Base field 3.3.1345.1
Generator \(w\), with minimal polynomial \(x^{3} - 7x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[8, 2, 2]$ |
| Dimension: | $10$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $17$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{10} - 3x^{9} - 31x^{8} + 86x^{7} + 314x^{6} - 840x^{5} - 1028x^{4} + 2920x^{3} - 104x^{2} - 1248x - 288\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, -w - 1]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w + 2]$ | $\phantom{-}\frac{1}{864}e^{9} - \frac{1}{96}e^{8} + \frac{23}{864}e^{7} + \frac{41}{216}e^{6} - \frac{443}{432}e^{5} - \frac{77}{72}e^{4} + \frac{1561}{216}e^{3} + \frac{68}{27}e^{2} - \frac{997}{108}e - \frac{91}{18}$ |
| 7 | $[7, 7, w + 3]$ | $\phantom{-}\frac{5}{324}e^{9} - \frac{1}{18}e^{8} - \frac{32}{81}e^{7} + \frac{221}{162}e^{6} + \frac{997}{324}e^{5} - \frac{1189}{108}e^{4} - \frac{430}{81}e^{3} + \frac{4819}{162}e^{2} - \frac{1195}{81}e - \frac{146}{27}$ |
| 7 | $[7, 7, -w + 2]$ | $\phantom{-}\frac{5}{324}e^{9} - \frac{1}{18}e^{8} - \frac{32}{81}e^{7} + \frac{221}{162}e^{6} + \frac{997}{324}e^{5} - \frac{1189}{108}e^{4} - \frac{430}{81}e^{3} + \frac{4819}{162}e^{2} - \frac{1195}{81}e - \frac{146}{27}$ |
| 7 | $[7, 7, -w + 1]$ | $-\frac{199}{2592}e^{9} + \frac{67}{288}e^{8} + \frac{6115}{2592}e^{7} - \frac{2203}{324}e^{6} - \frac{29779}{1296}e^{5} + \frac{14621}{216}e^{4} + \frac{40685}{648}e^{3} - \frac{19013}{81}e^{2} + \frac{25927}{324}e + \frac{3205}{54}$ |
| 8 | $[8, 2, 2]$ | $\phantom{-}1$ |
| 13 | $[13, 13, -w^{2} + w + 4]$ | $-\frac{49}{648}e^{9} + \frac{2}{9}e^{8} + \frac{773}{324}e^{7} - \frac{4283}{648}e^{6} - \frac{7777}{324}e^{5} + \frac{7195}{108}e^{4} + \frac{5671}{81}e^{3} - \frac{18809}{81}e^{2} + \frac{5734}{81}e + \frac{1466}{27}$ |
| 19 | $[19, 19, -w^{2} + 5]$ | $\phantom{-}\frac{271}{2592}e^{9} - \frac{103}{288}e^{8} - \frac{8023}{2592}e^{7} + \frac{6629}{648}e^{6} + \frac{37411}{1296}e^{5} - \frac{21353}{216}e^{4} - \frac{45581}{648}e^{3} + \frac{53731}{162}e^{2} - \frac{43279}{324}e - \frac{4627}{54}$ |
| 23 | $[23, 23, -w^{2} - w + 3]$ | $-\frac{7}{864}e^{9} + \frac{7}{96}e^{8} + \frac{55}{864}e^{7} - \frac{341}{216}e^{6} + \frac{401}{432}e^{5} + \frac{683}{72}e^{4} - \frac{1747}{216}e^{3} - \frac{314}{27}e^{2} + \frac{499}{108}e + \frac{97}{18}$ |
| 27 | $[27, 3, 3]$ | $\phantom{-}\frac{5}{36}e^{9} - \frac{1}{2}e^{8} - \frac{73}{18}e^{7} + \frac{257}{18}e^{6} + \frac{1339}{36}e^{5} - \frac{1645}{12}e^{4} - \frac{1571}{18}e^{3} + \frac{8185}{18}e^{2} - \frac{1636}{9}e - \frac{350}{3}$ |
