Base field 4.4.3981.1
Generator \(w\), with minimal polynomial \(x^4 - x^3 - 4 x^2 + 2 x + 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[37, 37, w^3 - 4 w + 1]$ |
| 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:
| \(x^9 - 3 x^8 - 19 x^7 + 59 x^6 + 101 x^5 - 353 x^4 - 104 x^3 + 670 x^2 - 99 x - 297\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w + 1]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w^3 - w^2 - 3 w + 1]$ | $\phantom{-}\frac{19}{10} e^8 - \frac{29}{10} e^7 - \frac{202}{5} e^6 + \frac{264}{5} e^5 + 270 e^4 - \frac{1381}{5} e^3 - 605 e^2 + \frac{787}{2} e + \frac{3909}{10}$ |
| 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 | $[13, 13, -w^3 + w^2 + 4 w]$ | $-\frac{1}{5} e^8 + \frac{1}{5} e^7 + \frac{21}{5} e^6 - \frac{17}{5} e^5 - 27 e^4 + \frac{78}{5} e^3 + 53 e^2 - 17 e - \frac{121}{5}$ |
| 16 | $[16, 2, 2]$ | $-\frac{23}{6} e^8 + 6 e^7 + \frac{244}{3} e^6 - \frac{655}{6} e^5 - \frac{3253}{6} e^4 + \frac{3421}{6} e^3 + \frac{3635}{3} e^2 - \frac{2425}{3} e - 782$ |
| 23 | $[23, 23, w^2 - 2 w - 2]$ | $-\frac{14}{15} e^8 + \frac{8}{5} e^7 + \frac{299}{15} e^6 - \frac{881}{30} e^5 - \frac{404}{3} e^4 + \frac{2332}{15} e^3 + \frac{1861}{6} e^2 - \frac{1337}{6} e - \frac{2121}{10}$ |
| 37 | $[37, 37, w^3 - 4 w + 1]$ | $-1$ |
| 37 | $[37, 37, w^3 - w^2 - 5 w + 1]$ | $-\frac{9}{5} e^8 + \frac{14}{5} e^7 + \frac{383}{10} e^6 - \frac{511}{10} e^5 - 256 e^4 + \frac{2679}{10} e^3 + 572 e^2 - 381 e - \frac{3643}{10}$ |
| 41 | $[41, 41, w^3 - 5 w + 1]$ | $-\frac{34}{15} e^8 + \frac{18}{5} e^7 + \frac{724}{15} e^6 - \frac{983}{15} e^5 - \frac{970}{3} e^4 + \frac{5147}{15} e^3 + \frac{2185}{3} e^2 - \frac{1478}{3} e - \frac{2373}{5}$ |
| 43 | $[43, 43, 2 w^3 - w^2 - 7 w]$ | $\phantom{-}\frac{17}{15} e^8 - \frac{9}{5} e^7 - \frac{362}{15} e^6 + \frac{499}{15} e^5 + \frac{485}{3} e^4 - \frac{2671}{15} e^3 - \frac{1094}{3} e^2 + \frac{784}{3} e + \frac{1199}{5}$ |
| 53 | $[53, 53, 2 w - 3]$ | $\phantom{-}\frac{17}{15} e^8 - \frac{9}{5} e^7 - \frac{362}{15} e^6 + \frac{983}{30} e^5 + \frac{485}{3} e^4 - \frac{2566}{15} e^3 - \frac{2191}{6} e^2 + \frac{1439}{6} e + \frac{2463}{10}$ |
| 59 | $[59, 59, -2 w^3 + w^2 + 9 w - 1]$ | $\phantom{-}\frac{21}{5} e^8 - \frac{31}{5} e^7 - \frac{446}{5} e^6 + \frac{562}{5} e^5 + 595 e^4 - \frac{2923}{5} e^3 - 1328 e^2 + 831 e + \frac{4236}{5}$ |
| 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 | $[67, 67, w^3 + w^2 - 5 w - 4]$ | $-\frac{7}{15} e^8 + \frac{4}{5} e^7 + \frac{299}{30} e^6 - \frac{433}{30} e^5 - \frac{202}{3} e^4 + \frac{2227}{30} e^3 + \frac{463}{3} e^2 - \frac{308}{3} e - \frac{1003}{10}$ |
| 71 | $[71, 71, 2 w^3 - 3 w^2 - 7 w + 5]$ | $\phantom{-}\frac{71}{15} e^8 - \frac{37}{5} e^7 - \frac{1511}{15} e^6 + \frac{2017}{15} e^5 + \frac{2024}{3} e^4 - \frac{10528}{15} e^3 - \frac{4556}{3} e^2 + \frac{2998}{3} e + \frac{4932}{5}$ |
| 73 | $[73, 73, w^3 - 6 w]$ | $\phantom{-}\frac{57}{10} e^8 - \frac{87}{10} e^7 - \frac{606}{5} e^6 + \frac{787}{5} e^5 + 809 e^4 - \frac{4078}{5} e^3 - 1803 e^2 + \frac{2299}{2} e + \frac{11527}{10}$ |
| 73 | $[73, 73, -w^3 - w^2 + 5 w + 3]$ | $-\frac{32}{5} e^8 + \frac{99}{10} e^7 + \frac{1359}{10} e^6 - \frac{1803}{10} e^5 - \frac{1815}{2} e^4 + \frac{4721}{5} e^3 + \frac{4073}{2} e^2 - 1347 e - \frac{13199}{10}$ |
| 79 | $[79, 79, w^3 - 3 w - 4]$ | $-\frac{82}{15} e^8 + \frac{83}{10} e^7 + \frac{3479}{30} e^6 - \frac{4513}{30} e^5 - \frac{4625}{6} e^4 + \frac{11726}{15} e^3 + \frac{10217}{6} e^2 - \frac{3320}{3} e - \frac{10723}{10}$ |
| 83 | $[83, 83, w^3 - 2 w^2 - 3 w + 1]$ | $-\frac{7}{5} e^8 + \frac{19}{10} e^7 + \frac{299}{10} e^6 - \frac{343}{10} e^5 - \frac{401}{2} e^4 + \frac{886}{5} e^3 + \frac{895}{2} e^2 - 248 e - \frac{2769}{10}$ |
| 83 | $[83, 83, -2 w^3 + 2 w^2 + 6 w - 3]$ | $-\frac{91}{15} e^8 + \frac{47}{5} e^7 + \frac{1936}{15} e^6 - \frac{2567}{15} e^5 - \frac{2593}{3} e^4 + \frac{13433}{15} e^3 + \frac{5842}{3} e^2 - \frac{3836}{3} e - \frac{6327}{5}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $37$ | $[37, 37, w^3 - 4 w + 1]$ | $1$ |