Base field 6.6.980125.1
Generator \(w\), with minimal polynomial \(x^{6} - x^{5} - 6x^{4} + 6x^{3} + 7x^{2} - 5x - 1\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2, 2]$ |
| Level: | $[31, 31, -w^{5} + w^{4} + 6w^{3} - 5w^{2} - 7w + 1]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $16$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 11x^{7} + 13x^{6} + 199x^{5} - 476x^{4} - 1193x^{3} + 3382x^{2} + 2394x - 7220\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 9 | $[9, 3, -w^{2} + 2]$ | $\phantom{-}e$ |
| 11 | $[11, 11, w^{4} + w^{3} - 4w^{2} - 3w + 2]$ | $\phantom{-}\frac{1537}{60154}e^{7} - \frac{18883}{60154}e^{6} + \frac{50989}{60154}e^{5} + \frac{176321}{60154}e^{4} - \frac{448209}{30077}e^{3} + \frac{171771}{60154}e^{2} + \frac{83195}{1583}e - \frac{81180}{1583}$ |
| 11 | $[11, 11, -w^{5} + 6w^{3} - w^{2} - 7w + 1]$ | $-\frac{443}{30077}e^{7} + \frac{4151}{30077}e^{6} - \frac{1370}{30077}e^{5} - \frac{67238}{30077}e^{4} + \frac{87026}{30077}e^{3} + \frac{284900}{30077}e^{2} - \frac{18171}{1583}e - \frac{3310}{1583}$ |
| 19 | $[19, 19, -w^{5} - w^{4} + 5w^{3} + 4w^{2} - 5w - 3]$ | $\phantom{-}\frac{739}{60154}e^{7} - \frac{15615}{60154}e^{6} + \frac{94689}{60154}e^{5} - \frac{30209}{60154}e^{4} - \frac{639026}{30077}e^{3} + \frac{2350007}{60154}e^{2} + \frac{105464}{1583}e - \frac{243862}{1583}$ |
| 29 | $[29, 29, -2w^{4} - w^{3} + 9w^{2} + 2w - 5]$ | $\phantom{-}\frac{1484}{30077}e^{7} - \frac{19269}{30077}e^{6} + \frac{58565}{30077}e^{5} + \frac{162981}{30077}e^{4} - \frac{962998}{30077}e^{3} + \frac{287193}{30077}e^{2} + \frac{173371}{1583}e - \frac{168552}{1583}$ |
| 31 | $[31, 31, -w^{5} + w^{4} + 6w^{3} - 5w^{2} - 7w + 1]$ | $-1$ |
| 41 | $[41, 41, 2w^{5} + w^{4} - 10w^{3} - 3w^{2} + 7w + 2]$ | $\phantom{-}\frac{2303}{30077}e^{7} - \frac{25789}{30077}e^{6} + \frac{53290}{30077}e^{5} + \frac{264136}{30077}e^{4} - \frac{930662}{30077}e^{3} - \frac{237142}{30077}e^{2} + \frac{160654}{1583}e - \frac{131230}{1583}$ |
| 41 | $[41, 41, w^{5} - w^{4} - 6w^{3} + 5w^{2} + 6w - 3]$ | $-\frac{443}{30077}e^{7} + \frac{4151}{30077}e^{6} - \frac{1370}{30077}e^{5} - \frac{67238}{30077}e^{4} + \frac{87026}{30077}e^{3} + \frac{314977}{30077}e^{2} - \frac{22920}{1583}e - \frac{15974}{1583}$ |
| 41 | $[41, 41, w^{5} - 6w^{3} + w^{2} + 7w - 2]$ | $\phantom{-}\frac{1719}{30077}e^{7} - \frac{16990}{30077}e^{6} + \frac{13056}{30077}e^{5} + \frac{255612}{30077}e^{4} - \frac{414684}{30077}e^{3} - \frac{1084332}{30077}e^{2} + \frac{84414}{1583}e + \frac{54538}{1583}$ |
| 59 | $[59, 59, -w^{5} - w^{4} + 4w^{3} + 4w^{2} - 2w - 3]$ | $-\frac{4249}{60154}e^{7} + \frac{26371}{60154}e^{6} + \frac{125567}{60154}e^{5} - \frac{815797}{60154}e^{4} - \frac{534521}{30077}e^{3} + \frac{7115623}{60154}e^{2} + \frac{78956}{1583}e - \frac{482430}{1583}$ |
