636x^{2} - 716x + 144\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w^{3} - 3w^{2} - w + 2]$ | $-1$ |
| 5 | $[5, 5, w^{2} - 2w - 3]$ | $-\frac{31}{2702}e^{8} + \frac{67}{772}e^{7} + \frac{97}{772}e^{6} - \frac{1495}{772}e^{5} + \frac{5063}{5404}e^{4} + \frac{67589}{5404}e^{3} - \frac{61553}{5404}e^{2} - \frac{24477}{1351}e + \frac{12153}{1351}$ |
| 5 | $[5, 5, w^{3} - 3w^{2} - 3w + 5]$ | $\phantom{-}e$ |
| 8 | $[8, 2, w^{3} - 2w^{2} - 3w + 3]$ | $-1$ |
| 13 | $[13, 13, -w^{2} + w + 3]$ | $-\frac{29}{1351}e^{8} + \frac{22}{193}e^{7} + \frac{153}{386}e^{6} - \frac{1025}{386}e^{5} - \frac{835}{1351}e^{4} + \frac{45709}{2702}e^{3} - \frac{36053}{2702}e^{2} - \frac{25705}{1351}e + \frac{15242}{1351}$ |
| 17 | $[17, 17, -w^{2} + 3w + 1]$ | $\phantom{-}\frac{97}{1351}e^{8} - \frac{301}{772}e^{7} - \frac{937}{772}e^{6} + \frac{6411}{772}e^{5} + \frac{14293}{5404}e^{4} - \frac{267437}{5404}e^{3} + \frac{140323}{5404}e^{2} + \frac{152997}{2702}e - \frac{31602}{1351}$ |
| 23 | $[23, 23, w^{2} - 3]$ | $\phantom{-}\frac{68}{1351}e^{8} - \frac{213}{772}e^{7} - \frac{631}{772}e^{6} + \frac{4361}{772}e^{5} + \frac{8251}{5404}e^{4} - \frac{170615}{5404}e^{3} + \frac{100641}{5404}e^{2} + \frac{40661}{1351}e - \frac{23115}{1351}$ |
| 25 | $[25, 5, -w^{2} + 2w + 1]$ | $-\frac{221}{2702}e^{8} + \frac{161}{386}e^{7} + \frac{573}{386}e^{6} - \frac{3483}{386}e^{5} - \frac{5674}{1351}e^{4} + \frac{145633}{2702}e^{3} - \frac{37719}{1351}e^{2} - \frac{155453}{2702}e + \frac{34691}{1351}$ |
| 37 | $[37, 37, w^{3} - 4w^{2} - 2w + 9]$ | $-\frac{27}{2702}e^{8} + \frac{21}{772}e^{7} + \frac{209}{772}e^{6} - \frac{555}{772}e^{5} - \frac{8403}{5404}e^{4} + \frac{23829}{5404}e^{3} - \frac{15957}{5404}e^{2} - \frac{1228}{1351}e + \frac{11195}{1351}$ |
| 41 | $[41, 41, w^{2} - w - 5]$ | $-\frac{229}{5404}e^{8} + \frac{207}{772}e^{7} + \frac{461}{772}e^{6} - \frac{4423}{772}e^{5} + \frac{3469}{5404}e^{4} + \frac{183989}{5404}e^{3} - \frac{68623}{2702}e^{2} - \frac{99231}{2702}e + \frac{21681}{1351}$ |
| 49 | $[49, 7, -w^{3} + 2w^{2} + 2w - 1]$ | $-\frac{345}{5404}e^{8} + \frac{295}{772}e^{7} + \frac{767}{772}e^{6} - \frac{6473}{772}e^{5} + \frac{129}{5404}e^{4} + \frac{275407}{5404}e^{3} - \frac{52338}{1351}e^{2} - \frac{147939}{2702}e + \frac{36923}{1351}$ |
