Base field 3.3.837.1
Generator \(w\), with minimal polynomial \(x^{3} - 6x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[13, 13, -2w - 5]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $15$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} - 2x^{8} - 10x^{7} + 18x^{6} + 31x^{5} - 49x^{4} - 29x^{3} + 42x^{2} + 4x - 8\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w^{2} - 2w - 1]$ | $\phantom{-}e$ |
| 3 | $[3, 3, w + 2]$ | $\phantom{-}\frac{3}{4}e^{8} - e^{7} - \frac{17}{2}e^{6} + \frac{17}{2}e^{5} + \frac{125}{4}e^{4} - \frac{81}{4}e^{3} - \frac{153}{4}e^{2} + 12e + 9$ |
| 4 | $[4, 2, w^{2} + w - 3]$ | $-\frac{1}{4}e^{8} + \frac{1}{2}e^{7} + \frac{5}{2}e^{6} - \frac{9}{2}e^{5} - \frac{31}{4}e^{4} + \frac{45}{4}e^{3} + \frac{33}{4}e^{2} - \frac{11}{2}e - 3$ |
| 5 | $[5, 5, -w + 2]$ | $-\frac{1}{2}e^{8} + \frac{1}{2}e^{7} + 6e^{6} - 4e^{5} - \frac{47}{2}e^{4} + 8e^{3} + 32e^{2} - \frac{5}{2}e - 10$ |
| 13 | $[13, 13, -2w - 5]$ | $\phantom{-}1$ |
| 25 | $[25, 5, -w^{2} - 2w + 2]$ | $\phantom{-}\frac{17}{4}e^{8} - 5e^{7} - \frac{93}{2}e^{6} + \frac{77}{2}e^{5} + \frac{655}{4}e^{4} - \frac{319}{4}e^{3} - \frac{767}{4}e^{2} + 42e + 49$ |
| 31 | $[31, 31, -w^{2} + 2]$ | $-\frac{11}{4}e^{8} + 4e^{7} + \frac{59}{2}e^{6} - \frac{63}{2}e^{5} - \frac{417}{4}e^{4} + \frac{265}{4}e^{3} + \frac{517}{4}e^{2} - 31e - 39$ |
| 31 | $[31, 31, -2w + 1]$ | $-3e^{8} + \frac{7}{2}e^{7} + 33e^{6} - 27e^{5} - 116e^{4} + \frac{107}{2}e^{3} + \frac{269}{2}e^{2} - \frac{39}{2}e - 36$ |
| 37 | $[37, 37, 2w + 3]$ | $\phantom{-}\frac{3}{4}e^{8} - e^{7} - \frac{17}{2}e^{6} + \frac{19}{2}e^{5} + \frac{117}{4}e^{4} - \frac{101}{4}e^{3} - \frac{113}{4}e^{2} + 14e + 1$ |
| 41 | $[41, 41, -w - 4]$ | $-\frac{1}{4}e^{8} + \frac{7}{2}e^{6} + \frac{1}{2}e^{5} - \frac{67}{4}e^{4} - \frac{9}{4}e^{3} + \frac{123}{4}e^{2} - e - 11$ |
| 43 | $[43, 43, 2w^{2} - 2w - 7]$ | $-\frac{3}{4}e^{8} + \frac{3}{2}e^{7} + \frac{13}{2}e^{6} - \frac{23}{2}e^{5} - \frac{61}{4}e^{4} + \frac{95}{4}e^{3} + \frac{15}{4}e^{2} - \frac{19}{2}e + 5$ |
| 47 | $[47, 47, -2w^{2} - w + 8]$ | $-\frac{1}{2}e^{8} + 6e^{6} + 2e^{5} - \frac{45}{2}e^{4} - \frac{29}{2}e^{3} + \frac{53}{2}e^{2} + 19e - 5$ |
| 53 | $[53, 53, 3w^{2} - 6w - 2]$ | $-\frac{5}{2}e^{8} + 3e^{7} + 27e^{6} - 23e^{5} - \frac{181}{2}e^{4} + \frac{89}{2}e^{3} + \frac{183}{2}e^{2} - 17e - 18$ |
| 53 | $[53, 53, -2w^{2} + 3w + 18]$ | $-\frac{15}{4}e^{8} + 6e^{7} + \frac{77}{2}e^{6} - \frac{97}{2}e^{5} - \frac{501}{4}e^{4} + \frac{425}{4}e^{3} + \frac{529}{4}e^{2} - 53e - 32$ |
| 53 | $[53, 53, 2w - 3]$ | $\phantom{-}\frac{9}{4}e^{8} - 2e^{7} - \frac{51}{2}e^{6} + \frac{27}{2}e^{5} + \frac{375}{4}e^{4} - \frac{83}{4}e^{3} - \frac{471}{4}e^{2} + 8e + 37$ |
| 59 | $[59, 59, 2w^{2} - 3w - 4]$ | $\phantom{-}7e^{8} - 10e^{7} - 74e^{6} + 79e^{5} + 250e^{4} - 168e^{3} - 276e^{2} + 84e + 70$ |
| 61 | $[61, 61, w^{2} - 2w - 4]$ | $-\frac{17}{4}e^{8} + 5e^{7} + \frac{93}{2}e^{6} - \frac{77}{2}e^{5} - \frac{647}{4}e^{4} + \frac{303}{4}e^{3} + \frac{735}{4}e^{2} - 28e - 49$ |
| 71 | $[71, 71, 4w + 9]$ | $\phantom{-}\frac{13}{2}e^{8} - 8e^{7} - 72e^{6} + 65e^{5} + \frac{509}{2}e^{4} - \frac{287}{2}e^{3} - \frac{589}{2}e^{2} + 75e + 76$ |
| 73 | $[73, 73, 2w^{2} - 9]$ | $\phantom{-}\frac{23}{4}e^{8} - 8e^{7} - \frac{125}{2}e^{6} + \frac{129}{2}e^{5} + \frac{869}{4}e^{4} - \frac{553}{4}e^{3} - \frac{993}{4}e^{2} + 64e + 67$ |
| 79 | $[79, 79, w^{2} - 4w + 2]$ | $\phantom{-}\frac{7}{4}e^{8} - 2e^{7} - \frac{41}{2}e^{6} + \frac{37}{2}e^{5} + \frac{309}{4}e^{4} - \frac{197}{4}e^{3} - \frac{397}{4}e^{2} + 30e + 29$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -2w - 5]$ | $-1$ |