Base field 4.4.3981.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 4x^{2} + 2x + 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[37, 37, w^{3} - 4w + 1]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $10$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} - 3x^{8} - 19x^{7} + 59x^{6} + 101x^{5} - 353x^{4} - 104x^{3} + 670x^{2} - 99x - 297\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w + 1]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w^{3} - w^{2} - 3w + 1]$ | $\phantom{-}\frac{19}{10}e^{8} - \frac{29}{10}e^{7} - \frac{202}{5}e^{6} + \frac{264}{5}e^{5} + 270e^{4} - \frac{1381}{5}e^{3} - 605e^{2} + \frac{787}{2}e + \frac{3909}{10}$ |
| 9 | $[9, 3, -w^{2} + 2]$ | $\phantom{-}\frac{37}{15}e^{8} - \frac{19}{5}e^{7} - \frac{787}{15}e^{6} + \frac{1034}{15}e^{5} + \frac{1051}{3}e^{4} - \frac{5381}{15}e^{3} - \frac{2341}{3}e^{2} + \frac{1526}{3}e + \frac{2479}{5}$ |
| 13 | $[13, 13, -w^{3} + w^{2} + 4w]$ | $-\frac{1}{5}e^{8} + \frac{1}{5}e^{7} + \frac{21}{5}e^{6} - \frac{17}{5}e^{5} - 27e^{4} + \frac{78}{5}e^{3} + 53e^{2} - 17e - \frac{121}{5}$ |
| 16 | $[16, 2, 2]$ | $-\frac{23}{6}e^{8} + 6e^{7} + \frac{244}{3}e^{6} - \frac{655}{6}e^{5} - \frac{3253}{6}e^{4} + \frac{3421}{6}e^{3} + \frac{3635}{3}e^{2} - \frac{2425}{3}e - 782$ |
| 23 | $[23, 23, w^{2} - 2w - 2]$ | $-\frac{14}{15}e^{8} + \frac{8}{5}e^{7} + \frac{299}{15}e^{6} - \frac{881}{30}e^{5} - \frac{404}{3}e^{4} + \frac{2332}{15}e^{3} + \frac{1861}{6}e^{2} - \frac{1337}{6}e - \frac{2121}{10}$ |
| 37 | $[37, 37, w^{3} - 4w + 1]$ | $-1$ |
| 37 | $[37, 37, w^{3} - w^{2} - 5w + 1]$ | $-\frac{9}{5}e^{8} + \frac{14}{5}e^{7} + \frac{383}{10}e^{6} - \frac{511}{10}e^{5} - 256e^{4} + \frac{2679}{10}e^{3} + 572e^{2} - 381e - \frac{3643}{10}$ |
| 41 | $[41, 41, w^{3} - 5w + 1]$ | $-\frac{34}{15}e^{8} + \frac{18}{5}e^{7} + \frac{724}{15}e^{6} - \frac{983}{15}e^{5} - \frac{970}{3}e^{4} + \frac{5147}{15}e^{3} + \frac{2185}{3}e^{2} - \frac{1478}{3}e - \frac{2373}{5}$ |
| 43 | $[43, 43, 2w^{3} - w^{2} - 7w]$ | $\phantom{-}\frac{17}{15}e^{8} - \frac{9}{5}e^{7} - \frac{362}{15}e^{6} + \frac{499}{15}e^{5} + \frac{485}{3}e^{4} - \frac{2671}{15}e^{3} - \frac{1094}{3}e^{2} + \frac{784}{3}e + \frac{1199}{5}$ |