| 49 | $[49, 7, w^{2} + w - 3]$ | $-\frac{51}{2702}e^{8} + \frac{26}{193}e^{7} + \frac{29}{193}e^{6} - \frac{1071}{386}e^{5} + \frac{3575}{1351}e^{4} + \frac{21948}{1351}e^{3} - \frac{33542}{1351}e^{2} - \frac{25887}{1351}e + \frac{29102}{1351}$ |
| 59 | $[59, 59, w^{3} - 3w^{2} - 4w + 5]$ | $-\frac{235}{5404}e^{8} + \frac{145}{772}e^{7} + \frac{763}{772}e^{6} - \frac{3391}{772}e^{5} - \frac{30339}{5404}e^{4} + \frac{154663}{5404}e^{3} - \frac{642}{1351}e^{2} - \frac{49827}{1351}e + \frac{8214}{1351}$ |
| 67 | $[67, 67, -w^{3} + 3w^{2} + w - 5]$ | $\phantom{-}\frac{93}{1351}e^{8} - \frac{209}{772}e^{7} - \frac{1161}{772}e^{6} + \frac{4145}{772}e^{5} + \frac{52033}{5404}e^{4} - \frac{152897}{5404}e^{3} - \frac{67055}{5404}e^{2} + \frac{60001}{2702}e + \frac{2738}{1351}$ |
| 71 | $[71, 71, 2w - 3]$ | $\phantom{-}\frac{41}{1351}e^{8} - \frac{171}{772}e^{7} - \frac{213}{772}e^{6} + \frac{3637}{772}e^{5} - \frac{19363}{5404}e^{4} - \frac{149977}{5404}e^{3} + \frac{190317}{5404}e^{2} + \frac{32801}{1351}e - \frac{30447}{1351}$ |
| 71 | $[71, 71, w^{3} - 4w^{2} + 2w + 5]$ | $-\frac{493}{5404}e^{8} + \frac{187}{386}e^{7} + \frac{301}{193}e^{6} - \frac{1961}{193}e^{5} - \frac{10475}{2702}e^{4} + \frac{80413}{1351}e^{3} - \frac{159867}{5404}e^{2} - \frac{92021}{1351}e + \frac{37083}{1351}$ |
| 79 | $[79, 79, -w^{3} + 5w^{2} - 4w - 5]$ | $-\frac{337}{5404}e^{8} + \frac{249}{772}e^{7} + \frac{879}{772}e^{6} - \frac{5533}{772}e^{5} - \frac{16039}{5404}e^{4} + \frac{237051}{5404}e^{3} - \frac{31482}{1351}e^{2} - \frac{60853}{1351}e + \frac{21104}{1351}$ |
| 79 | $[79, 79, -2w^{3} + 7w^{2} + 5w - 15]$ | $\phantom{-}\frac{587}{5404}e^{8} - \frac{108}{193}e^{7} - \frac{358}{193}e^{6} + \frac{2165}{193}e^{5} + \frac{7731}{1351}e^{4} - \frac{83315}{1351}e^{3} + \frac{132253}{5404}e^{2} + \frac{83851}{1351}e - \frac{40909}{1351}$ |
| 81 | $[81, 3, -3]$ | $\phantom{-}\frac{23}{2702}e^{8} + \frac{25}{772}e^{7} - \frac{321}{772}e^{6} - \frac{771}{772}e^{5} + \frac{32677}{5404}e^{4} + \frac{46951}{5404}e^{3} - \frac{140421}{5404}e^{2} - \frac{22021}{1351}e + \frac{8677}{1351}$ |
| 83 | $[83, 83, -2w^{3} + 6w^{2} + 2w - 5]$ | $\phantom{-}\frac{507}{5404}e^{8} - \frac{179}{386}e^{7} - \frac{697}{386}e^{6} + \frac{1938}{193}e^{5} + \frac{19295}{2702}e^{4} - \frac{162639}{2702}e^{3} + \frac{103209}{5404}e^{2} + \frac{90488}{1351}e - \frac{31329}{1351}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2, 2, w^{3} - 3w^{2} - w + 2]$ | $1$ |
| $8$ | $[8, 2, w^{3} - 2w^{2} - 3w + 3]$ | $1$ |