Base field 6.6.1397493.1
Generator \(w\), with minimal polynomial \(x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2, 2]$ |
| Level: | $[57,57,-w^{5} + 2w^{4} + 6w^{3} - 8w^{2} - 9w + 4]$ |
| Dimension: | $12$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $38$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{12} - 108x^{10} + 4164x^{8} - 66464x^{6} + 358272x^{4} - 112128x^{2} + 1024\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w - 1]$ | $-1$ |
| 17 | $[17, 17, -w^{2} + 2w + 1]$ | $-\frac{1}{3072}e^{11} + \frac{9}{256}e^{9} - \frac{3121}{2304}e^{7} + \frac{4141}{192}e^{5} - \frac{5515}{48}e^{3} + \frac{811}{36}e$ |
| 17 | $[17, 17, -w^{3} + w^{2} + 4w]$ | $\phantom{-}e$ |
| 19 | $[19, 19, w^{5} - 3w^{4} - 2w^{3} + 8w^{2} + w - 4]$ | $-\frac{1}{13824}e^{10} + \frac{29}{3456}e^{8} - \frac{1183}{3456}e^{6} + \frac{4909}{864}e^{4} - \frac{6695}{216}e^{2} + \frac{373}{54}$ |
| 19 | $[19, 19, w^{2} - w - 1]$ | $-1$ |
| 37 | $[37, 37, w^{4} - 2w^{3} - 3w^{2} + 3w + 2]$ | $-\frac{1}{6912}e^{10} + \frac{13}{864}e^{8} - \frac{973}{1728}e^{6} + \frac{3811}{432}e^{4} - \frac{5081}{108}e^{2} + \frac{253}{27}$ |
| 37 | $[37, 37, w^{4} - 4w^{3} + w^{2} + 7w - 3]$ | $-\frac{1}{13824}e^{10} + \frac{13}{1728}e^{8} - \frac{961}{3456}e^{6} + \frac{3613}{864}e^{4} - \frac{4217}{216}e^{2} - \frac{155}{54}$ |
| 53 | $[53, 53, 2w^{5} - 6w^{4} - 6w^{3} + 18w^{2} + 9w - 6]$ | $-\frac{7}{13824}e^{11} + \frac{47}{864}e^{9} - \frac{7207}{3456}e^{7} + \frac{28549}{864}e^{5} - \frac{37859}{216}e^{3} + \frac{1915}{54}e$ |
| 53 | $[53, 53, -w^{5} + 3w^{4} + 3w^{3} - 9w^{2} - 3w + 2]$ | $\phantom{-}\frac{11}{27648}e^{11} - \frac{149}{3456}e^{9} + \frac{11531}{6912}e^{7} - \frac{46187}{1728}e^{5} + \frac{62491}{432}e^{3} - \frac{4979}{108}e$ |
| 64 | $[64, 2, -2]$ | $\phantom{-}\frac{1}{13824}e^{10} - \frac{23}{3456}e^{8} + \frac{751}{3456}e^{6} - \frac{2587}{864}e^{4} + \frac{3287}{216}e^{2} - \frac{13}{54}$ |
| 71 | $[71, 71, -w^{5} + 4w^{4} - w^{3} - 8w^{2} + 3w - 1]$ | $\phantom{-}\frac{5}{13824}e^{11} - \frac{269}{6912}e^{9} + \frac{2587}{1728}e^{7} - \frac{20675}{864}e^{5} + \frac{14147}{108}e^{3} - \frac{3149}{54}e$ |
| 71 | $[71, 71, 2w^{5} - 5w^{4} - 8w^{3} + 15w^{2} + 12w - 6]$ | $\phantom{-}\frac{37}{27648}e^{11} - \frac{1001}{6912}e^{9} + \frac{38683}{6912}e^{7} - \frac{154849}{1728}e^{5} + \frac{210131}{432}e^{3} - \frac{18505}{108}e$ |
| 71 | $[71, 71, 2w^{5} - 6w^{4} - 6w^{3} + 18w^{2} + 9w - 7]$ | $-\frac{13}{13824}e^{11} + \frac{175}{1728}e^{9} - \frac{13453}{3456}e^{7} + \frac{53485}{864}e^{5} - \frac{71609}{216}e^{3} + \frac{4957}{54}e$ |
| 73 | $[73, 73, -2w^{5} + 6w^{4} + 5w^{3} - 16w^{2} - 7w + 3]$ | $\phantom{-}\frac{1}{13824}e^{10} - \frac{13}{1728}e^{8} + \frac{961}{3456}e^{6} - \frac{3685}{864}e^{4} + \frac{4793}{216}e^{2} + \frac{83}{54}$ |
| 73 | $[73, 73, -2w^{5} + 6w^{4} + 5w^{3} - 17w^{2} - 6w + 6]$ | $\phantom{-}\frac{1}{6912}e^{10} - \frac{13}{864}e^{8} + \frac{961}{1728}e^{6} - \frac{3649}{432}e^{4} + \frac{4613}{108}e^{2} - \frac{259}{27}$ |
| 89 | $[89, 89, w^{5} - 2w^{4} - 5w^{3} + 6w^{2} + 8w - 4]$ | $-\frac{13}{13824}e^{11} + \frac{175}{1728}e^{9} - \frac{13453}{3456}e^{7} + \frac{53485}{864}e^{5} - \frac{71609}{216}e^{3} + \frac{5011}{54}e$ |
| 89 | $[89, 89, w^{5} - w^{4} - 9w^{3} + 7w^{2} + 16w - 6]$ | $\phantom{-}\frac{1}{576}e^{7} - \frac{11}{96}e^{5} + \frac{9}{4}e^{3} - \frac{239}{18}e$ |
| 89 | $[89, 89, 2w^{5} - 6w^{4} - 4w^{3} + 15w^{2} + 3w - 3]$ | $\phantom{-}\frac{1}{576}e^{7} - \frac{11}{96}e^{5} + \frac{9}{4}e^{3} - \frac{239}{18}e$ |
| 89 | $[89, 89, w^{5} - 2w^{4} - 6w^{3} + 8w^{2} + 10w - 6]$ | $\phantom{-}\frac{7}{9216}e^{11} - \frac{21}{256}e^{9} + \frac{7285}{2304}e^{7} - \frac{29047}{576}e^{5} + \frac{13015}{48}e^{3} - \frac{2803}{36}e$ |
| 107 | $[107, 107, w^{5} - 2w^{4} - 7w^{3} + 10w^{2} + 12w - 6]$ | $\phantom{-}\frac{13}{9216}e^{11} - \frac{39}{256}e^{9} + \frac{1503}{256}e^{7} - \frac{53893}{576}e^{5} + \frac{8023}{16}e^{3} - \frac{1679}{12}e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3,3,-w^{5} + 3w^{4} + 3w^{3} - 9w^{2} - 5w + 3]$ | $1$ |
| $19$ | $[19,19,-w^{2} + w + 1]$ | $1$ |