| 53 | $[53, 53, 2w - 3]$ | $\phantom{-}\frac{17}{15}e^{8} - \frac{9}{5}e^{7} - \frac{362}{15}e^{6} + \frac{983}{30}e^{5} + \frac{485}{3}e^{4} - \frac{2566}{15}e^{3} - \frac{2191}{6}e^{2} + \frac{1439}{6}e + \frac{2463}{10}$ |
| 59 | $[59, 59, -2w^{3} + w^{2} + 9w - 1]$ | $\phantom{-}\frac{21}{5}e^{8} - \frac{31}{5}e^{7} - \frac{446}{5}e^{6} + \frac{562}{5}e^{5} + 595e^{4} - \frac{2923}{5}e^{3} - 1328e^{2} + 831e + \frac{4236}{5}$ |
| 67 | $[67, 67, -w - 3]$ | $-\frac{94}{15}e^{8} + \frac{48}{5}e^{7} + \frac{1999}{15}e^{6} - \frac{2618}{15}e^{5} - \frac{2671}{3}e^{4} + \frac{13682}{15}e^{3} + \frac{5968}{3}e^{2} - \frac{3905}{3}e - \frac{6343}{5}$ |
| 67 | $[67, 67, w^{3} + w^{2} - 5w - 4]$ | $-\frac{7}{15}e^{8} + \frac{4}{5}e^{7} + \frac{299}{30}e^{6} - \frac{433}{30}e^{5} - \frac{202}{3}e^{4} + \frac{2227}{30}e^{3} + \frac{463}{3}e^{2} - \frac{308}{3}e - \frac{1003}{10}$ |
| 71 | $[71, 71, 2w^{3} - 3w^{2} - 7w + 5]$ | $\phantom{-}\frac{71}{15}e^{8} - \frac{37}{5}e^{7} - \frac{1511}{15}e^{6} + \frac{2017}{15}e^{5} + \frac{2024}{3}e^{4} - \frac{10528}{15}e^{3} - \frac{4556}{3}e^{2} + \frac{2998}{3}e + \frac{4932}{5}$ |
| 73 | $[73, 73, w^{3} - 6w]$ | $\phantom{-}\frac{57}{10}e^{8} - \frac{87}{10}e^{7} - \frac{606}{5}e^{6} + \frac{787}{5}e^{5} + 809e^{4} - \frac{4078}{5}e^{3} - 1803e^{2} + \frac{2299}{2}e + \frac{11527}{10}$ |
| 73 | $[73, 73, -w^{3} - w^{2} + 5w + 3]$ | $-\frac{32}{5}e^{8} + \frac{99}{10}e^{7} + \frac{1359}{10}e^{6} - \frac{1803}{10}e^{5} - \frac{1815}{2}e^{4} + \frac{4721}{5}e^{3} + \frac{4073}{2}e^{2} - 1347e - \frac{13199}{10}$ |
| 79 | $[79, 79, w^{3} - 3w - 4]$ | $-\frac{82}{15}e^{8} + \frac{83}{10}e^{7} + \frac{3479}{30}e^{6} - \frac{4513}{30}e^{5} - \frac{4625}{6}e^{4} + \frac{11726}{15}e^{3} + \frac{10217}{6}e^{2} - \frac{3320}{3}e - \frac{10723}{10}$ |
| 83 | $[83, 83, w^{3} - 2w^{2} - 3w + 1]$ | $-\frac{7}{5}e^{8} + \frac{19}{10}e^{7} + \frac{299}{10}e^{6} - \frac{343}{10}e^{5} - \frac{401}{2}e^{4} + \frac{886}{5}e^{3} + \frac{895}{2}e^{2} - 248e - \frac{2769}{10}$ |
| 83 | $[83, 83, -2w^{3} + 2w^{2} + 6w - 3]$ | $-\frac{91}{15}e^{8} + \frac{47}{5}e^{7} + \frac{1936}{15}e^{6} - \frac{2567}{15}e^{5} - \frac{2593}{3}e^{4} + \frac{13433}{15}e^{3} + \frac{5842}{3}e^{2} - \frac{3836}{3}e - \frac{6327}{5}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $37$ | $[37, 37, w^{3} - 4w + 1]$ | $1$ |