| 59 | $[59, 59, w^{5} - w^{4} - 5w^{3} + 6w^{2} + 2w - 4]$ | $\phantom{-}\frac{5029}{60154}e^{7} - \frac{35445}{60154}e^{6} - \frac{107675}{60154}e^{5} + \frac{923593}{60154}e^{4} + \frac{380915}{30077}e^{3} - \frac{7454579}{60154}e^{2} - \frac{52221}{1583}e + \frac{485880}{1583}$ |
| 59 | $[59, 59, -w^{5} + 4w^{3} - w^{2} - w + 1]$ | $\phantom{-}\frac{939}{30077}e^{7} - \frac{7916}{30077}e^{6} - \frac{4836}{30077}e^{5} + \frac{147816}{30077}e^{4} - \frac{107472}{30077}e^{3} - \frac{715299}{30077}e^{2} + \frac{32527}{1583}e + \frac{28642}{1583}$ |
| 61 | $[61, 61, -w^{4} + 4w^{2} - 2w - 2]$ | $-\frac{1182}{30077}e^{7} + \frac{19766}{30077}e^{6} - \frac{96059}{30077}e^{5} - \frac{37029}{30077}e^{4} + \frac{1365078}{30077}e^{3} - \frac{2035030}{30077}e^{2} - \frac{232265}{1583}e + \frac{474916}{1583}$ |
| 64 | $[64, 2, -2]$ | $\phantom{-}\frac{983}{30077}e^{7} - \frac{10433}{30077}e^{6} + \frac{18384}{30077}e^{5} + \frac{111789}{30077}e^{4} - \frac{355238}{30077}e^{3} - \frac{121620}{30077}e^{2} + \frac{69801}{1583}e - \frac{53163}{1583}$ |
| 71 | $[71, 71, 2w^{5} + w^{4} - 10w^{3} - 3w^{2} + 9w + 3]$ | $-\frac{4293}{60154}e^{7} + \frac{58965}{60154}e^{6} - \frac{198423}{60154}e^{5} - \frac{448923}{60154}e^{4} + \frac{1574444}{30077}e^{3} - \frac{1508615}{60154}e^{2} - \frac{281609}{1583}e + \frac{312772}{1583}$ |
| 81 | $[81, 3, w^{5} + 2w^{4} - 4w^{3} - 8w^{2} + 2w + 3]$ | $-\frac{1625}{60154}e^{7} + \frac{23917}{60154}e^{6} - \frac{97429}{60154}e^{5} - \frac{104267}{60154}e^{4} + \frac{726052}{30077}e^{3} - \frac{1720053}{60154}e^{2} - \frac{126801}{1583}e + \frac{231054}{1583}$ |
| 89 | $[89, 89, -w^{5} + 6w^{3} - 7w]$ | $-\frac{1896}{30077}e^{7} + \frac{10026}{30077}e^{6} + \frac{71264}{30077}e^{5} - \frac{364289}{30077}e^{4} - \frac{744595}{30077}e^{3} + \frac{3600725}{30077}e^{2} + \frac{123185}{1583}e - \frac{546712}{1583}$ |
| 89 | $[89, 89, w^{5} + w^{4} - 5w^{3} - 4w^{2} + 6w + 2]$ | $-\frac{235}{60154}e^{7} - \frac{2279}{60154}e^{6} + \frac{45509}{60154}e^{5} - \frac{92631}{60154}e^{4} - \frac{274157}{30077}e^{3} + \frac{1371525}{60154}e^{2} + \frac{43687}{1583}e - \frac{103630}{1583}$ |
| 101 | $[101, 101, w^{5} - 4w^{3} + 2w^{2} + w - 3]$ | $\phantom{-}\frac{3185}{60154}e^{7} - \frac{42065}{60154}e^{6} + \frac{133213}{60154}e^{5} + \frac{319859}{60154}e^{4} - \frac{1033264}{30077}e^{3} + \frac{1042141}{60154}e^{2} + \frac{173939}{1583}e - \frac{224154}{1583}$ |
| 101 | $[101, 101, w^{5} + 2w^{4} - 4w^{3} - 8w^{2} + 3w + 4]$ | $-\frac{6951}{60154}e^{7} + \frac{83871}{60154}e^{6} - \frac{206643}{60154}e^{5} - \frac{852351}{60154}e^{4} + \frac{1805445}{30077}e^{3} + \frac{621863}{60154}e^{2} - \frac{329790}{1583}e + \frac{223692}{1583}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $31$ | $[31, 31, -w^{5} + w^{4} + 6w^{3} - 5w^{2} - 7w + 1]$ | $1$ |