Base field 4.4.13625.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 11x^{2} + 12x + 31\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16,4,-\frac{1}{2}w^{2} + \frac{3}{2}w + \frac{3}{2}]$ |
| Dimension: | $10$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $18$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{10} - 25x^{8} + 200x^{6} - 608x^{4} + 568x^{2} - 144\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -\frac{1}{2}w^{2} - \frac{1}{2}w + \frac{3}{2}]$ | $\phantom{-}0$ |
| 4 | $[4, 2, -\frac{1}{2}w^{2} + \frac{3}{2}w + \frac{1}{2}]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w - 3]$ | $-\frac{11}{204}e^{9} + \frac{251}{204}e^{7} - \frac{427}{51}e^{5} + \frac{1000}{51}e^{3} - \frac{479}{51}e$ |
| 11 | $[11, 11, \frac{1}{2}w^{3} - 4w - \frac{7}{2}]$ | $\phantom{-}\frac{1}{4}e^{6} - \frac{19}{4}e^{4} + \frac{41}{2}e^{2} - 10$ |
| 11 | $[11, 11, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{5}{2}w - 7]$ | $\phantom{-}\frac{3}{68}e^{8} - \frac{35}{34}e^{6} + \frac{489}{68}e^{4} - \frac{547}{34}e^{2} + \frac{92}{17}$ |
| 19 | $[19, 19, -w^{2} + 2w + 4]$ | $-\frac{1}{102}e^{9} + \frac{41}{204}e^{7} - \frac{241}{204}e^{5} + \frac{359}{102}e^{3} - \frac{407}{51}e$ |
| 19 | $[19, 19, -w^{2} + 5]$ | $\phantom{-}\frac{1}{102}e^{9} - \frac{41}{204}e^{7} + \frac{241}{204}e^{5} - \frac{359}{102}e^{3} + \frac{407}{51}e$ |
| 31 | $[31, 31, w]$ | $-\frac{5}{68}e^{8} + \frac{47}{34}e^{6} - \frac{407}{68}e^{4} + \frac{141}{34}e^{2} + \frac{45}{17}$ |
| 31 | $[31, 31, w + 3]$ | $\phantom{-}\frac{3}{68}e^{8} - \frac{9}{17}e^{6} - \frac{157}{68}e^{4} + \frac{813}{34}e^{2} - \frac{231}{17}$ |
| 31 | $[31, 31, -w + 4]$ | $-\frac{11}{68}e^{8} + \frac{117}{34}e^{6} - \frac{1385}{68}e^{4} + \frac{1303}{34}e^{2} - \frac{241}{17}$ |
| 31 | $[31, 31, w - 1]$ | $\phantom{-}\frac{1}{34}e^{8} - \frac{41}{68}e^{6} + \frac{241}{68}e^{4} - \frac{291}{34}e^{2} + \frac{169}{17}$ |
| 41 | $[41, 41, -w^{2} + 2]$ | $\phantom{-}\frac{3}{34}e^{8} - \frac{123}{68}e^{6} + \frac{655}{68}e^{4} - \frac{397}{34}e^{2} - \frac{54}{17}$ |
| 41 | $[41, 41, \frac{1}{2}w^{3} - \frac{5}{2}w^{2} - \frac{3}{2}w + 12]$ | $-\frac{2}{17}e^{8} + \frac{181}{68}e^{6} - \frac{1219}{68}e^{4} + \frac{1419}{34}e^{2} - \frac{302}{17}$ |
| 59 | $[59, 59, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{9}{2}w - 5]$ | $-\frac{16}{51}e^{9} + \frac{1465}{204}e^{7} - \frac{10007}{204}e^{5} + \frac{11743}{102}e^{3} - \frac{2926}{51}e$ |
| 59 | $[59, 59, -\frac{1}{2}w^{3} + 2w^{2} + 2w - \frac{17}{2}]$ | $\phantom{-}\frac{1}{34}e^{9} - \frac{41}{68}e^{7} + \frac{241}{68}e^{5} - \frac{325}{34}e^{3} + \frac{254}{17}e$ |
| 79 | $[79, 79, -w^{2} + 8]$ | $\phantom{-}\frac{1}{68}e^{9} - \frac{3}{17}e^{7} - \frac{75}{68}e^{5} + \frac{475}{34}e^{3} - \frac{468}{17}e$ |
| 79 | $[79, 79, w^{2} - 2w - 7]$ | $-\frac{5}{204}e^{9} + \frac{32}{51}e^{7} - \frac{985}{204}e^{5} + \frac{1127}{102}e^{3} - \frac{176}{51}e$ |
| 81 | $[81, 3, -3]$ | $\phantom{-}\frac{13}{68}e^{8} - \frac{73}{17}e^{6} + \frac{1949}{68}e^{4} - \frac{2291}{34}e^{2} + \frac{614}{17}$ |
| 101 | $[101, 101, -w^{3} + \frac{1}{2}w^{2} + \frac{15}{2}w + \frac{3}{2}]$ | $\phantom{-}\frac{5}{17}e^{8} - \frac{427}{68}e^{6} + \frac{2529}{68}e^{4} - \frac{2315}{34}e^{2} + \frac{449}{17}$ |
| 101 | $[101, 101, w^{3} - \frac{5}{2}w^{2} - \frac{11}{2}w + \frac{17}{2}]$ | $-\frac{1}{34}e^{8} + \frac{7}{68}e^{6} + \frac{405}{68}e^{4} - \frac{1171}{34}e^{2} + \frac{477}{17}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4,2,-\frac{1}{2}w^{2} - \frac{1}{2}w + \frac{3}{2}]$ | $1$ |