| 29 | $[29, 29, w^{2} - w - 3]$ | $\phantom{-}\frac{121}{864}e^{9} - \frac{49}{96}e^{8} - \frac{3481}{864}e^{7} + \frac{3125}{216}e^{6} + \frac{15625}{432}e^{5} - \frac{9947}{72}e^{4} - \frac{17291}{216}e^{3} + \frac{24691}{54}e^{2} - \frac{20629}{108}e - \frac{2263}{18}$ |
| 31 | $[31, 31, w^{2} - w - 8]$ | $-\frac{101}{1296}e^{9} + \frac{35}{144}e^{8} + \frac{3023}{1296}e^{7} - \frac{4529}{648}e^{6} - \frac{14225}{648}e^{5} + \frac{3713}{54}e^{4} + \frac{18001}{324}e^{3} - \frac{19136}{81}e^{2} + \frac{14459}{162}e + \frac{1577}{27}$ |
| 37 | $[37, 37, -w - 4]$ | $-\frac{655}{2592}e^{9} + \frac{235}{288}e^{8} + \frac{19603}{2592}e^{7} - \frac{3791}{162}e^{6} - \frac{92479}{1296}e^{5} + \frac{49085}{216}e^{4} + \frac{118133}{648}e^{3} - \frac{62090}{81}e^{2} + \frac{91435}{324}e + \frac{10681}{54}$ |
| 43 | $[43, 43, 2w^{2} - 4w - 3]$ | $\phantom{-}\frac{125}{1296}e^{9} - \frac{41}{144}e^{8} - \frac{3929}{1296}e^{7} + \frac{2803}{324}e^{6} + \frac{19307}{648}e^{5} - \frac{2413}{27}e^{4} - \frac{25411}{324}e^{3} + \frac{51847}{162}e^{2} - \frac{20891}{162}e - \frac{2168}{27}$ |
| 47 | $[47, 47, w^{2} - 3]$ | $-\frac{19}{54}e^{9} + \frac{7}{6}e^{8} + \frac{281}{27}e^{7} - \frac{1793}{54}e^{6} - \frac{5225}{54}e^{5} + \frac{2872}{9}e^{4} + \frac{6454}{27}e^{3} - \frac{28718}{27}e^{2} + \frac{10864}{27}e + \frac{2492}{9}$ |
| 53 | $[53, 53, 2w^{2} - w - 11]$ | $-\frac{37}{216}e^{9} + \frac{13}{24}e^{8} + \frac{1147}{216}e^{7} - \frac{1711}{108}e^{6} - \frac{1417}{27}e^{5} + \frac{2795}{18}e^{4} + \frac{8015}{54}e^{3} - \frac{14087}{27}e^{2} + \frac{4462}{27}e + \frac{1064}{9}$ |
| 59 | $[59, 59, w^{2} - 2w - 4]$ | $\phantom{-}\frac{197}{864}e^{9} - \frac{77}{96}e^{8} - \frac{5837}{864}e^{7} + \frac{4999}{216}e^{6} + \frac{27209}{432}e^{5} - \frac{16195}{72}e^{4} - \frac{32899}{216}e^{3} + \frac{40859}{54}e^{2} - \frac{32033}{108}e - \frac{3617}{18}$ |
| 67 | $[67, 67, w^{2} + w - 8]$ | $\phantom{-}\frac{5}{96}e^{9} - \frac{7}{32}e^{8} - \frac{149}{96}e^{7} + \frac{157}{24}e^{6} + \frac{701}{48}e^{5} - \frac{523}{8}e^{4} - \frac{787}{24}e^{3} + \frac{1349}{6}e^{2} - \frac{1193}{12}e - \frac{109}{2}$ |
| 71 | $[71, 71, w^{2} + w - 11]$ | $-\frac{289}{864}e^{9} + \frac{109}{96}e^{8} + \frac{8581}{864}e^{7} - \frac{3535}{108}e^{6} - \frac{39913}{432}e^{5} + \frac{22919}{72}e^{4} + \frac{48047}{216}e^{3} - \frac{28967}{27}e^{2} + \frac{46537}{108}e + \frac{5131}{18}$ |
| 73 | $[73, 73, -w^{2} + 4w - 2]$ | $\phantom{-}\frac{991}{2592}e^{9} - \frac{391}{288}e^{8} - \frac{29047}{2592}e^{7} + \frac{25133}{648}e^{6} + \frac{133819}{1296}e^{5} - \frac{80573}{216}e^{4} - \frac{158693}{648}e^{3} + \frac{201061}{162}e^{2} - \frac{160747}{324}e - \frac{17659}{54}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $8$ | $[8, 2, 2]$ | $-1$ |